From 32ab5b0005f0c23caa8d354ce03d2b24cdc1431a Mon Sep 17 00:00:00 2001 From: Beman Dawes Date: Sun, 25 Nov 2007 18:38:02 +0000 Subject: [PATCH] Full merge from trunk at revision 41356 of entire boost-root tree. 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create mode 100644 test/sinc_test.hpp create mode 100644 test/sinhc_test.hpp create mode 100644 test/special_functions_test.cpp create mode 100644 test/sph_bessel_data.ipp create mode 100644 test/sph_neumann_data.ipp create mode 100644 test/spherical_harmonic.ipp create mode 100644 test/std_real_concept_check.cpp create mode 100644 test/test_bernoulli.cpp create mode 100644 test/test_bessel_hooks.hpp create mode 100644 test/test_bessel_i.cpp create mode 100644 test/test_bessel_j.cpp create mode 100644 test/test_bessel_k.cpp create mode 100644 test/test_bessel_y.cpp create mode 100644 test/test_beta.cpp create mode 100644 test/test_beta_dist.cpp create mode 100644 test/test_beta_hooks.hpp create mode 100644 test/test_binomial.cpp create mode 100644 test/test_binomial_coeff.cpp create mode 100644 test/test_carlson.cpp create mode 100644 test/test_cauchy.cpp create mode 100644 test/test_cbrt.cpp create mode 100644 test/test_chi_squared.cpp create mode 100644 test/test_classify.cpp create 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tools/bessel_data.cpp create mode 100644 tools/beta_data.cpp create mode 100644 tools/carlson_ellint_data.cpp create mode 100644 tools/cbrt_data.cpp create mode 100644 tools/digamma_data.cpp create mode 100644 tools/ellint_e_data.cpp create mode 100644 tools/ellint_f_data.cpp create mode 100644 tools/ellint_k_data.cpp create mode 100644 tools/ellint_pi2_data.cpp create mode 100644 tools/ellint_pi3_data.cpp create mode 100644 tools/erf_data.cpp create mode 100644 tools/factorial_tables.cpp create mode 100644 tools/gamma_P_inva_data.cpp create mode 100644 tools/generate_rational_code.cpp create mode 100644 tools/generate_rational_test.cpp create mode 100644 tools/hermite_data.cpp create mode 100644 tools/ibeta_data.cpp create mode 100644 tools/ibeta_inv_data.cpp create mode 100644 tools/ibeta_invab_data.cpp create mode 100644 tools/igamma_data.cpp create mode 100644 tools/igamma_temme_large_coef.cpp create mode 100644 tools/laguerre_data.cpp create mode 100644 tools/lanczos_generator.cpp create mode 100644 tools/legendre_data.cpp create mode 100644 tools/log1p_expm1_data.cpp create mode 100644 tools/ntl_rr_digamma.hpp create mode 100644 tools/ntl_rr_lanczos.hpp create mode 100644 tools/process_perf_results.cpp create mode 100644 tools/rational_tests.cpp create mode 100644 tools/spherical_harmonic_data.cpp create mode 100644 tools/tgamma_ratio_data.cpp create mode 100644 vc71_fix/Jamfile.v2 create mode 100644 vc71_fix/instantiate_all.cpp diff --git a/doc/Jamfile.v2 b/doc/Jamfile.v2 index 150f48476..ee4e89619 100644 --- a/doc/Jamfile.v2 +++ b/doc/Jamfile.v2 @@ -10,8 +10,27 @@ boostbook standalone : math : - nav.layout=none - navig.graphics=0 + # Path for links to Boost: + boost.root=../../../.. + # Path for libraries index: + boost.libraries=../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=1 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=0 + # How far down sections get TOC's + toc.section.depth=1 ; + diff --git a/doc/common_factor.html b/doc/common_factor.html index 932356890..d0b6309db 100644 --- a/doc/common_factor.html +++ b/doc/common_factor.html @@ -1,11 +1,11 @@ - + Automatic redirection failed, please go to - ../../../doc/html/boost_math/gcd_lcm.html + gcd/html/index.html

Copyright Daryle Walker 2006

Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at www.boost.org/LICENSE_1_0.txt).

@@ -15,3 +15,4 @@ + diff --git a/doc/complex/Jamfile.v2 b/doc/complex/Jamfile.v2 new file mode 100644 index 000000000..f43fb240f --- /dev/null +++ b/doc/complex/Jamfile.v2 @@ -0,0 +1,80 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +path-constant images_location : html ; + +xml complex-tr1 : complex-tr1.qbk ; +boostbook standalone + : + complex-tr1 + : + # Path for links to Boost: + boost.root=../../../../.. + # Path for libraries index: + boost.libraries=../../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=10 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=1 + # How far down sections get TOC's + toc.section.depth=10 + # Max depth in each TOC: + toc.max.depth=4 + # How far down we go with TOC's + generate.section.toc.level=10 + #root.filename="sf_dist_and_tools" + + # PDF Options: + # TOC Generation: this is needed for FOP-0.9 and later: + # fop1.extensions=1 + pdf:xep.extensions=1 + # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9! + pdf:fop.extensions=0 + pdf:fop1.extensions=0 + # No indent on body text: + pdf:body.start.indent=0pt + # Margin size: + pdf:page.margin.inner=0.5in + # Margin size: + pdf:page.margin.outer=0.5in + # Paper type = A4 + pdf:paper.type=A4 + # Yes, we want graphics for admonishments: + admon.graphics=1 + # Set this one for PDF generation *only*: + # default pnd graphics are awful in PDF form, + # better use SVG's instead: + pdf:admon.graphics.extension=".svg" + pdf:use.role.for.mediaobject=1 + pdf:preferred.mediaobject.role=print + pdf:img.src.path=$(images_location)/ + pdf:admon.graphics.path=$(images_location)/../../svg-admon/ + pdf:draft.mode="no" + ; + + + + + + + + + + + + + diff --git a/doc/math-tr1.qbk b/doc/complex/complex-tr1.qbk similarity index 78% rename from doc/math-tr1.qbk rename to doc/complex/complex-tr1.qbk index 6287a7de3..f825f273b 100644 --- a/doc/math-tr1.qbk +++ b/doc/complex/complex-tr1.qbk @@ -1,17 +1,32 @@ +[article Complex Number TR1 Algorithms + [quickbook 1.4] + [copyright 2005 John Maddock] + [purpose Complex number arithmetic] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Maddock, John]] + [category math] + [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $] +] + [def __effects [*Effects: ]] [def __formula [*Formula: ]] [def __exm1 '''ex - 1'''] [def __ex '''ex'''] [def __te '''2ε'''] +[template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]] [section:inverse_complex Complex Number Inverse Trigonometric Functions] The following complex number algorithms are the inverses of trigonometric functions currently present in the C++ standard. Equivalents to these functions are part of the C99 standard, and -will be part of the forthcoming Technical Report on C++ Standard Library Extensions. +are part of the [tr1]. -[section Implementation and Accuracy] +[section:implementation Implementation and Accuracy] Although there are deceptively simple formulae available for all of these functions, a naive implementation that used these formulae would fail catastrophically for some input @@ -40,7 +55,7 @@ on the implementation methodology. __effects returns the inverse sine of the complex number z. -__formula [$../../libs/math/doc/images/asin.png] +__formula [$../../images/asin.png] [endsect] @@ -57,7 +72,7 @@ __formula [$../../libs/math/doc/images/asin.png] __effects returns the inverse cosine of the complex number z. -__formula [$../../libs/math/doc/images/acos.png] +__formula [$../../images/acos.png] [endsect] @@ -74,7 +89,7 @@ __formula [$../../libs/math/doc/images/acos.png] __effects returns the inverse tangent of the complex number z. -__formula [$../../libs/math/doc/images/atan.png] +__formula [$../../images/atan.png] [endsect] @@ -91,7 +106,7 @@ __formula [$../../libs/math/doc/images/atan.png] __effects returns the inverse hyperbolic sine of the complex number z. -__formula [$../../libs/math/doc/images/asinh.png] +__formula [$../../images/asinh.png] [endsect] @@ -108,7 +123,7 @@ __formula [$../../libs/math/doc/images/asinh.png] __effects returns the inverse hyperbolic cosine of the complex number z. -__formula [$../../libs/math/doc/images/acosh.png] +__formula [$../../images/acosh.png] [endsect] @@ -125,7 +140,7 @@ __formula [$../../libs/math/doc/images/acosh.png] __effects returns the inverse hyperbolic tangent of the complex number z. -__formula [$../../libs/math/doc/images/atanh.png] +__formula [$../../images/atanh.png] [endsect] @@ -140,3 +155,4 @@ __formula [$../../libs/math/doc/images/atanh.png] [endsect] + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex.html new file mode 100644 index 000000000..159b354ac --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex.html @@ -0,0 +1,68 @@ + + + + Complex + Number Inverse Trigonometric Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
+ Implementation and Accuracy
+
+ asin
+
+ acos
+
+ atan
+
+ asinh
+
+ acosh
+
+ atanh
+
History
+
+

+ The following complex number algorithms are the inverses of trigonometric functions + currently present in the C++ standard. Equivalents to these functions are part + of the C99 standard, and are part of the Technical + Report on C++ Library Extensions. +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acos.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acos.html new file mode 100644 index 000000000..fea581b6d --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acos.html @@ -0,0 +1,69 @@ + + + + + acos + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ + Header: +

+
+#include <boost/math/complex/acos.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> acos(const std::complex<T>& z);
+
+

+ Effects: returns the inverse cosine of + the complex number z. +

+

+ Formula: acos +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acosh.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acosh.html new file mode 100644 index 000000000..abd8957fd --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/acosh.html @@ -0,0 +1,69 @@ + + + + + acosh + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
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+ + Header: +

+
+#include <boost/math/complex/acosh.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> acosh(const std::complex<T>& z);
+
+

+ Effects: returns the inverse hyperbolic + cosine of the complex number z. +

+

+ Formula: acosh +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asin.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asin.html new file mode 100644 index 000000000..e784b48b5 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asin.html @@ -0,0 +1,69 @@ + + + + + asin + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ + Header: +

+
+#include <boost/math/complex/asin.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> asin(const std::complex<T>& z);
+
+

+ Effects: returns the inverse sine of the + complex number z. +

+

+ Formula: asin +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asinh.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asinh.html new file mode 100644 index 000000000..a7b1d67d2 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/asinh.html @@ -0,0 +1,69 @@ + + + + + asinh + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ + Header: +

+
+#include <boost/math/complex/asinh.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> asinh(const std::complex<T>& z);
+
+

+ Effects: returns the inverse hyperbolic + sine of the complex number z. +

+

+ Formula: asinh +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atan.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atan.html new file mode 100644 index 000000000..bab5195f0 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atan.html @@ -0,0 +1,69 @@ + + + + + atan + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ + Header: +

+
+#include <boost/math/complex/atan.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> atan(const std::complex<T>& z);
+
+

+ Effects: returns the inverse tangent of + the complex number z. +

+

+ Formula: atan +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atanh.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atanh.html new file mode 100644 index 000000000..66cde8890 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/atanh.html @@ -0,0 +1,68 @@ + + + + + atanh + + + + + + + + + + + + + + + +
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+ + Header: +

+
+#include <boost/math/complex/atanh.hpp>
+
+

+ + Synopsis: +

+
+template<class T> 
+std::complex<T> atanh(const std::complex<T>& z);
+
+

+ Effects: returns the inverse hyperbolic + tangent of the complex number z. +

+

+ Formula: atanh +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/history.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/history.html new file mode 100644 index 000000000..8d5402586 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/history.html @@ -0,0 +1,50 @@ + + + +History + + + + + + + + + + + + + + +
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    +
  • + 2005/12/17: Added support for platforms with no meaningful numeric_limits<>::infinity(). +
    • +
  • + 2005/12/01: Initial version, added as part of the TR1 library. +
    • +
+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/implementation.html b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/implementation.html new file mode 100644 index 000000000..43ba7b7d8 --- /dev/null +++ b/doc/complex/html/complex_number_tr1_algorithms/inverse_complex/implementation.html @@ -0,0 +1,64 @@ + + + + + Implementation and Accuracy + + + + + + + + + + + + + + + +
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+ Although there are deceptively simple formulae available for all of these + functions, a naive implementation that used these formulae would fail catastrophically + for some input values. The Boost versions of these functions have been implemented + using the methodology described in "Implementing the Complex Arcsine + and Arccosine Functions Using Exception Handling" by T. E. Hull Thomas + F. Fairgrieve and Ping Tak Peter Tang, ACM Transactions on Mathematical Software, + Vol. 23, No. 3, September 1997. This means that the functions are well defined + over the entire complex number range, and produce accurate values even at + the extremes of that range, where as a naive formula would cause overflow + or underflow to occur during the calculation, even though the result is actually + a representable value. The maximum theoretical relative error for all of + these functions is less than 9.5E for every machine-representable point in + the complex plane. Please refer to comments in the header files themselves + and to the above mentioned paper for more information on the implementation + methodology. +

+
+ + + +
Copyright © 2005 John Maddock

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/complex/html/index.html b/doc/complex/html/index.html new file mode 100644 index 000000000..7481583b5 --- /dev/null +++ b/doc/complex/html/index.html @@ -0,0 +1,72 @@ + + + +Complex Number TR1 Algorithms + + + + + + + + + + + + + +
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+

Copyright © 2005 John Maddock

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+
+

Table of Contents

+
+
Complex + Number Inverse Trigonometric Functions
+
+
+ Implementation and Accuracy
+
+ asin
+
+ acos
+
+ atan
+
+ asinh
+
+ acosh
+
+ atanh
+
History
+
+
+
+
+ + + +

Last revised: December 29, 2006 at 11:08:32 +0000

+
+
Next
+ + diff --git a/doc/gcd/Jamfile.v2 b/doc/gcd/Jamfile.v2 new file mode 100644 index 000000000..e7bcf5da1 --- /dev/null +++ b/doc/gcd/Jamfile.v2 @@ -0,0 +1,70 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +path-constant images_location : html ; + +xml math-gcd : math-gcd.qbk ; +boostbook standalone + : + math-gcd + : + # Path for links to Boost: + boost.root=../../../../.. + # Path for libraries index: + boost.libraries=../../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=10 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=1 + # How far down sections get TOC's + toc.section.depth=10 + # Max depth in each TOC: + toc.max.depth=4 + # How far down we go with TOC's + generate.section.toc.level=10 + #root.filename="sf_dist_and_tools" + + # PDF Options: + # TOC Generation: this is needed for FOP-0.9 and later: + # fop1.extensions=1 + pdf:xep.extensions=1 + # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9! + pdf:fop.extensions=0 + pdf:fop1.extensions=0 + # No indent on body text: + pdf:body.start.indent=0pt + # Margin size: + pdf:page.margin.inner=0.5in + # Margin size: + pdf:page.margin.outer=0.5in + # Paper type = A4 + pdf:paper.type=A4 + # Yes, we want graphics for admonishments: + admon.graphics=1 + # Set this one for PDF generation *only*: + # default pnd graphics are awful in PDF form, + # better use SVG's instead: + pdf:admon.graphics.extension=".svg" + pdf:use.role.for.mediaobject=1 + pdf:preferred.mediaobject.role=print + pdf:img.src.path=$(images_location)/ + pdf:admon.graphics.path=$(images_location)/../../svg-admon/ + pdf:draft.mode="no" + ; + + + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm.html new file mode 100644 index 000000000..7e6de349f --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm.html @@ -0,0 +1,61 @@ + + + + Greatest Common Divisor and Least + Common Multiple + + + + + + + + + + + + + + + +
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+
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+
Introduction
+
Synopsis
+
GCD Function + Object
+
LCM Function + Object
+
Run-time GCD & LCM + Determination
+
Compile time GCD and + LCM determination
+
Header <boost/math/common_factor.hpp>
+
Demonstration Program
+
Rationale
+
History
+
Credits
+
+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/compile_time.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/compile_time.html new file mode 100644 index 000000000..3f342cb77 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/compile_time.html @@ -0,0 +1,101 @@ + + + + Compile time GCD and + LCM determination + + + + + + + + + + + + + + + +
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+

+ Header: <boost/math/common_factor_ct.hpp> +

+
+template < unsigned long Value1, unsigned long Value2 >
+struct boost::math::static_gcd
+{
+   static unsigned long const  value = implementation_defined;
+};
+
+template < unsigned long Value1, unsigned long Value2 >
+struct boost::math::static_lcm
+{
+   static unsigned long const  value = implementation_defined;
+};
+
+

+ The boost::math::static_gcd and boost::math::static_lcm class templates take + two value-based template parameters of the unsigned long type and have a + single static constant data member, value, of that same type. The value of + that member is the greatest common factor or least common multiple, respectively, + of the template arguments. A compile-time error will occur if the least common + multiple is beyond the range of an unsigned long. +

+

+ + Example +

+
+#include <boost/math/common_factor.hpp>
+#include <algorithm>
+#include <iterator>
+
+
+int main()
+{
+   using std::cout;
+   using std::endl;
+
+   cout << "The GCD and LCM of 6 and 15 are "
+   << boost::math::gcd(6, 15) << " and "
+   << boost::math::lcm(6, 15) << ", respectively."
+   << endl;
+
+   cout << "The GCD and LCM of 8 and 9 are "
+   << boost::math::static_gcd<8, 9>::value
+   << " and "
+   << boost::math::static_lcm<8, 9>::value
+   << ", respectively." << endl;
+
+   int  a[] = { 4, 5, 6 }, b[] = { 7, 8, 9 }, c[3];
+   std::transform( a, a + 3, b, c, boost::math::gcd_evaluator<int>() );
+   std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
+}
+
+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/credits.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/credits.html new file mode 100644 index 000000000..1a7257a2a --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/credits.html @@ -0,0 +1,47 @@ + + + +Credits + + + + + + + + + + + + + + +
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+ The author of the Boost compilation of GCD and LCM computations is Daryle + Walker. The code was prompted by existing code hiding in the implementations + of Paul Moore's rational library and Steve Cleary's pool library. The code + had updates by Helmut Zeisel. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/demo.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/demo.html new file mode 100644 index 000000000..35375b82b --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/demo.html @@ -0,0 +1,49 @@ + + + + Demonstration Program + + + + + + + + + + + + + + + +
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+ The program common_factor_test.cpp + is a demonstration of the results from instantiating various examples of + the run-time GCD and LCM function templates and the compile-time GCD and + LCM class templates. (The run-time GCD and LCM class templates are tested + indirectly through the run-time function templates.) +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/gcd_function_object.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/gcd_function_object.html new file mode 100644 index 000000000..e83d7dd1e --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/gcd_function_object.html @@ -0,0 +1,77 @@ + + + +GCD Function + Object + + + + + + + + + + + + + + + +
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+

+ Header: <boost/math/common_factor_rt.hpp> +

+
+template < typename IntegerType >
+class boost::math::gcd_evaluator
+{
+public:
+   // Types
+   typedef IntegerType  result_type;
+   typedef IntegerType  first_argument_type;
+   typedef IntegerType  second_argument_type;
+
+   // Function object interface
+   result_type  operator ()( first_argument_type const &a,
+   second_argument_type const &b ) const;
+};
+
+

+ The boost::math::gcd_evaluator class template defines a function object class + to return the greatest common divisor of two integers. The template is parameterized + by a single type, called IntegerType here. This type should be a numeric + type that represents integers. The result of the function object is always + nonnegative, even if either of the operator arguments is negative. +

+

+ This function object class template is used in the corresponding version + of the GCD function template. If a numeric type wants to customize evaluations + of its greatest common divisors, then the type should specialize on the gcd_evaluator + class template. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/header.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/header.html new file mode 100644 index 000000000..c1021480a --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/header.html @@ -0,0 +1,52 @@ + + + + Header <boost/math/common_factor.hpp> + + + + + + + + + + + + + + + +
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+ This header simply includes the headers <boost/math/common_factor_ct.hpp> + and <boost/math/common_factor_rt.hpp>. +

+

+ Note this is a legacy header: it used to contain the actual implementation, + but the compile-time and run-time facilities were moved to separate headers + (since they were independent of each other). +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/history.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/history.html new file mode 100644 index 000000000..7f50fc731 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/history.html @@ -0,0 +1,53 @@ + + + +History + + + + + + + + + + + + + + + +
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    +
  • + 17 Dec 2005: Converted documentation to Quickbook Format. +
    • +
  • + 2 Jul 2002: Compile-time and run-time items separated to new headers. +
    • +
  • + 7 Nov 2001: Initial version +
    • +
+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/introduction.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/introduction.html new file mode 100644 index 000000000..125a18c06 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/introduction.html @@ -0,0 +1,49 @@ + + + +Introduction + + + + + + + + + + + + + + + +
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+

+ The class and function templates in <boost/math/common_factor.hpp> + provide run-time and compile-time evaluation of the greatest common divisor + (GCD) or least common multiple (LCM) of two integers. These facilities are + useful for many numeric-oriented generic programming problems. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/lcm_function_object.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/lcm_function_object.html new file mode 100644 index 000000000..61b16cc6f --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/lcm_function_object.html @@ -0,0 +1,80 @@ + + + +LCM Function + Object + + + + + + + + + + + + + + + +
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+

+ Header: <boost/math/common_factor_rt.hpp> +

+
+template < typename IntegerType >
+class boost::math::lcm_evaluator
+{
+public:
+   // Types
+   typedef IntegerType  result_type;
+   typedef IntegerType  first_argument_type;
+   typedef IntegerType  second_argument_type;
+
+   // Function object interface
+   result_type  operator ()( first_argument_type const &a,
+   second_argument_type const &b ) const;
+};
+
+

+ The boost::math::lcm_evaluator class template defines a function object class + to return the least common multiple of two integers. The template is parameterized + by a single type, called IntegerType here. This type should be a numeric + type that represents integers. The result of the function object is always + nonnegative, even if either of the operator arguments is negative. If the + least common multiple is beyond the range of the integer type, the results + are undefined. +

+

+ This function object class template is used in the corresponding version + of the LCM function template. If a numeric type wants to customize evaluations + of its least common multiples, then the type should specialize on the lcm_evaluator + class template. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/rationale.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/rationale.html new file mode 100644 index 000000000..deff5cdc0 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/rationale.html @@ -0,0 +1,48 @@ + + + +Rationale + + + + + + + + + + + + + + + +
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+ The greatest common divisor and least common multiple functions are greatly + used in some numeric contexts, including some of the other Boost libraries. + Centralizing these functions to one header improves code factoring and eases + maintainence. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/run_time.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/run_time.html new file mode 100644 index 000000000..10d621e07 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/run_time.html @@ -0,0 +1,66 @@ + + + + Run-time GCD & LCM + Determination + + + + + + + + + + + + + + + +
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+

+ Header: <boost/math/common_factor_rt.hpp> +

+
+template < typename IntegerType >
+IntegerType  boost::math::gcd( IntegerType const &a, IntegerType const &b );
+
+template < typename IntegerType >
+IntegerType  boost::math::lcm( IntegerType const &a, IntegerType const &b );
+
+

+ The boost::math::gcd function template returns the greatest common (nonnegative) + divisor of the two integers passed to it. The boost::math::lcm function template + returns the least common (nonnegative) multiple of the two integers passed + to it. The function templates are parameterized on the function arguments' + IntegerType, which is also the return type. Internally, these function templates + use an object of the corresponding version of the gcd_evaluator and lcm_evaluator + class templates, respectively. +

+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/gcd_and_lcm/gcd_lcm/synopsis.html b/doc/gcd/html/gcd_and_lcm/gcd_lcm/synopsis.html new file mode 100644 index 000000000..c8a4ea276 --- /dev/null +++ b/doc/gcd/html/gcd_and_lcm/gcd_lcm/synopsis.html @@ -0,0 +1,67 @@ + + + +Synopsis + + + + + + + + + + + + + + + +
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+
+namespace boost
+{
+namespace math
+{
+
+template < typename IntegerType >
+   class gcd_evaluator;
+template < typename IntegerType >
+   class lcm_evaluator;
+
+template < typename IntegerType >
+   IntegerType  gcd( IntegerType const &a, IntegerType const &b );
+template < typename IntegerType >
+   IntegerType  lcm( IntegerType const &a, IntegerType const &b );
+
+template < unsigned long Value1, unsigned long Value2 >
+   struct static_gcd;
+template < unsigned long Value1, unsigned long Value2 >
+   struct static_lcm;
+
+}
+}
+
+
+ + + +
Copyright © 2001 -2002 Daryle Walker

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/gcd/html/index.html b/doc/gcd/html/index.html new file mode 100644 index 000000000..27f350f62 --- /dev/null +++ b/doc/gcd/html/index.html @@ -0,0 +1,72 @@ + + + +GCD and LCM + + + + + + + + + + + + + +
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+
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+
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+

+GCD and LCM

+

+Daryle Walker +

+

Copyright © 2001 -2002 Daryle Walker

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+
+

Table of Contents

+
+
Greatest Common Divisor and Least + Common Multiple
+
+
Introduction
+
Synopsis
+
GCD Function + Object
+
LCM Function + Object
+
Run-time GCD & LCM + Determination
+
Compile time GCD and + LCM determination
+
Header <boost/math/common_factor.hpp>
+
Demonstration Program
+
Rationale
+
History
+
Credits
+
+
+
+
+ + + +

Last revised: December 29, 2006 at 11:08:32 +0000

+
+
Next
+ + diff --git a/doc/math-gcd.qbk b/doc/gcd/math-gcd.qbk similarity index 83% rename from doc/math-gcd.qbk rename to doc/gcd/math-gcd.qbk index 0f5e645bd..625b2a534 100644 --- a/doc/math-gcd.qbk +++ b/doc/gcd/math-gcd.qbk @@ -1,3 +1,16 @@ +[article GCD and LCM + [quickbook 1.3] + [copyright 2001-2002 Daryle Walker] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Walker, Daryle]] + [category math] + [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $] +] + [section:gcd_lcm Greatest Common Divisor and Least Common Multiple] @@ -40,7 +53,7 @@ programming problems. [section GCD Function Object] -[*Header: ] [@../../boost/math/common_factor_rt.hpp ] +[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp ] template < typename IntegerType > class boost::math::gcd_evaluator @@ -72,7 +85,7 @@ gcd_evaluator class template. [section LCM Function Object] -[*Header: ] [@../../boost/math/common_factor_rt.hpp ] +[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp ] template < typename IntegerType > class boost::math::lcm_evaluator @@ -103,9 +116,9 @@ specialize on the lcm_evaluator class template. [endsect] -[section Run-time GCD & LCM Determination] +[section:run_time Run-time GCD & LCM Determination] -[*Header: ] [@../../boost/math/common_factor_rt.hpp ] +[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp ] template < typename IntegerType > IntegerType boost::math::gcd( IntegerType const &a, IntegerType const &b ); @@ -124,9 +137,9 @@ gcd_evaluator and lcm_evaluator class templates, respectively. [endsect] -[section Compile time GCD and LCM determination] +[section:compile_time Compile time GCD and LCM determination] -[*Header: ] [@../../boost/math/common_factor_ct.hpp ] +[*Header: ] [@../../../../../boost/math/common_factor_ct.hpp ] template < unsigned long Value1, unsigned long Value2 > struct boost::math::static_gcd @@ -178,11 +191,11 @@ is beyond the range of an unsigned long. [endsect] -[section Header ] +[section:header Header ] This header simply includes the headers -[@../../boost/math/common_factor_ct.hpp ] -and [@../../boost/math/common_factor_rt.hpp ]. +[@../../../../../boost/math/common_factor_ct.hpp ] +and [@../../../../../boost/math/common_factor_rt.hpp ]. Note this is a legacy header: it used to contain the actual implementation, but the compile-time and run-time facilities @@ -190,9 +203,9 @@ were moved to separate headers (since they were independent of each other). [endsect] -[section Demonstration Program] +[section:demo Demonstration Program] -The program [@../../libs/math/test/common_factor_test.cpp common_factor_test.cpp] is a demonstration of the results from +The program [@../../../../../libs/math/test/common_factor_test.cpp common_factor_test.cpp] is a demonstration of the results from instantiating various examples of the run-time GCD and LCM function templates and the compile-time GCD and LCM class templates. (The run-time GCD and LCM class templates are tested indirectly through diff --git a/doc/html/index.html b/doc/html/index.html new file mode 100644 index 000000000..1948906a4 --- /dev/null +++ b/doc/html/index.html @@ -0,0 +1,372 @@ + + + +Boost.Math + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+
+
+
+

+Boost.Math

+

Copyright © 2007 Paul A. Bristow, Hubert Holin, John Maddock, Daryle + Walker and Xiaogang Zhang

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+

+ The following mathematical libraries are present in Boost: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Library +

+ 		+

+ Description +

+
+

+ Complex Number Inverse Trigonometric + Functions +

+ 		+

+ These complex number algorithms are the inverses of trigonometric functions + currently present in the C++ standard. Equivalents to these functions + are part of the C99 standard, and are part of the Technical + Report on C++ Library Extensions. +

+
+

+ Greatest Common Divisor and Least + Common Multiple +

+ 		+

+ The class and function templates in <boost/math/common_factor.hpp> + provide run-time and compile-time evaluation of the greatest common divisor + (GCD) or least common multiple (LCM) of two integers. These facilities + are useful for many numeric-oriented generic programming problems. +

+
+

+ Integer +

+ 		+

+ Headers to ease dealing with integral types. +

+
+

+ Interval +

+ 		+

+ As implied by its name, this library is intended to help manipulating + mathematical intervals. It consists of a single header <boost/numeric/interval.hpp> + and principally a type which can be used as interval<T>. +

+
+

+ Multi Array +

+ 		+

+ Boost.MultiArray provides a generic N-dimensional array concept definition + and common implementations of that interface. +

+
+

+ Numeric.Conversion +

+ 		+

+ The Boost Numeric Conversion library is a collection of tools to describe + and perform conversions between values of different numeric types. +

+
+

+ Octonions +

+ 		+

+ Octonions, like quaternions, + are a relative of complex numbers. +

+

+ Octonions see some use in theoretical physics. +

+

+ In practical terms, an octonion is simply an octuple of real numbers + (α,β,γ,δ,ε,ζ,η,θ), which we can write in the form o = α + βi + γj + δk + εe' + ζi' + ηj' + θk', + where i, j + and k are the same objects as + for quaternions, and e', i', + j' and k' + are distinct objects which play essentially the same kind of role as + i (or j + or k). +

+

+ Addition and a multiplication is defined on the set of octonions, which + generalize their quaternionic counterparts. The main novelty this time + is that the multiplication is not only not commutative, + is now not even associative (i.e. there are quaternions x, + y and z + such that x(yz) ≠ (xy)z). A way + of remembering things is by using the following multiplication table: +

+

+ octonion_blurb17 +

+

+ Octonions (and their kin) are described in far more details in this other + document (with errata + and addenda). +

+

+ Some traditional constructs, such as the exponential, carry over without + too much change into the realms of octonions, but other, such as taking + a square root, do not (the fact that the exponential has a closed form + is a result of the author, but the fact that the exponential exists at + all for octonions is known since quite a long time ago). +

+
+

+ Operators +

+ 		+

+ The header <boost/operators.hpp> supplies several sets of class + templates (in namespace boost). These templates define operators at namespace + scope in terms of a minimal number of fundamental operators provided + by the class. +

+
+

+ Special Functions +

+ 		+

+ Provides a number of high quality special functions, initially these + were concentrated on functions used in statistical applications along + with those in the Technical Report on C++ Library Extensions. +

+

+ The function families currently implemented are the gamma, beta & + erf functions along with the incomplete gamma and beta functions (four + variants of each) and all the possible inverses of these, plus digamma, + various factorial functions, Bessel functions, elliptic integrals, sinus + cardinals (along with their hyperbolic variants), inverse hyperbolic + functions, Legrendre/Laguerre/Hermite polynomials and various special + power and logarithmic functions. +

+

+ All the implementations are fully generic and support the use of arbitrary + "real-number" types, although they are optimised for use with + types with known-about significand (or mantissa) sizes: typically float, + double or long double. +

+
+

+ Statistical Distributions +

+ 		+

+ Provides a reasonably comprehensive set of statistical distributions, + upon which higher level statistical tests can be built. +

+

+ The initial focus is on the central univariate distributions. Both continuous + (like normal & Fisher) and discrete (like binomial & Poisson) + distributions are provided. +

+

+ A comprehensive tutorial is provided, along with a series of worked examples + illustrating how the library is used to conduct statistical tests. +

+
+

+ Quaternions +

+ 		+

+ Quaternions are a relative of complex numbers. +

+

+ Quaternions are in fact part of a small hierarchy of structures built + upon the real numbers, which comprise only the set of real numbers (traditionally + named R), the set + of complex numbers (traditionally named C), + the set of quaternions (traditionally named H) + and the set of octonions (traditionally named O), + which possess interesting mathematical properties (chief among which + is the fact that they are division algebras, i.e. + where the following property is true: if y + is an element of that algebra and is not equal + to zero, then yx = yx', + where x and x' + denote elements of that algebra, implies that x = + x'). Each member of the hierarchy is a super-set + of the former. +

+

+ One of the most important aspects of quaternions is that they provide + an efficient way to parameterize rotations in R3 + (the usual three-dimensional space) and R4. +

+

+ In practical terms, a quaternion is simply a quadruple of real numbers + (α,β,γ,δ), which we can write in the form q = α + βi + γj + δk, + where i is the same object as + for complex numbers, and j and + k are distinct objects which + play essentially the same kind of role as i. +

+

+ An addition and a multiplication is defined on the set of quaternions, + which generalize their real and complex counterparts. The main novelty + here is that the multiplication is not commutative + (i.e. there are quaternions x + and y such that xy + ≠ yx). A good mnemotechnical way of remembering things + is by using the formula i*i = j*j = k*k = -1. +

+

+ Quaternions (and their kin) are described in far more details in this + other document (with errata and addenda). +

+

+ Some traditional constructs, such as the exponential, carry over without + too much change into the realms of quaternions, but other, such as taking + a square root, do not. +

+
+

+ Random +

+ 		+

+ Random numbers are useful in a variety of applications. The Boost Random + Number Library (Boost.Random for short) provides a vast variety of generators + and distributions to produce random numbers having useful properties, + such as uniform distribution. +

+
+

+ Rational +

+ 		+

+ The header rational.hpp provides an implementation of rational numbers. + The implementation is template-based, in a similar manner to the standard + complex number class. +

+
+

+ uBLAS +

+ 		+

+ uBLAS is a C++ template class library that provides BLAS level 1, 2, + 3 functionality for dense, packed and sparse matrices. The design and + implementation unify mathematical notation via operator overloading and + efficient code generation via expression templates. +

+
+
+ + + +

Last revised: October 16, 2007 at 17:32:28 +0800

+
+
+ + diff --git a/doc/index.html b/doc/index.html index 419826279..20e2b748e 100644 --- a/doc/index.html +++ b/doc/index.html @@ -1,11 +1,11 @@ - + Automatic redirection failed, please go to - ../../../doc/html/boost_math.html + html/index.html

Copyright Daryle Walker, Hubert Holin and John Maddock 2006

Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at www.boost.org/LICENSE_1_0.txt).

@@ -14,3 +14,4 @@ + diff --git a/doc/math-background.qbk b/doc/math-background.qbk deleted file mode 100644 index 6a721979e..000000000 --- a/doc/math-background.qbk +++ /dev/null @@ -1,89 +0,0 @@ - -[def __form1 [^\]-1;1\[]] -[def __form2 [^\[0;+'''∞'''\[]] -[def __form3 [^\[+1;+'''∞'''\[]] -[def __form4 [^\]-'''∞''';0\]]] -[def __form5 [^x '''≥''' 0]] - - -[section Background Information and White Papers] - -[section The Inverse Hyperbolic Functions] - -The exponential funtion is defined, for all object for which this makes sense, -as the power series -[$../../libs/math/special_functions/graphics/special_functions_blurb1.jpeg], -with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]]. -In particular, the exponential function is well defined for real numbers, -complex number, quaternions, octonions, and matrices of complex numbers, -among others. - -[: ['[*Graph of exp on R]] ] - -[: [$../../libs/math/special_functions/graphics/exp_on_R.png] ] - -[: ['[*Real and Imaginary parts of exp on C]]] -[: [$../../libs/math/special_functions/graphics/Im_exp_on_C.png]] - -The hyperbolic functions are defined as power series which -can be computed (for reals, complex, quaternions and octonions) as: - -Hyperbolic cosine: [$../../libs/math/special_functions/graphics/special_functions_blurb5.jpeg] - -Hyperbolic sine: [$../../libs/math/special_functions/graphics/special_functions_blurb6.jpeg] - -Hyperbolic tangent: [$../../libs/math/special_functions/graphics/special_functions_blurb7.jpeg] - -[: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]] -[: [$../../libs/math/special_functions/graphics/trigonometric.png]] - -[: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]] -[: [$../../libs/math/special_functions/graphics/hyperbolic.png]] - -The hyperbolic sine is one to one on the set of real numbers, -with range the full set of reals, while the hyperbolic tangent is -also one to one on the set of real numbers but with range __form1, and -therefore both have inverses. The hyperbolic cosine is one to one from __form2 -onto __form3 (and from __form4 onto __form3); the inverse function we use -here is defined on __form3 with range __form2. - -The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent, -and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb15.jpeg]. - -The inverse of the hyperbolic sine is called the Argument hyperbolic sine, -and can be computed (for __form5) as [$../../libs/math/special_functions/graphics/special_functions_blurb17.jpeg]. - -The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine, -and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb18.jpeg]. - -[endsect] - -[section Sinus Cardinal and Hyperbolic Sinus Cardinal Functions] - -The Sinus Cardinal family of functions (indexed by the family of indices [^a > 0]) -is defined by -[$../../libs/math/special_functions/graphics/special_functions_blurb20.jpeg]; -it sees heavy use in signal processing tasks. - -By analogy, the Hyperbolic Sinus Cardinal family of functions -(also indexed by the family of indices [^a > 0]) is defined by -[$../../libs/math/special_functions/graphics/special_functions_blurb22.jpeg]. - -These two families of functions are composed of entire functions. - -[: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]] -[: [$../../libs/math/special_functions/graphics/sinc_pi_and_sinhc_pi_on_R.png]] - -[endsect] - -[section The Quaternionic Exponential] - -Please refer to the following PDF's: - -*[@../../libs/math/quaternion/TQE.pdf The Quaternionic Exponential (and beyond)] -*[@../../libs/math/quaternion/TQE_EA.pdf The Quaternionic Exponential (and beyond) ERRATA & ADDENDA] - -[endsect] - -[endsect] - diff --git a/doc/math-octonion.qbk b/doc/math-octonion.qbk deleted file mode 100644 index 8d7ecbd0e..000000000 --- a/doc/math-octonion.qbk +++ /dev/null @@ -1,983 +0,0 @@ - -[def __R ['[*R]]] -[def __C ['[*C]]] -[def __H ['[*H]]] -[def __O ['[*O]]] -[def __R3 ['[*'''R3''']]] -[def __R4 ['[*'''R4''']]] -[def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')] -[def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]] -[def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]] -[def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]] -[def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]] - - -[section Octonions] - -[section Overview] - -Octonions, like [link boost_math.quaternions quaternions], are a relative of complex numbers. - -Octonions see some use in theoretical physics. - -In practical terms, an octonion is simply an octuple of real numbers __octulple, -which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]] -are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']] -are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]). - -Addition and a multiplication is defined on the set of octonions, -which generalize their quaternionic counterparts. The main novelty this time -is that [*the multiplication is not only not commutative, is now not even -associative] (i.e. there are quaternions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal). -A way of remembering things is by using the following multiplication table: - -[$../../libs/math/octonion/graphics/octonion_blurb17.jpeg] - -Octonions (and their kin) are described in far more details in this other -[@../../libs/math/quaternion/TQE.pdf document] -(with [@../../libs/math/quaternion/TQE_EA.pdf errata and addenda]). - -Some traditional constructs, such as the exponential, carry over without too -much change into the realms of octonions, but other, such as taking a square root, -do not (the fact that the exponential has a closed form is a result of the -author, but the fact that the exponential exists at all for octonions is known -since quite a long time ago). - -[endsect] - -[section Header File] - -The interface and implementation are both supplied by the header file -[@../../boost/math/octonion.hpp octonion.hpp]. - -[endsect] - -[section Synopsis] - - namespace boost{ namespace math{ - - template class ``[link boost_math.octonions.template_class_octonion octonion]``; - template<> class ``[link boost_math.octonions.octonion_specializations octonion]``; - template<> class ``[link boost_math.octonion_double octonion]``; - template<> class ``[link boost_math.octonion_long_double octonion]``; - - // operators - - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (T const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (octonion const & lhs, T const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_addition_operators operator +]`` (octonion const & lhs, octonion const & rhs); - - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (T const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (octonion const & lhs, T const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_subtraction_operators operator -]`` (octonion const & lhs, octonion const & rhs); - - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (T const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (octonion const & lhs, T const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_multiplication_operators operator *]`` (octonion const & lhs, octonion const & rhs); - - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (T const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (octonion const & lhs, T const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.binary_division_operators operator /]`` (octonion const & lhs, octonion const & rhs); - - template octonion ``[link boost_math.octonions.octonion_non_member_operators.unary_plus_and_minus_operators operator +]`` (octonion const & o); - template octonion ``[link boost_math.octonions.octonion_non_member_operators.unary_plus_and_minus_operators operator -]`` (octonion const & o); - - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (T const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (octonion const & lhs, T const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (::std::complex const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (octonion const & lhs, ::std::complex const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_equality_operators operator ==]`` (octonion const & lhs, octonion const & rhs); - - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (T const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (octonion const & lhs, T const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (::std::complex const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (octonion const & lhs, ::std::complex const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool ``[link boost_math.octonions.octonion_non_member_operators.binary_inequality_operators operator !=]`` (octonion const & lhs, octonion const & rhs); - - template - ::std::basic_istream & ``[link boost_math.octonions.octonion_non_member_operators.stream_extractor operator >>]`` (::std::basic_istream & is, octonion & o); - - template - ::std::basic_ostream & ``[link boost_math.octonions.octonion_non_member_operators.stream_inserter operator <<]`` (::std::basic_ostream & os, octonion const & o); - - // values - - template T ``[link boost_math.octonions.octonion_value_operations.real_and_unreal real]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonion_value_operations.real_and_unreal unreal]``(octonion const & o); - - template T ``[link boost_math.octonions.octonion_value_operations.sup sup]``(octonion const & o); - template T ``[link boost_math.octonions.octonion_value_operations.l1 l1]``(octonionconst & o); - template T ``[link boost_math.octonions.octonion_value_operations.abs abs]``(octonion const & o); - template T ``[link boost_math.octonions.octonion_value_operations.norm norm]``(octonionconst & o); - template octonion ``[link boost_math.octonions.octonion_value_operations.conj conj]``(octonion const & o); - - template octonion ``[link boost_math.octonions.quaternion_creation_functions spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); - template octonion ``[link boost_math.octonions.quaternion_creation_functions multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); - template octonion ``[link boost_math.octonions.quaternion_creation_functions cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); - - // transcendentals - - template octonion ``[link boost_math.octonions.octonions_transcendentals.exp exp]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.cos cos]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.sin sin]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.tan tan]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.cosh cosh]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.sinh sinh]``(octonion const & o); - template octonion ``[link boost_math.octonions.octonions_transcendentals.tanh tanh]``(octonion const & o); - - template octonion ``[link boost_math.octonions.octonions_transcendentals.pow pow]``(octonion const & o, int n); - - } } // namespaces - -[endsect] - -[section Template Class octonion] - - namespace boost{ namespace math { - - template - class octonion - { - public: - typedef T value_type; - - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - template - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - - T ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; - octonion ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; - - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; - T ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; - - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; - - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; - - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (T const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (T const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (T const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (T const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (T const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); - }; - - } } // namespaces - -[endsect] - -[section Octonion Specializations] - - namespace boost{ namespace math{ - - template<> - class octonion - { - public: - typedef float value_type; - - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - - float ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; - octonion ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; - - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; - float ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; - - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; - - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; - - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (float const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (float const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (float const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (float const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (float const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); - }; - -[#boost_math.octonion_double] - - template<> - class octonion - { - public: - typedef double value_type; - - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - - double ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; - octonion ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; - - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; - double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; - - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; - - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; - - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (double const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); - }; - -[#boost_math.octonion_long_double] - - template<> - class octonion - { - public: - typedef long double value_type; - - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``( ::boost::math::quaternion const & q0, ::boost::math::quaternion const & z1 = ::boost::math::quaternion()); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - explicit ``[link boost_math.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); - - long double ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; - octonion ``[link boost_math.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; - - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; - long double ``[link boost_math.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; - - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; - ::std::complex ``[link boost_math.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; - - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link boost_math.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; - - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (long double const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link boost_math.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (long double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (long double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (long double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); - - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (long double const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); - template - octonion & ``[link boost_math.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); - }; - - } } // namespaces - -[endsect] - -[section Octonion Member Typedefs] - -[*value_type] - -Template version: - - typedef T value_type; - -Float specialization version: - - typedef float value_type; - -Double specialization version: - - typedef double value_type; - -Long double specialization version: - - typedef long double value_type; - -These provide easy acces to the type the template is built upon. - -[endsect] - -[section Octonion Member Functions] - -[h3 Constructors] - -Template version: - - explicit octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); - explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - template - explicit octonion(octonion const & a_recopier); - -Float specialization version: - - explicit octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); - explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit octonion(octonion const & a_recopier); - explicit octonion(octonion const & a_recopier); - -Double specialization version: - - explicit octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); - explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit octonion(octonion const & a_recopier); - explicit octonion(octonion const & a_recopier); - -Long double specialization version: - - explicit octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); - explicit octonion( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); - explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit octonion(octonion const & a_recopier); - explicit octonion(octonion const & a_recopier); - -A default constructor is provided for each form, which initializes each component -to the default values for their type (i.e. zero for floating numbers). -This constructor can also accept one to eight base type arguments. -A constructor is also provided to build octonions from one to four complex numbers -sharing the same base type, and another taking one or two quaternions -sharing the same base type. The unspecialized template also sports a -templarized copy constructor, while the specialized forms have copy -constructors from the other two specializations, which are explicit -when a risk of precision loss exists. For the unspecialized form, -the base type's constructors must not throw. - -Destructors and untemplated copy constructors (from the same type) -are provided by the compiler. Converting copy constructors make use -of a templated helper function in a "detail" subnamespace. - -[h3 Other member functions] - -[h4 Real and Unreal Parts] - - T real() const; - octonion unreal() const; - -Like complex number, octonions do have a meaningful notion of "real part", -but unlike them there is no meaningful notion of "imaginary part". -Instead there is an "unreal part" which itself is a octonion, -and usually nothing simpler (as opposed to the complex number case). -These are returned by the first two functions. - -[h4 Individual Real Components] - - T R_component_1() const; - T R_component_2() const; - T R_component_3() const; - T R_component_4() const; - T R_component_5() const; - T R_component_6() const; - T R_component_7() const; - T R_component_8() const; - -A octonion having eight real components, these are returned by -these eight functions. Hence real and R_component_1 return the same value. - -[h4 Individual Complex Components] - - ::std::complex C_component_1() const; - ::std::complex C_component_2() const; - ::std::complex C_component_3() const; - ::std::complex C_component_4() const; - -A octonion likewise has four complex components. Actually, octonions -are indeed a (left) vector field over the complexes, but beware, as -for any octonion __oct_formula we also have __oct_complex_formula -(note the [*minus] sign in the last factor). -What the C_component_n functions return, however, are the complexes -which could be used to build the octonion using the constructor, and -[*not] the components of the octonion on the basis ['[^(1, j, e', j')]]. - -[h4 Individual Quaternion Components] - - ::boost::math::quaternion H_component_1() const; - ::boost::math::quaternion H_component_2() const; - -Likewise, for any octonion __oct_formula we also have __oct_quat_formula, though there -is no meaningful vector-space-like structure based on the quaternions. -What the H_component_n functions return are the quaternions which -could be used to build the octonion using the constructor. - -[h3 Octonion Member Operators] -[h4 Assignment Operators] - - octonion & operator = (octonion const & a_affecter); - template - octonion & operator = (octonion const & a_affecter); - octonion & operator = (T const & a_affecter); - octonion & operator = (::std::complex const & a_affecter); - octonion & operator = (::boost::math::quaternion const & a_affecter); - -These perform the expected assignment, with type modification if -necessary (for instance, assigning from a base type will set the -real part to that value, and all other components to zero). -For the unspecialized form, the base type's assignment operators must not throw. - -[h4 Other Member Operators] - - octonion & operator += (T const & rhs) - octonion & operator += (::std::complex const & rhs); - octonion & operator += (::boost::math::quaternion const & rhs); - template - octonion & operator += (octonion const & rhs); - -These perform the mathematical operation `(*this)+rhs` and store the result in -`*this`. The unspecialized form has exception guards, which the specialized -forms do not, so as to insure exception safety. For the unspecialized form, -the base type's assignment operators must not throw. - - octonion & operator -= (T const & rhs) - octonion & operator -= (::std::complex const & rhs); - octonion & operator -= (::boost::math::quaternion const & rhs); - template - octonion & operator -= (octonion const & rhs); - -These perform the mathematical operation `(*this)-rhs` and store the result -in `*this`. The unspecialized form has exception guards, which the -specialized forms do not, so as to insure exception safety. -For the unspecialized form, the base type's assignment operators must not throw. - - octonion & operator *= (T const & rhs) - octonion & operator *= (::std::complex const & rhs); - octonion & operator *= (::boost::math::quaternion const & rhs); - template - octonion & operator *= (octonion const & rhs); - -These perform the mathematical operation `(*this)*rhs` in this order -(order is important as multiplication is not commutative for octonions) -and store the result in `*this`. The unspecialized form has exception guards, -which the specialized forms do not, so as to insure exception safety. -For the unspecialized form, the base type's assignment operators must -not throw. Also, for clarity's sake, you should always group the -factors in a multiplication by groups of two, as the multiplication is -not even associative on the octonions (though there are of course cases -where this does not matter, it usually does). - - octonion & operator /= (T const & rhs) - octonion & operator /= (::std::complex const & rhs); - octonion & operator /= (::boost::math::quaternion const & rhs); - template - octonion & operator /= (octonion const & rhs); - -These perform the mathematical operation `(*this)*inverse_of(rhs)` -in this order (order is important as multiplication is not commutative -for octonions) and store the result in `*this`. The unspecialized form -has exception guards, which the specialized forms do not, so as to -insure exception safety. For the unspecialized form, the base -type's assignment operators must not throw. As for the multiplication, -remember to group any two factors using parenthesis. - -[endsect] - -[section Octonion Non-Member Operators] - -[h4 Unary Plus and Minus Operators] - - template octonion operator + (octonion const & o); - -This unary operator simply returns o. - - template octonion operator - (octonion const & o); - -This unary operator returns the opposite of o. - -[h4 Binary Addition Operators] - - template octonion operator + (T const & lhs, octonion const & rhs); - template octonion operator + (octonion const & lhs, T const & rhs); - template octonion operator + (::std::complex const & lhs, octonion const & rhs); - template octonion operator + (octonion const & lhs, ::std::complex const & rhs); - template octonion operator + (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion operator + (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion operator + (octonion const & lhs, octonion const & rhs); - -These operators return `octonion(lhs) += rhs`. - -[h4 Binary Subtraction Operators] - - template octonion operator - (T const & lhs, octonion const & rhs); - template octonion operator - (octonion const & lhs, T const & rhs); - template octonion operator - (::std::complex const & lhs, octonion const & rhs); - template octonion operator - (octonion const & lhs, ::std::complex const & rhs); - template octonion operator - (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion operator - (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion operator - (octonion const & lhs, octonion const & rhs); - -These operators return `octonion(lhs) -= rhs`. - -[h4 Binary Multiplication Operators] - - template octonion operator * (T const & lhs, octonion const & rhs); - template octonion operator * (octonion const & lhs, T const & rhs); - template octonion operator * (::std::complex const & lhs, octonion const & rhs); - template octonion operator * (octonion const & lhs, ::std::complex const & rhs); - template octonion operator * (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion operator * (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion operator * (octonion const & lhs, octonion const & rhs); - -These operators return `octonion(lhs) *= rhs`. - -[h4 Binary Division Operators] - - template octonion operator / (T const & lhs, octonion const & rhs); - template octonion operator / (octonion const & lhs, T const & rhs); - template octonion operator / (::std::complex const & lhs, octonion const & rhs); - template octonion operator / (octonion const & lhs, ::std::complex const & rhs); - template octonion operator / (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion operator / (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion operator / (octonion const & lhs, octonion const & rhs); - -These operators return `octonion(lhs) /= rhs`. It is of course still an -error to divide by zero... - -[h4 Binary Equality Operators] - - template bool operator == (T const & lhs, octonion const & rhs); - template bool operator == (octonion const & lhs, T const & rhs); - template bool operator == (::std::complex const & lhs, octonion const & rhs); - template bool operator == (octonion const & lhs, ::std::complex const & rhs); - template bool operator == (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool operator == (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool operator == (octonion const & lhs, octonion const & rhs); - -These return true if and only if the four components of `octonion(lhs)` -are equal to their counterparts in `octonion(rhs)`. As with any -floating-type entity, this is essentially meaningless. - -[h4 Binary Inequality Operators] - - template bool operator != (T const & lhs, octonion const & rhs); - template bool operator != (octonion const & lhs, T const & rhs); - template bool operator != (::std::complex const & lhs, octonion const & rhs); - template bool operator != (octonion const & lhs, ::std::complex const & rhs); - template bool operator != (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool operator != (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool operator != (octonion const & lhs, octonion const & rhs); - -These return true if and only if `octonion(lhs) == octonion(rhs)` -is false. As with any floating-type entity, this is essentially meaningless. - -[h4 Stream Extractor] - - template - ::std::basic_istream & operator >> (::std::basic_istream & is, octonion & o); - -Extracts an octonion `o`. We accept any format which seems reasonable. -However, since this leads to a great many ambiguities, decisions were made -to lift these. In case of doubt, stick to lists of reals. - -The input values must be convertible to T. If bad input is encountered, -calls `is.setstate(ios::failbit)` (which may throw `ios::failure` (27.4.5.3)). - -Returns `is`. - -[h4 Stream Inserter] - - template - ::std::basic_ostream & operator << (::std::basic_ostream & os, octonion const & o); - -Inserts the octonion `o` onto the stream `os` as if it were implemented as follows: - - template - ::std::basic_ostream & operator << ( ::std::basic_ostream & os, - octonion const & o) - { - ::std::basic_ostringstream s; - - s.flags(os.flags()); - s.imbue(os.getloc()); - s.precision(os.precision()); - - s << '(' << o.R_component_1() << ',' - << o.R_component_2() << ',' - << o.R_component_3() << ',' - << o.R_component_4() << ',' - << o.R_component_5() << ',' - << o.R_component_6() << ',' - << o.R_component_7() << ',' - << o.R_component_8() << ')'; - - return os << s.str(); - } - -[endsect] - -[section Octonion Value Operations] - -[h4 Real and Unreal] - - template T real(octonion const & o); - template octonion unreal(octonion const & o); - -These return `o.real()` and `o.unreal()` respectively. - -[h4 conj] - - template octonion conj(octonion const & o); - -This returns the conjugate of the octonion. - -[h4 sup] - - template T sup(octonion const & o); - -This return the sup norm (the greatest among -`abs(o.R_component_1())...abs(o.R_component_8()))` of the octonion. - -[h4 l1] - - template T l1(octonion const & o); - -This return the l1 norm (`abs(o.R_component_1())+...+abs(o.R_component_8())`) -of the octonion. - -[h4 abs] - - template T abs(octonion const & o); - -This return the magnitude (Euclidian norm) of the octonion. - -[h4 norm] - - template T norm(octonionconst & o); - -This return the (Cayley) norm of the octonion. The term "norm" might -be confusing, as most people associate it with the Euclidian norm -(and quadratic functionals). For this version of (the mathematical -objects known as) octonions, the Euclidian norm (also known as -magnitude) is the square root of the Cayley norm. - -[endsect] - -[section Quaternion Creation Functions] - - template octonion spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); - template octonion multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); - template octonion cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); - -These build octonions in a way similar to the way polar builds -complex numbers, as there is no strict equivalent to -polar coordinates for octonions. - -`spherical` is a simple transposition of `polar`, it takes as inputs a -(positive) magnitude and a point on the hypersphere, given -by three angles. The first of these, ['theta] has a natural range of --pi to +pi, and the other two have natural ranges of --pi/2 to +pi/2 (as is the case with the usual spherical -coordinates in __R3). Due to the many symmetries and periodicities, -nothing untoward happens if the magnitude is negative or the angles are -outside their natural ranges. The expected degeneracies (a magnitude of -zero ignores the angles settings...) do happen however. - -`cylindrical` is likewise a simple transposition of the usual -cylindrical coordinates in __R3, which in turn is another derivative of -planar polar coordinates. The first two inputs are the polar -coordinates of the first __C component of the octonion. The third and -fourth inputs are placed into the third and fourth __R components of the -octonion, respectively. - -`multipolar` is yet another simple generalization of polar coordinates. -This time, both __C components of the octonion are given in polar coordinates. - -In this version of our implementation of octonions, there is no -analogue of the complex value operation arg as the situation is -somewhat more complicated. - -[endsect] - -[section Octonions Transcendentals] - -There is no `log` or `sqrt` provided for octonions in this implementation, -and `pow` is likewise restricted to integral powers of the exponent. -There are several reasons to this: on the one hand, the equivalent of -analytic continuation for octonions ("branch cuts") remains to be -investigated thoroughly (by me, at any rate...), and we wish to avoid -the nonsense introduced in the standard by exponentiations of -complexes by complexes (which is well defined, but not in the standard...). -Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is -just plain brain-dead... - -We do, however provide several transcendentals, chief among which is -the exponential. That it allows for a "closed formula" is a result -of the author (the existence and definition of the exponential, on the -octonions among others, on the other hand, is a few centuries old). -Basically, any converging power series with real coefficients which -allows for a closed formula in __C can be transposed to __O. More -transcendentals of this type could be added in a further revision upon -request. It should be noted that it is these functions which force the -dependency upon the -[@../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] -and the -[@../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] -headers. - -[h4 exp] - - template - octonion exp(octonion const & o); - -Computes the exponential of the octonion. - -[h4 cos] - - template - octonion cos(octonion const & o); - -Computes the cosine of the octonion - -[h4 sin] - - template - octonion sin(octonion const & o); - -Computes the sine of the octonion. - -[h4 tan] - - template - octonion tan(octonion const & o); - -Computes the tangent of the octonion. - -[h4 cosh] - - template - octonion cosh(octonion const & o); - -Computes the hyperbolic cosine of the octonion. - -[h4 sinh] - - template - octonion sinh(octonion const & o); - -Computes the hyperbolic sine of the octonion. - -[h4 tanh] - - template - octonion tanh(octonion const & o); - -Computes the hyperbolic tangent of the octonion. - -[h4 pow] - - template - octonion pow(octonion const & o, int n); - -Computes the n-th power of the octonion q. - -[endsect] - -[section Test Program] - -The [@../../libs/math/octonion/octonion_test.cpp octonion_test.cpp] -test program tests octonions specialisations for float, double and long double -([@../../libs/math/octonion/output.txt sample output]). - -If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional -output ([@../../libs/math/octonion/output_more.txt verbose output]); this will -only be helpfull if you enable message output at the same time, of course -(by uncommenting the relevant line in the test or by adding --log_level=messages -to your command line,...). In that case, and if you are running interactively, -you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to -interactively test the input operator with input of your choice from the -standard input (instead of hard-coding it in the test). - -[endsect] - -[section Acknowledgements] - -The mathematical text has been typeset with -[@http://www.nisus-soft.com/ Nisus Writer]. -Jens Maurer has helped with portability and standard adherence, and was the -Review Manager for this library. More acknowledgements in the -History section. Thank you to all who contributed to the discussion about this library. - -[endsect] - -[section History] - -* 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. -* 1.5.7 - 25/02/2003: transitionned to the unit test framework; now included by the library header (rather than the test files), via . -* 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). -* 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. -* 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes. -* 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). -* 1.5.2 - 07/07/2001: introduced namespace math. -* 1.5.1 - 07/06/2001: (end of Boost review) now includes and instead of ; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. -* 1.5.0 - 23/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version). -* 1.4.0 - 09/01/2001: added tan and tanh. -* 1.3.1 - 08/01/2001: cosmetic fixes. -* 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. -* 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. -* 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. -* 1.0.0 - 10/08/1999: first public version. - -[endsect] - -[section To Do] - -* Improve testing. -* Rewrite input operatore using Spirit (creates a dependency). -* Put in place an Expression Template mechanism (perhaps borrowing from uBlas). - -[endsect] - -[endsect] diff --git a/doc/math-quaternion.qbk b/doc/math-quaternion.qbk deleted file mode 100644 index 4971c35b9..000000000 --- a/doc/math-quaternion.qbk +++ /dev/null @@ -1,899 +0,0 @@ - -[def __R ['[*R]]] -[def __C ['[*C]]] -[def __H ['[*H]]] -[def __O ['[*O]]] -[def __R3 ['[*'''R3''']]] -[def __R4 ['[*'''R4''']]] -[def __quadrulple ('''α,β,γ,δ''')] -[def __quat_formula ['[^q = '''α + βi + γj + δk''']]] -[def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] -[def __not_equal ['[^xy '''≠''' yx]]] - - -[section Quaternions] - -[section Overview] - -Quaternions are a relative of complex numbers. - -Quaternions are in fact part of a small hierarchy of structures built -upon the real numbers, which comprise only the set of real numbers -(traditionally named __R), the set of complex numbers (traditionally named __C), -the set of quaternions (traditionally named __H) and the set of octonions -(traditionally named __O), which possess interesting mathematical properties -(chief among which is the fact that they are ['division algebras], -['i.e.] where the following property is true: if ['[^y]] is an element of that -algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] -denote elements of that algebra, implies that ['[^x = x']]). -Each member of the hierarchy is a super-set of the former. - -One of the most important aspects of quaternions is that they provide an -efficient way to parameterize rotations in __R3 (the usual three-dimensional space) -and __R4. - -In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, -which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, -and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]]. - -An addition and a multiplication is defined on the set of quaternions, -which generalize their real and complex counterparts. The main novelty -here is that [*the multiplication is not commutative] (i.e. there are -quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering -things is by using the formula ['[^i*i = j*j = k*k = -1]]. - -Quaternions (and their kin) are described in far more details in this -other [@../../libs/math/quaternion/TQE.pdf document] -(with [@../../libs/math/quaternion/TQE_EA.pdf errata and addenda]). - -Some traditional constructs, such as the exponential, carry over without -too much change into the realms of quaternions, but other, such as taking -a square root, do not. - -[endsect] - -[section Header File] - -The interface and implementation are both supplied by the header file -[@../../boost/math/quaternion.hpp quaternion.hpp]. - -[endsect] - -[section Synopsis] - - namespace boost{ namespace math{ - - template class ``[link boost_math.quaternions.template_class_quaternion quaternion]``; - template<> class ``[link boost_math.quaternions.quaternion_specializations quaternion]``; - template<> class ``[link boost_math.quaternion_double quaternion]``; - template<> class ``[link boost_math.quaternion_long_double quaternion]``; - - // operators - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion const & lhs, quaternion const & rhs); - - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion const & lhs, quaternion const & rhs); - - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion const & lhs, quaternion const & rhs); - - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion const & lhs, quaternion const & rhs); - - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.unary_plus operator +]`` (quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_non_member_operators.unary_minus operator -]`` (quaternion const & q); - - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (T const & lhs, quaternion const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion const & lhs, T const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (::std::complex const & lhs, quaternion const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion const & lhs, ::std::complex const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion const & lhs, quaternion const & rhs); - - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (T const & lhs, quaternion const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion const & lhs, T const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (::std::complex const & lhs, quaternion const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion const & lhs, ::std::complex const & rhs); - template bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion const & lhs, quaternion const & rhs); - - template - ::std::basic_istream& ``[link boost_math.quaternions.quaternion_non_member_operators.stream_extractor operator >>]`` (::std::basic_istream & is, quaternion & q); - - template - ::std::basic_ostream& operator ``[link boost_math.quaternions.quaternion_non_member_operators.stream_inserter operator <<]`` (::std::basic_ostream & os, quaternion const & q); - - // values - template T ``[link boost_math.quaternions.quaternion_value_operations.real_and_unreal real]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_value_operations.real_and_unreal unreal]``(quaternion const & q); - - template T ``[link boost_math.quaternions.quaternion_value_operations.sup sup]``(quaternion const & q); - template T ``[link boost_math.quaternions.quaternion_value_operations.l1 l1]``(quaternion const & q); - template T ``[link boost_math.quaternions.quaternion_value_operations.abs abs]``(quaternion const & q); - template T ``[link boost_math.quaternions.quaternion_value_operations.norm norm]``(quaternionconst & q); - template quaternion ``[link boost_math.quaternions.quaternion_value_operations.conj conj]``(quaternion const & q); - - template quaternion ``[link boost_math.quaternions.creation_spherical spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); - template quaternion ``[link boost_math.quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); - template quaternion ``[link boost_math.quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); - template quaternion ``[link boost_math.quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); - template quaternion ``[link boost_math.quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); - - // transcendentals - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.exp exp]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.cos cos]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.sin sin]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.tan tan]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.cosh cosh]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.sinh sinh]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.tanh tanh]``(quaternion const & q); - template quaternion ``[link boost_math.quaternions.quaternion_transcendentals.pow pow]``(quaternion const & q, int n); - - } // namespace math - } // namespace boost - -[endsect] - -[section Template Class quaternion] - - namespace boost{ namespace math{ - - template - class quaternion - { - public: - - typedef T ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; - - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - template - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - - T ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; - quaternion ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; - T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; - T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; - T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; - T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(T const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex const & a_affecter); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(T const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(T const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(T const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(T const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion const & rhs); - }; - - } // namespace math - } // namespace boost - -[endsect] - -[section Quaternion Specializations] - - namespace boost{ namespace math{ - - template<> - class quaternion - { - public: - typedef float ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; - - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - - float ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; - quaternion ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; - float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; - float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; - float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; - float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(float const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex const & a_affecter); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(float const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(float const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(float const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(float const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion const & rhs); - }; - -[#boost_math.quaternion_double] - - template<> - class quaternion - { - public: - typedef double ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; - - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - - double ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; - quaternion ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; - double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; - double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; - double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; - double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(double const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex const & a_affecter); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion const & rhs); - }; - -[#boost_math.quaternion_long_double] - - template<> - class quaternion - { - public: - typedef long double ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; - - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion const & a_recopier); - - long double ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; - quaternion ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; - long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; - long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; - long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; - long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; - ::std::complex ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(long double const & a_affecter); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex const & a_affecter); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(long double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(long double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(long double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion const & rhs); - - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(long double const & rhs); - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex const & rhs); - template - quaternion& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion const & rhs); - }; - - } // namespace math - } // namespace boost - -[endsect] - -[section Quaternion Member Typedefs] - -[*value_type] - -Template version: - - typedef T value_type; - -Float specialization version: - - typedef float value_type; - -Double specialization version: - - typedef double value_type; - -Long double specialization version: - - typedef long double value_type; - -These provide easy acces to the type the template is built upon. - -[endsect] - -[section Quaternion Member Functions] -[h3 Constructors] - -Template version: - - explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); - explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - template - explicit quaternion(quaternion const & a_recopier); - -Float specialization version: - - explicit quaternion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); - explicit quaternion(::std::complex const & z0,::std::complex const & z1 = ::std::complex()); - explicit quaternion(quaternion const & a_recopier); - explicit quaternion(quaternion const & a_recopier); - -Double specialization version: - - explicit quaternion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); - explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit quaternion(quaternion const & a_recopier); - explicit quaternion(quaternion const & a_recopier); - -Long double specialization version: - - explicit quaternion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L); - explicit quaternion( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit quaternion(quaternion const & a_recopier); - explicit quaternion(quaternion const & a_recopier); - -A default constructor is provided for each form, which initializes -each component to the default values for their type -(i.e. zero for floating numbers). This constructor can also accept -one to four base type arguments. A constructor is also provided to -build quaternions from one or two complex numbers sharing the same -base type. The unspecialized template also sports a templarized copy -constructor, while the specialized forms have copy constructors -from the other two specializations, which are explicit when a risk of -precision loss exists. For the unspecialized form, the base type's -constructors must not throw. - -Destructors and untemplated copy constructors (from the same type) are -provided by the compiler. Converting copy constructors make use of a -templated helper function in a "detail" subnamespace. - -[h3 Other member functions] -[h4 Real and Unreal Parts] - - T real() const; - quaternion unreal() const; - -Like complex number, quaternions do have a meaningful notion of "real part", -but unlike them there is no meaningful notion of "imaginary part". -Instead there is an "unreal part" which itself is a quaternion, -and usually nothing simpler (as opposed to the complex number case). -These are returned by the first two functions. - -[h4 Individual Real Components] - - T R_component_1() const; - T R_component_2() const; - T R_component_3() const; - T R_component_4() const; - -A quaternion having four real components, these are returned by these four -functions. Hence real and R_component_1 return the same value. - -[h4 Individual Complex Components] - - ::std::complex C_component_1() const; - ::std::complex C_component_2() const; - -A quaternion likewise has two complex components, and as we have seen above, -for any quaternion __quat_formula we also have __quat_complex_formula. These functions return them. -The real part of `q.C_component_1()` is the same as `q.real()`. - -[h3 Quaternion Member Operators] -[h4 Assignment Operators] - - quaternion& operator = (quaternion const & a_affecter); - template - quaternion& operator = (quaternion const& a_affecter); - quaternion& operator = (T const& a_affecter); - quaternion& operator = (::std::complex const& a_affecter); - -These perform the expected assignment, with type modification if necessary -(for instance, assigning from a base type will set the real part to that -value, and all other components to zero). For the unspecialized form, -the base type's assignment operators must not throw. - -[h4 Addition Operators] - - quaternion& operator += (T const & rhs) - quaternion& operator += (::std::complex const & rhs); - template - quaternion& operator += (quaternion const & rhs); - -These perform the mathematical operation `(*this)+rhs` and store the result in -`*this`. The unspecialized form has exception guards, which the specialized -forms do not, so as to insure exception safety. For the unspecialized form, -the base type's assignment operators must not throw. - -[h4 Subtraction Operators] - - quaternion& operator -= (T const & rhs) - quaternion& operator -= (::std::complex const & rhs); - template - quaternion& operator -= (quaternion const & rhs); - -These perform the mathematical operation `(*this)-rhs` and store the result -in `*this`. The unspecialized form has exception guards, which the -specialized forms do not, so as to insure exception safety. -For the unspecialized form, the base type's assignment operators -must not throw. - -[h4 Multiplication Operators] - - quaternion& operator *= (T const & rhs) - quaternion& operator *= (::std::complex const & rhs); - template - quaternion& operator *= (quaternion const & rhs); - -These perform the mathematical operation `(*this)*rhs` [*in this order] -(order is important as multiplication is not commutative for quaternions) -and store the result in `*this`. The unspecialized form has exception guards, -which the specialized forms do not, so as to insure exception safety. -For the unspecialized form, the base type's assignment operators must not throw. - -[h4 Division Operators] - - quaternion& operator /= (T const & rhs) - quaternion& operator /= (::std::complex const & rhs); - template - quaternion& operator /= (quaternion const & rhs); - -These perform the mathematical operation `(*this)*inverse_of(rhs)` [*in this -order] (order is important as multiplication is not commutative for quaternions) -and store the result in `*this`. The unspecialized form has exception guards, -which the specialized forms do not, so as to insure exception safety. -For the unspecialized form, the base type's assignment operators must not throw. - -[endsect] -[section Quaternion Non-Member Operators] - -[h4 Unary Plus] - - template - quaternion operator + (quaternion const & q); - -This unary operator simply returns q. - -[h4 Unary Minus] - - template - quaternion operator - (quaternion const & q); - -This unary operator returns the opposite of q. - -[h4 Binary Addition Operators] - - template quaternion operator + (T const & lhs, quaternion const & rhs); - template quaternion operator + (quaternion const & lhs, T const & rhs); - template quaternion operator + (::std::complex const & lhs, quaternion const & rhs); - template quaternion operator + (quaternion const & lhs, ::std::complex const & rhs); - template quaternion operator + (quaternion const & lhs, quaternion const & rhs); - -These operators return `quaternion(lhs) += rhs`. - -[h4 Binary Subtraction Operators] - - template quaternion operator - (T const & lhs, quaternion const & rhs); - template quaternion operator - (quaternion const & lhs, T const & rhs); - template quaternion operator - (::std::complex const & lhs, quaternion const & rhs); - template quaternion operator - (quaternion const & lhs, ::std::complex const & rhs); - template quaternion operator - (quaternion const & lhs, quaternion const & rhs); - -These operators return `quaternion(lhs) -= rhs`. - -[h4 Binary Multiplication Operators] - - template quaternion operator * (T const & lhs, quaternion const & rhs); - template quaternion operator * (quaternion const & lhs, T const & rhs); - template quaternion operator * (::std::complex const & lhs, quaternion const & rhs); - template quaternion operator * (quaternion const & lhs, ::std::complex const & rhs); - template quaternion operator * (quaternion const & lhs, quaternion const & rhs); - -These operators return `quaternion(lhs) *= rhs`. - -[h4 Binary Division Operators] - - template quaternion operator / (T const & lhs, quaternion const & rhs); - template quaternion operator / (quaternion const & lhs, T const & rhs); - template quaternion operator / (::std::complex const & lhs, quaternion const & rhs); - template quaternion operator / (quaternion const & lhs, ::std::complex const & rhs); - template quaternion operator / (quaternion const & lhs, quaternion const & rhs); - -These operators return `quaternion(lhs) /= rhs`. It is of course still an -error to divide by zero... - -[h4 Equality Operators] - - template bool operator == (T const & lhs, quaternion const & rhs); - template bool operator == (quaternion const & lhs, T const & rhs); - template bool operator == (::std::complex const & lhs, quaternion const & rhs); - template bool operator == (quaternion const & lhs, ::std::complex const & rhs); - template bool operator == (quaternion const & lhs, quaternion const & rhs); - -These return true if and only if the four components of `quaternion(lhs)` -are equal to their counterparts in `quaternion(rhs)`. As with any -floating-type entity, this is essentially meaningless. - -[h4 Inequality Operators] - - template bool operator != (T const & lhs, quaternion const & rhs); - template bool operator != (quaternion const & lhs, T const & rhs); - template bool operator != (::std::complex const & lhs, quaternion const & rhs); - template bool operator != (quaternion const & lhs, ::std::complex const & rhs); - template bool operator != (quaternion const & lhs, quaternion const & rhs); - -These return true if and only if `quaternion(lhs) == quaternion(rhs)` is -false. As with any floating-type entity, this is essentially meaningless. - -[h4 Stream Extractor] - - template - ::std::basic_istream& operator >> (::std::basic_istream & is, quaternion & q); - -Extracts a quaternion q of one of the following forms -(with a, b, c and d of type `T`): - -[^a (a), (a,b), (a,b,c), (a,b,c,d) (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d))] - -The input values must be convertible to `T`. If bad input is encountered, -calls `is.setstate(ios::failbit)` (which may throw ios::failure (27.4.5.3)). - -[*Returns:] `is`. - -The rationale for the list of accepted formats is that either we have a -list of up to four reals, or else we have a couple of complex numbers, -and in that case if it formated as a proper complex number, then it should -be accepted. Thus potential ambiguities are lifted (for instance (a,b) is -(a,b,0,0) and not (a,0,b,0), i.e. it is parsed as a list of two real numbers -and not two complex numbers which happen to have imaginary parts equal to zero). - -[h4 Stream Inserter] - - template - ::std::basic_ostream& operator << (::std::basic_ostream & os, quaternion const & q); - -Inserts the quaternion q onto the stream `os` as if it were implemented as follows: - - template - ::std::basic_ostream& operator << ( - ::std::basic_ostream & os, - quaternion const & q) - { - ::std::basic_ostringstream s; - - s.flags(os.flags()); - s.imbue(os.getloc()); - s.precision(os.precision()); - - s << '(' << q.R_component_1() << ',' - << q.R_component_2() << ',' - << q.R_component_3() << ',' - << q.R_component_4() << ')'; - - return os << s.str(); - } - -[endsect] - -[section Quaternion Value Operations] - -[h4 real and unreal] - - template T real(quaternion const & q); - template quaternion unreal(quaternion const & q); - -These return `q.real()` and `q.unreal()` respectively. - -[h4 conj] - - template quaternion conj(quaternion const & q); - -This returns the conjugate of the quaternion. - -[h4 sup] - -template T sup(quaternion const & q); - -This return the sup norm (the greatest among -`abs(q.R_component_1())...abs(q.R_component_4()))` of the quaternion. - -[h4 l1] - - template T l1(quaternion const & q); - -This return the l1 norm `(abs(q.R_component_1())+...+abs(q.R_component_4()))` -of the quaternion. - -[h4 abs] - - template T abs(quaternion const & q); - -This return the magnitude (Euclidian norm) of the quaternion. - -[h4 norm] - - template T norm(quaternionconst & q); - -This return the (Cayley) norm of the quaternion. -The term "norm" might be confusing, as most people associate it with the -Euclidian norm (and quadratic functionals). For this version of -(the mathematical objects known as) quaternions, the Euclidian norm -(also known as magnitude) is the square root of the Cayley norm. - -[endsect] - -[section Quaternion Creation Functions] - - template quaternion spherical(T const & rho, T const & theta, T const & phi1, T const & phi2); - template quaternion semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2); - template quaternion multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2); - template quaternion cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude); - template quaternion cylindrical(T const & r, T const & angle, T const & h1, T const & h2); - -These build quaternions in a way similar to the way polar builds complex -numbers, as there is no strict equivalent to polar coordinates for quaternions. - -[#boost_math.quaternions.creation_spherical] `spherical` is a simple transposition of `polar`, it takes as inputs -a (positive) magnitude and a point on the hypersphere, given by three angles. -The first of these, `theta` has a natural range of `-pi` to `+pi`, and the other -two have natural ranges of `-pi/2` to `+pi/2` (as is the case with the usual -spherical coordinates in __R3). Due to the many symmetries and periodicities, -nothing untoward happens if the magnitude is negative or the angles are -outside their natural ranges. The expected degeneracies (a magnitude of -zero ignores the angles settings...) do happen however. - -[#boost_math.quaternions.creation_cylindrical] `cylindrical` is likewise a simple transposition of the usual -cylindrical coordinates in __R3, which in turn is another derivative of -planar polar coordinates. The first two inputs are the polar coordinates of -the first __C component of the quaternion. The third and fourth inputs -are placed into the third and fourth __R components of the quaternion, -respectively. - -[#boost_math.quaternions.creation_multipolar] `multipolar` is yet another simple generalization of polar coordinates. -This time, both __C components of the quaternion are given in polar coordinates. - -[#boost_math.quaternions.creation_cylindrospherical] `cylindrospherical` is specific to quaternions. It is often interesting to -consider __H as the cartesian product of __R by __R3 (the quaternionic -multiplication as then a special form, as given here). This function -therefore builds a quaternion from this representation, with the __R3 -component given in usual __R3 spherical coordinates. - -[#boost_math.quaternions.creation_semipolar] `semipolar` is another generator which is specific to quaternions. -It takes as a first input the magnitude of the quaternion, as a -second input an angle in the range `0` to `+pi/2` such that magnitudes -of the first two __C components of the quaternion are the product of the -first input and the sine and cosine of this angle, respectively, and finally -as third and fourth inputs angles in the range `-pi/2` to `+pi/2` which -represent the arguments of the first and second __C components of -the quaternion, respectively. As usual, nothing untoward happens if -what should be magnitudes are negative numbers or angles are out of their -natural ranges, as symmetries and periodicities kick in. - -In this version of our implementation of quaternions, there is no -analogue of the complex value operation `arg` as the situation is -somewhat more complicated. Unit quaternions are linked both to -rotations in __R3 and in __R4, and the correspondences are not too complicated, -but there is currently a lack of standard (de facto or de jure) matrix -library with which the conversions could work. This should be remedied in -a further revision. In the mean time, an example of how this could be -done is presented here for -[@../../libs/math/quaternion/HSO3.hpp __R3], and here for -[@../../libs/math/quaternion/HSO4.hpp __R4] -([@../../libs/math/quaternion/HSO3SO4.cpp example test file]). - -[endsect] - -[section Quaternion Transcendentals] - -There is no `log` or `sqrt` provided for quaternions in this implementation, -and `pow` is likewise restricted to integral powers of the exponent. -There are several reasons to this: on the one hand, the equivalent of -analytic continuation for quaternions ("branch cuts") remains to be -investigated thoroughly (by me, at any rate...), and we wish to avoid the -nonsense introduced in the standard by exponentiations of complexes by -complexes (which is well defined, but not in the standard...). -Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is just -plain brain-dead... - -We do, however provide several transcendentals, chief among which is the -exponential. This author claims the complete proof of the "closed formula" -as his own, as well as its independant invention (there are claims to prior -invention of the formula, such as one by Professor Shoemake, and it is -possible that the formula had been known a couple of centuries back, but in -absence of bibliographical reference, the matter is pending, awaiting further -investigation; on the other hand, the definition and existence of the -exponential on the quaternions, is of course a fact known for a very long time). -Basically, any converging power series with real coefficients which allows for a -closed formula in __C can be transposed to __H. More transcendentals of this -type could be added in a further revision upon request. It should be -noted that it is these functions which force the dependency upon the -[@../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] and the -[@../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] headers. - -[h4 exp] - - template quaternion exp(quaternion const & q); - -Computes the exponential of the quaternion. - -[h4 cos] - - template quaternion cos(quaternion const & q); - -Computes the cosine of the quaternion - -[h4 sin] - - template quaternion sin(quaternion const & q); - -Computes the sine of the quaternion. - -[h4 tan] - - template quaternion tan(quaternion const & q); - -Computes the tangent of the quaternion. - -[h4 cosh] - - template quaternion cosh(quaternion const & q); - -Computes the hyperbolic cosine of the quaternion. - -[h4 sinh] - - template quaternion sinh(quaternion const & q); - -Computes the hyperbolic sine of the quaternion. - -[h4 tanh] - - template quaternion tanh(quaternion const & q); - -Computes the hyperbolic tangent of the quaternion. - -[h4 pow] - - template quaternion pow(quaternion const & q, int n); - -Computes the n-th power of the quaternion q. - -[endsect] - -[section Test Program] - -The [@../../libs/math/quaternion/quaternion_test.cpp quaternion_test.cpp] -test program tests quaternions specializations for float, double and long double -([@../../libs/math/quaternion/output.txt sample output], with message output -enabled). - -If you define the symbol BOOST_QUATERNION_TEST_VERBOSE, you will get -additional output ([@../../libs/math/quaternion/output_more.txt verbose output]); -this will only be helpfull if you enable message output at the same time, -of course (by uncommenting the relevant line in the test or by adding -[^--log_level=messages] to your command line,...). In that case, and if you -are running interactively, you may in addition define the symbol -BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to interactively test the input -operator with input of your choice from the standard input -(instead of hard-coding it in the test). - -[endsect] - -[section Acknowledgements] - -The mathematical text has been typeset with -[@http://www.nisus-soft.com/ Nisus Writer]. Jens Maurer has helped with -portability and standard adherence, and was the Review Manager -for this library. More acknowledgements in the History section. -Thank you to all who contributed to the discution about this library. - -[endsect] - -[section History] - -* 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. -* 1.5.7 - 24/02/2003: transitionned to the unit test framework; now included by the library header (rather than the test files). -* 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). -* 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. -* 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes. -* 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). -* 1.5.2 - 07/07/2001: introduced namespace math. -* 1.5.1 - 07/06/2001: (end of Boost review) now includes and instead of ; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. -* 1.5.0 - 26/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version) and output operator, added spherical, semipolar, multipolar, cylindrospherical and cylindrical. -* 1.4.0 - 09/01/2001: added tan and tanh. -* 1.3.1 - 08/01/2001: cosmetic fixes. -* 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. -* 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. -* 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. -* 1.0.0 - 10/08/1999: first public version. - -[endsect] -[section To Do] - -* Improve testing. -* Rewrite input operatore using Spirit (creates a dependency). -* Put in place an Expression Template mechanism (perhaps borrowing from uBlas). -* Use uBlas for the link with rotations (and move from the -[@../../libs/math/quaternion/HSO3SO4.cpp example] -implementation to an efficient one). - -[endsect] -[endsect] diff --git a/doc/math-sf.qbk b/doc/math-sf.qbk deleted file mode 100644 index 1f99fc5fe..000000000 --- a/doc/math-sf.qbk +++ /dev/null @@ -1,314 +0,0 @@ - -[def __asinh [link boost_math.math_special_functions.asinh asinh]] -[def __acosh [link boost_math.math_special_functions.acosh acosh]] -[def __atanh [link boost_math.math_special_functions.atanh atanh]] -[def __sinc_pi [link boost_math.math_special_functions.sinc_pi sinc_pi]] -[def __sinhc_pi [link boost_math.math_special_functions.sinhc_pi sinhc_pi]] - -[def __log1p [link boost_math.math_special_functions.log1p log1p]] -[def __expm1 [link boost_math.math_special_functions.expm1 expm1]] -[def __hypot [link boost_math.math_special_functions.hypot hypot]] - -[def __form1 [^\[0;+'''∞'''\[]] -[def __form2 [^\]-'''∞''';+1\[]] -[def __form3 [^\]-'''∞''';-1\[]] -[def __form4 [^\]+1;+'''∞'''\[]] -[def __form5 `[^\[-1;-1+'''ε'''\[]] -[def __form6 '''ε'''] -[def __form7 [^\]+1-'''ε''';+1\]]] - -[def __effects [*Effects: ]] -[def __formula [*Formula: ]] -[def __exm1 '''ex - 1'''] -[def __ex '''ex'''] -[def __te '''2ε'''] - - -[section Math Special Functions] - -[section Overview] - -The Special Functions library currently provides eight templated special functions, -in namespace boost. Two of these (__sinc_pi and __sinhc_pi) are needed by our -implementation of quaternions and octonions. - -The functions __acosh, __asinh and __atanh are entirely classical, -the function __sinc_pi sees heavy use in signal processing tasks, -and the function __sinhc_pi is an ad'hoc function whose naming is modelled on -__sinc_pi and hyperbolic functions. - -The functions __log1p, __expm1 and __hypot are all part of the C99 standard -but not yet C++. Two of these (__log1p and __hypot) were needed for the -[link boost_math.inverse_complex complex number inverse trigonometric functions]. - -The functions implemented here can throw standard exceptions, but no -exception specification has been made. - -[endsect] - -[section Header Files] - -The interface and implementation for each function (or forms of a function) -are both supplied by one header file: - -* [@../../boost/math/special_functions/acosh.hpp acosh.hpp] -* [@../../boost/math/special_functions/asinh.hpp asinh.hpp] -* [@../../boost/math/special_functions/atanh.hpp atanh.hpp] -* [@../../boost/math/special_functions/expm1.hpp expm1.hpp] -* [@../../boost/math/special_functions/hypot.hpp hypot.hpp] -* [@../../boost/math/special_functions/log1p.hpp log1p.hpp] -* [@../../boost/math/special_functions/sinc.hpp sinc.hpp] -* [@../../boost/math/special_functions/sinhc.hpp sinhc.hpp] - -[endsect] - -[section Synopsis] - - namespace boost{ namespace math{ - - template - T __acosh(const T x); - - template - T __asinh(const T x); - - template - T __atanh(const T x); - - template - T __expm1(const T x); - - template - T __hypot(const T x); - - template - T __log1p(const T x); - - template - T __sinc_pi(const T x); - - template class U> - U __sinc_pi(const U x); - - template - T __sinhc_pi(const T x); - - template class U> - U __sinhc_pi(const U x); - - } - } - -[endsect] - -[section acosh] - - template - T acosh(const T x); - -Computes the reciprocal of (the restriction to the range of __form1) -[link boost_math.background_information_and_white_papers.the_inverse_hyperbolic_functions -the hyperbolic cosine function], at x. Values returned are positive. Generalised -Taylor series are used near 1 and Laurent series are used near the -infinity to ensure accuracy. - -If x is in the range __form2 a quiet NaN is returned (if the system allows, -otherwise a `domain_error` exception is generated). - -[endsect] - -[section asinh] - - template - T asinh(const T x); - -Computes the reciprocal of -[link boost_math.background_information_and_white_papers.the_inverse_hyperbolic_functions -the hyperbolic sine function]. -Taylor series are used at the origin and Laurent series are used near the -infinity to ensure accuracy. - -[endsect] - -[section atanh] - - template - T atanh(const T x); - -Computes the reciprocal of -[link boost_math.background_information_and_white_papers.the_inverse_hyperbolic_functions -the hyperbolic tangent function], at x. -Taylor series are used at the origin to ensure accuracy. - -If x is in the range -__form3 -or in the range -__form4 -a quiet NaN is returned -(if the system allows, otherwise a `domain_error` exception is generated). - -If x is in the range -__form5, -minus infinity is returned -(if the system allows, otherwise an `out_of_range` exception -is generated), with -__form6 -denoting numeric_limits::epsilon(). - -If x is in the range -__form7, -plus infinity is returned (if the system allows, -otherwise an `out_of_range` exception is generated), with -__form6 -denoting -numeric_limits::epsilon(). - -[endsect] - -[section:expm1 expm1] - - template - T expm1(T t); - -__effects returns __exm1. - -For small x, then __ex is very close to 1, as a result calculating __exm1 results -in catastrophic cancellation errors when x is small. `expm1` calculates __exm1 using -a series expansion when x is small (giving an accuracy of less than __te). - -Finally when BOOST_HAS_EXPM1 is defined then the `float/double/long double` -specializations of this template simply forward to the platform's -native implementation of this function. - -[endsect] - -[section:hypot hypot] - - template - T hypot(T x, T y); - -__effects computes [$../../libs/math/doc/images/hypot.png] in such a way as to avoid undue underflow and overflow. - -When calculating [$../../libs/math/doc/images/hypot.png] it's quite easy for the intermediate terms to either -overflow or underflow, even though the result is in fact perfectly representable. -One possible alternative form is [$../../libs/math/doc/images/hypot2.png], but that can overflow or underflow -if x and y are of very differing magnitudes. The `hypot` function takes care of -all the special cases for you, so you don't have to. - -[endsect] - -[section:log1p log1p] - - template - T log1p(T x); - -__effects returns the natural logarithm of `x+1`. - -There are many situations where it is desirable to compute `log(x+1)`. -However, for small `x` then `x+1` suffers from catastrophic cancellation errors -so that `x+1 == 1` and `log(x+1) == 0`, when in fact for very small x, the -best approximation to `log(x+1)` would be `x`. `log1p` calculates the best -approximation to `log(1+x)` using a Taylor series expansion for accuracy -(less than __te). -Note that there are faster methods available, for example using the equivalence: - - log(1+x) == (log(1+x) * x) / ((1-x) - 1) - -However, experience has shown that these methods tend to fail quite spectacularly -once the compiler's optimizations are turned on. In contrast, the series expansion -method seems to be reasonably immune optimizer-induced errors. - -Finally when BOOST_HAS_LOG1P is defined then the `float/double/long double` -specializations of this template simply forward to the platform's -native implementation of this function. - -[endsect] - -[section sinc_pi] - - template - T sinc_pi(const T x); - - template class U> - U sinc_pi(const U x); - -Computes -[link boost_math.background_information_and_white_papers.sinus_cardinal_and_hyperbolic_sinus_cardinal_functions -the Sinus Cardinal] of x. The second form is for complexes, -quaternions, octonions... Taylor series are used at the origin -to ensure accuracy. - -[endsect] - -[section sinhc_pi] - - template - T sinhc_pi(const T x); - - template class U> - U sinhc_pi(const U x); - -Computes -[link boost_math.background_information_and_white_papers.sinus_cardinal_and_hyperbolic_sinus_cardinal_functions -the Hyperbolic Sinus Cardinal] of x. The second form is for -complexes, quaternions, octonions... Taylor series are used at the origin -to ensure accuracy. - -[endsect] - -[section Test Programs] - -The [@../../libs/math/special_functions/special_functions_test.cpp -special_functions_test.cpp] -and [@../../libs/math/test/log1p_expm1_test.cpp log1p_expm1_test.cpp] test programs test the functions for -float, double and long double arguments ([@../../libs/math/special_functions/output.txt sample output], with message -output enabled). - -If you define the symbol BOOST_SPECIAL_FUNCTIONS_TEST_VERBOSE, you will -get additional output -([@../../libs/math/special_functions/output_more.txt verbose output]), -which may prove useful for -tuning on your platform (the library use "reasonable" tolerances, -which may prove to be too strict for your platform); this will only be -helpfull if you enable message output at the same time, of course -(by uncommenting the relevant line in the test or by adding `--log_level=messages` -to your command line,...). - -[endsect] - -[section Acknowledgements] - -The mathematical text has been typeset with [@http://www.nisus-soft.com/ -Nisus Writer], and the -illustrations have been made with [@http://www.pacifict.com/ -Graphing Calculator]. Jens Maurer -was the Review Manager for this library. More acknowledgements in the -History section. Thank you to all who contributed to the discution -about this library. - -[endsect] - -[section History] - -* 1.5.0 - 17/12/2005: John Maddock added __log1p, __expm1 and __hypot. Converted documentation to Quickbook Format. -* 1.4.2 - 03/02/2003: transitionned to the unit test framework; now included by the library headers (rather than the test files). -* 1.4.1 - 18/09/2002: improved compatibility with Microsoft compilers. -* 1.4.0 - 14/09/2001: added (after rewrite) __acosh and __asinh, which were submited by Eric Ford; applied changes for Gcc 2.9.x suggested by John Maddock; improved accuracy; sanity check for test file, related to accuracy. -* 1.3.2 - 07/07/2001: introduced namespace math. -* 1.3.1 - 07/06/2001:(end of Boost review) split special_functions.hpp into atanh.hpp, sinc.hpp and sinhc.hpp; improved efficiency of __atanh with compile-time technique (Daryle Walker); improved accuracy of all functions near zero (Peter Schmitteckert). -* 1.3.0 - 26/03/2001: support for complexes & all, for cardinal functions. -* 1.2.0 - 31/01/2001: minor modifications for Koenig lookup. -* 1.1.0 - 23/01/2001: boostification. -* 1.0.0 - 10/08/1999: first public version. - -[endsect] - -[section To Do] - -* Add more functions. -* Improve test of each function. - -[endsect] -[endsect] - - diff --git a/doc/math.qbk b/doc/math.qbk index 492ab68ce..7ac058993 100644 --- a/doc/math.qbk +++ b/doc/math.qbk @@ -1,67 +1,190 @@ -[library Boost.Math - [quickbook 1.3] - [copyright 2001-2002 Daryle Walker, 2001-2003 Hubert Holin, 2005 John Maddock] - [purpose Various mathematical functions and data types] +[article Boost.Math + [quickbook 1.4] + [copyright 2007 Paul A. Bristow, Hubert Holin, John Maddock, Daryle Walker and Xiaogang Zhang] [license Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt]) ] - [authors [Holin, Hubert], [Maddock, John], [Walker, Daryle]] [category math] [last-revision $Date$] ] -[def __asinh [link boost_math.math_special_functions.asinh asinh]] -[def __acosh [link boost_math.math_special_functions.acosh acosh]] -[def __atanh [link boost_math.math_special_functions.atanh atanh]] -[def __sinc_pi [link boost_math.math_special_functions.sinc_pi sinc_pi]] -[def __sinhc_pi [link boost_math.math_special_functions.sinhc_pi sinhc_pi]] +[def __R ['[*R]]] +[def __C ['[*C]]] +[def __H ['[*H]]] +[def __O ['[*O]]] +[def __R3 ['[*'''R3''']]] +[def __R4 ['[*'''R4''']]] +[def __quadrulple ('''α,β,γ,δ''')] +[def __quat_formula ['[^q = '''α + βi + γj + δk''']]] +[def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] +[def __not_equal ['[^xy '''≠''' yx]]] +[def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')] +[def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]] +[def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]] +[def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]] +[def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]] +[template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]] -[def __log1p [link boost_math.math_special_functions.log1p log1p]] -[def __expm1 [link boost_math.math_special_functions.expm1 expm1]] -[def __hypot [link boost_math.math_special_functions.hypot hypot]] +The following mathematical libraries are present in Boost: +[table +[[Library][Description]] -[section Overview] + [[[@../complex/html/index.html Complex Number Inverse Trigonometric Functions]] + [ +These complex number algorithms are the inverses of trigonometric functions currently +present in the C++ standard. Equivalents to these functions are part of the C99 standard, and +are part of the [tr1]. + ]] + + [[[@../gcd/html/index.html Greatest Common Divisor and Least Common Multiple]] + [ +The class and function templates in +provide run-time and compile-time evaluation of the greatest common divisor +(GCD) or least common multiple (LCM) of two integers. +These facilities are useful for many numeric-oriented generic +programming problems. + ]] -This documentation is -[@http://boost-consulting.com/vault/index.php?action=downloadfile&filename=boost_math-1.34.pdf&directory=PDF%20Documentation& also available in printer-friendly PDF format]. + [[[@../../../integer/index.html Integer]] + [Headers to ease dealing with integral types.]] -The [link boost_math.gcd_lcm Greatest Common Divisor and Least Common Multiple library] -provides run-time and compile-time evaluation of the greatest common divisor (GCD) -or least common multiple (LCM) of two integers. + [[[@../../../numeric/interval/doc/interval.htm Interval]] + [As implied by its name, this library is intended to help manipulating + mathematical intervals. It consists of a single header + and principally a type which can be + used as interval. ]] + + [[[@../../../multi_array/doc/index.html Multi Array]] + [Boost.MultiArray provides a generic N-dimensional array concept + definition and common implementations of that interface.]] -The [link boost_math.math_special_functions Special Functions library] currently provides eight templated special functions, -in namespace boost. Two of these (__sinc_pi and __sinhc_pi) are needed by our -implementation of quaternions and octonions. The functions __acosh, -__asinh and __atanh are entirely classical, -the function __sinc_pi sees heavy use in signal processing tasks, -and the function __sinhc_pi is an ad'hoc function whose naming is modelled on -__sinc_pi and hyperbolic functions. The functions __log1p, __expm1 and __hypot are all part of the C99 standard -but not yet C++. Two of these (__log1p and __hypot) were needed for the -[link boost_math.inverse_complex complex number inverse trigonometric functions]. + [[[@../../../numeric/conversion/doc/index.html Numeric.Conversion]] + [The Boost Numeric Conversion library is a collection of tools to + describe and perform conversions between values of different numeric types.]] + +[[[@../octonion/html/index.html Octonions]] + [ +Octonions, like [@../quaternions/html/index.html quaternions], are a relative of complex numbers. -The [link boost_math.inverse_complex Complex Number Inverse Trigonometric Functions] -are the inverses of trigonometric functions currently present in the C++ standard. -Equivalents to these functions are part of the C99 standard, and they -will be part of the forthcoming Technical Report on C++ Standard Library Extensions. +Octonions see some use in theoretical physics. -[link boost_math.quaternions Quaternions] are a relative of -complex numbers often used to parameterise rotations in three -dimentional space. +In practical terms, an octonion is simply an octuple of real numbers __octulple, +which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]] +are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']] +are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]). -[link boost_math.octonions Octonions], like [link boost_math.quaternions quaternions], -are a relative of complex numbers. [link boost_math.octonions Octonions] -see some use in theoretical physics. +Addition and a multiplication is defined on the set of octonions, +which generalize their quaternionic counterparts. The main novelty this time +is that [*the multiplication is not only not commutative, is now not even +associative] (i.e. there are quaternions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal). +A way of remembering things is by using the following multiplication table: -[endsect] +[$../../octonion/graphics/octonion_blurb17.jpeg] -[include math-gcd.qbk] -[include math-sf.qbk] -[include math-tr1.qbk] -[include math-quaternion.qbk] -[include math-octonion.qbk] -[include math-background.qbk] +Octonions (and their kin) are described in far more details in this other +[@../../quaternion/TQE.pdf document] +(with [@../../quaternion/TQE_EA.pdf errata and addenda]). +Some traditional constructs, such as the exponential, carry over without too +much change into the realms of octonions, but other, such as taking a square root, +do not (the fact that the exponential has a closed form is a result of the +author, but the fact that the exponential exists at all for octonions is known +since quite a long time ago). + ]] + + [[[@../../../utility/operators.htm Operators]] + [The header supplies several sets of class + templates (in namespace boost). These templates define operators at + namespace scope in terms of a minimal number of fundamental + operators provided by the class.]] + +[[[@../sf_and_dist/html/index.html Special Functions]] + [ Provides a number of high quality special functions, initially + these were concentrated on functions used in statistical applications + along with those in the Technical Report on C++ Library Extensions. + + The function families currently implemented are the gamma, beta & erf + functions along with the incomplete gamma and beta functions + (four variants of each) and all the possible inverses of these, + plus digamma, various factorial functions, Bessel functions, + elliptic integrals, sinus cardinals (along with their + hyperbolic variants), inverse hyperbolic functions, + Legrendre/Laguerre/Hermite polynomials and various special power + and logarithmic functions. + + All the implementations are fully generic and support the use of + arbitrary "real-number" types, although they are optimised for + use with types with known-about significand (or mantissa) + sizes: typically float, double or long double. ]] + +[[[@../sf_and_dist/html/index.html Statistical Distributions]] + [Provides a reasonably comprehensive set of statistical distributions, + upon which higher level statistical tests can be built. + + The initial focus is on the central univariate distributions. + Both continuous (like normal & Fisher) and discrete (like binomial + & Poisson) distributions are provided. + + A comprehensive tutorial is provided, along with a series of worked + examples illustrating how the library is used to conduct statistical tests. ]] + +[[[@../quaternion/html/index.html Quaternions]] + [ +Quaternions are a relative of complex numbers. + +Quaternions are in fact part of a small hierarchy of structures built +upon the real numbers, which comprise only the set of real numbers +(traditionally named __R), the set of complex numbers (traditionally named __C), +the set of quaternions (traditionally named __H) and the set of octonions +(traditionally named __O), which possess interesting mathematical properties +(chief among which is the fact that they are ['division algebras], +['i.e.] where the following property is true: if ['[^y]] is an element of that +algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] +denote elements of that algebra, implies that ['[^x = x']]). +Each member of the hierarchy is a super-set of the former. + +One of the most important aspects of quaternions is that they provide an +efficient way to parameterize rotations in __R3 (the usual three-dimensional space) +and __R4. + +In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, +which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, +and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]]. + +An addition and a multiplication is defined on the set of quaternions, +which generalize their real and complex counterparts. The main novelty +here is that [*the multiplication is not commutative] (i.e. there are +quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering +things is by using the formula ['[^i*i = j*j = k*k = -1]]. + +Quaternions (and their kin) are described in far more details in this +other [@../../quaternion/TQE.pdf document] +(with [@../../quaternion/TQE_EA.pdf errata and addenda]). + +Some traditional constructs, such as the exponential, carry over without +too much change into the realms of quaternions, but other, such as taking +a square root, do not. + ]] + + [[[@../../../random/index.html Random]] + [Random numbers are useful in a variety of applications. The Boost + Random Number Library (Boost.Random for short) provides a vast variety + of generators and distributions to produce random numbers having useful + properties, such as uniform distribution.]] + + [[[@../../../rational/index.html Rational]] + [The header rational.hpp provides an implementation of rational numbers. + The implementation is template-based, in a similar manner to the + standard complex number class.]] + + [[[@../../../numeric/ublas/doc/index.htm uBLAS]] + [uBLAS is a C++ template class library that provides BLAS level 1, 2, 3 + functionality for dense, packed and sparse matrices. The design and + implementation unify mathematical notation via operator overloading and + efficient code generation via expression templates.]] + +] diff --git a/doc/octonion/Jamfile.v2 b/doc/octonion/Jamfile.v2 new file mode 100644 index 000000000..6490f5079 --- /dev/null +++ b/doc/octonion/Jamfile.v2 @@ -0,0 +1,70 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +path-constant images_location : html ; + +xml octonion : math-octonion.qbk ; +boostbook standalone + : + octonion + : + # Path for links to Boost: + boost.root=../../../../.. + # Path for libraries index: + boost.libraries=../../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=10 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=1 + # How far down sections get TOC's + toc.section.depth=10 + # Max depth in each TOC: + toc.max.depth=4 + # How far down we go with TOC's + generate.section.toc.level=10 + #root.filename="sf_dist_and_tools" + + # PDF Options: + # TOC Generation: this is needed for FOP-0.9 and later: + # fop1.extensions=1 + pdf:xep.extensions=1 + # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9! + pdf:fop.extensions=0 + pdf:fop1.extensions=0 + # No indent on body text: + pdf:body.start.indent=0pt + # Margin size: + pdf:page.margin.inner=0.5in + # Margin size: + pdf:page.margin.outer=0.5in + # Paper type = A4 + pdf:paper.type=A4 + # Yes, we want graphics for admonishments: + admon.graphics=1 + # Set this one for PDF generation *only*: + # default pnd graphics are awful in PDF form, + # better use SVG's instead: + pdf:admon.graphics.extension=".svg" + pdf:use.role.for.mediaobject=1 + pdf:preferred.mediaobject.role=print + pdf:img.src.path=$(images_location)/ + pdf:admon.graphics.path=$(images_location)/../../svg-admon/ + pdf:draft.mode="no" + ; + + + diff --git a/doc/octonion/html/boost_octonions/octonions.html b/doc/octonion/html/boost_octonions/octonions.html new file mode 100644 index 000000000..990e25ae1 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions.html @@ -0,0 +1,67 @@ + + + +Octonions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Overview
+
Header File
+
Synopsis
+
Template + Class octonion
+
Octonion + Specializations
+
Octonion + Member Typedefs
+
Octonion + Member Functions
+
Octonion Non-Member + Operators
+
Octonion + Value Operations
+
Octonion Creation + Functions
+
Octonions + Transcendentals
+
Test Program
+
Acknowledgements
+
History
+
To Do
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/acknowledgements.html b/doc/octonion/html/boost_octonions/octonions/acknowledgements.html new file mode 100644 index 000000000..d2165905e --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/acknowledgements.html @@ -0,0 +1,49 @@ + + + +Acknowledgements + + + + + + + + + + + + + + + +
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+

+ The mathematical text has been typeset with Nisus + Writer. Jens Maurer has helped with portability and standard adherence, + and was the Review Manager for this library. More acknowledgements in the + History section. Thank you to all who contributed to the discussion about + this library. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/header_file.html b/doc/octonion/html/boost_octonions/octonions/header_file.html new file mode 100644 index 000000000..b456859f6 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/header_file.html @@ -0,0 +1,45 @@ + + + +Header File + + + + + + + + + + + + + + + +
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+

+ The interface and implementation are both supplied by the header file octonion.hpp. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/history.html b/doc/octonion/html/boost_octonions/octonions/history.html new file mode 100644 index 000000000..23faf1561 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/history.html @@ -0,0 +1,104 @@ + + + +History + + + + + + + + + + + + + + + +
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+
+
    +
  • + 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. +
    • +
  • + 1.5.7 - 25/02/2003: transitionned to the unit test framework; <boost/config.hpp> + now included by the library header (rather than the test files), via <boost/math/quaternion.hpp>. +
    • +
  • + 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis + Evlogimenos (alkis@routescience.com). +
    • +
  • + 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens + (michael@acfr.usyd.edu.au); requires the /Za compiler option. +
    • +
  • + 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different + translation units); attempt at an improved compatibility with Microsoft + compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; + other compatibility fixes. +
    • +
  • + 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor + (gregod@cs.rpi.edu). +
    • +
  • + 1.5.2 - 07/07/2001: introduced namespace math. +
    • +
  • + 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> + and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; + corrected bug in sin (Daryle Walker); removed check for self-assignment + (Gary Powel); made converting functions explicit (Gary Powel); added overflow + guards for division operators and abs (Peter Schmitteckert); added sup + and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. +
    • +
  • + 1.5.0 - 23/03/2001: boostification, inlining of all operators except input, + output and pow, fixed exception safety of some members (template version). +
    • +
  • + 1.4.0 - 09/01/2001: added tan and tanh. +
    • +
  • + 1.3.1 - 08/01/2001: cosmetic fixes. +
    • +
  • + 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) + algorithm. +
    • +
  • + 1.2.0 - 25/05/2000: fixed the division operators and output; changed many + signatures. +
    • +
  • + 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. +
    • +
  • + 1.0.0 - 10/08/1999: first public version. +
    • +
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/non_mem.html b/doc/octonion/html/boost_octonions/octonions/non_mem.html new file mode 100644 index 000000000..35221ab8f --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/non_mem.html @@ -0,0 +1,231 @@ + + + +Octonion Non-Member Operators + + + + + + + + + + + + + + + +
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+
+
+ + Unary + Plus and Minus Operators +
+
+template<typename T> octonion<T> operator + (octonion<T> const & o);
+
+

+ This unary operator simply returns o. +

+
+template<typename T> octonion<T> operator - (octonion<T> const & o);
+
+

+ This unary operator returns the opposite of o. +

+
+ + Binary + Addition Operators +
+
+template<typename T> octonion<T> operator + (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator + (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator + (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These operators return octonion<T>(lhs) += + rhs. +

+
+ + Binary + Subtraction Operators +
+
+template<typename T> octonion<T> operator - (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator - (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator - (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These operators return octonion<T>(lhs) -= + rhs. +

+
+ + Binary + Multiplication Operators +
+
+template<typename T> octonion<T> operator * (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator * (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator * (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These operators return octonion<T>(lhs) *= + rhs. +

+
+ + Binary + Division Operators +
+
+template<typename T> octonion<T> operator / (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator / (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator / (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These operators return octonion<T>(lhs) /= + rhs. It is of course still an error + to divide by zero... +

+
+ + Binary + Equality Operators +
+
+template<typename T> bool operator == (T const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, T const & rhs);
+template<typename T> bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These return true if and only if the four components of octonion<T>(lhs) are + equal to their counterparts in octonion<T>(rhs). As + with any floating-type entity, this is essentially meaningless. +

+
+ + Binary + Inequality Operators +
+
+template<typename T> bool operator != (T const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, T const & rhs);
+template<typename T> bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, octonion<T> const & rhs);
+
+

+ These return true if and only if octonion<T>(lhs) == + octonion<T>(rhs) is + false. As with any floating-type entity, this is essentially meaningless. +

+
+ + Stream + Extractor +
+
+template<typename T, typename charT, class traits>
+::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o);
+
+

+ Extracts an octonion o. We + accept any format which seems reasonable. However, since this leads to a + great many ambiguities, decisions were made to lift these. In case of doubt, + stick to lists of reals. +

+

+ The input values must be convertible to T. If bad input is encountered, calls + is.setstate(ios::failbit) (which may throw ios::failure + (27.4.5.3)). +

+

+ Returns is. +

+
+ + Stream + Inserter +
+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);
+
+

+ Inserts the octonion o onto + the stream os as if it were + implemented as follows: +

+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits> & operator << ( ::std::basic_ostream<charT,traits> & os,
+octonion<T> const & o)
+{
+   ::std::basic_ostringstream<charT,traits> s;
+
+   s.flags(os.flags());
+   s.imbue(os.getloc());
+   s.precision(os.precision());
+
+   s << '(' << o.R_component_1() << ','
+      << o.R_component_2() << ','
+      << o.R_component_3() << ','
+      << o.R_component_4() << ','
+      << o.R_component_5() << ','
+      << o.R_component_6() << ','
+      << o.R_component_7() << ','
+      << o.R_component_8() << ')';
+
+   return os << s.str();
+}
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/oct_create.html b/doc/octonion/html/boost_octonions/octonions/oct_create.html new file mode 100644 index 000000000..64d52ff01 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/oct_create.html @@ -0,0 +1,82 @@ + + + +Octonion Creation Functions + + + + + + + + + + + + + + + +
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+
+
+template<typename T> octonion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6);
+template<typename T> octonion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4);
+template<typename T> octonion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6);
+
+

+ These build octonions in a way similar to the way polar builds complex numbers, + as there is no strict equivalent to polar coordinates for octonions. +

+

+ spherical is a simple transposition + of polar, it takes as inputs + a (positive) magnitude and a point on the hypersphere, given by three angles. + The first of these, theta has a natural range of -pi + to +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the + case with the usual spherical coordinates in R3). + Due to the many symmetries and periodicities, nothing untoward happens if + the magnitude is negative or the angles are outside their natural ranges. + The expected degeneracies (a magnitude of zero ignores the angles settings...) + do happen however. +

+

+ cylindrical is likewise a + simple transposition of the usual cylindrical coordinates in R3, which in turn is another derivative of + planar polar coordinates. The first two inputs are the polar coordinates + of the first C component + of the octonion. The third and fourth inputs are placed into the third and + fourth R components + of the octonion, respectively. +

+

+ multipolar is yet another + simple generalization of polar coordinates. This time, both C components of the octonion are given + in polar coordinates. +

+

+ In this version of our implementation of octonions, there is no analogue + of the complex value operation arg as the situation is somewhat more complicated. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/octonion_member_functions.html b/doc/octonion/html/boost_octonions/octonions/octonion_member_functions.html new file mode 100644 index 000000000..d2d626784 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/octonion_member_functions.html @@ -0,0 +1,272 @@ + + + +Octonion Member Functions + + + + + + + + + + + + + + + +
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+

+ + Constructors +

+

+ Template version: +

+
+explicit  octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T());
+explicit  octonion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>());
+explicit  octonion(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>());
+template<typename X> 
+explicit octonion(octonion<X> const & a_recopier);
+
+

+ Float specialization version: +

+
+explicit  octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f);
+explicit  octonion(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>());
+explicit  octonion(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>());
+explicit  octonion(octonion<double> const & a_recopier); 
+explicit  octonion(octonion<long double> const & a_recopier);
+
+

+ Double specialization version: +

+
+explicit  octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0);
+explicit  octonion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>());
+explicit  octonion(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>());
+explicit  octonion(octonion<float> const & a_recopier);
+explicit  octonion(octonion<long double> const & a_recopier);
+
+

+ Long double specialization version: +

+
+explicit  octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L);
+explicit  octonion( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>());
+explicit  octonion(::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & q1 = ::boost::math::quaternion<long double>());
+explicit  octonion(octonion<float> const & a_recopier);
+explicit  octonion(octonion<double> const & a_recopier);
+
+

+ A default constructor is provided for each form, which initializes each component + to the default values for their type (i.e. zero for floating numbers). This + constructor can also accept one to eight base type arguments. A constructor + is also provided to build octonions from one to four complex numbers sharing + the same base type, and another taking one or two quaternions sharing the + same base type. The unspecialized template also sports a templarized copy + constructor, while the specialized forms have copy constructors from the + other two specializations, which are explicit when a risk of precision loss + exists. For the unspecialized form, the base type's constructors must not + throw. +

+

+ Destructors and untemplated copy constructors (from the same type) are provided + by the compiler. Converting copy constructors make use of a templated helper + function in a "detail" subnamespace. +

+

+ + Other + member functions +

+
+ + Real + and Unreal Parts +
+
+T            real()   const;
+octonion<T>  unreal() const;
+
+

+ Like complex number, octonions do have a meaningful notion of "real + part", but unlike them there is no meaningful notion of "imaginary + part". Instead there is an "unreal part" which itself is a + octonion, and usually nothing simpler (as opposed to the complex number case). + These are returned by the first two functions. +

+
+ + Individual + Real Components +
+
+T   R_component_1() const;
+T   R_component_2() const;
+T   R_component_3() const;
+T   R_component_4() const;
+T   R_component_5() const;
+T   R_component_6() const;
+T   R_component_7() const;
+T   R_component_8() const;
+
+

+ A octonion having eight real components, these are returned by these eight + functions. Hence real and R_component_1 return the same value. +

+
+ + Individual + Complex Components +
+
+::std::complex<T> C_component_1() const;
+::std::complex<T> C_component_2() const;
+::std::complex<T> C_component_3() const;
+::std::complex<T> C_component_4() const;
+
+

+ A octonion likewise has four complex components. Actually, octonions are + indeed a (left) vector field over the complexes, but beware, as for any octonion + o = α + βi + γj + δk + εe' + ζi' + ηj' + θk' we also have o + = (α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' (note the minus + sign in the last factor). What the C_component_n functions return, however, + are the complexes which could be used to build the octonion using the constructor, + and not the components of the octonion on + the basis (1, j, e', j'). +

+
+ + Individual + Quaternion Components +
+
+::boost::math::quaternion<T> H_component_1() const;
+::boost::math::quaternion<T> H_component_2() const;
+
+

+ Likewise, for any octonion o = α + βi + γj + δk + εe' + ζi' + ηj' + θk' we + also have o = (α + βi + γj + δk) + (ε + ζi + ηj - θj)e', though there is + no meaningful vector-space-like structure based on the quaternions. What + the H_component_n functions return are the quaternions which could be used + to build the octonion using the constructor. +

+

+ + Octonion + Member Operators +

+
+ + Assignment + Operators +
+
+octonion<T> & operator = (octonion<T> const & a_affecter);
+template<typename X> 
+octonion<T> & operator = (octonion<X> const & a_affecter);
+octonion<T> & operator = (T const & a_affecter);
+octonion<T> & operator = (::std::complex<T> const & a_affecter);
+octonion<T> & operator = (::boost::math::quaternion<T> const & a_affecter);
+
+

+ These perform the expected assignment, with type modification if necessary + (for instance, assigning from a base type will set the real part to that + value, and all other components to zero). For the unspecialized form, the + base type's assignment operators must not throw. +

+
+ + Other + Member Operators +
+
+octonion<T> & operator += (T const & rhs)
+octonion<T> & operator += (::std::complex<T> const & rhs);
+octonion<T> & operator += (::boost::math::quaternion<T> const & rhs);
+template<typename X> 
+octonion<T> & operator += (octonion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)+rhs + and store the result in *this. + The unspecialized form has exception guards, which the specialized forms + do not, so as to insure exception safety. For the unspecialized form, the + base type's assignment operators must not throw. +

+
+octonion<T> & operator -= (T const & rhs)
+octonion<T> & operator -= (::std::complex<T> const & rhs);
+octonion<T> & operator -= (::boost::math::quaternion<T> const & rhs);
+template<typename X> 
+octonion<T> & operator -= (octonion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)-rhs + and store the result in *this. + The unspecialized form has exception guards, which the specialized forms + do not, so as to insure exception safety. For the unspecialized form, the + base type's assignment operators must not throw. +

+
+octonion<T> & operator *= (T const & rhs)
+octonion<T> & operator *= (::std::complex<T> const & rhs);
+octonion<T> & operator *= (::boost::math::quaternion<T> const & rhs);
+template<typename X> 
+octonion<T> & operator *= (octonion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)*rhs + in this order (order is important as multiplication is not commutative for + octonions) and store the result in *this. The unspecialized form has exception + guards, which the specialized forms do not, so as to insure exception safety. + For the unspecialized form, the base type's assignment operators must not + throw. Also, for clarity's sake, you should always group the factors in a + multiplication by groups of two, as the multiplication is not even associative + on the octonions (though there are of course cases where this does not matter, + it usually does). +

+
+octonion<T> & operator /= (T const & rhs)
+octonion<T> & operator /= (::std::complex<T> const & rhs);
+octonion<T> & operator /= (::boost::math::quaternion<T> const & rhs);
+template<typename X> 
+octonion<T> & operator /= (octonion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)*inverse_of(rhs) + in this order (order is important as multiplication is not commutative for + octonions) and store the result in *this. The unspecialized form has exception + guards, which the specialized forms do not, so as to insure exception safety. + For the unspecialized form, the base type's assignment operators must not + throw. As for the multiplication, remember to group any two factors using + parenthesis. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/octonion_member_typedefs.html b/doc/octonion/html/boost_octonions/octonions/octonion_member_typedefs.html new file mode 100644 index 000000000..51697808e --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/octonion_member_typedefs.html @@ -0,0 +1,73 @@ + + + +Octonion Member Typedefs + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ value_type +

+

+ Template version: +

+
+typedef T value_type;
+
+

+ Float specialization version: +

+
+typedef float value_type;
+
+

+ Double specialization version: +

+
+typedef double value_type;
+
+

+ Long double specialization version: +

+
+typedef long double value_type;
+
+

+ These provide easy acces to the type the template is built upon. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/octonion_specializations.html b/doc/octonion/html/boost_octonions/octonions/octonion_specializations.html new file mode 100644 index 000000000..3a4917daa --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/octonion_specializations.html @@ -0,0 +1,246 @@ + + + +Octonion Specializations + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+namespace boost{ namespace math{
+
+template<>
+class octonion<float>
+{
+public:
+   typedef float value_type;
+
+   explicit  octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f);
+   explicit  octonion(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>());
+   explicit  octonion(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>());
+   explicit  octonion(octonion<double> const & a_recopier);
+   explicit  octonion(octonion<long double> const & a_recopier);
+
+   float                             real() const;
+   octonion<float>                   unreal() const;
+
+   float                             R_component_1() const;
+   float                             R_component_2() const;
+   float                             R_component_3() const;
+   float                             R_component_4() const;
+   float                             R_component_5() const;
+   float                             R_component_6() const;
+   float                             R_component_7() const;
+   float                             R_component_8() const;
+
+   ::std::complex<float>             C_component_1() const;
+   ::std::complex<float>             C_component_2() const;
+   ::std::complex<float>             C_component_3() const;
+   ::std::complex<float>             C_component_4() const;
+
+   ::boost::math::quaternion<float>  H_component_1() const;
+   ::boost::math::quaternion<float>  H_component_2() const;
+
+   octonion<float> & operator = (octonion<float> const  & a_affecter);
+   template<typename X> 
+   octonion<float> & operator = (octonion<X> const  & a_affecter);
+   octonion<float> & operator = (float const  & a_affecter);
+   octonion<float> & operator = (::std::complex<float> const & a_affecter);
+   octonion<float> & operator = (::boost::math::quaternion<float> const & a_affecter);
+
+   octonion<float> & operator += (float const & rhs);
+   octonion<float> & operator += (::std::complex<float> const & rhs);
+   octonion<float> & operator += (::boost::math::quaternion<float> const & rhs);
+   template<typename X> 
+   octonion<float> & operator += (octonion<X> const & rhs);
+
+   octonion<float> & operator -= (float const & rhs);
+   octonion<float> & operator -= (::std::complex<float> const & rhs);
+   octonion<float> & operator -= (::boost::math::quaternion<float> const & rhs);
+   template<typename X> 
+   octonion<float> & operator -= (octonion<X> const & rhs);
+
+   octonion<float> & operator *= (float const & rhs);
+   octonion<float> & operator *= (::std::complex<float> const & rhs);
+   octonion<float> & operator *= (::boost::math::quaternion<float> const & rhs);
+   template<typename X> 
+   octonion<float> & operator *= (octonion<X> const & rhs);
+
+   octonion<float> & operator /= (float const & rhs);
+   octonion<float> & operator /= (::std::complex<float> const & rhs);
+   octonion<float> & operator /= (::boost::math::quaternion<float> const & rhs);
+   template<typename X> 
+   octonion<float> & operator /= (octonion<X> const & rhs);
+};
+
+

+

+
+template<>
+class octonion<double>
+{
+public:
+   typedef double value_type;
+
+   explicit  octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0);
+   explicit  octonion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>());
+   explicit  octonion(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>());
+   explicit  octonion(octonion<float> const & a_recopier);
+   explicit  octonion(octonion<long double> const & a_recopier);
+
+   double                             real() const;
+   octonion<double>                   unreal() const;
+
+   double                             R_component_1() const;
+   double                             R_component_2() const;
+   double                             R_component_3() const;
+   double                             R_component_4() const;
+   double                             R_component_5() const;
+   double                             R_component_6() const;
+   double                             R_component_7() const;
+   double                             R_component_8() const;
+
+   ::std::complex<double>             C_component_1() const;
+   ::std::complex<double>             C_component_2() const;
+   ::std::complex<double>             C_component_3() const;
+   ::std::complex<double>             C_component_4() const;
+
+   ::boost::math::quaternion<double>  H_component_1() const;
+   ::boost::math::quaternion<double>  H_component_2() const;
+
+   octonion<double> & operator = (octonion<double> const  & a_affecter);
+   template<typename X> 
+   octonion<double> & operator = (octonion<X> const  & a_affecter);
+   octonion<double> & operator = (double const  & a_affecter);
+   octonion<double> & operator = (::std::complex<double> const & a_affecter);
+   octonion<double> & operator = (::boost::math::quaternion<double> const & a_affecter);
+
+   octonion<double> & operator += (double const & rhs);
+   octonion<double> & operator += (::std::complex<double> const & rhs);
+   octonion<double> & operator += (::boost::math::quaternion<double> const & rhs);
+   template<typename X> 
+   octonion<double> & operator += (octonion<X> const & rhs);
+
+   octonion<double> & operator -= (double const & rhs);
+   octonion<double> & operator -= (::std::complex<double> const & rhs);
+   octonion<double> & operator -= (::boost::math::quaternion<double> const & rhs);
+   template<typename X> 
+   octonion<double> & operator -= (octonion<X> const & rhs);
+
+   octonion<double> & operator *= (double const & rhs);
+   octonion<double> & operator *= (::std::complex<double> const & rhs);
+   octonion<double> & operator *= (::boost::math::quaternion<double> const & rhs);
+   template<typename X> 
+   octonion<double> & operator *= (octonion<X> const & rhs);
+
+   octonion<double> & operator /= (double const & rhs);
+   octonion<double> & operator /= (::std::complex<double> const & rhs);
+   octonion<double> & operator /= (::boost::math::quaternion<double> const & rhs);
+   template<typename X> 
+   octonion<double> & operator /= (octonion<X> const & rhs);
+};
+
+

+

+
+template<>
+class octonion<long double>
+{
+public:
+   typedef long double value_type;
+
+   explicit   octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L);
+   explicit   octonion( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>());
+   explicit   octonion( ::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & z1 = ::boost::math::quaternion<long double>());
+   explicit   octonion(octonion<float> const & a_recopier);
+   explicit   octonion(octonion<double> const & a_recopier);
+
+   long double                             real() const;
+   octonion<long double>                   unreal() const;
+
+   long double                             R_component_1() const;
+   long double                             R_component_2() const;
+   long double                             R_component_3() const;
+   long double                             R_component_4() const;
+   long double                             R_component_5() const;
+   long double                             R_component_6() const;
+   long double                             R_component_7() const;
+   long double                             R_component_8() const;
+
+   ::std::complex<long double>             C_component_1() const;
+   ::std::complex<long double>             C_component_2() const;
+   ::std::complex<long double>             C_component_3() const;
+   ::std::complex<long double>             C_component_4() const;
+
+   ::boost::math::quaternion<long double>  H_component_1() const;
+   ::boost::math::quaternion<long double>  H_component_2() const;
+
+   octonion<long double> & operator = (octonion<long double> const  & a_affecter);
+   template<typename X> 
+   octonion<long double> & operator = (octonion<X> const  & a_affecter);
+   octonion<long double> & operator = (long double const  & a_affecter);
+   octonion<long double> & operator = (::std::complex<long double> const & a_affecter);
+   octonion<long double> & operator = (::boost::math::quaternion<long double> const & a_affecter);
+
+   octonion<long double> & operator += (long double const & rhs);
+   octonion<long double> & operator += (::std::complex<long double> const & rhs);
+   octonion<long double> & operator += (::boost::math::quaternion<long double> const & rhs);
+   template<typename X> 
+   octonion<long double> & operator += (octonion<X> const & rhs);
+
+   octonion<long double> & operator -= (long double const & rhs);
+   octonion<long double> & operator -= (::std::complex<long double> const & rhs);
+   octonion<long double> & operator -= (::boost::math::quaternion<long double> const & rhs);
+   template<typename X> 
+   octonion<long double> & operator -= (octonion<X> const & rhs);
+
+   octonion<long double> & operator *= (long double const & rhs);
+   octonion<long double> & operator *= (::std::complex<long double> const & rhs);
+   octonion<long double> & operator *= (::boost::math::quaternion<long double> const & rhs);
+   template<typename X> 
+   octonion<long double> & operator *= (octonion<X> const & rhs);
+
+   octonion<long double> & operator /= (long double const & rhs);
+   octonion<long double> & operator /= (::std::complex<long double> const & rhs);
+   octonion<long double> & operator /= (::boost::math::quaternion<long double> const & rhs);
+   template<typename X> 
+   octonion<long double> & operator /= (octonion<X> const & rhs);
+};
+
+} } // namespaces
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/octonion_value_operations.html b/doc/octonion/html/boost_octonions/octonions/octonion_value_operations.html new file mode 100644 index 000000000..7712b6f80 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/octonion_value_operations.html @@ -0,0 +1,111 @@ + + + +Octonion Value Operations + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Real + and Unreal +
+
+template<typename T> T  real(octonion<T> const & o);
+template<typename T> octonion<T> unreal(octonion<T> const & o);
+
+

+ These return o.real() + and o.unreal() + respectively. +

+
+ + conj +
+
+template<typename T> octonion<T> conj(octonion<T> const & o);
+
+

+ This returns the conjugate of the octonion. +

+
+ + sup +
+
+template<typename T> T   sup(octonion<T> const & o);
+
+

+ This return the sup norm (the greatest among abs(o.R_component_1())...abs(o.R_component_8())) of the octonion. +

+
+ + l1 +
+
+template<typename T> T   l1(octonion<T> const & o);
+
+

+ This return the l1 norm (abs(o.R_component_1())+...+abs(o.R_component_8())) of the octonion. +

+
+ + abs +
+
+template<typename T> T   abs(octonion<T> const & o);
+
+

+ This return the magnitude (Euclidian norm) of the octonion. +

+
+ + norm +
+
+template<typename T> T  norm(octonion<T>const  & o);
+
+

+ This return the (Cayley) norm of the octonion. The term "norm" + might be confusing, as most people associate it with the Euclidian norm (and + quadratic functionals). For this version of (the mathematical objects known + as) octonions, the Euclidian norm (also known as magnitude) is the square + root of the Cayley norm. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/octonions_transcendentals.html b/doc/octonion/html/boost_octonions/octonions/octonions_transcendentals.html new file mode 100644 index 000000000..dee687a81 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/octonions_transcendentals.html @@ -0,0 +1,156 @@ + + + +Octonions Transcendentals + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ There is no log or sqrt provided for octonions in this implementation, + and pow is likewise restricted + to integral powers of the exponent. There are several reasons to this: on + the one hand, the equivalent of analytic continuation for octonions ("branch + cuts") remains to be investigated thoroughly (by me, at any rate...), + and we wish to avoid the nonsense introduced in the standard by exponentiations + of complexes by complexes (which is well defined, but not in the standard...). + Talking of nonsense, saying that pow(0,0) is "implementation + defined" is just plain brain-dead... +

+

+ We do, however provide several transcendentals, chief among which is the + exponential. That it allows for a "closed formula" is a result + of the author (the existence and definition of the exponential, on the octonions + among others, on the other hand, is a few centuries old). Basically, any + converging power series with real coefficients which allows for a closed + formula in C can be + transposed to O. More + transcendentals of this type could be added in a further revision upon request. + It should be noted that it is these functions which force the dependency + upon the boost/math/special_functions/sinc.hpp + and the boost/math/special_functions/sinhc.hpp + headers. +

+
+ + exp +
+
+template<typename T> 
+octonion<T> exp(octonion<T> const & o);
+
+

+ Computes the exponential of the octonion. +

+
+ + cos +
+
+template<typename T> 
+octonion<T> cos(octonion<T> const & o);
+
+

+ Computes the cosine of the octonion +

+
+ + sin +
+
+template<typename T> 
+octonion<T> sin(octonion<T> const & o);
+
+

+ Computes the sine of the octonion. +

+
+ + tan +
+
+template<typename T> 
+octonion<T> tan(octonion<T> const & o);
+
+

+ Computes the tangent of the octonion. +

+
+ + cosh +
+
+template<typename T> 
+octonion<T> cosh(octonion<T> const & o);
+
+

+ Computes the hyperbolic cosine of the octonion. +

+
+ + sinh +
+
+template<typename T> 
+octonion<T> sinh(octonion<T> const & o);
+
+

+ Computes the hyperbolic sine of the octonion. +

+
+ + tanh +
+
+template<typename T> 
+octonion<T> tanh(octonion<T> const & o);
+
+

+ Computes the hyperbolic tangent of the octonion. +

+
+ + pow +
+
+template<typename T> 
+octonion<T>  pow(octonion<T> const & o, int n);
+
+

+ Computes the n-th power of the octonion q. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/octonion/html/boost_octonions/octonions/overview.html b/doc/octonion/html/boost_octonions/octonions/overview.html new file mode 100644 index 000000000..9f792ae84 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/overview.html @@ -0,0 +1,82 @@ + + + +Overview + + + + + + + + + + + + + + + +
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+ Octonions, like quaternions, + are a relative of complex numbers. +

+

+ Octonions see some use in theoretical physics. +

+

+ In practical terms, an octonion is simply an octuple of real numbers (α,β,γ,δ,ε,ζ,η,θ), + which we can write in the form o = α + βi + γj + δk + εe' + ζi' + ηj' + θk', + where i, j + and k are the same objects as for + quaternions, and e', i', + j' and k' + are distinct objects which play essentially the same kind of role as i + (or j or k). +

+

+ Addition and a multiplication is defined on the set of octonions, which generalize + their quaternionic counterparts. The main novelty this time is that the multiplication is not only not commutative, is now not even + associative (i.e. there are octonions x, + y and z + such that x(yz) ≠ (xy)z). A way of + remembering things is by using the following multiplication table: +

+

+ octonion_blurb17 +

+

+ Octonions (and their kin) are described in far more details in this other + document (with errata + and addenda). +

+

+ Some traditional constructs, such as the exponential, carry over without + too much change into the realms of octonions, but other, such as taking a + square root, do not (the fact that the exponential has a closed form is a + result of the author, but the fact that the exponential exists at all for + octonions is known since quite a long time ago). +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/synopsis.html b/doc/octonion/html/boost_octonions/octonions/synopsis.html new file mode 100644 index 000000000..a78965b2c --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/synopsis.html @@ -0,0 +1,138 @@ + + + +Synopsis + + + + + + + + + + + + + + + +
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+namespace boost{ namespace math{
+
+template<typename T> class octonion;
+template<>           class octonion<float>;
+template<>           class octonion<double>; 
+template<>           class octonion<long double>; 
+
+// operators
+
+template<typename T> octonion<T> operator + (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator + (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator + (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator + (octonion<T> const & lhs, octonion<T> const & rhs);
+
+template<typename T> octonion<T> operator - (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator - (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator - (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator - (octonion<T> const & lhs, octonion<T> const & rhs);
+
+template<typename T> octonion<T> operator * (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator * (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator * (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator * (octonion<T> const & lhs, octonion<T> const & rhs);
+
+template<typename T> octonion<T> operator / (T const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, T const & rhs);
+template<typename T> octonion<T> operator / (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> octonion<T> operator / (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> octonion<T> operator / (octonion<T> const & lhs, octonion<T> const & rhs); 
+
+template<typename T> octonion<T> operator + (octonion<T> const & o);
+template<typename T> octonion<T> operator - (octonion<T> const & o); 
+
+template<typename T> bool operator == (T const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, T const & rhs);
+template<typename T> bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> bool operator == (octonion<T> const & lhs, octonion<T> const & rhs);
+
+template<typename T> bool operator != (T const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, T const & rhs);
+template<typename T> bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
+template<typename T> bool operator != (octonion<T> const & lhs, octonion<T> const & rhs); 
+
+template<typename T, typename charT, class traits>
+::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o);
+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);
+
+// values
+
+template<typename T> T           real(octonion<T> const & o);
+template<typename T> octonion<T> unreal(octonion<T> const & o);
+
+template<typename T> T           sup(octonion<T> const & o);
+template<typename T> T           l1(octonion<T>const & o); 
+template<typename T> T           abs(octonion<T> const & o);
+template<typename T> T           norm(octonion<T>const  & o);
+template<typename T> octonion<T> conj(octonion<T> const & o);
+
+template<typename T> octonion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6);
+template<typename T> octonion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4);
+template<typename T> octonion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6);
+
+// transcendentals
+
+template<typename T> octonion<T> exp(octonion<T> const & o);
+template<typename T> octonion<T> cos(octonion<T> const & o);
+template<typename T> octonion<T> sin(octonion<T> const & o);
+template<typename T> octonion<T> tan(octonion<T> const & o);
+template<typename T> octonion<T> cosh(octonion<T> const & o);
+template<typename T> octonion<T> sinh(octonion<T> const & o);
+template<typename T> octonion<T> tanh(octonion<T> const & o);
+
+template<typename T> octonion<T> pow(octonion<T> const & o, int n);
+
+}  } // namespaces
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/template_class_octonion.html b/doc/octonion/html/boost_octonions/octonions/template_class_octonion.html new file mode 100644 index 000000000..d42a94b8e --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/template_class_octonion.html @@ -0,0 +1,112 @@ + + + +Template Class octonion + + + + + + + + + + + + + + + +
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+namespace boost{ namespace math {
+
+template<typename T>
+class octonion
+{
+public:
+   typedef T value_type;
+
+   explicit  octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T());
+   explicit  octonion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>());
+   explicit  octonion(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>());
+   template<typename X> 
+   explicit  octonion(octonion<X> const & a_recopier);
+
+   T                             real() const;
+   octonion<T>                   unreal() const;
+
+   T                             R_component_1() const;
+   T                             R_component_2() const;
+   T                             R_component_3() const;
+   T                             R_component_4() const;
+   T                             R_component_5() const;
+   T                             R_component_6() const;
+   T                             R_component_7() const;
+   T                             R_component_8() const;
+
+   ::std::complex<T>             C_component_1() const;
+   ::std::complex<T>             C_component_2() const;
+   ::std::complex<T>             C_component_3() const;
+   ::std::complex<T>             C_component_4() const;
+
+   ::boost::math::quaternion<T>  H_component_1() const;
+   ::boost::math::quaternion<T>  H_component_2() const;
+
+   octonion<T> & operator = (octonion<T> const  & a_affecter);
+   template<typename X> 
+   octonion<T> & operator = (octonion<X> const  & a_affecter);
+   octonion<T> & operator = (T const  & a_affecter);
+   octonion<T> & operator = (::std::complex<T> const & a_affecter);
+   octonion<T> & operator = (::boost::math::quaternion<T> const & a_affecter);
+
+   octonion<T> & operator += (T const & rhs);
+   octonion<T> & operator += (::std::complex<T> const & rhs);
+   octonion<T> & operator += (::boost::math::quaternion<T> const & rhs);
+   template<typename X> 
+   octonion<T> & operator += (octonion<X> const & rhs);
+
+   octonion<T> & operator -= (T const & rhs);
+   octonion<T> & operator -= (::std::complex<T> const & rhs);
+   octonion<T> & operator -= (::boost::math::quaternion<T> const & rhs);
+   template<typename X> 
+   octonion<T> & operator -= (octonion<X> const & rhs);
+
+   octonion<T> & operator *= (T const & rhs);
+   octonion<T> & operator *= (::std::complex<T> const & rhs);
+   octonion<T> & operator *= (::boost::math::quaternion<T> const & rhs);
+   template<typename X> 
+   octonion<T> & operator *= (octonion<X> const & rhs);
+
+   octonion<T> & operator /= (T const & rhs);
+   octonion<T> & operator /= (::std::complex<T> const & rhs);
+   octonion<T> & operator /= (::boost::math::quaternion<T> const & rhs);
+   template<typename X> 
+   octonion<T> & operator /= (octonion<X> const & rhs);
+};
+
+} } // namespaces
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/test_program.html b/doc/octonion/html/boost_octonions/octonions/test_program.html new file mode 100644 index 000000000..29e3de7dd --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/test_program.html @@ -0,0 +1,57 @@ + + + +Test Program + + + + + + + + + + + + + + + +
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+ The octonion_test.cpp + test program tests octonions specialisations for float, double and long double + (sample output). +

+

+ If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional + output (verbose output); + this will only be helpfull if you enable message output at the same time, + of course (by uncommenting the relevant line in the test or by adding --log_level=messages + to your command line,...). In that case, and if you are running interactively, + you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR + to interactively test the input operator with input of your choice from the + standard input (instead of hard-coding it in the test). +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/octonion/html/boost_octonions/octonions/to_do.html b/doc/octonion/html/boost_octonions/octonions/to_do.html new file mode 100644 index 000000000..7270fb4b9 --- /dev/null +++ b/doc/octonion/html/boost_octonions/octonions/to_do.html @@ -0,0 +1,52 @@ + + + +To Do + + + + + + + + + + + + + + +
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+To Do +

+
    +
  • + Improve testing. +
    • +
  • + Rewrite input operatore using Spirit (creates a dependency). +
    • +
  • + Put in place an Expression Template mechanism (perhaps borrowing from uBlas). +
    • +
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHome +
+ + diff --git a/doc/octonion/html/index.html b/doc/octonion/html/index.html new file mode 100644 index 000000000..4e4f54911 --- /dev/null +++ b/doc/octonion/html/index.html @@ -0,0 +1,78 @@ + + + +Boost.Octonions + + + + + + + + + + + + + +
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+Boost.Octonions

+

+Hubert Holin +

+

Copyright © 2001 -2003 Hubert Holin

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+
+

Table of Contents

+
+
Octonions
+
+
Overview
+
Header File
+
Synopsis
+
Template + Class octonion
+
Octonion + Specializations
+
Octonion + Member Typedefs
+
Octonion + Member Functions
+
Octonion Non-Member + Operators
+
Octonion + Value Operations
+
Octonion Creation + Functions
+
Octonions + Transcendentals
+
Test Program
+
Acknowledgements
+
History
+
To Do
+
+
+
+
+ + + +

Last revised: December 29, 2006 at 11:08:32 +0000

+
+
Next
+ + diff --git a/doc/octonion/math-octonion.qbk b/doc/octonion/math-octonion.qbk new file mode 100644 index 000000000..4896a6e07 --- /dev/null +++ b/doc/octonion/math-octonion.qbk @@ -0,0 +1,1005 @@ +[article Boost.Octonions + [quickbook 1.3] + [copyright 2001-2003 Hubert Holin] + [purpose Octonions] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Holin, Hubert]] + [category math] + [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $] +] + +[def __R ['[*R]]] +[def __C ['[*C]]] +[def __H ['[*H]]] +[def __O ['[*O]]] +[def __R3 ['[*'''R3''']]] +[def __R4 ['[*'''R4''']]] +[def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')] +[def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]] +[def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]] +[def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]] +[def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]] + +[def __asinh [link boost_quaternions.math_special_functions.asinh asinh]] +[def __acosh [link boost_quaternions.math_special_functions.acosh acosh]] +[def __atanh [link boost_quaternions.math_special_functions.atanh atanh]] +[def __sinc_pi [link boost_quaternions.math_special_functions.sinc_pi sinc_pi]] +[def __sinhc_pi [link boost_quaternions.math_special_functions.sinhc_pi sinhc_pi]] + +[def __log1p [link boost_quaternions.math_special_functions.log1p log1p]] +[def __expm1 [link boost_quaternions.math_special_functions.expm1 expm1]] +[def __hypot [link boost_quaternions.math_special_functions.hypot hypot]] + +[section Octonions] + +[section Overview] + +Octonions, like [@../../quaternions/html/index.html quaternions], are a relative of complex numbers. + +Octonions see some use in theoretical physics. + +In practical terms, an octonion is simply an octuple of real numbers __octulple, +which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]] +are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']] +are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]). + +Addition and a multiplication is defined on the set of octonions, +which generalize their quaternionic counterparts. The main novelty this time +is that [*the multiplication is not only not commutative, is now not even +associative] (i.e. there are octonions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal). +A way of remembering things is by using the following multiplication table: + +[$../../../octonion/graphics/octonion_blurb17.jpeg] + +Octonions (and their kin) are described in far more details in this other +[@../../../quaternion/TQE.pdf document] +(with [@../../../quaternion/TQE_EA.pdf errata and addenda]). + +Some traditional constructs, such as the exponential, carry over without too +much change into the realms of octonions, but other, such as taking a square root, +do not (the fact that the exponential has a closed form is a result of the +author, but the fact that the exponential exists at all for octonions is known +since quite a long time ago). + +[endsect] + +[section Header File] + +The interface and implementation are both supplied by the header file +[@../../../../../boost/math/octonion.hpp octonion.hpp]. + +[endsect] + +[section Synopsis] + + namespace boost{ namespace math{ + + template class ``[link boost_octonions.octonions.template_class_octonion octonion]``; + template<> class ``[link boost_octonions.octonions.octonion_specializations octonion]``; + template<> class ``[link boost_octonions.octonion_double octonion]``; + template<> class ``[link boost_octonions.octonion_long_double octonion]``; + + // operators + + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, T const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, octonion const & rhs); + + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, T const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, octonion const & rhs); + + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, T const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, octonion const & rhs); + + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (T const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, T const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, octonion const & rhs); + + template octonion ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator +]`` (octonion const & o); + template octonion ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator -]`` (octonion const & o); + + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, T const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::std::complex const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::std::complex const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, octonion const & rhs); + + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, T const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::std::complex const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::std::complex const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, octonion const & rhs); + + template + ::std::basic_istream & ``[link boost_octonions.octonions.non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, octonion & o); + + template + ::std::basic_ostream & ``[link boost_octonions.octonions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, octonion const & o); + + // values + + template T ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal real]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal unreal]``(octonion const & o); + + template T ``[link boost_octonions.octonions.octonion_value_operations.sup sup]``(octonion const & o); + template T ``[link boost_octonions.octonions.octonion_value_operations.l1 l1]``(octonionconst & o); + template T ``[link boost_octonions.octonions.octonion_value_operations.abs abs]``(octonion const & o); + template T ``[link boost_octonions.octonions.octonion_value_operations.norm norm]``(octonionconst & o); + template octonion ``[link boost_octonions.octonions.octonion_value_operations.conj conj]``(octonion const & o); + + template octonion ``[link boost_octonions.octonions.oct_create spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); + template octonion ``[link boost_octonions.octonions.oct_create multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); + template octonion ``[link boost_octonions.octonions.oct_create cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); + + // transcendentals + + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.exp exp]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.cos cos]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.sin sin]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.tan tan]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.cosh cosh]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.sinh sinh]``(octonion const & o); + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.tanh tanh]``(octonion const & o); + + template octonion ``[link boost_octonions.octonions.octonions_transcendentals.pow pow]``(octonion const & o, int n); + + } } // namespaces + +[endsect] + +[section Template Class octonion] + + namespace boost{ namespace math { + + template + class octonion + { + public: + typedef T value_type; + + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + template + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + + T ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; + octonion ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; + + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; + T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; + + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; + + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (T const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (T const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (T const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (T const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (T const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); + }; + + } } // namespaces + +[endsect] + +[section Octonion Specializations] + + namespace boost{ namespace math{ + + template<> + class octonion + { + public: + typedef float value_type; + + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + + float ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; + octonion ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; + + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; + float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; + + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; + + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (float const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (float const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (float const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (float const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (float const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); + }; + +[#boost_octonions.octonion_double] + + template<> + class octonion + { + public: + typedef double value_type; + + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + + double ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; + octonion ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; + + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; + double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; + + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; + + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (double const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); + }; + +[#boost_octonions.octonion_long_double] + + template<> + class octonion + { + public: + typedef long double value_type; + + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::boost::math::quaternion const & q0, ::boost::math::quaternion const & z1 = ::boost::math::quaternion()); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion const & a_recopier); + + long double ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; + octonion ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; + + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; + long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; + + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; + ::std::complex ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; + + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (long double const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (long double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (long double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (long double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion const & rhs); + + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (long double const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + template + octonion & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion const & rhs); + }; + + } } // namespaces + +[endsect] + +[section Octonion Member Typedefs] + +[*value_type] + +Template version: + + typedef T value_type; + +Float specialization version: + + typedef float value_type; + +Double specialization version: + + typedef double value_type; + +Long double specialization version: + + typedef long double value_type; + +These provide easy acces to the type the template is built upon. + +[endsect] + +[section Octonion Member Functions] + +[h3 Constructors] + +Template version: + + explicit octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); + explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + template + explicit octonion(octonion const & a_recopier); + +Float specialization version: + + explicit octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); + explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit octonion(octonion const & a_recopier); + explicit octonion(octonion const & a_recopier); + +Double specialization version: + + explicit octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); + explicit octonion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit octonion(octonion const & a_recopier); + explicit octonion(octonion const & a_recopier); + +Long double specialization version: + + explicit octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); + explicit octonion( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); + explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit octonion(octonion const & a_recopier); + explicit octonion(octonion const & a_recopier); + +A default constructor is provided for each form, which initializes each component +to the default values for their type (i.e. zero for floating numbers). +This constructor can also accept one to eight base type arguments. +A constructor is also provided to build octonions from one to four complex numbers +sharing the same base type, and another taking one or two quaternions +sharing the same base type. The unspecialized template also sports a +templarized copy constructor, while the specialized forms have copy +constructors from the other two specializations, which are explicit +when a risk of precision loss exists. For the unspecialized form, +the base type's constructors must not throw. + +Destructors and untemplated copy constructors (from the same type) +are provided by the compiler. Converting copy constructors make use +of a templated helper function in a "detail" subnamespace. + +[h3 Other member functions] + +[h4 Real and Unreal Parts] + + T real() const; + octonion unreal() const; + +Like complex number, octonions do have a meaningful notion of "real part", +but unlike them there is no meaningful notion of "imaginary part". +Instead there is an "unreal part" which itself is a octonion, +and usually nothing simpler (as opposed to the complex number case). +These are returned by the first two functions. + +[h4 Individual Real Components] + + T R_component_1() const; + T R_component_2() const; + T R_component_3() const; + T R_component_4() const; + T R_component_5() const; + T R_component_6() const; + T R_component_7() const; + T R_component_8() const; + +A octonion having eight real components, these are returned by +these eight functions. Hence real and R_component_1 return the same value. + +[h4 Individual Complex Components] + + ::std::complex C_component_1() const; + ::std::complex C_component_2() const; + ::std::complex C_component_3() const; + ::std::complex C_component_4() const; + +A octonion likewise has four complex components. Actually, octonions +are indeed a (left) vector field over the complexes, but beware, as +for any octonion __oct_formula we also have __oct_complex_formula +(note the [*minus] sign in the last factor). +What the C_component_n functions return, however, are the complexes +which could be used to build the octonion using the constructor, and +[*not] the components of the octonion on the basis ['[^(1, j, e', j')]]. + +[h4 Individual Quaternion Components] + + ::boost::math::quaternion H_component_1() const; + ::boost::math::quaternion H_component_2() const; + +Likewise, for any octonion __oct_formula we also have __oct_quat_formula, though there +is no meaningful vector-space-like structure based on the quaternions. +What the H_component_n functions return are the quaternions which +could be used to build the octonion using the constructor. + +[h3 Octonion Member Operators] +[h4 Assignment Operators] + + octonion & operator = (octonion const & a_affecter); + template + octonion & operator = (octonion const & a_affecter); + octonion & operator = (T const & a_affecter); + octonion & operator = (::std::complex const & a_affecter); + octonion & operator = (::boost::math::quaternion const & a_affecter); + +These perform the expected assignment, with type modification if +necessary (for instance, assigning from a base type will set the +real part to that value, and all other components to zero). +For the unspecialized form, the base type's assignment operators must not throw. + +[h4 Other Member Operators] + + octonion & operator += (T const & rhs) + octonion & operator += (::std::complex const & rhs); + octonion & operator += (::boost::math::quaternion const & rhs); + template + octonion & operator += (octonion const & rhs); + +These perform the mathematical operation `(*this)+rhs` and store the result in +`*this`. The unspecialized form has exception guards, which the specialized +forms do not, so as to insure exception safety. For the unspecialized form, +the base type's assignment operators must not throw. + + octonion & operator -= (T const & rhs) + octonion & operator -= (::std::complex const & rhs); + octonion & operator -= (::boost::math::quaternion const & rhs); + template + octonion & operator -= (octonion const & rhs); + +These perform the mathematical operation `(*this)-rhs` and store the result +in `*this`. The unspecialized form has exception guards, which the +specialized forms do not, so as to insure exception safety. +For the unspecialized form, the base type's assignment operators must not throw. + + octonion & operator *= (T const & rhs) + octonion & operator *= (::std::complex const & rhs); + octonion & operator *= (::boost::math::quaternion const & rhs); + template + octonion & operator *= (octonion const & rhs); + +These perform the mathematical operation `(*this)*rhs` in this order +(order is important as multiplication is not commutative for octonions) +and store the result in `*this`. The unspecialized form has exception guards, +which the specialized forms do not, so as to insure exception safety. +For the unspecialized form, the base type's assignment operators must +not throw. Also, for clarity's sake, you should always group the +factors in a multiplication by groups of two, as the multiplication is +not even associative on the octonions (though there are of course cases +where this does not matter, it usually does). + + octonion & operator /= (T const & rhs) + octonion & operator /= (::std::complex const & rhs); + octonion & operator /= (::boost::math::quaternion const & rhs); + template + octonion & operator /= (octonion const & rhs); + +These perform the mathematical operation `(*this)*inverse_of(rhs)` +in this order (order is important as multiplication is not commutative +for octonions) and store the result in `*this`. The unspecialized form +has exception guards, which the specialized forms do not, so as to +insure exception safety. For the unspecialized form, the base +type's assignment operators must not throw. As for the multiplication, +remember to group any two factors using parenthesis. + +[endsect] + +[section:non_mem Octonion Non-Member Operators] + +[h4 Unary Plus and Minus Operators] + + template octonion operator + (octonion const & o); + +This unary operator simply returns o. + + template octonion operator - (octonion const & o); + +This unary operator returns the opposite of o. + +[h4 Binary Addition Operators] + + template octonion operator + (T const & lhs, octonion const & rhs); + template octonion operator + (octonion const & lhs, T const & rhs); + template octonion operator + (::std::complex const & lhs, octonion const & rhs); + template octonion operator + (octonion const & lhs, ::std::complex const & rhs); + template octonion operator + (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion operator + (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion operator + (octonion const & lhs, octonion const & rhs); + +These operators return `octonion(lhs) += rhs`. + +[h4 Binary Subtraction Operators] + + template octonion operator - (T const & lhs, octonion const & rhs); + template octonion operator - (octonion const & lhs, T const & rhs); + template octonion operator - (::std::complex const & lhs, octonion const & rhs); + template octonion operator - (octonion const & lhs, ::std::complex const & rhs); + template octonion operator - (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion operator - (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion operator - (octonion const & lhs, octonion const & rhs); + +These operators return `octonion(lhs) -= rhs`. + +[h4 Binary Multiplication Operators] + + template octonion operator * (T const & lhs, octonion const & rhs); + template octonion operator * (octonion const & lhs, T const & rhs); + template octonion operator * (::std::complex const & lhs, octonion const & rhs); + template octonion operator * (octonion const & lhs, ::std::complex const & rhs); + template octonion operator * (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion operator * (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion operator * (octonion const & lhs, octonion const & rhs); + +These operators return `octonion(lhs) *= rhs`. + +[h4 Binary Division Operators] + + template octonion operator / (T const & lhs, octonion const & rhs); + template octonion operator / (octonion const & lhs, T const & rhs); + template octonion operator / (::std::complex const & lhs, octonion const & rhs); + template octonion operator / (octonion const & lhs, ::std::complex const & rhs); + template octonion operator / (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion operator / (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion operator / (octonion const & lhs, octonion const & rhs); + +These operators return `octonion(lhs) /= rhs`. It is of course still an +error to divide by zero... + +[h4 Binary Equality Operators] + + template bool operator == (T const & lhs, octonion const & rhs); + template bool operator == (octonion const & lhs, T const & rhs); + template bool operator == (::std::complex const & lhs, octonion const & rhs); + template bool operator == (octonion const & lhs, ::std::complex const & rhs); + template bool operator == (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool operator == (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool operator == (octonion const & lhs, octonion const & rhs); + +These return true if and only if the four components of `octonion(lhs)` +are equal to their counterparts in `octonion(rhs)`. As with any +floating-type entity, this is essentially meaningless. + +[h4 Binary Inequality Operators] + + template bool operator != (T const & lhs, octonion const & rhs); + template bool operator != (octonion const & lhs, T const & rhs); + template bool operator != (::std::complex const & lhs, octonion const & rhs); + template bool operator != (octonion const & lhs, ::std::complex const & rhs); + template bool operator != (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool operator != (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool operator != (octonion const & lhs, octonion const & rhs); + +These return true if and only if `octonion(lhs) == octonion(rhs)` +is false. As with any floating-type entity, this is essentially meaningless. + +[h4 Stream Extractor] + + template + ::std::basic_istream & operator >> (::std::basic_istream & is, octonion & o); + +Extracts an octonion `o`. We accept any format which seems reasonable. +However, since this leads to a great many ambiguities, decisions were made +to lift these. In case of doubt, stick to lists of reals. + +The input values must be convertible to T. If bad input is encountered, +calls `is.setstate(ios::failbit)` (which may throw `ios::failure` (27.4.5.3)). + +Returns `is`. + +[h4 Stream Inserter] + + template + ::std::basic_ostream & operator << (::std::basic_ostream & os, octonion const & o); + +Inserts the octonion `o` onto the stream `os` as if it were implemented as follows: + + template + ::std::basic_ostream & operator << ( ::std::basic_ostream & os, + octonion const & o) + { + ::std::basic_ostringstream s; + + s.flags(os.flags()); + s.imbue(os.getloc()); + s.precision(os.precision()); + + s << '(' << o.R_component_1() << ',' + << o.R_component_2() << ',' + << o.R_component_3() << ',' + << o.R_component_4() << ',' + << o.R_component_5() << ',' + << o.R_component_6() << ',' + << o.R_component_7() << ',' + << o.R_component_8() << ')'; + + return os << s.str(); + } + +[endsect] + +[section Octonion Value Operations] + +[h4 Real and Unreal] + + template T real(octonion const & o); + template octonion unreal(octonion const & o); + +These return `o.real()` and `o.unreal()` respectively. + +[h4 conj] + + template octonion conj(octonion const & o); + +This returns the conjugate of the octonion. + +[h4 sup] + + template T sup(octonion const & o); + +This return the sup norm (the greatest among +`abs(o.R_component_1())...abs(o.R_component_8()))` of the octonion. + +[h4 l1] + + template T l1(octonion const & o); + +This return the l1 norm (`abs(o.R_component_1())+...+abs(o.R_component_8())`) +of the octonion. + +[h4 abs] + + template T abs(octonion const & o); + +This return the magnitude (Euclidian norm) of the octonion. + +[h4 norm] + + template T norm(octonionconst & o); + +This return the (Cayley) norm of the octonion. The term "norm" might +be confusing, as most people associate it with the Euclidian norm +(and quadratic functionals). For this version of (the mathematical +objects known as) octonions, the Euclidian norm (also known as +magnitude) is the square root of the Cayley norm. + +[endsect] + +[section:oct_create Octonion Creation Functions] + + template octonion spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); + template octonion multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); + template octonion cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); + +These build octonions in a way similar to the way polar builds +complex numbers, as there is no strict equivalent to +polar coordinates for octonions. + +`spherical` is a simple transposition of `polar`, it takes as inputs a +(positive) magnitude and a point on the hypersphere, given +by three angles. The first of these, ['theta] has a natural range of +-pi to +pi, and the other two have natural ranges of +-pi/2 to +pi/2 (as is the case with the usual spherical +coordinates in __R3). Due to the many symmetries and periodicities, +nothing untoward happens if the magnitude is negative or the angles are +outside their natural ranges. The expected degeneracies (a magnitude of +zero ignores the angles settings...) do happen however. + +`cylindrical` is likewise a simple transposition of the usual +cylindrical coordinates in __R3, which in turn is another derivative of +planar polar coordinates. The first two inputs are the polar +coordinates of the first __C component of the octonion. The third and +fourth inputs are placed into the third and fourth __R components of the +octonion, respectively. + +`multipolar` is yet another simple generalization of polar coordinates. +This time, both __C components of the octonion are given in polar coordinates. + +In this version of our implementation of octonions, there is no +analogue of the complex value operation arg as the situation is +somewhat more complicated. + +[endsect] + +[section Octonions Transcendentals] + +There is no `log` or `sqrt` provided for octonions in this implementation, +and `pow` is likewise restricted to integral powers of the exponent. +There are several reasons to this: on the one hand, the equivalent of +analytic continuation for octonions ("branch cuts") remains to be +investigated thoroughly (by me, at any rate...), and we wish to avoid +the nonsense introduced in the standard by exponentiations of +complexes by complexes (which is well defined, but not in the standard...). +Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is +just plain brain-dead... + +We do, however provide several transcendentals, chief among which is +the exponential. That it allows for a "closed formula" is a result +of the author (the existence and definition of the exponential, on the +octonions among others, on the other hand, is a few centuries old). +Basically, any converging power series with real coefficients which +allows for a closed formula in __C can be transposed to __O. More +transcendentals of this type could be added in a further revision upon +request. It should be noted that it is these functions which force the +dependency upon the +[@../../../../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] +and the +[@../../../../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] +headers. + +[h4 exp] + + template + octonion exp(octonion const & o); + +Computes the exponential of the octonion. + +[h4 cos] + + template + octonion cos(octonion const & o); + +Computes the cosine of the octonion + +[h4 sin] + + template + octonion sin(octonion const & o); + +Computes the sine of the octonion. + +[h4 tan] + + template + octonion tan(octonion const & o); + +Computes the tangent of the octonion. + +[h4 cosh] + + template + octonion cosh(octonion const & o); + +Computes the hyperbolic cosine of the octonion. + +[h4 sinh] + + template + octonion sinh(octonion const & o); + +Computes the hyperbolic sine of the octonion. + +[h4 tanh] + + template + octonion tanh(octonion const & o); + +Computes the hyperbolic tangent of the octonion. + +[h4 pow] + + template + octonion pow(octonion const & o, int n); + +Computes the n-th power of the octonion q. + +[endsect] + +[section Test Program] + +The [@../../../octonion/octonion_test.cpp octonion_test.cpp] +test program tests octonions specialisations for float, double and long double +([@../../../octonion/output.txt sample output]). + +If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional +output ([@../../../octonion/output_more.txt verbose output]); this will +only be helpfull if you enable message output at the same time, of course +(by uncommenting the relevant line in the test or by adding --log_level=messages +to your command line,...). In that case, and if you are running interactively, +you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to +interactively test the input operator with input of your choice from the +standard input (instead of hard-coding it in the test). + +[endsect] + +[section Acknowledgements] + +The mathematical text has been typeset with +[@http://www.nisus-soft.com/ Nisus Writer]. +Jens Maurer has helped with portability and standard adherence, and was the +Review Manager for this library. More acknowledgements in the +History section. Thank you to all who contributed to the discussion about this library. + +[endsect] + +[section History] + +* 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. +* 1.5.7 - 25/02/2003: transitionned to the unit test framework; now included by the library header (rather than the test files), via . +* 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). +* 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. +* 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes. +* 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). +* 1.5.2 - 07/07/2001: introduced namespace math. +* 1.5.1 - 07/06/2001: (end of Boost review) now includes and instead of ; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. +* 1.5.0 - 23/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version). +* 1.4.0 - 09/01/2001: added tan and tanh. +* 1.3.1 - 08/01/2001: cosmetic fixes. +* 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. +* 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. +* 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. +* 1.0.0 - 10/08/1999: first public version. + +[endsect] + +[section To Do] + +* Improve testing. +* Rewrite input operatore using Spirit (creates a dependency). +* Put in place an Expression Template mechanism (perhaps borrowing from uBlas). + +[endsect] + +[endsect] diff --git a/doc/quaternion/Jamfile.v2 b/doc/quaternion/Jamfile.v2 new file mode 100644 index 000000000..67168f291 --- /dev/null +++ b/doc/quaternion/Jamfile.v2 @@ -0,0 +1,70 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +path-constant images_location : html ; + +xml quaternion : math-quaternion.qbk ; +boostbook standalone + : + quaternion + : + # Path for links to Boost: + boost.root=../../../../.. + # Path for libraries index: + boost.libraries=../../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=10 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=1 + # How far down sections get TOC's + toc.section.depth=10 + # Max depth in each TOC: + toc.max.depth=4 + # How far down we go with TOC's + generate.section.toc.level=10 + #root.filename="sf_dist_and_tools" + + # PDF Options: + # TOC Generation: this is needed for FOP-0.9 and later: + # fop1.extensions=1 + pdf:xep.extensions=1 + # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9! + pdf:fop.extensions=0 + pdf:fop1.extensions=0 + # No indent on body text: + pdf:body.start.indent=0pt + # Margin size: + pdf:page.margin.inner=0.5in + # Margin size: + pdf:page.margin.outer=0.5in + # Paper type = A4 + pdf:paper.type=A4 + # Yes, we want graphics for admonishments: + admon.graphics=1 + # Set this one for PDF generation *only*: + # default pnd graphics are awful in PDF form, + # better use SVG's instead: + pdf:admon.graphics.extension=".svg" + pdf:use.role.for.mediaobject=1 + pdf:preferred.mediaobject.role=print + pdf:img.src.path=$(images_location)/ + pdf:admon.graphics.path=$(images_location)/../../svg-admon/ + pdf:draft.mode="no" + ; + + + diff --git a/doc/quaternion/html/boost_quaternions/quaternions.html b/doc/quaternion/html/boost_quaternions/quaternions.html new file mode 100644 index 000000000..027c9977f --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions.html @@ -0,0 +1,65 @@ + + + +Quaternions + + + + + + + + + + + + + + + +
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Overview
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Header File
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Synopsis
+
Template Class quaternion
+
Quaternion Specializations
+
Quaternion + Member Typedefs
+
Quaternion Member + Functions
+
Quaternion Non-Member + Operators
+
Quaternion Value + Operations
+
Quaternion Creation + Functions
+
Quaternion Transcendentals
+
Test Program
+
The Quaternionic + Exponential
+
Acknowledgements
+
History
+
To Do
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/acknowledgements.html b/doc/quaternion/html/boost_quaternions/quaternions/acknowledgements.html new file mode 100644 index 000000000..640ec024e --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/acknowledgements.html @@ -0,0 +1,49 @@ + + + +Acknowledgements + + + + + + + + + + + + + + + +
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+ The mathematical text has been typeset with Nisus + Writer. Jens Maurer has helped with portability and standard adherence, + and was the Review Manager for this library. More acknowledgements in the + History section. Thank you to all who contributed to the discution about + this library. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/create.html b/doc/quaternion/html/boost_quaternions/quaternions/create.html new file mode 100644 index 000000000..035097883 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/create.html @@ -0,0 +1,118 @@ + + + + Quaternion Creation + Functions + + + + + + + + + + + + + + + +
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+template<typename T>	quaternion<T>	spherical(T const & rho, T const & theta, T const & phi1, T const & phi2);
+template<typename T>	quaternion<T>	semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2);
+template<typename T>	quaternion<T>	multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2);
+template<typename T>	quaternion<T>	cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude);
+template<typename T>	quaternion<T>	cylindrical(T const & r, T const & angle, T const & h1, T const & h2);
+
+

+ These build quaternions in a way similar to the way polar builds complex + numbers, as there is no strict equivalent to polar coordinates for quaternions. +

+

+ spherical is a simple transposition + of polar, it takes as inputs + a (positive) magnitude and a point on the hypersphere, given by three angles. + The first of these, theta + has a natural range of -pi + to +pi, + and the other two have natural ranges of -pi/2 + to +pi/2 (as is the + case with the usual spherical coordinates in R3). + Due to the many symmetries and periodicities, nothing untoward happens if + the magnitude is negative or the angles are outside their natural ranges. + The expected degeneracies (a magnitude of zero ignores the angles settings...) + do happen however. +

+

+ cylindrical is likewise a + simple transposition of the usual cylindrical coordinates in R3, which in turn is another derivative of + planar polar coordinates. The first two inputs are the polar coordinates + of the first C component + of the quaternion. The third and fourth inputs are placed into the third + and fourth R components + of the quaternion, respectively. +

+

+ multipolar is yet another + simple generalization of polar coordinates. This time, both C components of the quaternion are given + in polar coordinates. +

+

+ cylindrospherical is specific + to quaternions. It is often interesting to consider H + as the cartesian product of R + by R3 (the quaternionic + multiplication as then a special form, as given here). This function therefore + builds a quaternion from this representation, with the R3 component given in usual R3 spherical coordinates. +

+

+ semipolar is another generator + which is specific to quaternions. It takes as a first input the magnitude + of the quaternion, as a second input an angle in the range 0 to +pi/2 + such that magnitudes of the first two C + components of the quaternion are the product of the first input and the sine + and cosine of this angle, respectively, and finally as third and fourth inputs + angles in the range -pi/2 to +pi/2 which represent the arguments of the first + and second C components + of the quaternion, respectively. As usual, nothing untoward happens if what + should be magnitudes are negative numbers or angles are out of their natural + ranges, as symmetries and periodicities kick in. +

+

+ In this version of our implementation of quaternions, there is no analogue + of the complex value operation arg + as the situation is somewhat more complicated. Unit quaternions are linked + both to rotations in R3 + and in R4, and the correspondences + are not too complicated, but there is currently a lack of standard (de facto + or de jure) matrix library with which the conversions could work. This should + be remedied in a further revision. In the mean time, an example of how this + could be done is presented here for R3, and here for R4 (example + test file). +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/exp.html b/doc/quaternion/html/boost_quaternions/quaternions/exp.html new file mode 100644 index 000000000..7f99b8ec5 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/exp.html @@ -0,0 +1,53 @@ + + + + The Quaternionic + Exponential + + + + + + + + + + + + + + + +
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+ Please refer to the following PDF's: +

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+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/header_file.html b/doc/quaternion/html/boost_quaternions/quaternions/header_file.html new file mode 100644 index 000000000..4d929de28 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/header_file.html @@ -0,0 +1,44 @@ + + + +Header File + + + + + + + + + + + + + + + +
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+ The interface and implementation are both supplied by the header file quaternion.hpp. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/history.html b/doc/quaternion/html/boost_quaternions/quaternions/history.html new file mode 100644 index 000000000..18a5abe63 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/history.html @@ -0,0 +1,105 @@ + + + +History + + + + + + + + + + + + + + + +
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    +
  • + 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. +
    • +
  • + 1.5.7 - 24/02/2003: transitionned to the unit test framework; <boost/config.hpp> + now included by the library header (rather than the test files). +
    • +
  • + 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis + Evlogimenos (alkis@routescience.com). +
    • +
  • + 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens + (michael@acfr.usyd.edu.au); requires the /Za compiler option. +
    • +
  • + 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different + translation units); attempt at an improved compatibility with Microsoft + compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; + other compatibility fixes. +
    • +
  • + 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor + (gregod@cs.rpi.edu). +
    • +
  • + 1.5.2 - 07/07/2001: introduced namespace math. +
    • +
  • + 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> + and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; + corrected bug in sin (Daryle Walker); removed check for self-assignment + (Gary Powel); made converting functions explicit (Gary Powel); added overflow + guards for division operators and abs (Peter Schmitteckert); added sup + and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. +
    • +
  • + 1.5.0 - 26/03/2001: boostification, inlining of all operators except input, + output and pow, fixed exception safety of some members (template version) + and output operator, added spherical, semipolar, multipolar, cylindrospherical + and cylindrical. +
    • +
  • + 1.4.0 - 09/01/2001: added tan and tanh. +
    • +
  • + 1.3.1 - 08/01/2001: cosmetic fixes. +
    • +
  • + 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) + algorithm. +
    • +
  • + 1.2.0 - 25/05/2000: fixed the division operators and output; changed many + signatures. +
    • +
  • + 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. +
    • +
  • + 1.0.0 - 10/08/1999: first public version. +
    • +
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + Constructors +

+

+ Template version: +

+
+explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T());
+explicit quaternion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>());
+template<typename X>
+explicit quaternion(quaternion<X> const & a_recopier);
+
+

+ Float specialization version: +

+
+explicit quaternion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f);
+explicit quaternion(::std::complex<float> const & z0,::std::complex<float> const & z1 = ::std::complex<float>());
+explicit quaternion(quaternion<double> const & a_recopier); 
+explicit quaternion(quaternion<long double> const & a_recopier);
+
+

+ Double specialization version: +

+
+explicit quaternion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0);
+explicit quaternion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>());
+explicit quaternion(quaternion<float> const & a_recopier);
+explicit quaternion(quaternion<long double> const & a_recopier);
+
+

+ Long double specialization version: +

+
+explicit quaternion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L);
+explicit quaternion(	::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>());
+explicit quaternion(quaternion<float> const & a_recopier);
+explicit quaternion(quaternion<double> const & a_recopier);
+
+

+ A default constructor is provided for each form, which initializes each component + to the default values for their type (i.e. zero for floating numbers). This + constructor can also accept one to four base type arguments. A constructor + is also provided to build quaternions from one or two complex numbers sharing + the same base type. The unspecialized template also sports a templarized + copy constructor, while the specialized forms have copy constructors from + the other two specializations, which are explicit when a risk of precision + loss exists. For the unspecialized form, the base type's constructors must + not throw. +

+

+ Destructors and untemplated copy constructors (from the same type) are provided + by the compiler. Converting copy constructors make use of a templated helper + function in a "detail" subnamespace. +

+

+ + Other + member functions +

+

+ + Real + and Unreal Parts +

+
+T             real() const;
+quaternion<T> unreal() const;
+
+

+ Like complex number, quaternions do have a meaningful notion of "real + part", but unlike them there is no meaningful notion of "imaginary + part". Instead there is an "unreal part" which itself is a + quaternion, and usually nothing simpler (as opposed to the complex number + case). These are returned by the first two functions. +

+

+ + Individual + Real Components +

+
+T R_component_1() const;
+T R_component_2() const;
+T R_component_3() const;
+T R_component_4() const;
+
+

+ A quaternion having four real components, these are returned by these four + functions. Hence real and R_component_1 return the same value. +

+

+ + Individual + Complex Components +

+
+::std::complex<T>	C_component_1() const;
+::std::complex<T>	C_component_2() const;
+
+

+ A quaternion likewise has two complex components, and as we have seen above, + for any quaternion q = α + βi + γj + δk we also have + q = (α + βi) + (γ + δi)j . These functions return them. + The real part of q.C_component_1() + is the same as q.real(). +

+

+ + Quaternion + Member Operators +

+

+ + Assignment + Operators +

+
+quaternion<T>& operator = (quaternion<T> const & a_affecter);
+template<typename X> 
+quaternion<T>& operator = (quaternion<X> const& a_affecter);
+quaternion<T>& operator = (T const& a_affecter);
+quaternion<T>& operator = (::std::complex<T> const& a_affecter);
+
+

+ These perform the expected assignment, with type modification if necessary + (for instance, assigning from a base type will set the real part to that + value, and all other components to zero). For the unspecialized form, the + base type's assignment operators must not throw. +

+

+ + Addition + Operators +

+
+quaternion<T>& operator += (T const & rhs)
+quaternion<T>& operator += (::std::complex<T> const & rhs);
+template<typename X>	
+quaternion<T>& operator += (quaternion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)+rhs + and store the result in *this. + The unspecialized form has exception guards, which the specialized forms + do not, so as to insure exception safety. For the unspecialized form, the + base type's assignment operators must not throw. +

+

+ + Subtraction + Operators +

+
+quaternion<T>& operator -= (T const & rhs)
+quaternion<T>& operator -= (::std::complex<T> const & rhs);
+template<typename X>	
+quaternion<T>& operator -= (quaternion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)-rhs + and store the result in *this. + The unspecialized form has exception guards, which the specialized forms + do not, so as to insure exception safety. For the unspecialized form, the + base type's assignment operators must not throw. +

+

+ + Multiplication + Operators +

+
+quaternion<T>& operator *= (T const & rhs)
+quaternion<T>& operator *= (::std::complex<T> const & rhs);
+template<typename X>	
+quaternion<T>& operator *= (quaternion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)*rhs + in this order (order is important as multiplication + is not commutative for quaternions) and store the result in *this. The + unspecialized form has exception guards, which the specialized forms do not, + so as to insure exception safety. For the unspecialized form, the base type's + assignment operators must not throw. +

+

+ + Division + Operators +

+
+quaternion<T>& operator /= (T const & rhs)
+quaternion<T>& operator /= (::std::complex<T> const & rhs);
+template<typename X>	
+quaternion<T>& operator /= (quaternion<X> const & rhs);
+
+

+ These perform the mathematical operation (*this)*inverse_of(rhs) + in this order (order is important as multiplication + is not commutative for quaternions) and store the result in *this. The + unspecialized form has exception guards, which the specialized forms do not, + so as to insure exception safety. For the unspecialized form, the base type's + assignment operators must not throw. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/mem_typedef.html b/doc/quaternion/html/boost_quaternions/quaternions/mem_typedef.html new file mode 100644 index 000000000..0baab67c6 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/mem_typedef.html @@ -0,0 +1,75 @@ + + + + Quaternion + Member Typedefs + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ value_type +

+

+ Template version: +

+
+typedef T value_type;
+
+

+ Float specialization version: +

+
+typedef float value_type;
+
+

+ Double specialization version: +

+
+typedef double value_type;
+
+

+ Long double specialization version: +

+
+typedef long double value_type;
+
+

+ These provide easy acces to the type the template is built upon. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/non_mem.html b/doc/quaternion/html/boost_quaternions/quaternions/non_mem.html new file mode 100644 index 000000000..5f5dd4fe1 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/non_mem.html @@ -0,0 +1,233 @@ + + + + Quaternion Non-Member + Operators + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ + Unary Plus +

+
+template<typename T>	
+quaternion<T> operator + (quaternion<T> const & q);
+
+

+ This unary operator simply returns q. +

+

+ + Unary Minus +

+
+template<typename T>
+quaternion<T> operator - (quaternion<T> const & q);
+
+

+ This unary operator returns the opposite of q. +

+

+ + Binary + Addition Operators +

+
+template<typename T>	quaternion<T>	operator + (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	quaternion<T>	operator + (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These operators return quaternion<T>(lhs) += + rhs. +

+

+ + Binary + Subtraction Operators +

+
+template<typename T>	quaternion<T>	operator - (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	quaternion<T>	operator - (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These operators return quaternion<T>(lhs) -= + rhs. +

+

+ + Binary + Multiplication Operators +

+
+template<typename T>	quaternion<T>	operator * (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	quaternion<T>	operator * (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These operators return quaternion<T>(lhs) *= + rhs. +

+

+ + Binary + Division Operators +

+
+template<typename T>	quaternion<T>	operator / (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	quaternion<T>	operator / (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These operators return quaternion<T>(lhs) /= + rhs. It is of course still an error + to divide by zero... +

+

+ + Equality + Operators +

+
+template<typename T>	bool	operator == (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	bool	operator == (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	bool	operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	bool	operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	bool	operator == (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These return true if and only if the four components of quaternion<T>(lhs) are + equal to their counterparts in quaternion<T>(rhs). As + with any floating-type entity, this is essentially meaningless. +

+

+ + Inequality + Operators +

+
+template<typename T>	bool	operator != (T const & lhs, quaternion<T> const & rhs);
+template<typename T>	bool	operator != (quaternion<T> const & lhs, T const & rhs);
+template<typename T>	bool	operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T>	bool	operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T>	bool	operator != (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+

+ These return true if and only if quaternion<T>(lhs) == + quaternion<T>(rhs) is + false. As with any floating-type entity, this is essentially meaningless. +

+

+ + Stream + Extractor +

+
+template<typename T, typename charT, class traits>
+::std::basic_istream<charT,traits>& operator >> (::std::basic_istream<charT,traits> & is, quaternion<T> & q);
+
+

+ Extracts a quaternion q of one of the following forms (with a, b, c and d + of type T): +

+

+ a (a), (a,b), (a,b,c), (a,b,c,d) (a,(c)), (a,(c,d)), ((a)), ((a),c), + ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d)) +

+

+ The input values must be convertible to T. + If bad input is encountered, calls is.setstate(ios::failbit) + (which may throw ios::failure (27.4.5.3)). +

+

+ Returns: is. +

+

+ The rationale for the list of accepted formats is that either we have a list + of up to four reals, or else we have a couple of complex numbers, and in + that case if it formated as a proper complex number, then it should be accepted. + Thus potential ambiguities are lifted (for instance (a,b) is (a,b,0,0) and + not (a,0,b,0), i.e. it is parsed as a list of two real numbers and not two + complex numbers which happen to have imaginary parts equal to zero). +

+

+ + Stream + Inserter +

+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits>& operator << (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q);
+
+

+ Inserts the quaternion q onto the stream os + as if it were implemented as follows: +

+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits>& operator << (	
+               ::std::basic_ostream<charT,traits> & os,
+               quaternion<T> const & q)
+{
+   ::std::basic_ostringstream<charT,traits>	s;
+
+   s.flags(os.flags());
+   s.imbue(os.getloc());
+   s.precision(os.precision());
+
+   s << '(' << q.R_component_1() << ','
+            << q.R_component_2() << ','
+            << q.R_component_3() << ','
+            << q.R_component_4() << ')';
+
+   return os << s.str();
+}
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/overview.html b/doc/quaternion/html/boost_quaternions/quaternions/overview.html new file mode 100644 index 000000000..f6842ad1e --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/overview.html @@ -0,0 +1,91 @@ + + + +Overview + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Quaternions are a relative of complex numbers. +

+

+ Quaternions are in fact part of a small hierarchy of structures built upon + the real numbers, which comprise only the set of real numbers (traditionally + named R), the set of + complex numbers (traditionally named C), + the set of quaternions (traditionally named H) + and the set of octonions (traditionally named O), + which possess interesting mathematical properties (chief among which is the + fact that they are division algebras, i.e. + where the following property is true: if y + is an element of that algebra and is not equal to zero, + then yx = yx', where x + and x' denote elements of that algebra, + implies that x = x'). Each member + of the hierarchy is a super-set of the former. +

+

+ One of the most important aspects of quaternions is that they provide an + efficient way to parameterize rotations in R3 + (the usual three-dimensional space) and R4. +

+

+ In practical terms, a quaternion is simply a quadruple of real numbers (α,β,γ,δ), + which we can write in the form q = α + βi + γj + δk, + where i is the same object as for + complex numbers, and j and k + are distinct objects which play essentially the same kind of role as i. +

+

+ An addition and a multiplication is defined on the set of quaternions, which + generalize their real and complex counterparts. The main novelty here is + that the multiplication is not commutative + (i.e. there are quaternions x and + y such that xy + ≠ yx). A good mnemotechnical way of remembering things + is by using the formula i*i = j*j = k*k = -1. +

+

+ Quaternions (and their kin) are described in far more details in this other + document (with errata + and addenda). +

+

+ Some traditional constructs, such as the exponential, carry over without + too much change into the realms of quaternions, but other, such as taking + a square root, do not. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/quat.html b/doc/quaternion/html/boost_quaternions/quaternions/quat.html new file mode 100644 index 000000000..55e9f1dc9 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/quat.html @@ -0,0 +1,96 @@ + + + + Template Class quaternion + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+namespace boost{ namespace math{
+
+template<typename T>
+class quaternion
+{
+public:
+
+   typedef T value_type;
+
+   explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T());
+   explicit quaternion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>());
+   template<typename X> 
+   explicit quaternion(quaternion<X> const & a_recopier);
+
+   T                  real() const;
+   quaternion<T>      unreal() const;
+   T                  R_component_1() const;
+   T                  R_component_2() const;
+   T                  R_component_3() const;
+   T                  R_component_4() const;
+   ::std::complex<T>  C_component_1() const;
+   ::std::complex<T>  C_component_2() const;
+
+   quaternion<T>&     operator = (quaternion<T> const  & a_affecter);
+   template<typename X>	
+   quaternion<T>&     operator = (quaternion<X> const  & a_affecter);
+   quaternion<T>&     operator = (T const  & a_affecter);
+   quaternion<T>&     operator = (::std::complex<T> const & a_affecter);
+
+   quaternion<T>&     operator += (T const & rhs);
+   quaternion<T>&     operator += (::std::complex<T> const & rhs);
+   template<typename X>
+   quaternion<T>&     operator += (quaternion<X> const & rhs);
+
+   quaternion<T>&     operator -= (T const & rhs);
+   quaternion<T>&     operator -= (::std::complex<T> const & rhs);
+   template<typename X>
+   quaternion<T>&     operator -= (quaternion<X> const & rhs);
+
+   quaternion<T>&     operator *= (T const & rhs);
+   quaternion<T>&     operator *= (::std::complex<T> const & rhs);
+   template<typename X>
+   quaternion<T>&     operator *= (quaternion<X> const & rhs);
+
+   quaternion<T>&     operator /= (T const & rhs);
+   quaternion<T>&     operator /= (::std::complex<T> const & rhs);
+   template<typename X>
+   quaternion<T>&     operator /= (quaternion<X> const & rhs);
+};
+
+} // namespace math
+} // namespace boost
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/spec.html b/doc/quaternion/html/boost_quaternions/quaternions/spec.html new file mode 100644 index 000000000..75550064f --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/spec.html @@ -0,0 +1,198 @@ + + + + Quaternion Specializations + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+namespace boost{ namespace math{
+
+template<>
+class quaternion<float>
+{
+public:
+   typedef float value_type;
+	
+   explicit quaternion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f);
+   explicit quaternion(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>());
+   explicit quaternion(quaternion<double> const & a_recopier);
+   explicit quaternion(quaternion<long double> const & a_recopier);
+	
+   float                  real() const;
+   quaternion<float>      unreal() const;
+   float                  R_component_1() const;
+   float                  R_component_2() const;
+   float                  R_component_3() const;
+   float                  R_component_4() const;
+   ::std::complex<float>  C_component_1() const;
+   ::std::complex<float>  C_component_2() const;
+
+   quaternion<float>&     operator = (quaternion<float> const  & a_affecter);
+   template<typename X>	
+   quaternion<float>&     operator = (quaternion<X> const  & a_affecter);
+   quaternion<float>&     operator = (float const  & a_affecter);
+   quaternion<float>&     operator = (::std::complex<float> const & a_affecter);
+
+   quaternion<float>&     operator += (float const & rhs);
+   quaternion<float>&     operator += (::std::complex<float> const & rhs);
+   template<typename X>
+   quaternion<float>&     operator += (quaternion<X> const & rhs);
+
+   quaternion<float>&     operator -= (float const & rhs);
+   quaternion<float>&     operator -= (::std::complex<float> const & rhs);
+   template<typename X>
+   quaternion<float>&     operator -= (quaternion<X> const & rhs);
+
+   quaternion<float>&     operator *= (float const & rhs);
+   quaternion<float>&     operator *= (::std::complex<float> const & rhs);
+   template<typename X>
+   quaternion<float>&     operator *= (quaternion<X> const & rhs);
+
+   quaternion<float>&     operator /= (float const & rhs);
+   quaternion<float>&     operator /= (::std::complex<float> const & rhs);
+   template<typename X>
+   quaternion<float>&     operator /= (quaternion<X> const & rhs);
+};
+
+

+

+
+template<>
+class quaternion<double>
+{
+public:
+   typedef double value_type;
+	
+   explicit quaternion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0);
+   explicit quaternion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>());
+   explicit quaternion(quaternion<float> const & a_recopier);
+   explicit quaternion(quaternion<long double> const & a_recopier);
+	
+   double                  real() const;
+   quaternion<double>      unreal() const;
+   double                  R_component_1() const;
+   double                  R_component_2() const;
+   double                  R_component_3() const;
+   double                  R_component_4() const;
+   ::std::complex<double>  C_component_1() const;
+   ::std::complex<double>  C_component_2() const;
+
+   quaternion<double>&     operator = (quaternion<double> const  & a_affecter);
+   template<typename X>	
+   quaternion<double>&     operator = (quaternion<X> const  & a_affecter);
+   quaternion<double>&     operator = (double const  & a_affecter);
+   quaternion<double>&     operator = (::std::complex<double> const & a_affecter);
+
+   quaternion<double>&     operator += (double const & rhs);
+   quaternion<double>&     operator += (::std::complex<double> const & rhs);
+   template<typename X>
+   quaternion<double>&     operator += (quaternion<X> const & rhs);
+
+   quaternion<double>&     operator -= (double const & rhs);
+   quaternion<double>&     operator -= (::std::complex<double> const & rhs);
+   template<typename X>
+   quaternion<double>&     operator -= (quaternion<X> const & rhs);
+
+   quaternion<double>&     operator *= (double const & rhs);
+   quaternion<double>&     operator *= (::std::complex<double> const & rhs);
+   template<typename X>
+   quaternion<double>&     operator *= (quaternion<X> const & rhs);
+
+   quaternion<double>&     operator /= (double const & rhs);
+   quaternion<double>&     operator /= (::std::complex<double> const & rhs);
+   template<typename X>
+   quaternion<double>&     operator /= (quaternion<X> const & rhs);
+};
+
+

+

+
+template<>
+class quaternion<long double>
+{
+public:
+   typedef long double value_type;
+	
+   explicit quaternion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L);
+   explicit quaternion(::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>());
+   explicit quaternion(quaternion<float> const & a_recopier);
+   explicit quaternion(quaternion<double> const & a_recopier);
+	
+   long double                  real() const;
+   quaternion<long double>      unreal() const;
+   long double                  R_component_1() const;
+   long double                  R_component_2() const;
+   long double                  R_component_3() const;
+   long double                  R_component_4() const;
+   ::std::complex<long double>  C_component_1() const;
+   ::std::complex<long double>  C_component_2() const;
+
+   quaternion<long double>&     operator = (quaternion<long double> const  & a_affecter);
+   template<typename X>	
+   quaternion<long double>&     operator = (quaternion<X> const  & a_affecter);
+   quaternion<long double>&     operator = (long double const  & a_affecter);
+   quaternion<long double>&     operator = (::std::complex<long double> const & a_affecter);
+
+   quaternion<long double>&     operator += (long double const & rhs);
+   quaternion<long double>&     operator += (::std::complex<long double> const & rhs);
+   template<typename X>
+   quaternion<long double>&     operator += (quaternion<X> const & rhs);
+
+   quaternion<long double>&     operator -= (long double const & rhs);
+   quaternion<long double>&     operator -= (::std::complex<long double> const & rhs);
+   template<typename X>
+   quaternion<long double>&     operator -= (quaternion<X> const & rhs);
+
+   quaternion<long double>&     operator *= (long double const & rhs);
+   quaternion<long double>&     operator *= (::std::complex<long double> const & rhs);
+   template<typename X>
+   quaternion<long double>&     operator *= (quaternion<X> const & rhs);
+
+   quaternion<long double>&     operator /= (long double const & rhs);
+   quaternion<long double>&     operator /= (::std::complex<long double> const & rhs);
+   template<typename X>
+   quaternion<long double>&     operator /= (quaternion<X> const & rhs);
+};
+
+} // namespace math
+} // namespace boost
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/synopsis.html b/doc/quaternion/html/boost_quaternions/quaternions/synopsis.html new file mode 100644 index 000000000..6b1c18b62 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/synopsis.html @@ -0,0 +1,124 @@ + + + +Synopsis + + + + + + + + + + + + + + + +
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+namespace boost{ namespace math{
+
+template<typename T> class quaternion;
+template<>           class quaternion<float>;
+template<>           class quaternion<double>; 
+template<>           class quaternion<long double>; 
+
+// operators
+template<typename T> quaternion<T> operator + (T const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, T const & rhs);
+template<typename T> quaternion<T> operator + (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+template<typename T> quaternion<T> operator - (T const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, T const & rhs);
+template<typename T> quaternion<T> operator - (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+template<typename T> quaternion<T> operator * (T const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, T const & rhs);
+template<typename T> quaternion<T> operator * (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+template<typename T> quaternion<T> operator / (T const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, T const & rhs);
+template<typename T> quaternion<T> operator / (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, quaternion<T> const & rhs); 
+
+template<typename T> quaternion<T> operator + (quaternion<T> const & q);
+template<typename T> quaternion<T> operator - (quaternion<T> const & q); 
+
+template<typename T> bool operator == (T const & lhs, quaternion<T> const & rhs);
+template<typename T> bool operator == (quaternion<T> const & lhs, T const & rhs);
+template<typename T> bool operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator == (quaternion<T> const & lhs, quaternion<T> const & rhs);
+
+template<typename T> bool operator != (T const & lhs, quaternion<T> const & rhs);
+template<typename T> bool operator != (quaternion<T> const & lhs, T const & rhs);
+template<typename T> bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs);
+template<typename T> bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
+template<typename T> bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs); 
+
+template<typename T, typename charT, class traits>
+::std::basic_istream<charT,traits>& operator >> (::std::basic_istream<charT,traits> & is, quaternion<T> & q);
+
+template<typename T, typename charT, class traits>
+::std::basic_ostream<charT,traits>& operator operator << (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q);
+
+// values
+template<typename T>	T              real(quaternion<T> const & q);
+template<typename T>	quaternion<T>  unreal(quaternion<T> const & q);
+
+template<typename T>	T              sup(quaternion<T> const & q);
+template<typename T>	T              l1(quaternion<T> const & q);
+template<typename T>	T              abs(quaternion<T> const & q);
+template<typename T>	T              norm(quaternion<T>const  & q);
+template<typename T>	quaternion<T>  conj(quaternion<T> const & q);
+
+template<typename T>	quaternion<T>  spherical(T const & rho, T const & theta, T const & phi1, T const & phi2);
+template<typename T>	quaternion<T>  semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2);
+template<typename T>	quaternion<T>  multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2);
+template<typename T>	quaternion<T>  cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude);
+template<typename T>	quaternion<T>  cylindrical(T const & r, T const & angle, T const & h1, T const & h2);
+
+// transcendentals
+template<typename T>	quaternion<T>  exp(quaternion<T> const & q);
+template<typename T>	quaternion<T>  cos(quaternion<T> const & q);
+template<typename T>	quaternion<T>  sin(quaternion<T> const & q);
+template<typename T>	quaternion<T>  tan(quaternion<T> const & q);
+template<typename T>	quaternion<T>  cosh(quaternion<T> const & q);
+template<typename T>	quaternion<T>  sinh(quaternion<T> const & q);
+template<typename T>	quaternion<T>  tanh(quaternion<T> const & q);
+template<typename T>	quaternion<T>  pow(quaternion<T> const & q, int n);
+
+} // namespace math
+} // namespace boost
+
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/test_program.html b/doc/quaternion/html/boost_quaternions/quaternions/test_program.html new file mode 100644 index 000000000..ef18b4cfc --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/test_program.html @@ -0,0 +1,58 @@ + + + +Test Program + + + + + + + + + + + + + + + +
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+

+ The quaternion_test.cpp + test program tests quaternions specializations for float, double and long + double (sample output, + with message output enabled). +

+

+ If you define the symbol boost_quaternions_TEST_VERBOSE, you will get additional + output (verbose output); + this will only be helpfull if you enable message output at the same time, + of course (by uncommenting the relevant line in the test or by adding --log_level=messages + to your command line,...). In that case, and if you are running interactively, + you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR + to interactively test the input operator with input of your choice from the + standard input (instead of hard-coding it in the test). +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/to_do.html b/doc/quaternion/html/boost_quaternions/quaternions/to_do.html new file mode 100644 index 000000000..aba5a4c2f --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/to_do.html @@ -0,0 +1,55 @@ + + + +To Do + + + + + + + + + + + + + + +
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+
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+
    +
  • + Improve testing. +
    • +
  • + Rewrite input operatore using Spirit (creates a dependency). +
    • +
  • + Put in place an Expression Template mechanism (perhaps borrowing from uBlas). +
    • +
  • + Use uBlas for the link with rotations (and move from the example + implementation to an efficient one). +
    • +
+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/trans.html b/doc/quaternion/html/boost_quaternions/quaternions/trans.html new file mode 100644 index 000000000..92ad49201 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/trans.html @@ -0,0 +1,151 @@ + + + + Quaternion Transcendentals + + + + + + + + + + + + + + + +
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+

+ There is no log or sqrt provided for quaternions in this implementation, + and pow is likewise restricted + to integral powers of the exponent. There are several reasons to this: on + the one hand, the equivalent of analytic continuation for quaternions ("branch + cuts") remains to be investigated thoroughly (by me, at any rate...), + and we wish to avoid the nonsense introduced in the standard by exponentiations + of complexes by complexes (which is well defined, but not in the standard...). + Talking of nonsense, saying that pow(0,0) is "implementation + defined" is just plain brain-dead... +

+

+ We do, however provide several transcendentals, chief among which is the + exponential. This author claims the complete proof of the "closed formula" + as his own, as well as its independant invention (there are claims to prior + invention of the formula, such as one by Professor Shoemake, and it is possible + that the formula had been known a couple of centuries back, but in absence + of bibliographical reference, the matter is pending, awaiting further investigation; + on the other hand, the definition and existence of the exponential on the + quaternions, is of course a fact known for a very long time). Basically, + any converging power series with real coefficients which allows for a closed + formula in C can be + transposed to H. More + transcendentals of this type could be added in a further revision upon request. + It should be noted that it is these functions which force the dependency + upon the boost/math/special_functions/sinc.hpp + and the boost/math/special_functions/sinhc.hpp + headers. +

+

+ + exp +

+
+template<typename T>	quaternion<T> exp(quaternion<T> const & q);
+
+

+ Computes the exponential of the quaternion. +

+

+ + cos +

+
+template<typename T>	quaternion<T>  cos(quaternion<T> const & q);
+
+

+ Computes the cosine of the quaternion +

+

+ + sin +

+
+template<typename T>	quaternion<T>  sin(quaternion<T> const & q);
+
+

+ Computes the sine of the quaternion. +

+

+ + tan +

+
+template<typename T>	quaternion<T>  tan(quaternion<T> const & q);
+
+

+ Computes the tangent of the quaternion. +

+

+ + cosh +

+
+template<typename T>	quaternion<T>  cosh(quaternion<T> const & q);
+
+

+ Computes the hyperbolic cosine of the quaternion. +

+

+ + sinh +

+
+template<typename T>	quaternion<T>  sinh(quaternion<T> const & q);
+
+

+ Computes the hyperbolic sine of the quaternion. +

+

+ + tanh +

+
+template<typename T>	quaternion<T>  tanh(quaternion<T> const & q);
+
+

+ Computes the hyperbolic tangent of the quaternion. +

+

+ + pow +

+
+template<typename T>	quaternion<T>  pow(quaternion<T> const & q, int n);
+
+

+ Computes the n-th power of the quaternion q. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+PrevUpHomeNext +
+ + diff --git a/doc/quaternion/html/boost_quaternions/quaternions/value_op.html b/doc/quaternion/html/boost_quaternions/quaternions/value_op.html new file mode 100644 index 000000000..e297ed299 --- /dev/null +++ b/doc/quaternion/html/boost_quaternions/quaternions/value_op.html @@ -0,0 +1,114 @@ + + + + Quaternion Value + Operations + + + + + + + + + + + + + + + +
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+ + real + and unreal +

+
+template<typename T>	T              real(quaternion<T> const & q);
+template<typename T>	quaternion<T>  unreal(quaternion<T> const & q);
+
+

+ These return q.real() + and q.unreal() + respectively. +

+

+ + conj +

+
+template<typename T>	quaternion<T>  conj(quaternion<T> const & q);
+
+

+ This returns the conjugate of the quaternion. +

+

+ + sup +

+

+ template<typename T> T sup(quaternion<T> const & q); +

+

+ This return the sup norm (the greatest among abs(q.R_component_1())...abs(q.R_component_4())) of the quaternion. +

+

+ + l1 +

+
+template<typename T>	T  l1(quaternion<T> const & q);
+
+

+ This return the l1 norm (abs(q.R_component_1())+...+abs(q.R_component_4())) of the quaternion. +

+

+ + abs +

+
+template<typename T>	T  abs(quaternion<T> const & q);
+
+

+ This return the magnitude (Euclidian norm) of the quaternion. +

+

+ + norm +

+
+template<typename T>	T  norm(quaternion<T>const  & q);
+
+

+ This return the (Cayley) norm of the quaternion. The term "norm" + might be confusing, as most people associate it with the Euclidian norm (and + quadratic functionals). For this version of (the mathematical objects known + as) quaternions, the Euclidian norm (also known as magnitude) is the square + root of the Cayley norm. +

+
+ + + +
Copyright © 2001 -2003 Hubert Holin

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ + diff --git a/doc/quaternion/html/index.html b/doc/quaternion/html/index.html new file mode 100644 index 000000000..a3fcc39ef --- /dev/null +++ b/doc/quaternion/html/index.html @@ -0,0 +1,77 @@ + + + +Boost.Quaternions + + + + + + + + + + + + + +
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+Boost.Quaternions

+

+Hubert Holin +

+

Copyright © 2001 -2003 Hubert Holin

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+
+

Table of Contents

+
+
Quaternions
+
+
Overview
+
Header File
+
Synopsis
+
Template Class quaternion
+
Quaternion Specializations
+
Quaternion + Member Typedefs
+
Quaternion Member + Functions
+
Quaternion Non-Member + Operators
+
Quaternion Value + Operations
+
Quaternion Creation + Functions
+
Quaternion Transcendentals
+
Test Program
+
The Quaternionic + Exponential
+
Acknowledgements
+
History
+
To Do
+
+
+
+
+ + + +

Last revised: December 29, 2006 at 11:08:32 +0000

+
+
Next
+ + diff --git a/doc/quaternion/math-quaternion.qbk b/doc/quaternion/math-quaternion.qbk new file mode 100644 index 000000000..7f514d53f --- /dev/null +++ b/doc/quaternion/math-quaternion.qbk @@ -0,0 +1,931 @@ +[article Boost.Quaternions + [quickbook 1.3] + [copyright 2001-2003 Hubert Holin] + [purpose Quaternions] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Holin, Hubert]] + [category math] + [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $] +] + +[def __R ['[*R]]] +[def __C ['[*C]]] +[def __H ['[*H]]] +[def __O ['[*O]]] +[def __R3 ['[*'''R3''']]] +[def __R4 ['[*'''R4''']]] +[def __quadrulple ('''α,β,γ,δ''')] +[def __quat_formula ['[^q = '''α + βi + γj + δk''']]] +[def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] +[def __not_equal ['[^xy '''≠''' yx]]] + +[def __asinh [link boost_quaternions.math_special_functions.asinh asinh]] +[def __acosh [link boost_quaternions.math_special_functions.acosh acosh]] +[def __atanh [link boost_quaternions.math_special_functions.atanh atanh]] +[def __sinc_pi [link boost_quaternions.math_special_functions.sinc_pi sinc_pi]] +[def __sinhc_pi [link boost_quaternions.math_special_functions.sinhc_pi sinhc_pi]] + +[def __log1p [link boost_quaternions.math_special_functions.log1p log1p]] +[def __expm1 [link boost_quaternions.math_special_functions.expm1 expm1]] +[def __hypot [link boost_quaternions.math_special_functions.hypot hypot]] + + +[section Quaternions] + +[section Overview] + +Quaternions are a relative of complex numbers. + +Quaternions are in fact part of a small hierarchy of structures built +upon the real numbers, which comprise only the set of real numbers +(traditionally named __R), the set of complex numbers (traditionally named __C), +the set of quaternions (traditionally named __H) and the set of octonions +(traditionally named __O), which possess interesting mathematical properties +(chief among which is the fact that they are ['division algebras], +['i.e.] where the following property is true: if ['[^y]] is an element of that +algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] +denote elements of that algebra, implies that ['[^x = x']]). +Each member of the hierarchy is a super-set of the former. + +One of the most important aspects of quaternions is that they provide an +efficient way to parameterize rotations in __R3 (the usual three-dimensional space) +and __R4. + +In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, +which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, +and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]]. + +An addition and a multiplication is defined on the set of quaternions, +which generalize their real and complex counterparts. The main novelty +here is that [*the multiplication is not commutative] (i.e. there are +quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering +things is by using the formula ['[^i*i = j*j = k*k = -1]]. + +Quaternions (and their kin) are described in far more details in this +other [@../../../quaternion/TQE.pdf document] +(with [@../../../quaternion/TQE_EA.pdf errata and addenda]). + +Some traditional constructs, such as the exponential, carry over without +too much change into the realms of quaternions, but other, such as taking +a square root, do not. + +[endsect] + +[section Header File] + +The interface and implementation are both supplied by the header file +[@../../../../../boost/math/quaternion.hpp quaternion.hpp]. + +[endsect] + +[section Synopsis] + + namespace boost{ namespace math{ + + template class ``[link boost_quaternions.quaternions.quat quaternion]``; + template<> class ``[link boost_quaternions.quaternions.spec quaternion]``; + template<> class ``[link boost_quaternions.quaternion_double quaternion]``; + template<> class ``[link boost_quaternions.quaternion_long_double quaternion]``; + + // operators + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_addition_operators operator +]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, quaternion const & rhs); + + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, quaternion const & rhs); + + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, quaternion const & rhs); + + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_division_operators operator /]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link boost_quaternions.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, quaternion const & rhs); + + template quaternion ``[link boost_quaternions.quaternions.non_mem.unary_plus operator +]`` (quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.non_mem.unary_minus operator -]`` (quaternion const & q); + + template bool ``[link boost_quaternions.quaternions.non_mem.equality_operators operator ==]`` (T const & lhs, quaternion const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, T const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.equality_operators operator ==]`` (::std::complex const & lhs, quaternion const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, ::std::complex const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, quaternion const & rhs); + + template bool ``[link boost_quaternions.quaternions.non_mem.inequality_operators operator !=]`` (T const & lhs, quaternion const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, T const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.inequality_operators operator !=]`` (::std::complex const & lhs, quaternion const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, ::std::complex const & rhs); + template bool ``[link boost_quaternions.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, quaternion const & rhs); + + template + ::std::basic_istream& ``[link boost_quaternions.quaternions.non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, quaternion & q); + + template + ::std::basic_ostream& operator ``[link boost_quaternions.quaternions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, quaternion const & q); + + // values + template T ``[link boost_quaternions.quaternions.value_op.real_and_unreal real]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.value_op.real_and_unreal unreal]``(quaternion const & q); + + template T ``[link boost_quaternions.quaternions.value_op.sup sup]``(quaternion const & q); + template T ``[link boost_quaternions.quaternions.value_op.l1 l1]``(quaternion const & q); + template T ``[link boost_quaternions.quaternions.value_op.abs abs]``(quaternion const & q); + template T ``[link boost_quaternions.quaternions.value_op.norm norm]``(quaternionconst & q); + template quaternion ``[link boost_quaternions.quaternions.value_op.conj conj]``(quaternion const & q); + + template quaternion ``[link boost_quaternions.quaternions.creation_spherical spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); + template quaternion ``[link boost_quaternions.quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); + template quaternion ``[link boost_quaternions.quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); + template quaternion ``[link boost_quaternions.quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); + template quaternion ``[link boost_quaternions.quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); + + // transcendentals + template quaternion ``[link boost_quaternions.quaternions.trans.exp exp]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.cos cos]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.sin sin]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.tan tan]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.cosh cosh]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.sinh sinh]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.tanh tanh]``(quaternion const & q); + template quaternion ``[link boost_quaternions.quaternions.trans.pow pow]``(quaternion const & q, int n); + + } // namespace math + } // namespace boost + +[endsect] + +[section:quat Template Class quaternion] + + namespace boost{ namespace math{ + + template + class quaternion + { + public: + + typedef T ``[link boost_quaternions.quaternions.mem_typedef value_type]``; + + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + template + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + + T ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; + T ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_1]``() const; + T ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_2]``() const; + T ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_3]``() const; + T ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_1]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(T const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(T const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(T const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(T const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(T const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + }; + + } // namespace math + } // namespace boost + +[endsect] + +[section:spec Quaternion Specializations] + + namespace boost{ namespace math{ + + template<> + class quaternion + { + public: + typedef float ``[link boost_quaternions.quaternions.mem_typedef value_type]``; + + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + + float ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; + float ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_1]``() const; + float ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_2]``() const; + float ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_3]``() const; + float ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_1]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(float const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(float const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(float const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(float const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(float const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + }; + +[#boost_quaternions.quaternion_double] + + template<> + class quaternion + { + public: + typedef double ``[link boost_quaternions.quaternions.mem_typedef value_type]``; + + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + + double ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; + double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_1]``() const; + double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_2]``() const; + double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_3]``() const; + double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_1]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(double const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + }; + +[#boost_quaternions.quaternion_long_double] + + template<> + class quaternion + { + public: + typedef long double ``[link boost_quaternions.quaternions.mem_typedef value_type]``; + + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link boost_quaternions.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + + long double ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link boost_quaternions.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; + long double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_1]``() const; + long double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_2]``() const; + long double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_3]``() const; + long double ``[link boost_quaternions.quaternions.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_1]``() const; + ::std::complex ``[link boost_quaternions.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(long double const & a_affecter); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(long double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(long double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(long double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(long double const & rhs); + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + template + quaternion& ``[link boost_quaternions.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + }; + + } // namespace math + } // namespace boost + +[endsect] + +[section:mem_typedef Quaternion Member Typedefs] + +[*value_type] + +Template version: + + typedef T value_type; + +Float specialization version: + + typedef float value_type; + +Double specialization version: + + typedef double value_type; + +Long double specialization version: + + typedef long double value_type; + +These provide easy acces to the type the template is built upon. + +[endsect] + +[section:mem_fun Quaternion Member Functions] +[h3 Constructors] + +Template version: + + explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); + explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + template + explicit quaternion(quaternion const & a_recopier); + +Float specialization version: + + explicit quaternion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); + explicit quaternion(::std::complex const & z0,::std::complex const & z1 = ::std::complex()); + explicit quaternion(quaternion const & a_recopier); + explicit quaternion(quaternion const & a_recopier); + +Double specialization version: + + explicit quaternion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); + explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit quaternion(quaternion const & a_recopier); + explicit quaternion(quaternion const & a_recopier); + +Long double specialization version: + + explicit quaternion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L); + explicit quaternion( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit quaternion(quaternion const & a_recopier); + explicit quaternion(quaternion const & a_recopier); + +A default constructor is provided for each form, which initializes +each component to the default values for their type +(i.e. zero for floating numbers). This constructor can also accept +one to four base type arguments. A constructor is also provided to +build quaternions from one or two complex numbers sharing the same +base type. The unspecialized template also sports a templarized copy +constructor, while the specialized forms have copy constructors +from the other two specializations, which are explicit when a risk of +precision loss exists. For the unspecialized form, the base type's +constructors must not throw. + +Destructors and untemplated copy constructors (from the same type) are +provided by the compiler. Converting copy constructors make use of a +templated helper function in a "detail" subnamespace. + +[h3 Other member functions] +[h4 Real and Unreal Parts] + + T real() const; + quaternion unreal() const; + +Like complex number, quaternions do have a meaningful notion of "real part", +but unlike them there is no meaningful notion of "imaginary part". +Instead there is an "unreal part" which itself is a quaternion, +and usually nothing simpler (as opposed to the complex number case). +These are returned by the first two functions. + +[h4 Individual Real Components] + + T R_component_1() const; + T R_component_2() const; + T R_component_3() const; + T R_component_4() const; + +A quaternion having four real components, these are returned by these four +functions. Hence real and R_component_1 return the same value. + +[h4 Individual Complex Components] + + ::std::complex C_component_1() const; + ::std::complex C_component_2() const; + +A quaternion likewise has two complex components, and as we have seen above, +for any quaternion __quat_formula we also have __quat_complex_formula. These functions return them. +The real part of `q.C_component_1()` is the same as `q.real()`. + +[h3 Quaternion Member Operators] +[h4 Assignment Operators] + + quaternion& operator = (quaternion const & a_affecter); + template + quaternion& operator = (quaternion const& a_affecter); + quaternion& operator = (T const& a_affecter); + quaternion& operator = (::std::complex const& a_affecter); + +These perform the expected assignment, with type modification if necessary +(for instance, assigning from a base type will set the real part to that +value, and all other components to zero). For the unspecialized form, +the base type's assignment operators must not throw. + +[h4 Addition Operators] + + quaternion& operator += (T const & rhs) + quaternion& operator += (::std::complex const & rhs); + template + quaternion& operator += (quaternion const & rhs); + +These perform the mathematical operation `(*this)+rhs` and store the result in +`*this`. The unspecialized form has exception guards, which the specialized +forms do not, so as to insure exception safety. For the unspecialized form, +the base type's assignment operators must not throw. + +[h4 Subtraction Operators] + + quaternion& operator -= (T const & rhs) + quaternion& operator -= (::std::complex const & rhs); + template + quaternion& operator -= (quaternion const & rhs); + +These perform the mathematical operation `(*this)-rhs` and store the result +in `*this`. The unspecialized form has exception guards, which the +specialized forms do not, so as to insure exception safety. +For the unspecialized form, the base type's assignment operators +must not throw. + +[h4 Multiplication Operators] + + quaternion& operator *= (T const & rhs) + quaternion& operator *= (::std::complex const & rhs); + template + quaternion& operator *= (quaternion const & rhs); + +These perform the mathematical operation `(*this)*rhs` [*in this order] +(order is important as multiplication is not commutative for quaternions) +and store the result in `*this`. The unspecialized form has exception guards, +which the specialized forms do not, so as to insure exception safety. +For the unspecialized form, the base type's assignment operators must not throw. + +[h4 Division Operators] + + quaternion& operator /= (T const & rhs) + quaternion& operator /= (::std::complex const & rhs); + template + quaternion& operator /= (quaternion const & rhs); + +These perform the mathematical operation `(*this)*inverse_of(rhs)` [*in this +order] (order is important as multiplication is not commutative for quaternions) +and store the result in `*this`. The unspecialized form has exception guards, +which the specialized forms do not, so as to insure exception safety. +For the unspecialized form, the base type's assignment operators must not throw. + +[endsect] +[section:non_mem Quaternion Non-Member Operators] + +[h4 Unary Plus] + + template + quaternion operator + (quaternion const & q); + +This unary operator simply returns q. + +[h4 Unary Minus] + + template + quaternion operator - (quaternion const & q); + +This unary operator returns the opposite of q. + +[h4 Binary Addition Operators] + + template quaternion operator + (T const & lhs, quaternion const & rhs); + template quaternion operator + (quaternion const & lhs, T const & rhs); + template quaternion operator + (::std::complex const & lhs, quaternion const & rhs); + template quaternion operator + (quaternion const & lhs, ::std::complex const & rhs); + template quaternion operator + (quaternion const & lhs, quaternion const & rhs); + +These operators return `quaternion(lhs) += rhs`. + +[h4 Binary Subtraction Operators] + + template quaternion operator - (T const & lhs, quaternion const & rhs); + template quaternion operator - (quaternion const & lhs, T const & rhs); + template quaternion operator - (::std::complex const & lhs, quaternion const & rhs); + template quaternion operator - (quaternion const & lhs, ::std::complex const & rhs); + template quaternion operator - (quaternion const & lhs, quaternion const & rhs); + +These operators return `quaternion(lhs) -= rhs`. + +[h4 Binary Multiplication Operators] + + template quaternion operator * (T const & lhs, quaternion const & rhs); + template quaternion operator * (quaternion const & lhs, T const & rhs); + template quaternion operator * (::std::complex const & lhs, quaternion const & rhs); + template quaternion operator * (quaternion const & lhs, ::std::complex const & rhs); + template quaternion operator * (quaternion const & lhs, quaternion const & rhs); + +These operators return `quaternion(lhs) *= rhs`. + +[h4 Binary Division Operators] + + template quaternion operator / (T const & lhs, quaternion const & rhs); + template quaternion operator / (quaternion const & lhs, T const & rhs); + template quaternion operator / (::std::complex const & lhs, quaternion const & rhs); + template quaternion operator / (quaternion const & lhs, ::std::complex const & rhs); + template quaternion operator / (quaternion const & lhs, quaternion const & rhs); + +These operators return `quaternion(lhs) /= rhs`. It is of course still an +error to divide by zero... + +[h4 Equality Operators] + + template bool operator == (T const & lhs, quaternion const & rhs); + template bool operator == (quaternion const & lhs, T const & rhs); + template bool operator == (::std::complex const & lhs, quaternion const & rhs); + template bool operator == (quaternion const & lhs, ::std::complex const & rhs); + template bool operator == (quaternion const & lhs, quaternion const & rhs); + +These return true if and only if the four components of `quaternion(lhs)` +are equal to their counterparts in `quaternion(rhs)`. As with any +floating-type entity, this is essentially meaningless. + +[h4 Inequality Operators] + + template bool operator != (T const & lhs, quaternion const & rhs); + template bool operator != (quaternion const & lhs, T const & rhs); + template bool operator != (::std::complex const & lhs, quaternion const & rhs); + template bool operator != (quaternion const & lhs, ::std::complex const & rhs); + template bool operator != (quaternion const & lhs, quaternion const & rhs); + +These return true if and only if `quaternion(lhs) == quaternion(rhs)` is +false. As with any floating-type entity, this is essentially meaningless. + +[h4 Stream Extractor] + + template + ::std::basic_istream& operator >> (::std::basic_istream & is, quaternion & q); + +Extracts a quaternion q of one of the following forms +(with a, b, c and d of type `T`): + +[^a (a), (a,b), (a,b,c), (a,b,c,d) (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d))] + +The input values must be convertible to `T`. If bad input is encountered, +calls `is.setstate(ios::failbit)` (which may throw ios::failure (27.4.5.3)). + +[*Returns:] `is`. + +The rationale for the list of accepted formats is that either we have a +list of up to four reals, or else we have a couple of complex numbers, +and in that case if it formated as a proper complex number, then it should +be accepted. Thus potential ambiguities are lifted (for instance (a,b) is +(a,b,0,0) and not (a,0,b,0), i.e. it is parsed as a list of two real numbers +and not two complex numbers which happen to have imaginary parts equal to zero). + +[h4 Stream Inserter] + + template + ::std::basic_ostream& operator << (::std::basic_ostream & os, quaternion const & q); + +Inserts the quaternion q onto the stream `os` as if it were implemented as follows: + + template + ::std::basic_ostream& operator << ( + ::std::basic_ostream & os, + quaternion const & q) + { + ::std::basic_ostringstream s; + + s.flags(os.flags()); + s.imbue(os.getloc()); + s.precision(os.precision()); + + s << '(' << q.R_component_1() << ',' + << q.R_component_2() << ',' + << q.R_component_3() << ',' + << q.R_component_4() << ')'; + + return os << s.str(); + } + +[endsect] + +[section:value_op Quaternion Value Operations] + +[h4 real and unreal] + + template T real(quaternion const & q); + template quaternion unreal(quaternion const & q); + +These return `q.real()` and `q.unreal()` respectively. + +[h4 conj] + + template quaternion conj(quaternion const & q); + +This returns the conjugate of the quaternion. + +[h4 sup] + +template T sup(quaternion const & q); + +This return the sup norm (the greatest among +`abs(q.R_component_1())...abs(q.R_component_4()))` of the quaternion. + +[h4 l1] + + template T l1(quaternion const & q); + +This return the l1 norm `(abs(q.R_component_1())+...+abs(q.R_component_4()))` +of the quaternion. + +[h4 abs] + + template T abs(quaternion const & q); + +This return the magnitude (Euclidian norm) of the quaternion. + +[h4 norm] + + template T norm(quaternionconst & q); + +This return the (Cayley) norm of the quaternion. +The term "norm" might be confusing, as most people associate it with the +Euclidian norm (and quadratic functionals). For this version of +(the mathematical objects known as) quaternions, the Euclidian norm +(also known as magnitude) is the square root of the Cayley norm. + +[endsect] + +[section:create Quaternion Creation Functions] + + template quaternion spherical(T const & rho, T const & theta, T const & phi1, T const & phi2); + template quaternion semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2); + template quaternion multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2); + template quaternion cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude); + template quaternion cylindrical(T const & r, T const & angle, T const & h1, T const & h2); + +These build quaternions in a way similar to the way polar builds complex +numbers, as there is no strict equivalent to polar coordinates for quaternions. + +[#boost_quaternions.quaternions.creation_spherical] `spherical` is a simple transposition of `polar`, it takes as inputs +a (positive) magnitude and a point on the hypersphere, given by three angles. +The first of these, `theta` has a natural range of `-pi` to `+pi`, and the other +two have natural ranges of `-pi/2` to `+pi/2` (as is the case with the usual +spherical coordinates in __R3). Due to the many symmetries and periodicities, +nothing untoward happens if the magnitude is negative or the angles are +outside their natural ranges. The expected degeneracies (a magnitude of +zero ignores the angles settings...) do happen however. + +[#boost_quaternions.quaternions.creation_cylindrical] `cylindrical` is likewise a simple transposition of the usual +cylindrical coordinates in __R3, which in turn is another derivative of +planar polar coordinates. The first two inputs are the polar coordinates of +the first __C component of the quaternion. The third and fourth inputs +are placed into the third and fourth __R components of the quaternion, +respectively. + +[#boost_quaternions.quaternions.creation_multipolar] `multipolar` is yet another simple generalization of polar coordinates. +This time, both __C components of the quaternion are given in polar coordinates. + +[#boost_quaternions.quaternions.creation_cylindrospherical] `cylindrospherical` is specific to quaternions. It is often interesting to +consider __H as the cartesian product of __R by __R3 (the quaternionic +multiplication as then a special form, as given here). This function +therefore builds a quaternion from this representation, with the __R3 +component given in usual __R3 spherical coordinates. + +[#boost_quaternions.quaternions.creation_semipolar] `semipolar` is another generator which is specific to quaternions. +It takes as a first input the magnitude of the quaternion, as a +second input an angle in the range `0` to `+pi/2` such that magnitudes +of the first two __C components of the quaternion are the product of the +first input and the sine and cosine of this angle, respectively, and finally +as third and fourth inputs angles in the range `-pi/2` to `+pi/2` which +represent the arguments of the first and second __C components of +the quaternion, respectively. As usual, nothing untoward happens if +what should be magnitudes are negative numbers or angles are out of their +natural ranges, as symmetries and periodicities kick in. + +In this version of our implementation of quaternions, there is no +analogue of the complex value operation `arg` as the situation is +somewhat more complicated. Unit quaternions are linked both to +rotations in __R3 and in __R4, and the correspondences are not too complicated, +but there is currently a lack of standard (de facto or de jure) matrix +library with which the conversions could work. This should be remedied in +a further revision. In the mean time, an example of how this could be +done is presented here for +[@../../../quaternion/HSO3.hpp __R3], and here for +[@../../../quaternion/HSO4.hpp __R4] +([@../../../quaternion/HSO3SO4.cpp example test file]). + +[endsect] + +[section:trans Quaternion Transcendentals] + +There is no `log` or `sqrt` provided for quaternions in this implementation, +and `pow` is likewise restricted to integral powers of the exponent. +There are several reasons to this: on the one hand, the equivalent of +analytic continuation for quaternions ("branch cuts") remains to be +investigated thoroughly (by me, at any rate...), and we wish to avoid the +nonsense introduced in the standard by exponentiations of complexes by +complexes (which is well defined, but not in the standard...). +Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is just +plain brain-dead... + +We do, however provide several transcendentals, chief among which is the +exponential. This author claims the complete proof of the "closed formula" +as his own, as well as its independant invention (there are claims to prior +invention of the formula, such as one by Professor Shoemake, and it is +possible that the formula had been known a couple of centuries back, but in +absence of bibliographical reference, the matter is pending, awaiting further +investigation; on the other hand, the definition and existence of the +exponential on the quaternions, is of course a fact known for a very long time). +Basically, any converging power series with real coefficients which allows for a +closed formula in __C can be transposed to __H. More transcendentals of this +type could be added in a further revision upon request. It should be +noted that it is these functions which force the dependency upon the +[@../../../../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] and the +[@../../../../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] headers. + +[h4 exp] + + template quaternion exp(quaternion const & q); + +Computes the exponential of the quaternion. + +[h4 cos] + + template quaternion cos(quaternion const & q); + +Computes the cosine of the quaternion + +[h4 sin] + + template quaternion sin(quaternion const & q); + +Computes the sine of the quaternion. + +[h4 tan] + + template quaternion tan(quaternion const & q); + +Computes the tangent of the quaternion. + +[h4 cosh] + + template quaternion cosh(quaternion const & q); + +Computes the hyperbolic cosine of the quaternion. + +[h4 sinh] + + template quaternion sinh(quaternion const & q); + +Computes the hyperbolic sine of the quaternion. + +[h4 tanh] + + template quaternion tanh(quaternion const & q); + +Computes the hyperbolic tangent of the quaternion. + +[h4 pow] + + template quaternion pow(quaternion const & q, int n); + +Computes the n-th power of the quaternion q. + +[endsect] + +[section Test Program] + +The [@../../../quaternion/quaternion_test.cpp quaternion_test.cpp] +test program tests quaternions specializations for float, double and long double +([@../../../quaternion/output.txt sample output], with message output +enabled). + +If you define the symbol boost_quaternions_TEST_VERBOSE, you will get +additional output ([@../../../quaternion/output_more.txt verbose output]); +this will only be helpfull if you enable message output at the same time, +of course (by uncommenting the relevant line in the test or by adding +[^--log_level=messages] to your command line,...). In that case, and if you +are running interactively, you may in addition define the symbol +BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to interactively test the input +operator with input of your choice from the standard input +(instead of hard-coding it in the test). + +[endsect] + +[section:exp The Quaternionic Exponential] + +Please refer to the following PDF's: + +*[@../../../quaternion/TQE.pdf The Quaternionic Exponential (and beyond)] +*[@../../../quaternion/TQE_EA.pdf The Quaternionic Exponential (and beyond) ERRATA & ADDENDA] + +[endsect] + +[section Acknowledgements] + +The mathematical text has been typeset with +[@http://www.nisus-soft.com/ Nisus Writer]. Jens Maurer has helped with +portability and standard adherence, and was the Review Manager +for this library. More acknowledgements in the History section. +Thank you to all who contributed to the discution about this library. + +[endsect] + +[section History] + +* 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. +* 1.5.7 - 24/02/2003: transitionned to the unit test framework; now included by the library header (rather than the test files). +* 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). +* 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. +* 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes. +* 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). +* 1.5.2 - 07/07/2001: introduced namespace math. +* 1.5.1 - 07/06/2001: (end of Boost review) now includes and instead of ; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. +* 1.5.0 - 26/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version) and output operator, added spherical, semipolar, multipolar, cylindrospherical and cylindrical. +* 1.4.0 - 09/01/2001: added tan and tanh. +* 1.3.1 - 08/01/2001: cosmetic fixes. +* 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. +* 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. +* 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. +* 1.0.0 - 10/08/1999: first public version. + +[endsect] +[section To Do] + +* Improve testing. +* Rewrite input operatore using Spirit (creates a dependency). +* Put in place an Expression Template mechanism (perhaps borrowing from uBlas). +* Use uBlas for the link with rotations (and move from the +[@../../../quaternion/HSO3SO4.cpp example] +implementation to an efficient one). + +[endsect] +[endsect] diff --git a/doc/sf_and_dist/Jamfile.v2 b/doc/sf_and_dist/Jamfile.v2 new file mode 100644 index 000000000..809b96ea2 --- /dev/null +++ b/doc/sf_and_dist/Jamfile.v2 @@ -0,0 +1,70 @@ + +# Copyright John Maddock 2005. Use, modification, and distribution are +# subject to the Boost Software License, Version 1.0. (See accompanying +# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +using quickbook ; + +path-constant images_location : html ; + +xml math : math.qbk ; +boostbook standalone + : + math + : + # Path for links to Boost: + boost.root=../../../../.. + # Path for libraries index: + boost.libraries=../../../../../libs/libraries.htm + # Use the main Boost stylesheet: + html.stylesheet=../../../../../doc/html/boostbook.css + + # Some general style settings: + table.footnote.number.format=1 + footnote.number.format=1 + + # HTML options first: + # Use graphics not text for navigation: + navig.graphics=1 + # How far down we chunk nested sections, basically all of them: + chunk.section.depth=10 + # Don't put the first section on the same page as the TOC: + chunk.first.sections=1 + # How far down sections get TOC's + toc.section.depth=10 + # Max depth in each TOC: + toc.max.depth=4 + # How far down we go with TOC's + generate.section.toc.level=10 + #root.filename="sf_dist_and_tools" + + # PDF Options: + # TOC Generation: this is needed for FOP-0.9 and later: + # fop1.extensions=1 + pdf:xep.extensions=1 + # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9! + pdf:fop.extensions=0 + pdf:fop1.extensions=0 + # No indent on body text: + pdf:body.start.indent=0pt + # Margin size: + pdf:page.margin.inner=0.5in + # Margin size: + pdf:page.margin.outer=0.5in + # Paper type = A4 + pdf:paper.type=A4 + # Yes, we want graphics for admonishments: + admon.graphics=1 + # Set this one for PDF generation *only*: + # default pnd graphics are awful in PDF form, + # better use SVG's instead: + pdf:admon.graphics.extension=".svg" + pdf:use.role.for.mediaobject=1 + pdf:preferred.mediaobject.role=print + pdf:img.src.path=$(images_location)/ + pdf:admon.graphics.path=$(images_location)/../../svg-admon/ + pdf:draft.mode="no" + ; + + + diff --git a/doc/sf_and_dist/background.qbk b/doc/sf_and_dist/background.qbk new file mode 100644 index 000000000..34a0bfeb0 --- /dev/null +++ b/doc/sf_and_dist/background.qbk @@ -0,0 +1,85 @@ +[section:variates Random Variates and Distribution Parameters] + +[@http://en.wikipedia.org/wiki/Random_variate Random variates] +and [@http://en.wikipedia.org/wiki/Parameter distribution parameters] +are conventionally distinguished (for example in Wikipedia and Wolfram MathWorld +by placing a semi-colon after the random variate (whose value you 'choose'), +to separate the variate from the parameter(s) that defines the shape of the distribution. + +For example, the binomial distribution has two parameters: +n (the number of trials) and p (the probability of success on one trial). +It also has the random variate /k/: the number of successes observed. +This means the probability density\/mass function (pdf) is written as ['f(k; n, p)]. + +Translating this into code the `binomial_distribution` constructor +therefore has two parameters: + + binomial_distribution(RealType n, RealType p); + +While the function `pdf` has one argument specifying the distribution type +(which includes it's parameters, if any), +and a second argument for the random variate. So taking our binomial distribution +example, we would write: + + pdf(binomial_distribution(n, p), k); + +[endsect] + +[section:dist_params Discrete Probability Distributions] + +Note that the [@http://en.wikipedia.org/wiki/Discrete_probability_distribution +discrete distributions], including the binomial, negative binomial, Poisson & Bernoulli, +are all mathematically defined as discrete functions: +only integral values of the random variate are envisaged +and the functions are only defined at these integral values. +However because the method of calculation often uses continuous functions, +it is convenient to treat them as if they were continuous functions, +and permit non-integral values of their parameters. + +To enforce a strict mathematical model, +users may use floor or ceil functions on the random variate, +prior to calling the distribution function, +to enforce integral values. + +For similar reasons, in continuous distributions, parameters like degrees of freedom +that might appear to be integral, are treated as real values +(and are promoted from integer to floating-point if necessary). +In this case however, that there are a small number of situations where non-integral +degrees of freedom do have a genuine meaning. + +Generally speaking there is no loss of performance from allowing real-values +parameters: the underlying special functions contain optimizations for +integer-valued arguments when applicable. + +[caution +The quantile function of a discrete distribution will by +default return an integer result that has been +/rounded outwards/. That is to say lower quantiles (where the probability is +less than 0.5) are rounded downward, and upper quantiles (where the probability +is greater than 0.5) are rounded upwards. This behaviour +ensures that if an X% quantile is requested, then /at least/ the requested +coverage will be present in the central region, and /no more than/ +the requested coverage will be present in the tails. + +This behaviour can be changed so that the quantile functions are rounded +differently, or even return a real-valued result using +[link math_toolkit.policy.pol_overview Policies]. It is strongly +recommended that you read the tutorial +[link math_toolkit.policy.pol_tutorial.understand_dis_quant +Understanding Quantiles of Discrete Distributions] before +using the quantile function on a discrete distribution. The +[link math_toolkit.policy.pol_ref.discrete_quant_ref reference docs] +describe how to change the rounding policy +for these distributions. +] + +[endsect] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/bessel_ik.qbk b/doc/sf_and_dist/bessel_ik.qbk new file mode 100644 index 000000000..9c246323c --- /dev/null +++ b/doc/sf_and_dist/bessel_ik.qbk @@ -0,0 +1,197 @@ + +[section:mbessel Modified Bessel Functions of the First and Second Kinds] + +[h4 Synopsis] + + template + ``__sf_result`` cyl_bessel_i(T1 v, T2 x); + + template + ``__sf_result`` cyl_bessel_i(T1 v, T2 x, const ``__Policy``&); + + template + ``__sf_result`` cyl_bessel_k(T1 v, T2 x); + + template + ``__sf_result`` cyl_bessel_k(T1 v, T2 x, const ``__Policy``&); + + +[h4 Description] + +The functions __cyl_bessel_i and __cyl_bessel_k return the result of the +modified Bessel functions of the first and second kind respectively: + +cyl_bessel_i(v, x) = I[sub v](x) + +cyl_bessel_k(v, x) = K[sub v](x) + +where: + +[equation mbessel2] + +[equation mbessel3] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types. The functions are also optimised for the +relatively common case that T1 is an integer. + +[optional_policy] + +The functions return the result of __domain_error whenever the result is +undefined or complex. For __cyl_bessel_j this occurs when `x < 0` and v is not +an integer, or when `x == 0` and `v != 0`. For __cyl_neumann this occurs +when `x <= 0`. + +The following graph illustrates the exponential behaviour of I[sub v]. + +[$../graphs/bessel_i.png] + +The following graph illustrates the exponential decay of K[sub v]. + +[$../graphs/bessel_k.png] + +[h4 Testing] + +There are two sets of test values: spot values calculated using +[@http://functions.wolfram.com functions.wolfram.com], +and a much larger set of tests computed using +a simplified version of this implementation +(with all the special case handling removed). + +[h4 Accuracy] + +The following tables show how the accuracy of these functions +varies on various platforms, along with a comparison to the __gsl library. +Note that only results for the widest floating-point type on the +system are given, as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in cyl_bessel_i +[[Significand Size] [Platform and Compiler] [I[sub v]] ] +[[53] [Win32 / Visual C++ 8.0] [Peak=10 Mean=3.4 + GSL Peak=6000] ] +[[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=11 Mean=3] ] +[[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=11 Mean=4] ] +[[113] [HP-UX / HP aCC 6] [Peak=15 Mean=4] ] +] + +[table Errors Rates in cyl_bessel_k +[[Significand Size] [Platform and Compiler] [K[sub v]] ] +[[53] [Win32 / Visual C++ 8.0] [Peak=9 Mean=2 + +GSL Peak=9] ] +[[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=10 Mean=2] ] +[[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=10 Mean=2] ] +[[113] [HP-UX / HP aCC 6] [Peak=12 Mean=5] ] +] + +[h4 Implementation] + +The following are handled as special cases first: + +When computing I[sub v][space] for ['x < 0], then [nu][space] must be an integer +or a domain error occurs. If [nu][space] is an integer, then the function is +odd if [nu][space] is odd and even if [nu][space] is even, and we can reflect to +['x > 0]. + +For I[sub v][space] with v equal to 0, 1 or 0.5 are handled as special cases. + +The 0 and 1 cases use minimax rational approximations on +finite and infinite intervals. The coefficients are from: + +* J.M. Blair and C.A. Edwards, ['Stable rational minimax approximations + to the modified Bessel functions I_0(x) and I_1(x)], Atomic Energy of Canada + Limited Report 4928, Chalk River, 1974. +* S. Moshier, ['Methods and Programs for Mathematical Functions], + Ellis Horwood Ltd, Chichester, 1989. + +While the 0.5 case is a simple trigonometric function: + +I[sub 0.5](x) = sqrt(2 / [pi]x) * sinh(x) + +For K[sub v][space] with /v/ an integer, the result is calculated using the +recurrence relation: + +[equation mbessel5] + +starting from K[sub 0][space] and K[sub 1][space] which are calculated +using rational the approximations above. These rational approximations are +accurate to around 19 digits, and are therefore only used when T has +no more than 64 binary digits of precision. + +In the general case, we first normalize [nu][space] to \[[^0, [inf]]) +with the help of the reflection formulae: + +[equation mbessel9] + +[equation mbessel10] + +Let [mu][space] = [nu] - floor([nu] + 1/2), then [mu][space] is the fractional part of +[nu][space] such that |[mu]| <= 1/2 (we need this for convergence later). The idea is to +calculate K[sub [mu]](x) and K[sub [mu]+1](x), and use them to obtain +I[sub [nu]](x) and K[sub [nu]](x). + +The algorithm is proposed by Temme in +N.M. Temme, ['On the numerical evaluation of the modified bessel function + of the third kind], Journal of Computational Physics, vol 19, 324 (1975), +which needs two continued fractions as well as the Wronskian: + +[equation mbessel11] + +[equation mbessel12] + +[equation mbessel8] + +The continued fractions are computed using the modified Lentz's method +(W.J. Lentz, ['Generating Bessel functions in Mie scattering calculations + using continued fractions], Applied Optics, vol 15, 668 (1976)). +Their convergence rates depend on ['x], therefore we need +different strategies for large ['x] and small ['x]. + +['x > v], CF1 needs O(['x]) iterations to converge, CF2 converges rapidly. + +['x <= v], CF1 converges rapidly, CF2 fails to converge when ['x] [^->] 0. + +When ['x] is large (['x] > 2), both continued fractions converge (CF1 +may be slow for really large ['x]). K[sub [mu]][space] and K[sub [mu]+1][space] +can be calculated by + +[equation mbessel13] + +where + +[equation mbessel14] + +['S] is also a series that is summed along with CF2, see +I.J. Thompson and A.R. Barnett, ['Modified Bessel functions I_v and K_v + of real order and complex argument to selected accuracy], Computer Physics + Communications, vol 47, 245 (1987). + +When ['x] is small (['x] <= 2), CF2 convergence may fail (but CF1 +works very well). The solution here is Temme's series: + +[equation mbessel15] + +where + +[equation mbessel16] + +f[sub k][space] and h[sub k][space] +are also computed by recursions (involving gamma functions), but the +formulas are a little complicated, readers are referred to +N.M. Temme, ['On the numerical evaluation of the modified Bessel function + of the third kind], Journal of Computational Physics, vol 19, 324 (1975). +Note: Temme's series converge only for |[mu]| <= 1/2. + +K[sub [nu]](x) is then calculated from the forward +recurrence, as is K[sub [nu]+1](x). With these two values and +f[sub [nu]], the Wronskian yields I[sub [nu]](x) directly. + +[endsect] + +[/ + Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/bessel_introduction.qbk b/doc/sf_and_dist/bessel_introduction.qbk new file mode 100644 index 000000000..4cdfe629d --- /dev/null +++ b/doc/sf_and_dist/bessel_introduction.qbk @@ -0,0 +1,124 @@ + +[section:bessel_over Bessel Function Overview] + +[h4 Ordinary Bessel Functions] + +Bessel Functions are solutions to Bessel's ordinary differential +equation: + +[equation bessel1] + +where [nu][space] is the /order/ of the equation, and may be an arbitrary +real or complex number, although integer orders are the most common occurrence. + +This library supports either integer or real orders. + +Since this is a second order differential equation, there must be two +linearly independent solutions, the first of these is denoted J[sub v][space] +and known as a Bessel function of the first kind: + +[equation bessel2] + +This function is implemented in this library as __cyl_bessel_j. + +The second solution is denoted either Y[sub v][space] or N[sub v][space] +and is known as either a Bessel Function of the second kind, or as a +Neumann function: + +[equation bessel3] + +This function is implemented in this library as __cyl_neumann. + +The Bessel functions satisfy the recurrence relations: + +[equation bessel4] + +[equation bessel5] + +Have the derivatives: + +[equation bessel6] + +[equation bessel7] + +Have the Wronskian relation: + +[equation bessel8] + +and the reflection formulae: + +[equation bessel9] + +[equation bessel10] + + +[h4 Modified Bessel Functions] + +The Bessel functions are valid for complex argument /x/, and an important +special case is the situation where /x/ is purely imaginary: giving a real +valued result. In this case the functions are the two linearly +independent solutions to the modified Bessel equation: + +[equation mbessel1] + +The solutions are known as the modified Bessel functions of the first and +second kind (or occasionally as the hyperbolic Bessel functions of the first +and second kind). They are denoted I[sub v][space] and K[sub v][space] +respectively: + +[equation mbessel2] + +[equation mbessel3] + +These functions are implemented in this library as __cyl_bessel_i and +__cyl_bessel_k respectively. + +The modified Bessel functions satisfy the recurrence relations: + +[equation mbessel4] + +[equation mbessel5] + +Have the derivatives: + +[equation mbessel6] + +[equation mbessel7] + +Have the Wronskian relation: + +[equation mbessel8] + +and the reflection formulae: + +[equation mbessel9] + +[equation mbessel10] + +[h4 Spherical Bessel Functions] + +When solving the Helmholtz equation in spherical coordinates by +separation of variables, the radial equation has the form: + +[equation sbessel1] + +The two linearly independent solutions to this equation are called the +spherical Bessel functions j[sub n][space] and y[sub n][space], and are related to the +ordinary Bessel functions J[sub n][space] and Y[sub n][space] by: + +[equation sbessel2] + +The spherical Bessel function of the second kind y[sub n][space] +is also known as the spherical Neumann function n[sub n]. + +These functions are implemented in this library as __sph_bessel and +__sph_neumann. + +[endsect] + +[/ + Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/bessel_jy.qbk b/doc/sf_and_dist/bessel_jy.qbk new file mode 100644 index 000000000..04a6a5115 --- /dev/null +++ b/doc/sf_and_dist/bessel_jy.qbk @@ -0,0 +1,261 @@ + +[section:bessel Bessel Functions of the First and Second Kinds] + +[h4 Synopsis] + + template + ``__sf_result`` cyl_bessel_j(T1 v, T2 x); + + template + ``__sf_result`` cyl_bessel_j(T1 v, T2 x, const ``__Policy``&); + + template + ``__sf_result`` cyl_neumann(T1 v, T2 x); + + template + ``__sf_result`` cyl_neumann(T1 v, T2 x, const ``__Policy``&); + + +[h4 Description] + +The functions __cyl_bessel_j and __cyl_neumann return the result of the +Bessel functions of the first and second kinds respectively: + +cyl_bessel_j(v, x) = J[sub v](x) + +cyl_neumann(v, x) = Y[sub v](x) = N[sub v](x) + +where: + +[equation bessel2] + +[equation bessel3] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types. The functions are also optimised for the +relatively common case that T1 is an integer. + +[optional_policy] + +The functions return the result of __domain_error whenever the result is +undefined or complex. For __cyl_bessel_j this occurs when `x < 0` and v is not +an integer, or when `x == 0` and `v != 0`. For __cyl_neumann this occurs +when `x <= 0`. + +The following graph illustrates the cyclic nature of J[sub v]: + +[$../graphs/bessel_jn.png] + +The following graph shows the behaviour of Y[sub v]: this is also +cyclic for large /x/, but tends to -[infin][space] for small /x/: + +[$../graphs/bessel_yv.png] + +[h4 Testing] + +There are two sets of test values: spot values calculated using +[@http://functions.wolfram.com functions.wolfram.com], +and a much larger set of tests computed using +a simplified version of this implementation +(with all the special case handling removed). + +[h4 Accuracy] + +The following tables show how the accuracy of these functions +varies on various platforms, along with comparisons to the __gsl and +__cephes libraries. Note that the cyclic nature of these +functions means that they have an infinite number of irrational +roots: in general these functions have arbitrarily large /relative/ +errors when the arguments are sufficiently close to a root. Of +course the absolute error in such cases is always small. +Note that only results for the widest floating-point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in cyl_bessel_j +[[Significand Size] [Platform and Compiler] [J[sub 0][space] and J[sub 1]] [J[sub v]] [J[sub v][space] (large values of x > 1000)] ] +[[53] [Win32 / Visual C++ 8.0] + [Peak=2.5 Mean=1.1 + +GSL Peak=6.6 + +__cephes Peak=2.5 Mean=1.1] + [Peak=11 Mean=2.2 + +GSL Peak=11 + +__cephes Peak=17 Mean=2.5] + [Peak=413 Mean=110 + +GSL Peak=6x10[super 11] + +__cephes Peak=2x10[super 5] ] ] +[[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=7 Mean=3] [Peak=117 Mean=10] [Peak=2x10[super 4][space] Mean=6x10[super 3]] ] +[[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=7 Mean=3] [Peak=400 Mean=40] [Peak=2x10[super 4][space] Mean=1x10[super 4]] ] +[[113] [HP-UX / HP aCC 6] [Peak=14 Mean=6] [Peak=29 Mean=3] [Peak=2700 Mean=450] ] +] + +[table Errors Rates in cyl_neumann +[[Significand Size] [Platform and Compiler] [J[sub 0][space] and J[sub 1]] [J[sub n] (integer orders)] [J[sub v] (fractional orders)] ] +[[53] [Win32 / Visual C++ 8.0] + [Peak=330 Mean=54 + +GSL Peak=34 Mean=9 + +__cephes Peak=330 Mean=54] + [Peak=923 Mean=83 + +GSL Peak=500 Mean=54 + +__cephes Peak=923 Mean=83] + [Peak=561 Mean=36 + +GSL Peak=1.4x10[super 6][space] Mean\=7x10[super 4][space] + +__cephes Peak=+INF]] +[[64] [Red Hat Linux IA64 / G++ 3.4] [Peak=470 Mean=56] [Peak=843 Mean=51] [Peak=741 Mean=51] ] +[[64] [SUSE Linux AMD64 / G++ 4.1] [Peak=1300 Mean=424] [Peak=2x10[super 4][space] Mean=8x10[super 3]] [Peak=1x10[super 5][space] Mean=6x10[super 3]] ] +[[113] [HP-UX / HP aCC 6] [Peak=180 Mean=63] [Peak=340 Mean=150] [Peak=2x10[super 4][space] Mean=1200] ] +] + +Note that for large /x/ these functions are largely dependent on +the accuracy of the `std::sin` and `std::cos` functions. + +Comparison to GSL and __cephes is interesting: both __cephes and this library optimise +the integer order case - leading to identical results - simply using the general +case is for the most part slightly more accurate though, as noted by the +better accuracy of GSL in the integer argument cases. This implementation tends to +perform much better when the arguments become large, __cephes in particular produces +some remarkably inaccurate results with some of the test data (no significant figures +correct), and even GSL performs badly with some inputs to J[sub v]. Note that +by way of double-checking these results, the worst performing __cephes and GSL cases +were recomputed using [@http://functions.wolfram.com functions.wolfram.com], +and the result checked against our test data: no errors in the test data were found. + +[h4 Implementation] + +The implementation is mostly about filtering off various special cases: + +When /x/ is negative, then the order /v/ must be an integer or the +result is a domain error. If the order is an integer then the function +is odd for odd orders and even for even orders, so we reflect to /x > 0/. + +When the order /v/ is negative then the reflection formulae can be used to +move to /v > 0/: + +[equation bessel9] + +[equation bessel10] + +Note that if the order is an integer, then these formulae reduce to: + +J[sub -n] = (-1)[super n]J[sub n] + +Y[sub -n] = (-1)[super n]Y[sub n] + +However, in general, a negative order implies that we will need to compute +both J and Y. + +When /x/ is large compared to the order /v/ then the asymptotic expansions +for large /x/ in M. Abramowitz and I.A. Stegun, +['Handbook of Mathematical Functions] 9.2.19 are used +(these were found to be more reliable +than those in A&S 9.2.5). + +When the order /v/ is an integer the method first relates the result +to J[sub 0], J[sub 1], Y[sub 0][space] and Y[sub 1][space] using either +forwards or backwards recurrence (Miller's algorithm) depending upon which is stable. +The values for J[sub 0], J[sub 1], Y[sub 0][space] and Y[sub 1][space] are +calculated using the rational minimax approximations on +root-bracketing intervals for small ['|x|] and Hankel asymptotic +expansion for large ['|x|]. The coefficients are from: + +W.J. Cody, ['ALGORITHM 715: SPECFUN - A Portable FORTRAN Package of +Special Function Routines and Test Drivers], ACM Transactions on Mathematical +Software, vol 19, 22 (1993). + +and + +J.F. Hart et al, ['Computer Approximations], John Wiley & Sons, New York, 1968. + +These approximations are accurate to around 19 decimal digits: therefore +these methods are not used when type T has more than 64 binary digits. + +When /x/ is small, J[sub x][space] is best computed directly from the series: + +[equation bessel2] + +In the general case we compute J[sub v][space] and +Y[sub v][space] simultaneously. + +To get the initial values, let +[mu][space] = [nu] - floor([nu] + 1/2), then [mu][space] is the fractional part +of [nu][space] such that +|[mu]| <= 1/2 (we need this for convergence later). The idea is to +calculate J[sub [mu]](x), J[sub [mu]+1](x), Y[sub [mu]](x), Y[sub [mu]+1](x) +and use them to obtain J[sub [nu]](x), Y[sub [nu]](x). + +The algorithm is called Steed's method, which needs two +continued fractions as well as the Wronskian: + +[equation bessel8] + +[equation bessel11] + +[equation bessel12] + +See: F.S. Acton, ['Numerical Methods that Work], + The Mathematical Association of America, Washington, 1997. + +The continued fractions are computed using the modified Lentz's method +(W.J. Lentz, ['Generating Bessel functions in Mie scattering calculations +using continued fractions], Applied Optics, vol 15, 668 (1976)). +Their convergence rates depend on ['x], therefore we need +different strategies for large ['x] and small ['x]. + +['x > v], CF1 needs O(['x]) iterations to converge, CF2 converges rapidly + +['x <= v], CF1 converges rapidly, CF2 fails to converge when ['x] [^->] 0 + +When ['x] is large (['x] > 2), both continued fractions converge (CF1 +may be slow for really large ['x]). J[sub [mu]], J[sub [mu]+1], +Y[sub [mu]], Y[sub [mu]+1] can be calculated by + +[equation bessel13] + +where + +[equation bessel14] + +J[sub [nu]] and Y[sub [mu]] are then calculated using backward +(Miller's algorithm) and forward recurrence respectively. + +When ['x] is small (['x] <= 2), CF2 convergence may fail (but CF1 +works very well). The solution here is Temme's series: + +[equation bessel15] + +where + +[equation bessel16] + +g[sub k][space] and h[sub k][space] +are also computed by recursions (involving gamma functions), but the +formulas are a little complicated, readers are refered to +N.M. Temme, ['On the numerical evaluation of the ordinary Bessel function +of the second kind], Journal of Computational Physics, vol 21, 343 (1976). +Note Temme's series converge only for |[mu]| <= 1/2. + +As the previous case, Y[sub [nu]][space] is calculated from the forward +recurrence, so is Y[sub [nu]+1]. With these two +values and f[sub [nu]], the Wronskian yields J[sub [nu]](x) directly +without backward recurrence. + +[endsect] + +[/ + Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/bessel_spherical.qbk b/doc/sf_and_dist/bessel_spherical.qbk new file mode 100644 index 000000000..7be51d8a2 --- /dev/null +++ b/doc/sf_and_dist/bessel_spherical.qbk @@ -0,0 +1,87 @@ + +[section:sph_bessel Spherical Bessel Functions of the First and Second Kinds] + +[h4 Synopsis] + + template + ``__sf_result`` sph_bessel(unsigned v, T2 x); + + template + ``__sf_result`` sph_bessel(unsigned v, T2 x, const ``__Policy``&); + + template + ``__sf_result`` sph_neumann(unsigned v, T2 x); + + template + ``__sf_result`` sph_neumann(unsigned v, T2 x, const ``__Policy``&); + +[h4 Description] + +The functions __sph_bessel and __sph_neumann return the result of the +Spherical Bessel functions of the first and second kinds respectively: + +sph_bessel(v, x) = j[sub v](x) + +sph_neumann(v, x) = y[sub v](x) = n[sub v](x) + +where: + +[equation sbessel2] + +The return type of these functions is computed using the __arg_pomotion_rules +for the single argument type T. + +[optional_policy] + +The functions return the result of __domain_error whenever the result is +undefined or complex: this occurs when `x < 0`. + +The j[sub v][space] function is cyclic like J[sub v][space] but differs +in its behaviour at the origin: + +[$../graphs/sph_bessel_j.png] + +Likewise y[sub v][space] is also cyclic for large x, but tends to -[infin][space] +for small /x/: + +[$../graphs/sph_bessel_y.png] + +[h4 Testing] + +There are two sets of test values: spot values calculated using +[@http://functions.wolfram.com/ functions.wolfram.com], +and a much larger set of tests computed using +a simplified version of this implementation +(with all the special case handling removed). + +[h4 Accuracy] + +Other than for some special cases, these functions are computed in terms of +__cyl_bessel_j and __cyl_neumann: refer to these functions for accuracy data. + +[h4 Implementation] + +Other than error handling and a couple of special cases these functions +are implemented directly in terms of their definitions: + +[equation sbessel2] + +The special cases occur for: + +j[sub 0][space]= __sinc_pi(x) = sin(x) / x + +and for small ['x < 1], we can use the series: + +[equation sbessel5] + +which neatly avoids the problem of calculating 0/0 that can occur with the +main definition as x [rarr] 0. + +[endsect] + +[/ + Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/beta.qbk b/doc/sf_and_dist/beta.qbk new file mode 100644 index 000000000..8ec6ecc65 --- /dev/null +++ b/doc/sf_and_dist/beta.qbk @@ -0,0 +1,128 @@ +[section:beta_function Beta] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` beta(T1 a, T2 b); + + template + ``__sf_result`` beta(T1 a, T2 b, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +The beta function is defined by: + +[equation beta1] + +[$../graphs/beta.png] + +[optional_policy] + +There are effectively two versions of this function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the significand +of T correspond to a certain __lanczos. In practice any built-in +floating-point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp. + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types. + +[h4 Accuracy] + +The following table shows peak errors for various domains of input arguments, +along with comparisons to the __gsl and __cephes libraries. Note that +only results for the widest floating point type on the system are given as +narrower types have __zero_error. + +[table Peak Errors In the Beta Function +[[Significand Size] [Platform and Compiler] [Errors in range + +0.4 < a,b < 100] [Errors in range + +1e-6 < a,b < 36]] +[[53] [Win32, Visual C++ 8] [Peak=99 Mean=22 + +(GSL Peak=1178 Mean=238) + +(__cephes=1612)] [Peak=10.7 Mean=2.6 + +(GSL Peak=12 Mean=2.0) + +(__cephes=174)]] +[[64] [Red Hat Linux IA32, g++ 3.4.4] [Peak=112.1 Mean=26.9] [Peak=15.8 Mean=3.6]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=61.4 Mean=19.5] [Peak=12.2 Mean=3.6]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=42.03 Mean=13.94] [Peak=9.8 Mean=3.1]] +] + +Note that the worst errors occur when a or b are large, and that +when this is the case the result is very close to zero, so absolute +errors will be very small. + +[h4 Testing] + +A mixture of spot tests of exact values, and randomly generated test data are +used: the test data was computed using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision. + +[h4 Implementation] + +Traditional methods of evaluating the beta function either involve evaluating +the gamma functions directly, or taking logarithms and then +exponentiating the result. However, the former is prone to overflows +for even very modest arguments, while the latter is prone to cancellation +errors. As an alternative, if we regard the gamma function as a white-box +containing the __lanczos, then we can combine the power terms: + +[equation beta2] + +which is almost the ideal solution, however almost all of the error occurs +in evaluating the power terms when /a/ or /b/ are large. If we assume that /a > b/ +then the larger of the two power terms can be reduced by a factor of /b/, which +immediately cuts the maximum error in half: + +[equation beta3] + +This may not be the final solution, but it is very competitive compared to +other implementation methods. + +The generic implementation - where no __lanczos approximation is available - is +implemented in a very similar way to the generic version of the gamma function. +Again in order to avoid numerical overflow the power terms that prefix the series and +continued fraction parts are collected together into: + +[equation beta8] + +where la, lb and lc are the integration limits used for a, b, and a+b. + +There are a few special cases worth mentioning: + +When /a/ or /b/ are less than one, we can use the recurrence relations: + +[equation beta4] + +[equation beta5] + +to move to a more favorable region where they are both greater than 1. + +In addition: + +[equation beta7] + +[endsect][/section:beta_function The Beta Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/beta_derivative.qbk b/doc/sf_and_dist/beta_derivative.qbk new file mode 100644 index 000000000..6481df268 --- /dev/null +++ b/doc/sf_and_dist/beta_derivative.qbk @@ -0,0 +1,50 @@ +[section:beta_derivative Derivative of the Incomplete Beta Function] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` ibeta_derivative(T1 a, T2 b, T3 x); + + template + ``__sf_result`` ibeta_derivative(T1 a, T2 b, T3 x, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +This function finds some uses in statistical distributions: it +computes the partial derivative with respect to /x/ of the incomplete +beta function __ibeta. + +[equation derivative2] + +The return type of this function is computed using the __arg_pomotion_rules +when T1, T2 and T3 are different types. + +[optional_policy] + +[h4 Accuracy] + +Almost identical to the incomplete beta function __ibeta. + +[h4 Implementation] + +This function just expose some of the internals of the incomplete +beta function __ibeta: refer to the documentation for that function +for more information. + +[endsect][/section Derivatives of the Incomplete Beta and Gamma Functions] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/common_overviews.qbk b/doc/sf_and_dist/common_overviews.qbk new file mode 100644 index 000000000..9d42fae65 --- /dev/null +++ b/doc/sf_and_dist/common_overviews.qbk @@ -0,0 +1,223 @@ + + +[template policy_overview[] + +Policies are a powerful fine-grain mechanism that allow you to customise the +behaviour of this library according to your needs. There is more information +available in the [link math_toolkit.policy.pol_tutorial policy tutorial] +and the [link math_toolkit.policy.pol_ref policy reference]. + +Generally speaking unless you find that the +[link math_toolkit.policy.pol_tutorial.policy_tut_defaults + default policy behaviour] +when encountering 'bad' argument values does not meet your needs, +you should not need to worry about policies. + +Policies are a compile-time mechanism that allow you to change +error-handling or calculation precision either +program wide, or at the call site. + +Although the policy mechanism itself is rather complicated, +in practice it is easy to use, and very flexible. + +Using policies you can control: + +* [link math_toolkit.policy.pol_ref.error_handling_policies How results from 'bad' arguments are handled], + including those that cannot be fully evaluated. +* How [link math_toolkit.policy.pol_ref.internal_promotion accuracy is controlled by internal promotion] to use more precise types. +* What working [link math_toolkit.policy.pol_ref.precision_pol precision] should be used to calculate results. +* What to do when a [link math_toolkit.policy.pol_ref.assert_undefined mathematically undefined function] + is used: Should this raise a run-time or compile-time error? +* Whether [link math_toolkit.policy.pol_ref.discrete_quant_ref discrete functions], + like the binomial, should return real or only integral values, and how they are rounded. +* How many iterations a special function is permitted to perform in + a series evaluation or root finding algorithm before it gives up and raises an + __evaluation_error. + +You can control policies: + +* Using [link math_toolkit.policy.pol_ref.policy_defaults macros] to +change any default policy: the is the prefered method for installation +wide policies. +* At your chosen [link math_toolkit.policy.pol_ref.namespace_pol +namespace scope] for distributions and/or functions: this is the +prefered method for project, namespace, or translation unit scope +policies. +* In an ad-hoc manner [link math_toolkit.policy.pol_tutorial.ad_hoc_sf_policies +by passing a specific policy to a special function], or to a +[link math_toolkit.policy.pol_tutorial.ad_hoc_dist_policies +statistical distribution]. + +] + +[template performance_overview[] + +By and large the performance of this library should be acceptable +for most needs. However, you should note that the library's primary +emphasis is on accuracy and numerical stability, and /not/ speed. + +In terms of the algorithms used, this library aims to use the same "best +of breed" algorithms as many other libraries: the principle difference +is that this library is implemented in C++ - taking advantage of all +the abstraction mechanisms that C++ offers - where as most traditional +numeric libraries are implemented in C or FORTRAN. Traditionally +languages such as C or FORTAN are perceived as easier to optimise +than more complex languages like C++, so in a sense this library +provides a good test of current compiler technology, and the +"abstraction penalty" - if any - of C++ compared to other languages. + +The two most important things you can do to ensure the best performance +from this library are: + +# Turn on your compilers optimisations: the difference between "release" +and "debug" builds can easily be a [link math_toolkit.perf.getting_best factor of 20]. +# Pick your compiler carefully: [link math_toolkit.perf.comp_compilers +performance differences of up to +8 fold] have been found between some windows compilers for example. + +The [link math_toolkit.perf performance section] contains more +information on the performance +of this library, what you can do to fine tune it, and how this library +compares to some other open source alternatives. + +] + +[template compilers_overview[] + +This section contains some information about how various compilers +work with this library. +It is not comprehensive and updated experiences are always welcome. +Some effort has been made to suppress unhelpful warnings but it is +difficult to achieve this on all systems. + +[table Supported/Tested Compilers +[[Platform][Compiler][Has long double support][Notes]] +[[Windows][MSVC 7.1 and later][Yes] + [All tests OK. + + We aim to keep our headers warning free at level 4 with + this compiler.]] +[[Windows][Intel 8.1 and later][Yes] + [All tests OK. + + We aim to keep our headers warning free at level 4 with + this compiler. However, The tests cases tend to generate a lot of + warnings relating to numeric underflow of the test data: these are + harmless.]] +[[Windows][GNU Mingw32 C++][Yes] + [All tests OK. + + We aim to keep our headers warning free with -Wall with this compiler.]] +[[Windows][GNU Cygwin C++][No] + [All tests OK. + + We aim to keep our headers warning free with -Wall with this compiler. + + Long double support has been disabled because there are no native + long double C std library functions available.]] +[[Windows][Borland C++ 5.8.2 (Developer studio 2006)][No] + [We have only partial compatability with this compiler: + + Long double support has been disabled because the native + long double C standard library functions really only forward to the + double versions. This can result in unpredictable behaviour when + using the long double overloads: for example `sqrtl` applied to a + finite value, can result in an infinite result. + + Some functions still fail to compile, there are no known workarounds at present.]] + +[[Linux][GNU C++ 3.4 and later][Yes] + [All tests OK. + + We aim to keep our headers warning free with -Wall with this compiler.]] +[[Linux][Intel C++ 10.0 and later][Yes] + [All tests OK. + + We aim to keep our headers warning free with -Wall with this compiler. + However, The tests cases tend to generate a lot of + warnings relating to numeric underflow of the test data: these are + harmless.]] +[[Linux][Intel C++ 8.1 and 9.1][No] + [All tests OK. + + Long double support has been disabled with these compiler releases + because calling the standard library long double math functions + can result in a segfault. The issue is Linux distribution and + glibc version specific and is Intel bug report #409291. Fully up to date + releases of Intel 9.1 (post version l_cc_c_9.1.046) + shouldn't have this problem. If you need long + double support with this compiler, then comment out the define of + BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS at line 55 of + [@../../../../../boost/math/tools/config.hpp boost/math/tools/config.hpp]. + + We aim to keep our headers warning free with -Wall with this compiler. + However, The tests cases tend to generate a lot of + warnings relating to numeric underflow of the test data: these are + harmless.]] +[[Linux][QLogic PathScale 3.0][Yes] + [Some tests involving conceptual checks fail to build, otherwise + there appear to be no issues.]] +[[Linux][Sun Studio 12][Yes] + [Some tests involving function overload resolution fail to build, + these issues should be rairly encountered in practice.]] +[[Solaris][Sun Studio 12][Yes] + [Some tests involving function overload resolution fail to build, + these issues should be rairly encountered in practice.]] +[[Solaris][GNU C++ 4.x][Yes] + [All tests OK. + + We aim to keep our headers warning free with -Wall with this compiler.]] +[[HP Tru64][Compaq C++ 7.1][Yes] + [All tests OK.]] +[[HP-UX Itanium][HP aCC 6.x][Yes] + [All tests OK. + + Unfortunately this compiler emits quite a few warnings from libraries + upon which we depend (TR1, Array etc).]] +[[HP-UX PA-RISC][GNU C++ 3.4][No] + [All tests OK.]] +[[Apple Mac OS X, Intel][Darwin/GNU C++ 4.x][Yes][All tests OK.]] +[[Apple Mac OS X, PowerPC][Darwin/GNU C++ 4.x][No] + [All tests OK. + + Long double support has been disabled on this platform due to the + rather strange nature of Darwin's 106-bit long double + implementation. It should be possible to make this work if someone + is prepared to offer assistance.]] +[[IMB AIX][IBM xlc 5.3][Yes] + [Currently our accuracy tests are unable to be run on this platform + due to compiler errors in our /test/ code. The IBM compiler group + are aware of the problem.]] +] + +[table Unsupported Compilers +[[Platform][Compiler]] +[[Windows][Borland C++ 5.9.2 (Borland Developer Studio 2007)]] +[[Windows][MSVC 6 and 7]] +] + +If you're compiler or platform is not listed above, please try running the +regression tests: cd into boost-root/libs/math/test and do a: + + bjam mytoolset + +where "mytoolset" is the name of the +[@../../../../../tools/build/index.html Boost.Build] toolset used for your +compiler. The chances are that [*many of the accuracy tests will fail +at this stage] - don't panic - the default acceptable error tolerances +are quite tight, especially for long double types with an extended +exponent range (these cause more extreme test cases to be executed +for some functions). +You will need to cast an eye over the output from +the failing tests and make a judgement as to whether +the error rates are acceptable or not. + +] + +[/ math.qbk + Copyright 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/concepts.qbk b/doc/sf_and_dist/concepts.qbk new file mode 100644 index 000000000..ec64c6893 --- /dev/null +++ b/doc/sf_and_dist/concepts.qbk @@ -0,0 +1,369 @@ +[section:use_ntl Using With NTL - a High-Precision Floating-Point Library] + +The special functions and tools in this library can be used with +[@http://shoup.net/ntl/doc/RR.txt NTL::RR (an arbitrary precision number type)], +via the bindings in [@../../../../../boost/math/bindings/rr.hpp boost/math/bindings/rr.hpp]. +[@http://shoup.net/ntl/ See also NTL: A Library for doing Number Theory by +Victor Shoup] + +Unfortunately `NTL::RR` doesn't quite satisfy our conceptual requirements, +so there is a very thin wrapper class `boost::math::ntl::RR` defined in +[@../../../../../boost/math/bindings/rr.hpp boost/math/bindings/rr.hpp] that you +should use in place of `NTL::RR`. The class is intended to be a drop-in +replacement for the "real" NTL::RR that adds some syntactic sugar to keep +this library happy, plus some of the standard library functions not implemented +in NTL. + +Finally there is a high precision __lanczos suitable for use with `boost::math::ntl::RR`, +used at 1000-bit precision in +[@../../../tools/ntl_rr_lanczos.hpp libs/math/tools/ntl_rr_lanczos.hpp]. +The approximation has a theoretical precision of > 90 decimal digits, +and an experimental precision of > 100 decimal digits. To use that +approximation, just include that header before any of the special +function headers (if you don't use it, you'll get a slower, but +fully generic implementation for all of the gamma-like functions). + +[endsect][/section:use_ntl Using With NTL - a High Precision Floating-Point Library] + +[section:concepts Conceptual Requirements for Real Number Types] + +The functions, and statistical distributions in this library can be used with +any type /RealType/ that meets the conceptual requirements given below. All +the built in floating point types will meet these requirements. +User defined types that meet the requirements can also be used. For example, +with [link math_toolkit.using_udt.use_ntl a thin wrapper class] one of the types +provided with [@http://shoup.net/ntl/ NTL (RR)] can be used. Submissions +of binding to other extended precision types would also be most welcome! + +The guiding principal behind these requirements, is that a /RealType/ +behaves just like a built in floating point type. + +[h4 Basic Arithmetic Requirements] + +These requirements are common to all of the functions in this library. + +In the following table /r/ is an object of type `RealType`, /cr/ and +/cr2/ are objects +of type `const RealType`, and /ca/ is an object of type `const arithmetic-type` +(arithmetic types include all the built in integers and floating point types). + +[table +[[Expression][Result Type][Notes]] +[[`RealType(cr)`][RealType] + [RealType is copy constructible.]] +[[`RealType(ca)`][RealType] + [RealType is copy constructible from the arithmetic types.]] +[[`r = cr`][RealType&][Assignment operator.]] +[[`r = ca`][RealType&][Assignment operator from the arithmetic types.]] +[[`r += cr`][RealType&][Adds cr to r.]] +[[`r += ca`][RealType&][Adds ar to r.]] +[[`r -= cr`][RealType&][Subtracts cr from r.]] +[[`r -= ca`][RealType&][Subtracts ca from r.]] +[[`r *= cr`][RealType&][Multiplies r by cr.]] +[[`r *= ca`][RealType&][Multiplies r by ca.]] +[[`r /= cr`][RealType&][Divides r by cr.]] +[[`r /= ca`][RealType&][Divides r by ca.]] +[[`-r`][RealType][Unary Negation.]] +[[`+r`][RealType&][Identity Operation.]] +[[`cr + cr2`][RealType][Binary Addition]] +[[`cr + ca`][RealType][Binary Addition]] +[[`ca + cr`][RealType][Binary Addition]] +[[`cr - cr2`][RealType][Binary Subtraction]] +[[`cr - ca`][RealType][Binary Subtraction]] +[[`ca - cr`][RealType][Binary Subtraction]] +[[`cr * cr2`][RealType][Binary Multiplication]] +[[`cr * ca`][RealType][Binary Multiplication]] +[[`ca * cr`][RealType][Binary Multiplication]] +[[`cr / cr2`][RealType][Binary Subtraction]] +[[`cr / ca`][RealType][Binary Subtraction]] +[[`ca / cr`][RealType][Binary Subtraction]] +[[`cr == cr2`][bool][Equality Comparison]] +[[`cr == ca`][bool][Equality Comparison]] +[[`ca == cr`][bool][Equality Comparison]] +[[`cr != cr2`][bool][Inequality Comparison]] +[[`cr != ca`][bool][Inequality Comparison]] +[[`ca != cr`][bool][Inequality Comparison]] +[[`cr <= cr2`][bool][Less than equal to.]] +[[`cr <= ca`][bool][Less than equal to.]] +[[`ca <= cr`][bool][Less than equal to.]] +[[`cr >= cr2`][bool][Greater than equal to.]] +[[`cr >= ca`][bool][Greater than equal to.]] +[[`ca >= cr`][bool][Greater than equal to.]] +[[`cr < cr2`][bool][Less than comparison.]] +[[`cr < ca`][bool][Less than comparison.]] +[[`ca < cr`][bool][Less than comparison.]] +[[`cr > cr2`][bool][Greater than comparison.]] +[[`cr > ca`][bool][Greater than comparison.]] +[[`ca > cr`][bool][Greater than comparison.]] +[[`boost::math::tools::digits()`][int] + [The number of digits in the significand of RealType.]] +[[`boost::math::tools::max_value()`][RealType] + [The largest representable number by type RealType.]] +[[`boost::math::tools::min_value()`][RealType] + [The smallest representable number by type RealType.]] +[[`boost::math::tools::log_max_value()`][RealType] + [The natural logarithm of the largest representable number by type RealType.]] +[[`boost::math::tools::log_min_value()`][RealType] + [The natural logarithm of the smallest representable number by type RealType.]] +[[`boost::math::tools::epsilon()`][RealType] + [The machine epsilon of RealType.]] +] + +Note that: + +# The functions `log_max_value` and `log_min_value` can be +synthesised from the others, and so no explicit specialisation is required. +# The function `epsilon` can be synthesised from the others, so no +explicit specialisation is required provided the precision +of RealType does not vary at runtime (see the header +[@../../../../../boost/math/bindings/rr.hpp boost/math/bindings/rr.hpp] +for an example where the precision does vary at runtime). +# The functions `digits`, `max_value` and `min_value`, all get synthesised +automatically from `std::numeric_limits`. However, if `numeric_limits` +is not specialised for type RealType, then you will get a compiler error +when code tries to use these functions, /unless/ you explicitly specialise them. +For example if the precision of RealType varies at runtime, then +`numeric_limits` support may not be appropriate, see +[@../../../../../boost/math/tools/ntl.hpp boost/math/tools/ntl.hpp] for examples. + +[warning +If `std::numeric_limits<>` is *not specialized* +for type /RealType/ then the default float precision of 6 decimal digits +will be used by other Boost programs including: + +Boost.Test: giving misleading error messages like + +['"difference between {9.79796} and {9.79796} exceeds 5.42101e-19%".] + +Boost.LexicalCast and Boost.Serialization when converting the number +to a string, causing potentially serious loss of accuracy on output. + +Although it might seem obvious that RealType should require `std::numeric_limits` +to be specialized, this is not sensible for +`NTL::RR` and similar classes where the number of digits is a runtime +parameter (where as for `numeric_limits` it has to be fixed at compile time). +] + +[h4 Standard Library Support Requirements] + +Many (though not all) of the functions in this library make calls +to standard library functions, the following table summarises the +requirements. Note that most of the functions in this library +will only call a small subset of the functions listed here, so if in +doubt whether a user defined type has enough standard library +support to be useable the best advise is to try it and see! + +In the following table /r/ is an object of type `RealType`, +/cr1/ and /cr2/ are objects of type `const RealType`, and +/i/ is an object of type `int`. + +[table +[[Expression][Result Type]] +[[`fabs(cr1)`][RealType]] +[[`abs(cr1)`][RealType]] +[[`ceil(cr1)`][RealType]] +[[`floor(cr1)`][RealType]] +[[`exp(cr1)`][RealType]] +[[`pow(cr1, cr2)`][RealType]] +[[`sqrt(cr1)`][RealType]] +[[`log(cr1)`][RealType]] +[[`frexp(cr1, &i)`][RealType]] +[[`ldexp(cr1, i)`][RealType]] +[[`cos(cr1)`][RealType]] +[[`sin(cr1)`][RealType]] +[[`asin(cr1)`][RealType]] +[[`tan(cr1)`][RealType]] +[[`atan(cr1)`][RealType]] +] + +Note that the table above lists only those standard library functions known to +be used (or likely to be used in the near future) by this library. +The following functions: `acos`, `atan2`, `fmod`, `cosh`, `sinh`, `tanh`, `modf` and `log10` +are not currently used, but may be if further special functions are added. + +In addition, for efficient and accurate results, a __lanczos is highly desirable. +You may be able to adapt an existing approximation from +[@../../../../../boost/math/special_functions/lanczos.hpp +boost/math/special_functions/lanczos.hpp] or +[@../../../tools/ntl_rr_lanczos.hpp libs/math/tools/ntl_rr_lanczos.hpp]: +you will need change +static_cast's to lexical_cast's, and the constants to /strings/ +(in order to ensure the coefficients aren't truncated to long double) +and then specialise `lanczos_traits` for type T. Otherwise you may have to hack +[@../../../tools/lanczos_generator.cpp +libs/math/tools/lanczos_generator.cpp] to find a suitable +approximation for your RealType. The code will still compile if you don't do +this, but both accuracy and efficiency will be greatly compromised in any +function that makes use of the gamma\/beta\/erf family of functions. + +[endsect] + +[section:dist_concept Conceptual Requirements for Distribution Types] + +A /DistributionType/ is a type that implements the following conceptual +requirements, and encapsulates a statistical distribution. + +Please note that this documentation should not be used as a substitute +for the +[link math_toolkit.dist.dist_ref reference documentation], and +[link math_toolkit.dist.stat_tut tutorial] of the statistical +distributions. + +In the following table, /d/ is an object of type `DistributionType`, +/cd/ is an object of type `const DistributionType` and /cr/ is an +object of a type convertible to `RealType`. + +[table +[[Expression][Result Type][Notes]] +[[DistributionType::value_type][RealType] + [The real-number type /RealType/ upon which the distribution operates.]] +[[DistributionType::policy_type][RealType] + [The __Policy to use when evaluating functions that depend on this distribution.]] +[[d = cd][Distribution&][Distribution types are assignable.]] +[[Distribution(cd)][Distribution][Distribution types are copy constructible.]] +[[pdf(cd, cr)][RealType][Returns the PDF of the distribution.]] +[[cdf(cd, cr)][RealType][Returns the CDF of the distribution.]] +[[cdf(complement(cd, cr))][RealType] + [Returns the complement of the CDF of the distribution, + the same as: `1-cdf(cd, cr)`]] +[[quantile(cd, cr)][RealType][Returns the quantile of the distribution.]] +[[quantile(complement(cd, cr))][RealType] + [Returns the quantile of the distribution, starting from + the complement of the probability, the same as: `quantile(cd, 1-cr)`]] +[[chf(cd, cr)][RealType][Returns the cumulative hazard function of the distribution.]] +[[hazard(cd, cr)][RealType][Returns the hazard function of the distribution.]] +[[kurtosis(cd)][RealType][Returns the kurtosis of the distribution.]] +[[kurtosis_excess(cd)][RealType][Returns the kurtosis excess of the distribution.]] +[[mean(cd)][RealType][Returns the mean of the distribution.]] +[[mode(cd)][RealType][Returns the mode of the distribution.]] +[[skewness(cd)][RealType][Returns the skewness of the distribution.]] +[[standard_deviation(cd)][RealType][Returns the standard deviation of the distribution.]] +[[variance(cd)][RealType][Returns the variance of the distribution.]] +] + +[endsect] + +[section:archetypes Conceptual Archetypes and Testing] + +There are several concept archetypes available: + +``#include `` + + namespace boost{ + namespace math{ + namespace concepts{ + + class std_real_concept; + + }}} // namespaces + +`std_real_concept` is an archetype for the built-in Real types. + +The main purpose in providing this type is to verify +that standard library functions are found via a using declaration - +bringing those functions into the current scope - and not +just because they happen to be in global scope. + +In order to ensure that a call to say `pow` can be found +either via argument dependent lookup, or failing that then +in the std namespace: all calls to standard library functions +are unqualified, with the std:: versions found via a using declaration +to make them visible in the current scope. Unfortunately it's all +to easy to forget the using declaration, and call the double version of +the function that happens to be in the global scope by mistake. + +For example if the code calls ::pow rather than std::pow, +the code will cleanly compile, but truncation of long doubles to +double will cause a significant loss of precision. +In contrast a template instantiated with std_real_concept will *only* +compile if the all the standard library functions used have +been brought into the current scope with a using declaration. + +There is a test program +[@../../../test/std_real_concept_check.cpp libs/math/test/std_real_concept_check.cpp] +that instantiates every template in this library with type +`std_real_concept` to verify it's usage of standard library functions. + +``#include `` + + namespace boost{ + namespace math{ + namespace concepts{ + + class real_concept; + + }}} // namespaces + +`real_concept` is an archetype for [link math_toolkit.using_udt.concepts +user defined real types], it +declares it's standard library functions in it's own +namespace: these will only be found if they are called unqualified +allowing argument dependent lookup to locate them. In addition +this type is useable at runtime: +this allows code that would not otherwise be exercised by the built-in +floating point types to be tested. There is no std::numeric_limits<> +support for this type, since this is not a conceptual requirement +for [link math_toolkit.using_udt.concepts RealType]'s. + +NTL RR is an example of a type meeting the requirements that this type +models, but note that use of a thin wrapper class is required: refer to +[link math_toolkit.using_udt.use_ntl "Using With NTL - a High-Precision Floating-Point Library"]. + +There is no specific test case for type `real_concept`, instead, since this +type is usable at runtime, each individual test case as well as testing +`float`, `double` and `long double`, also tests `real_concept`. + +``#include `` + + namespace boost{ + namespace math{ + namespace concepts{ + + template + class distribution_archetype; + + template + struct DistributionConcept; + + }}} // namespaces + +The class template `distribution_archetype` is a model of the +[link math_toolkit.using_udt.dist_concept Distribution concept]. + +The class template `DistributionConcept` is a +[@../../../../../libs/concept_check/index.html concept checking class] +for distribution types. + +The test program +[@../../../test/compile_test/distribution_concept_check.cpp distribution_concept_check.cpp] +is responsible for using `DistributionConcept` to verify that all the +distributions in this library conform to the +[link math_toolkit.using_udt.dist_concept Distribution concept]. + +The class template `DistributionConcept` verifies the existence +(but not proper function) of the non-member accessors +required by the [link math_toolkit.using_udt.dist_concept Distribution concept]. +These are checked by calls like + +v = pdf(dist, x); // (Result v is ignored). + +And in addition, those that accept two arguments do the right thing when the +arguments are of different types (the result type is always the same as the +distribution's value_type). (This is implemented by some additional +forwarding-functions in derived_accessors.hpp, so that there is no need for +any code changes. Likewise boilerplate versions of the +hazard\/chf\/coefficient_of_variation functions are implemented in +there too.) + +[endsect][/section:archetypes Conceptual Archetypes and Testing] + + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + + diff --git a/doc/sf_and_dist/contact_info.qbk b/doc/sf_and_dist/contact_info.qbk new file mode 100644 index 000000000..ba8c40e18 --- /dev/null +++ b/doc/sf_and_dist/contact_info.qbk @@ -0,0 +1,25 @@ + +[section:contact Contact Info and Support] + +The main support for this library is via the Boost mailing lists: + +* Use the +[@http://www.boost.org/more/mailing_lists.htm#users boost-user list] +for general support questions. +* Use the +[@http://www.boost.org/more/mailing_lists.htm#main boost-developer list] +for discussion about implementation +and or submission of extensions. + +You can also find JM at john - at - johnmaddock.co.uk and PAB at +pbristow - at - hetp.u-net.com. + +[endsect] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/credits.qbk b/doc/sf_and_dist/credits.qbk new file mode 100644 index 000000000..37661dbcc --- /dev/null +++ b/doc/sf_and_dist/credits.qbk @@ -0,0 +1,48 @@ +[section:credits Credits and Acknowledgements] + +Hubert Holin started the Boost.Math library. The inverse +hyperbolic functions, and the sinus cardinal functions are his. + +John Maddock started this library, the beta, gamma, erf, polynomial, +and factorial functions are his, as is the "Toolkit" section, and many +of the statistical distributions. + +Paul A. Bristow threw down the challenge in +[@http://www2.open-std.org/JTC1/SC22/WG21/docs/papers/2004/n1668.pdf +A Proposal to add Mathematical Functions for Statistics to the C++ +Standard Library] to add the key math functions, especially those essential for +statistics. After JM accepted and solved the difficult problems, +not only numerically, but in full C++ template style, PAB +implemented a few of the statistical distributions. PAB also tirelessly +proof-read everything that JM threw at him (so that all +remaining editorial mistakes are his fault). + +Xiaogang Zhang worked on the Bessel functions and elliptic integrals for his +Google Summer of Code project 2006. + +Professor Nico Temme for advice on the inverse incomplete beta function. + +[@http:www.shoup.net Victor Shoup for NTL], +without which it would have much difficult to +produce high accuracy constants, and especially +the tables of accurate values for testing. + +We are grateful to Joel Guzman for helping us stress-test +his +[@http:www.boost.org/tools/quickbook/index.htm Boost.Quickbook] +program used to generate the html and pdf versions +of this document, adding several new features en route. + +We are also indebted to Matthias Schabel for managing the formal Boost-review +of this library, and to all the reviewers - including Guillaume Melquiond, +Arnaldur Gylfason, John Phillips, Stephan Tolksdorf and Jeff Garland +- for their many helpful comments. + +[endsect][/section:roadmap Roadmap] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/digamma.qbk b/doc/sf_and_dist/digamma.qbk new file mode 100644 index 000000000..c460c63b7 --- /dev/null +++ b/doc/sf_and_dist/digamma.qbk @@ -0,0 +1,144 @@ +[section:digamma Digamma] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` digamma(T z); + + template + ``__sf_result`` digamma(T z, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +Returns the digamma or psi function of /x/. Digamma is defined as the logarithmic +derivative of the gamma function: + +[equation digamma1] + +[$../graphs/digamma.png] + +[optional_policy] + +There is no fully generic version of this function: all the implementations +are tuned to specific accuracy levels, the most precise of which delivers +34-digits of precision. + +The return type of this function is computed using the __arg_pomotion_rules: +the result is of type `double` when T is an integer type, and type T otherwise. + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table +[[Significand Size] [Platform and Compiler] [Random Positive Values] [Values Near The Positive Root] [Values Near Zero] [Negative Values]] +[[53] [Win32 Visual C++ 8] [Peak=0.98 Mean=0.36] [Peak=0.99 Mean=0.5] [Peak=0.95 Mean=0.5] [Peak=214 Mean=16] ] +[[64] [Linux IA32 / GCC] [Peak=1.4 Mean=0.4] [Peak=1.3 Mean=0.45] [Peak=0.98 Mean=0.35] [Peak=180 Mean=13] ] +[[64] [Linux IA64 / GCC] [Peak=0.92 Mean=0.4] [Peak=1.3 Mean=0.45] [Peak=0.98 Mean=0.4] [Peak=180 Mean=13] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=0.9 Mean=0.4] [Peak=1.1 Mean=0.5] [Peak=0.99 Mean=0.4] [Peak=64 Mean=6] ] +] + +As shown above, error rates for positive arguments are generally very low. +For negative arguments there are an infinite number of irrational roots: +relative errors very close to these can be arbitrarily large, although +absolute error will remain very low. + +[h4 Testing] + +There are two sets of tests: spot values are computed using +the online calculator at functions.wolfram.com, while random test values +are generated using the high-precision reference implementation (a +differentiated __lanczos see below). + +[h4 Implementation] + +The implementation is divided up into the following domains: + +For Negative arguments the reflection formula: + + digamma(1-x) = digamma(x) + pi/tan(pi*x); + +is used to make /x/ positive. + +For arguments in the range [0,1] the recurrence relation: + + digamma(x) = digamma(x+1) - 1/x + +is used to shift the evaluation to [1,2]. + +For arguments in the range [1,2] a rational approximation [jm_rationals] is used (see below). + +For arguments in the range [2,BIG] the recurrence relation: + + digamma(x+1) = digamma(x) + 1/x; + +is used to shift the evaluation to the range [1,2]. + +For arguments > BIG the asymptotic expansion: + +[equation digamma2] + +can be used. However, this expansion is divergent after a few terms: +exactly how many terms depends on the size of /x/. Therefore the value +of /BIG/ must be chosen so that the series can be truncated at a term +that is too small to have any effect on the result when evaluated at /BIG/. +Choosing BIG=10 for up to 80-bit reals, and BIG=20 for 128-bit reals allows +the series to truncated after a suitably small number of terms and evaluated +as a polynomial in `1/(x*x)`. + +The rational approximation [jm_rationals] in the range [1,2] is derived as follows. + +First a high precision approximation to digamma was constructed using a 60-term +differentiated __lanczos, the form used is: + +[equation digamma3] + +Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos sum, +and P'(x) and Q'(x) are their first derivatives. The Lanzos part of this +approximation has a theoretical precision of ~100 decimal digits. However, +cancellation in the above sum will reduce that to around `99-(1/y)` digits +if /y/ is the result. This approximation was used to calculate the positive root +of digamma, and was found to agree with the value used by +Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher) +and with the value used by Morris to 35 digits (See TOMS Algorithm 708). + +Likewise a few spot tests agreed with values calculated using +functions.wolfram.com to >40 digits. +That's sufficiently precise to insure that the approximation below is +accurate to double precision. Achieving 128-bit long double precision requires that +the location of the root is known to ~70 digits, and it's not clear whether +the value calculated by this method meets that requirement: the difficulty +lies in independently verifying the value obtained. + +The rational approximation [jm_rationals] was optimised for absolute error using the form: + + digamma(x) = (x - X0)(Y + R(x - 1)); + +Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) is the +rational approximation. Note that since X0 is irrational, we need twice as many +digits in X0 as in x in order to avoid cancellation error during the subtraction +(this assumes that /x/ is an exact value, if it's not then all bets are off). That +means that even when x is the value of the root rounded to the nearest +representable value, the result of digamma(x) ['[*will not be zero]]. + + +[endsect][/section:digamma The Digamma Function] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/dist_algorithms.qbk b/doc/sf_and_dist/dist_algorithms.qbk new file mode 100644 index 000000000..94e2e4abc --- /dev/null +++ b/doc/sf_and_dist/dist_algorithms.qbk @@ -0,0 +1,78 @@ +[section:dist_algorithms Distribution Algorithms] + +[h4 Finding the Location and Scale for Normal and similar distributions] + +Two functions aid finding location and scale of random variable z +to give probability p (given a scale or location). +Only applies to distributions like normal, lognormal, extreme value, Cauchy, (and symmetrical triangular), +that have scale and location properties. + +These functions are useful to predict the mean and/or standard deviation that will be needed to meet a specified minimum weight or maximum dose. + +Complement versions are also provided, both with explicit and implicit (default) policy. + + using boost::math::policies::policy; // May be needed by users defining their own policies. + using boost::math::complement; // Will be needed by users who want to use complements. + +[h4 find_location function] + +``#include `` + + namespace boost{ namespace math{ + + template // explicit error handling policy + typename Dist::value_type find_location( // For example, normal mean. + typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p. + // For example, a nominal minimum acceptable z, so that p * 100 % are > z + typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z. + typename Dist::value_type scale, // scale parameter, for example, normal standard deviation. + const ``__Policy``& pol); + + template // with default policy. + typename Dist::value_type find_location( // For example, normal mean. + typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p. + // For example, a nominal minimum acceptable z, so that p * 100 % are > z + typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z. + typename Dist::value_type scale); // scale parameter, for example, normal standard deviation. + + }} // namespaces + +[h4 find_scale function] + +``#include `` + + namespace boost{ namespace math{ + + template + typename Dist::value_type find_scale( // For example, normal mean. + typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p. + // For example, a nominal minimum acceptable weight z, so that p * 100 % are > z + typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z. + typename Dist::value_type location, // location parameter, for example, normal distribution mean. + const ``__Policy``& pol); + + template // with default policy. + typename Dist::value_type find_scale( // For example, normal mean. + typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p. + // For example, a nominal minimum acceptable z, so that p * 100 % are > z + typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z. + typename Dist::value_type location) // location parameter, for example, normal distribution mean. + }} // namespaces + +All argument must be finite, otherwise __domain_error is called. + +Probability arguments must be [0, 1], otherwise __domain_error is called. + +If the choice of arguments would give a negative scale, __domain_error is called, unless the policy is to ignore, when the negative (impossible) value of scale is returned. + +[link math_toolkit.dist.stat_tut.weg.find_eg Find Mean and standard deviation examples] +gives simple examples of use of both find_scale and find_location, and a longer example finding means and standard deviations of normally distributed weights to meet a specification. + +[endsect] [/section:dist_algorithms dist_algorithms] + +[/ dist_algorithms.qbk + Copyright 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/dist_reference.qbk b/doc/sf_and_dist/dist_reference.qbk new file mode 100644 index 000000000..100bafee0 --- /dev/null +++ b/doc/sf_and_dist/dist_reference.qbk @@ -0,0 +1,128 @@ +[section:dist_ref Statistical Distributions Reference] + +[include distributions/non_members.qbk] + +[section:dists Distributions] + +[include distributions/bernoulli.qbk] +[include distributions/beta.qbk] +[include distributions/binomial.qbk] +[include distributions/cauchy.qbk] +[include distributions/chi_squared.qbk] +[include distributions/exponential.qbk] +[include distributions/extreme_value.qbk] +[include distributions/fisher.qbk] +[include distributions/gamma.qbk] +[include distributions/lognormal.qbk] +[include distributions/negative_binomial.qbk] +[include distributions/normal.qbk] +[include distributions/pareto.qbk] +[include distributions/poisson.qbk] +[include distributions/rayleigh.qbk] +[include distributions/students_t.qbk] +[include distributions/triangular.qbk] +[include distributions/weibull.qbk] +[include distributions/uniform.qbk] + +[endsect][/section:dists Distributions] + +[include dist_algorithms.qbk] + +[endsect][/section:dist_ref Statistical Distributions and Functions Reference] + + +[section:future Extras/Future Directions] + +[h4 Adding Additional Location and Scale Parameters] + +In some modelling applications we require a distribution with a specific +location and scale: +often this equates to a specific mean and standard deviation, although for many +distributions the relationship between these properties and the location and +scale parameters are non-trivial. +See [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm] for more +information. + +The obvious way to handle this is via an adapter template: + + template + class scaled_distribution + { + scaled_distribution( + const Dist dist, + typename Dist::value_type location, + typename Dist::value_type scale = 0); + }; + +Which would then have its own set of overloads for the non-member accessor functions. + +[h4 An "any_distribution" class] + +It would be fairly trivial to add a distribution object that virtualises +the actual type of the distribution, and can therefore hold "any" object +that conforms to the conceptual requirements of a distribution: + + template + class any_distribution + { + public: + template + any_distribution(const Distribution& d); + }; + + // Get the cdf of the underlying distribution: + template + RealType cdf(const any_distribution& d, RealType x); + // etc.... + +Such a class would facilitate the writing of non-template code that can +function with any distribution type. It's not clear yet whether there is a +compelling use case though. Possibly tests for goodness of fit might +provide such a use case: this needs more investigation. + +[h4 Higher Level Hypothesis Tests] + +Higher-level tests roughly corresponding to the +[@http://documents.wolfram.com/mathematica/Add-onsLinks/StandardPackages/Statistics/HypothesisTests.html Mathematica Hypothesis Tests] +package could be added reasonably easily, for example: + + template + typename std::iterator_traits::value_type + test_equal_mean( + InputIterator a, + InputIterator b, + typename std::iterator_traits::value_type expected_mean); + +Returns the probability that the data in the sequence [a,b) has the mean +/expected_mean/. + +[h4 Integration With Statistical Accumulators] + +[@http://boost-sandbox.sourceforge.net/libs/accumulators/doc/html/index.html +Eric Niebler's accumulator framework] - also work in progress - provides the means +to calculate various statistical properties from experimental data. There is an +opportunity to integrate the statistical tests with this framework at some later date: + + // Define an accumulator, all required statistics to calculate the test + // are calculated automatically: + accumulator_set > acc(expected_mean=4); + // Pass our data to the accumulator: + acc = std::for_each(mydata.begin(), mydata.end(), acc); + // Extract the result: + double p = probability(acc); + +[endsect][/section:future Extras Future Directions] + +[/ dist_reference.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + + + + + + diff --git a/doc/sf_and_dist/dist_tutorial.qbk b/doc/sf_and_dist/dist_tutorial.qbk new file mode 100644 index 000000000..c687b127e --- /dev/null +++ b/doc/sf_and_dist/dist_tutorial.qbk @@ -0,0 +1,391 @@ +[/ def names end in distrib to avoid clashes] +[def __binomial_distrib [link math_toolkit.dist.dist_ref.dists.binomial_dist Binomial Distribution]] +[def __chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.chi_squared_dist Chi Squared Distribution]] +[def __normal_distrib [link math_toolkit.dist.dist_ref.dists.normal_dist Normal Distribution]] +[def __F_distrib [link math_toolkit.dist.dist_ref.dists.f_dist Fisher F Distribution]] +[def __students_t_distrib [link math_toolkit.dist.dist_ref.dists.students_t_dist Students t Distribution]] + +[def __handbook [@http://www.itl.nist.gov/div898/handbook/ +NIST/SEMATECH e-Handbook of Statistical Methods.]] + +[section:stat_tut Statistical Distributions Tutorial] +This library is centred around statistical distributions, this tutorial +will give you an overview of what they are, how they can be used, and +provides a few worked examples of applying the library to statistical tests. + +[section:overview Overview] + +[h4 Headers and Namespaces] + +All the code in this library is inside namespace boost::math. + +In order to use a distribution /my_distribution/ you will need to include +either the header or +the "include everything" header: . + +For example, to use the Students-t distribution include either + or + + +[h4 Distributions are Objects] + +Each kind of distribution in this library is a class type. + +[link math_toolkit.policy Policies] provide fine-grained control +of the behaviour of these classes, allowing the user to customise +behaviour such as how errors are handled, or how the quantiles +of discrete distribtions behave. + +[tip If you are familiar with statistics libraries using functions, +and 'Distributions as Objects' seem alien, see +[link math_toolkit.dist.stat_tut.weg.nag_library the comparison to +other statistics libraries.] +] [/tip] + +Making distributions class types does two things: + +* It encapsulates the kind of distribution in the C++ type system; +so, for example, Students-t distributions are always a different C++ type from +Chi-Squared distributions. +* The distribution objects store any parameters associated with the +distribution: for example, the Students-t distribution has a +['degrees of freedom] parameter that controls the shape of the distribution. +This ['degrees of freedom] parameter has to be provided +to the Students-t object when it is constructed. + +Although the distribution classes in this library are templates, there +are typedefs on type /double/ that mostly take the usual name of the +distribution +(except where there is a clash with a function of the same name: beta and gamma, +in which case using the default template arguments - `RealType = double` - +is nearly as convenient). +Probably 95% of uses are covered by these typedefs: + + using namespace boost::math; + + // Construct a students_t distribution with 4 degrees of freedom: + students_t d1(4); + + // Construct a double-precision beta distribution + // with parameters a = 10, b = 20 + beta_distribution<> d2(10, 20); // Note: _distribution<> suffix ! + +If you need to use the distributions with a type other than `double`, +then you can instantiate the template directly: the names of the +templates are the same as the `double` typedef but with `_distribution` +appended, for example: __students_t_distrib or __binomial_distrib: + + // Construct a students_t distribution, of float type, + // with 4 degrees of freedom: + students_t_distribution d3(4); + + // Construct a binomial distribution, of long double type, + // with probability of success 0.3 + // and 20 trials in total: + binomial_distribution d4(20, 0.3); + +The parameters passed to the distributions can be accessed via getter member +functions: + + d1.degrees_of_freedom(); // returns 4.0 + +This is all well and good, but not very useful so far. What we often want +is to be able to calculate the /cumulative distribution functions/ and +/quantiles/ etc for these distributions. + +[h4 Generic operations common to all distributions are non-member functions] + +Want to calculate the PDF (Probability Density Function) of a distribution? +No problem, just use: + + pdf(my_dist, x); // Returns PDF (density) at point x of distribution my_dist. + +Or how about the CDF (Cumulative Distribution Function): + + cdf(my_dist, x); // Returns CDF (integral from -infinity to point x) + // of distribution my_dist. + +And quantiles are just the same: + + quantile(my_dist, p); // Returns the value of the random variable x + // such that cdf(my_dist, x) == p. + +If you're wondering why these aren't member functions, it's to +make the library more easily extensible: if you want to add additional +generic operations - let's say the /n'th moment/ - then all you have to +do is add the appropriate non-member functions, overloaded for each +implemented distribution type. + +[tip + +[*Random numbers that approximate Quantiles of Distributions] + +If you want random numbers that are distributed in a specific way, +for example in a uniform, normal or triangular, +see [@http://www.boost.org/libs/random/ Boost.Random]. + +Whilst in principal there's nothing to prevent you from using the +quantile function to convert a uniformly distributed random +number to another distribution, in practice there are much more +efficient algorithms available that are specific to random number generation. +] [/tip Random numbers that approximate Quantiles of Distributions] + +For example, the binomial distribution has two parameters: +n (the number of trials) and p (the probability of success on one trial). + +The `binomial_distribution` constructor therefore has two parameters: + +`binomial_distribution(RealType n, RealType p);` + +For this distribution the random variate is k: the number of successes observed. +The probability density\/mass function (pdf) is therefore written as ['f(k; n, p)]. + +[note + +[*Random Variates and Distribution Parameters] + +[@http://en.wikipedia.org/wiki/Random_variate Random variates] +and [@http://en.wikipedia.org/wiki/Parameter distribution parameters] +are conventionally distinguished (for example in Wikipedia and Wolfram MathWorld +by placing a semi-colon (or sometimes vertical bar) +after the random variate (whose value you 'choose'), +to separate the variate from the parameter(s) that defines the shape of the distribution. +] [/tip Random Variates and Distribution Parameters] + +As noted above the non-member function `pdf` has one parameter for the distribution object, +and a second for the random variate. So taking our binomial distribution +example, we would write: + +`pdf(binomial_distribution(n, p), k);` + +The distribution (effectively the random variate) is said to be 'supported' over a range that is +[@http://en.wikipedia.org/wiki/Probability_distribution + "the smallest closed set whose complement has probability zero"]. +MathWorld uses the word 'defined' for this range. +Non-mathematicians might say it means the 'interesting' smallest range +of random variate x that has the cdf going from zero to unity. +Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity. +Mathematically, the functions may make sense with an (+ or -) infinite value, +but except for a few special cases (in the Normal and Cauchy distributions) +this implementation limits random variates to finite values from the `max` +to `min` for the `RealType`. +(See [link math_toolkit.backgrounders.implementation.handling_of_floating_point_infinity +Handling of Floating-Point Infinity] for rationale). + +The range of random variate values that is permitted and supported can be +tested by using two functions `range` and `support`. + +[note + +[*Discrete Probability Distributions] + +Note that the [@http://en.wikipedia.org/wiki/Discrete_probability_distribution +discrete distributions], including the binomial, negative binomial, Poisson & Bernoulli, +are all mathematically defined as discrete functions: +that is to say the functions `cdf` and `pdf` are only defined for integral values +of the random variate. + +However, because the method of calculation often uses continuous functions +it is convenient to treat them as if they were continuous functions, +and permit non-integral values of their parameters. + +Users wanting to enforce a strict mathematical model may use `floor` +or `ceil` functions on the random variate prior to calling the distribution +function. + +The quantile functions for these distributions are hard to specify +in a manner that will satisfy everyone all of the time. The default +behaviour is to return an integer result, that has been rounded +/outwards/: that is to say, lower quantiles - where the probablity +is less than 0.5 are rounded down, while upper quantiles - where +the probability is greater than 0.5 - are rounded up. This behaviour +ensures that if an X% quantile is requested, then /at least/ the requested +coverage will be present in the central region, and /no more than/ +the requested coverage will be present in the tails. + +This behaviour can be changed so that the quantile functions are rounded +differently, or return a real-valued result using +[link math_toolkit.policy.pol_overview Policies]. It is strongly +recommended that you read the tutorial +[link math_toolkit.policy.pol_tutorial.understand_dis_quant +Understanding Quantiles of Discrete Distributions] before +using the quantile function on a discrete distribtion. The +[link math_toolkit.policy.pol_ref.discrete_quant_ref reference docs] +describe how to change the rounding policy +for these distributions. + +For similar reasons continuous distributions with parameters like +"degrees of freedom" +that might appear to be integral, are treated as real values +(and are promoted from integer to floating-point if necessary). +In this case however, there are a small number of situations where non-integral +degrees of freedom do have a genuine meaning. +] + +[#complements] +[h4 Complements are supported too] + +Often you don't want the value of the CDF, but its complement, which is +to say `1-p` rather than `p`. You could calculate the CDF and subtract +it from `1`, but if `p` is very close to `1` then cancellation error +will cause you to lose significant digits. In extreme cases, `p` may +actually be equal to `1`, even though the true value of the complement is non-zero. + +[link why_complements See also ['"Why complements?"]] + +In this library, whenever you want to receive a complement, just wrap +all the function arguments in a call to `complement(...)`, for example: + + students_t dist(5); + cout << "CDF at t = 1 is " << cdf(dist, 1.0) << endl; + cout << "Complement of CDF at t = 1 is " << cdf(complement(dist, 1.0)) << endl; + +But wait, now that we have a complement, we have to be able to use it as well. +Any function that accepts a probability as an argument can also accept a complement +by wrapping all of its arguments in a call to `complement(...)`, for example: + + students_t dist(5); + + for(double i = 10; i < 1e10; i *= 10) + { + // Calculate the quantile for a 1 in i chance: + double t = quantile(complement(dist, 1/i)); + // Print it out: + cout << "Quantile of students-t with 5 degrees of freedom\n" + "for a 1 in " << i << " chance is " << t << endl; + } + +[tip + +[*Critical values are just quantiles] + +Some texts talk about quantiles, others about critical values, the basic rule is: + +['Lower critical values] are the same as the quantile. + +['Upper critical values] are the same as the quantile from the complement +of the probability. + +For example, suppose we have a Bernoulli process, giving rise to a binomial +distribution with success ratio 0.1 and 100 trials in total. The +['lower critical value] for a probability of 0.05 is given by: + +`quantile(binomial(100, 0.1), 0.05)` + +and the ['upper critical value] is given by: + +`quantile(complement(binomial(100, 0.1), 0.05))` + +which return 4.82 and 14.63 respectively. +] + +[#why_complements] +[tip + +[*Why bother with complements anyway?] + +It's very tempting to dispense with complements, and simply subtract +the probability from 1 when required. However, consider what happens when +the probability is very close to 1: let's say the probability expressed at +float precision is `0.999999940f`, then `1 - 0.999999940f = 5.96046448e-008`, +but the result is actually accurate to just ['one single bit]: the only +bit that didn't cancel out! + +Or to look at this another way: consider that we want the risk of falsely +rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion, +for a sample size of 10,000. +This gives a probability of 1 - 10[super -9], which is exactly 1 when +calculated at float precision. In this case calculating the quantile from +the complement neatly solves the problem, so for example: + +`quantile(complement(students_t(10000), 1e-9))` + +returns the expected t-statistic `6.00336`, where as: + +`quantile(students_t(10000), 1-1e-9f)` + +raises an overflow error, since it is the same as: + +`quantile(students_t(10000), 1)` + +Which has no finite result. +] + +[h4 Parameters can be calculated] + +Sometimes it's the parameters that define the distribution that you +need to find. Suppose, for example, you have conducted a Students-t test +for equal means and the result is borderline. Maybe your two samples +differ from each other, or maybe they don't; based on the result +of the test you can't be sure. A legitimate question to ask then is +"How many more measurements would I have to take before I would get +an X% probability that the difference is real?" Parameter finders +can answer questions like this, and are necessarily different for +each distribution. They are implemented as static member functions +of the distributions, for example: + + students_t::find_degrees_of_freedom( + 1.3, // difference from true mean to detect + 0.05, // maximum risk of falsely rejecting the null-hypothesis. + 0.1, // maximum risk of falsely failing to reject the null-hypothesis. + 0.13); // sample standard deviation + +Returns the number of degrees of freedom required to obtain a 95% +probability that the observed differences in means is not down to +chance alone. In the case that a borderline Students-t test result +was previously obtained, this can be used to estimate how large the sample size +would have to become before the observed difference was considered +significant. It assumes, of course, that the sample mean and standard +deviation are invariant with sample size. + +[h4 Summary] + +* Distributions are objects, which are constructed from whatever +parameters the distribution may have. +* Member functions allow you to retrieve the parameters of a distribution. +* Generic non-member functions provide access to the properties that +are common to all the distributions (PDF, CDF, quantile etc). +* Complements of probabilities are calculated by wrapping the function's +arguments in a call to `complement(...)`. +* Functions that accept a probability can accept a complement of the +probability as well, by wrapping the function's +arguments in a call to `complement(...)`. +* Static member functions allow the parameters of a distribution +to be found from other information. + +Now that you have the basics, the next section looks at some worked examples. + +[endsect] [/section:overview Overview] + +[section:weg Worked Examples] +[include distributions/distribution_construction.qbk] +[include distributions/students_t_examples.qbk] +[include distributions/chi_squared_examples.qbk] +[include distributions/f_dist_example.qbk] +[include distributions/binomial_example.qbk] +[include distributions/negative_binomial_example.qbk] +[include distributions/normal_example.qbk] +[include distributions/error_handling_example.qbk] +[include distributions/find_location_and_scale.qbk] +[include distributions/nag_library.qbk] +[endsect] [/section:weg Worked Examples] + +[include background.qbk] + +[endsect] [/ section:stat_tut Statistical Distributions Tutorial] + +[/ dist_tutorial.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + + + + + + + + diff --git a/doc/sf_and_dist/distributions/bernoulli.qbk b/doc/sf_and_dist/distributions/bernoulli.qbk new file mode 100644 index 000000000..17200fb41 --- /dev/null +++ b/doc/sf_and_dist/distributions/bernoulli.qbk @@ -0,0 +1,118 @@ +[section:bernoulli_dist Bernoulli Distribution] + +``#include `` + + namespace boost{ namespace math{ + template + class bernoulli_distribution; + + typedef bernoulli_distribution<> bernoulli; + + template + class bernoulli_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + bernoulli_distribution(RealType p); // Constructor. + // Accessor function. + RealType success_fraction() const + // Probability of success (as a fraction). + }; + }} // namespaces + +The Bernoulli distribution is a discrete distribution of the outcome +of a single trial with only two results, 0 (failure) or 1 (success), +with a probability of success p. + +The Bernoulli distribution is the simplest building block +on which other discrete distributions of +sequences of independent Bernoulli trials can be based. + +The Bernoulli is the binomial distribution (k = 1, p) with only one trial. + +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf] +f(0) = 1 - p, f(1) = p. +[@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function] +D(k) = if (k == 0) 1 - p else 1. + +The following graph illustrates how the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf] +varies with the outcome of the single trial: + +[$../graphs/bernoulli_pdf.png] + +and the [@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function] + +[$../graphs/bernoulli_cdf.png] + +[h4 Member Functions] + + bernoulli_distribution(RealType p); + +Constructs a [@http://en.wikipedia.org/wiki/bernoulli_distribution +bernoulli distribution] with success_fraction /p/. + + RealType success_fraction() const + +Returns the /success_fraction/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random variable is 0 and 1, +and the useful supported range is only 0 or 1. + +Outside this range, functions are undefined, or may throw domain_error exception +and make an error message available. + +[h4 Accuracy] + +The Bernoulli distribution is implemented with simple arithmetic operators +and so should have errors within an epsilon or two. + +[h4 Implementation] + +In the following table /p/ is the probability of success and /q = 1-p/. +/k/ is the random variate, either 0 or 1. + +[note The Bernoulli distribution is implemented here as a /strict discrete/ distribution. +If a generalised version, allowing k to be any real, is required then +the binomial distribution with a single trial should be used, for example: + +`binomial_distribution(1, 0.25)` +] + +[table +[[Function][Implementation Notes]] +[[Supported range][{0, 1}]] +[[pdf][Using the relation: pdf = 1 - p for k = 0, else p ]] +[[cdf][Using the relation: cdf = 1 - p for k = 0, else 1]] +[[cdf complement][q = 1 - p]] +[[quantile][if x <= (1-p) 0 else 1]] +[[quantile from the complement][if x <= (1-p) 1 else 0]] +[[mean][p]] +[[variance][p * (1 - p)]] +[[mode][if (p < 0.5) 0 else 1]] +[[skewness][(1 - 2 * p) / sqrt(p * q)]] +[[kurtosis][6 * p * p - 6 * p +1/ p * q]] +[[kurtosis excess][kurtosis -3]] +] + +[h4 References] +* [@http://en.wikipedia.org/wiki/Bernoulli_distribution Wikpedia Bernoulli distribution] +* [@ Weisstein, Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram Web Resource.] + +[endsect][/section:bernoulli_dist bernoulli] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/beta.qbk b/doc/sf_and_dist/distributions/beta.qbk new file mode 100644 index 000000000..6686ea73c --- /dev/null +++ b/doc/sf_and_dist/distributions/beta.qbk @@ -0,0 +1,280 @@ +[section:beta_dist Beta Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class beta_distribution; + + // typedef beta_distribution beta; + // Note that this is deliberately NOT provided, + // to avoid a clash with the function name beta. + + template + class beta_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Constructor from two shape parameters, alpha & beta: + beta_distribution(RealType a, RealType b); + + // Parameter accessors: + RealType alpha() const; + RealType beta() const; + + // Parameter estimators of alpha or beta from mean and variance. + static RealType find_alpha( + RealType mean, // Expected value of mean. + RealType variance); // Expected value of variance. + + static RealType find_beta( + RealType mean, // Expected value of mean. + RealType variance); // Expected value of variance. + + // Parameter estimators from from + // either alpha or beta, and x and probability. + + static RealType find_alpha( + RealType beta, // from beta. + RealType x, // x. + RealType probability); // cdf + + static RealType find_beta( + RealType alpha, // alpha. + RealType x, // probability x. + RealType probability); // probability cdf. + }; + + }} // namespaces + +The class type `beta_distribution` represents a +[@http://en.wikipedia.org/wiki/Beta_distribution beta ] +[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function]. + +The [@http://mathworld.wolfram.com/BetaDistribution.htm beta distribution ] +is used as a [@http://en.wikipedia.org/wiki/Prior_distribution prior distribution] +for binomial proportions in +[@http://mathworld.wolfram.com/BayesianAnalysis.html Bayesian analysis]. + +See also: +[@http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html beta distribution] +and [@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics]. + +How the beta distribution is used for +[@http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt +Bayesian analysis of one parameter models] +is discussed by Jeff Grynaviski. + +The [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] +for the [@http://en.wikipedia.org/wiki/Beta_distribution beta distribution] +defined on the interval \[0,1\] is given by: + +f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta]) + +where B([alpha], [beta]) is the +[@http://en.wikipedia.org/wiki/Beta_function beta function], +implemented in this library as __beta. Division by the beta function +ensures that the pdf is normalized to the range zero to unity. + +The following graph illustrates examples of the pdf for various values +of the shape parameters. Note the [alpha] = [beta] = 2 (blue line) +is dome-shaped, and might be approximated by a symmetrical triangular +distribution. + +[$../graphs/beta_dist.png] + +If [alpha] = [beta] = 1, then it is a __space +[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution], +equal to unity in the entire interval x = 0 to 1. +If [alpha] __space and [beta] __space are < 1, then the pdf is U-shaped. +If [alpha] != [beta], then the shape is asymmetric +and could be approximated by a triangle +whose apex is away from the centre (where x = half). + +[h4 Member Functions] + +[h5 Constructor] + + beta_distribution(RealType alpha, RealType beta); + +Constructs a beta distribution with shape parameters /alpha/ and /beta/. + +Requires alpha,beta > 0,otherwise __domain_error is called. Note that +technically the beta distribution is defined for alpha,beta >= 0, but +it's not clear whether any program can actually make use of that latitude +or how many of the non-member functions can be usefully defined in that case. +Therefore for now, we regard it as an error if alpha or beta is zero. + +For example: + + beta_distribution<> mybeta(2, 5); + +Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow +in the graph above). + +[h5 Parameter Accessors] + + RealType alpha() const; + +Returns the parameter /alpha/ from which this distribution was constructed. + + RealType beta() const; + +Returns the parameter /beta/ from which this distribution was constructed. + +So for example: + + beta_distribution<> mybeta(2, 5); + assert(mybeta.alpha() == 2.); // mybeta.alpha() returns 2 + assert(mybeta.beta() == 5.); // mybeta.beta() returns 5 + +[h4 Parameter Estimators] + +Two pairs of parameter estimators are provided. + +One estimates either [alpha] __space or [beta] __space +from presumed-known mean and variance. + +The other pair estimates either [alpha] __space or [beta] __space from +the cdf and x. + +It is also possible to estimate [alpha] __space and [beta] __space from +'known' mode & quantile. For example, calculators are provided by the +[@http://www.ausvet.com.au/pprev/content.php?page=PPscript +Pooled Prevalence Calculator] and +[@http://www.epi.ucdavis.edu/diagnostictests/betabuster.html Beta Buster] +but this is not yet implemented here. + + static RealType find_alpha( + RealType mean, // Expected value of mean. + RealType variance); // Expected value of variance. + +Returns the unique value of [alpha][space] that corresponds to a +beta distribution with mean /mean/ and variance /variance/. + + static RealType find_beta( + RealType mean, // Expected value of mean. + RealType variance); // Expected value of variance. + +Returns the unique value of [beta][space] that corresponds to a +beta distribution with mean /mean/ and variance /variance/. + + static RealType find_alpha( + RealType beta, // from beta. + RealType x, // x. + RealType probability); // probability cdf + +Returns the value of [alpha][space] that gives: +`cdf(beta_distribution(alpha, beta), x) == probability`. + + static RealType find_beta( + RealType alpha, // alpha. + RealType x, // probability x. + RealType probability); // probability cdf. + +Returns the value of [beta][space] that gives: +`cdf(beta_distribution(alpha, beta), x) == probability`. + +[h4 Non-member Accessor Functions] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The formulae for calculating these are shown in the table below, and at +[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram Mathworld]. + +[h4 Applications] + +The beta distribution can be used to model events constrained +to take place within an interval defined by a minimum and maximum value: +so it is used in project management systems. + +It is also widely used in [@http://en.wikipedia.org/wiki/Bayesian_inference Bayesian statistical inference]. + +[h4 Related distributions] + +The beta distribution with both [alpha] __space and [beta] = 1 follows a +[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution]. + +The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular] +is used when less precise information is available. + +The [@http://en.wikipedia.org/wiki/Binomial_distribution binomial distribution] +is closely related when [alpha] __space and [beta] __space are integers. + +With integer values of [alpha] __space and [beta] __space the distribution B(i, j) is +that of the j-th highest of a sample of i + j + 1 independent random variables +uniformly distributed between 0 and 1. +The cumulative probability from 0 to x is thus +the probability that the j-th highest value is less than x. +Or it is the probability that that at least i of the random variables are less than x, +a probability given by summing over the __binomial_distrib +with its p parameter set to x. + +[h4 Accuracy] + +This distribution is implemented using the +[link math_toolkit.special.sf_beta.beta_function beta functions] __beta and +[link math_toolkit.special.sf_beta.ibeta_function incomplete beta functions] __ibeta and __ibetac; +please refer to these functions for information on accuracy. + +[h4 Implementation] + +In the following table /a/ and /b/ are the parameters [alpha][space] and [beta], +/x/ is the random variable, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf] + [f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta]) + + Implemented using __ibeta_derivative(a, b, x).]] + +[[cdf][Using the incomplete beta function __ibeta(a, b, x)]] +[[cdf complement][__ibetac(a, b, x)]] +[[quantile][Using the inverse incomplete beta function __ibeta_inv(a, b, p)]] +[[quantile from the complement][__ibetac_inv(a, b, q)]] +[[mean][`a/(a+b)`]] +[[variance][`a * b / (a+b)^2 * (a + b + 1)`]] +[[mode][`(a-1) / (a + b - 2)`]] +[[skewness][`2 (b-a) sqrt(a+b+1)/(a+b+2) * sqrt(a * b)`]] +[[kurtosis excess][ [equation beta_dist_kurtosis] ]] +[[kurtosis][`kurtosis + 3`]] +[[parameter estimation][ ]] +[[alpha + + from mean and variance][`mean * (( (mean * (1 - mean)) / variance)- 1)`]] +[[beta + + from mean and variance][`(1 - mean) * (((mean * (1 - mean)) /variance)-1)`]] +[[The member functions `find_alpha` and `find_beta` + + from cdf and probability x + + and *either* `alpha` or `beta`] + [Implemented in terms of the inverse incomplete beta functions + +__ibeta_inva, and __ibeta_invb respectively.]] +[[`find_alpha`][`ibeta_inva(beta, x, probability)`]] +[[`find_beta`][`ibeta_invb(alpha, x, probability)`]] +] + +[h4 References] + +[@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia Beta distribution] + +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm NIST Exploratory Data Analysis] + +[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld] + +[endsect][/section:beta_dist beta] + +[/ beta.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/distributions/binomial.qbk b/doc/sf_and_dist/distributions/binomial.qbk new file mode 100644 index 000000000..cad52611d --- /dev/null +++ b/doc/sf_and_dist/distributions/binomial.qbk @@ -0,0 +1,404 @@ +[section:binomial_dist Binomial Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class binomial_distribution; + + typedef binomial_distribution<> binomial; + + template + class binomial_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + static const ``['unspecified-type]`` cloppper_pearson_exact_interval; + static const ``['unspecified-type]`` jeffreys_prior_interval; + + // construct: + binomial_distribution(RealType n, RealType p); + + // parameter access:: + RealType success_fraction() const; + RealType trials() const; + + // Bounds on success fraction: + static RealType find_lower_bound_on_p( + RealType trials, + RealType successes, + RealType probability, + ``['unspecified-type]`` method = clopper_pearson_exact_interval); + static RealType find_upper_bound_on_p( + RealType trials, + RealType successes, + RealType probability, + ``['unspecified-type]`` method = clopper_pearson_exact_interval); + + // estimate min/max number of trials: + static RealType find_minimum_number_of_trials( + RealType k, // number of events + RealType p, // success fraction + RealType alpha); // risk level + + static RealType find_maximum_number_of_trials( + RealType k, // number of events + RealType p, // success fraction + RealType alpha); // risk level + }; + + }} // namespaces + +The class type `binomial_distribution` represents a +[@http://mathworld.wolfram.com/BinomialDistribution.html binomial distribution]: +it is used when there are exactly two mutually +exclusive outcomes of a trial. These outcomes are labelled +"success" and "failure". The +[@ binomial distribution] is used to obtain +the probability of observing k successes in N trials, with the +probability of success on a single trial denoted by p. The +binomial distribution assumes that p is fixed for all trials. + +[note The random variable for the binomial distribution is the number of successes, +(the number of trials is a fixed property of the distribution) +whereas for the negative binomial, +the random variable is the number of trials, for a fixed number of successes.] + +The PDF for the binomial distribution is given by: + +[equation binomial_ref2] + +The following two graphs illustrate how the PDF changes depending +upon the distributions parameters, first we'll keep the success +fraction /p/ fixed at 0.5, and vary the sample size: + +[$../graphs/binomial_pdf_1.png] + +Alternatively, we can keep the sample size fixed at N=20 and +vary the success fraction /p/: + +[$../graphs/binomial_pdf_2.png] + +[discrete_quantile_warning Binomial] + +[h4 Member Functions] + +[h5 Construct] + + binomial_distribution(RealType n, RealType p); + +Constructor: /n/ is the total number of trials, /p/ is the +probability of success of a single trial. + +Requires `0 <= p <= 1`, and `n >= 0`, otherwise calls __domain_error. + +[h5 Accessors] + + RealType success_fraction() const; + +Returns the parameter /p/ from which this distribution was constructed. + + RealType trials() const; + +Returns the parameter /n/ from which this distribution was constructed. + +[h5 Lower Bound on the Success Fraction] + + static RealType find_lower_bound_on_p( + RealType trials, + RealType successes, + RealType alpha, + ``['unspecified-type]`` method = clopper_pearson_exact_interval); + +Returns a lower bound on the success fraction: + +[variablelist +[[trials][The total number of trials conducted.]] +[[successes][The number of successes that occurred.]] +[[alpha][The largest acceptable probability that the true value of + the success fraction is [*less than] the value returned.]] +[[method][An optional parameter that specifies the method to be used + to compute the interval (See below).]] +] + +For example, if you observe /k/ successes from /n/ trials the +best estimate for the success fraction is simply ['k/n], but if you +want to be 95% sure that the true value is [*greater than] some value, +['p[sub min]], then: + + p``[sub min]`` = binomial_distribution::find_lower_bound_on_p( + n, k, 0.05); + +[link math_toolkit.dist.stat_tut.weg.binom_eg.binom_conf See worked example.] + +There are currently two possible values available for the /method/ +optional parameter: /clopper_pearson_exact_interval/ +or /jeffreys_prior_interval/. These constants are both members of +class template `binomial_distribution`, so usage is for example: + + p = binomial_distribution::find_lower_bound_on_p( + n, k, 0.05, binomial_distribution::jeffreys_prior_interval); + +The default method if this parameter is not specified is the Clopper Pearson +"exact" interval. This produces an interval that guarantees at least +`100(1-alpha)%` coverage, but which is known to be overly conservative, +sometimes producing intervals with much greater than the requested coverage. + +The alternative calculation method produces a non-informative +Jeffreys Prior interval. It produces `100(1-alpha)%` coverage only +['in the average case], though is typically very close to the requested +coverage level. It is one of the main methods of calculation recommended +in the review by Brown, Cai and DasGupta. + +Please note that the "textbook" calculation method using +a normal approximation (the Wald interval) is deliberately +not provided: it is known to produce consistently poor results, +even when the sample size is surprisingly large. +Refer to Brown, Cai and DasGupta for a full explanation. Many other methods +of calculation are available, and may be more appropriate for specific +situations. Unfortunately there appears to be no consensus amongst +statisticians as to which is "best": refer to the discussion at the end of +Brown, Cai and DasGupta for examples. + +The two methods provided here were chosen principally because they +can be used for both one and two sided intervals. +See also: + +Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), +Interval Estimation for a Binomial Proportion, +Statistical Science, Vol. 16, No. 2, 101-133. + +T. Tony Cai (2005), +One-sided confidence intervals in discrete distributions, +Journal of Statistical Planning and Inference 131, 63-88. + +Agresti, A. and Coull, B. A. (1998). Approximate is better than +"exact" for interval estimation of binomial proportions. Amer. +Statist. 52 119-126. + +Clopper, C. J. and Pearson, E. S. (1934). The use of confidence +or fiducial limits illustrated in the case of the binomial. +Biometrika 26 404-413. + +[h5 Upper Bound on the Success Fraction] + + static RealType find_upper_bound_on_p( + RealType trials, + RealType successes, + RealType alpha, + ``['unspecified-type]`` method = clopper_pearson_exact_interval); + +Returns an upper bound on the success fraction: + +[variablelist +[[trials][The total number of trials conducted.]] +[[successes][The number of successes that occurred.]] +[[alpha][The largest acceptable probability that the true value of + the success fraction is [*greater than] the value returned.]] +[[method][An optional parameter that specifies the method to be used + to compute the interval. Refer to the documentation for + `find_upper_bound_on_p` above for the meaning of the + method options.]] +] + +For example, if you observe /k/ successes from /n/ trials the +best estimate for the success fraction is simply ['k/n], but if you +want to be 95% sure that the true value is [*less than] some value, +['p[sub max]], then: + + p``[sub max]`` = binomial_distribution::find_upper_bound_on_p( + n, k, 0.05); + +[link math_toolkit.dist.stat_tut.weg.binom_eg.binom_conf See worked example.] + +[note +In order to obtain a two sided bound on the success fraction, you +call both `find_lower_bound_on_p` *and* `find_upper_bound_on_p` +each with the same arguments. + +If the desired risk level +that the true success fraction lies outside the bounds is [alpha], +then you pass [alpha]/2 to these functions. + +So for example a two sided 95% confidence interval would be obtained +by passing [alpha] = 0.025 to each of the functions. + +[link math_toolkit.dist.stat_tut.weg.binom_eg.binom_conf See worked example.] +] + + +[h5 Estimating the Number of Trials Required for a Certain Number of Successes] + + static RealType find_minimum_number_of_trials( + RealType k, // number of events + RealType p, // success fraction + RealType alpha); // probability threshold + +This function estimates the minimum number of trials required to ensure that +more than k events is observed with a level of risk /alpha/ that k or +fewer events occur. + +[variablelist +[[k][The number of success observed.]] +[[p][The probability of success for each trial.]] +[[alpha][The maximum acceptable probability that k events or fewer will be observed.]] +] + +For example: + + binomial_distribution::find_number_of_trials(10, 0.5, 0.05); + +Returns the smallest number of trials we must conduct to be 95% sure +of seeing 10 events that occur with frequency one half. + +[h5 Estimating the Maximum Number of Trials to Ensure no more than a Certain Number of Successes] + + static RealType find_maximum_number_of_trials( + RealType k, // number of events + RealType p, // success fraction + RealType alpha); // probability threshold + +This function estimates the maximum number of trials we can conduct +to ensure that k successes or fewer are observed, with a risk /alpha/ +that more than k occur. + +[variablelist +[[k][The number of success observed.]] +[[p][The probability of success for each trial.]] +[[alpha][The maximum acceptable probability that more than k events will be observed.]] +] + +For example: + + binomial_distribution::find_maximum_number_of_trials(0, 1e-6, 0.05); + +Returns the largest number of trials we can conduct and still be 95% certain +of not observing any events that occur with one in a million frequency. +This is typically used in failure analysis. + +[link math_toolkit.dist.stat_tut.weg.binom_eg.binom_size_eg See Worked Example.] + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain for the random variable /k/ is `0 <= k <= N`, otherwise a +__domain_error is returned. + +It's worth taking a moment to define what these accessors actually mean in +the context of this distribution: + +[table Meaning of the non-member accessors +[[Function][Meaning]] +[[__pdf] + [The probability of obtaining [*exactly k successes] from n trials + with success fraction p. For example: + +`pdf(binomial(n, p), k)`]] +[[__cdf] + [The probability of obtaining [*k successes or fewer] from n trials + with success fraction p. For example: + +`cdf(binomial(n, p), k)`]] +[[__ccdf] + [The probability of obtaining [*more than k successes] from n trials + with success fraction p. For example: + +`cdf(complement(binomial(n, p), k))`]] +[[__quantile] + [The [*greatest] number of successes that may be observed from n trials + with success fraction p, at probability P. Note that the value returned + is a real-number, and not an integer. Depending on the use case you may + want to take either the floor or ceiling of the result. For example: + +`quantile(binomial(n, p), P)`]] +[[__quantile_c] + [The [*smallest] number of successes that may be observed from n trials + with success fraction p, at probability P. Note that the value returned + is a real-number, and not an integer. Depending on the use case you may + want to take either the floor or ceiling of the result. For example: + +`quantile(complement(binomial(n, p), P))``]] +] + +[h4 Examples] + +Various [link math_toolkit.dist.stat_tut.weg.binom_eg worked examples] +are available illustrating the use of the binomial distribution. + +[h4 Accuracy] + +This distribution is implemented using the +incomplete beta functions __ibeta and __ibetac, +please refer to these functions for information on accuracy. + +[h4 Implementation] + +In the following table /p/ is the probability that one trial will +be successful (the success fraction), /n/ is the number of trials, +/k/ is the number of successes, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Implementation is in terms of __ibeta_derivative: if [sub n]C[sub k ] is the binomial + coefficient of a and b, then we have: + +[equation binomial_ref1] + +Which can be evaluated as `ibeta_derivative(k+1, n-k+1, p) / (n+1)` + +The function __ibeta_derivative is used here, since it has already + been optimised for the lowest possible error - indeed this is really + just a thin wrapper around part of the internals of the incomplete + beta function. + +There are also various special cases: refer to the code for details. + ]] +[[cdf][Using the relation: + +`` +p = I[sub 1-p](n - k, k + 1) + = 1 - I[sub p](k + 1, n - k) + = __ibetac(k + 1, n - k, p)`` + +There are also various special cases: refer to the code for details. +]] +[[cdf complement][Using the relation: q = __ibeta(k + 1, n - k, p) + +There are also various special cases: refer to the code for details. ]] +[[quantile][Since the cdf is non-linear in variate /k/ none of the inverse + incomplete beta functions can be used here. Instead the quantile + is found numerically using a derivative free method + ([link math_toolkit.toolkit.internals1.roots2 TOMS Algorithm 748]).]] +[[quantile from the complement][Found numerically as above.]] +[[mean][ `p * n` ]] +[[variance][ `p * n * (1-p)` ]] +[[mode][`floor(p * (n + 1))`]] +[[skewness][`(1 - 2 * p) / sqrt(n * p * (1 - p))`]] +[[kurtosis][`3 - (6 / n) + (1 / (n * p * (1 - p)))`]] +[[kurtosis excess][`(1 - 6 * p * q) / (n * p * q)`]] +[[parameter estimation][The member functions `find_upper_bound_on_p` + `find_lower_bound_on_p` and `find_number_of_trials` are + implemented in terms of the inverse incomplete beta functions + __ibetac_inv, __ibeta_inv, and __ibetac_invb respectively]] +] + +[h4 References] + +* [@http://mathworld.wolfram.com/BinomialDistribution.html Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource]. +* [@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia binomial distribution]. +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm NIST Explorary Data Analysis]. + +[endsect][/section:binomial_dist Binomial] + +[/ binomial.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/distributions/binomial_example.qbk b/doc/sf_and_dist/distributions/binomial_example.qbk new file mode 100644 index 000000000..41a462ab1 --- /dev/null +++ b/doc/sf_and_dist/distributions/binomial_example.qbk @@ -0,0 +1,332 @@ +[section:binom_eg Binomial Distribution Examples] + +See also the reference documentation for the __binomial_distrib. + +[section:binomial_coinflip_example Binomial Coin-Flipping Example] + +[import ../../../example/binomial_coinflip_example.cpp] +[binomial_coinflip_example1] + +See [@../../../example/binomial_coinflip_example.cpp binomial_coinflip_example.cpp] +for full source code, the program output looks like this: + +[binomial_coinflip_example_output] + +[endsect] [/section:binomial_coinflip_example Binomial coinflip example] + +[section:binomial_quiz_example Binomial Quiz Example] + +[import ../../../example/binomial_quiz_example.cpp] +[binomial_quiz_example1] +[binomial_quiz_example2] +[discrete_quantile_real] + +See [@../../../example/binomial_quiz_example.cpp binomial_quiz_example.cpp] +for full source code and output. + +[endsect] [/section:binomial_coinflip_quiz Binomial Coin-Flipping example] + +[section:binom_conf Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution] + +Imagine you have a process that follows a binomial distribution: for each +trial conducted, an event either occurs or does it does not, referred +to as "successes" and "failures". If, by experiment, you want to measure the +frequency with which successes occur, the best estimate is given simply +by /k/ \/ /N/, for /k/ successes out of /N/ trials. However our confidence in that +estimate will be shaped by how many trials were conducted, and how many successes +were observed. The static member functions +`binomial_distribution<>::find_lower_bound_on_p` and +`binomial_distribution<>::find_upper_bound_on_p` allow you to calculate +the confidence intervals for your estimate of the occurrence frequency. + +The sample program [@../../../example/binomial_confidence_limits.cpp +binomial_confidence_limits.cpp] illustrates their use. It begins by defining +a procedure that will print a table of confidence limits for various degrees +of certainty: + + #include + #include + #include + + void confidence_limits_on_frequency(unsigned trials, unsigned successes) + { + // + // trials = Total number of trials. + // successes = Total number of observed successes. + // + // Calculate confidence limits for an observed + // frequency of occurrence that follows a binomial + // distribution. + // + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "___________________________________________\n" + "2-Sided Confidence Limits For Success Ratio\n" + "___________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << trials << "\n"; + cout << setw(40) << left << "Number of successes" << "= " << successes << "\n"; + cout << setw(40) << left << "Sample frequency of occurrence" << "= " << double(successes) / trials << "\n"; + +The procedure now defines a table of significance levels: these are the +probabilities that the true occurrence frequency lies outside the calculated +interval: + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +Some pretty printing of the table header follows: + + cout << "\n\n" + "_______________________________________________________________________\n" + "Confidence Lower CP Upper CP Lower JP Upper JP\n" + " Value (%) Limit Limit Limit Limit\n" + "_______________________________________________________________________\n"; + + +And now for the important part - the intervals themselves - for each +value of /alpha/, we call `find_lower_bound_on_p` and +`find_lower_upper_on_p` to obtain lower and upper bounds +respectively. Note that since we are calculating a two-sided interval, +we must divide the value of alpha in two. + +Please note that calculating two separate /single sided bounds/, each with risk +level [alpha][space]is not the same thing as calculating a two sided interval. +Had we calculate two single-sided intervals each with a risk +that the true value is outside the interval of [alpha], then: + +* The risk that it is less than the lower bound is [alpha]. + +and + +* The risk that it is greater than the upper bound is also [alpha]. + +So the risk it is outside *upper or lower bound*, is *twice* alpha, and the +probability that it is inside the bounds is therefore not nearly as high as +one might have thought. This is why [alpha]/2 must be used in +the calculations below. + +In contrast, had we been calculating a +single-sided interval, for example: ['"Calculate a lower bound so that we are P% +sure that the true occurrence frequency is greater than some value"] +then we would *not* have divided by two. + +Finally note that `binomial_distribution` provides a choice of two +methods for the calculation, we print out the results from both +methods in this example: + + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // Calculate Clopper Pearson bounds: + double l = binomial_distribution<>::find_lower_bound_on_p( + trials, successes, alpha[i]/2); + double u = binomial_distribution<>::find_upper_bound_on_p( + trials, successes, alpha[i]/2); + // Print Clopper Pearson Limits: + cout << fixed << setprecision(5) << setw(15) << right << l; + cout << fixed << setprecision(5) << setw(15) << right << u; + // Calculate Jeffreys Prior Bounds: + l = binomial_distribution<>::find_lower_bound_on_p( + trials, successes, alpha[i]/2, + binomial_distribution<>::jeffreys_prior_interval); + u = binomial_distribution<>::find_upper_bound_on_p( + trials, successes, alpha[i]/2, + binomial_distribution<>::jeffreys_prior_interval); + // Print Jeffreys Prior Limits: + cout << fixed << setprecision(5) << setw(15) << right << l; + cout << fixed << setprecision(5) << setw(15) << right << u << std::endl; + } + cout << endl; + } + +And that's all there is to it. Let's see some sample output for a 2 in 10 +success ratio, first for 20 trials: + +[pre'''___________________________________________ +2-Sided Confidence Limits For Success Ratio +___________________________________________ + +Number of Observations = 20 +Number of successes = 4 +Sample frequency of occurrence = 0.2 + + +_______________________________________________________________________ +Confidence Lower CP Upper CP Lower JP Upper JP + Value (%) Limit Limit Limit Limit +_______________________________________________________________________ + 50.000 0.12840 0.29588 0.14974 0.26916 + 75.000 0.09775 0.34633 0.11653 0.31861 + 90.000 0.07135 0.40103 0.08734 0.37274 + 95.000 0.05733 0.43661 0.07152 0.40823 + 99.000 0.03576 0.50661 0.04655 0.47859 + 99.900 0.01905 0.58632 0.02634 0.55960 + 99.990 0.01042 0.64997 0.01530 0.62495 + 99.999 0.00577 0.70216 0.00901 0.67897 +'''] + +As you can see, even at the 95% confidence level the bounds are +really quite wide (this example is chosen to be easily compared to the one +in the __handbook +[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm +here]). Note also that the Clopper-Pearson calculation method (CP above) +produces quite noticeably more pessimistic estimates than the Jeffreys Prior +method (JP above). + + +Compare that with the program output for +2000 trials: + +[pre'''___________________________________________ +2-Sided Confidence Limits For Success Ratio +___________________________________________ + +Number of Observations = 2000 +Number of successes = 400 +Sample frequency of occurrence = 0.2000000 + + +_______________________________________________________________________ +Confidence Lower CP Upper CP Lower JP Upper JP + Value (%) Limit Limit Limit Limit +_______________________________________________________________________ + 50.000 0.19382 0.20638 0.19406 0.20613 + 75.000 0.18965 0.21072 0.18990 0.21047 + 90.000 0.18537 0.21528 0.18561 0.21503 + 95.000 0.18267 0.21821 0.18291 0.21796 + 99.000 0.17745 0.22400 0.17769 0.22374 + 99.900 0.17150 0.23079 0.17173 0.23053 + 99.990 0.16658 0.23657 0.16681 0.23631 + 99.999 0.16233 0.24169 0.16256 0.24143 +'''] + +Now even when the confidence level is very high, the limits are really +quite close to the experimentally calculated value of 0.2. Furthermore +the difference between the two calculation methods is now really quite small. + +[endsect] + +[section:binom_size_eg Estimating Sample Sizes for a Binomial Distribution.] + +Imagine you have a critical component that you know will fail in 1 in +N "uses" (for some suitable definition of "use"). You may want to schedule +routine replacement of the component so that its chance of failure between +routine replacements is less than P%. If the failures follow a binomial +distribution (each time the component is "used" it either fails or does not) +then the static member function `binomial_distibution<>::find_maximum_number_of_trials` +can be used to estimate the maximum number of "uses" of that component for some +acceptable risk level /alpha/. + +The example program +[@../../../example/binomial_sample_sizes.cpp binomial_sample_sizes.cpp] +demonstrates its usage. It centres on a routine that prints out +a table of maximum sample sizes for various probability thresholds: + + void find_max_sample_size( + double p, // success ratio. + unsigned successes) // Total number of observed successes permitted. + { + +The routine then declares a table of probability thresholds: these are the +maximum acceptable probability that /successes/ or fewer events will be +observed. In our example, /successes/ will be always zero, since we want +no component failures, but in other situations non-zero values may well +make sense. + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +Much of the rest of the program is pretty-printing, the important part +is in the calculation of maximum number of permitted trials for each +value of alpha: + + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate trials: + double t = binomial::find_maximum_number_of_trials( + successes, p, alpha[i]); + t = floor(t); + // Print Trials: + cout << fixed << setprecision(5) << setw(15) << right << t << endl; + } + +Note that since we're +calculating the maximum number of trials permitted, we'll err on the safe +side and take the floor of the result. Had we been calculating the +/minimum/ number of trials required to observe a certain number of /successes/ +using `find_minimum_number_of_trials` we would have taken the ceiling instead. + +We'll finish off by looking at some sample output, firstly for +a 1 in 1000 chance of component failure with each use: + +[pre +'''________________________ +Maximum Number of Trials +________________________ + +Success ratio = 0.001 +Maximum Number of "successes" permitted = 0 + + +____________________________ +Confidence Max Number + Value (%) Of Trials +____________________________ + 50.000 692 + 75.000 287 + 90.000 105 + 95.000 51 + 99.000 10 + 99.900 0 + 99.990 0 + 99.999 0''' +] + +So 51 "uses" of the component would yield a 95% chance that no +component failures would be observed. + +Compare that with a 1 in 1 million chance of component failure: + +[pre''' +________________________ +Maximum Number of Trials +________________________ + +Success ratio = 0.0000010 +Maximum Number of "successes" permitted = 0 + + +____________________________ +Confidence Max Number + Value (%) Of Trials +____________________________ + 50.000 693146 + 75.000 287681 + 90.000 105360 + 95.000 51293 + 99.000 10050 + 99.900 1000 + 99.990 100 + 99.999 10''' +] + +In this case, even 1000 uses of the component would still yield a +less than 1 in 1000 chance of observing a component failure +(i.e. a 99.9% chance of no failure). + +[endsect] [/section:binom_size_eg Estimating Sample Sizes for a Binomial Distribution.] + +[endsect][/section:binom_eg Binomial Distribution] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/cauchy.qbk b/doc/sf_and_dist/distributions/cauchy.qbk new file mode 100644 index 000000000..34ff68dd1 --- /dev/null +++ b/doc/sf_and_dist/distributions/cauchy.qbk @@ -0,0 +1,154 @@ +[section:cauchy_dist Cauchy-Lorentz Distribution] + +``#include `` + + template + class cauchy_distribution; + + typedef cauchy_distribution<> cauchy; + + template + class cauchy_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + cauchy_distribution(RealType location = 0, RealType scale = 1); + + RealType location()const; + RealType scale()const; + }; + +The [@http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy-Lorentz distribution] +is named after Augustin Cauchy and Hendrik Lorentz. +It is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution] +with [@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function PDF] +given by: + +[equation cauchy_ref1] + +The location parameter x[sub 0][space] is the location of the +peak of the distribution (the mode of the distribution), +while the scale parameter [gamma][space] specifies half the width +of the PDF at half the maximum height. If the location is +zero, and the scale 1, then the result is a standard Cauchy +distribution. + +The distribution is important in physics as it is the solution +to the differential equation describing forced resonance, +while in spectroscopy it is the description of the line shape +of spectral lines. + +The following graph shows how the distributions moves as the +location parameter changes: + +[$../graphs/cauchy1.png] + +While the following graph shows how the shape (scale) parameter alters +the distribution: + +[$../graphs/cauchy2.png] + +[h4 Member Functions] + + cauchy_distribution(RealType location = 0, RealType scale = 1); + +Constructs a Cauchy distribution, with location parameter /location/ +and scale parameter /scale/. When these parameters take their default +values (location = 0, scale = 1) +then the result is a Standard Cauchy Distribution. + +Requires scale > 0, otherwise calls __domain_error. + + RealType location()const; + +Returns the location parameter of the distribution. + + RealType scale()const; + +Returns the scale parameter of the distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +Note however that the Cauchy distribution does not have a mean, +standard deviation, etc. See __math_undefined +[/link math_toolkit.policy.pol_ref.assert_undefined mathematically undefined function] +to control whether these should fail to compile with a BOOST_STATIC_ASSERTION_FAILURE, +which is the default. + +Alternately, the functions __mean, __sd, +__variance, __skewness, __kurtosis and __kurtosis_excess will all +return a __domain_error if called. + +The domain of the random variable is \[-[max_value], +[min_value]\]. + +[h4 Accuracy] + +The Cauchy distribution is implemented in terms of the +standard library `tan` and `atan` functions, +and as such should have very low error rates. + +[h4 Implementation] + +[def __x0 x[sub 0 ]] + +In the following table __x0 is the location parameter of the distribution, +[gamma][space] is its scale parameter, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = 1 / ([pi] * [gamma] * (1 + ((x - __x0) / [gamma])[super 2]) ]] +[[cdf and its complement][ +The cdf is normally given by: + +p = 0.5 + atan(x)/[pi] + +But that suffers from cancellation error as x -> -[infin]. +So recall that for `x < 0`: + +atan(x) = -[pi]/2 - atan(1/x) + +Substituting into the above we get: + +p = -atan(1/x) ; x < 0 + +So the procedure is to calculate the cdf for -fabs(x) +using the above formula. Note that to factor in the location and scale +parameters you must substitute (x - __x0) / [gamma][space] for x in the above. + +This procedure yields the smaller of /p/ and /q/, so the result +may need subtracting from 1 depending on whether we want the complement +or not, and whether /x/ is less than __x0 or not. +]] +[[quantile][The same procedure is used irrespective of whether we're starting + from the probability or it's complement. First the argument /p/ is + reduced to the range \[-0.5, 0.5\], then the relation + +x = __x0 [plusminus] [gamma][space] / tan([pi] * p) + +is used to obtain the result. Whether we're adding + or subtracting from __x0 is determined by whether we're + starting from the complement or not.]] +[[mode][The location parameter.]] +] + +[h4 References] + +* [@http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy-Lorentz distribution] +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm NIST Exploratory Data Analysis] +* [@http://mathworld.wolfram.com/CauchyDistribution.html Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource.] + +[endsect][/section:cauchy_dist Cauchi] + +[/ cauchy.qbk + Copyright 2006, 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/distributions/chi_squared.qbk b/doc/sf_and_dist/distributions/chi_squared.qbk new file mode 100644 index 000000000..c4ff4c278 --- /dev/null +++ b/doc/sf_and_dist/distributions/chi_squared.qbk @@ -0,0 +1,156 @@ +[section:chi_squared_dist Chi Squared Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class chi_squared_distribution; + + typedef chi_squared_distribution<> chi_squared; + + template + class chi_squared_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + // Constructor: + chi_squared_distribution(RealType i); + + // Accessor to parameter: + RealType degrees_of_freedom()const; + + // Parameter estimation: + static RealType find_degrees_of_freedom( + RealType difference_from_mean, + RealType alpha, + RealType beta, + RealType sd, + RealType hint = 100); + }; + + }} // namespaces + +The Chi-Squared distribution is one of the most widely used distributions +in statistical tests. If [chi][sub i][space] are [nu][space] +independent, normally distributed +random variables with means [mu][sub i][space] and variances [sigma][sub i][super 2], +then the random variable: + +[equation chi_squ_ref1] + +is distributed according to the Chi-Squared distribution. + +The Chi-Squared distribution is a special case of the gamma distribution +and has a single parameter [nu][space] that specifies the number of degrees of +freedom. The following graph illustrates how the distribution changes +for different values of [nu]: + +[$../graphs/chi_square.png] + +[h4 Member Functions] + + chi_squared_distribution(RealType v); + +Constructs a Chi-Squared distribution with /v/ degrees of freedom. + +Requires v > 0, otherwise calls __domain_error. + + RealType degrees_of_freedom()const; + +Returns the parameter /v/ from which this object was constructed. + + static RealType find_degrees_of_freedom( + RealType difference_from_variance, + RealType alpha, + RealType beta, + RealType variance, + RealType hint = 100); + +Estimates the sample size required to detect a difference from a nominal +variance in a Chi-Squared test for equal standard deviations. + +[variablelist +[[difference_from_variance][The difference from the assumed nominal variance + that is to be detected: Note that the sign of this value is critical, see below.]] +[[alpha][The maximum acceptable risk of rejecting the null hypothesis when it is + in fact true.]] +[[beta][The maximum acceptable risk of falsely failing to reject the null hypothesis.]] +[[variance][The nominal variance being tested against.]] +[[hint][An optional hint on where to start looking for a result: the current sample + size would be a good choice.]] +] + +Note that this calculation works with /variances/ and not /standard deviations/. + +The sign of the parameter /difference_from_variance/ is important: the Chi +Squared distribution is asymmetric, and the caller must decide in advance +whether they are testing for a variance greater than a nominal value (positive +/difference_from_variance/) or testing for a variance less than a nominal value +(negative /difference_from_variance/). If the latter, then obviously it is +a requirement that `variance + difference_from_variance > 0`, since no sample +can have a negative variance! + +This procedure uses the method in Diamond, W. J. (1989). +Practical Experiment Designs, Van-Nostrand Reinhold, New York. + +See also section on Sample sizes required in +[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm the NIST Engineering Statistics Handbook, Section 7.2.3.2]. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, +[infin]\]. + +[h4 Examples] + +Various [link math_toolkit.dist.stat_tut.weg.cs_eg worked examples] +are available illustrating the use of the Chi Squared Distribution. + +[h4 Accuracy] + +The Chi-Squared distribution is implemented in terms of the +[link math_toolkit.special.sf_gamma.igamma incomplete gamma functions]: +please refer to the accuracy data for those functions. + +[h4 Implementation] + +In the following table /v/ is the number of degrees of freedom of the distribution, +/x/ is the random variate, /p/ is the probability, and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = __gamma_p_derivative(v / 2, x / 2) / 2 ]] +[[cdf][Using the relation: p = __gamma_p(v / 2, x / 2) ]] +[[cdf complement][Using the relation: q = __gamma_q(v / 2, x / 2) ]] +[[quantile][Using the relation: x = 2 * __gamma_p_inv(v / 2, p) ]] +[[quantile from the complement][Using the relation: x = 2 * __gamma_q_inv(v / 2, p) ]] +[[mean][v]] +[[variance][2v]] +[[mode][v - 2 (if v >= 2)]] +[[skewness][2 * sqrt(2 / v) == sqrt(8 / v)]] +[[kurtosis][3 + 12 / v]] +[[kurtosis excess][12 / v]] +] + +[h4 References] + +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm NIST Exploratory Data Analysis] +* [@http://en.wikipedia.org/wiki/Chi-square_distribution Chi-square distribution] +* [@http://mathworld.wolfram.com/Chi-SquaredDistribution.html Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.] + + +[endsect][/section:chi_squared_dist Chi Squared] + +[/ chi_squared.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/chi_squared_examples.qbk b/doc/sf_and_dist/distributions/chi_squared_examples.qbk new file mode 100644 index 000000000..cf071c264 --- /dev/null +++ b/doc/sf_and_dist/distributions/chi_squared_examples.qbk @@ -0,0 +1,500 @@ + +[section:cs_eg Chi Squared Distribution Examples] + +[section:chi_sq_intervals Confidence Intervals on the Standard Deviation] + +Once you have calculated the standard deviation for your data, a legitimate +question to ask is "How reliable is the calculated standard deviation?". +For this situation the Chi Squared distribution can be used to calculate +confidence intervals for the standard deviation. + +The full example code & sample output is in +[@../../../example/chi_square_std_dev_test.cpp chi_square_std_deviation_test.cpp]. + +We'll begin by defining the procedure that will calculate and print out the +confidence intervals: + + void confidence_limits_on_std_deviation( + double Sd, // Sample Standard Deviation + unsigned N) // Sample size + { + +We'll begin by printing out some general information: + + cout << + "________________________________________________\n" + "2-Sided Confidence Limits For Standard Deviation\n" + "________________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << N << "\n"; + cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n"; + +and then define a table of significance levels for which we'll calculate +intervals: + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +The distribution we'll need to calculate the confidence intervals is a +Chi Squared distribution, with N-1 degrees of freedom: + + chi_squared dist(N - 1); + +For each value of alpha, the formula for the confidence interval is given by: + +[equation chi_squ_tut1] + +Where [equation chi_squ_tut2] is the upper critical value, and +[equation chi_squ_tut3] is the lower critical value of the +Chi Squared distribution. + +In code we begin by printing out a table header: + + cout << "\n\n" + "_____________________________________________\n" + "Confidence Lower Upper\n" + " Value (%) Limit Limit\n" + "_____________________________________________\n"; + +and then loop over the values of alpha and calculate the intervals +for each: remember that the lower critical value is the same as the +quantile, and the upper critical value is the same as the quantile +from the complement of the probability: + + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // Calculate limits: + double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha[i] / 2))); + double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha[i] / 2)); + // Print Limits: + cout << fixed << setprecision(5) << setw(15) << right << lower_limit; + cout << fixed << setprecision(5) << setw(15) << right << upper_limit << endl; + } + cout << endl; + +To see some example output we'll use the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm +gear data] from the __handbook. +The data represents measurements of gear diameter from a manufacturing +process. + +[pre''' +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ + +Number of Observations = 100 +Standard Deviation = 0.006278908 + + +_____________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +_____________________________________________ + 50.000 0.00601 0.00662 + 75.000 0.00582 0.00685 + 90.000 0.00563 0.00712 + 95.000 0.00551 0.00729 + 99.000 0.00530 0.00766 + 99.900 0.00507 0.00812 + 99.990 0.00489 0.00855 + 99.999 0.00474 0.00895 +'''] + +So at the 95% confidence level we conclude that the standard deviation +is between 0.00551 and 0.00729. + +[h4 Confidence intervals as a function of the number of observations] + +Similarly, we can also list the confidence intervals for the standard deviation +for the common confidence levels 95%, for increasing numbers of observations. + +The standard deviation used to compute these values is unity, +so the limits listed are *multipliers* for any particular standard deviation. +For example, given a standard deviation of 0.0062789 as in the example +above; for 100 observations the multiplier is 0.8780 +giving the lower confidence limit of 0.8780 * 0.006728 = 0.00551. + +[pre''' +____________________________________________________ +Confidence level (two-sided) = 0.0500000 +Standard Deviation = 1.0000000 +________________________________________ +Observations Lower Upper + Limit Limit +________________________________________ + 2 0.4461 31.9102 + 3 0.5207 6.2847 + 4 0.5665 3.7285 + 5 0.5991 2.8736 + 6 0.6242 2.4526 + 7 0.6444 2.2021 + 8 0.6612 2.0353 + 9 0.6755 1.9158 + 10 0.6878 1.8256 + 15 0.7321 1.5771 + 20 0.7605 1.4606 + 30 0.7964 1.3443 + 40 0.8192 1.2840 + 50 0.8353 1.2461 + 60 0.8476 1.2197 + 100 0.8780 1.1617 + 120 0.8875 1.1454 + 1000 0.9580 1.0459 + 10000 0.9863 1.0141 + 50000 0.9938 1.0062 + 100000 0.9956 1.0044 + 1000000 0.9986 1.0014 +'''] + +With just 2 observations the limits are from *0.445* up to to *31.9*, +so the standard deviation might be about *half* +the observed value up to *30 times* the observed value! + +Estimating a standard deviation with just a handful of values leaves a very great uncertainty, +especially the upper limit. +Note especially how far the upper limit is skewed from the most likely standard deviation. + +Even for 10 observations, normally considered a reasonable number, +the range is still from 0.69 to 1.8, about a range of 0.7 to 2, +and is still highly skewed with an upper limit *twice* the median. + +When we have 1000 observations, the estimate of the standard deviation is starting to look convincing, +with a range from 0.95 to 1.05 - now near symmetrical, but still about + or - 5%. + +Only when we have 10000 or more repeated observations can we start to be reasonably confident +(provided we are sure that other factors like drift are not creeping in). + +For 10000 observations, the interval is 0.99 to 1.1 - finally a really convincing + or -1% confidence. + +[endsect][/section:chi_sq_intervals Confidence Intervals on the Standard Deviation] + +[section:chi_sq_test Chi-Square Test for the Standard Deviation] + +We use this test to determine whether the standard deviation of a sample +differs from a specified value. Typically this occurs in process change +situations where we wish to compare the standard deviation of a new +process to an established one. + +The code for this example is contained in +[@../../../example/chi_square_std_dev_test.cpp chi_square_std_dev_test.cpp], and +we'll begin by defining the procedure that will print out the test +statistics: + + void chi_squared_test( + double Sd, // Sample std deviation + double D, // True std deviation + unsigned N, // Sample size + double alpha) // Significance level + { + +The procedure begins by printing a summary of the input data: + + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "______________________________________________\n" + "Chi Squared test for sample standard deviation\n" + "______________________________________________\n\n"; + cout << setprecision(5); + cout << setw(55) << left << "Number of Observations" << "= " << N << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n"; + cout << setw(55) << left << "Expected True Standard Deviation" << "= " << D << "\n\n"; + +The test statistic (T) is simply the ratio of the sample and "true" standard +deviations squared, multiplied by the number of degrees of freedom (the +sample size less one): + + double t_stat = (N - 1) * (Sd / D) * (Sd / D); + cout << setw(55) << left << "Test Statistic" << "= " << t_stat << "\n"; + +The distribution we need to use, is a Chi Squared distribution with N-1 +degrees of freedom: + + chi_squared dist(N - 1); + +The various hypothesis that can be tested are summarised in the following table: + +[table +[[Hypothesis][Test]] +[[The null-hypothesis: there is no difference in standard deviation from the specified value] + [ Reject if T < [chi][super 2][sub (1-alpha/2; N-1)] or T > [chi][super 2][sub (alpha/2; N-1)] ]] +[[The alternative hypothesis: there is a difference in standard deviation from the specified value] + [ Reject if [chi][super 2][sub (1-alpha/2; N-1)] >= T >= [chi][super 2][sub (alpha/2; N-1)] ]] +[[The alternative hypothesis: the standard deviation is less than the specified value] + [ Reject if [chi][super 2][sub (1-alpha; N-1)] <= T ]] +[[The alternative hypothesis: the standard deviation is greater than the specified value] + [ Reject if [chi][super 2][sub (alpha; N-1)] >= T ]] +] + +Where [chi][super 2][sub (alpha; N-1)] is the upper critical value of the +Chi Squared distribution, and [chi][super 2][sub (1-alpha; N-1)] is the +lower critical value. + +Recall that the lower critical value is the same +as the quantile, and the upper critical value is the same as the quantile +from the complement of the probability, that gives us the following code +to calculate the critical values: + + double ucv = quantile(complement(dist, alpha)); + double ucv2 = quantile(complement(dist, alpha / 2)); + double lcv = quantile(dist, alpha); + double lcv2 = quantile(dist, alpha / 2); + cout << setw(55) << left << "Upper Critical Value at alpha: " << "= " + << setprecision(3) << scientific << ucv << "\n"; + cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << ucv2 << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha: " << "= " + << setprecision(3) << scientific << lcv << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << lcv2 << "\n\n"; + +Now that we have the critical values, we can compare these to our test +statistic, and print out the result of each hypothesis and test: + + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + + cout << "Standard Deviation != " << setprecision(3) << fixed << D << " "; + if((ucv2 < t_stat) || (lcv2 > t_stat)) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + + cout << "Standard Deviation < " << setprecision(3) << fixed << D << " "; + if(lcv > t_stat) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + + cout << "Standard Deviation > " << setprecision(3) << fixed << D << " "; + if(ucv < t_stat) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; + +To see some example output we'll use the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm +gear data] from the __handbook. +The data represents measurements of gear diameter from a manufacturing +process. The program output is deliberately designed to mirror +the DATAPLOT output shown in the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm +NIST Handbook Example]. + +[pre''' +______________________________________________ +Chi Squared test for sample standard deviation +______________________________________________ + +Number of Observations = 100 +Sample Standard Deviation = 0.00628 +Expected True Standard Deviation = 0.10000 + +Test Statistic = 0.39030 +CDF of test statistic: = 1.438e-099 +Upper Critical Value at alpha: = 1.232e+002 +Upper Critical Value at alpha/2: = 1.284e+002 +Lower Critical Value at alpha: = 7.705e+001 +Lower Critical Value at alpha/2: = 7.336e+001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion''' +Standard Deviation != 0.100 ACCEPTED +Standard Deviation < 0.100 ACCEPTED +Standard Deviation > 0.100 REJECTED +] + +In this case we are testing whether the sample standard deviation is 0.1, +and the null-hypothesis is rejected, so we conclude that the standard +deviation ['is not] 0.1. + +For an alternative example, consider the +[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm +silicon wafer data] again from the __handbook. +In this scenario a supplier of 100 ohm.cm silicon wafers claims +that his fabrication process can produce wafers with sufficient +consistency so that the standard deviation of resistivity for +the lot does not exceed 10 ohm.cm. A sample of N = 10 wafers taken +from the lot has a standard deviation of 13.97 ohm.cm, and the question +we ask ourselves is "Is the suppliers claim correct?". + +The program output now looks like this: + +[pre''' +______________________________________________ +Chi Squared test for sample standard deviation +______________________________________________ + +Number of Observations = 10 +Sample Standard Deviation = 13.97000 +Expected True Standard Deviation = 10.00000 + +Test Statistic = 17.56448 +CDF of test statistic: = 9.594e-001 +Upper Critical Value at alpha: = 1.692e+001 +Upper Critical Value at alpha/2: = 1.902e+001 +Lower Critical Value at alpha: = 3.325e+000 +Lower Critical Value at alpha/2: = 2.700e+000 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion''' +Standard Deviation != 10.000 REJECTED +Standard Deviation < 10.000 REJECTED +Standard Deviation > 10.000 ACCEPTED +] + +In this case, our null-hypothesis is that the standard deviation of +the sample is less than 10: this hypothesis is rejected in the analysis +above, and so we reject the manufacturers claim. + +[endsect][/section:chi_sq_test Chi-Square Test for the Standard Deviation] + +[section:chi_sq_size Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation] + +Suppose we conduct a Chi Squared test for standard deviation and the result +is borderline, a legitimate question to ask is "How large would the sample size +have to be in order to produce a definitive result?" + +The class template [link math_toolkit.dist.dist_ref.dists.chi_squared_dist +chi_squared_distribution] has a static method +`find_degrees_of_freedom` that will calculate this value for +some acceptable risk of type I failure /alpha/, type II failure +/beta/, and difference from the standard deviation /diff/. Please +note that the method used works on variance, and not standard deviation +as is usual for the Chi Squared Test. + +The code for this example is located in [@../../../example/chi_square_std_dev_test.cpp +chi_square_std_dev_test.cpp]. + +We begin by defining a procedure to print out the sample sizes required +for various risk levels: + + void chi_squared_sample_sized( + double diff, // difference from variance to detect + double variance) // true variance + { + +The procedure begins by printing out the input data: + + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "_____________________________________________________________\n" + "Estimated sample sizes required for various confidence levels\n" + "_____________________________________________________________\n\n"; + cout << setprecision(5); + cout << setw(40) << left << "True Variance" << "= " << variance << "\n"; + cout << setw(40) << left << "Difference to detect" << "= " << diff << "\n"; + +And defines a table of significance levels for which we'll calculate sample sizes: + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +For each value of alpha we can calculate two sample sizes: one where the +sample variance is less than the true value by /diff/ and one +where it is greater than the true value by /diff/. Thanks to the +asymmetric nature of the Chi Squared distribution these two values will +not be the same, the difference in their calculation differs only in the +sign of /diff/ that's passed to `find_degrees_of_freedom`. Finally +in this example we'll simply things, and let risk level /beta/ be the +same as /alpha/: + + cout << "\n\n" + "_______________________________________________________________\n" + "Confidence Estimated Estimated\n" + " Value (%) Sample Size Sample Size\n" + " (lower one (upper one\n" + " sided test) sided test)\n" + "_______________________________________________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate df for a lower single sided test: + double df = chi_squared::find_degrees_of_freedom( + -diff, alpha[i], alpha[i], variance); + // convert to sample size: + double size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size; + // calculate df for an upper single sided test: + df = chi_squared::find_degrees_of_freedom( + diff, alpha[i], alpha[i], variance); + // convert to sample size: + size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size << endl; + } + cout << endl; + +For some example output, consider the +[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm +silicon wafer data] from the __handbook. +In this scenario a supplier of 100 ohm.cm silicon wafers claims +that his fabrication process can produce wafers with sufficient +consistency so that the standard deviation of resistivity for +the lot does not exceed 10 ohm.cm. A sample of N = 10 wafers taken +from the lot has a standard deviation of 13.97 ohm.cm, and the question +we ask ourselves is "How large would our sample have to be to reliably +detect this difference?". + +To use our procedure above, we have to convert the +standard deviations to variance (square them), +after which the program output looks like this: + +[pre''' +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ + +True Variance = 100.00000 +Difference to detect = 95.16090 + + +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (lower one (upper one + sided test) sided test) +_______________________________________________________________ + 50.000 2 2 + 75.000 2 10 + 90.000 4 32 + 95.000 5 51 + 99.000 7 99 + 99.900 11 174 + 99.990 15 251 + 99.999 20 330''' +] + +In this case we are interested in a upper single sided test. +So for example, if the maximum acceptable risk of falsely rejecting +the null-hypothesis is 0.05 (Type I error), and the maximum acceptable +risk of failing to reject the null-hypothesis is also 0.05 +(Type II error), we estimate that we would need a sample size of 51. + +[endsect][/section:chi_sq_size Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation] + +[endsect][/section:cs_eg Chi Squared Distribution] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/distribution_construction.qbk b/doc/sf_and_dist/distributions/distribution_construction.qbk new file mode 100644 index 000000000..5aab16543 --- /dev/null +++ b/doc/sf_and_dist/distributions/distribution_construction.qbk @@ -0,0 +1,17 @@ +[section:dist_construct_eg Distribution Construction Example] + +See [@../../../example/distribution_construction.cpp distribution_construction.cpp] for full source code. + +[import ../../../example/distribution_construction.cpp] +[distribution_construction1] +[distribution_construction2] + +[endsect] [/section:dist_construct_eg Distribution Construction Example] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/error_handling_example.qbk b/doc/sf_and_dist/distributions/error_handling_example.qbk new file mode 100644 index 000000000..9cc3b4590 --- /dev/null +++ b/doc/sf_and_dist/distributions/error_handling_example.qbk @@ -0,0 +1,35 @@ +[section:error_eg Error Handling Example] + +See [link math_toolkit.main_overview.error_handling error handling documentation] +for a detailed explanation of the mechanism of handling errors, +including the common "bad" arguments to distributions and functions, +and how to use __policy_section to control it. + +But, by default, *exceptions will be raised*, for domain errors, +pole errors, numeric overflow, and internal evaluation errors. +To avoid the exceptions from getting thrown and instead get +an appropriate value returned, usually a NaN (domain errors +pole errors or internal errors), or infinity (from overflow), +you need to change the policy. + +[import ../../../example/error_handling_example.cpp] + +[error_handling_example] + +[caution If throwing of exceptions is enabled (the default) but +you do *not* have try & catch block, +then the program will terminate with an uncaught exception and probably abort. + +Therefore to get the benefit of helpful error messages, enabling *all exceptions +and using try & catch* is recommended for most applications. + +However, for simplicity, the is not done for most examples.] + +[endsect] [/section:error_eg Error Handling Example] +[/ + Copyright 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/exponential.qbk b/doc/sf_and_dist/distributions/exponential.qbk new file mode 100644 index 000000000..93286dfe9 --- /dev/null +++ b/doc/sf_and_dist/distributions/exponential.qbk @@ -0,0 +1,101 @@ +[section:exp_dist Exponential Distribution] + +``#include `` + + template + class exponential_distribution; + + typedef exponential_distribution<> exponential; + + template + class exponential_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + exponential_distribution(RealType lambda = 1); + + RealType lambda()const; + }; + + +The [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution] +is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution] +with PDF: + +[equation exponential_dist_ref1] + +It is often used to model the time between independent +events that happen at a constant average rate. + +The following graph shows how the distribution changes for different +values of the rate parameter lambda: + +[$../graphs/exponential_dist.png] + +[h4 Member Functions] + + exponential_distribution(RealType lambda = 1); + +Constructs an +[@http://en.wikipedia.org/wiki/Exponential_distribution Exponential distribution] +with parameter /lambda/. +Lambda is defined as the reciprocal of the scale parameter. + +Requires lambda > 0, otherwise calls __domain_error. + + RealType lambda()const; + +Accessor function returns the lambda parameter of the distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, +[infin]\]. + +[h4 Accuracy] + +The exponential distribution is implemented in terms of the +standard library functions `exp`, `log`, `log1p` and `expm1` +and as such should have very low error rates. + +[h4 Implementation] + +In the following table [lambda] is the parameter lambda of the distribution, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = [lambda] * exp(-[lambda] * x) ]] +[[cdf][Using the relation: p = 1 - exp(-x * [lambda]) = -expm1(-x * [lambda]) ]] +[[cdf complement][Using the relation: q = exp(-x * [lambda]) ]] +[[quantile][Using the relation: x = -log(1-p) / [lambda] = -log1p(-p) / [lambda]]] +[[quantile from the complement][Using the relation: x = -log(q) / [lambda]]] +[[mean][1/[lambda]]] +[[standard deviation][1/[lambda]]] +[[mode][0]] +[[skewness][2]] +[[kurtosis][9]] +[[kurtosis excess][6]] +] + +[h4 references] + +* [@http://mathworld.wolfram.com/ExponentialDistribution.html Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Web Resource] +* [@http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html Wolfram Mathematica calculator] +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST Exploratory Data Analysis] +* [@http://en.wikipedia.org/wiki/Exponential_distribution Wikipedia Exponential distribution] + +[endsect][/section:exp_dist Exponential] + +[/ exponential.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/extreme_value.qbk b/doc/sf_and_dist/distributions/extreme_value.qbk new file mode 100644 index 000000000..66c184088 --- /dev/null +++ b/doc/sf_and_dist/distributions/extreme_value.qbk @@ -0,0 +1,114 @@ +[section:extreme_dist Extreme Value Distribution] + +``#include `` + + template + class extreme_value_distribution; + + typedef extreme_value_distribution<> extreme_value; + + template + class extreme_value_distribution + { + public: + typedef RealType value_type; + + extreme_value_distribution(RealType location = 0, RealType scale = 1); + + RealType scale()const; + RealType location()const; + }; + +There are various +[@http://mathworld.wolfram.com/ExtremeValueDistribution.html extreme value distributions] +: this implementation represents the maximum case, +and is variously known as a Fisher-Tippett distribution, +a log-Weibull distribution or a Gumbel distribution. + +Extreme value theory is important for assessing risk for highly unusual events, +such as 100-year floods. + +More information can be found on the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm NIST], +[@http://en.wikipedia.org/wiki/Extreme_value_distribution Wikipedia], +[@http://mathworld.wolfram.com/ExtremeValueDistribution.html Mathworld], +and [@http://en.wikipedia.org/wiki/Extreme_value_theory Extreme value theory] +websites. + +The distribution has a PDF given by: + +f(x) = (1/scale) e[super -(x-location)/scale] e[super -e[super -(x-location)/scale]] + +Which in the standard case (scale = 1, location = 0) reduces to: + +f(x) = e[super -x]e[super -e[super -x]] + +The following graph illustrates how the PDF varies with the location parameter: + +[$../graphs/extreme_val_dist.png] + +And this graph illustrates how the PDF varies with the shape parameter: + +[$../graphs/extreme_val_dist2.png] + +[h4 Member Functions] + + extreme_value_distribution(RealType location = 0, RealType scale = 1); + +Constructs an Extreme Value distribution with the specified location and scale +parameters. + +Requires `scale > 0`, otherwise calls __domain_error. + + RealType location()const; + +Returns the location parameter of the distribution. + + RealType scale()const; + +Returns the scale parameter of the distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random parameter is \[-[infin], +[infin]\]. + +[h4 Accuracy] + +The extreme value distribution is implemented in terms of the +standard library `exp` and `log` functions and as such should have very low +error rates. + +[h4 Implementation] + +In the following table: +/a/ is the location parameter, /b/ is the scale parameter, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) / b ]] +[[cdf][Using the relation: p = exp(-exp((a-x)/b)) ]] +[[cdf complement][Using the relation: q = -expm1(-exp((a-x)/b)) ]] +[[quantile][Using the relation: a - log(-log(p)) * b]] +[[quantile from the complement][Using the relation: a - log(-log1p(-q)) * b]] +[[mean][a + [@http://en.wikipedia.org/wiki/Euler-Mascheroni_constant Euler-Mascheroni-constant] * b]] +[[standard deviation][pi * b / sqrt(6)]] +[[mode][The same as the location parameter /a/.]] +[[skewness][12 * sqrt(6) * zeta(3) / pi[super 3] ]] +[[kurtosis][27 / 5]] +[[kurtosis excess][kurtosis - 3 or 12 / 5]] +] + +[endsect][/section:extreme_dist Extreme Value] + +[/ extreme_value.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/f_dist_example.qbk b/doc/sf_and_dist/distributions/f_dist_example.qbk new file mode 100644 index 000000000..4d48cca47 --- /dev/null +++ b/doc/sf_and_dist/distributions/f_dist_example.qbk @@ -0,0 +1,220 @@ +[section:f_eg F Distribution Examples] + +Imagine that you want to compare the standard deviations of two +sample to determine if they differ in any significant way, in this +situation you use the F distribution and perform an F-test. This +situation commonly occurs when conducting a process change comparison: +"is a new process more consistent that the old one?". + +In this example we'll be using the data for ceramic strength from +[@http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm +http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm]. +The data for this case study were collected by Said Jahanmir of the +NIST Ceramics Division in 1996 in connection with a NIST/industry +ceramics consortium for strength optimization of ceramic strength. + +The example program is [@../../../example/f_test.cpp f_test.cpp], +program output has been deliberately made as similar as possible +to the DATAPLOT output in the corresponding +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm +NIST EngineeringStatistics Handbook example]. + +We'll begin by defining the procedure to conduct the test: + + void f_test( + double sd1, // Sample 1 std deviation + double sd2, // Sample 2 std deviation + double N1, // Sample 1 size + double N2, // Sample 2 size + double alpha) // Significance level + { + +The procedure begins by printing out a summary of our input data: + + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "____________________________________\n" + "F test for equal standard deviations\n" + "____________________________________\n\n"; + cout << setprecision(5); + cout << "Sample 1:\n"; + cout << setw(55) << left << "Number of Observations" << "= " << N1 << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd1 << "\n\n"; + cout << "Sample 2:\n"; + cout << setw(55) << left << "Number of Observations" << "= " << N2 << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd2 << "\n\n"; + +The test statistic for an F-test is simply the ratio of the square of +the two standard deviations: + +F = s[sub 1][super 2] / s[sub 2][super 2] + +where s[sub 1] is the standard deviation of the first sample and s[sub 2] +is the standard deviation of the second sample. Or in code: + + double F = (sd1 / sd2); + F *= F; + cout << setw(55) << left << "Test Statistic" << "= " << F << "\n\n"; + +At this point a word of caution: the F distribution is asymmetric, +so we have to be careful how we compute the tests, the following table +summarises the options available: + +[table +[[Hypothesis][Test]] +[[The null-hypothesis: there is no difference in standard deviations (two sided test)] + [Reject if F <= F[sub (1-alpha/2; N1-1, N2-1)] or F >= F[sub (alpha/2; N1-1, N2-1)] ]] +[[The alternative hypothesis: there is a difference in means (two sided test)] + [Reject if F[sub (1-alpha/2; N1-1, N2-1)] <= F <= F[sub (alpha/2; N1-1, N2-1)] ]] +[[The alternative hypothesis: Standard deviation of sample 1 is greater +than that of sample 2] + [Reject if F < F[sub (alpha; N1-1, N2-1)] ]] +[[The alternative hypothesis: Standard deviation of sample 1 is less +than that of sample 2] + [Reject if F > F[sub (1-alpha; N1-1, N2-1)] ]] +] + +Where F[sub (1-alpha; N1-1, N2-1)] is the lower critical value of the F distribution +with degrees of freedom N1-1 and N2-1, and F[sub (alpha; N1-1, N2-1)] is the upper +critical value of the F distribution with degrees of freedom N1-1 and N2-1. + +The upper and lower critical values can be computed using the quantile function: + +F[sub (1-alpha; N1-1, N2-1)] = `quantile(fisher_f(N1-1, N2-1), alpha)` + +F[sub (alpha; N1-1, N2-1)] = `quantile(complement(fisher_f(N1-1, N2-1), alpha))` + +In our example program we need both upper and lower critical values for alpha +and for alpha/2: + + double ucv = quantile(complement(dist, alpha)); + double ucv2 = quantile(complement(dist, alpha / 2)); + double lcv = quantile(dist, alpha); + double lcv2 = quantile(dist, alpha / 2); + cout << setw(55) << left << "Upper Critical Value at alpha: " << "= " + << setprecision(3) << scientific << ucv << "\n"; + cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << ucv2 << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha: " << "= " + << setprecision(3) << scientific << lcv << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << lcv2 << "\n\n"; + +The final step is to perform the comparisons given above, and print +out whether the hypothesis is rejected or not: + + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + + cout << "Standard deviations are unequal (two sided test) "; + if((ucv2 < F) || (lcv2 > F)) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + + cout << "Standard deviation 1 is less than standard deviation 2 "; + if(lcv > F) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + + cout << "Standard deviation 1 is greater than standard deviation 2 "; + if(ucv < F) + cout << "ACCEPTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; + +Using the ceramic strength data as an example we get the following +output: + +[pre +'''F test for equal standard deviations +____________________________________ + +Sample 1: +Number of Observations = 240 +Sample Standard Deviation = 65.549 + +Sample 2: +Number of Observations = 240 +Sample Standard Deviation = 61.854 + +Test Statistic = 1.123 + +CDF of test statistic: = 8.148e-001 +Upper Critical Value at alpha: = 1.238e+000 +Upper Critical Value at alpha/2: = 1.289e+000 +Lower Critical Value at alpha: = 8.080e-001 +Lower Critical Value at alpha/2: = 7.756e-001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Standard deviations are unequal (two sided test) REJECTED +Standard deviation 1 is less than standard deviation 2 REJECTED +Standard deviation 1 is greater than standard deviation 2 REJECTED''' +] + +In this case we are unable to reject the null-hypothesis, and must instead +reject the alternative hypothesis. + +By contrast let's see what happens when we use some different +[@http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm +sample data]:, once again from the NIST Engineering Statistics Handbook: +A new procedure to assemble a device is introduced and tested for +possible improvement in time of assembly. The question being addressed +is whether the standard deviation of the new assembly process (sample 2) is +better (i.e., smaller) than the standard deviation for the old assembly +process (sample 1). + +[pre +'''____________________________________ +F test for equal standard deviations +____________________________________ + +Sample 1: +Number of Observations = 11.00000 +Sample Standard Deviation = 4.90820 + +Sample 2: +Number of Observations = 9.00000 +Sample Standard Deviation = 2.58740 + +Test Statistic = 3.59847 + +CDF of test statistic: = 9.589e-001 +Upper Critical Value at alpha: = 3.347e+000 +Upper Critical Value at alpha/2: = 4.295e+000 +Lower Critical Value at alpha: = 3.256e-001 +Lower Critical Value at alpha/2: = 2.594e-001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Standard deviations are unequal (two sided test) REJECTED +Standard deviation 1 is less than standard deviation 2 REJECTED +Standard deviation 1 is greater than standard deviation 2 ACCEPTED''' +] + +In this case we take our null hypothesis as "standard deviation 1 is +less than or equal to standard deviation 2", since this represents the "no change" +situation. So we want to compare the upper critical value at /alpha/ +(a one sided test) with the test statistic, and since 3.35 < 3.6 this +hypothesis must be rejected. We therefore conclude that there is a change +for the better in our standard deviation. + +[endsect][/section:f_eg F Distribution] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/find_location_and_scale.qbk b/doc/sf_and_dist/distributions/find_location_and_scale.qbk new file mode 100644 index 000000000..c9d87a9e4 --- /dev/null +++ b/doc/sf_and_dist/distributions/find_location_and_scale.qbk @@ -0,0 +1,39 @@ +[section:find_eg Find Location and Scale Examples] + +[section:find_location_eg Find Location (Mean) Example] +[import ../../../example/find_location_example.cpp] +[find_location1] +[find_location2] +See [@../../../example/find_location_example.cpp find_location_example.cpp] +for full source code: the program output looks like this: +[find_location_example_output] +[endsect] [/section:find_location_eg Find Location (Mean) Example] + +[section:find_scale_eg Find Scale (Standard Deviation) Example] +[import ../../../example/find_scale_example.cpp] +[find_scale1] +[find_scale2] +See [@../../../example/find_scale_example.cpp find_scale_example.cpp] +for full source code: the program output looks like this: +[find_scale_example_output] +[endsect] [/section:find_scale_eg Scale (Standard Deviation) Example] +[section:find_mean_and_sd_eg Find mean and standard deviation example] + +[import ../../../example/find_mean_and_sd_normal.cpp] +[normal_std] +[normal_find_location_and_scale_eg] +See [@../../../example/find_mean_and_sd_normal.cpp find_mean_and_sd_normal.cpp] +for full source code & appended program output. +[endsect] [/find_mean_and_sd_eg Find mean and standard deviation example] + +[endsect] [/section:find_eg Find Location and Scale Examples] + + + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/fisher.qbk b/doc/sf_and_dist/distributions/fisher.qbk new file mode 100644 index 000000000..7887e2f94 --- /dev/null +++ b/doc/sf_and_dist/distributions/fisher.qbk @@ -0,0 +1,190 @@ +[section:f_dist F Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class fisher_f_distribution; + + typedef fisher_f_distribution<> fisher_f; + + template + class fisher_f_distribution + { + public: + typedef RealType value_type; + + // Construct: + fisher_f_distribution(const RealType& i, const RealType& j); + + // Accessors: + RealType degrees_of_freedom1()const; + RealType degrees_of_freedom2()const; + }; + + }} //namespaces + +The F distribution is a continuous distribution that arises when testing +whether two samples have the same variance. If [chi][super 2][sub m][space] and +[chi][super 2][sub n][space] are independent variates each distributed as +Chi-Squared with /m/ and /n/ degrees of freedom, then the test statistic: + +F[sub n,m][space] = ([chi][super 2][sub n][space] / n) / ([chi][super 2][sub m][space] / m) + +Is distributed over the range \[0, [infin]\] with an F distribution, and +has the PDF: + +[equation fisher_pdf] + +The following graph illustrates how the PDF varies depending on the +two degrees of freedom parameters. + +[$../graphs/fisher_f.png] + + +[h4 Member Functions] + + fisher_f_distribution(const RealType& df1, const RealType& df2); + +Constructs an F-distribution with numerator degrees of freedom /df1/ +and denominator degrees of freedom /df2/. + +Requires that /df1/ and /df2/ are both greater than zero, otherwise __domain_error +is called. + + RealType degrees_of_freedom1()const; + +Returns the numerator degrees of freedom parameter of the distribution. + + RealType degrees_of_freedom2()const; + +Returns the denominator degrees of freedom parameter of the distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, +[infin]\]. + +[h4 Examples] + +Various [link math_toolkit.dist.stat_tut.weg.f_eg worked examples] are +available illustrating the use of the F Distribution. + +[h4 Accuracy] + +The normal distribution is implemented in terms of the +[link math_toolkit.special.sf_beta.ibeta_function incomplete beta function] +and it's [link math_toolkit.special.sf_beta.ibeta_inv_function inverses], +refer to those functions for accuracy data. + +[h4 Implementation] + +In the following table /v1/ and /v2/ are the first and second +degrees of freedom parameters of the distribution, +/x/ is the random variate, /p/ is the probability, and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][The usual form of the PDF is given by: + +[equation fisher_pdf] + +However, that form is hard to evaluate directly without incurring problems with +either accuracy or numeric overflow. + +Direct differentiation of the CDF expressed in terms of the incomplete beta function + +led to the following two formulas: + +f[sub v1,v2](x) = y * __ibeta_derivative(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x)) + +with y = (v2 * v1) \/ ((v2 + v1 * x) * (v2 + v1 * x)) + +and + +f[sub v1,v2](x) = y * __ibeta_derivative(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x)) + +with y = (z * v1 - x * v1 * v1) \/ z[super 2] + +and z = v2 + v1 * x + +The first of these is used for v1 * x > v2, otherwise the second is used. + +The aim is to keep the /x/ argument to __ibeta_derivative away from 1 to avoid +rounding error. ]] +[[cdf][Using the relations: + +p = __ibeta(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x)) + +and + +p = __ibetac(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x)) + +The first is used for v1 * x > v2, otherwise the second is used. + +The aim is to keep the /x/ argument to __ibeta well away from 1 to +avoid rounding error. ]] + +[[cdf complement][Using the relations: + +p = __ibetac(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x)) + +and + +p = __ibeta(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x)) + +The first is used for v1 * x < v2, otherwise the second is used. + +The aim is to keep the /x/ argument to __ibeta well away from 1 to +avoid rounding error. ]] +[[quantile][Using the relation: + +x = v2 * a \/ (v1 * b) + +where: + +a = __ibeta_inv(v1 \/ 2, v2 \/ 2, p) + +and + +b = 1 - a + +Quantities /a/ and /b/ are both computed by __ibeta_inv without the +subtraction implied above.]] +[[quantile + +from the complement][Using the relation: + +x = v2 * a \/ (v1 * b) + +where + +a = __ibetac_inv(v1 \/ 2, v2 \/ 2, p) + +and + +b = 1 - a + +Quantities /a/ and /b/ are both computed by __ibetac_inv without the +subtraction implied above.]] +[[mean][v2 \/ (v2 - 2)]] +[[variance][2 * v2[super 2 ] * (v1 + v2 - 2) \/ (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))]] +[[mode][v2 * (v1 - 2) \/ (v1 * (v2 + 2))]] +[[skewness][2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) \/ (v1 * (v2 + v1 - 2))) \/ (v2 - 6)]] +[[kurtosis and kurtosis excess] + [Refer to, [@http://mathworld.wolfram.com/F-Distribution.html + Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource.] ]] +] + +[endsect][/section:f_dist F distribution] + +[/ fisher.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/distributions/gamma.qbk b/doc/sf_and_dist/distributions/gamma.qbk new file mode 100644 index 000000000..1bc233351 --- /dev/null +++ b/doc/sf_and_dist/distributions/gamma.qbk @@ -0,0 +1,138 @@ +[section:gamma_dist Gamma (and Erlang) Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class gamma_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + gamma_distribution(RealType shape, RealType scale = 1) + + RealType shape()const; + RealType scale()const; + }; + + }} // namespaces + +The gamma distribution is a continuous probability distribution. +When the shape parameter is an integer then it is known as the +Erlang Distribution. It is also closely related to the Poisson +and Chi Squared Distributions. + +When the shape parameter has an integer value, the distribution is the +[@http://en.wikipedia.org/wiki/Erlang_distribution Erlang distribution]. +Since this can be produced by ensuring that the shape parameter has an +integer value > 0, the Erlang distribution is not separately implemented. + +[note +To avoid potential confusion with the gamma functions, this +distribution does not provide the typedef: + +``typedef gamma_distibution gamma;`` + +Instead if you want +a double precision gamma distribution you can use + +``boost::gamma_distribution<>``] + +For shape parameter /k/ and scale parameter [theta][space] it is defined by the +probability density function: + +[equation gamma_dist_ref1] + +Sometimes an alternative formulation is used: given parameters +[alpha][space]= k and [beta][space]= 1 / [theta], then the +distribution can be defined by the PDF: + +[equation gamma_dist_ref2] + +In this form the inverse scale parameter is called a /rate parameter/. + +Both forms are in common usage: this library uses the first definition +throughout. Therefore to construct a Gamma Distribution from a ['rate +parameter], you should pass the reciprocal of the rate as the scale parameter. + +The following two graphs illustrate how the PDF of the gamma distribution +varies as the parameters vary: + +[$../graphs/gamma_dist1.png] + +[$../graphs/gamma_dist2.png] + +The [*Erlang Distribution] is the same as the Gamma, but with the shape parameter +an integer. It is often expressed using a /rate/ rather than a /scale/ as the +second parameter (remember that the rate is the reciprocal of the scale). + +Internally the functions used to implement the Gamma Distribution are +already optimised for small-integer arguments, so in general there should +be no great loss of performance from using a Gamma Distribution rather than +a dedicated Erlang Distribution. + +[h4 Member Functions] + + gamma_distribution(RealType shape, RealType scale = 1); + +Constructs a gamma distribution with shape /shape/ and +scale /scale/. + +Requires that the shape and scale parameters are greater than zero, otherwise calls +__domain_error. + + RealType shape()const; + +Returns the /shape/ parameter of this distribution. + + RealType scale()const; + +Returns the /scale/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[0,+[infin]\]. + +[h4 Accuracy] + +The lognormal distribution is implemented in terms of the +incomplete gamma functions __gamma_p and __gamma_q and their +inverses __gamma_p_inv and __gamma_q_inv: refer to the accuracy +data for those functions for more information. + +[h4 Implementation] + +In the following table /k/ is the shape parameter of the distribution, +[theta][space] is it's scale parameter, /x/ is the random variate, /p/ is the probability +and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = __gamma_p_derivative(k, x / [theta]) / [theta] ]] +[[cdf][Using the relation: p = __gamma_p(k, x / [theta]) ]] +[[cdf complement][Using the relation: q = __gamma_q(k, x / [theta]) ]] +[[quantile][Using the relation: x = [theta][space]* __gamma_p_inv(k, p) ]] +[[quantile from the complement][Using the relation: x = [theta][space]* __gamma_q_inv(k, p) ]] +[[mean][k[theta] ]] +[[variance][k[theta][super 2] ]] +[[mode][(k-1)[theta][space] for ['k>1] otherwise a __domain_error ]] +[[skewness][2 / sqrt(k) ]] +[[kurtosis][3 + 6 / k]] +[[kurtosis excess][6 / k ]] +] + +[endsect][/section:normal_dist Normal] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/lognormal.qbk b/doc/sf_and_dist/distributions/lognormal.qbk new file mode 100644 index 000000000..33ac076d6 --- /dev/null +++ b/doc/sf_and_dist/distributions/lognormal.qbk @@ -0,0 +1,119 @@ +[section:lognormal_dist Log Normal Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class lognormal_distribution; + + typedef lognormal_distribution<> lognormal; + + template + class lognormal_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Construct: + lognormal_distribution(RealType location = 0, RealType scale = 1); + // Accessors: + RealType location()const; + RealType scale()const; + }; + + }} // namespaces + +The lognormal distribution is the distribution that arises +when the logarithm of the random variable is normally distributed. +A lognormal distribution results when the variable is the product +of a large number of independent, identically-distributed variables. + +For location and scale parameters /m/ and /s/ it is defined by the +probability density function: + +[equation lognormal_ref] + +The location and scale parameters are equivalent to the mean and +standard deviation of the logarithm of the random variable. + +The following graph illustrates the effect of the location +parameter on the PDF, note that the range of the random +variable remains \[0,+[infin]\] irrespective of the value of the +location parameter: + +[$../graphs/lognormal1.png] + +The next graph illustrates the effect of the scale parameter on the PDF: + +[$../graphs/lognormal2.png] + +[h4 Member Functions] + + lognormal_distribution(RealType location = 0, RealType scale = 1); + +Constructs a lognormal distribution with location /location/ and +scale /scale/. + +The location parameter is the same as the mean of the logarithm of the +random variate. + +The scale parameter is the same as the standard deviation of the +logarithm of the random variate. + +Requires that the scale parameter is greater than zero, otherwise calls +__domain_error. + + RealType location()const; + +Returns the /location/ parameter of this distribution. + + RealType scale()const; + +Returns the /scale/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[0,+[infin]\]. + +[h4 Accuracy] + +The lognormal distribution is implemented in terms of the +standard library log and exp functions, plus the +[link math_toolkit.special.sf_erf.error_function error function], +and as such should have very low error rates. + +[h4 Implementation] + +In the following table /m/ is the location parameter of the distribution, +/s/ is it's scale parameter, /x/ is the random variate, /p/ is the probability +and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = e[super -(ln(x) - m)[super 2 ] \/ 2s[super 2 ] ] \/ (x * s * sqrt(2pi)) ]] +[[cdf][Using the relation: p = cdf(normal_distribtion(m, s), log(x)) ]] +[[cdf complement][Using the relation: q = cdf(complement(normal_distribtion(m, s), log(x))) ]] +[[quantile][Using the relation: x = exp(quantile(normal_distribtion(m, s), p))]] +[[quantile from the complement][Using the relation: x = exp(quantile(complement(normal_distribtion(m, s), q)))]] +[[mean][e[super m + s[super 2 ] / 2 ] ]] +[[variance][(e[super s[super 2] ] - 1) * e[super 2m + s[super 2 ] ] ]] +[[mode][e[super m + s[super 2 ] ] ]] +[[skewness][sqrt(e[super s[super 2] ] - 1) * (2 + e[super s[super 2] ]) ]] +[[kurtosis][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 3]] +[[kurtosis excess][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 6 ]] +] + +[endsect][/section:normal_dist Normal] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/nag_library.qbk b/doc/sf_and_dist/distributions/nag_library.qbk new file mode 100644 index 000000000..b0377871b --- /dev/null +++ b/doc/sf_and_dist/distributions/nag_library.qbk @@ -0,0 +1,60 @@ +[section:nag_library Comparison with C, R, FORTRAN-style Free Functions] + +You are probably familiar with a statistics library that has free functions, +for example the classic [@http://nag.com/numeric/CL/CLdescription.asp NAG C library] +and matching [@http://nag.com/numeric/FL/FLdescription.asp NAG FORTRAN Library], +[@http://office.microsoft.com/en-us/excel/HP052090051033.aspx Microsoft Excel BINOMDIST(number_s,trials,probability_s,cumulative)], +[@http://www.r-project.org/ R], [@http://www.ptc.com/products/mathcad/mathcad14/mathcad_func_chart.htm MathCAD pbinom] +and many others. + +If so, you may find 'Distributions as Objects' unfamiliar, if not alien. + +However, *do not panic*, both definition and usage are not really very different. + +A very simple example of generating the same values as the +[@http://nag.com/numeric/CL/CLdescription.asp NAG C library] +for the binomial distribution follows. +(If you find slightly different values, the Boost C++ version, using double or better, +is very likely to be the more accurate. +Of course, accuracy is not usually a concern for most applications of this function). + +The [@http://www.nag.co.uk/numeric/cl/manual/pdf/G01/g01bjc.pdf NAG function specification] is + + void nag_binomial_dist(Integer n, double p, Integer k, + double *plek, double *pgtk, double *peqk, NagError *fail) + +and is called + + g01bjc(n, p, k, &plek, &pgtk, &peqk, NAGERR_DEFAULT); + +The equivalent using this Boost C++ library is: + + using namespace boost::math; // Using declaration avoids very long names. + binomial my_dist(4, 0.5); // c.f. NAG n = 4, p = 0.5 + +and values can be output thus: + + cout + << my_dist.trials() << " " // Echo the NAG input n = 4 trials. + << my_dist.success_fraction() << " " // Echo the NAG input p = 0.5 + << cdf(my_dist, 2) << " " // NAG plek with k = 2 + << cdf(complement(my_dist, 2)) << " " // NAG pgtk with k = 2 + << pdf(my_dist, 2) << endl; // NAG peqk with k = 2 + +`cdf(dist, k)` is equivalent to NAG library `plek`, lower tail probability of <= k + +`cdf(complement(dist, k))` is equivalent to NAG library `pgtk`, upper tail probability of > k + +`pdf(dist, k)` is equivalent to NAG library `peqk`, point probability of == k + +See [@../../../example/binomial_example_nag.cpp binomial_example_nag.cpp] for details. + +[endsect] [/section:nag_library Comparison with C, R, FORTRAN-style Free Functions] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/negative_binomial.qbk b/doc/sf_and_dist/distributions/negative_binomial.qbk new file mode 100644 index 000000000..de27bbe7b --- /dev/null +++ b/doc/sf_and_dist/distributions/negative_binomial.qbk @@ -0,0 +1,372 @@ +[section:negative_binomial_dist Negative Binomial Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class negative_binomial_distribution; + + typedef negative_binomial_distribution<> negative_binomial; + + template + class negative_binomial_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Constructor from successes and success_fraction: + negative_binomial_distribution(RealType r, RealType p); + + // Parameter accessors: + RealType success_fraction() const; + RealType successes() const; + + // Bounds on success fraction: + static RealType find_lower_bound_on_p( + RealType trials, + RealType successes, + RealType probability); // alpha + static RealType find_upper_bound_on_p( + RealType trials, + RealType successes, + RealType probability); // alpha + + // Estimate min/max number of trials: + static RealType find_minimum_number_of_trials( + RealType k, // Number of failures. + RealType p, // Success fraction. + RealType probability); // Probability threshold alpha. + static RealType find_maximum_number_of_trials( + RealType k, // Number of failures. + RealType p, // Success fraction. + RealType probability); // Probability threshold alpha. + }; + + }} // namespaces + +The class type `negative_binomial_distribution` represents a +[@http://en.wikipedia.org/wiki/Negative_binomial_distribution negative_binomial distribution]: +it is used when there are exactly two mutually exclusive outcomes of a +[@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trial]: +these outcomes are labelled "success" and "failure". + +For k + r Bernoulli trials each with success fraction p, the +negative_binomial distribution gives the probability of observing +k failures and r successes with success on the last trial. +The negative_binomial distribution +assumes that success_fraction p is fixed for all (k + r) trials. + +[note The random variable for the negative binomial distribution is the number of trials, +(the number of successes is a fixed property of the distribution) +whereas for the binomial, +the random variable is the number of successes, for a fixed number of trials.] + + + +It has the PDF: + +[equation neg_binomial_ref] + +The following graph illustrate how the PDF varies as the success fraction +/p/ changes: + +[$../graphs/neg_binomial_pdf1.png] + +Alternatively, this graph shows how the shape of the PDF varies as +the number of successes changes: + +[$../graphs/neg_binomial_pdf2.png] + +[h4 Related Distributions] + +The name negative binomial distribution is reserved by some to the +case where the successes parameter r is an integer. +This integer version is also called the +[@http://mathworld.wolfram.com/PascalDistribution.html Pascal distribution]. + +This implementation uses real numbers for the computation throughout +(because it uses the *real-valued* incomplete beta function family of functions). +This real-valued version is also called the Polya Distribution. + +The Poisson distribution is a generalization of the Pascal distribution, +where the success parameter r is an integer: to obtain the Pascal +distribution you must ensure that an integer value is provided for r, +and take integer values (floor or ceiling) from functions that return +a number of successes. + +For large values of r (successes), the negative binomial distribution +converges to the Poisson distribution. + +The geometric distribution is a special case +where the successes parameter r = 1, +so only a first and only success is required. +geometric(p) = negative_binomial(1, p). + +The Poisson distribution is a special case for large successes + +poisson([lambda]) = lim [sub r [rarr] [infin]] [space] negative_binomial(r, r / ([lambda] + r))) + +[discrete_quantile_warning Negative Binomial] + +[h4 Member Functions] + +[h5 Construct] + + negative_binomial_distribution(RealType r, RealType p); + +Constructor: /r/ is the total number of successes, /p/ is the +probability of success of a single trial. + +Requires: `r > 0` and `0 <= p <= 1`. + +[h5 Accessors] + + RealType success_fraction() const; // successes / trials (0 <= p <= 1) + +Returns the parameter /p/ from which this distribution was constructed. + + RealType successes() const; // required successes (r > 0) + +Returns the parameter /r/ from which this distribution was constructed. + +[h5 Lower Bound on Parameter p] + + static RealType find_lower_bound_on_p( + RealType failures, + RealType successes, + RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. + +Returns a *lower bound* on the success fraction: + +[variablelist +[[failures][The total number of failures before the r th success.]] +[[successes][The number of successes required.]] +[[alpha][The largest acceptable probability that the true value of + the success fraction is [*less than] the value returned.]] +] + +For example, if you observe /k/ failures and /r/ successes from /n/ = k + r trials +the best estimate for the success fraction is simply ['r/n], but if you +want to be 95% sure that the true value is [*greater than] some value, +['p[sub min]], then: + + p``[sub min]`` = negative_binomial_distribution::find_lower_bound_on_p( + failures, successes, 0.05); + +[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.] + +This function uses the Clopper-Pearson method of computing the lower bound on the +success fraction, whilst many texts refer to this method as giving an "exact" +result in practice it produces an interval that guarantees ['at least] the +coverage required, and may produce pessimistic estimates for some combinations +of /failures/ and /successes/. See: + +[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf +Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions. +Computational statistics and data analysis, 2005, vol. 48, no3, 605-621]. + +[h5 Upper Bound on Parameter p] + + static RealType find_upper_bound_on_p( + RealType trials, + RealType successes, + RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. + +Returns an *upper bound* on the success fraction: + +[variablelist +[[trials][The total number of trials conducted.]] +[[successes][The number of successes that occurred.]] +[[alpha][The largest acceptable probability that the true value of + the success fraction is [*greater than] the value returned.]] +] + +For example, if you observe /k/ successes from /n/ trials the +best estimate for the success fraction is simply ['k/n], but if you +want to be 95% sure that the true value is [*less than] some value, +['p[sub max]], then: + + p``[sub max]`` = negative_binomial_distribution::find_upper_bound_on_p( + r, k, 0.05); + +[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.] + +This function uses the Clopper-Pearson method of computing the lower bound on the +success fraction, whilst many texts refer to this method as giving an "exact" +result in practice it produces an interval that guarantees ['at least] the +coverage required, and may produce pessimistic estimates for some combinations +of /failures/ and /successes/. See: + +[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf +Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions. +Computational statistics and data analysis, 2005, vol. 48, no3, 605-621]. + +[h5 Estimating Number of Trials to Ensure at Least a Certain Number of Failures] + + static RealType find_minimum_number_of_trials( + RealType k, // number of failures. + RealType p, // success fraction. + RealType alpha); // probability threshold (0.05 equivalent to 95%). + +This functions estimates the number of trials required to achieve a certain +probability that [*more than k failures will be observed]. + +[variablelist +[[k][The target number of failures to be observed.]] +[[p][The probability of ['success] for each trial.]] +[[alpha][The maximum acceptable risk that only k failures or fewer will be observed.]] +] + +For example: + + negative_binomial_distribution::find_minimum_number_of_trials(10, 0.5, 0.05); + +Returns the smallest number of trials we must conduct to be 95% sure +of seeing 10 failures that occur with frequency one half. + +[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.] + +This function uses numeric inversion of the negative binomial distribution +to obtain the result: another interpretation of the result, is that it finds +the number of trials (success+failures) that will lead to an /alpha/ probability +of observing k failures or fewer. + +[h5 Estimating Number of Trials to Ensure a Maximum Number of Failures or Less] + + static RealType find_maximum_number_of_trials( + RealType k, // number of failures. + RealType p, // success fraction. + RealType alpha); // probability threshold (0.05 equivalent to 95%). + +This functions estimates the maximum number of trials we can conduct and achieve +a certain probability that [*k failures or fewer will be observed]. + +[variablelist +[[k][The maximum number of failures to be observed.]] +[[p][The probability of ['success] for each trial.]] +[[alpha][The maximum acceptable ['risk] that more than k failures will be observed.]] +] + +For example: + + negative_binomial_distribution::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05); + +Returns the largest number of trials we can conduct and still be 95% sure +of seeing no failures that occur with frequency one in one million. + +This function uses numeric inversion of the negative binomial distribution +to obtain the result: another interpretation of the result, is that it finds +the number of trials (success+failures) that will lead to an /alpha/ probability +of observing more than k failures. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +However it's worth taking a moment to define what these actually mean in +the context of this distribution: + +[table Meaning of the non-member accessors. +[[Function][Meaning]] +[[__pdf] + [The probability of obtaining [*exactly k failures] from k+r trials + with success fraction p. For example: + +``pdf(negative_binomial(r, p), k)``]] +[[__cdf] + [The probability of obtaining [*k failures or fewer] from k+r trials + with success fraction p and success on the last trial. For example: + +``cdf(negative_binomial(r, p), k)``]] +[[__ccdf] + [The probability of obtaining [*more than k failures] from k+r trials + with success fraction p and success on the last trial. For example: + +``cdf(complement(negative_binomial(r, p), k))``]] +[[__quantile] + [The [*greatest] number of failures k expected to be observed from k+r trials + with success fraction p, at probability P. Note that the value returned + is a real-number, and not an integer. Depending on the use case you may + want to take either the floor or ceiling of the real result. For example: + +``quantile(negative_binomial(r, p), P)``]] +[[__quantile_c] + [The [*smallest] number of failures k expected to be observed from k+r trials + with success fraction p, at probability P. Note that the value returned + is a real-number, and not an integer. Depending on the use case you may + want to take either the floor or ceiling of the real result. For example: + ``quantile(complement(negative_binomial(r, p), P))``]] +] + +[h4 Accuracy] + +This distribution is implemented using the +incomplete beta functions __ibeta and __ibetac: +please refer to these functions for information on accuracy. + +[h4 Implementation] + +In the following table, /p/ is the probability that any one trial will +be successful (the success fraction), /r/ is the number of successes, +/k/ is the number of failures, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) * pow((1-p), k) + +Implementation is in terms of __ibeta_derivative: + +(p/(r + k)) * ibeta_derivative(r, static_cast(k+1), p) +The function __ibeta_derivative is used here, since it has already +been optimised for the lowest possible error - indeed this is really +just a thin wrapper around part of the internals of the incomplete +beta function. +]] +[[cdf][Using the relation: + +cdf = I[sub p](r, k+1) = ibeta(r, k+1, p) + += ibeta(r, static_cast(k+1), p)]] +[[cdf complement][Using the relation: + +1 - cdf = I[sub p](k+1, r) + += ibetac(r, static_cast(k+1), p) +]] +[[quantile][ibeta_invb(r, p, P) - 1]] +[[quantile from the complement][ibetac_invb(r, p, Q) -1)]] +[[mean][ `r(1-p)/p` ]] +[[variance][ `r (1-p) / p * p` ]] +[[mode][`floor((r-1) * (1 - p)/p)`]] +[[skewness][`(2 - p) / sqrt(r * (1 - p))`]] +[[kurtosis][`6 / r + (p * p) / r * (1 - p )`]] +[[kurtosis excess][`6 / r + (p * p) / r * (1 - p ) -3`]] +[[parameter estimation member functions][]] +[[`find_lower_bound_on_p`][ibeta_inv(successes, failures + 1, alpha)]] +[[`find_upper_bound_on_p`][ibetac_inv(successes, failures, alpha) plus see comments in code.]] +[[`find_minimum_number_of_trials`][ibeta_inva(k + 1, p, alpha)]] +[[`find_maximum_number_of_trials`][ibetac_inva(k + 1, p, alpha)]] +] + +Implementation notes: + +* The real concept type (that deliberately lacks the Lanczos approximation), +was found to take several minutes to evaluate some extreme test values, +so the test has been disabled for this type. + +* Much greater speed, and perhaps greater accuracy, +might be achieved for extreme values by using a normal approximation. +This is NOT been tested or implemented. + +[endsect][/section:negative_binomial_dist Negative Binomial] + +[/ negative_binomial.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/negative_binomial_example.qbk b/doc/sf_and_dist/distributions/negative_binomial_example.qbk new file mode 100644 index 000000000..c51817b88 --- /dev/null +++ b/doc/sf_and_dist/distributions/negative_binomial_example.qbk @@ -0,0 +1,192 @@ +[section:neg_binom_eg Negative Binomial Distribution Examples] + +(See also the reference documentation for the __negative_binomial_distrib.) + +[section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution] + +Imagine you have a process that follows a negative binomial distribution: +for each trial conducted, an event either occurs or does it does not, referred +to as "successes" and "failures". The frequency with which successes occur +is variously referred to as the +success fraction, success ratio, success percentage, occurrence frequency, or probability of occurrence. + +If, by experiment, you want to measure the + the best estimate of success fraction is given simply +by /k/ \/ /N/, for /k/ successes out of /N/ trials. + +However our confidence in that estimate will be shaped by how many trials were conducted, +and how many successes were observed. The static member functions +`negative_binomial_distribution<>::find_lower_bound_on_p` and +`negative_binomial_distribution<>::find_upper_bound_on_p` +allow you to calculate the confidence intervals for your estimate of the success fraction. + +The sample program [@../../../example/neg_binom_confidence_limits.cpp +neg_binom_confidence_limits.cpp] illustrates their use. + +[import ../../../example/neg_binom_confidence_limits.cpp] + +[neg_binomial_confidence_limits] +Let's see some sample output for a 1 in 10 +success ratio, first for a mere 20 trials: + +[pre'''______________________________________________ +2-Sided Confidence Limits For Success Fraction +______________________________________________ +Number of trials = 20 +Number of successes = 2 +Number of failures = 18 +Observed frequency of occurrence = 0.1 +___________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +___________________________________________ + 50.000 0.04812 0.13554 + 75.000 0.03078 0.17727 + 90.000 0.01807 0.22637 + 95.000 0.01235 0.26028 + 99.000 0.00530 0.33111 + 99.900 0.00164 0.41802 + 99.990 0.00051 0.49202 + 99.999 0.00016 0.55574 +'''] + +As you can see, even at the 95% confidence level the bounds (0.012 to 0.26) are +really very wide, and very asymmetric about the observed value 0.1. + +Compare that with the program output for a mass +2000 trials: + +[pre'''______________________________________________ +2-Sided Confidence Limits For Success Fraction +______________________________________________ +Number of trials = 2000 +Number of successes = 200 +Number of failures = 1800 +Observed frequency of occurrence = 0.1 +___________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +___________________________________________ + 50.000 0.09536 0.10445 + 75.000 0.09228 0.10776 + 90.000 0.08916 0.11125 + 95.000 0.08720 0.11352 + 99.000 0.08344 0.11802 + 99.900 0.07921 0.12336 + 99.990 0.07577 0.12795 + 99.999 0.07282 0.13206 +'''] + +Now even when the confidence level is very high, the limits (at 99.999%, 0.07 to 0.13) are really +quite close and nearly symmetric to the observed value of 0.1. + +[endsect][/section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence] + +[section:neg_binom_size_eg Estimating Sample Sizes for the Negative Binomial.] + +Imagine you have an event +(let's call it a "failure" - though we could equally well call it a success if we felt it was a 'good' event) +that you know will occur in 1 in N trials. You may want to know how many trials you need to +conduct to be P% sure of observing at least k such failures. +If the failure events follow a negative binomial +distribution (each trial either succeeds or fails) +then the static member function `negative_binomial_distibution<>::find_minimum_number_of_trials` +can be used to estimate the minimum number of trials required to be P% sure +of observing the desired number of failures. + +The example program +[@../../../example/neg_binomial_sample_sizes.cpp neg_binomial_sample_sizes.cpp] +demonstrates its usage. + +[import ../../../example/neg_binomial_sample_sizes.cpp] +[neg_binomial_sample_sizes] + +[note Since we're calculating the /minimum/ number of trials required, +we'll err on the safe side and take the ceiling of the result. +Had we been calculating the +/maximum/ number of trials permitted to observe less than a certain +number of /failures/ then we would have taken the floor instead. We +would also have called `find_minimum_number_of_trials` like this: +`` + floor(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i])) +`` +which would give us the largest number of trials we could conduct and +still be P% sure of observing /failures or less/ failure events, when the +probability of success is /p/.] + +We'll finish off by looking at some sample output, firstly suppose +we wish to observe at least 5 "failures" with a 50/50 (0.5) chance of +success or failure: + +[pre +'''Target number of failures = 5, Success fraction = 50% + +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 11 + 75.000 14 + 90.000 17 + 95.000 18 + 99.000 22 + 99.900 27 + 99.990 31 + 99.999 36 +''' +] + +So 18 trials or more would yield a 95% chance that at least our 5 +required failures would be observed. + +Compare that to what happens if the success ratio is 90%: + +[pre'''Target number of failures = 5.000, Success fraction = 90.000% + +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 57 + 75.000 73 + 90.000 91 + 95.000 103 + 99.000 127 + 99.900 159 + 99.990 189 + 99.999 217 +'''] + +So now 103 trials are required to observe at least 5 failures with +95% certainty. + +[endsect] [/section:neg_binom_size_eg Estimating Sample Sizes.] + +[section:negative_binomial_example1 Negative Binomial Sales Quota Example.] + +This example program +[@../../../example/negative_binomial_example1.cpp negative_binomial_example1.cpp (full source code)] +demonstrates a simple use to find the probability of meeting a sales quota. + +[import ../../../example/negative_binomial_example1.cpp] +[negative_binomial_eg1_1] +[negative_binomial_eg1_2] + +[endsect] [/section:negative_binomial_example1] + +[section:negative_binomial_example2 Negative Binomial Table Printing Example.] +Example program showing output of a table of values of cdf and pdf for various k failures. +[import ../../../example/negative_binomial_example2.cpp] +[neg_binomial_example2] +[neg_binomial_example2_1] +[endsect] [/section:negative_binomial_example1 Negative Binomial example 2.] + +[endsect] [/section:neg_binom_eg Negative Binomial Distribution Examples] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/non_members.qbk b/doc/sf_and_dist/distributions/non_members.qbk new file mode 100644 index 000000000..d94874226 --- /dev/null +++ b/doc/sf_and_dist/distributions/non_members.qbk @@ -0,0 +1,406 @@ +[section:nmp Non-Member Properties] + +Properties that are common to all distributions are accessed via non-member +getter functions. This allows more of these functions to be added over time +as the need arises. Unfortunately the literature uses many different and +confusing names to refer to a rather small number of actual concepts; refer +to the [link concept_index concept index] to find the property you +want by the name you are most familiar with. +Or use the [link function_index function index] +to go straight to the function you want if you already know its name. + +[h4 [#function_index]Function Index] + +* [link math.dist.cdf cdf]. +* [link math.dist.ccdf cdf complement]. +* [link math.dist.chf chf]. +* [link math.dist.hazard hazard]. +* __kurtosis. +* __kurtosis_excess +* __mean. +* [link math.dist.median median]. +* __mode. +* [link math.dist.pdf pdf]. +* [link math.dist.range range]. +* [link math.dist.quantile quantile]. +* [link math.dist.quantile_c quantile from the complement]. +* __skewness. +* [link math.dist.sd standard_deviation]. +* [link math.dist.support support]. +* __variance. + +[h4 [#concept_index]Conceptual Index] + +* __ccdf. +* __cdf. +* __chf. +* [link cdf_inv Inverse Cumulative Distribution Function]. +* [link survival_inv Inverse Survival Function]. +* __hazard +* [link lower_critical Lower Critical Value]. +* __kurtosis. +* __kurtosis_excess +* __mean. +* [link math.dist.median median]. +* __mode. +* [link cdfPQ P]. +* [link percent Percent Point Function]. +* __pdf. +* [link pmf Probability Mass Function]. +* [link math.dist.range range]. +* [link cdfPQ Q]. +* __quantile. +* [link math.dist.quantile_c Quantile from the complement of the probability]. +* __skewness. +* __sd +* [link survival Survival Function]. +* [link math.dist.support support]. +* [link upper_critical Upper Critical Value]. +* __variance. + +[h4 [#math.dist.cdf]Cumulative Distribution Function] + + template + RealType cdf(const ``['Distribution-Type]``& dist, const RealType& x); + +The __cdf is the probability that +the variable takes a value less than or equal to x. It is equivalent +to the integral from -infinity to x of the __pdf. + +This function may return a __domain_error if the random variable is outside +the defined range for the distribution. + +For example the following graph shows the cdf for the +normal distribution: + +[$../graphs/cdf.png] + +[h4 [#math.dist.ccdf]Complement of the Cumulative Distribution Function] + + template + RealType cdf(const ``['Unspecified-Complement-Type]``& comp); + +The complement of the __cdf +is the probability that +the variable takes a value greater than x. It is equivalent +to the integral from x to infinity of the __pdf, or 1 minus the __cdf of x. + +This is also known as the survival function. + +This function may return a __domain_error if the random variable is outside +the defined range for the distribution. + +In this library, it is obtained by wrapping the arguments to the `cdf` +function in a call to `complement`, for example: + + // standard normal distribution object: + boost::math::normal norm; + // print survival function for x=2.0: + std::cout << cdf(complement(norm, 2.0)) << std::endl; + +For example the following graph shows the __complement of the cdf for the +normal distribution: + +[$../graphs/survival.png] + +See __why_complements for why the complement is useful and when it should be used. + +[h4 [#math.dist.hazard]Hazard Function] + + template + RealType hazard(const ``['Distribution-Type]``& dist, const RealType& x); + +Returns the __hazard of /x/ and distibution /dist/. + +This function may return a __domain_error if the random variable is outside +the defined range for the distribution. + +[equation hazard] + +[caution +Some authors refer to this as the conditional failure +density function rather than the hazard function.] + +[h4 [#math.dist.chf]Cumulative Hazard Function] + + template + RealType chf(const ``['Distribution-Type]``& dist, const RealType& x); + +Returns the __chf of /x/ and distibution /dist/. + +This function may return a __domain_error if the random variable is outside +the defined range for the distribution. + +[equation chf] + +[caution +Some authors refer to this as simply the "Hazard Function".] + +[h4 [#math.dist.mean]mean] + + template + RealType mean(const ``['Distribution-Type]``& dist); + +Returns the mean of the distribution /dist/. + +This function may return a __domain_error if the distribution does not have +a defined mean (for example the Cauchy distribution). + +[h4 [#math.dist.median]median] + + template + RealType median(const ``['Distribution-Type]``& dist); + +Returns the median of the distribution /dist/. + +[h4 [#math.dist.mode]mode] + + template + RealType mode(const ``['Distribution-Type]``& dist); + +Returns the mode of the distribution /dist/. + +This function may return a __domain_error if the distribution does not have +a defined mode. + +[h4 [#math.dist.pdf]Probability Density Function] + + template + RealType pdf(const ``['Distribution-Type]``& dist, const RealType& x); + +For a continuous function, the probability density function (pdf) returns +the probability that the variate has the value x. +Since for continuous distributions the probability at a single point is actually zero, +the probability is better expressed as the integral of the pdf between two points: +see the __cdf. + +For a discrete distribution, the pdf is the probability that the +variate takes the value x. + +This function may return a __domain_error if the random variable is outside +the defined range for the distribution. + +For example for a standard normal distribution the pdf looks like this: + +[$../graphs/pdf.png] + +[h4 [#math.dist.range]range] + + template + std::pair range(const ``['Distribution-Type]``& dist); + +Returns the valid range of the random variable over distribution /dist/. + +[h4 [#math.dist.quantile]Quantile] + + template + RealType quantile(const ``['Distribution-Type]``& dist, const RealType& p); + +The quantile is best viewed as the inverse of the __cdf, it returns +a value /x/ such that `cdf(dist, x) == p`. + +This is also known as the /percent point function/, or a /percentile/, it is +also the same as calculating the ['lower critical value] of a distribution. + +This function returns a __domain_error if the probability lies outside [0,1]. +The function may return an __overflow_error if there is no finite value +that has the specified probability. + +The following graph shows the quantile function for a standard normal +distribution: + +[$../graphs/quantile.png] + +[h4 [#math.dist.quantile_c]Quantile from the complement of the probability.] +[link complements complements] + + + template + RealType quantile(const ``['Unspecified-Complement-Type]``& comp); + +This is the inverse of the __ccdf. It is calculated by wrapping +the arguments in a call to the quantile function in a call to +/complement/. For example: + + // define a standard normal distribution: + boost::math::normal norm; + // print the value of x for which the complement + // of the probability is 0.05: + std::cout << quantile(complement(norm, 0.05)) << std::endl; + +The function computes a value /x/ such that +`cdf(complement(dist, x)) == q` where /q/ is complement of the +probability. + +[link why_complements Why complements?] + +This function is also called the inverse survival function, and is the +same as calculating the ['upper critical value] of a distribution. + +This function returns a __domain_error if the probablity lies outside [0,1]. +The function may return an __overflow_error if there is no finite value +that has the specified probability. + +The following graph show the inverse survival function for the normal +distribution: + +[$../graphs/survival_inv.png] + +[h4 [#math.dist.sd]Standard Deviation] + + template + RealType standard_deviation(const ``['Distribution-Type]``& dist); + +Returns the standard deviation of distribution /dist/. + +This function may return a __domain_error if the distribution does not have +a defined standard deviation. + +[h4 [#math.dist.support]support] + + template + std::pair support(const ``['Distribution-Type]``& dist); + +Returns the supported range of random variable over the distribution /dist/. + +The distribution is said to be 'supported' over a range that is +[@http://en.wikipedia.org/wiki/Probability_distribution + "the smallest closed set whose complement has probability zero"]. +Non-mathematicians might say it means the 'interesting' smallest range +of random variate x that has the cdf going from zero to unity. +Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity. + +[h4 [#math.dist.variance]Variance] + + template + RealType variance(const ``['Distribution-Type]``& dist); + +Returns the variance of the distribution /dist/. + +This function may return a __domain_error if the distribution does not have +a defined variance. + +[h4 [#math.dist.skewness]Skewness] + + template + RealType skewness(const ``['Distribution-Type]``& dist); + +Returns the skewness of the distribution /dist/. + +This function may return a __domain_error if the distribution does not have +a defined skewness. + +[h4 [#math.dist.kurtosis]Kurtosis] + + template + RealType kurtosis(const ``['Distribution-Type]``& dist); + +Returns the 'proper' kurtosis (normalized fourth moment) of the distribution /dist/. + +kertosis = [beta][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2] + +Where [mu][sub i][space] is the i'th central moment of the distribution, and +in particular [mu][sub 2][space] is the variance of the distribution. + +The kurtosis is a measure of the "peakedness" of a distribution. + +Note that the literature definition of kurtosis is confusing. +The definition used here is that used by for example +[@http://mathworld.wolfram.com/Kurtosis.html Wolfram MathWorld] +(that includes a table of formulae for kurtosis excess for various distributions) +but NOT the definition of +[@http://en.wikipedia.org/wiki/Kurtosis kurtosis used by Wikipedia] +which treats "kurtosis" and "kurtosis excess" as the same quantity. + + kurtosis_excess = 'proper' kurtosis - 3 + +This subtraction of 3 is convenient so that the ['kurtosis excess] +of a normal distribution is zero. + +This function may return a __domain_error if the distribution does not have +a defined kurtosis. + +'Proper' kurtosis can have a value from zero to + infinity. + +[h4 [#math.dist.kurtosis_excess]Kurtosis excess] + + template + RealType kurtosis_excess(const ``['Distribution-Type]``& dist); + +Returns the kurtosis excess of the distribution /dist/. + +kurtosis excess = [gamma][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2][space]- 3 = kurtosis - 3 + +Where [mu][sub i][space] is the i'th central moment of the distribution, and +in particular [mu][sub 2][space] is the variance of the distribution. + +The kurtosis excess is a measure of the "peakedness" of a distribution, and +is more widely used than the "kurtosis proper". It is defined so that +the kurtosis excess of a normal distribution is zero. + +This function may return a __domain_error if the distribution does not have +a defined kurtosis excess. + +Kurtosis excess can have a value from -2 to + infinity. + + kurtosis = kurtosis_excess +3; + +The kurtosis excess of a normal distribution is zero. + +[h4 [#cdfPQ]P and Q] + +The terms P and Q are sometimes used to refer to the __cdf +and its [link math.dist.ccdf complement] respectively. +Lowercase p and q are sometimes used to refer to the values returned +by these functions. + +[h4 [#percent]Percent Point Function] + +The percent point function, also known as the percentile, is the same as +the __quantile. + +[h4 [#cdf_inv]Inverse CDF Function.] + +The inverse of the cumulative distribution function, is the same as the +__quantile. + +[h4 [#survival_inv]Inverse Survival Function.] + +The inverse of the survival function, is the same as computing the +[link math.dist.quantile_c quantile +from the complement of the probability]. + +[h4 [#pmf]Probability Mass Function] + +The Probability Mass Function is the same as the __pdf. + +The term Mass Function is usually applied to discrete distributions, +while the term __pdf applies to continuous distributions. + +[h4 [#lower_critical]Lower Critical Value.] + +The lower critical value calculates the value of the random variable +given the area under the left tail of the distribution. +It is equivalent to calculating the __quantile. + +[h4 [#upper_critical]Upper Critical Value.] + +The upper critical value calculates the value of the random variable +given the area under the right tail of the distribution. It is equivalent to +calculating the [link math.dist.quantile_c quantile from the complement of the +probability]. + +[h4 [#survival]Survival Function] + +Refer to the __ccdf. + +[endsect][/section:nmp Non-Member Properties] + + +[/ non_members.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/normal.qbk b/doc/sf_and_dist/distributions/normal.qbk new file mode 100644 index 000000000..1b057dd44 --- /dev/null +++ b/doc/sf_and_dist/distributions/normal.qbk @@ -0,0 +1,110 @@ +[section:normal_dist Normal (Gaussian) Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class normal_distribution; + + typedef normal_distribution<> normal; + + template + class normal_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Construct: + normal_distribution(RealType mean = 0, RealType sd = 1); + // Accessors: + RealType mean()const; // location. + RealType standard_deviation()const; // scale. + // Synonyms, provided to allow generic use of find_location and find_scale. + RealType location()const; + RealType scale()const; + }; + + }} // namespaces + +The normal distribution is probably the most well known statistical +distribution: it is also known as the Gaussian Distribution. +A normal distribution with mean zero and standard deviation one +is known as the ['Standard Normal Distribution]. + +Given mean [mu][space] and standard deviation [sigma][space] it has the PDF: + +[equation normal_ref1] + +The variation the PDF with its parameters is illustrated +in the following graph: + +[$../graphs/normal.png] + +[h4 Member Functions] + + normal_distribution(RealType mean = 0, RealType sd = 1); + +Constructs a normal distribution with mean /mean/ and +standard deviation /sd/. + +Requires sd > 0, otherwise __domain_error is called. + + RealType mean()const; + RealType location()const; + +both return the /mean/ of this distribution. + + RealType standard_deviation()const; + RealType scale()const; + +both return the /standard deviation/ of this distribution. +(Redundant location and scale function are provided to match other similar distributions, +allowing the functions find_location and find_scale to be used generically). + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[-[max_value], +[min_value]\]. +However, the pdf of +[infin] and -[infin] = 0 is also supported, +and cdf at -[infin] = 0, cdf at +[infin] = 1, +and complement cdf -[infin] = 1 and +[infin] = 0, +if RealType permits. + +[h4 Accuracy] + +The normal distribution is implemented in terms of the +[link math_toolkit.special.sf_erf.error_function error function], +and as such should have very low error rates. + +[h4 Implementation] + +In the following table /m/ is the mean of the distribution, +and /s/ is its standard deviation. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = e[super -(x-m)[super 2]\/(2s[super 2])] \/ (s * sqrt(2*pi)) ]] +[[cdf][Using the relation: p = 0.5 * __erfc(-(x-m)/(s*sqrt(2))) ]] +[[cdf complement][Using the relation: q = 0.5 * __erfc((x-m)/(s*sqrt(2))) ]] +[[quantile][Using the relation: x = m - s * sqrt(2) * __erfc_inv(2*p)]] +[[quantile from the complement][Using the relation: x = m + s * sqrt(2) * __erfc_inv(2*p)]] +[[mean and standard deviation][The same as `dist.mean()` and `dist.standard_deviation()`]] +[[mode][The same as the mean.]] +[[skewness][0]] +[[kurtosis][3]] +[[kurtosis excess][0]] +] + +[endsect][/section:normal_dist Normal] + +[/ normal.qbk + Copyright 2006, 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/normal_example.qbk b/doc/sf_and_dist/distributions/normal_example.qbk new file mode 100644 index 000000000..d5cafca11 --- /dev/null +++ b/doc/sf_and_dist/distributions/normal_example.qbk @@ -0,0 +1,36 @@ +[section:normal_example Normal Distribution Examples] + +(See also the reference documentation for the __normal_distrib.) + +[section:normal_misc Some Miscellaneous Examples of the Normal (Gaussian) Distribution] + +The sample program [@../../../example/normal_misc_examples.cpp +normal_misc_examples.cpp] illustrates their use. + +[import ../../../example/normal_misc_examples.cpp] + +[h4 Traditional Tables] +[normal_basic1] + +[h4 Standard deviations either side of the Mean] +[normal_basic2] +[h4 Some simple examples] +[h4 Life of light bulbs] +[normal_bulbs_example1] +[h4 How many onions?] +[normal_bulbs_example3] +[h4 Packing beef] +[normal_bulbs_example4] +[h4 Length of bolts] +[normal_bulbs_example5] + +[endsect] [/section:normal_misc Some Miscellaneous Examples of the Normal Distribution] +[endsect] [/section:normal_example Normal Distribution Examples] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/pareto.qbk b/doc/sf_and_dist/distributions/pareto.qbk new file mode 100644 index 000000000..040b3c622 --- /dev/null +++ b/doc/sf_and_dist/distributions/pareto.qbk @@ -0,0 +1,114 @@ +[section:pareto Pareto Distribution] + + +``#include `` + + namespace boost{ namespace math{ + + template + class pareto_distribution; + + typedef pareto_distribution<> pareto; + + template + class pareto_distribution + { + public: + typedef RealType value_type; + // Constructor: + pareto_distribution(RealType location = 1, RealType shape = 1) + // Accessors: + RealType location()const; + RealType shape()const; + }; + + }} // namespaces + +The [@http://en.wikipedia.org/wiki/pareto_distribution Pareto distribution] +is a continuous distribution with the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function (pdf)]: + +f(x; [alpha], [beta]) = [alpha][beta][super [alpha]] / x[super [alpha]+ 1] + +For shape parameter [alpha][space] > 0, and location parameter [beta][space] > 0, and [alpha][space] > 0. + +The [@http://mathworld.wolfram.com/paretoDistribution.html Pareto distribution] +often describes the larger compared to the smaller. +A classic example is that 80% of the wealth is owned by 20% of the population. + +The following graph illustrates how the PDF varies with the shape parameter [alpha]: + +[/$../graphs/paretoShape.png] +[/ TODO produce a graph as png or svg] +[@http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Pareto_distributionPDF.png/325px-Pareto_distributionPDF.png Pareto pdf] + +[h4 Related distributions] + + +[h4 Member Functions] + + pareto_distribution(RealType location = 1, RealType shape = 1); + +Constructs a [@http://en.wikipedia.org/wiki/pareto_distribution +pareto distribution] with shape /shape/ and scale /scale/. + +Requires that the /shape/ and /scale/ parameters are both greater than zero, +otherwise calls __domain_error. + + RealType location()const; + +Returns the /location/ parameter of this distribution. + + RealType shape()const; + +Returns the /shape/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The supported domain of the random variable is \[location, [infin]\]. + +[h4 Accuracy] + +The pareto distribution is implemented in terms of the +standard library `exp` functions plus __expm1 +and as such should have very low error rates +except when probability is very close to unity. + +[h4 Implementation] + +In the following table [alpha][space] is the shape parameter of the distribution, and +[beta][space] is its location parameter, /x/ is the random variate, /p/ is the probability +and its complement /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf p = [alpha][beta][super [alpha]]/x[super [alpha] +1] ]] +[[cdf][Using the relation: cdf p = 1 - ([beta][space] / x)[super [alpha]] ]] +[[cdf complement][Using the relation: q = 1 - p = -([beta][space] / x)[super [alpha]] ]] +[[quantile][Using the relation: x = [alpha] / (1 - p)[super 1/[beta]] ]] +[[quantile from the complement][Using the relation: x = [alpha] / (q)[super 1/[beta]] ]] +[[mean][[alpha][beta] / ([beta] - 1) ]] +[[variance][[beta][alpha][super 2] / ([beta] - 1)[super 2] ([beta] - 2) ]] +[[mode][[alpha]]] +[[skewness][Refer to [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis][Refer to [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "pareto Distribution." From MathWorld--A Wolfram Web Resource.] ]] +] + +[h4 References] +* [@http://en.wikipedia.org/wiki/pareto_distribution Pareto Distribution] +* [@http://mathworld.wolfram.com/paretoDistribution.html Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource.] + +[endsect][/section:pareto pareto] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/poisson.qbk b/doc/sf_and_dist/distributions/poisson.qbk new file mode 100644 index 000000000..e65638a97 --- /dev/null +++ b/doc/sf_and_dist/distributions/poisson.qbk @@ -0,0 +1,103 @@ +[section:poisson_dist Poisson Distribution] + +``#include `` + + namespace boost { namespace math { + + template + class poisson_distribution; + + typedef poisson_distribution<> poisson; + + template + class poisson_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + poisson_distribution(RealType mean = 1); // Constructor. + RealType mean()const; // Accessor. + } + + }} // namespaces boost::math + +The [@http://en.wikipedia.org/wiki/Poisson_distribution Poisson distribution] +is a well-known statistical discrete distribution. +It expresses the probability of a number of events +(or failures, arrivals, occurrences ...) +occurring in a fixed period of time, +provided these events occur with a known mean rate [lambda][space] +(events/time), and are independent of the time since the last event. + +The distribution was discovered by Sim__eacute on-Denis Poisson (1781 to 1840). + +It has the Probability Mass Function: + +[equation poisson_ref1] + +for k events, with an expected number of events [lambda]. + +The following graph illustrates how the PDF varies with the parameter [lambda]: + +[$../graphs/poisson.png] + +[discrete_quantile_warning Poisson] + +[h4 Member Functions] + + poisson_distribution(RealType mean = 1); + +Constructs a poisson distribution with mean /mean/. + + RealType mean()const; + +Returns the /mean/ of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, [infin]\]. + +[h4 Accuracy] + +The Poisson distribution is implemented in terms of the +incomplete gamma functions __gamma_p and __gamma_q +and as such should have low error rates: but refer to the documentation +of those functions for more information. +The quantile and its complement use the inverse gamma functions +and are therefore probably slightly less accurate: this is because the +inverse gamma functions are implemented using an iterative method with a +lower tolerance to avoid excessive computation. + +[h4 Implementation] + +In the following table [lambda][space] is the mean of the distribution, +/k/ is the random variable, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = e[super -[lambda]] [lambda][super k] \/ k! ]] +[[cdf][Using the relation: p = [Gamma](k+1, [lambda]) \/ k! = __gamma_q(k+1, [lambda])]] +[[cdf complement][Using the relation: q = __gamma_p(k+1, [lambda]) ]] +[[quantile][Using the relation: k = __gamma_q_inva([lambda], p) - 1]] +[[quantile from the complement][Using the relation: k = __gamma_p_inva([lambda], q) - 1]] +[[mean][[lambda]]] +[[mode][ floor ([lambda]) or [floorlr[lambda]] ]] +[[skewness][1/[radic][lambda]]] +[[kurtosis][3 + 1/[lambda]]] +[[kurtosis excess][1/[lambda]]] +] + +[/ poisson.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + +[endsect][/section:poisson_dist Poisson] + diff --git a/doc/sf_and_dist/distributions/rayleigh.qbk b/doc/sf_and_dist/distributions/rayleigh.qbk new file mode 100644 index 000000000..8fa768406 --- /dev/null +++ b/doc/sf_and_dist/distributions/rayleigh.qbk @@ -0,0 +1,119 @@ +[section:rayleigh Rayleigh Distribution] + + +``#include `` + + namespace boost{ namespace math{ + + template + class rayleigh_distribution; + + typedef rayleigh_distribution<> rayleigh; + + template + class rayleigh_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Construct: + rayleigh_distribution(RealType sigma = 1) + // Accessors: + RealType sigma()const; + }; + + }} // namespaces + +The [@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution] +is a continuous distribution with the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: + +f(x; sigma) = x * exp(-x[super 2]/2 [sigma][super 2]) / [sigma][super 2] + +For sigma parameter [sigma][space] > 0, and x > 0. + +The Rayleigh distribution is often used where two orthogonal components +have an absolute value, +for example, wind velocity and direction may be combined to yield a wind speed, +or real and imaginary components may have absolute values that are Rayleigh distributed. + +The following graph illustrates how the Probability density Function(pdf) varies with the shape parameter [sigma]: + +[$../graphs/rayleigh_pdf.png] + +and the Cumulative Distribution Function (cdf) + +[$../graphs/rayleigh_cdf.png] + +[h4 Related distributions] + +The absolute value of two independent normal distributions X and Y, [radic] (X[super 2] + Y[super 2]) +is a Rayleigh distribution. + +The [@http://en.wikipedia.org/wiki/Chi_distribution Chi], +[@http://en.wikipedia.org/wiki/Rice_distribution Rice] +and [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull] distributions are generalizations of the +[@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution]. + +[h4 Member Functions] + + rayleigh_distribution(RealType sigma = 1); + +Constructs a [@http://en.wikipedia.org/wiki/Rayleigh_distribution +Rayleigh distribution] with [sigma] /sigma/. + +Requires that the [sigma] parameter is greater than zero, +otherwise calls __domain_error. + + RealType sigma()const; + +Returns the /sigma/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, max_value\]. + +[h4 Accuracy] + +The Rayleigh distribution is implemented in terms of the +standard library `sqrt` and `exp` and as such should have very low error rates. +Some constants such as skewness and kurtosis were calculated using +NTL RR type with 150-bit accuracy, about 50 decimal digits. + +[h4 Implementation] + +In the following table [sigma][space] is the sigma parameter of the distribution, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = x * exp(-x[super 2])/2 [sigma][super 2] ]] +[[cdf][Using the relation: p = 1 - exp(-x[super 2]/2) [sigma][super 2][space] = -__expm1(-x[super 2]/2) [sigma][super 2]]] +[[cdf complement][Using the relation: q = exp(-x[super 2]/ 2) * [sigma][super 2] ]] +[[quantile][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(1 - p)) = sqrt(-2 * [sigma] [super 2]) * __log1p(-p))]] +[[quantile from the complement][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(q)) ]] +[[mean][[sigma] * sqrt([pi]/2) ]] +[[variance][[sigma][super 2] * (4 - [pi]/2) ]] +[[mode][[sigma] ]] +[[skewness][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis excess][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +] + +[h4 References] +* [@http://en.wikipedia.org/wiki/Rayleigh_distribution ] +* [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource.] + +[endsect][/section:Rayleigh Rayleigh] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/students_t.qbk b/doc/sf_and_dist/distributions/students_t.qbk new file mode 100644 index 000000000..25fcdf384 --- /dev/null +++ b/doc/sf_and_dist/distributions/students_t.qbk @@ -0,0 +1,171 @@ +[section:students_t_dist Students t Distribution] + +``#include `` + + namespace boost{ namespace math{ + + template + class students_t_distribution; + + typedef students_t_distribution<> students_t; + + template + class students_t_distribution + { + typedef RealType value_type; + typedef Policy policy_type; + + // Construct: + students_t_distribution(const RealType& v); + + // Accessor: + RealType degrees_of_freedom()const; + + // degrees of freedom estimation: + static RealType find_degrees_of_freedom( + RealType difference_from_mean, + RealType alpha, + RealType beta, + RealType sd, + RealType hint = 100); + }; + + }} // namespaces + +A statistical distribution published by William Gosset in 1908. +His employer, Guinness Breweries, required him to publish under a +pseudonym, so he chose "Student". Given N independent measurements, let + +[equation students_t_dist] + +where /M/ is the population mean,[' ''' μ '''] is the sample mean, and /s/ is the +sample variance. + +Student's t-distribution is defined as the distribution of the random +variable t which is - very loosely - the "best" that we can do not +knowing the true standard deviation of the sample. It has the PDF: + +[equation students_t_ref1] + +The Student's t-distribution takes a single parameter: the number of +degrees of freedom of the sample. When the degrees of freedom is +/one/ then this distribution is the same as the Cauchy-distribution. +As the number of degrees of freedom tends towards infinity, then this +distribution approaches the normal-distribution. The following graph +illustrates how the PDF varies with the degrees of freedom [nu]: + +[$../graphs/students_t.png] + +[h4 Member Functions] + + students_t_distribution(const RealType& v); + +Constructs a Student's t-distribution with /v/ degrees of freedom. + +Requires v > 0, otherwise calls __domain_error. Note that +non-integral degrees of freedom are supported, and +meaningful under certain circumstances. + + RealType degrees_of_freedom()const; + +Returns the number of degrees of freedom of this distribution. + + static RealType find_degrees_of_freedom( + RealType difference_from_mean, + RealType alpha, + RealType beta, + RealType sd, + RealType hint = 100); + +Returns the number of degrees of freedom required to observe a significant +result in the Student's t test when the mean differs from the "true" +mean by /difference_from_mean/. + +[variablelist +[[difference_from_mean][The difference between the true mean and the sample mean + that we wish to show is significant.]] +[[alpha][The maximum acceptable probability of rejecting the null hypothesis + when it is in fact true.]] +[[beta][The maximum acceptable probability of failing to reject the null hypothesis + when it is in fact false.]] +[[sd][The sample standard deviation.]] +[[hint][A hint for the location to start looking for the result, a good choice for this + would be the sample size of a previous borderline Student's t test.]] +] + +[note +Remember that for a two-sided test, you must divide alpha by two +before calling this function.] + +For more information on this function see the +[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm +NIST Engineering Statistics Handbook]. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[-[infin], +[infin]\]. + +[h4 Examples] + +Various [link math_toolkit.dist.stat_tut.weg.st_eg worked examples] are available illustrating the use of the Student's t +distribution. + +[h4 Accuracy] + +The normal distribution is implemented in terms of the +[link math_toolkit.special.sf_beta.ibeta_function incomplete beta function] +and [link math_toolkit.special.sf_beta.ibeta_inv_function it's inverses], +refer to accuracy data on those functions for more information. + +[h4 Implementation] + +In the following table /v/ is the degrees of freedom of the distribution, +/t/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = (v \/ (v + t[super 2]))[super (1+v)\/2 ] / (sqrt(v) * __beta(v\/2, 0.5)) ]] +[[cdf][Using the relations: + +p = 1 - z /iff t > 0/ + +p = z /otherwise/ + +where z is given by: + +__ibeta(v \/ 2, 0.5, v \/ (v + t[super 2])) \/ 2 ['iff v < 2t[super 2]] + +__ibetac(0.5, v \/ 2, t[super 2 ] / (v + t[super 2]) \/ 2 /otherwise/]] +[[cdf complement][Using the relation: q = cdf(-t) ]] +[[quantile][Using the relation: t = sign(p - 0.5) * sqrt(v * y \/ x) + +where: + +x = __ibeta_inv(v \/ 2, 0.5, 2 * min(p, q)) + +y = 1 - x + +The quantities /x/ and /y/ are both returned by __ibeta_inv +without the subtraction implied above.]] +[[quantile from the complement][Using the relation: t = -quantile(q)]] +[[mean][0]] +[[variance][v \/ (v - 2)]] +[[mode][0]] +[[skewness][0]] +[[kurtosis][3 * (v - 2) \/ (v - 4)]] +[[kurtosis excess][6 \/ (df - 4)]] +] + +[endsect][/section:students_t_dist Students t] + +[/ students_t.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/students_t_examples.qbk b/doc/sf_and_dist/distributions/students_t_examples.qbk new file mode 100644 index 000000000..bc4c35dea --- /dev/null +++ b/doc/sf_and_dist/distributions/students_t_examples.qbk @@ -0,0 +1,782 @@ + +[section:st_eg Student's t Distribution Examples] + +[section:tut_mean_intervals Calculating confidence intervals on the mean with the Students-t distribution] + +Let's say you have a sample mean, you may wish to know what confidence intervals +you can place on that mean. Colloquially: "I want an interval that I can be +P% sure contains the true mean". (On a technical point, note that +the interval either contains the true mean or it does not: the +meaning of the confidence level is subtly +different from this colloquialism. More background information can be found on the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm NIST site]). + +The formula for the interval can be expressed as: + +[equation dist_tutorial4] + +Where, ['Y[sub s]] is the sample mean, /s/ is the sample standard deviation, +/N/ is the sample size, /[alpha]/ is the desired significance level and +['t[sub ([alpha]/2,N-1)]] is the upper critical value of the Students-t +distribution with /N-1/ degrees of freedom. + +[note +The quantity [alpha][space] is the maximum acceptable risk of falsely rejecting +the null-hypothesis. The smaller the value of [alpha] the greater the +strength of the test. + +The confidence level of the test is defined as 1 - [alpha], and often expressed +as a percentage. So for example a significance level of 0.05, is equivalent +to a 95% confidence level. Refer to +[@http://www.itl.nist.gov/div898/handbook/prc/section1/prc14.htm +"What are confidence intervals?"] in __handbook for more information. +] [/ Note] + +[note +The usual assumptions of +[@http://en.wikipedia.org/wiki/Independent_and_identically-distributed_random_variables independent and identically distributed (i.i.d.)] +variables and [@http://en.wikipedia.org/wiki/Normal_distribution normal distribution] +of course apply here, as they do in other examples. +] + +From the formula, it should be clear that: + +* The width of the confidence interval decreases as the sample size increases. +* The width increases as the standard deviation increases. +* The width increases as the ['confidence level increases] (0.5 towards 0.99999 - stronger). +* The width increases as the ['significance level decreases] (0.5 towards 0.00000...01 - stronger). + +The following example code is taken from the example program +[@../../../example/students_t_single_sample.cpp students_t_single_sample.cpp]. + +We'll begin by defining a procedure to calculate intervals for +various confidence levels; the procedure will print these out +as a table: + + // Needed includes: + #include + #include + #include + // Bring everything into global namespace for ease of use: + using namespace boost::math; + using namespace std; + + void confidence_limits_on_mean( + double Sm, // Sm = Sample Mean. + double Sd, // Sd = Sample Standard Deviation. + unsigned Sn) // Sn = Sample Size. + { + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "__________________________________\n" + "2-Sided Confidence Limits For Mean\n" + "__________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n"; + cout << setw(40) << left << "Mean" << "= " << Sm << "\n"; + cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n"; + +We'll define a table of significance/risk levels for which we'll compute intervals: + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +Note that these are the complements of the confidence/probability levels: 0.5, 0.75, 0.9 .. 0.99999). + +Next we'll declare the distribution object we'll need, note that +the /degrees of freedom/ parameter is the sample size less one: + + students_t dist(Sn - 1); + +Most of what follows in the program is pretty printing, so let's focus +on the calculation of the interval. First we need the t-statistic, +computed using the /quantile/ function and our significance level. Note +that since the significance levels are the complement of the probability, +we have to wrap the arguments in a call to /complement(...)/: + + double T = quantile(complement(dist, alpha[i] / 2)); + +Note that alpha was divided by two, since we'll be calculating +both the upper and lower bounds: had we been interested in a single +sided interval then we would have omitted this step. + +Now to complete the picture, we'll get the (one-sided) width of the +interval from the t-statistic +by multiplying by the standard deviation, and dividing by the square +root of the sample size: + + double w = T * Sd / sqrt(double(Sn)); + +The two-sided interval is then the sample mean plus and minus this width. + +And apart from some more pretty-printing that completes the procedure. + +Let's take a look at some sample output, first using the +[@http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm +Heat flow data] from the NIST site. The data set was collected +by Bob Zarr of NIST in January, 1990 from a heat flow meter +calibration and stability analysis. +The corresponding dataplot +output for this test can be found in +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm +section 3.5.2] of the __handbook. + +[pre''' + __________________________________ + 2-Sided Confidence Limits For Mean + __________________________________ + + Number of Observations = 195 + Mean = 9.26146 + Standard Deviation = 0.02278881 + + + ___________________________________________________________________ + Confidence T Interval Lower Upper + Value (%) Value Width Limit Limit + ___________________________________________________________________ + 50.000 0.676 1.103e-003 9.26036 9.26256 + 75.000 1.154 1.883e-003 9.25958 9.26334 + 90.000 1.653 2.697e-003 9.25876 9.26416 + 95.000 1.972 3.219e-003 9.25824 9.26468 + 99.000 2.601 4.245e-003 9.25721 9.26571 + 99.900 3.341 5.453e-003 9.25601 9.26691 + 99.990 3.973 6.484e-003 9.25498 9.26794 + 99.999 4.537 7.404e-003 9.25406 9.26886 +'''] + +As you can see the large sample size (195) and small standard deviation (0.023) +have combined to give very small intervals, indeed we can be +very confident that the true mean is 9.2. + +For comparison the next example data output is taken from +['P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. +and from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 +J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907.] +The values result from the determination of mercury by cold-vapour +atomic absorption. + +[pre''' + __________________________________ + 2-Sided Confidence Limits For Mean + __________________________________ + + Number of Observations = 3 + Mean = 37.8000000 + Standard Deviation = 0.9643650 + + + ___________________________________________________________________ + Confidence T Interval Lower Upper + Value (%) Value Width Limit Limit + ___________________________________________________________________ + 50.000 0.816 0.455 37.34539 38.25461 + 75.000 1.604 0.893 36.90717 38.69283 + 90.000 2.920 1.626 36.17422 39.42578 + 95.000 4.303 2.396 35.40438 40.19562 + 99.000 9.925 5.526 32.27408 43.32592 + 99.900 31.599 17.594 20.20639 55.39361 + 99.990 99.992 55.673 -17.87346 93.47346 + 99.999 316.225 176.067 -138.26683 213.86683 +'''] + +This time the fact that there are only three measurements leads to +much wider intervals, indeed such large intervals that it's hard +to be very confident in the location of the mean. + +[endsect] + +[section:tut_mean_test Testing a sample mean for difference from a "true" mean] + +When calibrating or comparing a scientific instrument or measurement method of some kind, +we want to be answer the question "Does an observed sample mean differ from the +"true" mean in any significant way?". If it does, then we have evidence of +a systematic difference. This question can be answered with a Students-t test: +more information can be found +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm +on the NIST site]. + +Of course, the assignment of "true" to one mean may be quite arbitrary, +often this is simply a "traditional" method of measurement. + +The following example code is taken from the example program +[@../../../example/students_t_single_sample.cpp students_t_single_sample.cpp]. + +We'll begin by defining a procedure to determine which of the +possible hypothesis are rejected or not-rejected +at a given significance level: + +[note +Non-statisticians might say 'not-rejected' means 'accepted', +(often of the null-hypothesis) implying, wrongly, that there really *IS* no difference, +but statisticans eschew this to avoid implying that there is positive evidence of 'no difference'. +'Not-rejected' here means there is *no evidence* of difference, but there still might well be a difference. +For example, see [@http://en.wikipedia.org/wiki/Argument_from_ignorance argument from ignorance] and +[@http://www.bmj.com/cgi/content/full/311/7003/485 Absence of evidence does not constitute evidence of absence.] +] [/ note] + + + // Needed includes: + #include + #include + #include + // Bring everything into global namespace for ease of use: + using namespace boost::math; + using namespace std; + + void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha) + { + // + // M = true mean. + // Sm = Sample Mean. + // Sd = Sample Standard Deviation. + // Sn = Sample Size. + // alpha = Significance Level. + +Most of the procedure is pretty-printing, so let's just focus on the +calculation, we begin by calculating the t-statistic: + + // Difference in means: + double diff = Sm - M; + // Degrees of freedom: + unsigned v = Sn - 1; + // t-statistic: + double t_stat = diff * sqrt(double(Sn)) / Sd; + +Finally calculate the probability from the t-statistic. If we're interested +in simply whether there is a difference (either less or greater) or not, +we don't care about the sign of the t-statistic, +and we take the complement of the probability for comparison +to the significance level: + + students_t dist(v); + double q = cdf(complement(dist, fabs(t_stat))); + +The procedure then prints out the results of the various tests +that can be done, these +can be summarised in the following table: + +[table +[[Hypothesis][Test]] +[[The Null-hypothesis: there is +*no difference* in means] +[Reject if complement of CDF for |t| < significance level / 2: + +`cdf(complement(dist, fabs(t))) < alpha / 2`]] + +[[The Alternative-hypothesis: there +*is difference* in means] +[Reject if complement of CDF for |t| > significance level / 2: + +`cdf(complement(dist, fabs(t))) > alpha / 2`]] + +[[The Alternative-hypothesis: the sample mean *is less* than +the true mean.] +[Reject if CDF of t > significance level: + +`cdf(dist, t) > alpha`]] + +[[The Alternative-hypothesis: the sample mean *is greater* than +the true mean.] +[Reject if complement of CDF of t > significance level: + +`cdf(complement(dist, t)) > alpha`]] +] + +[note +Notice that the comparisons are against `alpha / 2` for a two-sided test +and against `alpha` for a one-sided test] + +Now that we have all the parts in place, let's take a look at some +sample output, first using the +[@http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm +Heat flow data] from the NIST site. The data set was collected +by Bob Zarr of NIST in January, 1990 from a heat flow meter +calibration and stability analysis. The corresponding dataplot +output for this test can be found in +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm +section 3.5.2] of the __handbook. + +[pre''' + __________________________________ + Student t test for a single sample + __________________________________ + + Number of Observations = 195 + Sample Mean = 9.26146 + Sample Standard Deviation = 0.02279 + Expected True Mean = 5.00000 + + Sample Mean - Expected Test Mean = 4.26146 + Degrees of Freedom = 194 + T Statistic = 2611.28380 + Probability that difference is due to chance = 0.000e+000 + + Results for Alternative Hypothesis and alpha = 0.0500''' + + Alternative Hypothesis Conclusion + Mean != 5.000 NOT REJECTED + Mean < 5.000 REJECTED + Mean > 5.000 NOT REJECTED +] + +You will note the line that says the probability that the difference is +due to chance is zero. From a philosophical point of view, of course, +the probability can never reach zero. However, in this case the calculated +probability is smaller than the smallest representable double precision number, +hence the appearance of a zero here. Whatever its "true" value is, we know it +must be extraordinarily small, so the alternative hypothesis - that there is +a difference in means - is not rejected. + +For comparison the next example data output is taken from +['P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. +and from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 +J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907.] +The values result from the determination of mercury by cold-vapour +atomic absorption. + +[pre''' + __________________________________ + Student t test for a single sample + __________________________________ + + Number of Observations = 3 + Sample Mean = 37.80000 + Sample Standard Deviation = 0.96437 + Expected True Mean = 38.90000 + + Sample Mean - Expected Test Mean = -1.10000 + Degrees of Freedom = 2 + T Statistic = -1.97566 + Probability that difference is due to chance = 1.869e-001 + + Results for Alternative Hypothesis and alpha = 0.0500''' + + Alternative Hypothesis Conclusion + Mean != 38.900 REJECTED + Mean < 38.900 REJECTED + Mean > 38.900 REJECTED +] + +As you can see the small number of measurements (3) has led to a large uncertainty +in the location of the true mean. So even though there appears to be a difference +between the sample mean and the expected true mean, we conclude that there +is no significant difference, and are unable to reject the null hypothesis. +However, if we were to lower the bar for acceptance down to alpha = 0.1 +(a 90% confidence level) we see a different output: + +[pre''' +__________________________________ +Student t test for a single sample +__________________________________ + +Number of Observations = 3 +Sample Mean = 37.80000 +Sample Standard Deviation = 0.96437 +Expected True Mean = 38.90000 + +Sample Mean - Expected Test Mean = -1.10000 +Degrees of Freedom = 2 +T Statistic = -1.97566 +Probability that difference is due to chance = 1.869e-001 + +Results for Alternative Hypothesis and alpha = 0.1000''' + +Alternative Hypothesis Conclusion +Mean != 38.900 REJECTED +Mean < 38.900 NOT REJECTED +Mean > 38.900 REJECTED +] + +In this case, we really have a borderline result, +and more data (and/or more accurate data), +is needed for a more convincing conclusion. + +[endsect] + +[section:tut_mean_size Estimating how large a sample size would have to become +in order to give a significant Students-t test result with a single sample test] + +Imagine you have conducted a Students-t test on a single sample in order +to check for systematic errors in your measurements. Imagine that the +result is borderline. At this point one might go off and collect more data, +but it might be prudent to first ask the question "How much more?". +The parameter estimators of the students_t_distribution class +can provide this information. + +This section is based on the example code in +[@../../../example/students_t_single_sample.cpp students_t_single_sample.cpp] +and we begin by defining a procedure that will print out a table of +estimated sample sizes for various confidence levels: + + // Needed includes: + #include + #include + #include + // Bring everything into global namespace for ease of use: + using namespace boost::math; + using namespace std; + + void single_sample_find_df( + double M, // M = true mean. + double Sm, // Sm = Sample Mean. + double Sd) // Sd = Sample Standard Deviation. + { + +Next we define a table of significance levels: + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +Printing out the table of sample sizes required for various confidence levels +begins with the table header: + + cout << "\n\n" + "_______________________________________________________________\n" + "Confidence Estimated Estimated\n" + " Value (%) Sample Size Sample Size\n" + " (one sided test) (two sided test)\n" + "_______________________________________________________________\n"; + + +And now the important part: the sample sizes required. Class +`students_t_distribution` has a static member function +`find_degrees_of_freedom` that will calculate how large +a sample size needs to be in order to give a definitive result. + +The first argument is the difference between the means that you +wish to be able to detect, here it's the absolute value of the +difference between the sample mean, and the true mean. + +Then come two probability values: alpha and beta. Alpha is the +maximum acceptable risk of rejecting the null-hypothesis when it is +in fact true. Beta is the maximum acceptable risk of failing to reject +the null-hypothesis when in fact it is false. +Also note that for a two-sided test, alpha must be divided by 2. + +The final parameter of the function is the standard deviation of the sample. + +In this example, we assume that alpha and beta are the same, and call +`find_degrees_of_freedom` twice: once with alpha for a one-sided test, +and once with alpha/2 for a two-sided test. + + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate df for single sided test: + double df = students_t::find_degrees_of_freedom( + fabs(M - Sm), alpha[i], alpha[i], Sd); + // convert to sample size: + double size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size; + // calculate df for two sided test: + df = students_t::find_degrees_of_freedom( + fabs(M - Sm), alpha[i]/2, alpha[i], Sd); + // convert to sample size: + size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size << endl; + } + cout << endl; + } + +Let's now look at some sample output using data taken from +['P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. +and from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 +J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907.] +The values result from the determination of mercury by cold-vapour +atomic absorption. + +Only three measurements were made, and the Students-t test above +gave a borderline result, so this example +will show us how many samples would need to be collected: + +[pre''' +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ + +True Mean = 38.90000 +Sample Mean = 37.80000 +Sample Standard Deviation = 0.96437 + + +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (one sided test) (two sided test) +_______________________________________________________________ + 50.000 2 3 + 75.000 4 5 + 90.000 8 10 + 95.000 12 14 + 99.000 21 23 + 99.900 36 38 + 99.990 51 54 + 99.999 67 69 +'''] + +So in this case, many more measurements would have had to be made, +for example at the 95% level, 14 measurements in total for a two-sided test. + +[endsect] +[section:two_sample_students_t Comparing the means of two samples with the Students-t test] + +Imagine that we have two samples, and we wish to determine whether +their means are different or not. This situation often arises when +determining whether a new process or treatment is better than an old one. + +In this example, we'll be using the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm +Car Mileage sample data] from the +[@http://www.itl.nist.gov NIST website]. The data compares +miles per gallon of US cars with miles per gallon of Japanese cars. + +The sample code is in +[@../../../example/students_t_two_samples.cpp students_t_two_samples.cpp]. + +There are two ways in which this test can be conducted: we can assume +that the true standard deviations of the two samples are equal or not. +If the standard deviations are assumed to be equal, then the calculation +of the t-statistic is greatly simplified, so we'll examine that case first. +In real life we should verify whether this assumption is valid with a +Chi-Squared test for equal variances. + +We begin by defining a procedure that will conduct our test assuming equal +variances: + + // Needed headers: + #include + #include + #include + // Simplify usage: + using namespace boost::math; + using namespace std; + + void two_samples_t_test_equal_sd( + double Sm1, // Sm1 = Sample 1 Mean. + double Sd1, // Sd1 = Sample 1 Standard Deviation. + unsigned Sn1, // Sn1 = Sample 1 Size. + double Sm2, // Sm2 = Sample 2 Mean. + double Sd2, // Sd2 = Sample 2 Standard Deviation. + unsigned Sn2, // Sn2 = Sample 2 Size. + double alpha) // alpha = Significance Level. + { + + +Our procedure will begin by calculating the t-statistic, assuming +equal variances the needed formulae are: + +[equation dist_tutorial1] + +where Sp is the "pooled" standard deviation of the two samples, +and /v/ is the number of degrees of freedom of the two combined +samples. We can now write the code to calculate the t-statistic: + + // Degrees of freedom: + double v = Sn1 + Sn2 - 2; + cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n"; + // Pooled variance: + double sp = sqrt(((Sn1-1) * Sd1 * Sd1 + (Sn2-1) * Sd2 * Sd2) / v); + cout << setw(55) << left << "Pooled Standard Deviation" << "= " << v << "\n"; + // t-statistic: + double t_stat = (Sm1 - Sm2) / (sp * sqrt(1.0 / Sn1 + 1.0 / Sn2)); + cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n"; + +The next step is to define our distribution object, and calculate the +complement of the probability: + + students_t dist(v); + double q = cdf(complement(dist, fabs(t_stat))); + cout << setw(55) << left << "Probability that difference is due to chance" << "= " + << setprecision(3) << scientific << 2 * q << "\n\n"; + +Here we've used the absolute value of the t-statistic, because we initially +want to know simply whether there is a difference or not (a two-sided test). +However, we can also test whether the mean of the second sample is greater +or is less (one-sided test) than that of the first: +all the possible tests are summed up in the following table: + +[table +[[Hypothesis][Test]] +[[The Null-hypothesis: there is +*no difference* in means] +[Reject if complement of CDF for |t| < significance level / 2: + +`cdf(complement(dist, fabs(t))) < alpha / 2`]] + +[[The Alternative-hypothesis: there is a +*difference* in means] +[Reject if complement of CDF for |t| > significance level / 2: + +`cdf(complement(dist, fabs(t))) < alpha / 2`]] + +[[The Alternative-hypothesis: Sample 1 Mean is *less* than +Sample 2 Mean.] +[Reject if CDF of t > significance level: + +`cdf(dist, t) > alpha`]] + +[[The Alternative-hypothesis: Sample 1 Mean is *greater* than +Sample 2 Mean.] + +[Reject if complement of CDF of t > significance level: + +`cdf(complement(dist, t)) > alpha`]] +] + +[note +For a two-sided test we must compare against alpha / 2 and not alpha.] + +Most of the rest of the sample program is pretty-printing, so we'll +skip over that, and take a look at the sample output for alpha=0.05 +(a 95% probability level). For comparison the dataplot output +for the same data is in +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm +section 1.3.5.3] of the __handbook. + +[pre''' + ________________________________________________ + Student t test for two samples (equal variances) + ________________________________________________ + + Number of Observations (Sample 1) = 249 + Sample 1 Mean = 20.14458 + Sample 1 Standard Deviation = 6.41470 + Number of Observations (Sample 2) = 79 + Sample 2 Mean = 30.48101 + Sample 2 Standard Deviation = 6.10771 + Degrees of Freedom = 326.00000 + Pooled Standard Deviation = 326.00000 + T Statistic = -12.62059 + Probability that difference is due to chance = 5.273e-030 + + Results for Alternative Hypothesis and alpha = 0.0500''' + + Alternative Hypothesis Conclusion + Sample 1 Mean != Sample 2 Mean NOT REJECTED + Sample 1 Mean < Sample 2 Mean NOT REJECTED + Sample 1 Mean > Sample 2 Mean REJECTED +] + +So with a probability that the difference is due to chance of just +5.273e-030, we can safely conclude that there is indeed a difference. + +The tests on the alternative hypothesis show that we must +also reject the hypothesis that Sample 1 Mean is +greater than that for Sample 2: in this case Sample 1 represents the +miles per gallon for Japanese cars, and Sample 2 the miles per gallon for +US cars, so we conclude that Japanese cars are on average more +fuel efficient. + +Now that we have the simple case out of the way, let's look for a moment +at the more complex one: that the standard deviations of the two samples +are not equal. In this case the formula for the t-statistic becomes: + +[equation dist_tutorial2] + +And for the combined degrees of freedom we use the +[@http://en.wikipedia.org/wiki/Welch-Satterthwaite_equation Welch-Satterthwaite] +approximation: + +[equation dist_tutorial3] + +Note that this is one of the rare situations where the degrees-of-freedom +parameter to the Student's t distribution is a real number, and not an +integer value. + +[note +Some statistical packages truncate the effective degrees of freedom to +an integer value: this may be necessary if you are relying on lookup tables, +but since our code fully supports non-integer degrees of freedom there is no +need to truncate in this case. Also note that when the degrees of freedom +is small then the Welch-Satterthwaite approximation may be a significant +source of error.] + +Putting these formulae into code we get: + + // Degrees of freedom: + double v = Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2; + v *= v; + double t1 = Sd1 * Sd1 / Sn1; + t1 *= t1; + t1 /= (Sn1 - 1); + double t2 = Sd2 * Sd2 / Sn2; + t2 *= t2; + t2 /= (Sn2 - 1); + v /= (t1 + t2); + cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n"; + // t-statistic: + double t_stat = (Sm1 - Sm2) / sqrt(Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2); + cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n"; + +Thereafter the code and the tests are performed the same as before. Using +are car mileage data again, here's what the output looks like: + +[pre''' + __________________________________________________ + Student t test for two samples (unequal variances) + __________________________________________________ + + Number of Observations (Sample 1) = 249 + Sample 1 Mean = 20.145 + Sample 1 Standard Deviation = 6.4147 + Number of Observations (Sample 2) = 79 + Sample 2 Mean = 30.481 + Sample 2 Standard Deviation = 6.1077 + Degrees of Freedom = 136.87 + T Statistic = -12.946 + Probability that difference is due to chance = 1.571e-025 + + Results for Alternative Hypothesis and alpha = 0.0500''' + + Alternative Hypothesis Conclusion + Sample 1 Mean != Sample 2 Mean NOT REJECTED + Sample 1 Mean < Sample 2 Mean NOT REJECTED + Sample 1 Mean > Sample 2 Mean REJECTED +] + +This time allowing the variances in the two samples to differ has yielded +a higher likelihood that the observed difference is down to chance alone +(1.571e-025 compared to 5.273e-030 when equal variances were assumed). +However, the conclusion remains the same: US cars are less fuel efficient +than Japanese models. + +[endsect] +[section:paired_st Comparing two paired samples with the Student's t distribution] + +Imagine that we have a before and after reading for each item in the sample: +for example we might have measured blood pressure before and after administration +of a new drug. We can't pool the results and compare the means before and after +the change, because each patient will have a different baseline reading. +Instead we calculate the difference between before and after measurements +in each patient, and calculate the mean and standard deviation of the differences. +To test whether a significant change has taken place, we can then test +the null-hypothesis that the true mean is zero using the same procedure +we used in the single sample cases previously discussed. + +That means we can: + +* [link math_toolkit.dist.stat_tut.weg.st_eg.tut_mean_intervals Calculate confidence intervals of the mean]. +If the endpoints of the interval differ in sign then we are unable to reject +the null-hypothesis that there is no change. +* [link math_toolkit.dist.stat_tut.weg.st_eg.tut_mean_test Test whether the true mean is zero]. If the +result is consistent with a true mean of zero, then we are unable to reject the +null-hypothesis that there is no change. +* [link math_toolkit.dist.stat_tut.weg.st_eg.tut_mean_size Calculate how many pairs of readings we would need +in order to obtain a significant result]. + +[endsect] + +[endsect][/section:st_eg Student's t] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/triangular.qbk b/doc/sf_and_dist/distributions/triangular.qbk new file mode 100644 index 000000000..eb04ee5a3 --- /dev/null +++ b/doc/sf_and_dist/distributions/triangular.qbk @@ -0,0 +1,168 @@ +[section:triangular_dist Triangular Distribution] + + +``#include `` + + namespace boost{ namespace math{ + template + class triangular_distribution; + + typedef triangular_distribution<> triangular; + + template + class triangular_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + + triangular_distribution(RealType lower = -1, RealType mode = 0) RealType upper = 1); // Constructor. + : m_lower(lower), m_mode(mode), m_upper(upper) // Default is -1, 0, +1 triangular distribution. + // Accessor functions. + RealType lower()const; + RealType mode()const; + RealType upper()const; + }; // class triangular_distribution + + }} // namespaces + +The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution] +is a [@http://en.wikipedia.org/wiki/Continuous_distribution continuous] +[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution] +with a lower limit a, +[@http://en.wikipedia.org/wiki/Mode_%28statistics%29 mode c], +and upper limit b. + +The triangular distribution is often used where the distribution is only vaguely known, +but, like the [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution], +upper and limits are 'known', but a 'best guess', the mode or center point, is also added. +It has been recommended as a +[@http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf proxy for the beta distribution.] +The distribution is used in business decision making and project planning. + +The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution] +is a distribution with the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: + +f(x) = + +* 2(x-a)/(b-a) (c-a) for a <= x <= c + +* 2(b-x)/(b-a)(b-c) for c < x <= b + +Parameter a (lower) can be any finite value. +Parameter b (upper) can be any finite value > a (lower). +Parameter c (mode) a <= c <= b. This is the most probable value. + +The [@http://en.wikipedia.org/wiki/Random_variate random variate] x must also be finite, and is supported lower <= x <= upper. + +The triangular distribution may be appropriate when an assumption of a normal distribution +is unjustified because uncertainty is caused by rounding and quantization from analog to digital conversion. +Upper and lower limits are known, and the most probable value lies midway. + +The distribution simplifies when the 'best guess' is either the lower or upper limit - a 90 degree angle triangle. +The default chosen is the 001 triangular distribution which expresses an estimate that the lowest value is the most likely; +for example, you believe that the next-day quoted delivery date is most likely +(knowing that a quicker delivery is impossible - the postman only comes once a day), +and that longer delays are decreasingly likely, +and delivery is assumed to never take more than your upper limit. + +The following graph illustrates how the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] +varies with the various parameters: + +[$../graphs/triangular_pdf.png] + +and cumulative distribution function + +[$../graphs/triangular_cdf.png] + +[h4 Member Functions] + + triangular_distribution(RealType lower = 0, RealType mode = 0 RealType upper = 1); + +Constructs a [@http://en.wikipedia.org/wiki/triangular_distribution triangular distribution] +with lower /lower/ (a) and upper /upper/ (b). + +Requires that the /lower/, /mode/ and /upper/ parameters are all finite, +otherwise calls __domain_error. + + RealType lower()const; + +Returns the /lower/ parameter of this distribution (default -1). + + RealType mode()const; + +Returns the /mode/ parameter of this distribution (default 0). + + RealType upper()const; + +Returns the /upper/ parameter of this distribution (default+1). + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \lower\ to \upper\, +and the supported range is lower <= x <= upper. + +[h4 Accuracy] + +The triangular distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two, +except quantiles with arguments nearing the extremes of zero and unity. + +[h4 Implementation] + +In the following table, a is the /lower/ parameter of the distribution, +c is the /mode/ parameter, +b is the /upper/ parameter, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = 0 for x < mode, 2(x-a)\/(b-a)(c-a) else 2*(b-x)\/((b-a)(b-c))]] +[[cdf][Using the relation: cdf = 0 for x < mode (x-a)[super 2]\/((b-a)(c-a)) else 1 - (b-x)[super 2]\/((b-a)(b-c))]] +[[cdf complement][Using the relation: q = 1 - p ]] +[[quantile][let p0 = (c-a)\/(b-a) the point of inflection on the cdf, +then given probability p and q = 1-p: + +x = sqrt((b-a)(c-a)p) + a ; for p < p0 + +x = c ; for p == p0 + +x = b - sqrt((b-a)(b-c)q) ; for p > p0 + +(See [@../../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]] +[[quantile from the complement][As quantile (See [@../../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]] +[[mean][(a + b + 3) \/ 3 ]] +[[variance][(a[super 2]+b[super 2]+c[super 2] - ab - ac - bc)\/18]] +[[mode][c]] +[[skewness][(See [@../../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details). ]] +[[kurtosis][12\/5]] +[[kurtosis excess][-3\/5]] +] + +Some 'known good' test values were obtained from +[@http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm Statlet: Calculate and plot probability distributions] + +[h4 References] + +* [@http://en.wikipedia.org/wiki/Triangular_distribution Wikpedia triangular distribution] +* [@http://mathworld.wolfram.com/TriangularDistribution.html Weisstein, Eric W. "Triangular Distribution." From MathWorld--A Wolfram Web Resource.] +* Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, 2000, ISBN - 0471371246] +* [@http://www.brighton-webs.co.uk/distributions/triangular.asp Brighton Webs Ltd. BW D-Calc 1.0 Distribution Calculator] +* [@http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf The Triangular Distribution including its history.] +* [@http://www.measurement.sk/2002/S1/Wimmer2.pdf Gejza Wimmer, Viktor Witkovsky and Tomas Duby, +Measurement Science Review, Volume 2, Section 1, 2002, Proper Rounding Of The Measurement Results Under The Assumption Of Triangular Distribution.] + +[endsect][/section:triangular_dist triangular] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/uniform.qbk b/doc/sf_and_dist/distributions/uniform.qbk new file mode 100644 index 000000000..098a8d150 --- /dev/null +++ b/doc/sf_and_dist/distributions/uniform.qbk @@ -0,0 +1,134 @@ +[section:uniform_dist Uniform Distribution] + + +``#include `` + + namespace boost{ namespace math{ + template + class uniform_distribution; + + typedef uniform_distribution<> uniform; + + template + class uniform_distribution + { + public: + typedef RealType value_type; + + uniform_distribution(RealType lower = 0, RealType upper = 1); // Constructor. + : m_lower(lower), m_upper(upper) // Default is standard uniform distribution. + // Accessor functions. + RealType lower()const; + RealType upper()const; + }; // class uniform_distribution + + }} // namespaces + +The uniform distribution, also known as a rectangular distribution, +is a probability distribution that has constant probability. + +The [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 continuous uniform distribution] +is a distribution with the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: + +f(x) = + +* 1 / (upper - lower) for lower < x < upper + +* zero for x < lower or x > upper + +and in this implementation: + +* 1 / (upper - lower) for x = lower or x = upper + +The choice of x = lower or x = upper is made because statistical use of this distribution judged is most likely: +the method of maximum likelihood uses this definition. + +There is also a [@http://en.wikipedia.org/wiki/Discrete_uniform_distribution *discrete* uniform distribution]. + +Parameters lower and upper can be any finite value. + +The [@http://en.wikipedia.org/wiki/Random_variate random variate] +x must also be finite, and is supported lower <= x <= upper. + +The lower parameter is also called the +[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm location parameter], +[@http://en.wikipedia.org/wiki/Location_parameter that is where the origin of a plot will lie], +and (upper - lower) is also called the [@http://en.wikipedia.org/wiki/Scale_parameter scale parameter]. + +The following graph illustrates how the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] +varies with the shape parameter: + +[$../graphs/uniform_pdf.png] + +Likewise for the CDF: + +[$../graphs/uniform_cdf.png] + +[h4 Member Functions] + + uniform_distribution(RealType lower = 0, RealType upper = 1); + +Constructs a [@http://en.wikipedia.org/wiki/uniform_distribution +uniform distribution] with lower /lower/ (a) and upper /upper/ (b). + +Requires that the /lower/ and /upper/ parameters are both finite; +otherwise if infinity or NaN then calls __domain_error. + + RealType lower()const; + +Returns the /lower/ parameter of this distribution. + + RealType upper()const; + +Returns the /upper/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +that are generic to all distributions are supported: __usual_accessors. + +The domain of the random variable is any finite value, +but the supported range is only /lower/ <= x <= /upper/. + +[h4 Accuracy] + +The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two. + +[h4 Implementation] + +In the following table a is the /lower/ parameter of the distribution, +b is the /upper/ parameter, +/x/ is the random variate, /p/ is the probability and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = 0 for x < a, 1 / (b - a) for a <= x <= b, 0 for x > b ]] +[[cdf][Using the relation: cdf = 0 for x < a, (x - a) / (b - a) for a <= x <= b, 1 for x > b]] +[[cdf complement][Using the relation: q = 1 - p, (b - x) / (b - a) ]] +[[quantile][Using the relation: x = p * (b - a) + a; ]] +[[quantile from the complement][x = -q * (b - a) + b ]] +[[mean][(a + b) / 2 ]] +[[variance][(b - a) [super 2] / 12 ]] +[[mode][any value in \[a, b\] but a is chosen. (Would NaN be better?) ]] +[[skewness][0]] +[[kurtosis excess][-6/5 = -1.2 exactly. (kurtosis - 3)]] +[[kurtosis][9/5]] +] + +[h4 References] +* [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 Wikpedia continuous uniform distribution] +* [@http://mathworld.wolfram.com/UniformDistribution.html Weisstein, Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource.] +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm] + +[endsect][/section:uniform_dist Uniform] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/distributions/weibull.qbk b/doc/sf_and_dist/distributions/weibull.qbk new file mode 100644 index 000000000..140f13792 --- /dev/null +++ b/doc/sf_and_dist/distributions/weibull.qbk @@ -0,0 +1,127 @@ +[section:weibull Weibull Distribution] + + +``#include `` + + namespace boost{ namespace math{ + + template + class weibull_distribution; + + typedef weibull_distribution<> weibull; + + template + class weibull_distribution + { + public: + typedef RealType value_type; + typedef Policy policy_type; + // Construct: + weibull_distribution(RealType shape, RealType scale = 1) + // Accessors: + RealType shape()const; + RealType scale()const; + }; + + }} // namespaces + +The [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] +is a continuous distribution +with the +[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: + +f(x; [alpha], [beta]) = ([alpha]\/[beta]) * (x \/ [beta])[super [alpha] - 1] * e[super -(x\/[beta])[super [alpha]]] + +For shape parameter [alpha][space] > 0, and scale parameter [beta][space] > 0, and x > 0. + +The Weibull distribution is often used in the field of failure analysis; +in particular it can mimic distributions where the failure rate varies over time. +If the failure rate is: + +* constant over time, then [alpha][space] = 1, suggests that items are failing from random events. +* decreases over time, then [alpha][space] < 1, suggesting "infant mortality". +* increases over time, then [alpha][space] > 1, suggesting "wear out" - more likely to fail as time goes by. + +The following graph illustrates how the PDF varies with the shape parameter [alpha]: + +[$../graphs/weibull.png] + +While this graph illustrates how the PDF varies with the scale parameter [beta]: + +[$../graphs/weibull2.png] + +[h4 Related distributions] + +When [alpha][space] = 3, the +[@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] appears similar to the +[@http://en.wikipedia.org/wiki/Normal_distribution normal distribution]. +When [alpha][space] = 1, the Weibull distribution reduces to the +[@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution]. + +[h4 Member Functions] + + weibull_distribution(RealType shape, RealType scale = 1); + +Constructs a [@http://en.wikipedia.org/wiki/Weibull_distribution +Weibull distribution] with shape /shape/ and scale /scale/. + +Requires that the /shape/ and /scale/ parameters are both greater than zero, +otherwise calls __domain_error. + + RealType shape()const; + +Returns the /shape/ parameter of this distribution. + + RealType scale()const; + +Returns the /scale/ parameter of this distribution. + +[h4 Non-member Accessors] + +All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +distributions are supported: __usual_accessors. + +The domain of the random variable is \[0, [infin]\]. + +[h4 Accuracy] + +The Weibull distribution is implemented in terms of the +standard library `log` and `exp` functions plus __expm1 and __log1p +and as such should have very low error rates. + +[h4 Implementation] + + +In the following table [alpha][space] is the shape parameter of the distribution, +[beta][space] is it's scale parameter, /x/ is the random variate, /p/ is the probability +and /q = 1-p/. + +[table +[[Function][Implementation Notes]] +[[pdf][Using the relation: pdf = [alpha][beta][super -[alpha] ]x[super [alpha] - 1] e[super -(x/beta)[super alpha]] ]] +[[cdf][Using the relation: p = -__expm1(-(x\/[beta])[super [alpha]]) ]] +[[cdf complement][Using the relation: q = e[super -(x\/[beta])[super [alpha]]] ]] +[[quantile][Using the relation: x = [beta] * (-__log1p(-p))[super 1\/[alpha]] ]] +[[quantile from the complement][Using the relation: x = [beta] * (-log(q))[super 1\/[alpha]] ]] +[[mean][[beta] * [Gamma](1 + 1\/[alpha]) ]] +[[variance][[beta][super 2]([Gamma](1 + 2\/[alpha]) - [Gamma][super 2](1 + 1\/[alpha])) ]] +[[mode][[beta](([alpha] - 1) \/ [alpha])[super 1\/[alpha]] ]] +[[skewness][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]] +] + +[h4 References] +* [@http://en.wikipedia.org/wiki/Weibull_distribution ] +* [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] +* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm Weibull in NIST Exploratory Data Analysis] + +[endsect][/section:weibull Weibull] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/ellint_carlson.qbk b/doc/sf_and_dist/ellint_carlson.qbk new file mode 100644 index 000000000..de09a536d --- /dev/null +++ b/doc/sf_and_dist/ellint_carlson.qbk @@ -0,0 +1,213 @@ +[/ +Copyright (c) 2006 Xiaogang Zhang +Copyright (c) 2006 John Maddock +Use, modification and distribution are subject to the +Boost Software License, Version 1.0. (See accompanying file +LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +] + +[section:ellint_carlson Elliptic Integrals - Carlson Form] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_rf(T1 x, T2 y, T3 z) + + template + ``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_rd(T1 x, T2 y, T3 z) + + template + ``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p) + + template + ``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_rc(T1 x, T2 y) + + template + ``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&) + + }} // namespaces + + +[heading Description] + +These functions return Carlson's symmetrical elliptic integrals, the functions +have complicated behavior over all their possible domains, but the following +graph gives an idea of their behavior: + +[$../graphs/ellint_c.png] + +The return type of these functions is computed using the __arg_pomotion_rules +when the arguments are of different types: otherwise the return is the same type +as the arguments. + + template + ``__sf_result`` ellint_rf(T1 x, T2 y, T3 z) + + template + ``__sf_result`` ellint_rf(T1 x, T2 y, T3 z, const ``__Policy``&) + +Returns Carlson's Elliptic Integral R[sub F]: + +[equation ellint9] + +Requires that all of the arguments are non-negative, and at most +one may be zero. Otherwise returns the result of __domain_error. + +[optional_policy] + + template + ``__sf_result`` ellint_rd(T1 x, T2 y, T3 z) + + template + ``__sf_result`` ellint_rd(T1 x, T2 y, T3 z, const ``__Policy``&) + +Returns Carlson's elliptic integral R[sub D]: + +[equation ellint10] + +Requires that x and y are non-negative, with at most one of them +zero, and that z >= 0. Otherwise returns the result of __domain_error. + +[optional_policy] + + template + ``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p) + + template + ``__sf_result`` ellint_rj(T1 x, T2 y, T3 z, T4 p, const ``__Policy``&) + +Returns Carlson's elliptic integral R[sub J]: + +[equation ellint11] + +Requires that x, y and z are non-negative, with at most one of them +zero, and that ['p != 0]. Otherwise returns the result of __domain_error. + +[optional_policy] + +When ['p < 0] the function returns the +[@http://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value] +using the relation: + +[equation ellint17] + + template + ``__sf_result`` ellint_rc(T1 x, T2 y) + + template + ``__sf_result`` ellint_rc(T1 x, T2 y, const ``__Policy``&) + +Returns Carlson's elliptic integral R[sub C]: + +[equation ellint12] + +Requires that ['x > 0] and that ['y != 0]. +Otherwise returns the result of __domain_error. + +[optional_policy] + +When ['y < 0] the function returns the +[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal value] +using the relation: + +[equation ellint18] + +[heading Testing] + +There are two sets of tests. + +Spot tests compare selected values with test data given in: + +[:B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227 +Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms, +Volume 10, Number 1 / March, 1995, pp 13-26.] + +Random test data generated using NTL::RR at 1000-bit precision and our +implementation checks for rounding-errors and/or regressions. + +There are also sanity checks that use the inter-relations between the integrals +to verify their correctness: see the above Carlson paper for details. + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating-point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Carlson Elliptic Integrals +[[Significand Size] [Platform and Compiler] [R[sub F]] [R[sub D]] [R[sub J]] [R[sub C]]] +[[53] [Win32 / Visual C++ 8.0] [Peak=2.9 Mean=0.75] [Peak=2.6 Mean=0.9] [Peak=108 Mean=6.9] [Peak=2.4 Mean=0.6] ] +[[64] [Red Hat Linux / G++ 3.4] [Peak=2.5 Mean=0.75] [Peak=2.7 Mean=0.9] [Peak=105 Mean=8] [Peak=1.9 Mean=0.7] ] +[[113] [HP-UX / HP aCC 6] [Peak=5.3 Mean=1.6] [Peak=2.9 Mean=0.99] [Peak=180 Mean=12] [Peak=1.8 Mean=0.7] ] +] + +[heading Implementation] + +The key of Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] is the +duplication theorem: + +[equation ellint13] + +By applying it repeatedly, ['x], ['y], ['z] get +closer and closer. When they are nearly equal, the special case equation + +[equation ellint16] + +is used. More specifically, ['[R F]] is evaluated from a Taylor series +expansion to the fifth order. The calculations of the other three integrals +are analogous. + +For ['p < 0] in ['R[sub J](x, y, z, p)] and ['y < 0] in ['R[sub C](x, y)], +the integrals are singular and their +[@http://mathworld.wolfram.com/CauchyPrincipalValue.html Cauchy principal values] +are returned via the relations: + +[equation ellint17] + +[equation ellint18] + +[endsect] diff --git a/doc/sf_and_dist/ellint_introduction.qbk b/doc/sf_and_dist/ellint_introduction.qbk new file mode 100644 index 000000000..c93dd5fec --- /dev/null +++ b/doc/sf_and_dist/ellint_introduction.qbk @@ -0,0 +1,224 @@ +[/ +Copyright (c) 2006 Xiaogang Zhang +Use, modification and distribution are subject to the +Boost Software License, Version 1.0. (See accompanying file +LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +] + +[section:ellint_intro Elliptic Integral Overview] + +The main reference for the elliptic integrals is: + +[:M. Abramowitz and I. A. Stegun (Eds.) (1964) +Handbook of Mathematical Functions with Formulas, Graphs, and +Mathematical Tables, +National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.] + +Mathworld also contain a lot of useful background information: + +[:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W. +"Elliptic Integral." From MathWorld--A Wolfram Web Resource.]] + +As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic integral]. + +[h4 Notation] + +All variables are real numbers unless otherwise noted. + +[h4 [#ellint_def]Definition] + +[equation ellint1] + +is called elliptic integral if ['R(t, s)] is a rational function +of ['t] and ['s], and ['s[super 2]] is a cubic or quartic polynomial +in ['t]. + +Elliptic integrals generally can not be expressed in terms of +elementary functions. However, Legendre showed that all elliptic +integrals can be reduced to the following three canonical forms: + +Elliptic Integral of the First Kind (Legendre form) + +[equation ellint2] + +Elliptic Integral of the Second Kind (Legendre form) + +[equation ellint3] + +Elliptic Integral of the Third Kind (Legendre form) + +[equation ellint4] + +where + +[equation ellint5] + +[note ['[phi]] is called the amplitude. + +['k] is called the modulus. + +['[alpha]] is called the modular angle. + +['n] is called the characteristic.] + +[caution Perhaps more than any other special functions the elliptic +integrals are expressed in a variety of different ways. In particular, +the final parameter /k/ (the modulus) may be expressed using a modular +angle [alpha], or a parameter /m/. These are related by: + +k = sin[alpha] + +m = k[super 2] = sin[super 2][alpha] + +So that the integral of the third kind (for example) may be expressed as +either: + +[Pi](n, [phi], k) + +[Pi](n, [phi] \\ [alpha]) + +[Pi](n, [phi]| m) + +To further complicate matters, some texts refer to the ['complement +of the parameter m], or 1 - m, where: + +1 - m = 1 - k[super 2] = cos[super 2][alpha] + +This implementation uses /k/ throughout: this matches the requirements +of the [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf +Technical Report on C++ Library Extensions]. However, you should +be extra careful when using these functions!] + +When ['[phi]] = ['[pi]] / 2, the elliptic integrals are called ['complete]. + +Complete Elliptic Integral of the First Kind (Legendre form) + +[equation ellint6] + +Complete Elliptic Integral of the Second Kind (Legendre form) + +[equation ellint7] + +Complete Elliptic Integral of the Third Kind (Legendre form) + +[equation ellint8] + +Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78 Carlson78]] gives an alternative definition of +elliptic integral's canonical forms: + +Carlson's Elliptic Integral of the First Kind + +[equation ellint9] + +where ['x], ['y], ['z] are nonnegative and at most one of them +may be zero. + +Carlson's Elliptic Integral of the Second Kind + +[equation ellint10] + +where ['x], ['y] are nonnegative, at most one of them may be zero, +and ['z] must be positive. + +Carlson's Elliptic Integral of the Third Kind + +[equation ellint11] + +where ['x], ['y], ['z] are nonnegative, at most one of them may be +zero, and ['p] must be nonzero. + +Carlson's Degenerate Elliptic Integral + +[equation ellint12] + +where ['x] is nonnegative and ['y] is nonzero. + +[note ['R[sub C](x, y) = R[sub F](x, y, y)] + +['R[sub D](x, y, z) = R[sub J](x, y, z, z)]] + + +[h4 [#ellint_theorem]Duplication Theorem] + +Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that + +[equation ellint13] + +[h4 [#ellint_formula]Carlson's Formulas] + +The Legendre form and Carlson form of elliptic integrals are related +by equations: + +[equation ellint14] + +In particular, + +[equation ellint15] + +[h4 Numerical Algorithms] + +The conventional methods for computing elliptic integrals are Gauss +and Landen transformations, which converge quadratically and work +well for elliptic integrals of the first and second kinds. +Unfortunately they suffer from loss of significant digits for the +third kind. Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] [[link ellint_ref_carlson78 Carlson78]], by contrast, +provides a unified method for all three kinds of elliptic integrals +with satisfactory precisions. + +[h4 [#ellint_refs]References] + +Special mention goes to: + +[:A. M. Legendre, ['Traitd des Fonctions Elliptiques et des Integrales +Euleriennes], Vol. 1. Paris (1825).] + +However the main references are: + +# [#ellint_ref_AS]M. Abramowitz and I. A. Stegun (Eds.) (1964) +Handbook of Mathematical Functions with Formulas, Graphs, and +Mathematical Tables, +National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C. +# [#ellint_ref_carlson79]B.C. Carlson, ['Computing elliptic integrals by duplication], + Numerische Mathematik, vol 33, 1 (1979). +# [#ellint_ref_carlson77]B.C. Carlson, ['Elliptic Integrals of the First Kind], + SIAM Journal on Mathematical Analysis, vol 8, 231 (1977). +# [#ellint_ref_carlson78]B.C. Carlson, ['Short Proofs of Three Theorems on Elliptic Integrals], + SIAM Journal on Mathematical Analysis, vol 9, 524 (1978). +# [#ellint_ref_carlson81]B.C. Carlson and E.M. Notis, ['ALGORITHM 577: Algorithms for Incomplete + Elliptic Integrals], ACM Transactions on Mathematmal Software, + vol 7, 398 (1981). +# B. C. Carlson, ['On computing elliptic integrals and functions]. J. Math. and +Phys., 44 (1965), pp. 36-51. +# B. C. Carlson, ['A table of elliptic integrals of the second kind]. Math. Comp., 49 +(1987), pp. 595-606. (Supplement, ibid., pp. S13-S17.) +# B. C. Carlson, ['A table of elliptic integrals of the third kind]. Math. Comp., 51 (1988), +pp. 267-280. (Supplement, ibid., pp. S1-S5.) +# B. C. Carlson, ['A table of elliptic integrals: cubic cases]. Math. Comp., 53 (1989), pp. +327-333. +# B. C. Carlson, ['A table of elliptic integrals: one quadratic factor]. Math. Comp., 56 (1991), +pp. 267-280. +# B. C. Carlson, ['A table of elliptic integrals: two quadratic factors]. Math. Comp., 59 +(1992), pp. 165-180. +# B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227 +Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms, +Volume 10, Number 1 / March, 1995, p13-26. +# B. C. Carlson and John L. Gustafson, ['[@http://arxiv.org/abs/math.CA/9310223 +Asymptotic Approximations for Symmetric Elliptic Integrals]], +SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303. + + +The following references, while not directly relevent to our implementation, +may also be of interest: + +# R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions.] +Numerical Mathematik 7, 78-90. +# R. Burlisch, ['An extension of the Bartky Transformation to Incomplete +Elliptic Integrals of the Third Kind]. Numerical Mathematik 13, 266-284. +# R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions. III]. +Numerical Mathematik 13, 305-315. +# T. Fukushima and H. Ishizaki, ['[@http://adsabs.harvard.edu/abs/1994CeMDA..59..237F +Numerical Computation of Incomplete Elliptic Integrals of a General Form.]] +Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 1994, +237-251. + +[endsect] diff --git a/doc/sf_and_dist/ellint_legendre.qbk b/doc/sf_and_dist/ellint_legendre.qbk new file mode 100644 index 000000000..d461d8b8e --- /dev/null +++ b/doc/sf_and_dist/ellint_legendre.qbk @@ -0,0 +1,343 @@ +[/ +Copyright (c) 2006 Xiaogang Zhang +Copyright (c) 2006 John Maddock +Use, modification and distribution are subject to the +Boost Software License, Version 1.0. (See accompanying file +LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +] + +[section:ellint_1 Elliptic Integrals of the First Kind - Legendre Form] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_1(T1 k, T2 phi); + + template + ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&); + + template + ``__sf_result`` ellint_1(T k); + + template + ``__sf_result`` ellint_1(T k, const ``__Policy``&); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the first kind +['F([phi], k)] and its complete counterpart ['K(k) = F([pi]/2, k)]. + +[$../graphs/ellint_1.png] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types: when they are the same type then the result +is the same type as the arguments. + + template + ``__sf_result`` ellint_1(T1 k, T2 phi); + + template + ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&); + +Returns the incomplete elliptic integral of the first kind ['F([phi], k)]: + +[equation ellint2] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[optional_policy] + + template + ``__sf_result`` ellint_1(T k); + + template + ``__sf_result`` ellint_1(T k, const ``__Policy``&); + +Returns the complete elliptic integral of the first kind ['K(k)]: + +[equation ellint6] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[optional_policy] + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Elliptic Integrals of the First Kind +[[Significand Size] [Platform and Compiler] [F([phi], k)] [K(k)] ] +[[53] [Win32 / Visual C++ 8.0] [Peak=3 Mean=0.8] [Peak=1.8 Mean=0.7] ] +[[64] [Red Hat Linux / G++ 3.4] [Peak=2.6 Mean=1.7] [Peak=2.2 Mean=1.8] ] +[[113] [HP-UX / HP aCC 6] [Peak=4.6 Mean=1.5] [Peak=3.7 Mean=1.5] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [@http://functions.wolfram.com/ functions.wolfram.com], +and random test data generated using +NTL::RR at 1000-bit precision and this implementation. + +[heading Implementation] + +These functions are implemented in terms of Carlson's integrals +using the relations: + +[equation ellint19] + +and + +[equation ellint20] + + +[endsect] + +[section:ellint_2 Elliptic Integrals of the Second Kind - Legendre Form] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_2(T1 k, T2 phi); + + template + ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&); + + template + ``__sf_result`` ellint_2(T k); + + template + ``__sf_result`` ellint_2(T k, const ``__Policy``&); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the second kind +['E([phi], k)] and its complete counterpart ['E(k) = E([pi]/2, k)]. + +[$../graphs/ellint_2.png] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types: when they are the same type then the result +is the same type as the arguments. + + template + ``__sf_result`` ellint_2(T1 k, T2 phi); + + template + ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&); + +Returns the incomplete elliptic integral of the second kind ['E([phi], k)]: + +[equation ellint3] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[optional_policy] + + template + ``__sf_result`` ellint_2(T k); + + template + ``__sf_result`` ellint_2(T k, const ``__Policy``&); + +Returns the complete elliptic integral of the first kind ['E(k)]: + +[equation ellint7] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[optional_policy] + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Elliptic Integrals of the Second Kind +[[Significand Size] [Platform and Compiler] [F([phi], k)] [K(k)] ] +[[53] [Win32 / Visual C++ 8.0] [Peak=4.6 Mean=1.2] [Peak=3.5 Mean=1.0] ] +[[64] [Red Hat Linux / G++ 3.4] [Peak=4.3 Mean=1.1] [Peak=4.6 Mean=1.2] ] +[[113] [HP-UX / HP aCC 6] [Peak=5.8 Mean=2.2] [Peak=10.8 Mean=2.3] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [http://@functions.wolfram.com +functions.wolfram.com], and random test data generated using +NTL::RR at 1000-bit precision and this implementation. + +[heading Implementation] + +These functions are implemented in terms of Carlson's integrals +using the relations: + +[equation ellint21] + +and + +[equation ellint22] + + +[endsect] + +[section:ellint_3 Elliptic Integrals of the Third Kind - Legendre Form] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi); + + template + ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&); + + template + ``__sf_result`` ellint_3(T1 k, T2 n); + + template + ``__sf_result`` ellint_3(T1 k, T2 n, const ``__Policy``&); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the third kind +['[Pi](n, [phi], k)] and its complete counterpart ['[Pi](n, k) = E(n, [pi]/2, k)]. + +[$../graphs/ellint_3.png] + +The return type of these functions is computed using the __arg_pomotion_rules +when the arguments are of different types: when they are the same type then the result +is the same type as the arguments. + + template + ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi); + + template + ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&); + +Returns the incomplete elliptic integral of the third kind ['[Pi](n, [phi], k)]: + +[equation ellint4] + +Requires ['-1 <= k <= 1] and ['n < 1/sin[super 2]([phi])], otherwise +returns the result of __domain_error (outside this range the result +would be complex). + +[optional_policy] + +[caution In addition, the region where ['n > 1] and [phi][space] ['is not in the range] +\[0, [pi]\/2\] is currently unsupported and returns the result of __domain_error. +For this reason it is recomended that you keep [phi][space] inside its "natural" range +of \[0, [pi]\/2\].] + + template + ``__sf_result`` ellint_3(T1 k, T2 n); + + template + ``__sf_result`` ellint_3(T1 k, T2 n, const ``__Policy``&); + +Returns the complete elliptic integral of the first kind ['[Pi](n, k)]: + +[equation ellint8] + +Requires ['-1 <= k <= 1] and ['n < 1], otherwise returns the +result of __domain_error (outside this range the result would be complex). + +[opitonal_policy] + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Elliptic Integrals of the Third Kind +[[Significand Size] [Platform and Compiler] [[Pi](n, [phi], k)] [[Pi](n, k)] ] +[[53] [Win32 / Visual C++ 8.0] [Peak=29 Mean=2.2] [Peak=3 Mean=0.8] ] +[[64] [Red Hat Linux / G++ 3.4] [Peak=14 Mean=1.3] [Peak=2.3 Mean=0.8] ] +[[113] [HP-UX / HP aCC 6] [Peak=10 Mean=1.4] [Peak=4.2 Mean=1.1] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [@http://functions.wolfram.com +functions.wolfram.com], and random test data generated using +NTL::RR at 1000-bit precision and this implementation. + +[heading Implementation] + +The implementation for [Pi](n, [phi], k) first siphons off the special cases: + +['[Pi](0, [phi], k) = F([phi], k)] + +['[Pi](n, [pi]/2, k) = [Pi](n, k)] + +and + +[equation ellint23] + +Then if n < 0 the relations (A&S 17.7.15/16): + +[equation ellint24] + +are used to shift /n/ to the range \[0, 1\]. + +Then the relations: + +['[Pi](n, -[phi], k) = -[Pi](n, [phi], k)] + +['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) + 2m[Pi](n, k)] + +are used to move [phi][space] to the range \[0, [pi]\/2\]. + +The functions are then implemented in terms of Carlson's integrals +using the relations: + +[equation ellint25] + +and + +[equation ellint26] + +The remaining problem area occurs when n > 1 and [phi][space] is outside +the range \[0, [pi]\/2\]. In this range the reduction formula for +large [phi][space] can no longer be applied. 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+f(x)=Γ(ν+12)νπΓ(ν2)(1+x2ν)ν+12 \ No newline at end of file diff --git a/doc/sf_and_dist/erf.qbk b/doc/sf_and_dist/erf.qbk new file mode 100644 index 000000000..046346336 --- /dev/null +++ b/doc/sf_and_dist/erf.qbk @@ -0,0 +1,213 @@ +[section:error_function Error Functions] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` erf(T z); + + template + ``__sf_result`` erf(T z, const ``__Policy``&); + + template + ``__sf_result`` erfc(T z); + + template + ``__sf_result`` erfc(T z, const ``__Policy``&); + + }} // namespaces + +The return type of these functions is computed using the __arg_pomotion_rules: +the return type is `double` if T is an integer type, and T otherwise. + +[optional_policy] + +[h4 Description] + + template + ``__sf_result`` erf(T z); + + template + ``__sf_result`` erf(T z, const ``__Policy``&); + +Returns the [@http://en.wikipedia.org/wiki/Error_function error function] +[@http://functions.wolfram.com/GammaBetaErf/Erf/ erf] of z: + +[equation erf1] + +[$../graphs/erf1.png] + + template + ``__sf_result`` erfc(T z); + + template + ``__sf_result`` erfc(T z, const ``__Policy``&); + +Returns the complement of the [@http://functions.wolfram.com/GammaBetaErf/Erfc/ error function] of z: + +[equation erf2] + +[$../graphs/erf2.png] + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types, +along with comparisons to the __gsl, __glibc, __hpc and __cephes libraries. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table Errors In the Function erf(z) +[[Significand Size] [Platform and Compiler] [z < 0.5] [0.5 < z < 8] [z > 8]] +[[53] [Win32, Visual C++ 8] [Peak=0 Mean=0 + +GSL Peak=2.0 Mean=0.3 + +__cephes Peak=1.1 Mean=0.7] [Peak=0.9 Mean=0.09 + +GSL Peak=2.3 Mean=0.3 + +__cephes Peak=1.3 Mean=0.2] [Peak=0 Mean=0 + +GSL Peak=0 Mean=0 + +__cephes Peak=0 Mean=0]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=0.7 Mean=0.07 + +__glibc Peak=0.9 Mean=0.2] [Peak=0.9 Mean=0.2 + +__glibc Peak=0.9 Mean=0.07] [Peak=0 Mean=0 + +__glibc Peak=0 Mean=0]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=0.7 Mean=0.07 + +__glibc Peak=0 Mean=0] [Peak=0.9 Mean=0.1 + +__glibc Peak=0.5 Mean=0.03] [Peak=0 Mean=0 + +__glibc Peak=0 Mean=0]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=0.8 Mean=0.1 + +__hpc Lib Peak=0.9 Mean=0.2] [Peak=0.9 Mean=0.1 + +__hpc Lib Peak=0.5 Mean=0.02] [Peak=0 Mean=0 + +__hpc Lib Peak=0 Mean=0]] +] + +[table Errors In the Function erfc(z) +[[Significand Size] [Platform and Compiler] [z < 0.5] [0.5 < z < 8] [z > 8]] +[[53] [Win32, Visual C++ 8] [Peak=0.7 Mean=0.06 + +GSL Peak=1.0 Mean=0.4 + +__cephes Peak=0.7 Mean=0.06] [Peak=0.99 Mean=0.3 + +GSL Peak=2.6 Mean=0.6 + +__cephes Peak=3.6 Mean=0.7] [Peak=1.0 Mean=0.2 + +GSL Peak=3.9 Mean=0.4 + +__cephes Peak=2.7 Mean=0.4]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=0 Mean=0 + +__glibc Peak=0 Mean=0] [Peak=1.4 Mean=0.3 + +__glibc Peak=1.3 Mean=0.3] [Peak=1.6 Mean=0.4 + +__glibc Peak=1.3 Mean=0.4]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=0 Mean=0 + +__glibc Peak=0 Mean=0] [Peak=1.4 Mean=0.3 + +__glibc Peak=0 Mean=0] [Peak=1.5 Mean=0.4 + +__glibc Peak=0 Mean=0] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=0 Mean=0 + +__hpc Peak=0 Mean=0] [Peak=1.5 Mean=0.3 + +__hpc Peak=0.9 Mean=0.08] [Peak=1.6 Mean=0.4 + +__hpc Peak=0.9 Mean=0.1]] +] + +[h4 Testing] + +The tests for these functions come in two parts: +basic sanity checks use spot values calculated using +[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf Mathworld's online evaluator], +while accuracy checks use high-precision test values calculated at 1000-bit precision with +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation. +Note that the generic and type-specific +versions of these functions use differing implementations internally, so this +gives us reasonably independent test data. Using our test data to test other +"known good" implementations also provides an additional sanity check. + +[h4 Implementation] + +All versions of these functions first use the usual reflection formulas +to make their arguments positive: + + erf(-z) = 1 - erf(z); + + erfc(-z) = 2 - erfc(z); // preferred when -z < -0.5 + + erfc(-z) = 1 + erf(z); // preferred when -0.5 <= -z < 0 + +The generic versions of these functions are implemented in terms of +the incomplete gamma function. + +When the significand (mantissa) size is recognised +(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double) +then a series of rational approximations [jm_rationals] are used. + +For `z <= 0.5` then a rational approximation to erf is used, based on the +observation that: + + erf(z)/z ~ 1.12.... + +Therefore erf is calculated using: + + erf(z) = z * (1.125F + R(z)); + +where the rational approximation R(z) is optimised for absolute error: +as long as its absolute error is small enough compared to 1.125, then any +round-off error incurred during the computation of R(z) will effectively +disappear from the result. As a result the error for erf and erfc in this +region is very low: the last bit is incorrect in only a very small number of +cases. + +For `z > 0.5` we observe that over a small interval [a, b) then: + + erfc(z) * exp(z*z) * z ~ c + +for some constant c. + +Therefore for `z > 0.5` we calculate erfc using: + + erfc(z) = exp(-z*z) * (c + R(z)) / z; + +Again R(z) is optimised for absolute error, and the constant `c` is +the average of `erfc(z) * exp(z*z) * z` taken at the endpoints of the range. +Once again, as long as the absolute error in R(z) is small +compared to `c` then `c + R(z)` will be correctly rounded, and the error +in the result will depend only on the accuracy of the exp function. In practice, +in all but a very small number of cases, the error is confined to the last bit +of the result. + +[endsect] +[/ :error_function The Error Functions] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/erf_inv.qbk b/doc/sf_and_dist/erf_inv.qbk new file mode 100644 index 000000000..fe79369e2 --- /dev/null +++ b/doc/sf_and_dist/erf_inv.qbk @@ -0,0 +1,132 @@ +[section:error_inv Error Function Inverses] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` erf_inv(T p); + + template + ``__sf_result`` erf_inv(T p, const ``__Policy``&); + + template + ``__sf_result`` erfc_inv(T p); + + template + ``__sf_result`` erfc_inv(T p, const ``__Policy``&); + + }} // namespaces + +The return type of these functions is computed using the __arg_pomotion_rules: +the return type is `double` if T is an integer type, and T otherwise. + +[optional_policy] + +[h4 Description] + + template + ``__sf_result`` erf_inv(T z); + + template + ``__sf_result`` erf_inv(T z, const ``__Policy``&); + +Returns the [@http://functions.wolfram.com/GammaBetaErf/InverseErf/ inverse error function] +of z, that is a value x such that: + + p = erf(x); + +[$../graphs/erf_inv.png] + + template + ``__sf_result`` erfc_inv(T z); + + template + ``__sf_result`` erfc_inv(T z, const ``__Policy``&); + +Returns the inverse of the complement of the error function of z, that is a +value x such that: + + p = erfc(x); + +[$../graphs/erfc_inv.png] + +[h4 Accuracy] + +For types up to and including 80-bit long doubles the approximations used +are accurate to less than ~ 2 epsilon. For higher precision types these +functions have the same accuracy as the +[link math_toolkit.special.sf_erf.error_function forward error functions]. + +[h4 Testing] + +There are two sets of tests: + +* Basic sanity checks attempt to "round-trip" from +/x/ to /p/ and back again. These tests have quite +generous tolerances: in general both the error functions and their +inverses change so rapidly in some places that round tripping to more than a couple +of significant digits isn't possible. This is especially true when +/p/ is very near one: in this case there isn't enough +"information content" in the input to the inverse function to get +back where you started. +* Accuracy checks using high-precision test values. These measure +the accuracy of the result, given /exact/ input values. + +[h4 Implementation] + +These functions use a rational approximation [jm_rationals] +to calculate an initial +approximation to the result that is accurate to ~10[super -19], +then only if that has insufficient accuracy compared to the epsilon for T, +do we clean up the result using +[@http://en.wikipedia.org/wiki/Simple_rational_approximation Halley iteration]. + +Constructing rational approximations to the erf/erfc functions is actually +surprisingly hard, especially at high precision. For this reason no attempt +has been made to achieve 10[super -34 ] accuracy suitable for use with 128-bit +reals. + +In the following discussion, /p/ is the value passed to erf_inv, and /q/ is +the value passed to erfc_inv, so that /p = 1 - q/ and /q = 1 - p/ and in both +cases we want to solve for the same result /x/. + +For /p < 0.5/ the inverse erf function is reasonably smooth and the approximation: + + x = p(p + 10)(Y + R(p)) + +Gives a good result for a constant Y, and R(p) optimised for low absolute error +compared to |Y|. + +For q < 0.5 things get trickier, over the interval /0.5 > q > 0.25/ +the following approximation works well: + + x = sqrt(-2log(q)) / (Y + R(q)) + +While for q < 0.25, let + + z = sqrt(-log(q)) + +Then the result is given by: + + x = z(Y + R(z - B)) + +As before Y is a constant and the rational function R is optimised for low +absolute error compared to |Y|. B is also a constant: it is the smallest value +of /z/ for which each approximation is valid. There are several approximations +of this form each of which reaches a little further into the tail of the erfc +function (at `long double` precision the extended exponent range compared to +`double` means that the tail goes on for a very long way indeed). + +[endsect][/ :error_inv The Error Function Inverses] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/error.qbk b/doc/sf_and_dist/error.qbk new file mode 100644 index 000000000..6bdc2d35a --- /dev/null +++ b/doc/sf_and_dist/error.qbk @@ -0,0 +1,71 @@ +[section:relative_error Relative Error] + +Given an actual value /a/ and a found value /v/ the relative error can be +calculated from: + +[equation error2] + +However the test programs in the library use the symmetrical form: + +[equation error1] + +which measures /relative difference/ and happens to be less error +prone in use since we don't have to worry which value is the "true" +result, and which is the experimental one. It guarantees to return a value +at least as large as the relative error. + +Special care needs to be taken when one value is zero: we could either take the +absolute error in this case (but that's cheating as the absolute error is likely +to be very small), or we could assign a value of either 1 or infinity to the +relative error in this special case. In the test cases for the special functions +in this library, everything below a threshold is regarded as "effectively zero", +otherwise the relative error is assigned the value of 1 if only one of the terms +is zero. The threshold is currently set at `std::numeric_limits<>::min()`: +in other words all denormalised numbers are regarded as a zero. + +All the test programs calculate /quantized relative error/, whereas the graphs +in this manual are produced with the /actual error/. The difference is as +follows: in the test programs, the test data is rounded to the target real type +under test when the program is compiled, +so the error observed will then be a whole number of /units in the last place/ +either rounded up from the actual error, or rounded down (possibly to zero). +In contrast the /true error/ is obtained by extending +the precision of the calculated value, and then comparing to the actual value: +in this case the calculated error may be some fraction of /units in the last place/. + +Note that throughout this manual and the test programs the relative error is +usually quoted in units of epsilon. However, remember that /units in the last place/ +more accurately reflect the number of contaminated digits, and that relative +error can /"wobble"/ by a factor of 2 compared to /units in the last place/. +In other words: two implementations of the same function, whose +maximum relative errors differ by a factor of 2, can actually be accurate +to the same number of binary digits. You have been warned! + +[#zero_error][h4 The Impossibility of Zero Error] + +For many of the functions in this library, it is assumed that the error is +"effectively zero" if the computation can be done with a number of guard +digits. However it should be remembered that if the result is a +/transcendental number/ +then as a point of principle we can never be sure that the result is accurate +to more than 1 ulp. This is an example of /the table makers dilemma/: consider what +happens if the first guard digit is a one, and the remaining guard digits are all zero. +Do we have a tie or not? Since the only thing we can tell about a transcendental number +is that its digits have no particular pattern, we can never tell if we have a tie, +no matter how many guard digits we have. Therefore, we can never be completely sure +that the result has been rounded in the right direction. Of course, transcendental +numbers that just happen to be a tie - for however many guard digits we have - are +extremely rare, and get rarer the more guard digits we have, but even so.... + +Refer to the classic text +[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic] +for more information. + +[endsect][/section:relative_error Relative Error] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/error_handling.qbk b/doc/sf_and_dist/error_handling.qbk new file mode 100644 index 000000000..eca907b6b --- /dev/null +++ b/doc/sf_and_dist/error_handling.qbk @@ -0,0 +1,318 @@ +[section:error_handling Error Handling] + +[def __format [@../../../../format/index.html Boost.Format]] + +[heading Quick Reference] + +Handling of errors by this library is split into two orthogonal parts: + +* What kind of error has been raised? +* What should be done when the error is raised? + +The kinds of errors that can be raised are: + +[variablelist +[[Domain Error][Occurs when one or more arguments to a function + are out of range.]] +[[Pole Error][Occurs when the particular arguments cause the function to be + evaluated at a pole with no well defined residual value. For example if + __tgamma is evaluated at exactly -2, the function approaches different limiting + values depending upon whether you approach from just above or just below + -2. Hence the function has no well defined value at this point and a + Pole Error will be raised.]] +[[Overflow Error][Occurs when the result is either infinite, or too large + to represent in the numeric type being returned by the function.]] +[[Underflow Error][Occurs when the result is not zero, but is too small + to be represented by any other value in the type being returned by + the function.]] +[[Denormalisation Error][Occurs when the returned result would be a denormalised value.]] +[[Evaluation Error][Occurs when an internal error occured that prevented the + result from being evaluated: this should never occur, but if it does, then + it's likely to be due to an iterative method not converging fast enough.]] +] + +The action undertaken by each error condition is determined by the current +__Policy in effect. This can be changed program-wide by setting some +configuration macros, or at namespace scope, or at the call site (by +specifying a specific policy in the function call). + +The available actions are: + +[variablelist +[[throw_on_error][Throws the exception most appropriate to the error condition.]] +[[errno_on_error][Sets ::errno to an appropriate value, and then returns the most +appropriate result]] +[[ignore_error][Ignores the error and simply the returns the most appropriate result.]] +[[user_error][Calls a + [link math_toolkit.policy.pol_tutorial.user_def_err_pol user-supplied error handler].]] +] + +The following tables show all the permutations of errors and actions, +with the default action for each error shown in bold: + +[table Possible Actions for Domain Errors +[[Action] [Behaviour]] +[[throw_on_error][[*Throws `std::domain_error`]]] +[[errno_on_error][Sets `::errno` to `EDOM` and returns `std::numeric_limits::quiet_NaN()`]] +[[ignore_error][Returns `std::numeric_limits::quiet_NaN()`]] +[[user_error][Returns the result of `boost::math::policies::user_domain_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[table Possible Actions for Pole Errors +[[Action] [Behaviour]] +[[throw_on_error] [[*Throws `std::domain_error`]]] +[[errno_on_error][Sets `::errno` to `EDOM` and returns `std::numeric_limits::quiet_NaN()`]] +[[ignore_error][Returns `std::numeric_limits::quiet_NaN()`]] +[[user_error][Returns the result of `boost::math::policies::user_pole_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[table Possible Actions for Overflow Errors +[[Action] [Behaviour]] +[[throw_on_error][[*Throws `std::overflow_error`]]] +[[errno_on_error][Sets `::errno` to `ERANGE` and returns `std::numeric_limits::infinity()`]] +[[ignore_error][Returns `std::numeric_limits::infinity()`]] +[[user_error][Returns the result of `boost::math::policies::user_overflow_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[table Possible Actions for Underflow Errors +[[Action] [Behaviour]] +[[throw_on_error][Throws `std::underflow_error`]] +[[errno_on_error][Sets `::errno` to `ERANGE` and returns 0.]] +[[ignore_error][[*Returns 0]]] +[[user_error][Returns the result of `boost::math::policies::user_underflow_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[table Possible Actions for Denorm Errors +[[Action] [Behaviour]] +[[throw_on_error][Throws `std::underflow_error`]] +[[errno_on_error][Sets `::errno` to `ERANGE` and returns the denormalised value.]] +[[ignore_error][[*Returns the denormalised value.]]] +[[user_error][Returns the result of `boost::math::policies::user_denorm_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[table Possible Actions for Internal Evaluation Errors +[[Action] [Behaviour]] +[[throw_on_error][[*Throws `boost::math::evaluation_error`]]] +[[errno_on_error][Sets `::errno` to `EDOM` and returns `std::numeric_limits::infinity()`.]] +[[ignore_error][Returns `std::numeric_limits::infinity()`.]] +[[user_error][Returns the result of `boost::math::policies::user_evaluation_error`: + [link math_toolkit.policy.pol_tutorial.user_def_err_pol + this function must be defined by the user].]] +] + +[heading Rationale] + +The flexibility of the current implementation should be reasonably obvious, the +default behaviours were chosen based on feedback during the formal review of +this library. It was felt that: + +* Genuine errors should be flagged with exceptions +rather than following C-compatible behaviour and setting ::errno. +* Numeric underflow and denormalised results were not considered to be +fatal errors in most cases, so it was felt that these should be ignored. + +[heading Finding More Information] + +There are some pre-processor macro defines that can be used to +[link math_toolkit.policy.pol_ref.policy_defaults +change the policy defaults]. See also the [link math_toolkit.policy +policy section]. + +An example is at the Policy tutorial in +[link math_toolkit.policy.pol_tutorial.changing_policy_defaults +Changing the Policy Defaults]. + +Full source code of this typical example of passing a 'bad' argument +(negative degrees of freedom) to Student's t distribution +is [link math_toolkit.dist.stat_tut.weg.error_eg in the error handling example]. + +The various kind of errors are described in more detail below. + +[heading [#domain_error]Domain Errors] + +When a special function is passed an argument that is outside the range +of values for which that function is defined, then the function returns +the result of: + + boost::math::policies::raise_domain_error(FunctionName, Message, Val, __Policy); + +Where +`T` is the floating-point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +Val is the value that was out of range, and __Policy is the current policy +in use for the function that was called. + +The default policy behaviour of this function is to throw a +std::domain_error C++ exception. But if the __Policy is to ignore +the error, or set global ::errno, then a NaN will be returned. + +This behaviour is chosen to assist compatibility with the behaviour of +['ISO/IEC 9899:1999 Programming languages - C] +and with the +[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Draft Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 6]: + +[:['"Each of the functions declared above shall return a NaN (Not a Number) +if any argument value is a NaN, but it shall not report a domain error. +Otherwise, each of the functions declared above shall report a domain error +for just those argument values for which:]] + +[:['"the function description's Returns clause explicitly specifies a domain, and those arguments fall outside the specified domain; or] + +['"the corresponding mathematical function value has a non-zero imaginary component; or] + +['"the corresponding mathematical function is not mathematically defined.]] + +[:['"Note 2: A mathematical function is mathematically defined +for a given set of argument values if it is explicitly defined +for that set of argument values or +if its limiting value exists and does not depend on the direction of approach."]] + +Note that in order to support information-rich error messages when throwing +exceptions, `Message` must contain +a __format recognised format specifier: the argument `Val` is inserted into +the error message according to the specifier used. + +For example if `Message` contains a "%1%" then it is replaced by the value of +`Val` to the full precision of T, where as "%.3g" would contain the value of +`Val` to 3 digits. See the __format documentation for more details. + +[heading [#pole_error]Evaluation at a pole] + +When a special function is passed an argument that is at a pole +without a well defined residual value, then the function returns +the result of: + + boost::math::policies::raise_pole_error(FunctionName, Message, Val, __Policy); + +Where +`T` is the floating point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +`Val` is the value of the argument that is at a pole, and __Policy is the +current policy in use for the function that was called. + +The default behaviour of this function is to throw a std::domain_error exception. +But __error_policy can be used to change this, for example to `ignore_error` +and return NaN. + +Note that in order to support information-rich error messages when throwing +exceptions, `Message` must contain +a __format recognised format specifier: the argument `val` is inserted into +the error message according to the specifier used. + +For example if `Message` contains a "%1%" then it is replaced by the value of +`val` to the full precision of T, where as "%.3g" would contain the value of +`val` to 3 digits. See the __format documentation for more details. + +[heading [#overflow_error]Numeric Overflow] + +When the result of a special function is too large to fit in the argument +floating-point type, then the function returns the result of: + + boost::math::policies::raise_overflow_error(FunctionName, Message, __Policy); + +Where +`T` is the floating-point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +and __Policy is the current policy +in use for the function that was called. + +The default policy for this function is that `std::overflow_error` +C++ exception is thrown. But if, for example, an `ignore_error` policy +is used, then returns `std::numeric_limits::infinity()`. +In this situation if the type `T` doesn't support infinities, +the maximum value for the type is returned. + +[heading [#underflow_error]Numeric Underflow] + +If the result of a special function is known to be non-zero, but the +calculated result underflows to zero, then the function returns the result of: + + boost::math::policies::raise_underflow_error(FunctionName, Message, __Policy); + +Where +`T` is the floating point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +and __Policy is the current policy +in use for the called function. + +The default version of this function returns zero. +But with another policy, like `throw_on_error`, +throws an `std::underflow_error` C++ exception. + +[heading [#denorm_error]Denormalisation Errors] + +If the result of a special function is a denormalised value /z/ then the function +returns the result of: + + boost::math::policies::raise_denorm_error(z, FunctionName, Message, __Policy); + +Where +`T` is the floating point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +and __Policy is the current policy +in use for the called function. + +The default version of this function returns /z/. +But with another policy, like `throw_on_error` +throws an `std::underflow_error` C++ exception. + +[heading [#evaluation_error]Evaluation Errors] + +When a special function calculates a result that is known to be erroneous, +or where the result is incalculable then it calls: + + boost::math::policies::raise_evaluation_error(FunctionName, Message, Val, __Policy); + +Where +`T` is the floating point type passed to the function, `FunctionName` is the +name of the function, `Message` is an error message describing the problem, +`Val` is the erroneous value, +and __Policy is the current policy +in use for the called function. + +The default behaviour of this function is to throw a `boost::math::evaluation_error`. + +Note that in order to support information rich error messages when throwing +exceptions, `Message` must contain +a __format recognised format specifier: the argument `val` is inserted into +the error message according to the specifier used. + +For example if `Message` contains a "%1%" then it is replaced by the value of +`val` to the full precision of T, where as "%.3g" would contain the value of +`val` to 3 digits. See the __format documentation for more details. + +[heading [#checked_narrowing_cast]Errors from typecasts] + +Many special functions evaluate their results at a higher precision +than their arguments in order to ensure full machine precision in +the result: for example, a function passed a float argument may evaluate +its result using double precision internally. Many of the errors listed +above may therefore occur not during evaluation, but when converting +the result to the narrower result type. The function: + + template + T checked_narrowing_cast(U const& val, const char* function); + +Is used to perform these conversions, and will call the error handlers +listed above on [link overflow_error overflow], +[link underflow_error underflow] or [link denorm_error denormalisation]. + +[endsect][/section:error_handling Error Handling] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/factorials.qbk b/doc/sf_and_dist/factorials.qbk new file mode 100644 index 000000000..6827db461 --- /dev/null +++ b/doc/sf_and_dist/factorials.qbk @@ -0,0 +1,301 @@ +[section:factorials Factorials and Binomial Coefficients] + +[section:sf_factorial Factorial] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T factorial(unsigned i); + + template + T factorial(unsigned i, const ``__Policy``&); + + template + T unchecked_factorial(unsigned i); + + template + struct max_factorial; + + }} // namespaces + +[h4 Description] + + template + T factorial(unsigned i); + + template + T factorial(unsigned i, const ``__Policy``&); + +Returns [^i!]. + +[optional_policy] + +For [^i <= max_factorial::value] this is implemented by table lookup, +for larger values of [^i], this function is implemented in terms of __tgamma. + +If [^i] is so large that the result can not be represented in type T, then +calls __overflow_error. + + template + T unchecked_factorial(unsigned i); + +Returns [^i!]. + +Internally this function performs table lookup of the result. Further it performs +no range checking on the value of i: it is up to the caller to ensure +that [^i <= max_factorial::value]. This function is intended to be used +inside inner loops that require fast table lookup of factorials, but requires +care to ensure that argument [^i] never grows too large. + + template + struct max_factorial + { + static const unsigned value = X; + }; + +This traits class defines the largest value that can be passed to +[^unchecked_factorial]. The member `value` can be used where integral +constant expressions are required: for example to define the size of +further tables that depend on the factorials. + +[h4 Accuracy] + +For arguments smaller than `max_factorial::value` +the result should be +correctly rounded. For larger arguments the accuracy will be the same +as for __tgamma. + +[h4 Testing] + +Basic sanity checks and spot values to verify the data tables: +the main tests for the __tgamma function handle those cases already. + +[h4 Implementation] + +The factorial function is table driven for small arguments, and is +implemented in terms of __tgamma for larger arguments. + +[endsect] + +[section:sf_double_factorial Double Factorial] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T double_factorial(unsigned i); + + template + T double_factorial(unsigned i, const ``__Policy``&); + + }} // namespaces + +Returns [^i!!]. + +[optional_policy] + +May return the result of __overflow_error if the result is too large +to represent in type T. The implementation is designed to be optimised +for small /i/ where table lookup of i! is possible. + +[h4 Accuracy] + +The implementation uses a trivial adaptation of +the factorial function, so error rates should be no more than a couple +of epsilon higher. + +[h4 Testing] + +The spot tests for the double factorial use data +generated by functions.wolfram.com. + +[h4 Implementation] + +The double factorial is implemented in terms of the factorial and gamma +functions using the relations: + +(2n)!! = 2[super n ] * n! + +(2n+1)!! = (2n+1)! / (2[super n ] n!) + +and + +(2n-1)!! = [Gamma]((2n+1)/2) * 2[super n ] / sqrt(pi) + +[endsect] + +[section:sf_rising_factorial Rising Factorial] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` rising_factorial(T x, int i); + + template + ``__sf_result`` rising_factorial(T x, int i, const ``__Policy``&); + + }} // namespaces + +Returns the rising factorial of /x/ and /i/: + +rising_factorial(x, i) = [Gamma](x + i) / [Gamma](x); + +or + +rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i) + +Note that both /x/ and /i/ can be negative as well as positive. + +[optional_policy] + +May return the result of __overflow_error if the result is too large +to represent in type T. + +The return type of these functions is computed using the __arg_pomotion_rules: +the type of the result is `double` if T is an integer type, otherwise the type +of the result is T. + +[h4 Accuracy] + +The accuracy will be the same as +the __tgamma_delta_ratio function. + +[h4 Testing] + +The spot tests for the rising factorials use data generated +by functions.wolfram.com. + +[h4 Implementation] + +Rising and falling factorials are implemented as ratios of gamma functions +using __tgamma_delta_ratio. Optimisations for +small integer arguments are handled internally by that function. + +[endsect] + +[section:sf_falling_factorial Falling Factorial] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` falling_factorial(T x, unsigned i); + + template + ``__sf_result`` falling_factorial(T x, unsigned i, const ``__Policy``&); + + }} // namespaces + +Returns the falling factorial of /x/ and /i/: + +falling_factorial(x, i) = x(x-1)(x-2)(x-3)...(x-i+1) + +Note that this function is only defined for positive /i/, hence the +`unsigned` second argument. Argument /x/ can be either positive or +negative however. + +[optional_policy] + +May return the result of __overflow_error if the result is too large +to represent in type T. + +The return type of these functions is computed using the __arg_pomotion_rules: +the type of the result is `double` if T is an integer type, otherwise the type +of the result is T. + +[h4 Accuracy] + +The accuracy will be the same as +the __tgamma_delta_ratio function. + +[h4 Testing] + +The spot tests for the falling factorials use data generated by +functions.wolfram.com. + +[h4 Implementation] + +Rising and falling factorials are implemented as ratios of gamma functions +using __tgamma_delta_ratio. Optimisations for +small integer arguments are handled internally by that function. + +[endsect] + +[section:sf_binomial Binomial Coefficients] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + T binomial_coefficient(unsigned n, unsigned k); + + template + T binomial_coefficient(unsigned n, unsigned k, const ``__Policy``&); + + }} // namespaces + +Returns the binomial coefficient: [sub n]C[sub k]. + +Requires k <= n. + +[optional_policy] + +May return the result of __overflow_error if the result is too large +to represent in type T. + +[h4 Accuracy] + +The accuracy will be the same as for the +factorials for small arguments (i.e. no more than one or two epsilon), +and the __beta function for larger arguments. + +[h4 Testing] + +The spot tests for the binomial coefficients use data +generated by functions.wolfram.com. + +[h4 Implementation] + +Binomial coefficients are calculated using table lookup of factorials +where possible using: + +[sub n]C[sub k] = n! / (k!(n-k)!) + +Otherwise it is implemented in terms of the beta function using the relations: + +[sub n]C[sub k] = 1 / (k * __beta(k, n-k+1)) + +and + +[sub n]C[sub k] = 1 / ((n-k) * __beta(k+1, n-k)) + +[endsect] + + +[endsect][/section:factorials Factorials] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/fpclassify.qbk b/doc/sf_and_dist/fpclassify.qbk new file mode 100644 index 000000000..a663576f0 --- /dev/null +++ b/doc/sf_and_dist/fpclassify.qbk @@ -0,0 +1,99 @@ +[section:fpclass Floating Point Classification: Infinities and NaN's] + +[h4 Synopsis] + + #define FP_ZERO /* implementation specific value */ + #define FP_NORMAL /* implementation specific value */ + #define FP_INFINITE /* implementation specific value */ + #define FP_NAN /* implementation specific value */ + #define FP_SUBNORMAL /* implementation specific value */ + + template + int fpclassify(T t); + + template + bool isfinite(T z); + + template + bool isinf(T t); + + template + bool isnan(T t); + + template + bool isnormal(T t); + +[h4 Description] + +These functions provide the same functionality as the macros with the same +name in C99, indeed if the C99 macros are available, then these functions +are implemented in terms of them, otherwise they rely on std::numeric_limits<> +to function. + +Note that the definition of these functions ['does not suppress the definition +of these names as macros by math.h] on those platforms that already provide +these as macros. That mean that the following have differing meanings: + + using namespace boost::math; + + // This might call a global macro if defined, + // but might not work if the type of z is unsupported + // by the std lib macro: + isnan(z); + // + // This calls the Boost version + // (found via the "using namespace boost::math" declaration) + // it works for any type that has numeric_limits support for type z: + (isnan)(z); + // + // As above but with namespace qualification. + (boost::math::isnan)(z); + // + // This will cause a compiler error is isnan is a native macro: + boost::math::isnan(z); + // So always use (boost::math::isnan)(z); instead. + +Detailed descriptions for each of these functions follows: + + template + int fpclassify(T t); + +Returns an integer value that classifies the value /t/: + +[table +[[fpclassify value] [class of t.]] +[[FP_ZERO] [If /t/ is zero.]] +[[FP_NORMAL] [If /t/ is a non-zero, non-denormalised finite value.]] +[[FP_INFINITE] [If /t/ is plus or minus infinity.]] +[[FP_NAN] [If /t/ is a NaN.]] +[[FP_SUBNORMAL] [If /t/ is a denormalised number.]] +] + + template + bool isfinite(T z); + +Returns true only if /z/ is not an infinity or a NaN. + + template + bool isinf(T t); + +Returns true only if /z/ is plus or minus infinity. + + template + bool isnan(T t); + +Returns true only if /z/ is a NaN. + + template + bool isnormal(T t); + +Returns true only if /z/ is a normal number (not zero, infinite, NaN, or denormalised). + +[endsect] [/section:fpclass Floating Point Classification: Infinities and NaN's] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/fraction.qbk b/doc/sf_and_dist/fraction.qbk new file mode 100644 index 000000000..452e203ff --- /dev/null +++ b/doc/sf_and_dist/fraction.qbk @@ -0,0 +1,157 @@ +[section:cf Continued Fraction Evaluation] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + typename detail::fraction_traits::result_type + continued_fraction_b(Gen& g, int bits); + + template + typename detail::fraction_traits::result_type + continued_fraction_b(Gen& g, int bits, boost::uintmax_t& max_terms); + + template + typename detail::fraction_traits::result_type + continued_fraction_a(Gen& g, int bits); + + template + typename detail::fraction_traits::result_type + continued_fraction_a(Gen& g, int bits, boost::uintmax_t& max_terms); + + }}} // namespaces + +[h4 Description] + +[@http://en.wikipedia.org/wiki/Continued_fraction Continued fractions are a common method of approximation. ] +These functions all evaluate the continued fraction described by the /generator/ +type argument. The functions with an "_a" suffix evaluate the fraction: + +[equation fraction2] + +and those with a "_b" suffix evaluate the fraction: + +[equation fraction1] + +This latter form is somewhat more natural in that it corresponds with the usual +definition of a continued fraction, but note that the first /a/ value returned by +the generator is discarded. Further, often the first /a/ and /b/ values in a +continued fraction have different defining equations to the remaining terms, which +may make the "_a" suffixed form more appropriate. + +The generator type should be a function object which supports the following +operations: + +[table +[[Expression] [Description]] +[[Gen::result_type] [The type that is the result of invoking operator(). + This can be either an arithmetic type, or a std::pair<> of arithmetic types.]] +[[g()] [Returns an object of type Gen::result_type. + +Each time this operator is called then the next pair of /a/ and /b/ + values is returned. Or, if result_type is an arithmetic type, + then the next /b/ value is returned and all the /a/ values + are assumed to 1.]] +] + +In all the continued fraction evaluation functions the /bits/ parameter is the +number of bits precision desired in the result, evaluation of the fraction will +continue until the last term evaluated leaves the first /bits/ bits in the result +unchanged. + +If the optional /max_terms/ parameter is specified then no more than /max_terms/ +calls to the generator will be made, and on output, +/max_terms/ will be set to actual number of +calls made. This facility is particularly useful when profiling a continued +fraction for convergence. + +[h4 Implementation] + +Internally these algorithms all use the modified Lentz algorithm: refer to +Numeric Recipes in C++, W. H. Press et all, chapter 5, +(especially 5.2 Evaluation of continued fractions, p 175 - 179) +for more information, also +Lentz, W.J. 1976, Applied Optics, vol. 15, pp. 668-671. + +[h4 Examples] + +The [@http://en.wikipedia.org/wiki/Golden_ratio golden ratio phi = 1.618033989...] +can be computed from the simplest continued fraction of all: + +[equation fraction3] + +We begin by defining a generator function: + + template + struct golden_ratio_fraction + { + typedef T result_type; + + result_type operator() + { + return 1; + } + }; + +The golden ratio can then be computed to double precision using: + + continued_fraction_a( + golden_ratio_fraction(), + std::numeric_limits::digits); + +It's more usual though to have to define both the /a/'s and the /b/'s +when evaluating special functions by continued fractions, for example +the tan function is defined by: + +[equation fraction4] + +So it's generator object would look like: + + template + struct tan_fraction + { + private: + T a, b; + public: + tan_fraction(T v) + : a(-v*v), b(-1) + {} + + typedef std::pair result_type; + + std::pair operator()() + { + b += 2; + return std::make_pair(a, b); + } + }; + +Notice that if the continuant is subtracted from the /b/ terms, +as is the case here, then all the /a/ terms returned by the generator +will be negative. The tangent function can now be evaluated using: + + template + T tan(T a) + { + tan_fraction fract(a); + return a / continued_fraction_b(fract, std::numeric_limits::digits); + } + +Notice that this time we're using the "_b" suffixed version to evaluate +the fraction: we're removing the leading /a/ term during fraction evaluation +as it's different from all the others. + +[endsect][/section:cf Continued Fraction Evaluation] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/gamma_derivatives.qbk b/doc/sf_and_dist/gamma_derivatives.qbk new file mode 100644 index 000000000..9bfb82e03 --- /dev/null +++ b/doc/sf_and_dist/gamma_derivatives.qbk @@ -0,0 +1,54 @@ +[section:gamma_derivatives Derivative of the Incomplete Gamma Function] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` gamma_p_derivative(T1 a, T2 x); + + template + ``__sf_result`` gamma_p_derivative(T1 a, T2 x, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +This function find some uses in statistical distributions: it +implements the partial derivative with respect to /x/ of the incomplete +gamma function. + +[equation derivative1] + +[optional_policy] + +Note that the derivative of the function __gamma_q can be obtained by negating +the result of this function. + +The return type of this function is computed using the __arg_pomotion_rules +when T1 and T2 are different types, otherwise the return type is simply T1. + +[h4 Accuracy] + +Almost identical to the incomplete gamma function __gamma_p: refer to +the documentation for that function for more information. + +[h4 Implementation] + +This function just expose some of the internals of the incomplete +gamma function __gamma_p: refer to the documentation for that function +for more information. + +[endsect][/section Derivative of the Incomplete Gamma Functions] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/gamma_ratios.qbk b/doc/sf_and_dist/gamma_ratios.qbk new file mode 100644 index 000000000..c459cee73 --- /dev/null +++ b/doc/sf_and_dist/gamma_ratios.qbk @@ -0,0 +1,112 @@ +[section:gamma_ratios Ratios of Gamma Functions] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` tgamma_ratio(T1 a, T2 b); + + template + ``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&); + + template + ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta); + + template + ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + + template + ``__sf_result`` tgamma_ratio(T1 a, T2 b); + + template + ``__sf_result`` tgamma_ratio(T1 a, T2 b, const ``__Policy``&); + +Returns the ratio of gamma functions: + +[equation gamma_ratio0] + +[optional_policy] + +Internally this just calls `tgamma_delta_ratio(a, b-a)`. + + template + ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta); + + template + ``__sf_result`` tgamma_delta_ratio(T1 a, T2 delta, const ``__Policy``&); + +Returns the ratio of gamma functions: + +[equation gamma_ratio1] + +[optional_policy] + +Note that the result is calculated accurately even when /delta/ is +small compared to /a/: indeed even if /a+delta ~ a/. The function is +typically used when /a/ is large and /delta/ is very small. + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types, otherwise the result type is simple T1. + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table Errors In the Function tgamma_delta_ratio(a, delta) +[[Significand Size] [Platform and Compiler] [20 < a < 80 + +and + +delta < 1]] +[[53] [Win32, Visual C++ 8] [Peak=16.9 Mean=1.7] ] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=24 Mean=2.7]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=12.8 Mean=1.8]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=21.4 Mean=2.3] ] +] + +[table Errors In the Function tgamma_ratio(a, b) +[[Significand Size] [Platform and Compiler] [6 < a,b < 50]] +[[53] [Win32, Visual C++ 8] [Peak=34 Mean=9] ] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=91 Mean=23]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=35.6 Mean=9.3]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=43.9 Mean=13.2] ] +] + +[h4 Testing] + +Accuracy tests use data generated at very high precision +(with [@http://shoup.net/ntl/doc/RR.txt NTL RR class] +set at 1000-bit precision: about 300 decimal digits) +and a deliberately naive calculation of [Gamma](x)/[Gamma](y). + +[h4 Implementation] + +The implementation of these functions is very similar to that of +__beta, and is based on combining similar power terms +to improve accuracy and avoid spurious overflow/underflow. + +In addition there are optimisations for the situation where /delta/ +is a small integer: in which case this function is basically +the reciprocal of a rising factorial, or where both arguments +are smallish integers: in which case table lookup of factorials +can be used to calculate the ratio. + +[endsect][/section:gamma_ratios Ratios of Gamma Functions] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/graphs/bernoulli_cdf.png b/doc/sf_and_dist/graphs/bernoulli_cdf.png new file mode 100644 index 0000000000000000000000000000000000000000..8643e136db2e6f4057d945db393bdc91b06c1583 GIT binary patch literal 16479 zcmeAS@N?(olHy`uVBq!ia0y~yV2Wj6U|hq&#=yYve2EZ9g0sLQvY3H^?+6GpPSxg< zW?uUajuCoBoj24`HW3ADlTFYbak1fBqTM~O2`TCP8RI-57+gj2#Fnr-I16r31X@`j;- zQ7)lDlffsB{M<8PI;jx zN{^m0Jn-O*FT4}3^B{%$$n+vl@AD6fg@ss4WT)xfy>e@oyoSMHj}N}wwRb08~t4s@W1qPzSo`Gu2qshcBTug{onm| z{hv!x3<+mCvo?S3)?LTda88%u#NFRXmn|3?maNE^y}o}JzpsscCzIo@-i?J}VbcVs zA2t#a|HA**zkUD0_50FQ?B4Z%x)5vm(I>I1L2sS3wyvu0iRI}_j3Z@sJ)GbEL!SL# zsT`Z*oC?1AhlO2RI+yjF_7>PW`}i}i9ho*()4$K3^@IQLza>X^igjN4AsCR{({QC! z%c1Tqy9Sq_!o}Z5WB)JC-}l2np|0|w*5vNu|L@KJrXO7TNVR$P2g`G+8ZGC9J{gOD zKXP7Z0VAUalhy?GQx38(7@QN>)->oAu;d+({?RNTz!}CY=E1VEfq#>O+zl2kN5dPO 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zJ9{C;?Ml!_`3J{JJ?3`I*~?&lj?4KCTi%_iZEu)&eC!D3TrP32*FSEK;0kIFmEj$P hTfq79KdS^ok+!?~+ew!t7#J8BJYD@<);T3K0RWUy*M$H8 literal 0 HcmV?d00001 diff --git a/doc/sf_and_dist/hermite.qbk b/doc/sf_and_dist/hermite.qbk new file mode 100644 index 000000000..c19c1f2b3 --- /dev/null +++ b/doc/sf_and_dist/hermite.qbk @@ -0,0 +1,115 @@ +[section:hermite Hermite Polynomials] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` hermite(unsigned n, T x); + + template + ``__sf_result`` hermite(unsigned n, T x, const ``__Policy``&); + + template + ``__sf_result`` hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1); + + }} // namespaces + +[h4 Description] + +The return type of these functions is computed using the __arg_pomotion_rules: +note than when there is a single template argument the result is the same type +as that argument or `double` if the template argument is an integer type. + + template + ``__sf_result`` hermite(unsigned n, T x); + + template + ``__sf_result`` hermite(unsigned n, T x, const ``__Policy``&); + +Returns the value of the Hermite Polynomial of order /n/ at point /x/: + +[equation hermite_0] + +[optional_policy] + +The following graph illustrates the behaviour of the first few +Hermite Polynomials: + +[$../graphs/hermite.png] + + template + ``__sf_result`` hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1); + +Implements the three term recurrence relation for the Hermite +polynomials, this function can be used to create a sequence of +values evaluated at the same /x/, and for rising /n/. + +[equation hermite_1] + +For example we could produce a vector of the first 10 polynomial +values using: + + double x = 0.5; // Abscissa value + vector v; + v.push_back(hermite(0, x)).push_back(hermite(1, x)); + for(unsigned l = 1; l < 10; ++l) + v.push_back(hermite_next(l, x, v[l], v[l-1])); + +Formally the arguments are: + +[variablelist +[[n][The degree /n/ of the last polynomial calculated.]] +[[x][The abscissa value]] +[[Hn][The value of the polynomial evaluated at degree /n/.]] +[[Hnm1][The value of the polynomial evaluated at degree /n-1/.]] +] + +[h4 Accuracy] + +The following table shows peak errors (in units of epsilon) +for various domains of input arguments. +Note that only results for the widest floating point type on the system are +given as narrower types have __zero_error. + +[table Peak Errors In the Hermite Polynomial +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] ] +[[53] [Win32, Visual C++ 8] [Peak=4.5 Mean=1.5] ] +[[64] [Red Hat Linux IA32, g++ 4.1] [Peak=6 Mean=2]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=6 Mean=2] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=6 Mean=4]] +] + +Note that the worst errors occur when the degree increases, values greater than +~120 are very unlikely to produce sensible results, especially in the associated +polynomial case when the order is also large. Further the relative errors +are likely to grow arbitrarily large when the function is very close to a root. + +[h4 Testing] + +A mixture of spot tests of values calculated using functions.wolfram.com, +and randomly generated test data are +used: the test data was computed using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision. + +[h4 Implementation] + +These functions are implemented using the stable three term +recurrence relations. These relations guarentee low absolute error +but cannot guarentee low relative error near one of the roots of the +polynomials. + +[endsect][/section:beta_function The Beta Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/html/index.html b/doc/sf_and_dist/html/index.html new file mode 100644 index 000000000..366c74497 --- /dev/null +++ b/doc/sf_and_dist/html/index.html @@ -0,0 +1,395 @@ + + + +Math Toolkit + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+

+
Next
+
+
+
+

+Math Toolkit

+
+

+John Maddock +

+

+Paul A. Bristow +

+

+Hubert Holin +

+

+Xiaogang Zhang +

+
+

Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+
+

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+
+
+

Table of Contents

+
+
Overview
+
+
About the Math Toolkit
+
Navigation
+
Directory and + File Structure
+
Namespaces
+
Calculation + of the Type of the Result
+
Error Handling
+
Compilers
+
Configuration + and Policies
+
Thread Safety
+
Performance
+
History and What's + New
+
Contact Info and + Support
+
+
Statistical Distributions and Functions
+
+
Statistical Distributions + Tutorial
+
+
Overview
+
Worked Examples
+
+
+ Distribution Construction Example
+
Student's + t Distribution Examples
+
Chi Squared + Distribution Examples
+
F Distribution + Examples
+
Binomial + Distribution Examples
+
Negative + Binomial Distribution Examples
+
Normal + Distribution Examples
+
Error Handling + Example
+
Find Location + and Scale Examples
+
Comparison + with C, R, FORTRAN-style Free Functions
+
+
Random Variates + and Distribution Parameters
+
Discrete Probability + Distributions
+
+
Statistical Distributions + Reference
+
+
Non-Member Properties
+
Distributions
+
+
+ Bernoulli Distribution
+
Beta + Distribution
+
+ Binomial Distribution
+
Cauchy-Lorentz + Distribution
+
+ Chi Squared Distribution
+
Exponential + Distribution
+
Extreme + Value Distribution
+
F Distribution
+
Gamma + (and Erlang) Distribution
+
+ Log Normal Distribution
+
+ Negative Binomial Distribution
+
Normal + (Gaussian) Distribution
+
Pareto + Distribution
+
Poisson + Distribution
+
Rayleigh + Distribution
+
+ Students t Distribution
+
+ Triangular Distribution
+
Weibull + Distribution
+
Uniform + Distribution
+
+
Distribution + Algorithms
+
+
Extras/Future Directions
+
+
Special Functions
+
+
Gamma Functions
+
+
Gamma
+
Log Gamma
+
Digamma
+
Ratios + of Gamma Functions
+
Incomplete Gamma + Functions
+
Incomplete + Gamma Function Inverses
+
Derivative + of the Incomplete Gamma Function
+
+
Factorials and Binomial + Coefficients
+
+
Factorial
+
+ Double Factorial
+
+ Rising Factorial
+
+ Falling Factorial
+
Binomial + Coefficients
+
+
Beta Functions
+
+
Beta
+
Incomplete + Beta Functions
+
The + Incomplete Beta Function Inverses
+
Derivative + of the Incomplete Beta Function
+
+
Error Functions
+
+
Error + Functions
+
Error Function + Inverses
+
+
Polynomials
+
+
Legendre (and + Associated) Polynomials
+
Laguerre (and + Associated) Polynomials
+
Hermite Polynomials
+
Spherical Harmonics
+
+
Bessel Functions
+
+
Bessel Function + Overview
+
Bessel Functions + of the First and Second Kinds
+
Modified Bessel + Functions of the First and Second Kinds
+
Spherical + Bessel Functions of the First and Second Kinds
+
+
Elliptic Integrals
+
+
Elliptic + Integral Overview
+
Elliptic + Integrals - Carlson Form
+
Elliptic Integrals + of the First Kind - Legendre Form
+
Elliptic Integrals + of the Second Kind - Legendre Form
+
Elliptic Integrals + of the Third Kind - Legendre Form
+
+
Logs, Powers, Roots and + Exponentials
+
+
log1p
+
expm1
+
cbrt
+
sqrt1pm1
+
powm1
+
hypot
+
+
Sinus Cardinal and Hyperbolic + Sinus Cardinal Functions
+
+
Sinus Cardinal + and Hyperbolic Sinus Cardinal Functions Overview
+
sinc_pi
+
sinhc_pi
+
+
Inverse Hyperbolic Functions
+
+
Inverse + Hyperbolic Functions Overview
+
acosh
+
asinh
+
atanh
+
+
Floating Point Classification: + Infinities and NaN's
+
+
Internal Details and Tools (Experimental)
+
+
Overview
+
Reused Utilities
+
+
+ Series Evaluation
+
Continued Fraction + Evaluation
+
Polynomial + and Rational Function Evaluation
+
Root Finding + With Derivatives
+
Root Finding + Without Derivatives
+
Locating Function + Minima
+
+
Testing and Development
+
+
Polynomials
+
Minimax Approximations + and the Remez Algorithm
+
Relative + Error and Testing
+
Graphing, + Profiling, and Generating Test Data for Special Functions
+
+
+
Use with User Defined Floating-Point + Types
+
+
Using With NTL - a High-Precision + Floating-Point Library
+
Conceptual Requirements + for Real Number Types
+
Conceptual Requirements + for Distribution Types
+
Conceptual Archetypes + and Testing
+
+
Policies
+
+
Policy Overview
+
Policy Tutorial
+
+
+ So Just What is a Policy Anyway?
+
+ Policies Have Sensible Defaults
+
So + How are Policies Used Anyway?
+
+ Changing the Policy Defaults
+
+ Setting Policies for Distributions on an Ad Hoc Basis
+
+ Changing the Policy on an Ad Hoc Basis for the Special Functions
+
+ Setting Policies at Namespace or Translation Unit Scope
+
+ Calling User Defined Error Handlers
+
+ Understanding Quantiles of Discrete Distributions
+
+
Policy Reference
+
+
+ Error Handling Policies
+
Internal + Promotion Policies
+
Mathematically + Undefined Function Policies
+
Discrete + Quantile Policies
+
Precision + Policies
+
Iteration + Limits Policies
+
Using + macros to Change the Policy Defaults
+
Setting + Polices at Namespace Scope
+
Policy Class + Reference
+
+
+
Performance
+
+
Performance Overview
+
Interpreting these Results
+
Getting the Best Performance + from this Library
+
Comparing Compilers
+
Performance Tuning Macros
+
Comparisons to Other + Open Source Libraries
+
The Performance Test + Application
+
+
Backgrounders
+
+
Additional + Implementation Notes
+
Relative + Error
+
The Lanczos Approximation
+
The Remez Method
+
References
+
+
Library Status
+
+
History and What's New
+
Compilers
+
Known Issues, and Todo List
+
Credits and Acknowledgements
+
+
+
+

+ ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22 +

+
+ + + +

Last revised: November 13, 2007 at 17:03:47 GMT

+
+
Next
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders.html b/doc/sf_and_dist/html/math_toolkit/backgrounders.html new file mode 100644 index 000000000..e6421dc1b --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders.html @@ -0,0 +1,53 @@ + + + +Backgrounders + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Additional + Implementation Notes
+
Relative + Error
+
The Lanczos Approximation
+
The Remez Method
+
References
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/implementation.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/implementation.html new file mode 100644 index 000000000..b8aafcc13 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/implementation.html @@ -0,0 +1,752 @@ + + + +Additional Implementation Notes + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ The majority of the implementation notes are included with the documentation + of each function or distribution. The notes here are of a more general nature, + and reflect more the general implementation philosophy used. +

+
+ + Implemention + philosophy +
+

+ "First be right, then be fast." +

+

+ There will always be potential compromises to be made between speed and accuracy. + It may be possible to find faster methods, particularly for certain limited + ranges of arguments, but for most applications of math functions and distributions, + we judge that speed is rarely as important as accuracy. +

+

+ So our priority is accuracy. +

+

+ To permit evaluation of accuracy of the special functions, production of + extremely accurate tables of test values has received considerable effort. +

+

+ (It also required much CPU effort - there was some danger of molten plastic + dripping from the bottom of JM's laptop, so instead, PAB's Dual-core desktop + was kept 50% busy for days calculating some + tables of test values!) +

+

+ For a specific RealType, say float or double, it may be possible to find + approximations for some functions that are simpler and thus faster, but less + accurate (perhaps because there are no refining iterations, for example, + when calculating inverse functions). +

+

+ If these prove accurate enough to be "fit for his purpose", then + a user may substitute his custom specialization. +

+

+ For example, there are approximations dating back from times when computation + was a lot more expensive: +

+

+ H Goldberg and H Levine, Approximate formulas for percentage points and normalisation + of t and chi squared, Ann. Math. Stat., 17(4), 216 - 225 (Dec 1946). +

+

+ A H Carter, Approximations to percentage points of the z-distribution, Biometrika + 34(2), 352 - 358 (Dec 1947). +

+

+ These could still provide sufficient accuracy for some speed-critical applications. +

+
+ + Accuracy + and Representation of Test Values +
+

+ In order to be accurate enough for as many as possible real types, constant + values are given to 50 decimal digits if available (though many sources proved + only accurate near to 64-bit double precision). Values are specified as long + double types by appending L, unless they are exactly representable, for example + integers, or binary fractions like 0.125. This avoids the risk of loss of + accuracy converting from double, the default type. Values are used after + static_cast<RealType>(1.2345L) to provide the appropriate RealType + for spot tests. +

+

+ Functions that return constants values, like kurtosis for example, are written + as +

+

+ static_cast<RealType>(-3) / + 5; +

+

+ to provide the most accurate value that the compiler can compute for the + real type. (The denominator is an integer and so will be promoted exactly). +

+

+ So tests for one third, not exactly representable + with radix two floating-point, (should) use, for example: +

+

+ static_cast<RealType>(1) / + 3; +

+

+ If a function is very sensitive to changes in input, specifying an inexact + value as input (such as 0.1) can throw the result off by a noticeable amount: + 0.1f is "wrong" by ~1e-7 for example (because 0.1 has no exact + binary representation). That is why exact binary values - halves, quarters, + and eighths etc - are used in test code along with the occasional fraction + a/b with b + a power of two (in order to ensure that the result is an exactly representable + binary value). +

+
+ + Tolerance + of Tests +
+

+ The tolerances need to be set to the maximum of: +

+
    +
  • + Some epsilon value. +
    • +
  • + The accuracy of the data (often only near 64-bit double). +
    • +
+

+ Otherwise when long double has more digits than the test data, then no amount + of tweaking an epsilon based tolerance will work. +

+

+ A common problem is when tolerances that are suitable for implementations + like Microsoft VS.NET where double and long double are the same size: tests + fail on other systems where long double is more accurate than double. Check + first that the suffix L is present, and then that the tolerance is big enough. +

+
+ + Handling + Unsuitable Arguments +
+

+ In Errors + in Mathematical Special Functions, J. Marraffino & M. Paterno + it is proposed that signalling a domain error is mandatory when the argument + would give an mathematically undefined result. +

+
  • + Guideline 1 +
+
+

+

+

+ A mathematical function is said to be defined at a point a = (a1, a2, + . . .) if the limits as x = (x1, x2, . . .) 'approaches a from all directions + agree'. The defined value may be any number, or +infinity, or -infinity. +

+

+

+
+

+ Put crudely, if the function goes to + infinity and then emerges 'round-the-back' + with - infinity, it is NOT defined. +

+
+

+

+

+ The library function which approximates a mathematical function shall + signal a domain error whenever evaluated with argument values for which + the mathematical function is undefined. +

+

+

+
+
  • + Guideline 2 +
+
+

+

+

+ The library function which approximates a mathematical function shall + signal a domain error whenever evaluated with argument values for which + the mathematical function obtains a non-real value. +

+

+

+
+

+ This implementation is believed to follow these proposals and to assist compatibility + with ISO/IEC 9899:1999 Programming languages - C and + with the Draft + Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph + 5. See + also domain_error. +

+

+ See policy reference for + details of the error handling policies that should allow a user to comply + with any of these recommendations, as well as other behaviour. +

+

+ See error handling + for a detailed explanation of the mechanism, and error_handling + example and error_handling_example.cpp +

+
+ + + + + +
[Caution]Caution

+ If you enable throw but do NOT have try & catch block, then the program + will terminate with an uncaught exception and probably abort. Therefore + to get the benefit of helpful error messages, enabling all + exceptions and using try&catch is + recommended for all applications. However, for simplicity, this is not + done for most examples. +

+
+ + Handling + of Functions that are Not Mathematically defined +
+

+ Functions that are not mathematically defined, like the Cauchy mean, fail + to compile by default. A + policy allows control of this. +

+

+ If the policy is to permit undefined functions, then calling them throws + a domain error, by default. But the error policy can be set to not throw, + and to return NaN instead. For example, +

+

+ #define BOOST_MATH_DOMAIN_ERROR_POLICY + ignore_error +

+

+ appears before the first Boost include, then if the un-implemented function + is called, mean(cauchy<>()) will return std::numeric_limits<T>::quiet_NaN(). +

+
+ + + + + +
[Warning]Warning

+ If std::numeric_limits<T>::has_quiet_NaN is false (for example T + is a User-defined type), then an exception will always be thrown when a + domain error occurs. Catching exceptions is therefore strongly recommended. +

+
+ + Median + of distributions +
+

+ There are many distributions for which we have been unable to find an analytic + formula, and this has deterred us from implementing median + functions, the mid-point in a list of values. +

+

+ However a useful median approximation for distribution dist + may be available from +

+

+ quantile(dist, 0.5). +

+

+ Mean, + Median, and Skew, Paul T von Hippel +

+

+ Descriptive + Statistics, +

+

+ and + +

+

+ Mathematica + Basic Statistics. give more detail, in particular for discrete distributions. +

+
+ + Handling + of Floating-Point Infinity +
+

+ Some functions and distributions are well defined with + or - infinity as + argument(s), but after some experiments with handling infinite arguments + as special cases, we concluded that it was generally more useful to forbid + this, and instead to return the result of domain_error. +

+

+ Handling infinity as special cases is additionally complicated because, unlike + built-in types on most - but not all - platforms, not all User-Defined Types + are specialized to provide std::numeric_limits<RealType>::infinity() and would return zero rather than any representation + of infinity. +

+

+ The rationale is that non-finiteness may happen because of error or overflow + in the users code, and it will be more helpful for this to be diagnosed promptly + rather than just continuing. The code also became much more complicated, + more error-prone, much more work to test, and much less readable. +

+

+ However in a few cases, for example normal, where we felt it obvious, we + have permitted argument(s) to be infinity, provided infinity is implemented + for the realType on that implementation. +

+

+ Overflow, underflow, denorm can be handled using error + handling policies. +

+

+ We have also tried to catch boundary cases where the mathematical specification + would result in divide by zero or overflow and signalling these similarly. + What happens at (and near), poles can be controlled through error + handling policies. +

+
+ + Scale, + Shape and Location +
+

+ We considered adding location and scale to the list of functions, for example: +

+
+template <class RealType>
+inline RealType scale(const triangular_distribution<RealType>& dist)
+{
+  RealType lower = dist.lower();
+  RealType mode = dist.mode();
+  RealType upper = dist.upper();
+  RealType result;  // of checks.
+  if(false == detail::check_triangular(BOOST_CURRENT_FUNCTION, lower, mode, upper, &result))
+  {
+    return result;
+  }
+  return (upper - lower);
+}
+
+

+ but found that these concepts are not defined (or their definition too contentious) + for too many distributions to be generally applicable. Because they are non-member + functions, they can be added if required. +

+
+ + Notes + on Implementation of Specific Functions & Distributions +
+
  • + Default parameters for the Triangular Distribution. We are uncertain about + the best default parameters. Some sources suggest that the Standard Triangular + Distribution has lower = 0, mode = half and upper = 1. However as a approximation + for the normal distribution, the most common usage, lower = -1, mode = + 0 and upper = 1 would be more suitable. +
+
+ + Rational + Approximations Used +
+

+ Some of the special functions in this library are implemented via rational + approximations. These are either taken from the literature, or devised by + John Maddock using our + Remez code. +

+

+ Rational rather than Polynomial approximations are used to ensure accuracy: + polynomial approximations are often wonderful up to a certain level of accuracy, + but then quite often fail to provide much greater accuracy no matter how + many more terms are added. +

+

+ Our own approximations were devised either for added accuracy (to support + 128-bit long doubles for example), or because literature methods were unavailable + or under non-BSL compatible license. Our Remez code is known to produce good + agreement with literature results in fairly simple "toy" cases. + All approximations were checked for convergence and to ensure that they were + not ill-conditioned (the coefficients can give a theoretically good solution, + but the resulting rational function may be un-computable at fixed precision). +

+

+ Recomputing using different Remez implementations may well produce differing + coefficients: the problem is well known to be ill conditioned in general, + and our Remez implementation often found a broad and ill-defined minima for + many of these approximations (of course for simple "toy" examples + like approximating exp the + minima is well defined, and the coeffiecents should agree no matter whose + Remez implementation is used). This should not in general effect the validity + of the approximations: there's good literature supporting the idea that coefficients + can be "in error" without necessarily adversely effecting the result. + Note that "in error" has a special meaning in this context, see + "Approximate construction + of rational approximations and the effect of error autocorrection.", + Grigori Litvinov, eprint arXiv:math/0101042. Therefore the coefficients + still need to be accurately calculated, even if they can be in error compared + to the "true" minimax solution. +

+
+ + Representation + of Mathematical Constants +
+

+ A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used to provide high + accuracy constants to mathematical functions and distributions, since it + is important to provide values uniformly for both built-in float, double + and long double types, and for User Defined types like NTL::quad_float and + NTL::RR. +

+

+ To permit calculations in this Math ToolKit and its tests, (and elsewhere) + at about 100 decimal digits with NTL::RR type, it is obviously necessary + to define constants to this accuracy. +

+

+ However, some compilers do not accept decimal digits strings as long as this. + So the constant is split into two parts, with the 1st containing at least + long double precision, and the 2nd zero if not needed or known. The 3rd part + permits an exponent to be provided if necessary (use zero if none) - the + other two parameters may only contain decimal digits (and sign and decimal + point), and may NOT include an exponent like 1.234E99 (nor a trailing F or + L). The second digit string is only used if T is a User-Defined Type, when + the constant is converted to a long string literal and lexical_casted to + type T. (This is necessary because you can't use a numeric constant since + even a long double might not have enough digits). +

+

+ For example, pi is defined: +

+
+BOOST_DEFINE_MATH_CONSTANT(pi,
+  3.141592653589793238462643383279502884197169399375105820974944,
+  5923078164062862089986280348253421170679821480865132823066470938446095505,
+  0)                                              
+
+

+ And used thus: +

+
+using namespace boost::math::constants;
+
+double diameter = 1.;
+double radius = diameter * pi<double>();
+
+or boost::math::constants::pi<NTL::RR>()
+
+

+ Note that it is necessary (if inconvenient) to specify the type explicitly. +

+

+ So you cannot write +

+
+double p = boost::math::constants::pi<>();  // could not deduce template argument for 'T'  
+
+

+ Neither can you write: +

+
+double p = boost::math::constants::pi; // Context does not allow for disambiguation of overloaded function     
+double p = boost::math::constants::pi(); // Context does not allow for disambiguation of overloaded function     
+
+
+ + Thread + safety +
+

+ Reporting of error by setting errno should be thread safe already (otherwise + none of the std lib math functions would be thread safe?). If you turn on + reporting of errors via exceptions, errno gets left unused anyway. +

+

+ Other than that, the code is intended to be thread safe for + built in real-number types : so float, double and long double + are all thread safe. +

+

+ For non-built-in types - NTL::RR for example - initialisation of the various + constants used in the implementation is potentially not + thread safe. This most undesiable, but it would be a signficant challenge + to fix it. Some compilers may offer the option of having static-constants + initialised in a thread safe manner (Commeau, and maybe others?), if that's + the case then the problem is solved. This is a topic of hot debate for the + next C++ std revision, so hopefully all compilers will be required to do + the right thing here at some point. +

+
+ + Sources + of Test Data +
+

+ We found a large number of sources of test data. We have assumed that these + are "known good" if they agree with the results + from our test and only consulted other sources for their 'vote' + in the case of serious disagreement. The accuracy, actual and claimed, vary + very widely. Only Wolfram Mathematica + functions provided a higher accuracy than C++ double (64-bit floating-point) + and was regarded as the most-trusted source by far. +

+

+ A useful index of sources is: Web-oriented + Teaching Resources in Probability and Statistics +

+

+ Statlet: + Is a Javascript application that calculates and plots probability distributions, + and provides the most complete range of distributions: +

+
+

+

+

+ Bernoulli, Binomial, discrete uniform, geometric, hypergeometric, negative + binomial, Poisson, beta, Cauchy-Lorentz, chi-sequared, Erlang, exponential, + extreme value, Fisher, gamma, Laplace, logistic, lognormal, normal, Parteo, + Student's t, triangular, uniform, and Weibull. +

+

+

+
+

+ It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, & + critical values for 5 tail values. +

+

+ It is also the only independent source found for the Weibull distribution; + unfortunately it appears to suffer from very poor accuracy in areas where + the underlying special function is known to be difficult to implement. +

+
+ + Creating + and Managing the Equations +
+

+ The primary source for the equations is now MathML: + see the *.mml files in libs/math/doc/sf_and_dist/equations/. +

+

+ These are most easily edited by a GUI editor such as Mathcast, + please note that the equation editor supplied with Open Office currently + mangles these files and should not currently be used. +

+

+ Convertion to SVG was achieved using SVGMath + and a command line such as: +

+
$for file in *.mml; do 
+>/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \
+>>$file > $(basename $file .mml).svg
+>done
+
+

+ Note that SVGMath requires that the mml files are not + wrapped in an XHTML XML wrapper - this is added by Mathcast by default - + one workaround is to copy an existing mml file and then edit it with Mathcast: + the existing format should then be preserved. This is a bug in the XML parser + used by SVGMath which the author is aware of. +

+

+ If neccessary the XHTML wrapper can be removed with: +

+
cat filename | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*</math>\).*/\1/' > newfile
+

+ Setting up fonts for SVGMath is currently rather tricky, on a Windows XP + system JM's font setup is the same as the sample config file provided with + SVGMath but with: +

+
<!-- Double-struck -->
+    <mathvariant name="double-struck" family="Mathematica7, Lucida Sans Unicode"/>
+
+

+ changed to: +

+
<!-- Double-struck -->
+    <mathvariant name="double-struck" family="Lucida Sans Unicode"/>
+
+

+ Note that unlike the sample config file supplied with SVGMath, this does + not make use of the Mathematica 7 font as this lacks sufficient Unicode information + for it to be used with either SVGMath or XEP "as is". +

+

+ Also note that the SVG files in the repository are almost certainly Windows-specific + since they reference various Windows Fonts. +

+

+ PNG files can be created from the SVG's using Batik + and a command such as: +

+
java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg
+

+ The PDF is generated into \pdf\math.pdf using a command from a shell or command + window with current directory \math_toolkit\libs\math\doc\sf_and_dist, typically: +

+
bjam -a pdf
+

+ Note that XEP will have to be configured to use and + embed whatever fonts are used by the SVG equations (if necessary + editing the sample xep.xml provided by the XEP installation). +

+

+ (html is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html + using just bjam -a). +

+

+ JM's XEP config file has the following font configuration section added: +

+
<font-group xml:base="file:/C:/Windows/Fonts/" label="Windows TrueType" embed="true" subset="true"> 
+      <font-family name="Arial">
+        <font><font-data ttf="arial.ttf"/></font>
+        <font style="oblique"><font-data ttf="ariali.ttf"/></font>
+        <font weight="bold"><font-data ttf="arialbd.ttf"/></font>
+        <font weight="bold" style="oblique"><font-data ttf="arialbi.ttf"/></font>
+      </font-family>
+
+      <font-family name="Times New Roman" ligatures="&#xFB01; &#xFB02;">
+        <font><font-data ttf="times.ttf"/></font>
+        <font style="italic"><font-data ttf="timesi.ttf"/></font>
+        <font weight="bold"><font-data ttf="timesbd.ttf"/></font>
+        <font weight="bold" style="italic"><font-data ttf="timesbi.ttf"/></font>
+      </font-family>
+
+      <font-family name="Courier New">
+        <font><font-data ttf="cour.ttf"/></font>
+        <font style="oblique"><font-data ttf="couri.ttf"/></font>
+        <font weight="bold"><font-data ttf="courbd.ttf"/></font>
+        <font weight="bold" style="oblique"><font-data ttf="courbi.ttf"/></font>
+      </font-family>
+
+      <font-family name="Tahoma" embed="true">
+        <font><font-data ttf="tahoma.ttf"/></font>
+        <font weight="bold"><font-data ttf="tahomabd.ttf"/></font>
+      </font-family>
+
+      <font-family name="Verdana" embed="true">
+        <font><font-data ttf="verdana.ttf"/></font>
+        <font style="oblique"><font-data ttf="verdanai.ttf"/></font>
+        <font weight="bold"><font-data ttf="verdanab.ttf"/></font>
+        <font weight="bold" style="oblique"><font-data ttf="verdanaz.ttf"/></font>
+      </font-family>
+
+      <font-family name="Palatino" embed="true" ligatures="&#xFB00; &#xFB01; &#xFB02; &#xFB03; &#xFB04;">
+        <font><font-data ttf="pala.ttf"/></font>
+        <font style="italic"><font-data ttf="palai.ttf"/></font>
+        <font weight="bold"><font-data ttf="palab.ttf"/></font>
+        <font weight="bold" style="italic"><font-data ttf="palabi.ttf"/></font>
+      </font-family>
+      
+    <font-family name="Lucida Sans Unicode">
+         <font><font-data ttf="lsansuni.ttf"/></font>
+    </font-family>
+
+

+ PAB had to alter his because the Lucida Sans Unicode font had a different + name. Changes are very likely to be required if you are not using Windows. +

+

+ XZ authored his equations using the venerable Latex, JM converted these to + MathML using mxlatex. + This process is currently unreliable and required some manual intervention: + consequently Latex source is not considered a viable route for the automatic + production of SVG versions of equations. +

+

+ Equations are embedded in the quickbook source using the equation + template defined in math.qbk. This outputs Docbook XML that looks like: +

+
<inlinemediaobject>
+<imageobject role"html">
+<imagedata fileref"../equations/myfile.png"></imagedata>
+</imageobject>
+<imageobject role"print">
+<imagedata fileref"../equations/myfile.svg"></imagedata>
+</imageobject>
+</inlinemediaobject>
+
+

+ MathML is not currently present in the Docbook output, or in the generated + HTML: this needs further investigation. +

+
+ + Producing + Graphs +
+

+ Graphs were mostly produced by a very laborious process entailing output + of columns of values from C++ programs to a .csv file, use of RJS + Graph to arrange the display and axes, and output to a .ps file, + followed by conversion to .png using Adobe Photoshop, or similar utility. + This rigmarole is not recommended! +

+

+ We plan to carry out this process in a single step using the Google + Summer of Code 2007 project of Jacob Voytko (whose work so far is + at .\boost-sandbox\SOC\2007\visualization) that should, when completed, allow + output of annotated graphs as Scalable Vector Graphic .svg files directly + from C++ programs. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/lanczos.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/lanczos.html new file mode 100644 index 000000000..b9f8d0ad0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/lanczos.html @@ -0,0 +1,578 @@ + + + +The Lanczos Approximation + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Motivation +
+

+ Why base gamma and gamma-like functions on the Lanczos approximation? +

+

+ First of all I should make clear that for the gamma function over real numbers + (as opposed to complex ones) the Lanczos approximation (See Wikipedia + or Mathworld) + appears to offer no clear advantage over more traditional methods such as + Stirling's + approximation. Pugh carried out an extensive + comparison of the various methods available and discovered that they were + all very similar in terms of complexity and relative error. However, the + Lanczos approximation does have a couple of properties that make it worthy + of further consideration: +

+
    +
  • + The approximation has an easy to compute truncation error that holds for + all z > 0. In practice that means we can use the + same approximation for all z > 0, and be certain + that no matter how large or small z is, the truncation + error will at worst be bounded by some finite value. +
    • +
  • + The approximation has a form that is particularly amenable to analytic + manipulation, in particular ratios of gamma or gamma-like functions are + particularly easy to compute without resorting to logarithms. +
    • +
+

+ It is the combination of these two properties that make the approximation + attractive: Stirling's approximation is highly accurate for large z, and + has some of the same analytic properties as the Lanczos approximation, but + can't easily be used across the whole range of z. +

+

+ As the simplest example, consider the ratio of two gamma functions: one could + compute the result via lgamma: +

+
+exp(lgamma(a) - lgamma(b));
+
+

+ However, even if lgamma is uniformly accurate to 0.5ulp, the worst case relative + error in the above can easily be shown to be: +

+
+Erel > a * log(a)/2 + b * log(b)/2
+
+

+ For small a and b that's not a + problem, but to put the relationship another way: each time a and + b increase in magnitude by a factor of 10, at least one decimal digit of + precision will be lost. +

+

+ In contrast, by analytically combining like power terms in a ratio of Lanczos + approximation's, these errors can be virtually eliminated for small a + and b, and kept under control for very large (or very + small for that matter) a and b. + Of course, computing large powers is itself a notoriously hard problem, but + even so, analytic combinations of Lanczos approximations can make the difference + between obtaining a valid result, or simply garbage. Refer to the implementation + notes for the beta + function for an example of this method in practice. The incomplete gamma_p gamma and + beta functions + use similar analytic combinations of power terms, to combine gamma and beta + functions divided by large powers into single (simpler) expressions. +

+
+ + The + Approximation +
+

+ The Lanczos Approximation to the Gamma Function is given by: +

+

+ +

+

+ Where Sg(z) is an infinite sum, that is convergent for all z > 0, and + g is an arbitrary parameter that controls the "shape" + of the terms in the sum which is given by: +

+

+ +

+

+ With individual coefficients defined in closed form by: +

+

+ +

+

+ However, evaluation of the sum in that form can lead to numerical instability + in the computation of the ratios of rising and falling factorials (effectively + we're multiplying by a series of numbers very close to 1, so roundoff errors + can accumulate quite rapidly). +

+

+ The Lanczos approximation is therefore often written in partial fraction + form with the leading constants absorbed by the coefficients in the sum: +

+

+ +

+

+ where: +

+

+ +

+

+ Again parameter g is an arbitrarily chosen constant, + and N is an arbitrarily chosen number of terms to evaluate + in the "Lanczos sum" part. +

+
+ + + + + +
[Note]Note

+ Some authors choose to define the sum from k=1 to N, and hence end up with + N+1 coefficients. This happens to confuse both the following discussion + and the code (since C++ deals with half open array ranges, rather than + the closed range of the sum). This convention is consistent with Godfrey, but not Pugh, + so take care when referring to the literature in this field. +

+
+ + Computing + the Coefficients +
+

+ The coefficients C0..CN-1 need to be computed from N + and g at high precision, and then stored as part of + the program. Calculation of the coefficients is performed via the method + of Godfrey; let the constants be contained + in a column vector P, then: +

+

+ P = B D C F +

+

+ where B is an NxN matrix: +

+

+ +

+

+ D is an NxN matrix: +

+

+ +

+

+ C is an NxN matrix: +

+

+ +

+

+ and F is an N element column vector: +

+

+ +

+

+ Note than the matrices B, D and C contain all integer terms and depend only + on N, this product should be computed first, and then + multiplied by F as the last step. +

+
+ + Choosing + the Right Parameters +
+

+ The trick is to choose N and g + to give the desired level of accuracy: choosing a small value for g + leads to a strictly convergent series, but one which converges only slowly. + Choosing a larger value of g causes the terms in the + series to be large and/or divergent for about the first g-1 + terms, and to then suddenly converge with a "crunch". +

+

+ Pugh has determined the optimal value of g + for N in the range 1 <= N <= 60: + unfortunately in practice choosing these values leads to cancellation errors + in the Lanczos sum as the largest term in the (alternating) series is approximately + 1000 times larger than the result. These optimal values appear not to be + useful in practice unless the evaluation can be done with a number of guard + digits and the coefficients are stored at higher precision + than that desired in the result. These values are best reserved for say, + computing to float precision with double precision arithmetic. +

+
+

Table 45. Optimal choices for N and g when computing with + guard digits (source: Pugh)

+
++++++ + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ N +

+ 		+

+ g +

+ 		+

+ Max Error +

+
+

+ 24 +

+ 		+

+ 6 +

+ 		+

+ 5.581 +

+ 		+

+ 9.51e-12 +

+
+

+ 53 +

+ 		+

+ 13 +

+ 		+

+ 13.144565 +

+ 		+

+ 9.2213e-23 +

+
+
+

+ The alternative described by Godfrey is to + perform an exhaustive search of the N and g + parameter space to determine the optimal combination for a given p + digit floating-point type. Repeating this work found a good approximation + for double precision arithmetic (close to the one Godfrey + found), but failed to find really good approximations for 80 or 128-bit long + doubles. Further it was observed that the approximations obtained tended + to optimised for the small values of z (1 < z < 200) used to test the + implementation against the factorials. Computing ratios of gamma functions + with large arguments were observed to suffer from error resulting from the + truncation of the Lancozos series. +

+

+ Pugh identified all the locations where the theoretical + error of the approximation were at a minimum, but unfortunately has published + only the largest of these minima. However, he makes the observation that + the minima coincide closely with the location where the first neglected term + (aN) in the Lanczos series Sg(z) changes sign. These locations are quite + easy to locate, albeit with considerable computer time. These "sweet + spots" need only be computed once, tabulated, and then searched when + required for an approximation that delivers the required precision for some + fixed precision type. +

+

+ Unfortunately, following this path failed to find a really good approximation + for 128-bit long doubles, and those found for 64 and 80-bit reals required + an excessive number of terms. There are two competing issues here: high precision + requires a large value of g, but avoiding cancellation + errors in the evaluation requires a small g. +

+

+ At this point note that the Lanczos sum can be converted into rational form + (a ratio of two polynomials, obtained from the partial-fraction form using + polynomial arithmetic), and doing so changes the coefficients so that they + are all positive. That means that the sum in rational form can + be evaluated without cancellation error, albeit with double the number of + coefficients for a given N. Repeating the search of the "sweet spots", + this time evaluating the Lanczos sum in rational form, and testing only those + "sweet spots" whose theoretical error is less than the machine + epsilon for the type being tested, yielded good approximations for all the + types tested. The optimal values found were quite close to the best cases + reported by Pugh (just slightly larger N + and slightly smaller g for a given precision than Pugh reports), and even though converting to rational + form doubles the number of stored coefficients, it should be noted that half + of them are integers (and therefore require less storage space) and the approximations + require a smaller N than would otherwise be required, + so fewer floating point operations may be required overall. +

+

+ The following table shows the optimal values for N and + g when computing at fixed precision. These should be + taken as work in progress: there are no values for 106-bit significand machines + (Darwin long doubles & NTL quad_float), and further optimisation of the + values of g may be possible. Errors given in the table + are estimates of the error due to truncation of the Lanczos infinite series + to N terms. They are calculated from the sum of the + first five neglected terms - and are known to be rather pessimistic estimates + - although it is noticeable that the best combinations of N + and g occurred when the estimated truncation error almost + exactly matches the machine epsilon for the type in question. +

+
+

Table 46. Optimum value for N and g when computing at fixed + precision

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform/Compiler Used +

+ 		+

+ N +

+ 		+

+ g +

+ 		+

+ Max Truncation Error +

+
+

+ 24 +

+ 		+

+ Win32, VC++ 7.1 +

+ 		+

+ 6 +

+ 		+

+ 1.428456135094165802001953125 +

+ 		+

+ 9.41e-007 +

+
+

+ 53 +

+ 		+

+ Win32, VC++ 7.1 +

+ 		+

+ 13 +

+ 		+

+ 6.024680040776729583740234375 +

+ 		+

+ 3.23e-016 +

+
+

+ 64 +

+ 		+

+ Suse Linux 9 IA64, gcc-3.3.3 +

+ 		+

+ 17 +

+ 		+

+ 12.2252227365970611572265625 +

+ 		+

+ 2.34e-024 +

+
+

+ 116 +

+ 		+

+ HP Tru64 Unix 5.1B / Alpha, Compaq C++ V7.1-006 +

+ 		+

+ 24 +

+ 		+

+ 20.3209821879863739013671875 +

+ 		+

+ 4.75e-035 +

+
+
+

+ Finally note that the Lanczos approximation can be written as follows by + removing a factor of exp(g) from the denominator, and then dividing all the + coefficients by exp(g): +

+

+ +

+

+ This form is more convenient for calculating lgamma, but for the gamma function + the division by e turns a possibly exact quality into + an inexact value: this reduces accuracy in the common case that the input + is exact, and so isn't used for the gamma function. +

+
+ + References +
+
    +
  1. + Paul Godfrey, "A note + on the computation of the convergent Lanczos complex Gamma approximation". +
    2. +
  2. + Glendon Ralph Pugh, "An + Analysis of the Lanczos Gamma Approximation", PhD Thesis November + 2004. +
    3. +
  3. + Viktor T. Toth, "Calculators + and the Gamma Function". +
    4. +
  4. + Mathworld, The + Lanczos Approximation. +
    5. +
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/refs.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/refs.html new file mode 100644 index 000000000..b8837d1da --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/refs.html @@ -0,0 +1,170 @@ + + + +References + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + General + references +
+

+ (Specific detailed sources for individual functions and distributions are + given at the end of each individual section). +

+

+ DLMF (NIST Digital Library of Mathematical + Functions) is intended to be a replacement for the legendary Abramowitz + and Stegun's Handbook of Mathematical Functions, now scheduled to be completed + in 2007. +

+

+ M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions + with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards + Applied Mathematics Series, U.S. Government Printing Office, Washington, + D.C.. +

+

+ The Wolfram Functions Site + The Wolfram Functions Site - Providing the mathematical and scientific community + with the world's largest (and most authorititive) collection of formulas + and graphics about mathematical functions. +

+

+ NIST/SEMATECH + e-Handbook of Statistical Methods +

+

+ Mathematica + Documentation: DiscreteDistributions The Wolfram Research Documentation + Center is a collection of online reference materials about Mathematica, CalculationCenter, + and other Wolfram Research products. +

+

+ Mathematica + Documentation: ContinuousDistributions The Wolfram Research Documentation + Center is a collection of online reference materials about Mathematica, CalculationCenter, + and other Wolfram Research products. +

+

+ Statistical Distributions (Wiley Series in Probability & Statistics) + (Paperback) by N.A.J. Hastings, Brian Peacock, Merran Evans, ISBN: 0471371246, + Wiley 2000. +

+

+ pugh.pdf + (application/pdf Object) Pugh Msc Thesis on the Lanczzos approximation + to the gamma function. +

+

+ N1514, + 03-0097, A Proposal to Add Mathematical Special Functions to the C++ Standard + Library (version 2), Walter E. Brown +

+
+ + Calculators* + that we found (and used to cross-check - as far as their widely-varying accuracy + allowed). +
+

+ http://www.adsciengineering.combpdcalc + Binomial Probability Distribution Calculator. +

+
+ + Other Libraries +
+

+ Cephes library by Shephen + Moshier and his book: +

+

+ Methods and programs for mathematical functions, Stephen L B Moshier, Ellis + Horwood (1989) ISBN 0745802893 0470216093 provided inspiration. +

+

+ 100-decimal digit calculator + provided some spot values. +

+

+ C++ + version. +

+

+ CDFLIB Library of Fortran Routines + for Cumulative Distribution functions. +

+

+ DCDFLIB + C++ version DCDFLIB is a library of C++ routines, using double precision + arithmetic, for evaluating cumulative probability density functions. +

+

+ http://www.softintegration.com/docs/packagechnagstat +

+

+ NAG libraries. +

+

+ MathCAD +

+

+ JMSL + Numerical Library (Java). +

+

+ John F Hart, Computer Approximations, (1978) ISBN 0 088275 642-7. +

+

+ William J Cody, Software Manual for the Elementary Functions, Prentice-Hall + (1980) ISBN 0138220646. +

+

+ Nico Temme, Special Functions, An Introduction to the Classical Functions + of Mathematical Physics, Wiley, ISBN: 0471-11313-1 (1996) who also gave valueable + advice. +

+

+ Statistics + Glossary, Valerie Easton and John H. McColl. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html new file mode 100644 index 000000000..f939b2315 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html @@ -0,0 +1,124 @@ + + + +Relative Error + + + + + + + + + + + + + + + +
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+
+PrevUpHomeNext +
+
+
+

+ Given an actual value a and a found value v + the relative error can be calculated from: +

+

+ +

+

+ However the test programs in the library use the symmetrical form: +

+

+ +

+

+ which measures relative difference and happens to be + less error prone in use since we don't have to worry which value is the "true" + result, and which is the experimental one. It guarantees to return a value + at least as large as the relative error. +

+

+ Special care needs to be taken when one value is zero: we could either take + the absolute error in this case (but that's cheating as the absolute error + is likely to be very small), or we could assign a value of either 1 or infinity + to the relative error in this special case. In the test cases for the special + functions in this library, everything below a threshold is regarded as "effectively + zero", otherwise the relative error is assigned the value of 1 if only + one of the terms is zero. The threshold is currently set at std::numeric_limits<>::min(): in other words all denormalised numbers + are regarded as a zero. +

+

+ All the test programs calculate quantized relative error, + whereas the graphs in this manual are produced with the actual + error. The difference is as follows: in the test programs, the + test data is rounded to the target real type under test when the program + is compiled, so the error observed will then be a whole number of units + in the last place either rounded up from the actual error, or + rounded down (possibly to zero). In contrast the true error + is obtained by extending the precision of the calculated value, and then + comparing to the actual value: in this case the calculated error may be some + fraction of units in the last place. +

+

+ Note that throughout this manual and the test programs the relative error + is usually quoted in units of epsilon. However, remember that units + in the last place more accurately reflect the number of contaminated + digits, and that relative error can "wobble" + by a factor of 2 compared to units in the last place. + In other words: two implementations of the same function, whose maximum relative + errors differ by a factor of 2, can actually be accurate to the same number + of binary digits. You have been warned! +

+

+

+
+ + The + Impossibility of Zero Error +
+

+ For many of the functions in this library, it is assumed that the error is + "effectively zero" if the computation can be done with a number + of guard digits. However it should be remembered that if the result is a + transcendental number then as a point of principle we + can never be sure that the result is accurate to more than 1 ulp. This is + an example of the table makers dilemma: consider what + happens if the first guard digit is a one, and the remaining guard digits + are all zero. Do we have a tie or not? Since the only thing we can tell about + a transcendental number is that its digits have no particular pattern, we + can never tell if we have a tie, no matter how many guard digits we have. + Therefore, we can never be completely sure that the result has been rounded + in the right direction. Of course, transcendental numbers that just happen + to be a tie - for however many guard digits we have - are extremely rare, + and get rarer the more guard digits we have, but even so.... +

+

+ Refer to the classic text What + Every Computer Scientist Should Know About Floating-Point Arithmetic + for more information. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/remez.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/remez.html new file mode 100644 index 000000000..5189a94e3 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/remez.html @@ -0,0 +1,542 @@ + + + +The Remez Method + + + + + + + + + + + + + + + +
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+
+
+

+ The Remez algorithm + is a methodology for locating the minimax rational approximation to a function. + This short article gives a brief overview of the method, but it should not + be regarded as a thorough theoretical treatment, for that you should consult + your favorite textbook. +

+

+ Imagine that you want to approximate some function f(x) by way of a rational + function R(x), where R(x) may be either a polynomial P(x) or a ratio of two + polynomials P(x)/Q(x) (a rational function). Initially we'll concentrate + on the polynomial case, as it's by far the easier to deal with, later we'll + extend to the full rational function case. +

+

+ We want to find the "best" rational approximation, where "best" + is defined to be the approximation that has the least deviation from f(x). + We can measure the deviation by way of an error function: +

+

+ Eabs(x) = f(x) - R(x) +

+

+ which is expressed in terms of absolute error, but we can equally use relative + error: +

+

+ Erel(x) = (f(x) - R(x)) / |f(x)| +

+

+ And indeed in general we can scale the error function in any way we want, + it makes no difference to the maths, although the two forms above cover almost + every practical case that you're likely to encounter. +

+

+ The minimax rational function R(x) is then defined to be the function that + yields the smallest maximal value of the error function. Chebyshev showed + that there is a unique minimax solution for R(x) that has the following properties: +

+
    +
  • + If R(x) is a polynomial of degree N, then there are N+2 unknowns: the N+1 + coefficients of the polynomial, and maximal value of the error function. +
    • +
  • + The error function has N+1 roots, and N+2 extrema (minima and maxima). +
    • +
  • + The extrema alternate in sign, and all have the same magnitude. +
    • +
+

+ That means that if we know the location of the extrema of the error function + then we can write N+2 simultaneous equations: +

+

+ R(xi) + (-1)iE = f(xi) +

+

+ where E is the maximal error term, and xi are the abscissa values of the N+2 + extrema of the error function. It is then trivial to solve the simultaneous + equations to obtain the polynomial coefficients and the error term. +

+

+ Unfortunately we don't know where the extrema of the error function + are located! +

+
+ + The Remez + Method +
+

+ The Remez method is an iterative technique which, given a broad range of + assumptions, will converge on the extrema of the error function, and therefore + the minimax solution. +

+

+ In the following discussion we'll use a concrete example to illustrate the + Remez method: an approximation to the function ex over the range [-1, 1]. +

+

+ Before we can begin the Remez method, we must obtain an initial value for + the location of the extrema of the error function. We could "guess" + these, but a much closer first approximation can be obtained by first constructing + an interpolated polynomial approximation to f(x). +

+

+ In order to obtain the N+1 coefficients of the interpolated polynomial we + need N+1 points (x0...xN): with our interpolated form passing through each + of those points that yields N+1 simultaneous equations: +

+

+ f(xi) = P(xi) = c0 + c1xi ... + cNxiN +

+

+ Which can be solved for the coefficients c0...cN in P(x). +

+

+ Obviously this is not a minimax solution, indeed our only guarantee is that + f(x) and P(x) touch at N+1 locations, away from those points the error may + be arbitrarily large. However, we would clearly like this initial approximation + to be as close to f(x) as possible, and it turns out that using the zeros + of an orthogonal polynomial as the initial interpolation points is a good + choice. In our example we'll use the zeros of a Chebyshev polynomial as these + are particularly easy to calculate, interpolating for a polynomial of degree + 4, and measuring relative error we get the following + error function: +

+

+ remez-2 +

+

+ Which has a peak relative error of 1.2x10-3. +

+

+ While this is a pretty good approximation already, judging by the shape of + the error function we can clearly do better. Before starting on the Remez + method propper, we have one more step to perform: locate all the extrema + of the error function, and store these locations as our initial Chebyshev + control points. +

+
+ + + + + +
[Note]Note
+

+ In the simple case of a polynomial approximation, by interpolating through + the roots of a Chebyshev polynomial we have in fact created a Chebyshev + approximation to the function: in terms of absolute + error this is the best a priori choice for the interpolated + form we can achieve, and typically is very close to the minimax solution. +

+

+ However, if we want to optimise for relative error, + or if the approximation is a rational function, then the initial Chebyshev + solution can be quite far from the ideal minimax solution. +

+

+ A more technical discussion of the theory involved can be found in this + online + course. +

+
+
+ + Remez Step + 1 +
+

+ The first step in the Remez method, given our current set of N+2 Chebyshev + control points xi, is to solve the N+2 simultaneous equations: +

+

+ P(xi) + (-1)iE = f(xi) +

+

+ To obtain the error term E, and the coefficients of the polynomial P(x). +

+

+ This gives us a new approximation to f(x) that has the same error E + at each of the control points, and whose error function alternates + in sign at the control points. This is still not necessarily the + minimax solution though: since the control points may not be at the extrema + of the error function. After this first step here's what our approximation's + error function looks like: +

+

+ remez-3 +

+

+ Clearly this is still not the minimax solution since the control points are + not located at the extrema, but the maximum relative error has now dropped + to 5.6x10-4. +

+
+ + Remez Step + 2 +
+

+ The second step is to locate the extrema of the new approximation, which + we do in two stages: first, since the error function changes sign at each + control point, we must have N+1 roots of the error function located between + each pair of N+2 control points. Once these roots are found by standard root + finding techniques, we know that N extrema are bracketed between each pair + of roots, plus two more between the endpoints of the range and the first + and last roots. The N+2 extrema can then be found using standard function + minimisation techniques. +

+

+ We now have a choice: multi-point exchange, or single point exchange. +

+

+ In single point exchange, we move the control point nearest to the largest + extrema to the absissa value of the extrema. +

+

+ In multi-point exchange we swap all the current control points, for the locations + of the extrema. +

+

+ In our example we perform multi-point exchange. +

+
+ + Iteration +
+

+ The Remez method then performs steps 1 and 2 above iteratively until the + control points are located at the extrema of the error function: this is + then the minimax solution. +

+

+ For our current example, two more iterations converges on a minimax solution + with a peak relative error of 5x10-4 and an error function that looks like: +

+

+ remez-4 +

+
+ + Rational + Approximations +
+

+ If we wish to extend the Remez method to a rational approximation of the + form +

+

+ f(x) = R(x) = P(x) / Q(x) +

+

+ where P(x) and Q(x) are polynomials, then we proceed as before, except that + now we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M. + This assumes that Q(x) is normalised so that it's leading coefficient is + 1, giving N+M+1 polynomial coefficients in total, plus the error term E. +

+

+ The simultaneous equations to be solved are now: +

+

+ P(xi) / Q(xi) + (-1)iE = f(xi) +

+

+ Evaluated at the N+M+2 control points xi. +

+

+ Unfortunately these equations are non-linear in the error term E: we can + only solve them if we know E, and yet E is one of the unknowns! +

+

+ The method usually adopted to solve these equations is an iterative one: + we guess the value of E, solve the equations to obtain a new value for E + (as well as the polynomial coefficients), then use the new value of E as + the next guess. The method is repeated until E converges on a stable value. +

+

+ These complications extend the running time required for the development + of rational approximations quite considerably. It is often desirable to obtain + a rational rather than polynomial approximation none the less: rational approximations + will often match more difficult to approximate functions, to greater accuracy, + and with greater efficiency, than their polynomial alternatives. For example, + if we takes our previous example of an approximation to ex, we obtained 5x10-4 accuracy + with an order 4 polynomial. If we move two of the unknowns into the denominator + to give a pair of order 2 polynomials, and re-minimise, then the peak relative + error drops to 8.7x10-5. That's a 5 fold increase in accuracy, for the same + number of terms overall. +

+
+ + Practical + Considerations +
+

+ Most treatises on approximation theory stop at this point. However, from + a practical point of view, most of the work involves finding the right approximating + form, and then persuading the Remez method to converge on a solution. +

+

+ So far we have used a direct approximation: +

+

+ f(x) = R(x) +

+

+ But this will converge to a useful approximation only if f(x) is smooth. + In addition round-off errors when evaluating the rational form mean that + this will never get closer than within a few epsilon of machine precision. + Therefore this form of direct approximation is often reserved for situations + where we want efficiency, rather than accuracy. +

+

+ The first step in improving the situation is generally to split f(x) into + a dominant part that we can compute accurately by another method, and a slowly + changing remainder which can be approximated by a rational approximation. + We might be tempted to write: +

+

+ f(x) = g(x) + R(x) +

+

+ where g(x) is the dominant part of f(x), but if f(x)/g(x) is approximately + constant over the interval of interest then: +

+

+ f(x) = g(x)(c + R(x)) +

+

+ Will yield a much better solution: here c is a constant + that is the approximate value of f(x)/g(x) and R(x) is typically tiny compared + to c. In this situation if R(x) is optimised for absolute + error, then as long as its error is small compared to the constant c, + that error will effectively get wiped out when R(x) is added to c. +

+

+ The difficult part is obviously finding the right g(x) to extract from your + function: often the asymptotic behaviour of the function will give a clue, + so for example the function erfc + becomes proportional to e-x2/x as x becomes large. Therefore using: +

+

+ erfc(z) = (C + R(x)) e-x2/x +

+

+ as the approximating form seems like an obvious thing to try, and does indeed + yield a useful approximation. +

+

+ However, the difficulty then becomes one of converging the minimax solution. + Unfortunately, it is known that for some functions the Remez method can lead + to divergent behaviour, even when the initial starting approximation is quite + good. Furthermore, it is not uncommon for the solution obtained in the first + Remez step above to be a bad one: the equations to be solved are generally + "stiff", often very close to being singular, and assuming a solution + is found at all, round-off errors and a rapidly changing error function, + can lead to a situation where the error function does not in fact change + sign at each control point as required. If this occurs, it is fatal to the + Remez method. It is also possible to obtain solutions that are perfectly + valid mathematically, but which are quite useless computationally: either + because there is an unavoidable amount of roundoff error in the computation + of the rational function, or because the denominator has one or more roots + over the interval of the approximation. In the latter case while the approximation + may have the correct limiting value at the roots, the approximation is nonetheless + useless. +

+

+ Assuming that the approximation does not have any fatal errors, and that + the only issue is converging adequately on the minimax solution, the aim + is to get as close as possible to the minimax solution before beginning the + Remez method. Using the zeros of a Chebyshev polynomial for the initial interpolation + is a good start, but may not be ideal when dealing with relative errors and/or + rational (rather than polynomial) approximations. One approach is to skew + the initial interpolation points to one end: for example if we raise the + roots of the Chebyshev polynomial to a positive power greater than 1 then + the roots will be skewed towards the middle of the [-1,1] interval, while + a positive power less than one will skew them towards either end. More usefully, + if we initially rescale the points over [0,1] and then raise to a positive + power, we can skew them to the left or right. Returning to our example of + ex over [-1,1], the initial interpolated form was some way from the minimax + solution: +

+

+ remez-2 +

+

+ However, if we first skew the interpolation points to the left (rescale them + to [0, 1], raise to the power 1.3, and then rescale back to [-1,1]) we reduce + the error from 1.3x10-3to 6x10-4: +

+

+ remez-5 +

+

+ It's clearly still not ideal, but it is only a few percent away from our + desired minimax solution (5x10-4). +

+
+ + Remez + Method Checklist +
+

+ The following lists some of the things to check if the Remez method goes + wrong, it is by no means an exhaustive list, but is provided in the hopes + that it will prove useful. +

+
    +
  • + Is the function smooth enough? Can it be better separated into a rapidly + changing part, and an asymptotic part? +
    • +
  • + Does the function being approximated have any "blips" in it? + Check for problems as the function changes computation method, or if a + root, or an infinity has been divided out. The telltale sign is if there + is a narrow region where the Remez method will not converge. +
    • +
  • + Check you have enough accuracy in your calculations: remember that the + Remez method works on the difference between the approximation and the + function being approximated: so you must have more digits of precision + available than the precision of the approximation being constructed. So + for example at double precision, you shouldn't expect to be able to get + better than a float precision approximation. +
    • +
  • + Try skewing the initial interpolated approximation to minimise the error + before you begin the Remez steps. +
    • +
  • + If the approximation won't converge or is ill-conditioned from one starting + location, try starting from a different location. +
    • +
  • + If a rational function won't converge, one can minimise a polynomial (which + presents no problems), then rotate one term from the numerator to the denominator + and minimise again. In theory one can continue moving terms one at a time + from numerator to denominator, and then re-minimising, retaining the last + set of control points at each stage. +
    • +
  • + Try using a smaller interval. It may also be possible to optimise over + one (small) interval, rescale the control points over a larger interval, + and then re-minimise. +
    • +
  • + Keep absissa values small: use a change of variable to keep the abscissa + over, say [0, b], for some smallish value b. +
    • +
+
+ + References +
+

+ The original references for the Remez Method and it's extension to rational + functions are unfortunately in Russian: +

+

+ Remez, E.Ya., Fundamentals of numerical methods for Chebyshev approximations, + "Naukova Dumka", Kiev, 1969. +

+

+ Remez, E.Ya., Gavrilyuk, V.T., Computer development of certain + approaches to the approximate construction of solutions of Chebyshev problems + nonlinearly depending on parameters, Ukr. Mat. Zh. 12 (1960), + 324-338. +

+

+ Gavrilyuk, V.T., Generalization of the first polynomial algorithm + of E.Ya.Remez for the problem of constructing rational-fractional Chebyshev + approximations, Ukr. Mat. Zh. 16 (1961), 575-585. +

+

+ Some English language sources include: +

+

+ Fraser, W., Hart, J.F., On the computation of rational approximations + to continuous functions, Comm. of the ACM 5 (1962), 401-403, 414. +

+

+ Ralston, A., Rational Chebyshev approximation by Remes' algorithms, + Numer.Math. 7 (1965), no. 4, 322-330. +

+

+ A. Ralston, Rational Chebyshev approximation, Mathematical Methods + for Digital Computers v. 2 (Ralston A., Wilf H., eds.), Wiley, + New York, 1967, pp. 264-284. +

+

+ Hart, J.F. e.a., Computer approximations, Wiley, New + York a.o., 1968. +

+

+ Cody, W.J., Fraser, W., Hart, J.F., Rational Chebyshev approximation + using linear equations, Numer.Math. 12 (1968), 242-251. +

+

+ Cody, W.J., A survey of practical rational and polynomial approximation + of functions, SIAM Review 12 (1970), no. 3, 400-423. +

+

+ Barrar, R.B., Loeb, H.J., On the Remez algorithm for non-linear + families, Numer.Math. 15 (1970), 382-391. +

+

+ Dunham, Ch.B., Convergence of the Fraser-Hart algorithm for rational + Chebyshev approximation, Math. Comp. 29 (1975), no. 132, 1078-1082. +

+

+ G. L. Litvinov, Approximate construction of rational approximations + and the effect of error autocorrection, Russian Journal of Mathematical + Physics, vol.1, No. 3, 1994. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist.html b/doc/sf_and_dist/html/math_toolkit/dist.html new file mode 100644 index 000000000..cfc39beb7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist.html @@ -0,0 +1,180 @@ + + + +Statistical Distributions and Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Statistical Distributions + Tutorial
+
+
Overview
+
Worked Examples
+
+
+ Distribution Construction Example
+
Student's + t Distribution Examples
+
+
+ Calculating confidence intervals on the mean with the Students-t distribution
+
+ Testing a sample mean for difference from a "true" mean
+
+ Estimating how large a sample size would have to become in order to give + a significant Students-t test result with a single sample test
+
+ Comparing the means of two samples with the Students-t test
+
+ Comparing two paired samples with the Student's t distribution
+
+
Chi Squared + Distribution Examples
+
+
+ Confidence Intervals on the Standard Deviation
+
+ Chi-Square Test for the Standard Deviation
+
+ Estimating the Required Sample Sizes for a Chi-Square Test for the Standard + Deviation
+
+
F Distribution + Examples
+
Binomial + Distribution Examples
+
+
+ Binomial Coin-Flipping Example
+
+ Binomial Quiz Example
+
+ Calculating Confidence Limits on the Frequency of Occurrence for a Binomial + Distribution
+
+ Estimating Sample Sizes for a Binomial Distribution.
+
+
Negative + Binomial Distribution Examples
+
+
+ Calculating Confidence Limits on the Frequency of Occurrence for the + Negative Binomial Distribution
+
+ Estimating Sample Sizes for the Negative Binomial.
+
+ Negative Binomial Sales Quota Example.
+
+ Negative Binomial Table Printing Example.
+
+
Normal + Distribution Examples
+
+ Some Miscellaneous Examples of the Normal (Gaussian) Distribution
+
Error Handling + Example
+
Find Location + and Scale Examples
+
+
+ Find Location (Mean) Example
+
+ Find Scale (Standard Deviation) Example
+
+ Find mean and standard deviation example
+
+
Comparison + with C, R, FORTRAN-style Free Functions
+
+
Random Variates + and Distribution Parameters
+
Discrete Probability + Distributions
+
+
Statistical Distributions + Reference
+
+
Non-Member Properties
+
Distributions
+
+
+ Bernoulli Distribution
+
Beta + Distribution
+
+ Binomial Distribution
+
Cauchy-Lorentz + Distribution
+
+ Chi Squared Distribution
+
Exponential + Distribution
+
Extreme + Value Distribution
+
F Distribution
+
Gamma + (and Erlang) Distribution
+
+ Log Normal Distribution
+
+ Negative Binomial Distribution
+
Normal + (Gaussian) Distribution
+
Pareto + Distribution
+
Poisson + Distribution
+
Rayleigh + Distribution
+
+ Students t Distribution
+
+ Triangular Distribution
+
Weibull + Distribution
+
Uniform + Distribution
+
+
Distribution + Algorithms
+
+
Extras/Future Directions
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref.html new file mode 100644 index 000000000..bf4dd5f91 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref.html @@ -0,0 +1,90 @@ + + + +Statistical Distributions Reference + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Non-Member Properties
+
Distributions
+
+
+ Bernoulli Distribution
+
Beta + Distribution
+
+ Binomial Distribution
+
Cauchy-Lorentz + Distribution
+
+ Chi Squared Distribution
+
Exponential + Distribution
+
Extreme + Value Distribution
+
F Distribution
+
Gamma + (and Erlang) Distribution
+
+ Log Normal Distribution
+
+ Negative Binomial Distribution
+
Normal + (Gaussian) Distribution
+
Pareto + Distribution
+
Poisson + Distribution
+
Rayleigh + Distribution
+
+ Students t Distribution
+
+ Triangular Distribution
+
Weibull + Distribution
+
Uniform + Distribution
+
+
Distribution + Algorithms
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dist_algorithms.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dist_algorithms.html new file mode 100644 index 000000000..a8fe8283a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dist_algorithms.html @@ -0,0 +1,150 @@ + + + +Distribution Algorithms + + + + + + + + + + + + + + + +
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+
+
+ + Finding + the Location and Scale for Normal and similar distributions +
+

+ Two functions aid finding location and scale of random variable z to give + probability p (given a scale or location). Only applies to distributions + like normal, lognormal, extreme value, Cauchy, (and symmetrical triangular), + that have scale and location properties. +

+

+ These functions are useful to predict the mean and/or standard deviation + that will be needed to meet a specified minimum weight or maximum dose. +

+

+ Complement versions are also provided, both with explicit and implicit + (default) policy. +

+
+using boost::math::policies::policy; // May be needed by users defining their own policies.
+using boost::math::complement; // Will be needed by users who want to use complements.
+
+
+ + find_location + function +
+

+ +

+
+#include <boost/math/distributions/find_location.hpp>
+

+

+
+namespace boost{ namespace math{
+
+template <class Dist, class Policy> // explicit error handling policy
+  typename Dist::value_type find_location( // For example, normal mean.
+  typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p.
+  // For example, a nominal minimum acceptable z, so that p * 100 % are > z
+  typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z.
+  typename Dist::value_type scale, // scale parameter, for example, normal standard deviation.
+  const Policy& pol);
+
+template <class Dist>  // with default policy.
+  typename Dist::value_type find_location( // For example, normal mean.
+  typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p.
+  // For example, a nominal minimum acceptable z, so that p * 100 % are > z
+  typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z.
+  typename Dist::value_type scale); // scale parameter, for example, normal standard deviation.
+
+  }} // namespaces
+
+
+ + find_scale + function +
+

+ +

+
+#include <boost/math/distributions/find_scale.hpp>
+

+

+
+namespace boost{ namespace math{ 
+
+template <class Dist, class Policy>
+  typename Dist::value_type find_scale( // For example, normal mean.
+  typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p.
+  // For example, a nominal minimum acceptable weight z, so that p * 100 % are > z
+  typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z.
+  typename Dist::value_type location, // location parameter, for example, normal distribution mean.
+  const Policy& pol);
+
+ template <class Dist> // with default policy.
+   typename Dist::value_type find_scale( // For example, normal mean.
+   typename Dist::value_type z, // location of random variable z to give probability, P(X > z) == p.
+   // For example, a nominal minimum acceptable z, so that p * 100 % are > z
+   typename Dist::value_type p, // probability value desired at x, say 0.95 for 95% > z.
+   typename Dist::value_type location) // location parameter, for example, normal distribution mean.
+}} // namespaces
+
+

+ All argument must be finite, otherwise domain_error + is called. +

+

+ Probability arguments must be [0, 1], otherwise domain_error + is called. +

+

+ If the choice of arguments would give a negative scale, domain_error + is called, unless the policy is to ignore, when the negative (impossible) + value of scale is returned. +

+

+ Find Mean and standard + deviation examples gives simple examples of use of both find_scale + and find_location, and a longer example finding means and standard deviations + of normally distributed weights to meet a specification. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists.html new file mode 100644 index 000000000..8f6e34ddc --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists.html @@ -0,0 +1,83 @@ + + + +Distributions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ Bernoulli Distribution
+
Beta + Distribution
+
+ Binomial Distribution
+
Cauchy-Lorentz + Distribution
+
+ Chi Squared Distribution
+
Exponential + Distribution
+
Extreme + Value Distribution
+
F Distribution
+
Gamma + (and Erlang) Distribution
+
+ Log Normal Distribution
+
+ Negative Binomial Distribution
+
Normal + (Gaussian) Distribution
+
Pareto + Distribution
+
Poisson + Distribution
+
Rayleigh + Distribution
+
+ Students t Distribution
+
+ Triangular Distribution
+
Weibull + Distribution
+
Uniform + Distribution
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html new file mode 100644 index 000000000..dc1840cab --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html @@ -0,0 +1,358 @@ + + + +Bernoulli Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/bernoulli.hpp>
+

+

+
+namespace boost{ namespace math{
+ template <class RealType = double, 
+           class Policy   = policies::policy<> >
+ class bernoulli_distribution;
+   
+ typedef bernoulli_distribution<> bernoulli;
+
+ template <class RealType, class Policy>
+ class bernoulli_distribution
+ {
+ public:
+    typedef RealType  value_type;
+    typedef Policy    policy_type;
+
+    bernoulli_distribution(RealType p); // Constructor.
+    // Accessor function.
+    RealType success_fraction() const
+    // Probability of success (as a fraction).
+ }; 
+}} // namespaces
+
+

+ The Bernoulli distribution is a discrete distribution of the outcome + of a single trial with only two results, 0 (failure) or 1 (success), + with a probability of success p. +

+

+ The Bernoulli distribution is the simplest building block on which other + discrete distributions of sequences of independent Bernoulli trials can + be based. +

+

+ The Bernoulli is the binomial distribution (k = 1, p) with only one trial. +

+

+ probability + density function pdf f(0) = 1 - p, f(1) = p. Cumulative + distribution function D(k) = if (k == 0) 1 - p else 1. +

+

+ The following graph illustrates how the probability + density function pdf varies with the outcome of the single trial: +

+

+ bernoulli_pdf +

+

+ and the Cumulative + distribution function +

+

+ bernoulli_cdf +

+
+ + Member + Functions +
+
+bernoulli_distribution(RealType p);
+
+

+ Constructs a bernoulli + distribution with success_fraction p. +

+
+RealType success_fraction() const
+
+

+ Returns the success_fraction parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is 0 and 1, and the useful supported + range is only 0 or 1. +

+

+ Outside this range, functions are undefined, or may throw domain_error + exception and make an error message available. +

+
+ + Accuracy +
+

+ The Bernoulli distribution is implemented with simple arithmetic operators + and so should have errors within an epsilon or two. +

+
+ + Implementation +
+

+ In the following table p is the probability of success + and q = 1-p. k is the random + variate, either 0 or 1. +

+
+ + + + + +
[Note]Note
+

+ The Bernoulli distribution is implemented here as a strict + discrete distribution. If a generalised version, allowing + k to be any real, is required then the binomial distribution with a + single trial should be used, for example: +

+

+ binomial_distribution(1, + 0.25) +

+
+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ Supported range +

+ 		+

+ {0, 1} +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = 1 - p for k = 0, else p +

+
+

+ cdf +

+ 		+

+ Using the relation: cdf = 1 - p for k = 0, else 1 +

+
+

+ cdf complement +

+ 		+

+ q = 1 - p +

+
+

+ quantile +

+ 		+

+ if x <= (1-p) 0 else 1 +

+
+

+ quantile from the complement +

+ 		+

+ if x <= (1-p) 1 else 0 +

+
+

+ mean +

+ 		+

+ p +

+
+

+ variance +

+ 		+

+ p * (1 - p) +

+
+

+ mode +

+ 		+

+ if (p < 0.5) 0 else 1 +

+
+

+ skewness +

+ 		+

+ (1 - 2 * p) / sqrt(p * q) +

+
+

+ kurtosis +

+ 		+

+ 6 * p * p - 6 * p +1/ p * q +

+
+

+ kurtosis excess +

+ 		+

+ kurtosis -3 +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/beta_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/beta_dist.html new file mode 100644 index 000000000..8ddd465f1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/beta_dist.html @@ -0,0 +1,641 @@ + + + +Beta Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/beta.hpp>
+

+

+
+namespace boost{ namespace math{ 
+
+ template <class RealType = double, 
+           class Policy   = policies::policy<> >
+class beta_distribution;
+
+// typedef beta_distribution<double> beta;
+// Note that this is deliberately NOT provided,
+// to avoid a clash with the function name beta.
+
+template <class RealType, class Policy>
+class beta_distribution
+{
+public:
+   typedef RealType  value_type;
+   typedef Policy    policy_type;
+   // Constructor from two shape parameters, alpha & beta:
+   beta_distribution(RealType a, RealType b);
+   
+   // Parameter accessors:
+   RealType alpha() const;
+   RealType beta() const;
+   
+   // Parameter estimators of alpha or beta from mean and variance.
+   static RealType find_alpha(
+     RealType mean, // Expected value of mean.
+     RealType variance); // Expected value of variance.
+   
+   static RealType find_beta(
+     RealType mean, // Expected value of mean.
+     RealType variance); // Expected value of variance.
+
+   // Parameter estimators from from
+   // either alpha or beta, and x and probability.
+   
+   static RealType find_alpha(
+     RealType beta, // from beta.
+     RealType x, //  x.
+     RealType probability); // cdf
+   
+   static RealType find_beta(
+     RealType alpha, // alpha.
+     RealType x, // probability x.
+     RealType probability); // probability cdf.
+};
+
+}} // namespaces
+
+

+ The class type beta_distribution + represents a beta + probability + distribution function. +

+

+ The beta + distribution is used as a prior + distribution for binomial proportions in Bayesian + analysis. +

+

+ See also: beta + distribution and Bayesian + statistics. +

+

+ How the beta distribution is used for Bayesian + analysis of one parameter models is discussed by Jeff Grynaviski. +

+

+ The probability + density function PDF for the beta + distribution defined on the interval [0,1] is given by: +

+

+ f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) +

+

+ where B(α, β) is the beta + function, implemented in this library as beta. + Division by the beta function ensures that the pdf is normalized to the + range zero to unity. +

+

+ The following graph illustrates examples of the pdf for various values + of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and + might be approximated by a symmetrical triangular distribution. +

+

+ beta_dist +

+

+ If α = β = 1, then it is a ​ +uniform + distribution, equal to unity in the entire interval x = 0 to + 1. If α ​ and β ​ are < 1, then the pdf is U-shaped. If α != β, then the shape + is asymmetric and could be approximated by a triangle whose apex is away + from the centre (where x = half). +

+
+ + Member + Functions +
+
+ + Constructor +
+
+beta_distribution(RealType alpha, RealType beta);
+
+

+ Constructs a beta distribution with shape parameters alpha + and beta. +

+

+ Requires alpha,beta > 0,otherwise domain_error + is called. Note that technically the beta distribution is defined for + alpha,beta >= 0, but it's not clear whether any program can actually + make use of that latitude or how many of the non-member functions can + be usefully defined in that case. Therefore for now, we regard it as + an error if alpha or beta is zero. +

+

+ For example: +

+
+beta_distribution<> mybeta(2, 5);
+
+

+ Constructs a the beta distribution with alpha=2 and beta=5 (shown in + yellow in the graph above). +

+
+ + Parameter + Accessors +
+
+RealType alpha() const;
+
+

+ Returns the parameter alpha from which this distribution + was constructed. +

+
+RealType beta() const;
+
+

+ Returns the parameter beta from which this distribution + was constructed. +

+

+ So for example: +

+
+beta_distribution<> mybeta(2, 5);
+assert(mybeta.alpha() == 2.);  // mybeta.alpha() returns 2
+assert(mybeta.beta() == 5.);   // mybeta.beta()  returns 5
+
+
+ + Parameter + Estimators +
+

+ Two pairs of parameter estimators are provided. +

+

+ One estimates either α ​ or β ​ +from presumed-known mean and variance. +

+

+ The other pair estimates either α ​ or β ​ from the cdf and x. +

+

+ It is also possible to estimate α ​ and β ​ from 'known' mode & quantile. + For example, calculators are provided by the Pooled + Prevalence Calculator and Beta + Buster but this is not yet implemented here. +

+
+static RealType find_alpha(
+  RealType mean, // Expected value of mean.
+  RealType variance); // Expected value of variance.
+
+

+ Returns the unique value of α that corresponds to a beta distribution with + mean mean and variance variance. +

+
+static RealType find_beta(
+  RealType mean, // Expected value of mean.
+  RealType variance); // Expected value of variance.
+
+

+ Returns the unique value of β that corresponds to a beta distribution with + mean mean and variance variance. +

+
+static RealType find_alpha(
+  RealType beta, // from beta.
+  RealType x, //  x.
+  RealType probability); // probability cdf
+
+

+ Returns the value of α that gives: cdf(beta_distribution<RealType>(alpha, beta), x) == probability. +

+
+static RealType find_beta(
+  RealType alpha, // alpha.
+  RealType x, // probability x.
+  RealType probability); // probability cdf.
+
+

+ Returns the value of β that gives: cdf(beta_distribution<RealType>(alpha, beta), x) == probability. +

+
+ + Non-member + Accessor Functions +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The formulae for calculating these are shown in the table below, and + at Wolfram + Mathworld. +

+
+ + Applications +
+

+ The beta distribution can be used to model events constrained to take + place within an interval defined by a minimum and maximum value: so it + is used in project management systems. +

+

+ It is also widely used in Bayesian + statistical inference. +

+
+ + Related + distributions +
+

+ The beta distribution with both α ​ and β = 1 follows a uniform + distribution. +

+

+ The triangular + is used when less precise information is available. +

+

+ The binomial + distribution is closely related when α ​ and β ​ are integers. +

+

+ With integer values of α ​ and β ​ the distribution B(i, j) is that of the j-th + highest of a sample of i + j + 1 independent random variables uniformly + distributed between 0 and 1. The cumulative probability from 0 to x is + thus the probability that the j-th highest value is less than x. Or it + is the probability that that at least i of the random variables are less + than x, a probability given by summing over the Binomial + Distribution with its p parameter set to x. +

+
+ + Accuracy +
+

+ This distribution is implemented using the beta + functions beta + and incomplete + beta functions ibeta + and ibetac; + please refer to these functions for information on accuracy. +

+
+ + Implementation +
+

+ In the following table a and b + are the parameters α and β, x is the random variable, + p is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) +

+

+ Implemented using ibeta_derivative(a, + b, x). +

+
+

+ cdf +

+ 		+

+ Using the incomplete beta function ibeta(a, + b, x) +

+
+

+ cdf complement +

+ 		+

+ ibetac(a, + b, x) +

+
+

+ quantile +

+ 		+

+ Using the inverse incomplete beta function ibeta_inv(a, + b, p) +

+
+

+ quantile from the complement +

+ 		+

+ ibetac_inv(a, + b, q) +

+
+

+ mean +

+ 		+

+ a/(a+b) +

+
+

+ variance +

+ 		+

+ a * + b / + (a+b)^2 * (a + + b + + 1) +

+
+

+ mode +

+ 		+

+ (a-1) / + (a + + b + - 2) +

+
+

+ skewness +

+ 		+

+ 2 (b-a) + sqrt(a+b+1)/(a+b+2) * sqrt(a + * b) +

+
+

+ kurtosis excess +

+ 		+

+ +

+
+

+ kurtosis +

+ 		+

+ kurtosis + + 3 +

+
+

+ parameter estimation +

+ 		+

+

+
+

+ alpha +

+

+ from mean and variance +

+ 		+

+ mean * + (( (mean * + (1 + - mean)) / + variance)- + 1) +

+
+

+ beta +

+

+ from mean and variance +

+ 		+

+ (1 + - mean) * + (((mean + * (1 - mean)) + /variance)-1) +

+
+

+ The member functions find_alpha + and find_beta +

+

+ from cdf and probability x +

+

+ and either alpha + or beta +

+ 		+

+ Implemented in terms of the inverse incomplete beta functions +

+

+ ibeta_inva, + and ibeta_invb + respectively. +

+
+

+ find_alpha +

+ 		+

+ ibeta_inva(beta, + x, + probability) +

+
+

+ find_beta +

+ 		+

+ ibeta_invb(alpha, + x, + probability) +

+
+
+ + References +
+

+ Wikipedia + Beta distribution +

+

+ NIST + Exploratory Data Analysis +

+

+ Wolfram + MathWorld +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html new file mode 100644 index 000000000..3933a411e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html @@ -0,0 +1,937 @@ + + + +Binomial Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/binomial.hpp>
+

+

+
+namespace boost{ namespace math{ 
+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class binomial_distribution;
+
+typedef binomial_distribution<> binomial;
+
+template <class RealType, class Policy>
+class binomial_distribution
+{
+public:
+   typedef RealType  value_type;
+   typedef Policy    policy_type;
+   
+   static const unspecified-type cloppper_pearson_exact_interval;
+   static const unspecified-type jeffreys_prior_interval;
+   
+   // construct:
+   binomial_distribution(RealType n, RealType p);
+   
+   // parameter access::
+   RealType success_fraction() const;
+   RealType trials() const;
+   
+   // Bounds on success fraction:
+   static RealType find_lower_bound_on_p(
+      RealType trials, 
+      RealType successes,
+      RealType probability,
+      unspecified-type method = clopper_pearson_exact_interval);
+   static RealType find_upper_bound_on_p(
+      RealType trials, 
+      RealType successes,
+      RealType probability,
+      unspecified-type method = clopper_pearson_exact_interval);
+      
+   // estimate min/max number of trials:
+   static RealType find_minimum_number_of_trials(
+      RealType k,     // number of events
+      RealType p,     // success fraction
+      RealType alpha); // risk level
+      
+   static RealType find_maximum_number_of_trials(
+      RealType k,     // number of events
+      RealType p,     // success fraction
+      RealType alpha); // risk level
+};
+
+}} // namespaces
+
+

+ The class type binomial_distribution + represents a binomial + distribution: it is used when there are exactly two mutually + exclusive outcomes of a trial. These outcomes are labelled "success" + and "failure". The binomial distribution + is used to obtain the probability of observing k successes in N trials, + with the probability of success on a single trial denoted by p. The binomial + distribution assumes that p is fixed for all trials. +

+
+ + + + + +
[Note]Note

+ The random variable for the binomial distribution is the number of + successes, (the number of trials is a fixed property of the distribution) + whereas for the negative binomial, the random variable is the number + of trials, for a fixed number of successes. +

+

+ The PDF for the binomial distribution is given by: +

+

+ +

+

+ The following two graphs illustrate how the PDF changes depending upon + the distributions parameters, first we'll keep the success fraction + p fixed at 0.5, and vary the sample size: +

+

+ binomial_pdf_1 +

+

+ Alternatively, we can keep the sample size fixed at N=20 and vary the + success fraction p: +

+

+ binomial_pdf_2 +

+

+

+
+ + + + + +
[Caution]Caution
+

+ The Binomial distribution is a discrete distribution: internally + functions like the cdf + and pdf are treated + "as if" they are continuous functions, but in reality the + results returned from these functions only have meaning if an integer + value is provided for the random variate argument. +

+

+ The quantile function will by default return an integer result that + has been rounded outwards. That is to say lower + quantiles (where the probability is less than 0.5) are rounded downward, + and upper quantiles (where the probability is greater than 0.5) are + rounded upwards. This behaviour ensures that if an X% quantile is + requested, then at least the requested coverage + will be present in the central region, and no more than + the requested coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are + rounded differently, or even return a real-valued result using Policies. It is + strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on the Binomial distribution. The reference + docs describe how to change the rounding policy for these + distributions. +

+
+

+

+
+ + Member + Functions +
+
+ + Construct +
+
+binomial_distribution(RealType n, RealType p);
+
+

+ Constructor: n is the total number of trials, p + is the probability of success of a single trial. +

+

+ Requires 0 <= + p <= + 1, and n + >= 0, + otherwise calls domain_error. +

+
+ + Accessors +
+
+RealType success_fraction() const;
+
+

+ Returns the parameter p from which this distribution + was constructed. +

+
+RealType trials() const;
+
+

+ Returns the parameter n from which this distribution + was constructed. +

+
+ + Lower + Bound on the Success Fraction +
+
+static RealType find_lower_bound_on_p(
+   RealType trials, 
+   RealType successes,
+   RealType alpha,
+   unspecified-type method = clopper_pearson_exact_interval);
+
+

+ Returns a lower bound on the success fraction: +

+
+

+
+
trials
+

+ The total number of trials conducted. +

+
successes
+

+ The number of successes that occurred. +

+
alpha
+

+ The largest acceptable probability that the true value of the success + fraction is less than the value + returned. +

+
method
+

+ An optional parameter that specifies the method to be used to compute + the interval (See below). +

+
+
+

+ For example, if you observe k successes from n + trials the best estimate for the success fraction is simply k/n, + but if you want to be 95% sure that the true value is greater + than some value, pmin, then: +

+
+pmin = binomial_distribution<RealType>::find_lower_bound_on_p(
+                    n, k, 0.05);
+
+

+ See + worked example. +

+

+ There are currently two possible values available for the method + optional parameter: clopper_pearson_exact_interval + or jeffreys_prior_interval. These constants are + both members of class template binomial_distribution, + so usage is for example: +

+
+p = binomial_distribution<RealType>::find_lower_bound_on_p(
+    n, k, 0.05, binomial_distribution<RealType>::jeffreys_prior_interval);
+
+

+ The default method if this parameter is not specified is the Clopper + Pearson "exact" interval. This produces an interval that guarantees + at least 100(1-alpha)% coverage, but which is known to be + overly conservative, sometimes producing intervals with much greater + than the requested coverage. +

+

+ The alternative calculation method produces a non-informative Jeffreys + Prior interval. It produces 100(1-alpha)% + coverage only in the average case, though is typically + very close to the requested coverage level. It is one of the main methods + of calculation recommended in the review by Brown, Cai and DasGupta. +

+

+ Please note that the "textbook" calculation method using a + normal approximation (the Wald interval) is deliberately not provided: + it is known to produce consistently poor results, even when the sample + size is surprisingly large. Refer to Brown, Cai and DasGupta for a full + explanation. Many other methods of calculation are available, and may + be more appropriate for specific situations. Unfortunately there appears + to be no consensus amongst statisticians as to which is "best": + refer to the discussion at the end of Brown, Cai and DasGupta for examples. +

+

+ The two methods provided here were chosen principally because they can + be used for both one and two sided intervals. See also: +

+

+ Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), Interval + Estimation for a Binomial Proportion, Statistical Science, Vol. 16, No. + 2, 101-133. +

+

+ T. Tony Cai (2005), One-sided confidence intervals in discrete distributions, + Journal of Statistical Planning and Inference 131, 63-88. +

+

+ Agresti, A. and Coull, B. A. (1998). Approximate is better than "exact" + for interval estimation of binomial proportions. Amer. Statist. 52 119-126. +

+

+ Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial + limits illustrated in the case of the binomial. Biometrika 26 404-413. +

+
+ + Upper + Bound on the Success Fraction +
+
+static RealType find_upper_bound_on_p(
+   RealType trials, 
+   RealType successes,
+   RealType alpha,
+   unspecified-type method = clopper_pearson_exact_interval);
+
+

+ Returns an upper bound on the success fraction: +

+
+

+
+
trials
+

+ The total number of trials conducted. +

+
successes
+

+ The number of successes that occurred. +

+
alpha
+

+ The largest acceptable probability that the true value of the success + fraction is greater than the value + returned. +

+
method
+

+ An optional parameter that specifies the method to be used to compute + the interval. Refer to the documentation for find_upper_bound_on_p + above for the meaning of the method options. +

+
+
+

+ For example, if you observe k successes from n + trials the best estimate for the success fraction is simply k/n, + but if you want to be 95% sure that the true value is less + than some value, pmax, then: +

+
+pmax = binomial_distribution<RealType>::find_upper_bound_on_p(
+                    n, k, 0.05);
+
+

+ See + worked example. +

+
+ + + + + +
[Note]Note
+

+ In order to obtain a two sided bound on the success fraction, you call + both find_lower_bound_on_p + and find_upper_bound_on_p + each with the same arguments. +

+

+ If the desired risk level that the true success fraction lies outside + the bounds is α, then you pass α/2 to these functions. +

+

+ So for example a two sided 95% confidence interval would be obtained + by passing α = 0.025 to each of the functions. +

+

+ See + worked example. +

+
+
+ + Estimating + the Number of Trials Required for a Certain Number of Successes +
+
+static RealType find_minimum_number_of_trials(
+   RealType k,     // number of events
+   RealType p,     // success fraction
+   RealType alpha); // probability threshold
+
+

+ This function estimates the minimum number of trials required to ensure + that more than k events is observed with a level of risk alpha + that k or fewer events occur. +

+
+

+
+
k
+

+ The number of success observed. +

+
p
+

+ The probability of success for each trial. +

+
alpha
+

+ The maximum acceptable probability that k events or fewer will be + observed. +

+
+
+

+ For example: +

+
+binomial_distribution<RealType>::find_number_of_trials(10, 0.5, 0.05);
+
+

+ Returns the smallest number of trials we must conduct to be 95% sure + of seeing 10 events that occur with frequency one half. +

+
+ + Estimating + the Maximum Number of Trials to Ensure no more than a Certain Number + of Successes +
+
+static RealType find_maximum_number_of_trials(
+   RealType k,     // number of events
+   RealType p,     // success fraction
+   RealType alpha); // probability threshold
+
+

+ This function estimates the maximum number of trials we can conduct to + ensure that k successes or fewer are observed, with a risk alpha + that more than k occur. +

+
+

+
+
k
+

+ The number of success observed. +

+
p
+

+ The probability of success for each trial. +

+
alpha
+

+ The maximum acceptable probability that more than k events will be + observed. +

+
+
+

+ For example: +

+
+binomial_distribution<RealType>::find_maximum_number_of_trials(0, 1e-6, 0.05);
+
+

+ Returns the largest number of trials we can conduct and still be 95% + certain of not observing any events that occur with one in a million + frequency. This is typically used in failure analysis. +

+

+ See + Worked Example. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain for the random variable k is 0 <= k <= N, otherwise a domain_error + is returned. +

+

+ It's worth taking a moment to define what these accessors actually mean + in the context of this distribution: +

+
+

Table 9. Meaning of the non-member accessors

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Meaning +

+
+

+ Probability Density Function +

+ 		+

+ The probability of obtaining exactly k successes + from n trials with success fraction p. For example: +

+

+ pdf(binomial(n, + p), + k) +

+
+

+ Cumulative Distribution Function +

+ 		+

+ The probability of obtaining k successes + or fewer from n trials with success fraction p. For + example: +

+

+ cdf(binomial(n, + p), + k) +

+
+

+ Complement of the Cumulative Distribution + Function +

+ 		+

+ The probability of obtaining more than k + successes from n trials with success fraction p. For + example: +

+

+ cdf(complement(binomial(n, + p), + k)) +

+
+

+ Quantile +

+ 		+

+ The greatest number of successes + that may be observed from n trials with success fraction p, at + probability P. Note that the value returned is a real-number, and + not an integer. Depending on the use case you may want to take + either the floor or ceiling of the result. For example: +

+

+ quantile(binomial(n, + p), + P) +

+
+

+ Quantile from the complement + of the probability +

+ 		+

+ The smallest number of successes + that may be observed from n trials with success fraction p, at + probability P. Note that the value returned is a real-number, and + not an integer. Depending on the use case you may want to take + either the floor or ceiling of the result. For example: +

+

+ quantile(complement(binomial(n, + p), + P))` +

+
+
+
+ + Examples +
+

+ Various worked + examples are available illustrating the use of the binomial distribution. +

+
+ + Accuracy +
+

+ This distribution is implemented using the incomplete beta functions + ibeta + and ibetac, + please refer to these functions for information on accuracy. +

+
+ + Implementation +
+

+ In the following table p is the probability that + one trial will be successful (the success fraction), n + is the number of trials, k is the number of successes, + p is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Implementation is in terms of ibeta_derivative: + if nCk is the binomial coefficient of a and b, then we have: +

+

+ +

+

+ Which can be evaluated as ibeta_derivative(k+1, n-k+1, p) / + (n+1) +

+

+ The function ibeta_derivative + is used here, since it has already been optimised for the lowest + possible error - indeed this is really just a thin wrapper around + part of the internals of the incomplete beta function. +

+

+ There are also various special cases: refer to the code for details. +

+
+

+ cdf +

+ 		+

+ Using the relation: +

+

+ +

+
+p = I[sub 1-p](n - k, k + 1)
+  = 1 - I[sub p](k + 1, n - k)
+  = ibetac(k + 1, n - k, p)
+

+

+

+ There are also various special cases: refer to the code for details. +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = ibeta(k + + 1, n - k, p) +

+

+ There are also various special cases: refer to the code for details. +

+
+

+ quantile +

+ 		+

+ Since the cdf is non-linear in variate k + none of the inverse incomplete beta functions can be used here. + Instead the quantile is found numerically using a derivative + free method (TOMS + Algorithm 748). +

+
+

+ quantile from the complement +

+ 		+

+ Found numerically as above. +

+
+

+ mean +

+ 		+

+ p * + n +

+
+

+ variance +

+ 		+

+ p * + n * + (1-p) +

+
+

+ mode +

+ 		+

+ floor(p * + (n + + 1)) +

+
+

+ skewness +

+ 		+

+ (1 + - 2 + * p) / + sqrt(n * + p * + (1 + - p)) +

+
+

+ kurtosis +

+ 		+

+ 3 - + (6 + / n) + + (1 + / (n * + p * + (1 + - p))) +

+
+

+ kurtosis excess +

+ 		+

+ (1 + - 6 + * p + * q) / + (n + * p + * q) +

+
+

+ parameter estimation +

+ 		+

+ The member functions find_upper_bound_on_p + find_lower_bound_on_p + and find_number_of_trials + are implemented in terms of the inverse incomplete beta functions + ibetac_inv, + ibeta_inv, + and ibetac_invb + respectively +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html new file mode 100644 index 000000000..ee9dc03db --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html @@ -0,0 +1,311 @@ + + + +Cauchy-Lorentz Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/cauchy.hpp>
+

+

+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class cauchy_distribution;
+
+typedef cauchy_distribution<> cauchy;
+
+template <class RealType, class Policy>
+class cauchy_distribution
+{
+public:
+   typedef RealType  value_type;
+   typedef Policy    policy_type;
+
+   cauchy_distribution(RealType location = 0, RealType scale = 1);
+   
+   RealType location()const;
+   RealType scale()const;
+};
+
+

+ The Cauchy-Lorentz + distribution is named after Augustin Cauchy and Hendrik Lorentz. + It is a continuous + probability distribution with probability + distribution function PDF given by: +

+

+ +

+

+ The location parameter x0 is the location of the peak of the distribution + (the mode of the distribution), while the scale parameter γ specifies half + the width of the PDF at half the maximum height. If the location is zero, + and the scale 1, then the result is a standard Cauchy distribution. +

+

+ The distribution is important in physics as it is the solution to the + differential equation describing forced resonance, while in spectroscopy + it is the description of the line shape of spectral lines. +

+

+ The following graph shows how the distributions moves as the location + parameter changes: +

+

+ cauchy1 +

+

+ While the following graph shows how the shape (scale) parameter alters + the distribution: +

+

+ cauchy2 +

+
+ + Member + Functions +
+
+cauchy_distribution(RealType location = 0, RealType scale = 1);
+
+

+ Constructs a Cauchy distribution, with location parameter location + and scale parameter scale. When these parameters + take their default values (location = 0, scale = 1) then the result is + a Standard Cauchy Distribution. +

+

+ Requires scale > 0, otherwise calls domain_error. +

+
+RealType location()const;
+
+

+ Returns the location parameter of the distribution. +

+
+RealType scale()const;
+
+

+ Returns the scale parameter of the distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ Note however that the Cauchy distribution does not have a mean, standard + deviation, etc. See mathematically + undefined function +

+

+ to control whether these should fail to compile with a BOOST_STATIC_ASSERTION_FAILURE, + which is the default. +

+

+ Alternately, the functions mean, + standard deviation, variance, + skewness, kurtosis + and kurtosis_excess + will all return a domain_error if + called. +

+

+ The domain of the random variable is [-[max_value], +[min_value]]. +

+
+ + Accuracy +
+

+ The Cauchy distribution is implemented in terms of the standard library + tan and atan functions, and as such should + have very low error rates. +

+
+ + Implementation +
+

+ In the following table x0 is the location parameter of the distribution, + γ is its scale parameter, x is the random variate, + p is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = 1 / (π * γ * (1 + ((x - x0 ) / γ)2) +

+
+

+ cdf and its complement +

+ 		+

+ The cdf is normally given by: +

+

+ p = 0.5 + atan(x)/π +

+

+ But that suffers from cancellation error as x -> -∞. So recall + that for x < + 0: +

+

+ atan(x) = -π/2 - atan(1/x) +

+

+ Substituting into the above we get: +

+

+ p = -atan(1/x) ; x < 0 +

+

+ So the procedure is to calculate the cdf for -fabs(x) using the + above formula. Note that to factor in the location and scale + parameters you must substitute (x - x0 ) / γ for x in the above. +

+

+ This procedure yields the smaller of p and + q, so the result may need subtracting from + 1 depending on whether we want the complement or not, and whether + x is less than x0 or not. +

+
+

+ quantile +

+ 		+

+ The same procedure is used irrespective of whether we're starting + from the probability or it's complement. First the argument + p is reduced to the range [-0.5, 0.5], then + the relation +

+

+ x = x0 ± γ / tan(π * p) +

+

+ is used to obtain the result. Whether we're adding or subtracting + from x0 is determined by whether we're starting from the complement + or not. +

+
+

+ mode +

+ 		+

+ The location parameter. +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html new file mode 100644 index 000000000..8de41d8c9 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html @@ -0,0 +1,408 @@ + + + +Chi Squared Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/chi_squared.hpp>
+

+

+
+namespace boost{ namespace math{ 
+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class chi_squared_distribution;
+
+typedef chi_squared_distribution<> chi_squared;
+
+template <class RealType, class Policy>
+class chi_squared_distribution
+{
+public:
+   typedef RealType  value_type;
+   typedef Policy    policy_type;
+
+   // Constructor:
+   chi_squared_distribution(RealType i);
+
+   // Accessor to parameter:
+   RealType degrees_of_freedom()const;
+
+   // Parameter estimation:
+   static RealType find_degrees_of_freedom(
+      RealType difference_from_mean,
+      RealType alpha,
+      RealType beta,
+      RealType sd,
+      RealType hint = 100);
+};
+
+}} // namespaces
+
+

+ The Chi-Squared distribution is one of the most widely used distributions + in statistical tests. If χi are ν +independent, normally distributed random + variables with means μi and variances σi2, then the random variable: +

+

+ +

+

+ is distributed according to the Chi-Squared distribution. +

+

+ The Chi-Squared distribution is a special case of the gamma distribution + and has a single parameter ν that specifies the number of degrees of freedom. + The following graph illustrates how the distribution changes for different + values of ν: +

+

+ chi_square +

+
+ + Member + Functions +
+
+chi_squared_distribution(RealType v);
+
+

+ Constructs a Chi-Squared distribution with v degrees + of freedom. +

+

+ Requires v > 0, otherwise calls domain_error. +

+
+RealType degrees_of_freedom()const;
+
+

+ Returns the parameter v from which this object was + constructed. +

+
+static RealType find_degrees_of_freedom(
+   RealType difference_from_variance,
+   RealType alpha,
+   RealType beta,
+   RealType variance,
+   RealType hint = 100);
+
+

+ Estimates the sample size required to detect a difference from a nominal + variance in a Chi-Squared test for equal standard deviations. +

+
+

+
+
difference_from_variance
+

+ The difference from the assumed nominal variance that is to be detected: + Note that the sign of this value is critical, see below. +

+
alpha
+

+ The maximum acceptable risk of rejecting the null hypothesis when + it is in fact true. +

+
beta
+

+ The maximum acceptable risk of falsely failing to reject the null + hypothesis. +

+
variance
+

+ The nominal variance being tested against. +

+
hint
+

+ An optional hint on where to start looking for a result: the current + sample size would be a good choice. +

+
+
+

+ Note that this calculation works with variances + and not standard deviations. +

+

+ The sign of the parameter difference_from_variance + is important: the Chi Squared distribution is asymmetric, and the caller + must decide in advance whether they are testing for a variance greater + than a nominal value (positive difference_from_variance) + or testing for a variance less than a nominal value (negative difference_from_variance). + If the latter, then obviously it is a requirement that variance + + difference_from_variance > + 0, since no sample can have a negative + variance! +

+

+ This procedure uses the method in Diamond, W. J. (1989). Practical Experiment + Designs, Van-Nostrand Reinhold, New York. +

+

+ See also section on Sample sizes required in the + NIST Engineering Statistics Handbook, Section 7.2.3.2. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, +∞]. +

+
+ + Examples +
+

+ Various worked examples + are available illustrating the use of the Chi Squared Distribution. +

+
+ + Accuracy +
+

+ The Chi-Squared distribution is implemented in terms of the incomplete + gamma functions: please refer to the accuracy data for those functions. +

+
+ + Implementation +
+

+ In the following table v is the number of degrees + of freedom of the distribution, x is the random + variate, p is the probability, and q = + 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = gamma_p_derivative(v + / 2, x / 2) / 2 +

+
+

+ cdf +

+ 		+

+ Using the relation: p = gamma_p(v + / 2, x / 2) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = gamma_q(v + / 2, x / 2) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = 2 * gamma_p_inv(v + / 2, p) +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = 2 * gamma_q_inv(v + / 2, p) +

+
+

+ mean +

+ 		+

+ v +

+
+

+ variance +

+ 		+

+ 2v +

+
+

+ mode +

+ 		+

+ v - 2 (if v >= 2) +

+
+

+ skewness +

+ 		+

+ 2 * sqrt(2 / v) == sqrt(8 / v) +

+
+

+ kurtosis +

+ 		+

+ 3 + 12 / v +

+
+

+ kurtosis excess +

+ 		+

+ 12 / v +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html new file mode 100644 index 000000000..a38019ee1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html @@ -0,0 +1,319 @@ + + + +Exponential Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/exponential.hpp>
+

+

+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class exponential_distribution;
+
+typedef exponential_distribution<> exponential;
+
+template <class RealType, class Policy>
+class exponential_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+
+   exponential_distribution(RealType lambda = 1);
+
+   RealType lambda()const;
+};
+
+

+ The exponential + distribution is a continuous + probability distribution with PDF: +

+

+ +

+

+ It is often used to model the time between independent events that happen + at a constant average rate. +

+

+ The following graph shows how the distribution changes for different + values of the rate parameter lambda: +

+

+ exponential_dist +

+
+ + Member + Functions +
+
+exponential_distribution(RealType lambda = 1);
+
+

+ Constructs an Exponential + distribution with parameter lambda. Lambda + is defined as the reciprocal of the scale parameter. +

+

+ Requires lambda > 0, otherwise calls domain_error. +

+
+RealType lambda()const;
+
+

+ Accessor function returns the lambda parameter of the distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, +∞]. +

+
+ + Accuracy +
+

+ The exponential distribution is implemented in terms of the standard + library functions exp, + log, log1p + and expm1 and as such + should have very low error rates. +

+
+ + Implementation +
+

+ In the following table λ is the parameter lambda of the distribution, + x is the random variate, p + is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = λ * exp(-λ * x) +

+
+

+ cdf +

+ 		+

+ Using the relation: p = 1 - exp(-x * λ) = -expm1(-x * λ) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = exp(-x * λ) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = -log(1-p) / λ = -log1p(-p) / λ +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = -log(q) / λ +

+
+

+ mean +

+ 		+

+ 1/λ +

+
+

+ standard deviation +

+ 		+

+ 1/λ +

+
+

+ mode +

+ 		+

+ 0 +

+
+

+ skewness +

+ 		+

+ 2 +

+
+

+ kurtosis +

+ 		+

+ 9 +

+
+

+ kurtosis excess +

+ 		+

+ 6 +

+
+
+ + references +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html new file mode 100644 index 000000000..6768671b8 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html @@ -0,0 +1,334 @@ + + + +Extreme Value Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/extreme.hpp>
+

+

+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class extreme_value_distribution;
+
+typedef extreme_value_distribution<> extreme_value;
+
+template <class RealType, class Policy>
+class extreme_value_distribution
+{
+public:
+   typedef RealType value_type;
+
+   extreme_value_distribution(RealType location = 0, RealType scale = 1);
+
+   RealType scale()const;
+   RealType location()const;
+};
+
+

+ There are various extreme + value distributions : this implementation represents the maximum + case, and is variously known as a Fisher-Tippett distribution, a log-Weibull + distribution or a Gumbel distribution. +

+

+ Extreme value theory is important for assessing risk for highly unusual + events, such as 100-year floods. +

+

+ More information can be found on the NIST, + Wikipedia, + Mathworld, + and Extreme + value theory websites. +

+

+ The distribution has a PDF given by: +

+

+ f(x) = (1/scale) e-(x-location)/scale e-e-(x-location)/scale +

+

+ Which in the standard case (scale = 1, location = 0) reduces to: +

+

+ f(x) = e-xe-e-x +

+

+ The following graph illustrates how the PDF varies with the location + parameter: +

+

+ extreme_val_dist +

+

+ And this graph illustrates how the PDF varies with the shape parameter: +

+

+ extreme_val_dist2 +

+
+ + Member + Functions +
+
+extreme_value_distribution(RealType location = 0, RealType scale = 1);
+
+

+ Constructs an Extreme Value distribution with the specified location + and scale parameters. +

+

+ Requires scale > + 0, otherwise calls domain_error. +

+
+RealType location()const;
+
+

+ Returns the location parameter of the distribution. +

+
+RealType scale()const;
+
+

+ Returns the scale parameter of the distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random parameter is [-∞, +∞]. +

+
+ + Accuracy +
+

+ The extreme value distribution is implemented in terms of the standard + library exp and log functions and as such should have + very low error rates. +

+
+ + Implementation +
+

+ In the following table: a is the location parameter, + b is the scale parameter, x + is the random variate, p is the probability and + q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) / + b +

+
+

+ cdf +

+ 		+

+ Using the relation: p = exp(-exp((a-x)/b)) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = -expm1(-exp((a-x)/b)) +

+
+

+ quantile +

+ 		+

+ Using the relation: a - log(-log(p)) * b +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: a - log(-log1p(-q)) * b +

+
+

+ mean +

+ 		+

+ a + Euler-Mascheroni-constant + * b +

+
+

+ standard deviation +

+ 		+

+ pi * b / sqrt(6) +

+
+

+ mode +

+ 		+

+ The same as the location parameter a. +

+
+

+ skewness +

+ 		+

+ 12 * sqrt(6) * zeta(3) / pi3 +

+
+

+ kurtosis +

+ 		+

+ 27 / 5 +

+
+

+ kurtosis excess +

+ 		+

+ kurtosis - 3 or 12 / 5 +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html new file mode 100644 index 000000000..9e012cc72 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html @@ -0,0 +1,443 @@ + + + +F Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/fisher_f.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class fisher_f_distribution;
+
+typedef fisher_f_distribution<> fisher_f;
+
+template <class RealType, class Policy>
+class fisher_f_distribution
+{
+public:
+   typedef RealType value_type;
+   
+   // Construct:
+   fisher_f_distribution(const RealType& i, const RealType& j);
+   
+   // Accessors:
+   RealType degrees_of_freedom1()const;
+   RealType degrees_of_freedom2()const;
+};
+
+}} //namespaces
+
+

+ The F distribution is a continuous distribution that arises when testing + whether two samples have the same variance. If χ2m and χ2n are independent + variates each distributed as Chi-Squared with m + and n degrees of freedom, then the test statistic: +

+

+ Fn,m = (χ2n / n) / (χ2m / m) +

+

+ Is distributed over the range [0, ∞] with an F distribution, and has the + PDF: +

+

+ +

+

+ The following graph illustrates how the PDF varies depending on the two + degrees of freedom parameters. +

+

+ fisher_f +

+
+ + Member + Functions +
+
+fisher_f_distribution(const RealType& df1, const RealType& df2);
+
+

+ Constructs an F-distribution with numerator degrees of freedom df1 + and denominator degrees of freedom df2. +

+

+ Requires that df1 and df2 are + both greater than zero, otherwise domain_error + is called. +

+
+RealType degrees_of_freedom1()const;
+
+

+ Returns the numerator degrees of freedom parameter of the distribution. +

+
+RealType degrees_of_freedom2()const;
+
+

+ Returns the denominator degrees of freedom parameter of the distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, +∞]. +

+
+ + Examples +
+

+ Various worked examples + are available illustrating the use of the F Distribution. +

+
+ + Accuracy +
+

+ The normal distribution is implemented in terms of the incomplete + beta function and it's inverses, + refer to those functions for accuracy data. +

+
+ + Implementation +
+

+ In the following table v1 and v2 + are the first and second degrees of freedom parameters of the distribution, + x is the random variate, p + is the probability, and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ The usual form of the PDF is given by: +

+

+ +

+

+ However, that form is hard to evaluate directly without incurring + problems with either accuracy or numeric overflow. +

+

+ Direct differentiation of the CDF expressed in terms of the incomplete + beta function +

+

+ led to the following two formulas: +

+

+ fv1,v2(x) = y * ibeta_derivative(v2 + / 2, v1 / 2, v2 / (v2 + v1 * x)) +

+

+ with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x)) +

+

+ and +

+

+ fv1,v2(x) = y * ibeta_derivative(v1 + / 2, v2 / 2, v1 * x / (v2 + v1 * x)) +

+

+ with y = (z * v1 - x * v1 * v1) / z2 +

+

+ and z = v2 + v1 * x +

+

+ The first of these is used for v1 * x > v2, otherwise the + second is used. +

+

+ The aim is to keep the x argument to ibeta_derivative + away from 1 to avoid rounding error. +

+
+

+ cdf +

+ 		+

+ Using the relations: +

+

+ p = ibeta(v1 + / 2, v2 / 2, v1 * x / (v2 + v1 * x)) +

+

+ and +

+

+ p = ibetac(v2 + / 2, v1 / 2, v2 / (v2 + v1 * x)) +

+

+ The first is used for v1 * x > v2, otherwise the second is + used. +

+

+ The aim is to keep the x argument to ibeta + well away from 1 to avoid rounding error. +

+
+

+ cdf complement +

+ 		+

+ Using the relations: +

+

+ p = ibetac(v1 + / 2, v2 / 2, v1 * x / (v2 + v1 * x)) +

+

+ and +

+

+ p = ibeta(v2 + / 2, v1 / 2, v2 / (v2 + v1 * x)) +

+

+ The first is used for v1 * x < v2, otherwise the second is + used. +

+

+ The aim is to keep the x argument to ibeta + well away from 1 to avoid rounding error. +

+
+

+ quantile +

+ 		+

+ Using the relation: +

+

+ x = v2 * a / (v1 * b) +

+

+ where: +

+

+ a = ibeta_inv(v1 + / 2, v2 / 2, p) +

+

+ and +

+

+ b = 1 - a +

+

+ Quantities a and b + are both computed by ibeta_inv + without the subtraction implied above. +

+
+

+ quantile +

+

+ from the complement +

+ 		+

+ Using the relation: +

+

+ x = v2 * a / (v1 * b) +

+

+ where +

+

+ a = ibetac_inv(v1 + / 2, v2 / 2, p) +

+

+ and +

+

+ b = 1 - a +

+

+ Quantities a and b + are both computed by ibetac_inv + without the subtraction implied above. +

+
+

+ mean +

+ 		+

+ v2 / (v2 - 2) +

+
+

+ variance +

+ 		+

+ 2 * v22 * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4)) +

+
+

+ mode +

+ 		+

+ v2 * (v1 - 2) / (v1 * (v2 + 2)) +

+
+

+ skewness +

+ 		+

+ 2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 - + 2))) / (v2 - 6) +

+
+

+ kurtosis and kurtosis excess +

+ 		+

+ Refer to, Weisstein, + Eric W. "F-Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html new file mode 100644 index 000000000..38d3e699f --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html @@ -0,0 +1,385 @@ + + + +Gamma (and Erlang) Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/gamma.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class gamma_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+
+   gamma_distribution(RealType shape, RealType scale = 1)
+
+   RealType shape()const;
+   RealType scale()const;
+};
+
+}} // namespaces
+
+

+ The gamma distribution is a continuous probability distribution. When + the shape parameter is an integer then it is known as the Erlang Distribution. + It is also closely related to the Poisson and Chi Squared Distributions. +

+

+ When the shape parameter has an integer value, the distribution is the + Erlang + distribution. Since this can be produced by ensuring that the + shape parameter has an integer value > 0, the Erlang distribution + is not separately implemented. +

+
+ + + + + +
[Note]Note
+

+ To avoid potential confusion with the gamma functions, this distribution + does not provide the typedef: +

+

+ +

+
+typedef gamma_distibution<double> gamma;
+

+

+

+ Instead if you want a double precision gamma distribution you can use +

+

+ +

+
+boost::gamma_distribution<>
+

+

+
+

+ For shape parameter k and scale parameter θ it is + defined by the probability density function: +

+

+ +

+

+ Sometimes an alternative formulation is used: given parameters α= k and + β= 1 / θ, then the distribution can be defined by the PDF: +

+

+ +

+

+ In this form the inverse scale parameter is called a rate parameter. +

+

+ Both forms are in common usage: this library uses the first definition + throughout. Therefore to construct a Gamma Distribution from a rate + parameter, you should pass the reciprocal of the rate as the + scale parameter. +

+

+ The following two graphs illustrate how the PDF of the gamma distribution + varies as the parameters vary: +

+

+ gamma_dist1 +

+

+ gamma_dist2 +

+

+ The Erlang Distribution is the same + as the Gamma, but with the shape parameter an integer. It is often expressed + using a rate rather than a scale + as the second parameter (remember that the rate is the reciprocal of + the scale). +

+

+ Internally the functions used to implement the Gamma Distribution are + already optimised for small-integer arguments, so in general there should + be no great loss of performance from using a Gamma Distribution rather + than a dedicated Erlang Distribution. +

+
+ + Member + Functions +
+
+gamma_distribution(RealType shape, RealType scale = 1);
+
+

+ Constructs a gamma distribution with shape shape + and scale scale. +

+

+ Requires that the shape and scale parameters are greater than zero, otherwise + calls domain_error. +

+
+RealType shape()const;
+
+

+ Returns the shape parameter of this distribution. +

+
+RealType scale()const;
+
+

+ Returns the scale parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0,+∞]. +

+
+ + Accuracy +
+

+ The lognormal distribution is implemented in terms of the incomplete + gamma functions gamma_p + and gamma_q + and their inverses gamma_p_inv + and gamma_q_inv: + refer to the accuracy data for those functions for more information. +

+
+ + Implementation +
+

+ In the following table k is the shape parameter + of the distribution, θ is it's scale parameter, x + is the random variate, p is the probability and + q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = gamma_p_derivative(k, + x / θ) / θ +

+
+

+ cdf +

+ 		+

+ Using the relation: p = gamma_p(k, + x / θ) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = gamma_q(k, + x / θ) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = θ* gamma_p_inv(k, + p) +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = θ* gamma_q_inv(k, + p) +

+
+

+ mean +

+ 		+

+ kθ +

+
+

+ variance +

+ 		+

+ kθ2 +

+
+

+ mode +

+ 		+

+ (k-1)θ for k>1 otherwise a domain_error +

+
+

+ skewness +

+ 		+

+ 2 / sqrt(k) +

+
+

+ kurtosis +

+ 		+

+ 3 + 6 / k +

+
+

+ kurtosis excess +

+ 		+

+ 6 / k +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html new file mode 100644 index 000000000..66921d43b --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html @@ -0,0 +1,338 @@ + + + +Log Normal Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/lognormal.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class lognormal_distribution;
+
+typedef lognormal_distribution<> lognormal;
+
+template <class RealType, class Policy>
+class lognormal_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   // Construct:
+   lognormal_distribution(RealType location = 0, RealType scale = 1);
+   // Accessors:
+   RealType location()const;
+   RealType scale()const;
+};
+
+}} // namespaces
+
+

+ The lognormal distribution is the distribution that arises when the logarithm + of the random variable is normally distributed. A lognormal distribution + results when the variable is the product of a large number of independent, + identically-distributed variables. +

+

+ For location and scale parameters m and s + it is defined by the probability density function: +

+

+ +

+

+ The location and scale parameters are equivalent to the mean and standard + deviation of the logarithm of the random variable. +

+

+ The following graph illustrates the effect of the location parameter + on the PDF, note that the range of the random variable remains [0,+∞] + irrespective of the value of the location parameter: +

+

+ lognormal1 +

+

+ The next graph illustrates the effect of the scale parameter on the PDF: +

+

+ lognormal2 +

+
+ + Member + Functions +
+
+lognormal_distribution(RealType location = 0, RealType scale = 1);
+
+

+ Constructs a lognormal distribution with location location + and scale scale. +

+

+ The location parameter is the same as the mean of the logarithm of the + random variate. +

+

+ The scale parameter is the same as the standard deviation of the logarithm + of the random variate. +

+

+ Requires that the scale parameter is greater than zero, otherwise calls + domain_error. +

+
+RealType location()const;
+
+

+ Returns the location parameter of this distribution. +

+
+RealType scale()const;
+
+

+ Returns the scale parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0,+∞]. +

+
+ + Accuracy +
+

+ The lognormal distribution is implemented in terms of the standard library + log and exp functions, plus the error + function, and as such should have very low error rates. +

+
+ + Implementation +
+

+ In the following table m is the location parameter + of the distribution, s is it's scale parameter, + x is the random variate, p + is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = e-(ln(x) - m)2 / 2s2 / (x * s * sqrt(2pi)) +

+
+

+ cdf +

+ 		+

+ Using the relation: p = cdf(normal_distribtion<RealType>(m, + s), log(x)) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = cdf(complement(normal_distribtion<RealType>(m, + s), log(x))) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = exp(quantile(normal_distribtion<RealType>(m, + s), p)) +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = exp(quantile(complement(normal_distribtion<RealType>(m, + s), q))) +

+
+

+ mean +

+ 		+

+ em + s2 / 2 +

+
+

+ variance +

+ 		+

+ (es2 - 1) * e2m + s2 +

+
+

+ mode +

+ 		+

+ em + s2 +

+
+

+ skewness +

+ 		+

+ sqrt(es2 - 1) * (2 + es2 ) +

+
+

+ kurtosis +

+ 		+

+ e4s2 + 2e3s2 + 3e2s2 - 3 +

+
+

+ kurtosis excess +

+ 		+

+ e4s2 + 2e3s2 + 3e2s2 - 6 +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html new file mode 100644 index 000000000..579dccac0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html @@ -0,0 +1,927 @@ + + + +Negative Binomial Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+

+

+
+namespace boost{ namespace math{ 
+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class negative_binomial_distribution;
+
+typedef negative_binomial_distribution<> negative_binomial;
+
+template <class RealType, class Policy>
+class negative_binomial_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   // Constructor from successes and success_fraction:
+   negative_binomial_distribution(RealType r, RealType p);
+   
+   // Parameter accessors:
+   RealType success_fraction() const;
+   RealType successes() const;
+   
+   // Bounds on success fraction:
+   static RealType find_lower_bound_on_p(
+      RealType trials, 
+      RealType successes,
+      RealType probability); // alpha
+   static RealType find_upper_bound_on_p(
+      RealType trials, 
+      RealType successes,
+      RealType probability); // alpha
+      
+   // Estimate min/max number of trials:
+   static RealType find_minimum_number_of_trials(
+      RealType k,     // Number of failures.
+      RealType p,     // Success fraction.
+      RealType probability); // Probability threshold alpha.
+   static RealType find_maximum_number_of_trials(
+      RealType k,     // Number of failures.
+      RealType p,     // Success fraction.
+      RealType probability); // Probability threshold alpha.
+};
+
+}} // namespaces
+
+

+ The class type negative_binomial_distribution + represents a negative_binomial + distribution: it is used when there are exactly two mutually + exclusive outcomes of a Bernoulli + trial: these outcomes are labelled "success" and "failure". +

+

+ For k + r Bernoulli trials each with success fraction p, the negative_binomial + distribution gives the probability of observing k failures and r successes + with success on the last trial. The negative_binomial distribution assumes + that success_fraction p is fixed for all (k + r) trials. +

+
+ + + + + +
[Note]Note

+ The random variable for the negative binomial distribution is the number + of trials, (the number of successes is a fixed property of the distribution) + whereas for the binomial, the random variable is the number of successes, + for a fixed number of trials. +

+

+ It has the PDF: +

+

+ +

+

+ The following graph illustrate how the PDF varies as the success fraction + p changes: +

+

+ neg_binomial_pdf1 +

+

+ Alternatively, this graph shows how the shape of the PDF varies as the + number of successes changes: +

+

+ neg_binomial_pdf2 +

+
+ + Related + Distributions +
+

+ The name negative binomial distribution is reserved by some to the case + where the successes parameter r is an integer. This integer version is + also called the Pascal + distribution. +

+

+ This implementation uses real numbers for the computation throughout + (because it uses the real-valued incomplete + beta function family of functions). This real-valued version is also + called the Polya Distribution. +

+

+ The Poisson distribution is a generalization of the Pascal distribution, + where the success parameter r is an integer: to obtain the Pascal distribution + you must ensure that an integer value is provided for r, and take integer + values (floor or ceiling) from functions that return a number of successes. +

+

+ For large values of r (successes), the negative binomial distribution + converges to the Poisson distribution. +

+

+ The geometric distribution is a special case where the successes parameter + r = 1, so only a first and only success is required. geometric(p) = negative_binomial(1, + p). +

+

+ The Poisson distribution is a special case for large successes +

+

+ poisson(λ) = lim r → ∞ negative_binomial(r, r / (λ + r))) +

+

+

+
+ + + + + +
[Caution]Caution
+

+ The Negative Binomial distribution is a discrete distribution: internally + functions like the cdf + and pdf are treated + "as if" they are continuous functions, but in reality the + results returned from these functions only have meaning if an integer + value is provided for the random variate argument. +

+

+ The quantile function will by default return an integer result that + has been rounded outwards. That is to say lower + quantiles (where the probability is less than 0.5) are rounded downward, + and upper quantiles (where the probability is greater than 0.5) are + rounded upwards. This behaviour ensures that if an X% quantile is + requested, then at least the requested coverage + will be present in the central region, and no more than + the requested coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are + rounded differently, or even return a real-valued result using Policies. It is + strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on the Negative Binomial distribution. The reference + docs describe how to change the rounding policy for these + distributions. +

+
+

+

+
+ + Member + Functions +
+
+ + Construct +
+
+negative_binomial_distribution(RealType r, RealType p);
+
+

+ Constructor: r is the total number of successes, + p is the probability of success of a single trial. +

+

+ Requires: r > + 0 and 0 + <= p + <= 1. +

+
+ + Accessors +
+
+RealType success_fraction() const; // successes / trials (0 <= p <= 1)
+
+

+ Returns the parameter p from which this distribution + was constructed. +

+
+RealType successes() const; // required successes (r > 0)
+
+

+ Returns the parameter r from which this distribution + was constructed. +

+
+ + Lower + Bound on Parameter p +
+
+static RealType find_lower_bound_on_p(
+  RealType failures, 
+  RealType successes,
+  RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
+
+

+ Returns a lower bound on the success + fraction: +

+
+

+
+
failures
+

+ The total number of failures before the r th success. +

+
successes
+

+ The number of successes required. +

+
alpha
+

+ The largest acceptable probability that the true value of the success + fraction is less than the value + returned. +

+
+
+

+ For example, if you observe k failures and r + successes from n = k + r trials the best estimate + for the success fraction is simply r/n, but if you + want to be 95% sure that the true value is greater + than some value, pmin, then: +

+
+pmin = negative_binomial_distribution<RealType>::find_lower_bound_on_p(
+                    failures, successes, 0.05);
+
+

+ See + negative binomial confidence interval example. +

+

+ This function uses the Clopper-Pearson method of computing the lower + bound on the success fraction, whilst many texts refer to this method + as giving an "exact" result in practice it produces an interval + that guarantees at least the coverage required, + and may produce pessimistic estimates for some combinations of failures + and successes. See: +

+

+ Yong + Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some + Discrete Distributions. Computational statistics and data analysis, 2005, + vol. 48, no3, 605-621. +

+
+ + Upper + Bound on Parameter p +
+
+static RealType find_upper_bound_on_p(
+   RealType trials, 
+   RealType successes,
+   RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
+
+

+ Returns an upper bound on the success + fraction: +

+
+

+
+
trials
+

+ The total number of trials conducted. +

+
successes
+

+ The number of successes that occurred. +

+
alpha
+

+ The largest acceptable probability that the true value of the success + fraction is greater than the value + returned. +

+
+
+

+ For example, if you observe k successes from n + trials the best estimate for the success fraction is simply k/n, + but if you want to be 95% sure that the true value is less + than some value, pmax, then: +

+
+pmax = negative_binomial_distribution<RealType>::find_upper_bound_on_p(
+                    r, k, 0.05);
+
+

+ See + negative binomial confidence interval example. +

+

+ This function uses the Clopper-Pearson method of computing the lower + bound on the success fraction, whilst many texts refer to this method + as giving an "exact" result in practice it produces an interval + that guarantees at least the coverage required, + and may produce pessimistic estimates for some combinations of failures + and successes. See: +

+

+ Yong + Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some + Discrete Distributions. Computational statistics and data analysis, 2005, + vol. 48, no3, 605-621. +

+
+ + Estimating + Number of Trials to Ensure at Least a Certain Number of Failures +
+
+static RealType find_minimum_number_of_trials(
+   RealType k,     // number of failures.
+   RealType p,     // success fraction.
+   RealType alpha); // probability threshold (0.05 equivalent to 95%).
+
+

+ This functions estimates the number of trials required to achieve a certain + probability that more than k failures will be observed. +

+
+

+
+
k
+

+ The target number of failures to be observed. +

+
p
+

+ The probability of success for each trial. +

+
alpha
+

+ The maximum acceptable risk that only k failures or fewer will be + observed. +

+
+
+

+ For example: +

+
+negative_binomial_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
+
+

+ Returns the smallest number of trials we must conduct to be 95% sure + of seeing 10 failures that occur with frequency one half. +

+

+ Worked + Example. +

+

+ This function uses numeric inversion of the negative binomial distribution + to obtain the result: another interpretation of the result, is that it + finds the number of trials (success+failures) that will lead to an alpha + probability of observing k failures or fewer. +

+
+ + Estimating + Number of Trials to Ensure a Maximum Number of Failures or Less +
+
+static RealType find_maximum_number_of_trials(
+   RealType k,     // number of failures.
+   RealType p,     // success fraction.
+   RealType alpha); // probability threshold (0.05 equivalent to 95%).
+
+

+ This functions estimates the maximum number of trials we can conduct + and achieve a certain probability that k failures + or fewer will be observed. +

+
+

+
+
k
+

+ The maximum number of failures to be observed. +

+
p
+

+ The probability of success for each trial. +

+
alpha
+

+ The maximum acceptable risk that more than k + failures will be observed. +

+
+
+

+ For example: +

+
+negative_binomial_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05);
+
+

+ Returns the largest number of trials we can conduct and still be 95% + sure of seeing no failures that occur with frequency one in one million. +

+

+ This function uses numeric inversion of the negative binomial distribution + to obtain the result: another interpretation of the result, is that it + finds the number of trials (success+failures) that will lead to an alpha + probability of observing more than k failures. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ However it's worth taking a moment to define what these actually mean + in the context of this distribution: +

+
+

Table 10. Meaning of the non-member accessors.

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Meaning +

+
+

+ Probability Density Function +

+ 		+

+ The probability of obtaining exactly k failures + from k+r trials with success fraction p. For example: +

+

+ +

+
+pdf(negative_binomial(r, p), k)
+

+

+
+

+ Cumulative Distribution Function +

+ 		+

+ The probability of obtaining k failures or + fewer from k+r trials with success fraction p and success + on the last trial. For example: +

+

+ +

+
+cdf(negative_binomial(r, p), k)
+

+

+
+

+ Complement of the Cumulative Distribution + Function +

+ 		+

+ The probability of obtaining more than k + failures from k+r trials with success fraction p and + success on the last trial. For example: +

+

+ +

+
+cdf(complement(negative_binomial(r, p), k))
+

+

+
+

+ Quantile +

+ 		+

+ The greatest number of failures + k expected to be observed from k+r trials with success fraction + p, at probability P. Note that the value returned is a real-number, + and not an integer. Depending on the use case you may want to take + either the floor or ceiling of the real result. For example: +

+

+ +

+
+quantile(negative_binomial(r, p), P)
+

+

+
+

+ Quantile from the complement + of the probability +

+ 		+

+ The smallest number of failures + k expected to be observed from k+r trials with success fraction + p, at probability P. Note that the value returned is a real-number, + and not an integer. Depending on the use case you may want to take + either the floor or ceiling of the real result. For example: +

+
+quantile(complement(negative_binomial(r, p), P))
+

+

+
+
+
+ + Accuracy +
+

+ This distribution is implemented using the incomplete beta functions + ibeta + and ibetac: + please refer to these functions for information on accuracy. +

+
+ + Implementation +
+

+ In the following table, p is the probability that + any one trial will be successful (the success fraction), r + is the number of successes, k is the number of failures, + p is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) + * pow((1-p), k) +

+

+ Implementation is in terms of ibeta_derivative: +

+

+ (p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), + p) The function ibeta_derivative + is used here, since it has already been optimised for the lowest + possible error - indeed this is really just a thin wrapper around + part of the internals of the incomplete beta function. +

+
+

+ cdf +

+ 		+

+ Using the relation: +

+

+ cdf = Ip(r, k+1) = ibeta(r, k+1, p) +

+

+ = ibeta(r, static_cast<RealType>(k+1), p) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: +

+

+ 1 - cdf = Ip(k+1, r) +

+

+ = ibetac(r, static_cast<RealType>(k+1), p) +

+
+

+ quantile +

+ 		+

+ ibeta_invb(r, p, P) - 1 +

+
+

+ quantile from the complement +

+ 		+

+ ibetac_invb(r, p, Q) -1) +

+
+

+ mean +

+ 		+

+ r(1-p)/p +

+
+

+ variance +

+ 		+

+ r (1-p) + / p + * p +

+
+

+ mode +

+ 		+

+ floor((r-1) * (1 - p)/p) +

+
+

+ skewness +

+ 		+

+ (2 + - p) / + sqrt(r * + (1 + - p)) +

+
+

+ kurtosis +

+ 		+

+ 6 / + r + + (p + * p) / + r * + (1 + - p + ) +

+
+

+ kurtosis excess +

+ 		+

+ 6 / + r + + (p + * p) / + r * + (1 + - p + ) -3 +

+
+

+ parameter estimation member functions +

+ 		+

+

+
+

+ find_lower_bound_on_p +

+ 		+

+ ibeta_inv(successes, failures + 1, alpha) +

+
+

+ find_upper_bound_on_p +

+ 		+

+ ibetac_inv(successes, failures, alpha) plus see comments in code. +

+
+

+ find_minimum_number_of_trials +

+ 		+

+ ibeta_inva(k + 1, p, alpha) +

+
+

+ find_maximum_number_of_trials +

+ 		+

+ ibetac_inva(k + 1, p, alpha) +

+
+

+ Implementation notes: +

+
    +
  • + The real concept type (that deliberately lacks the Lanczos approximation), + was found to take several minutes to evaluate some extreme test values, + so the test has been disabled for this type. +
    • +
  • + Much greater speed, and perhaps greater accuracy, might be achieved + for extreme values by using a normal approximation. This is NOT been + tested or implemented. +
    • +
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html new file mode 100644 index 000000000..d5a83fc45 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html @@ -0,0 +1,309 @@ + + + +Normal (Gaussian) Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/normal.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class normal_distribution;
+
+typedef normal_distribution<> normal;
+
+template <class RealType, class Policy>
+class normal_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   // Construct:
+   normal_distribution(RealType mean = 0, RealType sd = 1);
+   // Accessors:
+   RealType mean()const; // location.
+   RealType standard_deviation()const; // scale.
+   // Synonyms, provided to allow generic use of find_location and find_scale.
+   RealType location()const;
+   RealType scale()const;
+};
+
+}} // namespaces
+
+

+ The normal distribution is probably the most well known statistical distribution: + it is also known as the Gaussian Distribution. A normal distribution + with mean zero and standard deviation one is known as the Standard + Normal Distribution. +

+

+ Given mean μ and standard deviation σ it has the PDF: +

+

+ +

+

+ The variation the PDF with its parameters is illustrated in the following + graph: +

+

+ normal +

+
+ + Member + Functions +
+
+normal_distribution(RealType mean = 0, RealType sd = 1);
+
+

+ Constructs a normal distribution with mean mean + and standard deviation sd. +

+

+ Requires sd > 0, otherwise domain_error + is called. +

+
+RealType mean()const;
+RealType location()const;   
+
+

+ both return the mean of this distribution. +

+
+RealType standard_deviation()const;
+RealType scale()const;
+
+

+ both return the standard deviation of this distribution. + (Redundant location and scale function are provided to match other similar + distributions, allowing the functions find_location and find_scale to + be used generically). +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [-[max_value], +[min_value]]. However, + the pdf of +∞ and -∞ = 0 is also supported, and cdf at -∞ = 0, cdf at +∞ = 1, + and complement cdf -∞ = 1 and +∞ = 0, if RealType permits. +

+
+ + Accuracy +
+

+ The normal distribution is implemented in terms of the error + function, and as such should have very low error rates. +

+
+ + Implementation +
+

+ In the following table m is the mean of the distribution, + and s is its standard deviation. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = e-(x-m)2/(2s2) / (s * sqrt(2*pi)) +

+
+

+ cdf +

+ 		+

+ Using the relation: p = 0.5 * erfc(-(x-m)/(s*sqrt(2))) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = 0.5 * erfc((x-m)/(s*sqrt(2))) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = m - s * sqrt(2) * erfc_inv(2*p) +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = m + s * sqrt(2) * erfc_inv(2*p) +

+
+

+ mean and standard deviation +

+ 		+

+ The same as dist.mean() and dist.standard_deviation() +

+
+

+ mode +

+ 		+

+ The same as the mean. +

+
+

+ skewness +

+ 		+

+ 0 +

+
+

+ kurtosis +

+ 		+

+ 3 +

+
+

+ kurtosis excess +

+ 		+

+ 0 +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html new file mode 100644 index 000000000..a7c9e9471 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html @@ -0,0 +1,342 @@ + + + +Pareto Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/pareto.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class pareto_distribution;
+
+typedef pareto_distribution<> pareto;
+
+template <class RealType, class Policy>
+class pareto_distribution
+{
+public:
+   typedef RealType value_type;
+   // Constructor:
+   pareto_distribution(RealType location = 1, RealType shape = 1)
+   // Accessors:
+   RealType location()const;
+   RealType shape()const;
+};
+
+}} // namespaces
+
+

+ The Pareto + distribution is a continuous distribution with the probability + density function (pdf): +

+

+ f(x; α, β) = αβα / xα+ 1 +

+

+ For shape parameter α > 0, and location parameter β > 0, and α > 0. +

+

+ The Pareto + distribution often describes the larger compared to the smaller. + A classic example is that 80% of the wealth is owned by 20% of the population. +

+

+ The following graph illustrates how the PDF varies with the shape parameter + α: +

+

+ Pareto + pdf +

+
+ + Related + distributions +
+
+ + Member + Functions +
+
+pareto_distribution(RealType location = 1, RealType shape = 1);
+
+

+ Constructs a pareto + distribution with shape shape and scale + scale. +

+

+ Requires that the shape and scale + parameters are both greater than zero, otherwise calls domain_error. +

+
+RealType location()const;
+
+

+ Returns the location parameter of this distribution. +

+
+RealType shape()const;
+
+

+ Returns the shape parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The supported domain of the random variable is [location, ∞]. +

+
+ + Accuracy +
+

+ The pareto distribution is implemented in terms of the standard library + exp functions plus expm1 and as such + should have very low error rates except when probability is very close + to unity. +

+
+ + Implementation +
+

+ In the following table α is the shape parameter of the distribution, and + β is its location parameter, x is the random variate, + p is the probability and its complement q + = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf p = αβα/xα +1 +

+
+

+ cdf +

+ 		+

+ Using the relation: cdf p = 1 - (β / x)α +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = 1 - p = -(β / x)α +

+
+

+ quantile +

+ 		+

+ Using the relation: x = α / (1 - p)1/β +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = α / (q)1/β +

+
+

+ mean +

+ 		+

+ αβ / (β - 1) +

+
+

+ variance +

+ 		+

+ βα2 / (β - 1)2 (β - 2) +

+
+

+ mode +

+ 		+

+ α +

+
+

+ skewness +

+ 		+

+ Refer to Weisstein, + Eric W. "Pareto Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis +

+ 		+

+ Refer to Weisstein, + Eric W. "Pareto Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis excess +

+ 		+

+ Refer to Weisstein, + Eric W. "pareto Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html new file mode 100644 index 000000000..2d6a1caf5 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html @@ -0,0 +1,345 @@ + + + +Poisson Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/poisson.hpp>
+

+

+
+namespace boost { namespace math {
+
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class poisson_distribution;
+
+typedef poisson_distribution<> poisson;
+
+template <class RealType, class Policy>
+class poisson_distribution
+{ 
+public:
+  typedef RealType value_type;
+  typedef Policy   policy_type;
+  
+  poisson_distribution(RealType mean = 1); // Constructor.
+  RealType mean()const; // Accessor.
+}
+ 
+}} // namespaces boost::math
+
+

+ The Poisson + distribution is a well-known statistical discrete distribution. + It expresses the probability of a number of events (or failures, arrivals, + occurrences ...) occurring in a fixed period of time, provided these + events occur with a known mean rate λ +(events/time), and are independent + of the time since the last event. +

+

+ The distribution was discovered by Simé on-Denis Poisson (1781 to 1840). +

+

+ It has the Probability Mass Function: +

+

+ +

+

+ for k events, with an expected number of events λ. +

+

+ The following graph illustrates how the PDF varies with the parameter + λ: +

+

+ poisson +

+

+

+
+ + + + + +
[Caution]Caution
+

+ The Poisson distribution is a discrete distribution: internally functions + like the cdf and + pdf are treated "as + if" they are continuous functions, but in reality the results + returned from these functions only have meaning if an integer value + is provided for the random variate argument. +

+

+ The quantile function will by default return an integer result that + has been rounded outwards. That is to say lower + quantiles (where the probability is less than 0.5) are rounded downward, + and upper quantiles (where the probability is greater than 0.5) are + rounded upwards. This behaviour ensures that if an X% quantile is + requested, then at least the requested coverage + will be present in the central region, and no more than + the requested coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are + rounded differently, or even return a real-valued result using Policies. It is + strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on the Poisson distribution. The reference + docs describe how to change the rounding policy for these + distributions. +

+
+

+

+
+ + Member + Functions +
+
+poisson_distribution(RealType mean = 1);
+
+

+ Constructs a poisson distribution with mean mean. +

+
+RealType mean()const;
+
+

+ Returns the mean of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, ∞]. +

+
+ + Accuracy +
+

+ The Poisson distribution is implemented in terms of the incomplete gamma + functions gamma_p + and gamma_q + and as such should have low error rates: but refer to the documentation + of those functions for more information. The quantile and its complement + use the inverse gamma functions and are therefore probably slightly less + accurate: this is because the inverse gamma functions are implemented + using an iterative method with a lower tolerance to avoid excessive computation. +

+
+ + Implementation +
+

+ In the following table λ is the mean of the distribution, k + is the random variable, p is the probability and + q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = e-λ λk / k! +

+
+

+ cdf +

+ 		+

+ Using the relation: p = Γ(k+1, λ) / k! = gamma_q(k+1, + λ) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = gamma_p(k+1, + λ) +

+
+

+ quantile +

+ 		+

+ Using the relation: k = gamma_q_inva(λ, + p) - 1 +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: k = gamma_p_inva(λ, + q) - 1 +

+
+

+ mean +

+ 		+

+ λ +

+
+

+ mode +

+ 		+

+ floor (λ) or ⌊λ⌋ +

+
+

+ skewness +

+ 		+

+ 1/√λ +

+
+

+ kurtosis +

+ 		+

+ 3 + 1/λ +

+
+

+ kurtosis excess +

+ 		+

+ 1/λ +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html new file mode 100644 index 000000000..f212a3f3e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html @@ -0,0 +1,351 @@ + + + +Rayleigh Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/rayleigh.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class rayleigh_distribution;
+
+typedef rayleigh_distribution<> rayleigh;
+
+template <class RealType, class Policy>
+class rayleigh_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   // Construct:
+   rayleigh_distribution(RealType sigma = 1)
+   // Accessors:
+   RealType sigma()const;
+};
+
+}} // namespaces
+
+

+ The Rayleigh + distribution is a continuous distribution with the probability + density function: +

+

+ f(x; sigma) = x * exp(-x2/2 σ2) / σ2 +

+

+ For sigma parameter σ > 0, and x > 0. +

+

+ The Rayleigh distribution is often used where two orthogonal components + have an absolute value, for example, wind velocity and direction may + be combined to yield a wind speed, or real and imaginary components may + have absolute values that are Rayleigh distributed. +

+

+ The following graph illustrates how the Probability density Function(pdf) + varies with the shape parameter σ: +

+

+ rayleigh_pdf +

+

+ and the Cumulative Distribution Function (cdf) +

+

+ rayleigh_cdf +

+
+ + Related + distributions +
+

+ The absolute value of two independent normal distributions X and Y, √ (X2 + + Y2) is a Rayleigh distribution. +

+

+ The Chi, + Rice + and Weibull + distributions are generalizations of the Rayleigh + distribution. +

+
+ + Member + Functions +
+
+rayleigh_distribution(RealType sigma = 1);
+
+

+ Constructs a Rayleigh + distribution with σ sigma. +

+

+ Requires that the σ parameter is greater than zero, otherwise calls domain_error. +

+
+RealType sigma()const;
+
+

+ Returns the sigma parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, max_value]. +

+
+ + Accuracy +
+

+ The Rayleigh distribution is implemented in terms of the standard library + sqrt and exp and as such should have very low + error rates. Some constants such as skewness and kurtosis were calculated + using NTL RR type with 150-bit accuracy, about 50 decimal digits. +

+
+ + Implementation +
+

+ In the following table σ is the sigma parameter of the distribution, x + is the random variate, p is the probability and + q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = x * exp(-x2)/2 σ2 +

+
+

+ cdf +

+ 		+

+ Using the relation: p = 1 - exp(-x2/2) σ2 = -expm1(-x2/2) + σ2 +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = exp(-x2/ 2) * σ2 +

+
+

+ quantile +

+ 		+

+ Using the relation: x = sqrt(-2 * σ 2) * log(1 - p)) = sqrt(-2 + * σ 2) * log1p(-p)) +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = sqrt(-2 * σ 2) * log(q)) +

+
+

+ mean +

+ 		+

+ σ * sqrt(π/2) +

+
+

+ variance +

+ 		+

+ σ2 * (4 - π/2) +

+
+

+ mode +

+ 		+

+ σ +

+
+

+ skewness +

+ 		+

+ Constant from Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis +

+ 		+

+ Constant from Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis excess +

+ 		+

+ Constant from Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html new file mode 100644 index 000000000..f6c111ea3 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html @@ -0,0 +1,428 @@ + + + +Students t Distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/students_t.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class students_t_distribution;
+
+typedef students_t_distribution<> students_t;
+
+template <class RealType, class Policy>
+class students_t_distribution
+{
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   
+   // Construct:
+   students_t_distribution(const RealType& v);
+   
+   // Accessor:
+   RealType degrees_of_freedom()const;
+   
+   // degrees of freedom estimation:
+   static RealType find_degrees_of_freedom(
+      RealType difference_from_mean,
+      RealType alpha,
+      RealType beta,
+      RealType sd,
+      RealType hint = 100);
+};
+
+}} // namespaces
+
+

+ A statistical distribution published by William Gosset in 1908. His employer, + Guinness Breweries, required him to publish under a pseudonym, so he + chose "Student". Given N independent measurements, let +

+

+ +

+

+ where M is the population mean,μ + is the sample mean, and s is the sample variance. +

+

+ Student's t-distribution is defined as the distribution of the random + variable t which is - very loosely - the "best" that we can + do not knowing the true standard deviation of the sample. It has the + PDF: +

+

+ +

+

+ The Student's t-distribution takes a single parameter: the number of + degrees of freedom of the sample. When the degrees of freedom is one + then this distribution is the same as the Cauchy-distribution. As the + number of degrees of freedom tends towards infinity, then this distribution + approaches the normal-distribution. The following graph illustrates how + the PDF varies with the degrees of freedom ν: +

+

+ students_t +

+
+ + Member + Functions +
+
+students_t_distribution(const RealType& v);
+
+

+ Constructs a Student's t-distribution with v degrees + of freedom. +

+

+ Requires v > 0, otherwise calls domain_error. + Note that non-integral degrees of freedom are supported, and meaningful + under certain circumstances. +

+
+RealType degrees_of_freedom()const;
+
+

+ Returns the number of degrees of freedom of this distribution. +

+
+static RealType find_degrees_of_freedom(
+   RealType difference_from_mean,
+   RealType alpha,
+   RealType beta,
+   RealType sd,
+   RealType hint = 100);
+
+

+ Returns the number of degrees of freedom required to observe a significant + result in the Student's t test when the mean differs from the "true" + mean by difference_from_mean. +

+
+

+
+
difference_from_mean
+

+ The difference between the true mean and the sample mean that we + wish to show is significant. +

+
alpha
+

+ The maximum acceptable probability of rejecting the null hypothesis + when it is in fact true. +

+
beta
+

+ The maximum acceptable probability of failing to reject the null + hypothesis when it is in fact false. +

+
sd
+

+ The sample standard deviation. +

+
hint
+

+ A hint for the location to start looking for the result, a good choice + for this would be the sample size of a previous borderline Student's + t test. +

+
+
+
+ + + + + +
[Note]Note

+ Remember that for a two-sided test, you must divide alpha by two before + calling this function. +

+

+ For more information on this function see the NIST + Engineering Statistics Handbook. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [-∞, +∞]. +

+
+ + Examples +
+

+ Various worked examples + are available illustrating the use of the Student's t distribution. +

+
+ + Accuracy +
+

+ The normal distribution is implemented in terms of the incomplete + beta function and it's + inverses, refer to accuracy data on those functions for more information. +

+
+ + Implementation +
+

+ In the following table v is the degrees of freedom + of the distribution, t is the random variate, p + is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = (v / (v + t2))(1+v)/2 / (sqrt(v) * + beta(v/2, + 0.5)) +

+
+

+ cdf +

+ 		+

+ Using the relations: +

+

+ p = 1 - z iff t > 0 +

+

+ p = z otherwise +

+

+ where z is given by: +

+

+ ibeta(v + / 2, 0.5, v / (v + t2)) / 2 iff v < 2t2 +

+

+ ibetac(0.5, + v / 2, t2 / (v + t2) / 2 otherwise +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = cdf(-t) +

+
+

+ quantile +

+ 		+

+ Using the relation: t = sign(p - 0.5) * sqrt(v * y / x) +

+

+ where: +

+

+ x = ibeta_inv(v + / 2, 0.5, 2 * min(p, q)) +

+

+ y = 1 - x +

+

+ The quantities x and y + are both returned by ibeta_inv + without the subtraction implied above. +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: t = -quantile(q) +

+
+

+ mean +

+ 		+

+ 0 +

+
+

+ variance +

+ 		+

+ v / (v - 2) +

+
+

+ mode +

+ 		+

+ 0 +

+
+

+ skewness +

+ 		+

+ 0 +

+
+

+ kurtosis +

+ 		+

+ 3 * (v - 2) / (v - 4) +

+
+

+ kurtosis excess +

+ 		+

+ 6 / (df - 4) +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html new file mode 100644 index 000000000..c998ccc2b --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html @@ -0,0 +1,425 @@ + + + +Triangular Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/triangular.hpp>
+

+

+
+namespace boost{ namespace math{
+ template <class RealType = double, 
+           class Policy   = policies::policy<> >
+ class triangular_distribution;
+   
+ typedef triangular_distribution<> triangular;
+
+ template <class RealType, class Policy>
+ class triangular_distribution
+ {
+ public:
+    typedef RealType value_type;
+    typedef Policy   policy_type;
+
+    triangular_distribution(RealType lower = -1, RealType mode = 0) RealType upper = 1); // Constructor.
+       : m_lower(lower), m_mode(mode), m_upper(upper) // Default is -1, 0, +1 triangular distribution.
+    // Accessor functions.
+    RealType lower()const;
+    RealType mode()const;
+    RealType upper()const;
+ }; // class triangular_distribution
+
+}} // namespaces
+
+

+ The triangular + distribution is a continuous + probability + distribution with a lower limit a, mode + c, and upper limit b. +

+

+ The triangular distribution is often used where the distribution is only + vaguely known, but, like the uniform + distribution, upper and limits are 'known', but a 'best guess', + the mode or center point, is also added. It has been recommended as a + proxy + for the beta distribution. The distribution is used in business + decision making and project planning. +

+

+ The triangular + distribution is a distribution with the probability + density function: +

+

+ f(x) = +

+
    +
  • + 2(x-a)/(b-a) (c-a) for a <= x <= c +
    • +
  • + 2(b-x)/(b-a)(b-c) for c < x <= b +
    • +
+

+ Parameter a (lower) can be any finite value. Parameter b (upper) can + be any finite value > a (lower). Parameter c (mode) a <= c <= + b. This is the most probable value. +

+

+ The random variate + x must also be finite, and is supported lower <= x <= upper. +

+

+ The triangular distribution may be appropriate when an assumption of + a normal distribution is unjustified because uncertainty is caused by + rounding and quantization from analog to digital conversion. Upper and + lower limits are known, and the most probable value lies midway. +

+

+ The distribution simplifies when the 'best guess' is either the lower + or upper limit - a 90 degree angle triangle. The default chosen is the + 001 triangular distribution which expresses an estimate that the lowest + value is the most likely; for example, you believe that the next-day + quoted delivery date is most likely (knowing that a quicker delivery + is impossible - the postman only comes once a day), and that longer delays + are decreasingly likely, and delivery is assumed to never take more than + your upper limit. +

+

+ The following graph illustrates how the probability + density function PDF varies with the various parameters: +

+

+ triangular_pdf +

+

+ and cumulative distribution function +

+

+ triangular_cdf +

+
+ + Member + Functions +
+
+triangular_distribution(RealType lower = 0, RealType mode = 0 RealType upper = 1);
+
+

+ Constructs a triangular + distribution with lower lower (a) and upper + upper (b). +

+

+ Requires that the lower, mode + and upper parameters are all finite, otherwise calls + domain_error. +

+
+RealType lower()const;
+
+

+ Returns the lower parameter of this distribution + (default -1). +

+
+RealType mode()const;
+
+

+ Returns the mode parameter of this distribution + (default 0). +

+
+RealType upper()const;
+
+

+ Returns the upper parameter of this distribution + (default+1). +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is \lowerto \upper, and the supported + range is lower <= x <= upper. +

+
+ + Accuracy +
+

+ The triangular distribution is implemented with simple arithmetic operators + and so should have errors within an epsilon or two, except quantiles + with arguments nearing the extremes of zero and unity. +

+
+ + Implementation +
+

+ In the following table, a is the lower parameter + of the distribution, c is the mode parameter, b + is the upper parameter, x is + the random variate, p is the probability and q + = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = 0 for x < mode, 2(x-a)/(b-a)(c-a) + else 2*(b-x)/((b-a)(b-c)) +

+
+

+ cdf +

+ 		+

+ Using the relation: cdf = 0 for x < mode (x-a)2/((b-a)(c-a)) + else 1 - (b-x)2/((b-a)(b-c)) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = 1 - p +

+
+

+ quantile +

+ 		+

+ let p0 = (c-a)/(b-a) the point of inflection on the cdf, then + given probability p and q = 1-p: +

+

+ x = sqrt((b-a)(c-a)p) + a ; for p < p0 +

+

+ x = c ; for p == p0 +

+

+ x = b - sqrt((b-a)(b-c)q) ; for p > p0 +

+

+ (See /boost/math/distributions/triangular.hpp + for details.) +

+
+

+ quantile from the complement +

+ 		+

+ As quantile (See /boost/math/distributions/triangular.hpp + for details.) +

+
+

+ mean +

+ 		+

+ (a + b + 3) / 3 +

+
+

+ variance +

+ 		+

+ (a2+b2+c2 - ab - ac - bc)/18 +

+
+

+ mode +

+ 		+

+ c +

+
+

+ skewness +

+ 		+

+ (See /boost/math/distributions/triangular.hpp + for details). +

+
+

+ kurtosis +

+ 		+

+ 12/5 +

+
+

+ kurtosis excess +

+ 		+

+ -3/5 +

+
+

+ Some 'known good' test values were obtained from Statlet: + Calculate and plot probability distributions +

+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html new file mode 100644 index 000000000..bb0f465e7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html @@ -0,0 +1,371 @@ + + + +Uniform Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/uniform.hpp>
+

+

+
+namespace boost{ namespace math{
+ template <class RealType = double, 
+           class Policy   = policies::policy<> >
+ class uniform_distribution;
+   
+ typedef uniform_distribution<> uniform;
+
+ template <class RealType, class Policy>
+ class uniform_distribution
+ {
+ public:
+    typedef RealType value_type;
+
+    uniform_distribution(RealType lower = 0, RealType upper = 1); // Constructor.
+       : m_lower(lower), m_upper(upper) // Default is standard uniform distribution.
+    // Accessor functions.
+    RealType lower()const;
+    RealType upper()const;
+ }; // class uniform_distribution
+
+}} // namespaces
+
+

+ The uniform distribution, also known as a rectangular distribution, is + a probability distribution that has constant probability. +

+

+ The continuous + uniform distribution is a distribution with the probability + density function: +

+

+ f(x) = +

+
    +
  • + 1 / (upper - lower) for lower < x < upper +
    • +
  • + zero for x < lower or x > upper +
    • +
+

+ and in this implementation: +

+
  • + 1 / (upper - lower) for x = lower or x = upper +
+

+ The choice of x = lower or x = upper is made because statistical use + of this distribution judged is most likely: the method of maximum likelihood + uses this definition. +

+

+ There is also a discrete uniform distribution. +

+

+ Parameters lower and upper can be any finite value. +

+

+ The random variate + x must also be finite, and is supported lower <= x <= upper. +

+

+ The lower parameter is also called the location + parameter, that + is where the origin of a plot will lie, and (upper - lower) is + also called the scale + parameter. +

+

+ The following graph illustrates how the probability + density function PDF varies with the shape parameter: +

+

+ uniform_pdf +

+

+ Likewise for the CDF: +

+

+ uniform_cdf +

+
+ + Member + Functions +
+
+uniform_distribution(RealType lower = 0, RealType upper = 1);
+
+

+ Constructs a uniform + distribution with lower lower (a) and upper + upper (b). +

+

+ Requires that the lower and upper + parameters are both finite; otherwise if infinity or NaN then calls + domain_error. +

+
+RealType lower()const;
+
+

+ Returns the lower parameter of this distribution. +

+
+RealType upper()const;
+
+

+ Returns the upper parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is any finite value, but the supported + range is only lower <= x <= upper. +

+
+ + Accuracy +
+

+ The uniform distribution is implemented with simple arithmetic operators + and so should have errors within an epsilon or two. +

+
+ + Implementation +
+

+ In the following table a is the lower parameter + of the distribution, b is the upper parameter, + x is the random variate, p + is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = 0 for x < a, 1 / (b - a) for a <= + x <= b, 0 for x > b +

+
+

+ cdf +

+ 		+

+ Using the relation: cdf = 0 for x < a, (x - a) / (b - a) for + a <= x <= b, 1 for x > b +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = 1 - p, (b - x) / (b - a) +

+
+

+ quantile +

+ 		+

+ Using the relation: x = p * (b - a) + a; +

+
+

+ quantile from the complement +

+ 		+

+ x = -q * (b - a) + b +

+
+

+ mean +

+ 		+

+ (a + b) / 2 +

+
+

+ variance +

+ 		+

+ (b - a) 2 / 12 +

+
+

+ mode +

+ 		+

+ any value in [a, b] but a is chosen. (Would NaN be better?) +

+
+

+ skewness +

+ 		+

+ 0 +

+
+

+ kurtosis excess +

+ 		+

+ -6/5 = -1.2 exactly. (kurtosis - 3) +

+
+

+ kurtosis +

+ 		+

+ 9/5 +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html new file mode 100644 index 000000000..1e462f269 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html @@ -0,0 +1,369 @@ + + + +Weibull Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/distributions/weibull.hpp>
+

+

+
+namespace boost{ namespace math{ 
+   
+template <class RealType = double, 
+          class Policy   = policies::policy<> >
+class weibull_distribution;
+
+typedef weibull_distribution<> weibull;
+
+template <class RealType, class Policy>
+class weibull_distribution
+{
+public:
+   typedef RealType value_type;
+   typedef Policy   policy_type;
+   // Construct:
+   weibull_distribution(RealType shape, RealType scale = 1)
+   // Accessors:
+   RealType shape()const;
+   RealType scale()const;
+};
+
+}} // namespaces
+
+

+ The Weibull + distribution is a continuous distribution with the probability + density function: +

+

+ f(x; α, β) = (α/β) * (x / β)α - 1 * e-(x/β)α +

+

+ For shape parameter α > 0, and scale parameter β > 0, and x > 0. +

+

+ The Weibull distribution is often used in the field of failure analysis; + in particular it can mimic distributions where the failure rate varies + over time. If the failure rate is: +

+
    +
  • + constant over time, then α = 1, suggests that items are failing from + random events. +
    • +
  • + decreases over time, then α < 1, suggesting "infant mortality". +
    • +
  • + increases over time, then α > 1, suggesting "wear out" - + more likely to fail as time goes by. +
    • +
+

+ The following graph illustrates how the PDF varies with the shape parameter + α: +

+

+ weibull +

+

+ While this graph illustrates how the PDF varies with the scale parameter + β: +

+

+ weibull2 +

+
+ + Related + distributions +
+

+ When α = 3, the Weibull + distribution appears similar to the normal + distribution. When α = 1, the Weibull distribution reduces to the + exponential + distribution. +

+
+ + Member + Functions +
+
+weibull_distribution(RealType shape, RealType scale = 1);
+
+

+ Constructs a Weibull + distribution with shape shape and scale + scale. +

+

+ Requires that the shape and scale + parameters are both greater than zero, otherwise calls domain_error. +

+
+RealType shape()const;
+
+

+ Returns the shape parameter of this distribution. +

+
+RealType scale()const;
+
+

+ Returns the scale parameter of this distribution. +

+
+ + Non-member + Accessors +
+

+ All the usual non-member + accessor functions that are generic to all distributions are supported: + Cumulative Distribution Function, + Probability Density Function, Quantile, Hazard + Function, Cumulative Hazard Function, + mean, median, + mode, variance, + standard deviation, skewness, + kurtosis, kurtosis_excess, + range and support. +

+

+ The domain of the random variable is [0, ∞]. +

+
+ + Accuracy +
+

+ The Weibull distribution is implemented in terms of the standard library + log and exp functions plus expm1 + and log1p and + as such should have very low error rates. +

+
+ + Implementation +
+

+ In the following table α is the shape parameter of the distribution, β is + it's scale parameter, x is the random variate, + p is the probability and q = 1-p. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Implementation Notes +

+
+

+ pdf +

+ 		+

+ Using the relation: pdf = αβ-α xα - 1 e-(x/beta)alpha +

+
+

+ cdf +

+ 		+

+ Using the relation: p = -expm1(-(x/β)α) +

+
+

+ cdf complement +

+ 		+

+ Using the relation: q = e-(x/β)α +

+
+

+ quantile +

+ 		+

+ Using the relation: x = β * (-log1p(-p))1/α +

+
+

+ quantile from the complement +

+ 		+

+ Using the relation: x = β * (-log(q))1/α +

+
+

+ mean +

+ 		+

+ β * Γ(1 + 1/α) +

+
+

+ variance +

+ 		+

+ β2(Γ(1 + 2/α) - Γ2(1 + 1/α)) +

+
+

+ mode +

+ 		+

+ β((α - 1) / α)1/α +

+
+

+ skewness +

+ 		+

+ Refer to Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis +

+ 		+

+ Refer to Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+

+ kurtosis excess +

+ 		+

+ Refer to Weisstein, + Eric W. "Weibull Distribution." From MathWorld--A Wolfram + Web Resource. +

+
+
+ + References +
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html new file mode 100644 index 000000000..15f1410cb --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html @@ -0,0 +1,703 @@ + + + +Non-Member Properties + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Properties that are common to all distributions are accessed via non-member + getter functions. This allows more of these functions to be added over + time as the need arises. Unfortunately the literature uses many different + and confusing names to refer to a rather small number of actual concepts; + refer to the concept index to find + the property you want by the name you are most familiar with. Or use the + function index to go straight to + the function you want if you already know its name. +

+
+ + Function + Index +
+
+
+ + Conceptual + Index +
+
+
+ + Cumulative + Distribution Function +
+
+template <class RealType, class Policy>
+RealType cdf(const Distribution-Type<RealType, Policy>& dist, const RealType& x);
+
+

+ The Cumulative Distribution Function + is the probability that the variable takes a value less than or equal to + x. It is equivalent to the integral from -infinity to x of the Probability + Density Function. +

+

+ This function may return a domain_error + if the random variable is outside the defined range for the distribution. +

+

+ For example the following graph shows the cdf for the normal distribution: +

+

+ cdf +

+
+ + Complement + of the Cumulative Distribution Function +
+
+template <class Distribution, class RealType>
+RealType cdf(const Unspecified-Complement-Type<Distribution, RealType>& comp);
+
+

+ The complement of the Cumulative Distribution + Function is the probability that the variable takes a value greater + than x. It is equivalent to the integral from x to infinity of the Probability Density Function, or 1 minus + the Cumulative Distribution Function + of x. +

+

+ This is also known as the survival function. +

+

+ This function may return a domain_error + if the random variable is outside the defined range for the distribution. +

+

+ In this library, it is obtained by wrapping the arguments to the cdf function in a call to complement, for example: +

+
+// standard normal distribution object:
+boost::math::normal norm;
+// print survival function for x=2.0:
+std::cout << cdf(complement(norm, 2.0)) << std::endl;
+
+

+ For example the following graph shows the __complement of the cdf for the + normal distribution: +

+

+ survival +

+

+ See why complements? for why the + complement is useful and when it should be used. +

+
+ + Hazard Function +
+
+template <class RealType, class Policy>
+RealType hazard(const Distribution-Type<RealType, Policy>& dist, const RealType& x);
+
+

+ Returns the Hazard Function of + x and distibution dist. +

+

+ This function may return a domain_error + if the random variable is outside the defined range for the distribution. +

+

+ +

+
+ + + + + +
[Caution]Caution

+ Some authors refer to this as the conditional failure density function + rather than the hazard function. +

+
+ + Cumulative + Hazard Function +
+
+template <class RealType, class Policy>
+RealType chf(const Distribution-Type<RealType, Policy>& dist, const RealType& x);
+
+

+ Returns the Cumulative Hazard Function + of x and distibution dist. +

+

+ This function may return a domain_error + if the random variable is outside the defined range for the distribution. +

+

+ +

+
+ + + + + +
[Caution]Caution

+ Some authors refer to this as simply the "Hazard Function". +

+
+ + mean +
+
+template<class RealType, class Policy>
+RealType mean(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the mean of the distribution dist. +

+

+ This function may return a domain_error + if the distribution does not have a defined mean (for example the Cauchy + distribution). +

+
+ + median +
+
+template<class RealType, class Policy>
+RealType median(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the median of the distribution dist. +

+
+ + mode +
+
+template<class RealType, Policy>
+RealType mode(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the mode of the distribution dist. +

+

+ This function may return a domain_error + if the distribution does not have a defined mode. +

+
+ + Probability + Density Function +
+
+template <class RealType, class Policy>
+RealType pdf(const Distribution-Type<RealType, Policy>& dist, const RealType& x);
+
+

+ For a continuous function, the probability density function (pdf) returns + the probability that the variate has the value x. Since for continuous + distributions the probability at a single point is actually zero, the probability + is better expressed as the integral of the pdf between two points: see + the Cumulative Distribution Function. +

+

+ For a discrete distribution, the pdf is the probability that the variate + takes the value x. +

+

+ This function may return a domain_error + if the random variable is outside the defined range for the distribution. +

+

+ For example for a standard normal distribution the pdf looks like this: +

+

+ pdf +

+
+ + range +
+
+template<class RealType, class Policy>
+std::pair<RealType, RealType> range(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the valid range of the random variable over distribution dist. +

+
+ + Quantile +
+
+template <class RealType, class Policy>
+RealType quantile(const Distribution-Type<RealType, Policy>& dist, const RealType& p);
+
+

+ The quantile is best viewed as the inverse of the Cumulative + Distribution Function, it returns a value x + such that cdf(dist, x) == + p. +

+

+ This is also known as the percent point function, + or a percentile, it is also the same as calculating + the lower critical value of a distribution. +

+

+ This function returns a domain_error + if the probability lies outside [0,1]. The function may return an overflow_error if there is no finite value + that has the specified probability. +

+

+ The following graph shows the quantile function for a standard normal distribution: +

+

+ quantile +

+
+ + Quantile + from the complement of the probability. +
+

+ complements +

+
+template <class Distribution, class RealType>
+RealType quantile(const Unspecified-Complement-Type<Distribution, RealType>& comp);
+
+

+ This is the inverse of the Complement of + the Cumulative Distribution Function. It is calculated by wrapping + the arguments in a call to the quantile function in a call to complement. + For example: +

+
+// define a standard normal distribution:
+boost::math::normal norm;
+// print the value of x for which the complement
+// of the probability is 0.05:
+std::cout << quantile(complement(norm, 0.05)) << std::endl;
+
+

+ The function computes a value x such that cdf(complement(dist, x)) == q + where q is complement of the probability. +

+

+ Why complements? +

+

+ This function is also called the inverse survival function, and is the + same as calculating the upper critical value of a + distribution. +

+

+ This function returns a domain_error + if the probablity lies outside [0,1]. The function may return an overflow_error if there is no finite value + that has the specified probability. +

+

+ The following graph show the inverse survival function for the normal distribution: +

+

+ survival_inv +

+
+ + Standard + Deviation +
+
+template <class RealType, class Policy>
+RealType standard_deviation(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the standard deviation of distribution dist. +

+

+ This function may return a domain_error + if the distribution does not have a defined standard deviation. +

+
+ + support +
+
+template<class RealType, class Policy>
+std::pair<RealType, RealType> support(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the supported range of random variable over the distribution dist. +

+

+ The distribution is said to be 'supported' over a range that is "the smallest + closed set whose complement has probability zero". Non-mathematicians + might say it means the 'interesting' smallest range of random variate x + that has the cdf going from zero to unity. Outside are uninteresting zones + where the pdf is zero, and the cdf zero or unity. +

+
+ + Variance +
+
+template <class RealType, class Policy>
+RealType variance(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the variance of the distribution dist. +

+

+ This function may return a domain_error + if the distribution does not have a defined variance. +

+
+ + Skewness +
+
+template <class RealType, class Policy>
+RealType skewness(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the skewness of the distribution dist. +

+

+ This function may return a domain_error + if the distribution does not have a defined skewness. +

+
+ + Kurtosis +
+
+template <class RealType, class Policy>
+RealType kurtosis(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the 'proper' kurtosis (normalized fourth moment) of the distribution + dist. +

+

+ kertosis = β2= μ4 / μ22 +

+

+ Where μi is the i'th central moment of the distribution, and in particular + μ2 is the variance of the distribution. +

+

+ The kurtosis is a measure of the "peakedness" of a distribution. +

+

+ Note that the literature definition of kurtosis is confusing. The definition + used here is that used by for example Wolfram + MathWorld (that includes a table of formulae for kurtosis excess + for various distributions) but NOT the definition of kurtosis + used by Wikipedia which treats "kurtosis" and "kurtosis + excess" as the same quantity. +

+
+kurtosis_excess = 'proper' kurtosis - 3
+
+

+ This subtraction of 3 is convenient so that the kurtosis excess + of a normal distribution is zero. +

+

+ This function may return a domain_error + if the distribution does not have a defined kurtosis. +

+

+ 'Proper' kurtosis can have a value from zero to + infinity. +

+
+ + Kurtosis + excess +
+
+template <class RealType, Policy>
+RealType kurtosis_excess(const Distribution-Type<RealType, Policy>& dist);
+
+

+ Returns the kurtosis excess of the distribution dist. +

+

+ kurtosis excess = γ2= μ4 / μ22- 3 = kurtosis - 3 +

+

+ Where μi is the i'th central moment of the distribution, and in particular + μ2 is the variance of the distribution. +

+

+ The kurtosis excess is a measure of the "peakedness" of a distribution, + and is more widely used than the "kurtosis proper". It is defined + so that the kurtosis excess of a normal distribution is zero. +

+

+ This function may return a domain_error + if the distribution does not have a defined kurtosis excess. +

+

+ Kurtosis excess can have a value from -2 to + infinity. +

+
+kurtosis = kurtosis_excess +3;
+
+

+ The kurtosis excess of a normal distribution is zero. +

+
+ + P and Q +
+

+ The terms P and Q are sometimes used to refer to the Cumulative + Distribution Function and its complement + respectively. Lowercase p and q are sometimes used to refer to the values + returned by these functions. +

+
+ + Percent + Point Function +
+

+ The percent point function, also known as the percentile, is the same as + the Quantile. +

+
+ + Inverse + CDF Function. +
+

+ The inverse of the cumulative distribution function, is the same as the + Quantile. +

+
+ + Inverse + Survival Function. +
+

+ The inverse of the survival function, is the same as computing the quantile from the complement of the probability. +

+
+ + Probability + Mass Function +
+

+ The Probability Mass Function is the same as the Probability + Density Function. +

+

+ The term Mass Function is usually applied to discrete distributions, while + the term Probability Density Function + applies to continuous distributions. +

+
+ + Lower + Critical Value. +
+

+ The lower critical value calculates the value of the random variable given + the area under the left tail of the distribution. It is equivalent to calculating + the Quantile. +

+
+ + Upper + Critical Value. +
+

+ The upper critical value calculates the value of the random variable given + the area under the right tail of the distribution. It is equivalent to + calculating the quantile from the + complement of the probability. +

+
+ + Survival + Function +
+

+ Refer to the Complement of the Cumulative + Distribution Function. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/future.html b/doc/sf_and_dist/html/math_toolkit/dist/future.html new file mode 100644 index 000000000..0eb9f6d13 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/future.html @@ -0,0 +1,145 @@ + + + +Extras/Future Directions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Adding + Additional Location and Scale Parameters +
+

+ In some modelling applications we require a distribution with a specific + location and scale: often this equates to a specific mean and standard deviation, + although for many distributions the relationship between these properties + and the location and scale parameters are non-trivial. See http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm + for more information. +

+

+ The obvious way to handle this is via an adapter template: +

+
+template <class Dist>
+class scaled_distribution
+{
+   scaled_distribution(
+     const Dist dist, 
+     typename Dist::value_type location,
+     typename Dist::value_type scale = 0);
+};
+
+

+ Which would then have its own set of overloads for the non-member accessor + functions. +

+
+ + An + "any_distribution" class +
+

+ It would be fairly trivial to add a distribution object that virtualises + the actual type of the distribution, and can therefore hold "any" + object that conforms to the conceptual requirements of a distribution: +

+
+template <class RealType>
+class any_distribution
+{
+public:
+   template <class Distribution>
+   any_distribution(const Distribution& d);
+};
+
+// Get the cdf of the underlying distribution:
+template <class RealType>
+RealType cdf(const any_distribution<RealType>& d, RealType x);
+// etc....
+
+

+ Such a class would facilitate the writing of non-template code that can function + with any distribution type. It's not clear yet whether there is a compelling + use case though. Possibly tests for goodness of fit might provide such a + use case: this needs more investigation. +

+
+ + Higher + Level Hypothesis Tests +
+

+ Higher-level tests roughly corresponding to the Mathematica + Hypothesis Tests package could be added reasonably easily, for example: +

+
+template <class InputIterator>
+typename std::iterator_traits<InputIterator>::value_type
+   test_equal_mean(
+     InputIterator a,
+     InputIterator b,
+     typename std::iterator_traits<InputIterator>::value_type expected_mean);
+
+

+ Returns the probability that the data in the sequence [a,b) has the mean + expected_mean. +

+
+ + Integration + With Statistical Accumulators +
+

+ Eric + Niebler's accumulator framework - also work in progress - provides + the means to calculate various statistical properties from experimental data. + There is an opportunity to integrate the statistical tests with this framework + at some later date: +

+
+// Define an accumulator, all required statistics to calculate the test
+// are calculated automatically:
+accumulator_set<double, features<tag::test_expected_mean> > acc(expected_mean=4);
+// Pass our data to the accumulator:
+acc = std::for_each(mydata.begin(), mydata.end(), acc);
+// Extract the result:
+double p = probability(acc);
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html new file mode 100644 index 000000000..4cb9d5570 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html @@ -0,0 +1,134 @@ + + + +Statistical Distributions Tutorial + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Overview
+
Worked Examples
+
+
+ Distribution Construction Example
+
Student's + t Distribution Examples
+
+
+ Calculating confidence intervals on the mean with the Students-t distribution
+
+ Testing a sample mean for difference from a "true" mean
+
+ Estimating how large a sample size would have to become in order to give + a significant Students-t test result with a single sample test
+
+ Comparing the means of two samples with the Students-t test
+
+ Comparing two paired samples with the Student's t distribution
+
+
Chi Squared + Distribution Examples
+
+
+ Confidence Intervals on the Standard Deviation
+
+ Chi-Square Test for the Standard Deviation
+
+ Estimating the Required Sample Sizes for a Chi-Square Test for the Standard + Deviation
+
+
F Distribution + Examples
+
Binomial + Distribution Examples
+
+
+ Binomial Coin-Flipping Example
+
+ Binomial Quiz Example
+
+ Calculating Confidence Limits on the Frequency of Occurrence for a Binomial + Distribution
+
+ Estimating Sample Sizes for a Binomial Distribution.
+
+
Negative + Binomial Distribution Examples
+
+
+ Calculating Confidence Limits on the Frequency of Occurrence for the + Negative Binomial Distribution
+
+ Estimating Sample Sizes for the Negative Binomial.
+
+ Negative Binomial Sales Quota Example.
+
+ Negative Binomial Table Printing Example.
+
+
Normal + Distribution Examples
+
+ Some Miscellaneous Examples of the Normal (Gaussian) Distribution
+
Error Handling + Example
+
Find Location + and Scale Examples
+
+
+ Find Location (Mean) Example
+
+ Find Scale (Standard Deviation) Example
+
+ Find mean and standard deviation example
+
+
Comparison + with C, R, FORTRAN-style Free Functions
+
+
Random Variates + and Distribution Parameters
+
Discrete Probability + Distributions
+
+

+ This library is centred around statistical distributions, this tutorial will + give you an overview of what they are, how they can be used, and provides + a few worked examples of applying the library to statistical tests. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/dist_params.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/dist_params.html new file mode 100644 index 000000000..a5ab02f83 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/dist_params.html @@ -0,0 +1,98 @@ + + + +Discrete Probability Distributions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Note that the discrete + distributions, including the binomial, negative binomial, Poisson + & Bernoulli, are all mathematically defined as discrete functions: + only integral values of the random variate are envisaged and the functions + are only defined at these integral values. However because the method of + calculation often uses continuous functions, it is convenient to treat + them as if they were continuous functions, and permit non-integral values + of their parameters. +

+

+ To enforce a strict mathematical model, users may use floor or ceil functions + on the random variate, prior to calling the distribution function, to enforce + integral values. +

+

+ For similar reasons, in continuous distributions, parameters like degrees + of freedom that might appear to be integral, are treated as real values + (and are promoted from integer to floating-point if necessary). In this + case however, that there are a small number of situations where non-integral + degrees of freedom do have a genuine meaning. +

+

+ Generally speaking there is no loss of performance from allowing real-values + parameters: the underlying special functions contain optimizations for + integer-valued arguments when applicable. +

+
+ + + + + +
[Caution]Caution
+

+ The quantile function of a discrete distribution will by default return + an integer result that has been rounded outwards. + That is to say lower quantiles (where the probability is less than 0.5) + are rounded downward, and upper quantiles (where the probability is greater + than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile + is requested, then at least the requested coverage + will be present in the central region, and no more than + the requested coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are rounded + differently, or even return a real-valued result using Policies. + It is strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on a discrete distribution. The reference + docs describe how to change the rounding policy for these distributions. +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/overview.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/overview.html new file mode 100644 index 000000000..ebc337527 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/overview.html @@ -0,0 +1,531 @@ + + + +Overview + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Headers + and Namespaces +
+

+ All the code in this library is inside namespace boost::math. +

+

+ In order to use a distribution my_distribution you + will need to include either the header <boost/math/distributions/my_distribution.hpp> + or the "include everything" header: <boost/math/distributions.hpp>. +

+

+ For example, to use the Students-t distribution include either <boost/math/distributions/students_t.hpp> + or <boost/math/distributions.hpp> +

+
+ + Distributions + are Objects +
+

+ Each kind of distribution in this library is a class type. +

+

+ Policies provide fine-grained + control of the behaviour of these classes, allowing the user to customise + behaviour such as how errors are handled, or how the quantiles of discrete + distribtions behave. +

+
+ + + + + +
[Tip]Tip

+ If you are familiar with statistics libraries using functions, and 'Distributions + as Objects' seem alien, see the + comparison to other statistics libraries. +

+

+ Making distributions class types does two things: +

+
    +
  • + It encapsulates the kind of distribution in the C++ type system; so, + for example, Students-t distributions are always a different C++ type + from Chi-Squared distributions. +
    • +
  • + The distribution objects store any parameters associated with the distribution: + for example, the Students-t distribution has a degrees of freedom + parameter that controls the shape of the distribution. This degrees + of freedom parameter has to be provided to the Students-t + object when it is constructed. +
    • +
+

+ Although the distribution classes in this library are templates, there + are typedefs on type double that mostly take the usual + name of the distribution (except where there is a clash with a function + of the same name: beta and gamma, in which case using the default template + arguments - RealType = + double - is nearly as convenient). + Probably 95% of uses are covered by these typedefs: +

+
+using namespace boost::math;
+
+// Construct a students_t distribution with 4 degrees of freedom:
+students_t d1(4);
+
+// Construct a double-precision beta distribution 
+// with parameters a = 10, b = 20
+beta_distribution<> d2(10, 20); // Note: _distribution<> suffix !
+
+

+ If you need to use the distributions with a type other than double, then you can instantiate the template + directly: the names of the templates are the same as the double typedef but with _distribution + appended, for example: Students + t Distribution or Binomial + Distribution: +

+
+// Construct a students_t distribution, of float type,
+// with 4 degrees of freedom:
+students_t_distribution<float> d3(4);
+
+// Construct a binomial distribution, of long double type,
+// with probability of success 0.3
+// and 20 trials in total:
+binomial_distribution<long double> d4(20, 0.3);
+
+

+ The parameters passed to the distributions can be accessed via getter member + functions: +

+
+d1.degrees_of_freedom();  // returns 4.0 
+
+

+ This is all well and good, but not very useful so far. What we often want + is to be able to calculate the cumulative distribution functions + and quantiles etc for these distributions. +

+
+ + Generic + operations common to all distributions are non-member functions +
+

+ Want to calculate the PDF (Probability Density Function) of a distribution? + No problem, just use: +

+
+pdf(my_dist, x);  // Returns PDF (density) at point x of distribution my_dist.
+
+

+ Or how about the CDF (Cumulative Distribution Function): +

+
+cdf(my_dist, x);  // Returns CDF (integral from -infinity to point x)
+                  // of distribution my_dist.
+
+

+ And quantiles are just the same: +

+
+quantile(my_dist, p);  // Returns the value of the random variable x
+                       // such that cdf(my_dist, x) == p.
+
+

+ If you're wondering why these aren't member functions, it's to make the + library more easily extensible: if you want to add additional generic operations + - let's say the n'th moment - then all you have to + do is add the appropriate non-member functions, overloaded for each implemented + distribution type. +

+
+ + + + + +
[Tip]Tip
+

+

+

+ Random numbers that approximate Quantiles of Distributions +

+

+ If you want random numbers that are distributed in a specific way, for + example in a uniform, normal or triangular, see Boost.Random. +

+

+ Whilst in principal there's nothing to prevent you from using the quantile + function to convert a uniformly distributed random number to another + distribution, in practice there are much more efficient algorithms available + that are specific to random number generation. +

+
+

+ For example, the binomial distribution has two parameters: n (the number + of trials) and p (the probability of success on one trial). +

+

+ The binomial_distribution + constructor therefore has two parameters: +

+

+ binomial_distribution(RealType n, RealType + p); +

+

+ For this distribution the random variate is k: the number of successes + observed. The probability density/mass function (pdf) is therefore written + as f(k; n, p). +

+
+ + + + + +
[Note]Note
+

+

+

+ Random Variates and Distribution Parameters +

+

+ Random variates + and distribution + parameters are conventionally distinguished (for example in Wikipedia + and Wolfram MathWorld by placing a semi-colon (or sometimes vertical + bar) after the random variate (whose value you 'choose'), to separate + the variate from the parameter(s) that defines the shape of the distribution. +

+
+

+ As noted above the non-member function pdf + has one parameter for the distribution object, and a second for the random + variate. So taking our binomial distribution example, we would write: +

+

+ pdf(binomial_distribution<RealType>(n, p), k); +

+

+ The distribution (effectively the random variate) is said to be 'supported' + over a range that is "the + smallest closed set whose complement has probability zero". + MathWorld uses the word 'defined' for this range. Non-mathematicians might + say it means the 'interesting' smallest range of random variate x that + has the cdf going from zero to unity. Outside are uninteresting zones where + the pdf is zero, and the cdf zero or unity. Mathematically, the functions + may make sense with an (+ or -) infinite value, but except for a few special + cases (in the Normal and Cauchy distributions) this implementation limits + random variates to finite values from the max + to min for the RealType. (See Handling + of Floating-Point Infinity for rationale). +

+

+ The range of random variate values that is permitted and supported can + be tested by using two functions range + and support. +

+
+ + + + + +
[Note]Note
+

+

+

+ Discrete Probability Distributions +

+

+ Note that the discrete + distributions, including the binomial, negative binomial, Poisson + & Bernoulli, are all mathematically defined as discrete functions: + that is to say the functions cdf + and pdf are only defined + for integral values of the random variate. +

+

+ However, because the method of calculation often uses continuous functions + it is convenient to treat them as if they were continuous functions, + and permit non-integral values of their parameters. +

+

+ Users wanting to enforce a strict mathematical model may use floor or ceil + functions on the random variate prior to calling the distribution function. +

+

+ The quantile functions for these distributions are hard to specify in + a manner that will satisfy everyone all of the time. The default behaviour + is to return an integer result, that has been rounded outwards: + that is to say, lower quantiles - where the probablity is less than 0.5 + are rounded down, while upper quantiles - where the probability is greater + than 0.5 - are rounded up. This behaviour ensures that if an X% quantile + is requested, then at least the requested coverage + will be present in the central region, and no more than + the requested coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are rounded + differently, or return a real-valued result using Policies. + It is strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on a discrete distribtion. The reference + docs describe how to change the rounding policy for these distributions. +

+

+ For similar reasons continuous distributions with parameters like "degrees + of freedom" that might appear to be integral, are treated as real + values (and are promoted from integer to floating-point if necessary). + In this case however, there are a small number of situations where non-integral + degrees of freedom do have a genuine meaning. +

+
+

+

+
+ + Complements + are supported too +
+

+ Often you don't want the value of the CDF, but its complement, which is + to say 1-p rather than p. + You could calculate the CDF and subtract it from 1, + but if p is very close + to 1 then cancellation error + will cause you to lose significant digits. In extreme cases, p may actually be equal to 1, even though the true value of the complement + is non-zero. +

+

+ See also "Why complements?" +

+

+ In this library, whenever you want to receive a complement, just wrap all + the function arguments in a call to complement(...), for example: +

+
+students_t dist(5);
+cout << "CDF at t = 1 is " << cdf(dist, 1.0) << endl;
+cout << "Complement of CDF at t = 1 is " << cdf(complement(dist, 1.0)) << endl;
+
+

+ But wait, now that we have a complement, we have to be able to use it as + well. Any function that accepts a probability as an argument can also accept + a complement by wrapping all of its arguments in a call to complement(...), + for example: +

+
+students_t dist(5);
+
+for(double i = 10; i < 1e10; i *= 10)
+{
+   // Calculate the quantile for a 1 in i chance:
+   double t = quantile(complement(dist, 1/i));
+   // Print it out:
+   cout << "Quantile of students-t with 5 degrees of freedom\n"
+           "for a 1 in " << i << " chance is " << t << endl;
+}
+
+
+ + + + + +
[Tip]Tip
+

+

+

+ Critical values are just quantiles +

+

+ Some texts talk about quantiles, others about critical values, the basic + rule is: +

+

+ Lower critical values are the same as the quantile. +

+

+ Upper critical values are the same as the quantile + from the complement of the probability. +

+

+ For example, suppose we have a Bernoulli process, giving rise to a binomial + distribution with success ratio 0.1 and 100 trials in total. The lower + critical value for a probability of 0.05 is given by: +

+

+ quantile(binomial(100, 0.1), 0.05) +

+

+ and the upper critical value is given by: +

+

+ quantile(complement(binomial(100, 0.1), 0.05)) +

+

+ which return 4.82 and 14.63 respectively. +

+
+

+

+
+ + + + + +
[Tip]Tip
+

+

+

+ Why bother with complements anyway? +

+

+ It's very tempting to dispense with complements, and simply subtract + the probability from 1 when required. However, consider what happens + when the probability is very close to 1: let's say the probability expressed + at float precision is 0.999999940f, + then 1 - + 0.999999940f = + 5.96046448e-008, but the result + is actually accurate to just one single bit: the + only bit that didn't cancel out! +

+

+ Or to look at this another way: consider that we want the risk of falsely + rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion, + for a sample size of 10,000. This gives a probability of 1 - 10-9, which + is exactly 1 when calculated at float precision. In this case calculating + the quantile from the complement neatly solves the problem, so for example: +

+

+ quantile(complement(students_t(10000), 1e-9)) +

+

+ returns the expected t-statistic 6.00336, + where as: +

+

+ quantile(students_t(10000), 1-1e-9f) +

+

+ raises an overflow error, since it is the same as: +

+

+ quantile(students_t(10000), 1) +

+

+ Which has no finite result. +

+
+
+ + Parameters + can be calculated +
+

+ Sometimes it's the parameters that define the distribution that you need + to find. Suppose, for example, you have conducted a Students-t test for + equal means and the result is borderline. Maybe your two samples differ + from each other, or maybe they don't; based on the result of the test you + can't be sure. A legitimate question to ask then is "How many more + measurements would I have to take before I would get an X% probability + that the difference is real?" Parameter finders can answer questions + like this, and are necessarily different for each distribution. They are + implemented as static member functions of the distributions, for example: +

+
+students_t::find_degrees_of_freedom(
+   1.3,        // difference from true mean to detect
+   0.05,       // maximum risk of falsely rejecting the null-hypothesis.
+   0.1,        // maximum risk of falsely failing to reject the null-hypothesis.
+   0.13);      // sample standard deviation
+
+

+ Returns the number of degrees of freedom required to obtain a 95% probability + that the observed differences in means is not down to chance alone. In + the case that a borderline Students-t test result was previously obtained, + this can be used to estimate how large the sample size would have to become + before the observed difference was considered significant. It assumes, + of course, that the sample mean and standard deviation are invariant with + sample size. +

+
+ + Summary +
+
    +
  • + Distributions are objects, which are constructed from whatever parameters + the distribution may have. +
    • +
  • + Member functions allow you to retrieve the parameters of a distribution. +
    • +
  • + Generic non-member functions provide access to the properties that are + common to all the distributions (PDF, CDF, quantile etc). +
    • +
  • + Complements of probabilities are calculated by wrapping the function's + arguments in a call to complement(...). +
    • +
  • + Functions that accept a probability can accept a complement of the probability + as well, by wrapping the function's arguments in a call to complement(...). +
    • +
  • + Static member functions allow the parameters of a distribution to be + found from other information. +
    • +
+

+ Now that you have the basics, the next section looks at some worked examples. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/variates.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/variates.html new file mode 100644 index 000000000..65b72ddbd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/variates.html @@ -0,0 +1,76 @@ + + + +Random Variates and Distribution Parameters + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Random variates + and distribution parameters + are conventionally distinguished (for example in Wikipedia and Wolfram + MathWorld by placing a semi-colon after the random variate (whose value + you 'choose'), to separate the variate from the parameter(s) that defines + the shape of the distribution. +

+

+ For example, the binomial distribution has two parameters: n (the number + of trials) and p (the probability of success on one trial). It also has + the random variate k: the number of successes observed. + This means the probability density/mass function (pdf) is written as f(k; + n, p). +

+

+ Translating this into code the binomial_distribution + constructor therefore has two parameters: +

+
+binomial_distribution(RealType n, RealType p);
+
+

+ While the function pdf + has one argument specifying the distribution type (which includes it's + parameters, if any), and a second argument for the random variate. So taking + our binomial distribution example, we would write: +

+
+pdf(binomial_distribution<RealType>(n, p), k);
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html new file mode 100644 index 000000000..e68359f8b --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html @@ -0,0 +1,124 @@ + + + +Worked Examples + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ Distribution Construction Example
+
Student's + t Distribution Examples
+
+
+ Calculating confidence intervals on the mean with the Students-t distribution
+
+ Testing a sample mean for difference from a "true" mean
+
+ Estimating how large a sample size would have to become in order to give + a significant Students-t test result with a single sample test
+
+ Comparing the means of two samples with the Students-t test
+
+ Comparing two paired samples with the Student's t distribution
+
+
Chi Squared + Distribution Examples
+
+
+ Confidence Intervals on the Standard Deviation
+
+ Chi-Square Test for the Standard Deviation
+
+ Estimating the Required Sample Sizes for a Chi-Square Test for the Standard + Deviation
+
+
F Distribution + Examples
+
Binomial + Distribution Examples
+
+
+ Binomial Coin-Flipping Example
+
+ Binomial Quiz Example
+
+ Calculating Confidence Limits on the Frequency of Occurrence for a Binomial + Distribution
+
+ Estimating Sample Sizes for a Binomial Distribution.
+
+
Negative + Binomial Distribution Examples
+
+
+ Calculating Confidence Limits on the Frequency of Occurrence for the + Negative Binomial Distribution
+
+ Estimating Sample Sizes for the Negative Binomial.
+
+ Negative Binomial Sales Quota Example.
+
+ Negative Binomial Table Printing Example.
+
+
Normal + Distribution Examples
+
+ Some Miscellaneous Examples of the Normal (Gaussian) Distribution
+
Error Handling + Example
+
Find Location + and Scale Examples
+
+
+ Find Location (Mean) Example
+
+ Find Scale (Standard Deviation) Example
+
+ Find mean and standard deviation example
+
+
Comparison + with C, R, FORTRAN-style Free Functions
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg.html new file mode 100644 index 000000000..cdddbac0d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg.html @@ -0,0 +1,60 @@ + + + +Binomial Distribution Examples + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ Binomial Coin-Flipping Example
+
+ Binomial Quiz Example
+
+ Calculating Confidence Limits on the Frequency of Occurrence for a Binomial + Distribution
+
+ Estimating Sample Sizes for a Binomial Distribution.
+
+

+ See also the reference documentation for the Binomial + Distribution. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html new file mode 100644 index 000000000..37a91f07f --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html @@ -0,0 +1,246 @@ + + + +Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution + + + + + + + + + + + + + + + +
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+
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+
+
+

+ Imagine you have a process that follows a binomial distribution: for + each trial conducted, an event either occurs or does it does not, referred + to as "successes" and "failures". If, by experiment, + you want to measure the frequency with which successes occur, the best + estimate is given simply by k / N, + for k successes out of N + trials. However our confidence in that estimate will be shaped by how + many trials were conducted, and how many successes were observed. The + static member functions binomial_distribution<>::find_lower_bound_on_p + and binomial_distribution<>::find_upper_bound_on_p + allow you to calculate the confidence intervals for your estimate of + the occurrence frequency. +

+

+ The sample program binomial_confidence_limits.cpp + illustrates their use. It begins by defining a procedure that will + print a table of confidence limits for various degrees of certainty: +

+
+#include <iostream>
+#include <iomanip>
+#include <boost/math/distributions/binomial.hpp>
+
+void confidence_limits_on_frequency(unsigned trials, unsigned successes)
+{
+   //
+   // trials = Total number of trials.
+   // successes = Total number of observed successes.
+   //
+   // Calculate confidence limits for an observed
+   // frequency of occurrence that follows a binomial
+   // distribution.
+   //
+   using namespace std;
+   using namespace boost::math;
+
+   // Print out general info:
+   cout <<
+      "___________________________________________\n"
+      "2-Sided Confidence Limits For Success Ratio\n"
+      "___________________________________________\n\n";
+   cout << setprecision(7);
+   cout << setw(40) << left << "Number of Observations" << "=  " << trials << "\n";
+   cout << setw(40) << left << "Number of successes" << "=  " << successes << "\n";
+   cout << setw(40) << left << "Sample frequency of occurrence" << "=  " << double(successes) / trials << "\n";
+
+

+ The procedure now defines a table of significance levels: these are + the probabilities that the true occurrence frequency lies outside the + calculated interval: +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ Some pretty printing of the table header follows: +

+
+cout << "\n\n"
+        "_______________________________________________________________________\n"
+        "Confidence        Lower CP       Upper CP       Lower JP       Upper JP\n"
+        " Value (%)        Limit          Limit          Limit          Limit\n"
+        "_______________________________________________________________________\n";
+
+

+ And now for the important part - the intervals themselves - for each + value of alpha, we call find_lower_bound_on_p + and find_lower_upper_on_p + to obtain lower and upper bounds respectively. Note that since we are + calculating a two-sided interval, we must divide the value of alpha + in two. +

+

+ Please note that calculating two separate single sided bounds, + each with risk level αis not the same thing as calculating a two sided + interval. Had we calculate two single-sided intervals each with a risk + that the true value is outside the interval of α, then: +

+
  • + The risk that it is less than the lower bound is α. +
+

+ and +

+
  • + The risk that it is greater than the upper bound is also α. +
+

+ So the risk it is outside upper or lower bound, + is twice alpha, and the probability + that it is inside the bounds is therefore not nearly as high as one + might have thought. This is why α/2 must be used in the calculations + below. +

+

+ In contrast, had we been calculating a single-sided interval, for example: + "Calculate a lower bound so that we are P% sure that + the true occurrence frequency is greater than some value" + then we would not have divided by + two. +

+

+ Finally note that binomial_distribution + provides a choice of two methods for the calculation, we print out + the results from both methods in this example: +

+
+   for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+   {
+      // Confidence value:
+      cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+      // Calculate Clopper Pearson bounds:
+      double l = binomial_distribution<>::find_lower_bound_on_p(
+                     trials, successes, alpha[i]/2);
+      double u = binomial_distribution<>::find_upper_bound_on_p(
+                     trials, successes, alpha[i]/2);
+      // Print Clopper Pearson Limits:
+      cout << fixed << setprecision(5) << setw(15) << right << l;
+      cout << fixed << setprecision(5) << setw(15) << right << u;
+      // Calculate Jeffreys Prior Bounds:
+      l = binomial_distribution<>::find_lower_bound_on_p(
+            trials, successes, alpha[i]/2, 
+            binomial_distribution<>::jeffreys_prior_interval);
+      u = binomial_distribution<>::find_upper_bound_on_p(
+            trials, successes, alpha[i]/2, 
+            binomial_distribution<>::jeffreys_prior_interval);
+      // Print Jeffreys Prior Limits:
+      cout << fixed << setprecision(5) << setw(15) << right << l;
+      cout << fixed << setprecision(5) << setw(15) << right << u << std::endl;
+   }
+   cout << endl;
+}
+
+

+ And that's all there is to it. Let's see some sample output for a 2 + in 10 success ratio, first for 20 trials: +

+
___________________________________________
+2-Sided Confidence Limits For Success Ratio
+___________________________________________
+
+Number of Observations                  =  20
+Number of successes                     =  4
+Sample frequency of occurrence          =  0.2
+
+
+_______________________________________________________________________
+Confidence        Lower CP       Upper CP       Lower JP       Upper JP
+ Value (%)        Limit          Limit          Limit          Limit
+_______________________________________________________________________
+    50.000        0.12840        0.29588        0.14974        0.26916
+    75.000        0.09775        0.34633        0.11653        0.31861
+    90.000        0.07135        0.40103        0.08734        0.37274
+    95.000        0.05733        0.43661        0.07152        0.40823
+    99.000        0.03576        0.50661        0.04655        0.47859
+    99.900        0.01905        0.58632        0.02634        0.55960
+    99.990        0.01042        0.64997        0.01530        0.62495
+    99.999        0.00577        0.70216        0.00901        0.67897
+
+

+ As you can see, even at the 95% confidence level the bounds are really + quite wide (this example is chosen to be easily compared to the one + in the NIST/SEMATECH + e-Handbook of Statistical Methods. here). + Note also that the Clopper-Pearson calculation method (CP above) produces + quite noticeably more pessimistic estimates than the Jeffreys Prior + method (JP above). +

+

+ Compare that with the program output for 2000 trials: +

+
___________________________________________
+2-Sided Confidence Limits For Success Ratio
+___________________________________________
+
+Number of Observations                  =  2000
+Number of successes                     =  400
+Sample frequency of occurrence          =  0.2000000
+
+
+_______________________________________________________________________
+Confidence        Lower CP       Upper CP       Lower JP       Upper JP
+ Value (%)        Limit          Limit          Limit          Limit
+_______________________________________________________________________
+    50.000        0.19382        0.20638        0.19406        0.20613
+    75.000        0.18965        0.21072        0.18990        0.21047
+    90.000        0.18537        0.21528        0.18561        0.21503
+    95.000        0.18267        0.21821        0.18291        0.21796
+    99.000        0.17745        0.22400        0.17769        0.22374
+    99.900        0.17150        0.23079        0.17173        0.23053
+    99.990        0.16658        0.23657        0.16681        0.23631
+    99.999        0.16233        0.24169        0.16256        0.24143
+
+

+ Now even when the confidence level is very high, the limits are really + quite close to the experimentally calculated value of 0.2. Furthermore + the difference between the two calculation methods is now really quite + small. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html new file mode 100644 index 000000000..4050311a0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html @@ -0,0 +1,161 @@ + + + +Estimating Sample Sizes for a Binomial Distribution. + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Imagine you have a critical component that you know will fail in 1 + in N "uses" (for some suitable definition of "use"). + You may want to schedule routine replacement of the component so that + its chance of failure between routine replacements is less than P%. + If the failures follow a binomial distribution (each time the component + is "used" it either fails or does not) then the static member + function binomial_distibution<>::find_maximum_number_of_trials + can be used to estimate the maximum number of "uses" of that + component for some acceptable risk level alpha. +

+

+ The example program binomial_sample_sizes.cpp + demonstrates its usage. It centres on a routine that prints out a table + of maximum sample sizes for various probability thresholds: +

+
+void find_max_sample_size(
+   double p,              // success ratio.
+   unsigned successes)    // Total number of observed successes permitted.
+{
+
+

+ The routine then declares a table of probability thresholds: these + are the maximum acceptable probability that successes + or fewer events will be observed. In our example, successes + will be always zero, since we want no component failures, but in other + situations non-zero values may well make sense. +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ Much of the rest of the program is pretty-printing, the important part + is in the calculation of maximum number of permitted trials for each + value of alpha: +

+
+for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+{
+   // Confidence value:
+   cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+   // calculate trials:
+   double t = binomial::find_maximum_number_of_trials(
+                  successes, p, alpha[i]);
+   t = floor(t);
+   // Print Trials:
+   cout << fixed << setprecision(5) << setw(15) << right << t << endl;
+}
+
+

+ Note that since we're calculating the maximum number of trials permitted, + we'll err on the safe side and take the floor of the result. Had we + been calculating the minimum number of trials + required to observe a certain number of successes + using find_minimum_number_of_trials + we would have taken the ceiling instead. +

+

+ We'll finish off by looking at some sample output, firstly for a 1 + in 1000 chance of component failure with each use: +

+
________________________
+Maximum Number of Trials
+________________________
+
+Success ratio                           =  0.001
+Maximum Number of "successes" permitted =  0
+
+
+____________________________
+Confidence        Max Number
+ Value (%)        Of Trials
+____________________________
+    50.000            692
+    75.000            287
+    90.000            105
+    95.000             51
+    99.000             10
+    99.900              0
+    99.990              0
+    99.999              0
+
+

+ So 51 "uses" of the component would yield a 95% chance that + no component failures would be observed. +

+

+ Compare that with a 1 in 1 million chance of component failure: +

+
________________________
+Maximum Number of Trials
+________________________
+
+Success ratio                           =  0.0000010
+Maximum Number of "successes" permitted =  0
+
+
+____________________________
+Confidence        Max Number
+ Value (%)        Of Trials
+____________________________
+    50.000         693146
+    75.000         287681
+    90.000         105360
+    95.000          51293
+    99.000          10050
+    99.900           1000
+    99.990            100
+    99.999             10
+
+

+ In this case, even 1000 uses of the component would still yield a less + than 1 in 1000 chance of observing a component failure (i.e. a 99.9% + chance of no failure). +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html new file mode 100644 index 000000000..6ddc49d7d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html @@ -0,0 +1,420 @@ + + + +Binomial Coin-Flipping Example + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+

+

+ An example of a Bernoulli + process is coin flipping. A variable in such a sequence may + be called a Bernoulli variable. +

+

+

+

+ This example shows using the Binomial distribution to predict the + probability of heads and tails when throwing a coin. +

+

+

+

+ The number of correct answers (say heads), X, is distributed as a + binomial random variable with binomial distribution parameters number + of trials (flips) n = 10 and probability (success_fraction) of getting + a head p = 0.5 (a 'fair' coin). +

+

+

+

+ (Our coin is assumed fair, but we could easily change the success_fraction + parameter p from 0.5 to some other value to simulate an unfair coin, + say 0.6 for one with chewing gum on the tail, so it is more likely + to fall tails down and heads up). +

+

+

+

+ First we need some includes and using statements to be able to use + the binomial distribution, some std input and output, and get started: +

+

+

+

+ +

+
+#include <boost/math/distributions/binomial.hpp>
+  using boost::math::binomial;
+
+#include <iostream>
+  using std::cout;  using std::endl;  using std::left;
+#include <iomanip>
+  using std::setw;
+
+int main()
+{
+  cout << "Using Binomial distribution to predict how many heads and tails." << endl;
+  try
+  {
+

+

+

+

+

+ See note with the catch block + about why a try and catch block is always a good idea. +

+

+

+

+ First, construct a binomial distribution with parameters success_fraction + 1/2, and how many flips. +

+

+

+

+ +

+
+const double success_fraction = 0.5; // = 50% = 1/2 for a 'fair' coin.
+int flips = 10;
+binomial flip(flips, success_fraction);
+
+cout.precision(4);
+

+

+

+

+

+ Then some examples of using Binomial moments (and echoing the parameters). +

+

+

+

+ +

+
+cout << "From " << flips << " one can expect to get on average "
+  << mean(flip) << " heads (or tails)." << endl;
+cout << "Mode is " << mode(flip) << endl;
+cout << "Standard deviation is " << standard_deviation(flip) << endl;
+cout << "So about 2/3 will lie within 1 standard deviation and get between "
+  <<  ceil(mean(flip) - standard_deviation(flip))  << " and "
+  << floor(mean(flip) + standard_deviation(flip)) << " correct." << endl;
+cout << "Skewness is " << skewness(flip) << endl;
+// Skewness of binomial distributions is only zero (symmetrical)
+// if success_fraction is exactly one half,
+// for example, when flipping 'fair' coins.
+cout << "Skewness if success_fraction is " << flip.success_fraction()
+  << " is " << skewness(flip) << endl << endl; // Expect zero for a 'fair' coin.
+

+

+

+

+

+ Now we show a variety of predictions on the probability of heads: +

+

+

+

+ +

+
+cout << "For " << flip.trials() << " coin flips: " << endl;
+cout << "Probability of getting no heads is " << pdf(flip, 0) << endl;
+cout << "Probability of getting at least one head is " << 1. - pdf(flip, 0) << endl;
+

+

+

+

+

+ When we want to calculate the probability for a range or values we + can sum the PDF's: +

+

+

+

+ +

+
+cout << "Probability of getting 0 or 1 heads is "
+  << pdf(flip, 0) + pdf(flip, 1) << endl; // sum of exactly == probabilities
+

+

+

+

+

+ Or we can use the cdf. +

+

+

+

+ +

+
+cout << "Probability of getting 0 or 1 (<= 1) heads is " << cdf(flip, 1) << endl;
+cout << "Probability of getting 9 or 10 heads is " << pdf(flip, 9) + pdf(flip, 10) << endl;
+

+

+

+

+

+ Note that using +

+

+

+

+ +

+
+cout << "Probability of getting 9 or 10 heads is " << 1. - cdf(flip, 8) << endl;
+

+

+

+

+

+ is less accurate than using the complement +

+

+

+

+ +

+
+cout << "Probability of getting 9 or 10 heads is " << cdf(complement(flip, 8)) << endl;
+

+

+

+

+

+ Since the subtraction may involve cancellation + error, where as cdf(complement(flip, 8)) does not use such a subtraction + internally, and so does not exhibit the problem. +

+

+

+

+ To get the probability for a range of heads, we can either add the + pdfs for each number of heads +

+

+

+

+ +

+
+cout << "Probability of between 4 and 6 heads (4 or 5 or 6) is "
+  //  P(X == 4) + P(X == 5) + P(X == 6)
+  << pdf(flip, 4) + pdf(flip, 5) + pdf(flip, 6) << endl;
+

+

+

+

+

+ But this is probably less efficient than using the cdf +

+

+

+

+ +

+
+cout << "Probability of between 4 and 6 heads (4 or 5 or 6) is "
+  // P(X <= 6) - P(X <= 3) == P(X < 4)
+  << cdf(flip, 6) - cdf(flip, 3) << endl;
+

+

+

+

+

+ Certainly for a bigger range like, 3 to 7 +

+

+

+

+ +

+
+cout << "Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is "
+  // P(X <= 7) - P(X <= 2) == P(X < 3)
+  << cdf(flip, 7) - cdf(flip, 2) << endl;
+cout << endl;
+

+

+

+

+

+ Finally, print two tables of probability for the exactly + and at least a number of heads. +

+

+

+

+ +

+
+// Print a table of probability for the exactly a number of heads.
+cout << "Probability of getting exactly (==) heads" << endl;
+for (int successes = 0; successes <= flips; successes++)
+{ // Say success means getting a head (or equally success means getting a tail).
+  double probability = pdf(flip, successes);
+  cout << left << setw(2) << successes << "     " << setw(10)
+    << probability << " or 1 in " << 1. / probability
+    << ", or " << probability * 100. << "%" << endl;
+} // for i
+cout << endl;
+
+// Tabulate the probability of getting between zero heads and 0 upto 10 heads.
+cout << "Probability of getting upto (<=) heads" << endl;
+for (int successes = 0; successes <= flips; successes++)
+{ // Say success means getting a head
+  // (equally success could mean getting a tail).
+  double probability = cdf(flip, successes); // P(X <= heads)
+  cout << setw(2) << successes << "        " << setw(10) << left
+    << probability << " or 1 in " << 1. / probability << ", or "
+    << probability * 100. << "%"<< endl;
+} // for i
+

+

+

+

+

+ The last (0 to 10 heads) must, of course, be 100% probability. +

+

+

+

+ +

+
+}
+catch(const std::exception& e)
+{
+  //
+

+

+

+ +

+

+ It is always essential to include try & catch blocks because + default policies are to throw exceptions on arguments that are out + of domain or cause errors like numeric-overflow. +

+

+

+

+ Lacking try & catch blocks, the program will abort, whereas the + message below from the thrown exception will give some helpful clues + as to the cause of the problem. +

+

+

+

+ +

+
+  std::cout <<
+    "\n""Message from thrown exception was:\n   " << e.what() << std::endl;
+}
+
+

+

+

+

+

+ See binomial_coinflip_example.cpp + for full source code, the program output looks like this: +

+

+ +

+
Using Binomial distribution to predict how many heads and tails.
+From 10 one can expect to get on average 5 heads (or tails).
+Mode is 5
+Standard deviation is 1.581
+So about 2/3 will lie within 1 standard deviation and get between 4 and 6 correct.
+Skewness is 0
+Skewness if success_fraction is 0.5 is 0
+
+For 10 coin flips:
+Probability of getting no heads is 0.0009766
+Probability of getting at least one head is 0.999
+Probability of getting 0 or 1 heads is 0.01074
+Probability of getting 0 or 1 (<= 1) heads is 0.01074
+Probability of getting 9 or 10 heads is 0.01074
+Probability of getting 9 or 10 heads is 0.01074
+Probability of getting 9 or 10 heads is 0.01074
+Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6562
+Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6563
+Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is 0.8906
+
+Probability of getting exactly (=) heads
+0      0.0009766  or 1 in 1024, or 0.09766%
+1      0.009766   or 1 in 102.4, or 0.9766%
+2      0.04395    or 1 in 22.76, or 4.395%
+3      0.1172     or 1 in 8.533, or 11.72%
+4      0.2051     or 1 in 4.876, or 20.51%
+5      0.2461     or 1 in 4.063, or 24.61%
+6      0.2051     or 1 in 4.876, or 20.51%
+7      0.1172     or 1 in 8.533, or 11.72%
+8      0.04395    or 1 in 22.76, or 4.395%
+9      0.009766   or 1 in 102.4, or 0.9766%
+10     0.0009766  or 1 in 1024, or 0.09766%
+
+Probability of getting upto (<) heads
+0         0.0009766  or 1 in 1024, or 0.09766%
+1         0.01074    or 1 in 93.09, or 1.074%
+2         0.05469    or 1 in 18.29, or 5.469%
+3         0.1719     or 1 in 5.818, or 17.19%
+4         0.377      or 1 in 2.653, or 37.7%
+5         0.623      or 1 in 1.605, or 62.3%
+6         0.8281     or 1 in 1.208, or 82.81%
+7         0.9453     or 1 in 1.058, or 94.53%
+8         0.9893     or 1 in 1.011, or 98.93%
+9         0.999      or 1 in 1.001, or 99.9%
+10        1          or 1 in 1, or 100%
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html new file mode 100644 index 000000000..c61ed99ef --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html @@ -0,0 +1,770 @@ + + + +Binomial Quiz Example + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+

+

+ A multiple choice test has four possible answers to each of 16 questions. + A student guesses the answer to each question, so the probability + of getting a correct answer on any given question is one in four, + a quarter, 1/4, 25% or fraction 0.25. The conditions of the binomial + experiment are assumed to be met: n = 16 questions constitute the + trials; each question results in one of two possible outcomes (correct + or incorrect); the probability of being correct is 0.25 and is constant + if no knowledge about the subject is assumed; the questions are answered + independently if the student's answer to a question in no way influences + his/her answer to another question. +

+

+

+

+ First, we need to be able to use the binomial distribution constructor + (and some std input/output, of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/binomial.hpp>
+  using boost::math::binomial;
+
+#include <iostream>
+  using std::cout; using std::endl;
+  using std::ios; using std::flush; using std::left; using std::right; using std::fixed;
+#include <iomanip>
+  using std::setw; using std::setprecision;
+
+

+

+

+

+

+ The number of correct answers, X, is distributed as a binomial random + variable with binomial distribution parameters: questions n = 16 + and success fraction probability p = 0.25. So we construct a binomial + distribution: +

+

+

+

+ +

+
+int questions = 16; // All the questions in the quiz.
+int answers = 4; // Possible answers to each question.
+double success_fraction = (double)answers / (double)questions; // If a random guess.
+// Caution:  = answers / questions would be zero (because they are integers)!
+binomial quiz(questions, success_fraction);
+

+

+

+

+

+ and display the distribution parameters we used thus: +

+

+

+

+ +

+
+cout << "In a quiz with " << quiz.trials()
+  << " questions and with a probability of guessing right of "
+  << quiz.success_fraction() * 100 << " %" 
+  << " or 1 in " << static_cast<int>(1. / quiz.success_fraction()) << endl;
+

+

+

+

+

+ Show a few probabilities of just guessing: +

+

+

+

+ +

+
+cout << "Probability of getting none right is " << pdf(quiz, 0) << endl; // 0.010023
+cout << "Probability of getting exactly one right is " << pdf(quiz, 1) << endl;
+cout << "Probability of getting exactly two right is " << pdf(quiz, 2) << endl;
+int pass_score = 11;
+cout << "Probability of getting exactly " << pass_score << " answers right by chance is " 
+  << pdf(quiz, questions) << endl;
+

+

+

+ +

+
Probability of getting none right is 0.0100226
+Probability of getting exactly one right is 0.0534538
+Probability of getting exactly two right is 0.133635
+Probability of getting exactly 11 answers right by chance is 2.32831e-010
+
+

+

+

+ These don't give any encouragement to guessers! +

+

+

+

+ We can tabulate the 'getting exactly right' ( == ) probabilities + thus: +

+

+

+

+ +

+
+cout << "\n" "Guessed Probability" << right << endl;
+for (int successes = 0; successes <= questions; successes++)
+{
+  double probability = pdf(quiz, successes);
+  cout << setw(2) << successes << "      " << probability << endl;
+}
+cout << endl;
+

+

+

+ +

+
Guessed Probability
+ 0      0.0100226
+ 1      0.0534538
+ 2      0.133635
+ 3      0.207876
+ 4      0.225199
+ 5      0.180159
+ 6      0.110097
+ 7      0.0524273
+ 8      0.0196602
+ 9      0.00582526
+10      0.00135923
+11      0.000247132
+12      3.43239e-005
+13      3.5204e-006
+14      2.51457e-007
+15      1.11759e-008
+16      2.32831e-010
+
+

+

+

+ Then we can add the probabilities of some 'exactly right' like this: +

+

+

+

+ +

+
+cout << "Probability of getting none or one right is " << pdf(quiz, 0) + pdf(quiz, 1) << endl;
+

+

+

+ +

+
Probability of getting none or one right is 0.0634764
+
+

+

+

+ But if more than a couple of scores are involved, it is more convenient + (and may be more accurate) to use the Cumulative Distribution Function + (cdf) instead: +

+

+

+

+ +

+
+cout << "Probability of getting none or one right is " << cdf(quiz, 1) << endl;
+

+

+

+ +

+
Probability of getting none or one right is 0.0634764
+
+

+

+

+ Since the cdf is inclusive, we can get the probability of getting + up to 10 right ( <= ) +

+

+

+

+ +

+
+cout << "Probability of getting <= 10 right (to fail) is " << cdf(quiz, 10) << endl;
+

+

+

+ +

+
Probability of getting <= 10 right (to fail) is 0.999715
+
+

+

+

+ To get the probability of getting 11 or more right (to pass), it + is tempting to use +

+
+1 - cdf(quiz, 10)
+

+ to get the probability of > 10 +

+

+

+

+ +

+
+cout << "Probability of getting > 10 right (to pass) is " << 1 - cdf(quiz, 10) << endl;
+

+

+

+ +

+
Probability of getting > 10 right (to pass) is 0.000285239
+
+

+

+

+ But this should be resisted in favor of using the complement function. + Why complements? +

+

+

+

+ +

+
+cout << "Probability of getting > 10 right (to pass) is " << cdf(complement(quiz, 10)) << endl;
+

+

+

+ +

+
Probability of getting > 10 right (to pass) is 0.000285239
+
+

+

+

+ And we can check that these two, <= 10 and > 10, add up to + unity. +

+

+

+

+ +

+
+BOOST_ASSERT((cdf(quiz, 10) + cdf(complement(quiz, 10))) == 1.);
+

+

+

+

+

+ If we want a < rather than a <= test, because the CDF is inclusive, + we must subtract one from the score. +

+

+

+

+ +

+
+cout << "Probability of getting less than " << pass_score
+  << " (< " << pass_score << ") answers right by guessing is "
+  << cdf(quiz, pass_score -1) << endl;
+

+

+

+ +

+
Probability of getting less than 11 (< 11) answers right by guessing is 0.999715
+
+

+

+

+ and similarly to get a >= rather than a > test we also need + to subtract one from the score (and can again check the sum is unity). + This is because if the cdf is inclusive, then + its complement must be exclusive otherwise there + would be one possible outcome counted twice! +

+

+

+

+ +

+
+cout << "Probability of getting at least " << pass_score 
+  << "(>= " << pass_score << ") answers right by guessing is "
+  << cdf(complement(quiz, pass_score-1))
+  << ", only 1 in " << 1/cdf(complement(quiz, pass_score-1)) << endl;
+
+BOOST_ASSERT((cdf(quiz, pass_score -1) + cdf(complement(quiz, pass_score-1))) == 1);
+

+

+

+ +

+
Probability of getting at least 11 (>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83
+
+

+

+

+ Finally we can tabulate some probabilities: +

+

+

+

+ +

+
+cout << "\n" "At most (<=)""\n""Guessed OK   Probability" << right << endl;
+for (int score = 0; score <= questions; score++)
+{
+  cout << setw(2) << score << "           " << setprecision(10)
+    << cdf(quiz, score) << endl;
+}
+cout << endl;
+

+

+

+ +

+
At most (<=)
+Guessed OK   Probability
+ 0           0.01002259576
+ 1           0.0634764398
+ 2           0.1971110499
+ 3           0.4049871101
+ 4           0.6301861752
+ 5           0.8103454274
+ 6           0.9204427481
+ 7           0.9728700437
+ 8           0.9925302796
+ 9           0.9983555346
+10           0.9997147608
+11           0.9999618928
+12           0.9999962167
+13           0.9999997371
+14           0.9999999886
+15           0.9999999998
+16           1
+
+

+

+

+ +

+
+cout << "\n" "At least (>)""\n""Guessed OK   Probability" << right << endl;
+for (int score = 0; score <= questions; score++)
+{
+  cout << setw(2) << score << "           "  << setprecision(10)
+    << cdf(complement(quiz, score)) << endl;
+}
+

+

+

+ +

+
At least (>)
+Guessed OK   Probability
+ 0           0.9899774042
+ 1           0.9365235602
+ 2           0.8028889501
+ 3           0.5950128899
+ 4           0.3698138248
+ 5           0.1896545726
+ 6           0.07955725188
+ 7           0.02712995629
+ 8           0.00746972044
+ 9           0.001644465374
+10           0.0002852391917
+11           3.810715862e-005
+12           3.783265129e-006
+13           2.628657967e-007
+14           1.140870154e-008
+15           2.328306437e-010
+16           0
+
+

+

+

+ We now consider the probabilities of ranges + of correct guesses. +

+

+

+

+ First, calculate the probability of getting a range of guesses right, + by adding the exact probabilities of each from low ... high. +

+

+

+

+ +

+
+int low = 3; // Getting at least 3 right.
+int high = 5; // Getting as most 5 right.
+double sum = 0.;
+for (int i = low; i <= high; i++)
+{
+  sum += pdf(quiz, i);
+}
+cout.precision(4);
+cout << "Probability of getting between "
+  << low << " and " << high << " answers right by guessing is "
+  << sum  << endl; // 0.61323
+

+

+

+ +

+
Probability of getting between 3 and 5 answers right by guessing is 0.6132
+
+

+

+

+ Or, usually better, we can use the difference of cdfs instead: +

+

+

+

+ +

+
+cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+  <<  cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 0.61323
+

+

+

+ +

+
Probability of getting between 3 and 5 answers right by guessing is 0.6132
+
+

+

+

+ And we can also try a few more combinations of high and low choices: +

+

+

+

+ +

+
+low = 1; high = 6; 
+cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+  <<  cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 and 6 P= 0.91042
+low = 1; high = 8; 
+cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+  <<  cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 <= x 8 P = 0.9825
+low = 4; high = 4; 
+cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+  <<  cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 4 <= x 4 P = 0.22520
+

+

+

+ +

+
Probability of getting between 1 and 6 answers right by guessing is 0.9104
+Probability of getting between 1 and 8 answers right by guessing is 0.9825
+Probability of getting between 4 and 4 answers right by guessing is 0.2252
+
+

+ +

+
+ + Using + Binomial distribution moments +
+

+

+

+ Using moments of the distribution, we can say more about the spread + of results from guessing. +

+

+

+

+ +

+
+cout << "By guessing, on average, one can expect to get " << mean(quiz) << " correct answers." << endl;
+cout << "Standard deviation is " << standard_deviation(quiz) << endl;
+cout << "So about 2/3 will lie within 1 standard deviation and get between "
+  <<  ceil(mean(quiz) - standard_deviation(quiz))  << " and "
+  << floor(mean(quiz) + standard_deviation(quiz)) << " correct." << endl; 
+cout << "Mode (the most frequent) is " << mode(quiz) << endl;
+cout << "Skewness is " << skewness(quiz) << endl;
+

+

+

+ +

+
By guessing, on average, one can expect to get 4 correct answers.
+Standard deviation is 1.732
+So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
+Mode (the most frequent) is 4
+Skewness is 0.2887
+
+

+ +

+
+ + Quantiles +
+

+

+

+ The quantiles (percentiles or percentage points) for a few probability + levels: +

+

+

+

+ +

+
+cout << "Quartiles " << quantile(quiz, 0.25) << " to "
+  << quantile(complement(quiz, 0.25)) << endl; // Quartiles 
+cout << "1 standard deviation " << quantile(quiz, 0.33) << " to " 
+  << quantile(quiz, 0.67) << endl; // 1 sd 
+cout << "Deciles " << quantile(quiz, 0.1)  << " to "
+  << quantile(complement(quiz, 0.1))<< endl; // Deciles 
+cout << "5 to 95% " << quantile(quiz, 0.05)  << " to "
+  << quantile(complement(quiz, 0.05))<< endl; // 5 to 95%
+cout << "2.5 to 97.5% " << quantile(quiz, 0.025) << " to "
+  <<  quantile(complement(quiz, 0.025)) << endl; // 2.5 to 97.5% 
+cout << "2 to 98% " << quantile(quiz, 0.02)  << " to "
+  << quantile(complement(quiz, 0.02)) << endl; //  2 to 98%
+
+cout << "If guessing then percentiles 1 to 99% will get " << quantile(quiz, 0.01) 
+  << " to " << quantile(complement(quiz, 0.01)) << " right." << endl;
+

+

+

+

+

+ Notice that these output integral values because the default policy + is integer_round_outwards. +

+

+ +

+
Quartiles 2 to 5
+1 standard deviation 2 to 5
+Deciles 1 to 6
+5 to 95% 0 to 7
+2.5 to 97.5% 0 to 8
+2 to 98% 0 to 8
+
+

+

+

+ Quantiles values are controlled by the discrete + quantile policy chosen. The default is integer_round_outwards, + so the lower quantile is rounded down, and the upper quantile is + rounded up. +

+

+

+

+ But we might believe that the real values tell us a little more - + see Understanding + Discrete Quantile Policy. +

+

+

+

+ We could control the policy for all + distributions by +

+

+ +

+
+#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+
+at the head of the program would make this policy apply
+
+

+

+

+ #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real +

+

+ +

+
+at the head of the program would make this policy apply
+
+

+

+

+ at the head of the program would make this policy apply to this + one, and only, translation unit. +

+

+

+

+ Or we can now create a (typedef for) policy that has discrete quantiles + real (here avoiding any 'using namespaces ...' statements): +

+

+

+

+ +

+
+using boost::math::policies::policy;
+using boost::math::policies::discrete_quantile;
+using boost::math::policies::real;
+using boost::math::policies::integer_round_outwards; // Default.
+typedef boost::math::policies::policy<discrete_quantile<real> > real_quantile_policy;
+

+

+

+

+

+ Add a custom binomial distribution called +

+
+real_quantile_binomial
+

+ that uses +

+
+real_quantile_policy
+

+

+

+

+

+ +

+
+using boost::math::binomial_distribution;
+typedef binomial_distribution<double, real_quantile_policy> real_quantile_binomial;
+

+

+

+

+

+ Construct an object of this custom distribution: +

+

+

+

+ +

+
+real_quantile_binomial quiz_real(questions, success_fraction);
+

+

+

+

+

+ And use this to show some quantiles - that now have real rather than + integer values. +

+

+

+

+ +

+
+cout << "Quartiles " << quantile(quiz, 0.25) << " to "
+  << quantile(complement(quiz_real, 0.25)) << endl; // Quartiles 2 to 4.6212
+cout << "1 standard deviation " << quantile(quiz_real, 0.33) << " to " 
+  << quantile(quiz_real, 0.67) << endl; // 1 sd 2.6654 4.194
+cout << "Deciles " << quantile(quiz_real, 0.1)  << " to "
+  << quantile(complement(quiz_real, 0.1))<< endl; // Deciles 1.3487 5.7583
+cout << "5 to 95% " << quantile(quiz_real, 0.05)  << " to "
+  << quantile(complement(quiz_real, 0.05))<< endl; // 5 to 95% 0.83739 6.4559
+cout << "2.5 to 97.5% " << quantile(quiz_real, 0.025) << " to "
+  <<  quantile(complement(quiz_real, 0.025)) << endl; // 2.5 to 97.5% 0.42806 7.0688
+cout << "2 to 98% " << quantile(quiz_real, 0.02)  << " to "
+  << quantile(complement(quiz_real, 0.02)) << endl; //  2 to 98% 0.31311 7.7880
+
+cout << "If guessing, then percentiles 1 to 99% will get " << quantile(quiz_real, 0.01) 
+  << " to " << quantile(complement(quiz_real, 0.01)) << " right." << endl;
+

+

+

+ +

+
Real Quantiles
+Quartiles 2 to 4.621
+1 standard deviation 2.665 to 4.194
+Deciles 1.349 to 5.758
+5 to 95% 0.8374 to 6.456
+2.5 to 97.5% 0.4281 to 7.069
+2 to 98% 0.3131 to 7.252
+If guessing then percentiles 1 to 99% will get 0 to 7.788 right.
+
+

+

+

+ See binomial_quiz_example.cpp + for full source code and output. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg.html new file mode 100644 index 000000000..6f2e25e9e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg.html @@ -0,0 +1,54 @@ + + + +Chi Squared Distribution Examples + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
+ Confidence Intervals on the Standard Deviation
+
+ Chi-Square Test for the Standard Deviation
+
+ Estimating the Required Sample Sizes for a Chi-Square Test for the Standard + Deviation
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html new file mode 100644 index 000000000..1067088c0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html @@ -0,0 +1,241 @@ + + + +Confidence Intervals on the Standard Deviation + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Once you have calculated the standard deviation for your data, a legitimate + question to ask is "How reliable is the calculated standard deviation?". + For this situation the Chi Squared distribution can be used to calculate + confidence intervals for the standard deviation. +

+

+ The full example code & sample output is in chi_square_std_deviation_test.cpp. +

+

+ We'll begin by defining the procedure that will calculate and print + out the confidence intervals: +

+
+void confidence_limits_on_std_deviation(
+     double Sd,    // Sample Standard Deviation
+     unsigned N)   // Sample size
+{
+
+

+ We'll begin by printing out some general information: +

+
+cout <<
+   "________________________________________________\n"
+   "2-Sided Confidence Limits For Standard Deviation\n"
+   "________________________________________________\n\n";
+cout << setprecision(7);
+cout << setw(40) << left << "Number of Observations" << "=  " << N << "\n";
+cout << setw(40) << left << "Standard Deviation" << "=  " << Sd << "\n";
+
+

+ and then define a table of significance levels for which we'll calculate + intervals: +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ The distribution we'll need to calculate the confidence intervals is + a Chi Squared distribution, with N-1 degrees of freedom: +

+
+chi_squared dist(N - 1);
+
+

+ For each value of alpha, the formula for the confidence interval is + given by: +

+

+ +

+

+ Where is the upper critical value, and is + the lower critical value of the Chi Squared distribution. +

+

+ In code we begin by printing out a table header: +

+
+cout << "\n\n"
+        "_____________________________________________\n"
+        "Confidence          Lower          Upper\n"
+        " Value (%)          Limit          Limit\n"
+        "_____________________________________________\n";
+
+

+ and then loop over the values of alpha and calculate the intervals + for each: remember that the lower critical value is the same as the + quantile, and the upper critical value is the same as the quantile + from the complement of the probability: +

+
+for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+{
+   // Confidence value:
+   cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+   // Calculate limits:
+   double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha[i] / 2)));
+   double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha[i] / 2));
+   // Print Limits:
+   cout << fixed << setprecision(5) << setw(15) << right << lower_limit;
+   cout << fixed << setprecision(5) << setw(15) << right << upper_limit << endl;
+}
+cout << endl;
+
+

+ To see some example output we'll use the gear + data from the NIST/SEMATECH + e-Handbook of Statistical Methods.. The data represents measurements + of gear diameter from a manufacturing process. +

+
________________________________________________
+2-Sided Confidence Limits For Standard Deviation
+________________________________________________
+
+Number of Observations                  =  100
+Standard Deviation                      =  0.006278908
+
+
+_____________________________________________
+Confidence          Lower          Upper
+ Value (%)          Limit          Limit
+_____________________________________________
+    50.000        0.00601        0.00662
+    75.000        0.00582        0.00685
+    90.000        0.00563        0.00712
+    95.000        0.00551        0.00729
+    99.000        0.00530        0.00766
+    99.900        0.00507        0.00812
+    99.990        0.00489        0.00855
+    99.999        0.00474        0.00895
+
+

+ So at the 95% confidence level we conclude that the standard deviation + is between 0.00551 and 0.00729. +

+
+ + Confidence + intervals as a function of the number of observations +
+

+ Similarly, we can also list the confidence intervals for the standard + deviation for the common confidence levels 95%, for increasing numbers + of observations. +

+

+ The standard deviation used to compute these values is unity, so the + limits listed are multipliers for + any particular standard deviation. For example, given a standard deviation + of 0.0062789 as in the example above; for 100 observations the multiplier + is 0.8780 giving the lower confidence limit of 0.8780 * 0.006728 = + 0.00551. +

+
____________________________________________________
+Confidence level (two-sided)            =  0.0500000
+Standard Deviation                      =  1.0000000
+________________________________________
+Observations        Lower          Upper
+                    Limit          Limit
+________________________________________
+         2         0.4461        31.9102
+         3         0.5207         6.2847
+         4         0.5665         3.7285
+         5         0.5991         2.8736
+         6         0.6242         2.4526
+         7         0.6444         2.2021
+         8         0.6612         2.0353
+         9         0.6755         1.9158
+        10         0.6878         1.8256
+        15         0.7321         1.5771
+        20         0.7605         1.4606
+        30         0.7964         1.3443
+        40         0.8192         1.2840
+        50         0.8353         1.2461
+        60         0.8476         1.2197
+       100         0.8780         1.1617
+       120         0.8875         1.1454
+      1000         0.9580         1.0459
+     10000         0.9863         1.0141
+     50000         0.9938         1.0062
+    100000         0.9956         1.0044
+   1000000         0.9986         1.0014
+
+

+ With just 2 observations the limits are from 0.445 + up to to 31.9, so the standard deviation + might be about half the observed value + up to 30 times the observed value! +

+

+ Estimating a standard deviation with just a handful of values leaves + a very great uncertainty, especially the upper limit. Note especially + how far the upper limit is skewed from the most likely standard deviation. +

+

+ Even for 10 observations, normally considered a reasonable number, + the range is still from 0.69 to 1.8, about a range of 0.7 to 2, and + is still highly skewed with an upper limit twice + the median. +

+

+ When we have 1000 observations, the estimate of the standard deviation + is starting to look convincing, with a range from 0.95 to 1.05 - now + near symmetrical, but still about + or - 5%. +

+

+ Only when we have 10000 or more repeated observations can we start + to be reasonably confident (provided we are sure that other factors + like drift are not creeping in). +

+

+ For 10000 observations, the interval is 0.99 to 1.1 - finally a really + convincing + or -1% confidence. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html new file mode 100644 index 000000000..8523d0dc2 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html @@ -0,0 +1,185 @@ + + + +Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Suppose we conduct a Chi Squared test for standard deviation and the + result is borderline, a legitimate question to ask is "How large + would the sample size have to be in order to produce a definitive result?" +

+

+ The class template chi_squared_distribution + has a static method find_degrees_of_freedom + that will calculate this value for some acceptable risk of type I failure + alpha, type II failure beta, + and difference from the standard deviation diff. + Please note that the method used works on variance, and not standard + deviation as is usual for the Chi Squared Test. +

+

+ The code for this example is located in chi_square_std_dev_test.cpp. +

+

+ We begin by defining a procedure to print out the sample sizes required + for various risk levels: +

+
+void chi_squared_sample_sized(
+     double diff,      // difference from variance to detect
+     double variance)  // true variance
+{
+
+

+ The procedure begins by printing out the input data: +

+
+using namespace std;
+using namespace boost::math;
+
+// Print out general info:
+cout <<
+   "_____________________________________________________________\n"
+   "Estimated sample sizes required for various confidence levels\n"
+   "_____________________________________________________________\n\n";
+cout << setprecision(5);
+cout << setw(40) << left << "True Variance" << "=  " << variance << "\n";
+cout << setw(40) << left << "Difference to detect" << "=  " << diff << "\n";
+
+

+ And defines a table of significance levels for which we'll calculate + sample sizes: +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ For each value of alpha we can calculate two sample sizes: one where + the sample variance is less than the true value by diff + and one where it is greater than the true value by diff. + Thanks to the asymmetric nature of the Chi Squared distribution these + two values will not be the same, the difference in their calculation + differs only in the sign of diff that's passed + to find_degrees_of_freedom. + Finally in this example we'll simply things, and let risk level beta + be the same as alpha: +

+
+cout << "\n\n"
+        "_______________________________________________________________\n"
+        "Confidence       Estimated          Estimated\n"
+        " Value (%)      Sample Size        Sample Size\n"
+        "                (lower one         (upper one\n"
+        "                 sided test)        sided test)\n"
+        "_______________________________________________________________\n";
+//
+// Now print out the data for the table rows.
+//
+for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+{
+   // Confidence value:
+   cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+   // calculate df for a lower single sided test:
+   double df = chi_squared::find_degrees_of_freedom(
+      -diff, alpha[i], alpha[i], variance);
+   // convert to sample size:
+   double size = ceil(df) + 1;
+   // Print size:
+   cout << fixed << setprecision(0) << setw(16) << right << size;
+   // calculate df for an upper single sided test:
+   df = chi_squared::find_degrees_of_freedom(
+      diff, alpha[i], alpha[i], variance);
+   // convert to sample size:
+   size = ceil(df) + 1;
+   // Print size:
+   cout << fixed << setprecision(0) << setw(16) << right << size << endl;
+}
+cout << endl;
+
+

+ For some example output, consider the silicon + wafer data from the NIST/SEMATECH + e-Handbook of Statistical Methods.. In this scenario a supplier + of 100 ohm.cm silicon wafers claims that his fabrication process can + produce wafers with sufficient consistency so that the standard deviation + of resistivity for the lot does not exceed 10 ohm.cm. A sample of N + = 10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm, + and the question we ask ourselves is "How large would our sample + have to be to reliably detect this difference?". +

+

+ To use our procedure above, we have to convert the standard deviations + to variance (square them), after which the program output looks like + this: +

+
_____________________________________________________________
+Estimated sample sizes required for various confidence levels
+_____________________________________________________________
+
+True Variance                           =  100.00000
+Difference to detect                    =  95.16090
+
+
+_______________________________________________________________
+Confidence       Estimated          Estimated
+ Value (%)      Sample Size        Sample Size
+                (lower one         (upper one
+                 sided test)        sided test)
+_______________________________________________________________
+    50.000               2               2
+    75.000               2              10
+    90.000               4              32
+    95.000               5              51
+    99.000               7              99
+    99.900              11             174
+    99.990              15             251
+    99.999              20             330
+
+

+ In this case we are interested in a upper single sided test. So for + example, if the maximum acceptable risk of falsely rejecting the null-hypothesis + is 0.05 (Type I error), and the maximum acceptable risk of failing + to reject the null-hypothesis is also 0.05 (Type II error), we estimate + that we would need a sample size of 51. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html new file mode 100644 index 000000000..9afdb9cea --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html @@ -0,0 +1,301 @@ + + + +Chi-Square Test for the Standard Deviation + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ We use this test to determine whether the standard deviation of a sample + differs from a specified value. Typically this occurs in process change + situations where we wish to compare the standard deviation of a new + process to an established one. +

+

+ The code for this example is contained in chi_square_std_dev_test.cpp, + and we'll begin by defining the procedure that will print out the test + statistics: +

+
+void chi_squared_test(
+    double Sd,     // Sample std deviation
+    double D,      // True std deviation
+    unsigned N,    // Sample size
+    double alpha)  // Significance level
+{
+
+

+ The procedure begins by printing a summary of the input data: +

+
+using namespace std;
+using namespace boost::math;
+
+// Print header:
+cout <<
+   "______________________________________________\n"
+   "Chi Squared test for sample standard deviation\n"
+   "______________________________________________\n\n";
+cout << setprecision(5);
+cout << setw(55) << left << "Number of Observations" << "=  " << N << "\n";
+cout << setw(55) << left << "Sample Standard Deviation" << "=  " << Sd << "\n";
+cout << setw(55) << left << "Expected True Standard Deviation" << "=  " << D << "\n\n";
+
+

+ The test statistic (T) is simply the ratio of the sample and "true" + standard deviations squared, multiplied by the number of degrees of + freedom (the sample size less one): +

+
+double t_stat = (N - 1) * (Sd / D) * (Sd / D);
+cout << setw(55) << left << "Test Statistic" << "=  " << t_stat << "\n";
+
+

+ The distribution we need to use, is a Chi Squared distribution with + N-1 degrees of freedom: +

+
+chi_squared dist(N - 1);
+
+

+ The various hypothesis that can be tested are summarised in the following + table: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Hypothesis +

+ 		+

+ Test +

+
+

+ The null-hypothesis: there is no difference in standard deviation + from the specified value +

+ 		+

+ Reject if T < χ2(1-alpha/2; N-1) or T > χ2(alpha/2; N-1) +

+
+

+ The alternative hypothesis: there is a difference in standard + deviation from the specified value +

+ 		+

+ Reject if χ2(1-alpha/2; N-1) >= T >= χ2(alpha/2; N-1) +

+
+

+ The alternative hypothesis: the standard deviation is less + than the specified value +

+ 		+

+ Reject if χ2(1-alpha; N-1) <= T +

+
+

+ The alternative hypothesis: the standard deviation is greater + than the specified value +

+ 		+

+ Reject if χ2(alpha; N-1) >= T +

+
+

+ Where χ2(alpha; N-1) is the upper critical value of the Chi Squared distribution, + and χ2(1-alpha; N-1) is the lower critical value. +

+

+ Recall that the lower critical value is the same as the quantile, and + the upper critical value is the same as the quantile from the complement + of the probability, that gives us the following code to calculate the + critical values: +

+
+double ucv = quantile(complement(dist, alpha));
+double ucv2 = quantile(complement(dist, alpha / 2));
+double lcv = quantile(dist, alpha);
+double lcv2 = quantile(dist, alpha / 2);
+cout << setw(55) << left << "Upper Critical Value at alpha: " << "=  "
+   << setprecision(3) << scientific << ucv << "\n";
+cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "=  "
+   << setprecision(3) << scientific << ucv2 << "\n";
+cout << setw(55) << left << "Lower Critical Value at alpha: " << "=  "
+   << setprecision(3) << scientific << lcv << "\n";
+cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "=  "
+   << setprecision(3) << scientific << lcv2 << "\n\n";
+
+

+ Now that we have the critical values, we can compare these to our test + statistic, and print out the result of each hypothesis and test: +

+
+cout << setw(55) << left <<
+   "Results for Alternative Hypothesis and alpha" << "=  "
+   << setprecision(4) << fixed << alpha << "\n\n";
+cout << "Alternative Hypothesis              Conclusion\n";
+
+cout << "Standard Deviation != " << setprecision(3) << fixed << D << "            ";
+if((ucv2 < t_stat) || (lcv2 > t_stat))
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+
+cout << "Standard Deviation  < " << setprecision(3) << fixed << D << "            ";
+if(lcv > t_stat)
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+
+cout << "Standard Deviation  > " << setprecision(3) << fixed << D << "            ";
+if(ucv < t_stat)
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+cout << endl << endl;
+
+

+ To see some example output we'll use the gear + data from the NIST/SEMATECH + e-Handbook of Statistical Methods.. The data represents measurements + of gear diameter from a manufacturing process. The program output is + deliberately designed to mirror the DATAPLOT output shown in the NIST + Handbook Example. +

+
______________________________________________
+Chi Squared test for sample standard deviation
+______________________________________________
+
+Number of Observations                                 =  100
+Sample Standard Deviation                              =  0.00628
+Expected True Standard Deviation                       =  0.10000
+
+Test Statistic                                         =  0.39030
+CDF of test statistic:                                 =  1.438e-099
+Upper Critical Value at alpha:                         =  1.232e+002
+Upper Critical Value at alpha/2:                       =  1.284e+002
+Lower Critical Value at alpha:                         =  7.705e+001
+Lower Critical Value at alpha/2:                       =  7.336e+001
+
+Results for Alternative Hypothesis and alpha           =  0.0500
+
+Alternative Hypothesis              Conclusion
+Standard Deviation != 0.100            ACCEPTED
+Standard Deviation  < 0.100            ACCEPTED
+Standard Deviation  > 0.100            REJECTED
+
+

+ In this case we are testing whether the sample standard deviation is + 0.1, and the null-hypothesis is rejected, so we conclude that the standard + deviation is not 0.1. +

+

+ For an alternative example, consider the silicon + wafer data again from the NIST/SEMATECH + e-Handbook of Statistical Methods.. In this scenario a supplier + of 100 ohm.cm silicon wafers claims that his fabrication process can + produce wafers with sufficient consistency so that the standard deviation + of resistivity for the lot does not exceed 10 ohm.cm. A sample of N + = 10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm, + and the question we ask ourselves is "Is the suppliers claim correct?". +

+

+ The program output now looks like this: +

+
______________________________________________
+Chi Squared test for sample standard deviation
+______________________________________________
+
+Number of Observations                                 =  10
+Sample Standard Deviation                              =  13.97000
+Expected True Standard Deviation                       =  10.00000
+
+Test Statistic                                         =  17.56448
+CDF of test statistic:                                 =  9.594e-001
+Upper Critical Value at alpha:                         =  1.692e+001
+Upper Critical Value at alpha/2:                       =  1.902e+001
+Lower Critical Value at alpha:                         =  3.325e+000
+Lower Critical Value at alpha/2:                       =  2.700e+000
+
+Results for Alternative Hypothesis and alpha           =  0.0500
+
+Alternative Hypothesis              Conclusion
+Standard Deviation != 10.000            REJECTED
+Standard Deviation  < 10.000            REJECTED
+Standard Deviation  > 10.000            ACCEPTED
+
+

+ In this case, our null-hypothesis is that the standard deviation of + the sample is less than 10: this hypothesis is rejected in the analysis + above, and so we reject the manufacturers claim. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html new file mode 100644 index 000000000..705f14aaa --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html @@ -0,0 +1,431 @@ + + + +Distribution Construction Example + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ See distribution_construction.cpp + for full source code. +

+

+

+

+ The structure of distributions is rather different from some other + statistical libraries, for example in less object-oriented language + like FORTRAN and C, that provide a few arguments to each free function. + This library provides each distribution as a template C++ class. A + distribution is constructed with a few arguments, and then member and + non-member functions are used to find values of the distribution, often + a function of a random variate. +

+

+

+

+ First we need some includes to access the negative binomial distribution + (and the binomial, beta and gamma too). +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp> // for negative_binomial_distribution
+  using boost::math::negative_binomial_distribution; // default type is double.
+  using boost::math::negative_binomial; // typedef provides default type is double.
+#include <boost/math/distributions/binomial.hpp> // for binomial_distribution.
+#include <boost/math/distributions/beta.hpp> // for beta_distribution.
+#include <boost/math/distributions/gamma.hpp> // for gamma_distribution.
+#include <boost/math/distributions/normal.hpp> // for normal_distribution.
+

+

+

+

+

+ Several examples of constructing distributions follow: +

+

+

+

+ First, a negative binomial distribution with 8 successes and a success + fraction 0.25, 25% or 1 in 4, is constructed like this: +

+

+

+

+ +

+
+boost::math::negative_binomial_distribution<double> mydist0(8., 0.25);
+

+

+

+

+

+ But this is inconveniently long, so by writing +

+

+

+

+ +

+
+using namespace boost::math;
+

+

+

+

+

+ or +

+

+

+

+ +

+
+using boost::math::negative_binomial_distribution;
+

+

+

+

+

+ we can reduce typing. +

+

+

+

+ Since the vast majority of applications use will be using double precision, + the template argument to the distribution (RealType) defaults to type + double, so we can also write: +

+

+

+

+ +

+
+negative_binomial_distribution<> mydist9(8., 0.25); // Uses default RealType = double.
+

+

+

+

+

+ But the name "negative_binomial_distribution" is still inconveniently + long, so for most distributions, a convenience typedef is provided, + for example: +

+

+ +

+
+typedef negative_binomial_distribution<double> negative_binomial; // Reserved name of type double.
+
+

+

+
+ + + + + +
[Caution]Caution

+ This convenience typedef is not provided if + a clash would occur with the name of a function: currently only "beta" + and "gamma" fall into this category. +

+

+

+

+ So, after a using statement, +

+

+

+

+ +

+
+using boost::math::negative_binomial;
+

+

+

+

+

+ we have a convenient typedef to negative_binomial_distribution<double>: +

+

+

+

+ +

+
+negative_binomial mydist(8., 0.25);
+

+

+

+

+

+ Some more examples using the convenience typedef: +

+

+

+

+ +

+
+negative_binomial mydist10(5., 0.4); // Both arguments double.
+

+

+

+

+

+ And automatic conversion takes place, so you can use integers and floats: +

+

+

+

+ +

+
+negative_binomial mydist11(5, 0.4); // Using provided typedef double, int and double arguments.
+

+

+

+

+

+ This is probably the most common usage. +

+

+

+

+ +

+
+negative_binomial mydist12(5., 0.4F); // Double and float arguments.
+negative_binomial mydist13(5, 1); // Both arguments integer.
+

+

+

+

+

+ Similarly for most other distributions like the binomial. +

+

+

+

+ +

+
+binomial mybinomial(1, 0.5); // is more concise than
+binomial_distribution<> mybinomd1(1, 0.5);
+

+

+

+

+

+ For cases when the typdef distribution name would clash with a math + special function (currently only beta and gamma) the typedef is deliberately + not provided, and the longer version of the name must be used. For + example do not use: +

+

+ +

+
+using boost::math::beta;
+beta mybetad0(1, 0.5); // Error beta is a math FUNCTION!
+
+

+

+

+ Which produces the error messages: +

+

+ +

+
error C2146: syntax error : missing ';' before identifier 'mybetad0'
+warning C4551: function call missing argument list
+error C3861: 'mybetad0': identifier not found
+
+

+

+

+ Instead you should use: +

+

+

+

+ +

+
+using boost::math::beta_distribution;
+beta_distribution<> mybetad1(1, 0.5);
+

+

+

+

+

+ or for the gamma distribution: +

+

+

+

+ +

+
+gamma_distribution<> mygammad1(1, 0.5);
+

+

+

+

+

+ We can, of course, still provide the type explicitly thus: +

+

+

+

+ +

+
+// Explicit double precision:
+negative_binomial_distribution<double>        mydist1(8., 0.25); 
+
+// Explicit float precision, double arguments are truncated to float:
+negative_binomial_distribution<float>         mydist2(8., 0.25);
+
+// Explicit float precision, integer & double arguments converted to float.
+negative_binomial_distribution<float>         mydist3(8, 0.25); 
+
+// Explicit float precision, float arguments, so no conversion:
+negative_binomial_distribution<float>         mydist4(8.F, 0.25F); 
+
+// Explicit float precision, integer arguments promoted to float.
+negative_binomial_distribution<float>         mydist5(8, 1); 
+
+// Explicit double precision:
+negative_binomial_distribution<double>        mydist6(8., 0.25); 
+
+// Explicit long double precision:
+negative_binomial_distribution<long double>   mydist7(8., 0.25);
+

+

+

+

+

+ And if you have your own RealType called MyFPType, for example NTL + RR (an arbitrary precision type), then we can write: +

+

+ +

+
+negative_binomial_distribution<MyFPType>  mydist6(8, 1); // Integer arguments -> MyFPType.
+
+

+ +

+
+ + Default + arguments to distribution constructors. +
+

+

+

+ Note that default constructor arguments are only provided for some + distributions. So if you wrongly assume a default argument you will + get an error message, for example: +

+

+ +

+
+negative_binomial_distribution<> mydist8;
+
+

+ +

+
error C2512 no appropriate default constructor available.
+

+

+

+ No default constructors are provided for the negative binomial, because + it is difficult to chose any sensible default values for this distribution. + For other distributions, like the normal distribution, it is obviously + very useful to provide 'standard' defaults for the mean and standard + deviation thus: +

+

+ +

+
+normal_distribution(RealType mean = 0, RealType sd = 1);
+
+

+

+

+ So in this case we can write: +

+

+

+

+ +

+
+  using boost::math::normal;
+  
+  normal norm1;       // Standard normal distribution.
+  normal norm2(2);    // Mean = 2, std deviation = 1.
+  normal norm3(2, 3); // Mean = 2, std deviation = 3.
+
+  return 0;
+}  // int main()
+

+

+

+

+

+ There is no useful output from this program, of course. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/error_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/error_eg.html new file mode 100644 index 000000000..172e08913 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/error_eg.html @@ -0,0 +1,271 @@ + + + +Error Handling Example + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ See error handling + documentation for a detailed explanation of the mechanism of handling + errors, including the common "bad" arguments to distributions + and functions, and how to use Policies + to control it. +

+

+ But, by default, exceptions will be raised, + for domain errors, pole errors, numeric overflow, and internal evaluation + errors. To avoid the exceptions from getting thrown and instead get an + appropriate value returned, usually a NaN (domain errors pole errors + or internal errors), or infinity (from overflow), you need to change + the policy. +

+

+

+

+ The following example demonstrates the effect of setting the macro + BOOST_MATH_DOMAIN_ERROR_POLICY when an invalid argument is encountered. + For the purposes of this example, we'll pass a negative degrees of + freedom parameter to the student's t distribution. +

+

+

+

+ Since we know that this is a single file program we could just add: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
+
+

+

+

+ to the top of the source file to change the default policy to one that + simply returns a NaN when a domain error occurs. Alternatively we could + use: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
+
+

+

+

+ To ensure the ::errno + is set when a domain error occurs as well as returning a NaN. +

+

+

+

+ This is safe provided the program consists of a single translation + unit and we place the define before + any #includes. Note that should we add the define after the includes + then it will have no effect! A warning such as: +

+

+ +

+
warning C4005: 'BOOST_MATH_OVERFLOW_ERROR_POLICY' : macro redefinition
+

+

+

+ is a certain sign that it will not have the desired + effect. +

+

+

+

+ We'll begin our sample program with the needed includes: +

+

+

+

+ +

+
+// Boost
+#include <boost/math/distributions/students_t.hpp>
+   using boost::math::students_t;  // Probability of students_t(df, t).
+
+// std
+#include <iostream>
+   using std::cout;
+   using std::endl;
+
+#include <stdexcept>
+   using std::exception;
+

+

+

+

+

+ Next we'll define the program's main() to call the student's t distribution + with an invalid degrees of freedom parameter, the program is set up + to handle either an exception or a NaN: +

+

+

+

+ +

+
+int main()
+{
+   cout << "Example error handling using Student's t function. " << endl;
+   cout << "BOOST_MATH_DOMAIN_ERROR_POLICY is set to: "
+      << BOOST_STRINGIZE(BOOST_MATH_DOMAIN_ERROR_POLICY) << endl;
+
+   double degrees_of_freedom = -1; // A bad argument!
+   double t = 10;
+
+   try
+   {
+      errno = 0;
+      students_t dist(degrees_of_freedom); // exception is thrown here if enabled
+      double p = cdf(dist, t);
+      // test for error reported by other means:
+      if((boost::math::isnan)(p))
+      {
+         cout << "cdf returned a NaN!" << endl;
+         cout << "errno is set to: " << errno << endl;
+      }
+      else
+         cout << "Probability of Student's t is " << p << endl;
+   }
+   catch(const std::exception& e)
+   {
+      std::cout <<
+         "\n""Message from thrown exception was:\n   " << e.what() << std::endl;
+   }
+
+   return 0;
+} // int main()
+

+

+

+

+

+ Here's what the program output looks like with a default build (one + that does throw exceptions): +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: throw_on_error
+
+Message from thrown exception was:
+   Error in function boost::math::students_t_distribution<double>::students_t_distribution:
+   Degrees of freedom argument is -1, but must be > 0 !
+
+

+

+

+ Alternatively let's build with: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
+
+

+

+

+ Now the program output is: +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: ignore_error
+cdf returned a NaN!
+errno is set to: 0
+
+

+

+

+ And finally let's build with: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
+
+

+

+

+ Which gives the output: +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: errno_on_error
+cdf returned a NaN!
+errno is set to: 33
+
+

+

+
+ + + + + +
[Caution]Caution
+

+ If throwing of exceptions is enabled (the default) but you do not have try & catch block, then the program + will terminate with an uncaught exception and probably abort. +

+

+ Therefore to get the benefit of helpful error messages, enabling all exceptions and using try & catch is + recommended for most applications. +

+

+ However, for simplicity, the is not done for most examples. +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/f_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/f_eg.html new file mode 100644 index 000000000..50d025682 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/f_eg.html @@ -0,0 +1,335 @@ + + + +F Distribution Examples + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Imagine that you want to compare the standard deviations of two sample + to determine if they differ in any significant way, in this situation + you use the F distribution and perform an F-test. This situation commonly + occurs when conducting a process change comparison: "is a new process + more consistent that the old one?". +

+

+ In this example we'll be using the data for ceramic strength from http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm. + The data for this case study were collected by Said Jahanmir of the NIST + Ceramics Division in 1996 in connection with a NIST/industry ceramics + consortium for strength optimization of ceramic strength. +

+

+ The example program is f_test.cpp, + program output has been deliberately made as similar as possible to the + DATAPLOT output in the corresponding NIST + EngineeringStatistics Handbook example. +

+

+ We'll begin by defining the procedure to conduct the test: +

+
+void f_test(
+    double sd1,     // Sample 1 std deviation
+    double sd2,     // Sample 2 std deviation
+    double N1,      // Sample 1 size
+    double N2,      // Sample 2 size
+    double alpha)  // Significance level
+{
+
+

+ The procedure begins by printing out a summary of our input data: +

+
+using namespace std;
+using namespace boost::math;
+
+// Print header:
+cout <<
+   "____________________________________\n"
+   "F test for equal standard deviations\n"
+   "____________________________________\n\n";
+cout << setprecision(5);
+cout << "Sample 1:\n";
+cout << setw(55) << left << "Number of Observations" << "=  " << N1 << "\n";
+cout << setw(55) << left << "Sample Standard Deviation" << "=  " << sd1 << "\n\n";
+cout << "Sample 2:\n";
+cout << setw(55) << left << "Number of Observations" << "=  " << N2 << "\n";
+cout << setw(55) << left << "Sample Standard Deviation" << "=  " << sd2 << "\n\n";
+
+

+ The test statistic for an F-test is simply the ratio of the square of + the two standard deviations: +

+

+ F = s12 / s22 +

+

+ where s1 is the standard deviation of the first sample and s2 +is the standard + deviation of the second sample. Or in code: +

+
+double F = (sd1 / sd2);
+F *= F;
+cout << setw(55) << left << "Test Statistic" << "=  " << F << "\n\n";
+
+

+ At this point a word of caution: the F distribution is asymmetric, so + we have to be careful how we compute the tests, the following table summarises + the options available: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Hypothesis +

+ 		+

+ Test +

+
+

+ The null-hypothesis: there is no difference in standard deviations + (two sided test) +

+ 		+

+ Reject if F <= F(1-alpha/2; N1-1, N2-1) or F >= F(alpha/2; + N1-1, N2-1) +

+
+

+ The alternative hypothesis: there is a difference in means (two + sided test) +

+ 		+

+ Reject if F(1-alpha/2; N1-1, N2-1) <= F <= F(alpha/2; N1-1, + N2-1) +

+
+

+ The alternative hypothesis: Standard deviation of sample 1 is + greater than that of sample 2 +

+ 		+

+ Reject if F < F(alpha; N1-1, N2-1) +

+
+

+ The alternative hypothesis: Standard deviation of sample 1 is + less than that of sample 2 +

+ 		+

+ Reject if F > F(1-alpha; N1-1, N2-1) +

+
+

+ Where F(1-alpha; N1-1, N2-1) is the lower critical value of the F distribution + with degrees of freedom N1-1 and N2-1, and F(alpha; N1-1, N2-1) is the + upper critical value of the F distribution with degrees of freedom N1-1 + and N2-1. +

+

+ The upper and lower critical values can be computed using the quantile + function: +

+

+ F(1-alpha; N1-1, N2-1) = quantile(fisher_f(N1-1, + N2-1), alpha) +

+

+ F(alpha; N1-1, N2-1) = quantile(complement(fisher_f(N1-1, + N2-1), alpha)) +

+

+ In our example program we need both upper and lower critical values for + alpha and for alpha/2: +

+
+double ucv = quantile(complement(dist, alpha));
+double ucv2 = quantile(complement(dist, alpha / 2));
+double lcv = quantile(dist, alpha);
+double lcv2 = quantile(dist, alpha / 2);
+cout << setw(55) << left << "Upper Critical Value at alpha: " << "=  "
+   << setprecision(3) << scientific << ucv << "\n";
+cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "=  "
+   << setprecision(3) << scientific << ucv2 << "\n";
+cout << setw(55) << left << "Lower Critical Value at alpha: " << "=  "
+   << setprecision(3) << scientific << lcv << "\n";
+cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "=  "
+   << setprecision(3) << scientific << lcv2 << "\n\n";
+
+

+ The final step is to perform the comparisons given above, and print out + whether the hypothesis is rejected or not: +

+
+cout << setw(55) << left <<
+   "Results for Alternative Hypothesis and alpha" << "=  "
+   << setprecision(4) << fixed << alpha << "\n\n";
+cout << "Alternative Hypothesis                                    Conclusion\n";
+
+cout << "Standard deviations are unequal (two sided test)          ";
+if((ucv2 < F) || (lcv2 > F))
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+
+cout << "Standard deviation 1 is less than standard deviation 2    ";
+if(lcv > F)
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+
+cout << "Standard deviation 1 is greater than standard deviation 2 ";
+if(ucv < F)
+   cout << "ACCEPTED\n";
+else
+   cout << "REJECTED\n";
+cout << endl << endl;
+
+

+ Using the ceramic strength data as an example we get the following output: +

+
F test for equal standard deviations
+____________________________________
+
+Sample 1:
+Number of Observations                                 =  240
+Sample Standard Deviation                              =  65.549
+
+Sample 2:
+Number of Observations                                 =  240
+Sample Standard Deviation                              =  61.854
+
+Test Statistic                                         =  1.123
+
+CDF of test statistic:                                 =  8.148e-001
+Upper Critical Value at alpha:                         =  1.238e+000
+Upper Critical Value at alpha/2:                       =  1.289e+000
+Lower Critical Value at alpha:                         =  8.080e-001
+Lower Critical Value at alpha/2:                       =  7.756e-001
+
+Results for Alternative Hypothesis and alpha           =  0.0500
+
+Alternative Hypothesis                                    Conclusion
+Standard deviations are unequal (two sided test)          REJECTED
+Standard deviation 1 is less than standard deviation 2    REJECTED
+Standard deviation 1 is greater than standard deviation 2 REJECTED
+
+

+ In this case we are unable to reject the null-hypothesis, and must instead + reject the alternative hypothesis. +

+

+ By contrast let's see what happens when we use some different sample + data:, once again from the NIST Engineering Statistics Handbook: + A new procedure to assemble a device is introduced and tested for possible + improvement in time of assembly. The question being addressed is whether + the standard deviation of the new assembly process (sample 2) is better + (i.e., smaller) than the standard deviation for the old assembly process + (sample 1). +

+
____________________________________
+F test for equal standard deviations
+____________________________________
+
+Sample 1:
+Number of Observations                                 =  11.00000
+Sample Standard Deviation                              =  4.90820
+
+Sample 2:
+Number of Observations                                 =  9.00000
+Sample Standard Deviation                              =  2.58740
+
+Test Statistic                                         =  3.59847
+
+CDF of test statistic:                                 =  9.589e-001
+Upper Critical Value at alpha:                         =  3.347e+000
+Upper Critical Value at alpha/2:                       =  4.295e+000
+Lower Critical Value at alpha:                         =  3.256e-001
+Lower Critical Value at alpha/2:                       =  2.594e-001
+
+Results for Alternative Hypothesis and alpha           =  0.0500
+
+Alternative Hypothesis                                    Conclusion
+Standard deviations are unequal (two sided test)          REJECTED
+Standard deviation 1 is less than standard deviation 2    REJECTED
+Standard deviation 1 is greater than standard deviation 2 ACCEPTED
+
+

+ In this case we take our null hypothesis as "standard deviation + 1 is less than or equal to standard deviation 2", since this represents + the "no change" situation. So we want to compare the upper + critical value at alpha (a one sided test) with + the test statistic, and since 3.35 < 3.6 this hypothesis must be rejected. + We therefore conclude that there is a change for the better in our standard + deviation. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg.html new file mode 100644 index 000000000..1e0ab7e5c --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg.html @@ -0,0 +1,53 @@ + + + +Find Location and Scale Examples + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ Find Location (Mean) Example
+
+ Find Scale (Standard Deviation) Example
+
+ Find mean and standard deviation example
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html new file mode 100644 index 000000000..837f46238 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html @@ -0,0 +1,290 @@ + + + +Find Location (Mean) Example + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+

+

+ First we need some includes to access the normal distribution, the + algorithms to find location (and some std output of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp> // for normal_distribution
+  using boost::math::normal; // typedef provides default type is double.
+#include <boost/math/distributions/cauchy.hpp> // for cauchy_distribution
+  using boost::math::cauchy; // typedef provides default type is double.
+#include <boost/math/distributions/find_location.hpp>
+  using boost::math::find_location;
+  using boost::math::complement; // Needed if you want to use the complement version.
+  using boost::math::policies::policy;
+
+#include <iostream>
+  using std::cout; using std::endl;
+#include <iomanip>
+  using std::setw; using std::setprecision;
+#include <limits>
+  using std::numeric_limits;
+
+

+

+

+

+

+ For this example, we will use the standard normal distribution, with + mean (location) zero and standard deviation (scale) unity. This is + also the default for this implementation. +

+

+

+

+ +

+
+normal N01;  // Default 'standard' normal distribution with zero mean and 
+double sd = 1.; // normal default standard deviation is 1.
+

+

+

+

+

+ Suppose we want to find a different normal distribution whose mean + is shifted so that only fraction p (here 0.001 or 0.1%) are below + a certain chosen limit (here -2, two standard deviations). +

+

+

+

+ +

+
+double z = -2.; // z to give prob p
+double p = 0.001; // only 0.1% below z
+
+cout << "Normal distribution with mean = " << N01.location()
+  << ", standard deviation " << N01.scale()
+  << ", has " << "fraction <= " << z 
+  << ", p = "  << cdf(N01, z) << endl;
+cout << "Normal distribution with mean = " << N01.location()
+  << ", standard deviation " << N01.scale()
+  << ", has " << "fraction > " << z
+  << ", p = "  << cdf(complement(N01, z)) << endl; // Note: uses complement.
+

+

+

+ +

+
Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501
+Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725
+
+

+

+

+ We can now use ''find_location'' to give a new offset mean. +

+

+

+

+ +

+
+double l = find_location<normal>(z, p, sd);
+cout << "offset location (mean) = " << l << endl;
+

+

+

+

+

+ that outputs: +

+

+ +

+
offset location (mean) = 1.09023
+
+

+

+

+ showing that we need to shift the mean just over one standard deviation + from its previous value of zero. +

+

+

+

+ Then we can check that we have achieved our objective by constructing + a new distribution with the offset mean (but same standard deviation): +

+

+

+

+ +

+
+normal np001pc(l, sd); // Same standard_deviation (scale) but with mean (location) shifted.
+

+

+

+

+

+ And re-calculating the fraction below our chosen limit. +

+

+

+

+ +

+
+cout << "Normal distribution with mean = " << l 
+    << " has " << "fraction <= " << z 
+    << ", p = "  << cdf(np001pc, z) << endl;
+  cout << "Normal distribution with mean = " << l 
+    << " has " << "fraction > " << z 
+    << ", p = "  << cdf(complement(np001pc, z)) << endl;
+

+

+

+ +

+
Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001
+Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999
+
+

+ +

+
+ + Controlling + Error Handling from find_location +
+

+

+

+ We can also control the policy for handling various errors. For example, + we can define a new (possibly unwise) policy to ignore domain errors + ('bad' arguments). +

+

+

+

+ Unless we are using the boost::math namespace, we will need: +

+

+

+

+ +

+
+using boost::math::policies::policy;
+using boost::math::policies::domain_error;
+using boost::math::policies::ignore_error;
+

+

+

+

+

+ Using a typedef is often convenient, especially if it is re-used, + although it is not required, as the various examples below show. +

+

+

+

+ +

+
+typedef policy<domain_error<ignore_error> > ignore_domain_policy;
+// find_location with new policy, using typedef.
+l = find_location<normal>(z, p, sd, ignore_domain_policy());
+// Default policy policy<>, needs "using boost::math::policies::policy;"
+l = find_location<normal>(z, p, sd, policy<>());
+// Default policy, fully specified.
+l = find_location<normal>(z, p, sd, boost::math::policies::policy<>());
+// A new policy, ignoring domain errors, without using a typedef.
+l = find_location<normal>(z, p, sd, policy<domain_error<ignore_error> >());
+

+

+

+

+

+ If we want to use a probability that is the complement + of our probability, we should not even think of writing find_location<normal>(z, 1 - p, sd), + but, to avoid loss of accuracy, + use the complement version. +

+

+

+

+ +

+
+z = 2.;
+double q = 0.95; // = 1 - p; // complement.
+l = find_location<normal>(complement(z, q, sd));
+
+normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(location) shifted
+cout << "Normal distribution with mean = " << l << " has " 
+  << "fraction <= " << z << " = "  << cdf(np95pc, z) << endl;
+cout << "Normal distribution with mean = " << l << " has " 
+  << "fraction > " << z << " = "  << cdf(complement(np95pc, z)) << endl;
+
+

+

+

+ See find_location_example.cpp + for full source code: the program output looks like this: +

+
Example: Find location (mean).
+Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501
+Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725
+offset location (mean) = 1.09023
+Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001
+Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999
+Normal distribution with mean = 0.355146 has fraction <= 2 = 0.95
+Normal distribution with mean = 0.355146 has fraction > 2 = 0.05
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html new file mode 100644 index 000000000..980163d0c --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html @@ -0,0 +1,750 @@ + + + +Find mean and standard deviation example + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+

+

+ First we need some includes to access the normal distribution, the + algorithms to find location and scale (and some std output of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp> // for normal_distribution
+  using boost::math::normal; // typedef provides default type is double.
+#include <boost/math/distributions/cauchy.hpp> // for cauchy_distribution
+  using boost::math::cauchy; // typedef provides default type is double.
+#include <boost/math/distributions/find_location.hpp>
+  using boost::math::find_location;
+#include <boost/math/distributions/find_scale.hpp>
+  using boost::math::find_scale;
+  using boost::math::complement;
+  using boost::math::policies::policy;
+
+#include <iostream>
+  using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
+#include <iomanip>
+  using std::setw; using std::setprecision;
+#include <limits>
+  using std::numeric_limits;
+
+

+

+

+ +

+
+ + Using + find_location and find_scale to meet dispensing and measurement specifications +
+

+

+

+ Consider an example from K Krishnamoorthy, Handbook of Statistical + Distributions with Applications, ISBN 1-58488-635-8, (2006) p 126, + example 10.3.7. +

+

+

+

+ "A machine is set to pack 3 kg of ground beef per pack. Over + a long period of time it is found that the average packed was 3 kg + with a standard deviation of 0.1 kg. Assume the packing is normally + distributed." +

+

+

+

+ We start by constructing a normal distribution with the given parameters: +

+

+

+

+ +

+
+double mean = 3.; // kg
+double standard_deviation = 0.1; // kg
+normal packs(mean, standard_deviation);
+

+

+

+

+

+ We can then find the fraction (or %) of packages that weigh more + than 3.1 kg. +

+

+

+

+ +

+
+double max_weight = 3.1; // kg
+cout << "Percentage of packs > " << max_weight << " is "
+<< cdf(complement(packs, max_weight)) * 100. << endl; // P(X > 3.1)
+

+

+

+

+

+ We might want to ensure that 95% of packs are over a minimum weight + specification, then we want the value of the mean such that P(X < + 2.9) = 0.05. +

+

+

+

+ Using the mean of 3 kg, we can estimate the fraction of packs that + fail to meet the specification of 2.9 kg. +

+

+

+

+ +

+
+double minimum_weight = 2.9;
+cout <<"Fraction of packs <= " << minimum_weight << " with a mean of " << mean
+  << " is " << cdf(complement(packs, minimum_weight)) << endl;
+// fraction of packs <= 2.9 with a mean of 3 is 0.841345
+

+

+

+

+

+ This is 0.84 - more than the target fraction of 0.95. If we want + 95% to be over the minimum weight, what should we set the mean weight + to be? +

+

+

+

+ Using the KK StatCalc program supplied with the book and the method + given on page 126 gives 3.06449. +

+

+

+

+ We can confirm this by constructing a new distribution which we call + 'xpacks' with a safety margin mean of 3.06449 thus: +

+

+

+

+ +

+
+double over_mean = 3.06449;
+normal xpacks(over_mean, standard_deviation);
+cout << "Fraction of packs >= " << minimum_weight
+<< " with a mean of " << xpacks.mean()
+  << " is " << cdf(complement(xpacks, minimum_weight)) << endl;
+// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005
+

+

+

+

+

+ Using this Math Toolkit, we can calculate the required mean directly + thus: +

+

+

+

+ +

+
+double under_fraction = 0.05;  // so 95% are above the minimum weight mean - sd = 2.9
+double low_limit = standard_deviation;
+double offset = mean - low_limit - quantile(packs, under_fraction);
+double nominal_mean = mean + offset;
+// mean + (mean - low_limit - quantile(packs, under_fraction));
+
+normal nominal_packs(nominal_mean, standard_deviation);
+cout << "Setting the packer to " << nominal_mean << " will mean that "
+  << "fraction of packs >= " << minimum_weight
+  << " is " << cdf(complement(nominal_packs, minimum_weight)) << endl;
+// Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95
+

+

+

+

+

+ This calculation is generalized as the free function called find_location. +

+

+

+

+ To use this we will need to +

+

+

+

+ +

+
+#include <boost/math/distributions/find_location.hpp>
+  using boost::math::find_location;
+

+

+

+

+

+ and then use find_location function to find safe_mean, & construct + a new normal distribution called 'goodpacks'. +

+

+

+

+ +

+
+double safe_mean = find_location<normal>(minimum_weight, under_fraction, standard_deviation);
+normal good_packs(safe_mean, standard_deviation);
+

+

+

+

+

+ with the same confirmation as before: +

+

+

+

+ +

+
+cout << "Setting the packer to " << nominal_mean << " will mean that "
+  << "fraction of packs >= " << minimum_weight
+  << " is " << cdf(complement(good_packs, minimum_weight)) << endl;
+// Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95
+

+

+

+ +

+
+ + Using + Cauchy-Lorentz instead of normal distribution +
+

+

+

+ After examining the weight distribution of a large number of packs, + we might decide that, after all, the assumption of a normal distribution + is not really justified. We might find that the fit is better to + a Cauchy + Distribution. This distribution has wider 'wings', so that + whereas most of the values are closer to the mean than the normal, + there are also more values than 'normal' that lie further from the + mean than the normal. +

+

+

+

+ This might happen because a larger than normal lump of meat is either + included or excluded. +

+

+

+

+ We first create a Cauchy + Distribution with the original mean and standard deviation, + and estimate the fraction that lie below our minimum weight specification. +

+

+

+

+ +

+
+cauchy cpacks(mean, standard_deviation);
+cout << "Cauchy Setting the packer to " << mean << " will mean that "
+  << "fraction of packs >= " << minimum_weight
+  << " is " << cdf(complement(cpacks, minimum_weight)) << endl;
+// Cauchy Setting the packer to 3 will mean that fraction of packs >= 2.9 is 0.75
+

+

+

+

+

+ Note that far fewer of the packs meet the specification, only 75% + instead of 95%. Now we can repeat the find_location, using the cauchy + distribution as template parameter, in place of the normal used above. +

+

+

+

+ +

+
+double lc = find_location<cauchy>(minimum_weight, under_fraction, standard_deviation);
+cout << "find_location<cauchy>(minimum_weight, over fraction, standard_deviation); " << lc << endl;
+// find_location<cauchy>(minimum_weight, over fraction, packs.standard_deviation()); 3.53138
+

+

+

+

+

+ Note that the safe_mean setting needs to be much higher, 3.53138 + instead of 3.06449, so we will make rather less profit. +

+

+

+

+ And again confirm that the fraction meeting specification is as expected. +

+

+

+

+ +

+
+cauchy goodcpacks(lc, standard_deviation);
+cout << "Cauchy Setting the packer to " << lc << " will mean that "
+  << "fraction of packs >= " << minimum_weight
+  << " is " << cdf(complement(goodcpacks, minimum_weight)) << endl;
+// Cauchy Setting the packer to 3.53138 will mean that fraction of packs >= 2.9 is 0.95
+

+

+

+

+

+ Finally we could estimate the effect of a much tighter specification, + that 99% of packs met the specification. +

+

+

+

+ +

+
+cout << "Cauchy Setting the packer to "
+  << find_location<cauchy>(minimum_weight, 0.99, standard_deviation)
+  << " will mean that "
+  << "fraction of packs >= " << minimum_weight
+  << " is " << cdf(complement(goodcpacks, minimum_weight)) << endl;
+

+

+

+

+

+ Setting the packer to 3.13263 will mean that fraction of packs >= + 2.9 is 0.99, but will more than double the mean loss from 0.0644 + to 0.133 kg per pack. +

+

+

+

+ Of course, this calculation is not limited to packs of meat, it applies + to dispensing anything, and it also applies to a 'virtual' material + like any measurement. +

+

+

+

+ The only caveat is that the calculation assumes that the standard + deviation (scale) is known with a reasonably low uncertainty, something + that is not so easy to ensure in practice. And that the distribution + is well defined, Normal + Distribution or Cauchy + Distribution, or some other. +

+

+

+

+ If one is simply dispensing a very large number of packs, then it + may be feasible to measure the weight of hundreds or thousands of + packs. With a healthy 'degrees of freedom', the confidence intervals + for the standard deviation are not too wide, typically about + and + - 10% for hundreds of observations. +

+

+

+

+ For other applications, where it is more difficult or expensive to + make many observations, the confidence intervals are depressingly + wide. +

+

+

+

+ See Confidence + Intervals on the standard deviation for a worked example + chi_square_std_dev_test.cpp + of estimating these intervals. +

+

+ +

+
+ + Changing + the scale or standard deviation +
+

+

+

+ Alternatively, we could invest in a better (more precise) packer + (or measuring device) with a lower standard deviation, or scale. +

+

+

+

+ This might cost more, but would reduce the amount we have to 'give + away' in order to meet the specification. +

+

+

+

+ To estimate how much better (how much smaller standard deviation) + it would have to be, we need to get the 5% quantile to be located + at the under_weight limit, 2.9 +

+

+

+

+ +

+
+double p = 0.05; // wanted p th quantile.
+cout << "Quantile of " << p << " = " << quantile(packs, p)
+  << ", mean = " << packs.mean() << ", sd = " << packs.standard_deviation() << endl;
+

+

+

+

+

+ Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 +

+

+

+

+ With the current packer (mean = 3, sd = 0.1), the 5% quantile is + at 2.8551 kg, a little below our target of 2.9 kg. So we know that + the standard deviation is going to have to be smaller. +

+

+

+

+ Let's start by guessing that it (now 0.1) needs to be halved, to + a standard deviation of 0.05 kg. +

+

+

+

+ +

+
+normal pack05(mean, 0.05);
+cout << "Quantile of " << p << " = " << quantile(pack05, p)
+  << ", mean = " << pack05.mean() << ", sd = " << pack05.standard_deviation() << endl;
+// Quantile of 0.05 = 2.91776, mean = 3, sd = 0.05
+
+cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean
+  << " and standard deviation of " << pack05.standard_deviation()
+  << " is " << cdf(complement(pack05, minimum_weight)) << endl;
+// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.97725
+

+

+

+

+

+ So 0.05 was quite a good guess, but we are a little over the 2.9 + target, so the standard deviation could be a tiny bit more. So we + could do some more guessing to get closer, say by increasing standard + deviation to 0.06 kg, constructing another new distribution called + pack06. +

+

+

+

+ +

+
+normal pack06(mean, 0.06);
+cout << "Quantile of " << p << " = " << quantile(pack06, p)
+  << ", mean = " << pack06.mean() << ", sd = " << pack06.standard_deviation() << endl;
+// Quantile of 0.05 = 2.90131, mean = 3, sd = 0.06
+
+cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean
+  << " and standard deviation of " << pack06.standard_deviation()
+  << " is " << cdf(complement(pack06, minimum_weight)) << endl;
+// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95221
+

+

+

+

+

+ Now we are getting really close, but to do the job properly, we might + need to use root finding method, for example the tools provided, + and used elsewhere, in the Math Toolkit, see Root + Finding Without Derivatives. +

+

+

+

+ But in this (normal) distribution case, we can and should be even + smarter and make a direct calculation. +

+

+

+

+ Our required limit is minimum_weight = 2.9 kg, often called the random + variate z. For a standard normal distribution, then probability p + = N((minimum_weight - mean) / sd). +

+

+

+

+ We want to find the standard deviation that would be required to + meet this limit, so that the p th quantile is located at z (minimum_weight). + In this case, the 0.05 (5%) quantile is at 2.9 kg pack weight, when + the mean is 3 kg, ensuring that 0.95 (95%) of packs are above the + minimum weight. +

+

+

+

+ Rearranging, we can directly calculate the required standard deviation: +

+

+

+

+ +

+
+normal N01; // standard normal distribution with meamn zero and unit standard deviation.
+p = 0.05;
+double qp = quantile(N01, p);
+double sd95 = (minimum_weight - mean) / qp;
+
+cout << "For the "<< p << "th quantile to be located at "
+  << minimum_weight << ", would need a standard deviation of " << sd95 << endl;
+// For the 0.05th quantile to be located at 2.9, would need a standard deviation of 0.0607957
+

+

+

+

+

+ We can now construct a new (normal) distribution pack95 for the 'better' + packer, and check that our distribution will meet the specification. +

+

+

+

+ +

+
+normal pack95(mean, sd95);
+cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean
+  << " and standard deviation of " << pack95.standard_deviation()
+  << " is " << cdf(complement(pack95, minimum_weight)) << endl;
+// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95
+

+

+

+

+

+ This calculation is generalized in the free function find_scale, + as shown below, giving the same standard deviation. +

+

+

+

+ +

+
+double ss = find_scale<normal>(minimum_weight, under_fraction, packs.mean());
+cout << "find_scale<normal>(minimum_weight, under_fraction, packs.mean()); " << ss << endl;
+// find_scale<normal>(minimum_weight, under_fraction, packs.mean()); 0.0607957
+

+

+

+

+

+ If we had defined an over_fraction, or percentage that must pass + specification +

+

+

+

+ +

+
+double over_fraction = 0.95;
+

+

+

+

+

+ And (wrongly) written +

+

+ +

+
+double sso = find_scale<normal>(minimum_weight, over_fraction, packs.mean());
+
+

+

+

+ With the default policy, we would get a message like +

+

+ +

+
Message from thrown exception was:
+   Error in function boost::math::find_scale<Dist, Policy>(double, double, double, Policy):
+   Computed scale (-0.060795683191176959) is <= 0! Was the complement intended?
+
+

+

+

+ But this would return a negative + standard deviation - obviously impossible. The probability should + be 1 - over_fraction, not over_fraction, thus: +

+

+

+

+ +

+
+double ss1o = find_scale<normal>(minimum_weight, 1 - over_fraction, packs.mean());
+cout << "find_scale<normal>(minimum_weight, under_fraction, packs.mean()); " << ss1o << endl;
+// find_scale<normal>(minimum_weight, under_fraction, packs.mean()); 0.0607957
+

+

+

+

+

+ But notice that using '1 - over_fraction' - will lead to a loss of accuracy, especially if over_fraction + was close to unity. In this (very common) case, we should + instead use the complements, giving + the most accurate result. +

+

+

+

+ +

+
+double ssc = find_scale<normal>(complement(minimum_weight, over_fraction, packs.mean()));
+cout << "find_scale<normal>(complement(minimum_weight, over_fraction, packs.mean())); " << ssc << endl;
+// find_scale<normal>(complement(minimum_weight, over_fraction, packs.mean())); 0.0607957
+

+

+

+

+

+ Note that our guess of 0.06 was close to the accurate value of 0.060795683191176959. +

+

+

+

+ We can again confirm our prediction thus: +

+

+

+

+ +

+
+normal pack95c(mean, ssc);
+cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean
+  << " and standard deviation of " << pack95c.standard_deviation()
+  << " is " << cdf(complement(pack95c, minimum_weight)) << endl;
+// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95
+

+

+

+

+

+ Notice that these two deceptively simple questions: +

+

+

+
  • + Do we over-fill to make sure we meet a minimum specification (or + under-fill to avoid an overdose)? +
+

+

+

+ and/or +

+

+

+
  • + Do we measure better? +
+

+

+

+ are actually extremely common. +

+

+

+

+ The weight of beef might be replaced by a measurement of more or + less anything, from drug tablet content, Apollo landing rocket firing, + X-ray treatment doses... +

+

+

+

+ The scale can be variation in dispensing or uncertainty in measurement. +

+

+ See find_mean_and_sd_normal.cpp + for full source code & appended program output. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html new file mode 100644 index 000000000..5b679fa4d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html @@ -0,0 +1,315 @@ + + + +Find Scale (Standard Deviation) Example + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+

+

+ First we need some includes to access the Normal + Distribution, the algorithms to find scale (and some std output + of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp> // for normal_distribution
+  using boost::math::normal; // typedef provides default type is double.
+#include <boost/math/distributions/find_scale.hpp>
+  using boost::math::find_scale; 
+  using boost::math::complement; // Needed if you want to use the complement version.
+  using boost::math::policies::policy; // Needed to specify the error handling policy.
+
+#include <iostream>
+  using std::cout; using std::endl;
+#include <iomanip>
+  using std::setw; using std::setprecision;
+#include <limits>
+  using std::numeric_limits;
+
+

+

+

+

+

+ For this example, we will use the standard Normal + Distribution, with location (mean) zero and standard deviation + (scale) unity. Conveniently, this is also the default for this implementation's + constructor. +

+

+

+

+ +

+
+normal N01;  // Default 'standard' normal distribution with zero mean
+double sd = 1.; // and standard deviation is 1.
+

+

+

+

+

+ Suppose we want to find a different normal distribution with standard + deviation so that only fraction p (here 0.001 or 0.1%) are below + a certain chosen limit (here -2. standard deviations). +

+

+

+

+ +

+
+double z = -2.; // z to give prob p
+double p = 0.001; // only 0.1% below z = -2
+
+cout << "Normal distribution with mean = " << N01.location()  // aka N01.mean()
+  << ", standard deviation " << N01.scale() // aka N01.standard_deviation()
+  << ", has " << "fraction <= " << z 
+  << ", p = "  << cdf(N01, z) << endl;
+cout << "Normal distribution with mean = " << N01.location()
+  << ", standard deviation " << N01.scale()
+  << ", has " << "fraction > " << z
+  << ", p = "  << cdf(complement(N01, z)) << endl; // Note: uses complement.
+

+

+

+ +

+
Normal distribution with mean = 0 has fraction <= -2, p = 0.0227501
+Normal distribution with mean = 0 has fraction > -2, p = 0.97725
+
+

+

+

+ Noting that p = 0.02 instead of our target of 0.001, we can now use + find_scale to give + a new standard deviation. +

+

+

+

+ +

+
+double l = N01.location();
+double s = find_scale<normal>(z, p, l);
+cout << "scale (standard deviation) = " << s << endl;
+

+

+

+

+

+ that outputs: +

+

+ +

+
scale (standard deviation) = 0.647201
+
+

+

+

+ showing that we need to reduce the standard deviation from 1. to + 0.65. +

+

+

+

+ Then we can check that we have achieved our objective by constructing + a new distribution with the new standard deviation (but same zero + mean): +

+

+

+

+ +

+
+normal np001pc(N01.location(), s);
+

+

+

+

+

+ And re-calculating the fraction below (and above) our chosen limit. +

+

+

+

+ +

+
+cout << "Normal distribution with mean = " << l 
+  << " has " << "fraction <= " << z 
+  << ", p = "  << cdf(np001pc, z) << endl;
+cout << "Normal distribution with mean = " << l 
+  << " has " << "fraction > " << z 
+  << ", p = "  << cdf(complement(np001pc, z)) << endl;
+

+

+

+ +

+
Normal distribution with mean = 0 has fraction <= -2, p = 0.001
+Normal distribution with mean = 0 has fraction > -2, p = 0.999
+
+

+ +

+
+ + Controlling + how Errors from find_scale are handled +
+

+

+

+ We can also control the policy for handling various errors. For example, + we can define a new (possibly unwise) policy to ignore domain errors + ('bad' arguments). +

+

+

+

+ Unless we are using the boost::math namespace, we will need: +

+

+

+

+ +

+
+using boost::math::policies::policy;
+using boost::math::policies::domain_error;
+using boost::math::policies::ignore_error;
+

+

+

+

+

+ Using a typedef is convenient, especially if it is re-used, although + it is not required, as the various examples below show. +

+

+

+

+ +

+
+typedef policy<domain_error<ignore_error> > ignore_domain_policy;
+// find_scale with new policy, using typedef.
+l = find_scale<normal>(z, p, l, ignore_domain_policy());
+// Default policy policy<>, needs using boost::math::policies::policy;
+
+l = find_scale<normal>(z, p, l, policy<>());
+// Default policy, fully specified.
+l = find_scale<normal>(z, p, l, boost::math::policies::policy<>());
+// New policy, without typedef.
+l = find_scale<normal>(z, p, l, policy<domain_error<ignore_error> >());
+

+

+

+

+

+ If we want to express a probability, say 0.999, that is a complement, + 1 - + p we should not even think + of writing find_scale<normal>(z, 1 - p, l), but instead, + use the complements version. +

+

+

+

+ +

+
+z = -2.;
+double q = 0.999; // = 1 - p; // complement of 0.001.
+sd = find_scale<normal>(complement(z, q, l));
+
+normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(scale) shifted
+cout << "Normal distribution with mean = " << l << " has " 
+  << "fraction <= " << z << " = "  << cdf(np95pc, z) << endl;
+cout << "Normal distribution with mean = " << l << " has " 
+  << "fraction > " << z << " = "  << cdf(complement(np95pc, z)) << endl;
+

+

+

+

+

+ Sadly, it is all too easy to get probabilities the wrong way round, + when you may get a warning like this: +

+

+ +

+
Message from thrown exception was:
+   Error in function boost::math::find_scale<Dist, Policy>(complement(double, double, double, Policy)):
+   Computed scale (-0.48043523852179076) is <= 0! Was the complement intended?
+
+

+

+

+ The default error handling policy is to throw an exception with this + message, but if you chose a policy to ignore the error, the (impossible) + negative scale is quietly returned. +

+

+ See find_scale_example.cpp + for full source code: the program output looks like this: +

+
Example: Find scale (standard deviation).
+Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501
+Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725
+scale (standard deviation) = 0.647201
+Normal distribution with mean = 0 has fraction <= -2, p = 0.001
+Normal distribution with mean = 0 has fraction > -2, p = 0.999
+Normal distribution with mean = 0.946339 has fraction <= -2 = 0.001
+Normal distribution with mean = 0.946339 has fraction > -2 = 0.999
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/nag_library.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/nag_library.html new file mode 100644 index 000000000..24fed1d00 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/nag_library.html @@ -0,0 +1,118 @@ + + + +Comparison with C, R, FORTRAN-style Free Functions + + + + + + + + + + + + + + + +
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+
+

+ You are probably familiar with a statistics library that has free functions, + for example the classic NAG + C library and matching NAG + FORTRAN Library, Microsoft + Excel BINOMDIST(number_s,trials,probability_s,cumulative), R, MathCAD + pbinom and many others. +

+

+ If so, you may find 'Distributions as Objects' unfamiliar, if not alien. +

+

+ However, do not panic, both definition + and usage are not really very different. +

+

+ A very simple example of generating the same values as the NAG + C library for the binomial distribution follows. (If you find + slightly different values, the Boost C++ version, using double or better, + is very likely to be the more accurate. Of course, accuracy is not usually + a concern for most applications of this function). +

+

+ The NAG + function specification is +

+
+void nag_binomial_dist(Integer n, double p, Integer k,
+double *plek, double *pgtk, double *peqk, NagError *fail)
+
+

+ and is called +

+
+g01bjc(n, p, k, &plek, &pgtk, &peqk, NAGERR_DEFAULT);
+
+

+ The equivalent using this Boost C++ library is: +

+
+using namespace boost::math;  // Using declaration avoids very long names.
+binomial my_dist(4, 0.5); // c.f. NAG n = 4, p = 0.5
+
+

+ and values can be output thus: +

+
+cout
+  << my_dist.trials() << " "             // Echo the NAG input n = 4 trials.
+  << my_dist.success_fraction() << " "   // Echo the NAG input p = 0.5
+  << cdf(my_dist, 2) << "  "             // NAG plek with k = 2
+  << cdf(complement(my_dist, 2)) << "  " // NAG pgtk with k = 2
+  << pdf(my_dist, 2) << endl;            // NAG peqk with k = 2
+
+

+ cdf(dist, k) + is equivalent to NAG library plek, + lower tail probability of <= k +

+

+ cdf(complement(dist, k)) + is equivalent to NAG library pgtk, + upper tail probability of > k +

+

+ pdf(dist, k) + is equivalent to NAG library peqk, + point probability of == k +

+

+ See binomial_example_nag.cpp + for details. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html new file mode 100644 index 000000000..7d855f48d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html @@ -0,0 +1,60 @@ + + + +Negative Binomial Distribution Examples + + + + + + + + + + + + + + + +
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+
+
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+
+
+
+
+ Calculating Confidence Limits on the Frequency of Occurrence for the + Negative Binomial Distribution
+
+ Estimating Sample Sizes for the Negative Binomial.
+
+ Negative Binomial Sales Quota Example.
+
+ Negative Binomial Table Printing Example.
+
+

+ (See also the reference documentation for the Negative + Binomial Distribution.) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html new file mode 100644 index 000000000..7c2523b1d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html @@ -0,0 +1,271 @@ + + + +Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution + + + + + + + + + + + + + + + +
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+
+

+ Imagine you have a process that follows a negative binomial distribution: + for each trial conducted, an event either occurs or does it does not, + referred to as "successes" and "failures". The + frequency with which successes occur is variously referred to as the + success fraction, success ratio, success percentage, occurrence frequency, + or probability of occurrence. +

+

+ If, by experiment, you want to measure the the best estimate of success + fraction is given simply by k / N, + for k successes out of N + trials. +

+

+ However our confidence in that estimate will be shaped by how many + trials were conducted, and how many successes were observed. The static + member functions negative_binomial_distribution<>::find_lower_bound_on_p + and negative_binomial_distribution<>::find_upper_bound_on_p + allow you to calculate the confidence intervals for your estimate of + the success fraction. +

+

+ The sample program neg_binom_confidence_limits.cpp + illustrates their use. +

+

+

+

+ First we need some includes to access the negative binomial distribution + (and some basic std output of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+using boost::math::negative_binomial;
+
+#include <iostream>
+using std::cout; using std::endl;
+#include <iomanip>
+using std::setprecision;
+using std::setw; using std::left; using std::fixed; using std::right;
+

+

+

+

+

+ First define a table of significance levels: these are the probabilities + that the true occurrence frequency lies outside the calculated interval: +

+

+

+

+ +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+

+

+

+

+

+ Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% + confidence that the true occurence frequency lies inside + the calculated interval. +

+

+

+

+ We need a function to calculate and print confidence limits for an + observed frequency of occurrence that follows a negative binomial + distribution. +

+

+

+

+ +

+
+void confidence_limits_on_frequency(unsigned trials, unsigned successes)
+{
+   // trials = Total number of trials.
+   // successes = Total number of observed successes.
+   // failures = trials - successes.
+   // success_fraction = successes /trials.
+   // Print out general info:
+   cout <<
+      "______________________________________________\n"
+      "2-Sided Confidence Limits For Success Fraction\n"
+      "______________________________________________\n\n";
+   cout << setprecision(7);
+   cout << setw(40) << left << "Number of trials" << " =  " << trials << "\n";
+   cout << setw(40) << left << "Number of successes" << " =  " << successes << "\n";
+   cout << setw(40) << left << "Number of failures" << " =  " << trials - successes << "\n";
+   cout << setw(40) << left << "Observed frequency of occurrence" << " =  " << double(successes) / trials << "\n";
+
+   // Print table header:
+   cout << "\n\n"
+           "___________________________________________\n"
+           "Confidence        Lower          Upper\n"
+           " Value (%)        Limit          Limit\n"
+           "___________________________________________\n";
+

+

+

+

+

+ And now for the important part - the bounds themselves. For each + value of alpha, we call find_lower_bound_on_p + and find_upper_bound_on_p + to obtain lower and upper bounds respectively. Note that since we + are calculating a two-sided interval, we must divide the value of + alpha in two. Had we been calculating a single-sided interval, for + example: "Calculate a lower bound so that we are P% + sure that the true occurrence frequency is greater than some value" + then we would not have divided by + two. +

+

+

+

+ +

+
+   // Now print out the upper and lower limits for the alpha table values.
+   for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+   {
+      // Confidence value:
+      cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+      // Calculate bounds:
+      double lower = negative_binomial::find_lower_bound_on_p(trials, successes, alpha[i]/2);
+      double upper = negative_binomial::find_upper_bound_on_p(trials, successes, alpha[i]/2);
+      // Print limits:
+      cout << fixed << setprecision(5) << setw(15) << right << lower;
+      cout << fixed << setprecision(5) << setw(15) << right << upper << endl;
+   }
+   cout << endl;
+} // void confidence_limits_on_frequency(unsigned trials, unsigned successes)
+

+

+

+

+

+ And then call confidence_limits_on_frequency with increasing numbers + of trials, but always the same success fraction 0.1, or 1 in 10. +

+

+

+

+ +

+
+int main()
+{
+  confidence_limits_on_frequency(20, 2); // 20 trials, 2 successes, 2 in 20, = 1 in 10 = 0.1 success fraction.
+  confidence_limits_on_frequency(200, 20); // More trials, but same 0.1 success fraction.
+  confidence_limits_on_frequency(2000, 200); // Many more trials, but same 0.1 success fraction.
+
+  return 0;
+} // int main()
+
+
+

+

+

+ Let's see some sample output for a 1 in 10 success ratio, first for + a mere 20 trials: +

+
______________________________________________
+2-Sided Confidence Limits For Success Fraction
+______________________________________________
+Number of trials                         =  20
+Number of successes                      =  2
+Number of failures                       =  18
+Observed frequency of occurrence         =  0.1
+___________________________________________
+Confidence        Lower          Upper
+ Value (%)        Limit          Limit
+___________________________________________
+    50.000        0.04812        0.13554
+    75.000        0.03078        0.17727
+    90.000        0.01807        0.22637
+    95.000        0.01235        0.26028
+    99.000        0.00530        0.33111
+    99.900        0.00164        0.41802
+    99.990        0.00051        0.49202
+    99.999        0.00016        0.55574
+
+

+ As you can see, even at the 95% confidence level the bounds (0.012 + to 0.26) are really very wide, and very asymmetric about the observed + value 0.1. +

+

+ Compare that with the program output for a mass 2000 trials: +

+
______________________________________________
+2-Sided Confidence Limits For Success Fraction
+______________________________________________
+Number of trials                         =  2000
+Number of successes                      =  200
+Number of failures                       =  1800
+Observed frequency of occurrence         =  0.1
+___________________________________________
+Confidence        Lower          Upper
+ Value (%)        Limit          Limit
+___________________________________________
+    50.000        0.09536        0.10445
+    75.000        0.09228        0.10776
+    90.000        0.08916        0.11125
+    95.000        0.08720        0.11352
+    99.000        0.08344        0.11802
+    99.900        0.07921        0.12336
+    99.990        0.07577        0.12795
+    99.999        0.07282        0.13206
+
+

+ Now even when the confidence level is very high, the limits (at 99.999%, + 0.07 to 0.13) are really quite close and nearly symmetric to the observed + value of 0.1. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html new file mode 100644 index 000000000..cb8c5d512 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html @@ -0,0 +1,254 @@ + + + +Estimating Sample Sizes for the Negative Binomial. + + + + + + + + + + + + + + + +
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+
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+
+
+

+ Imagine you have an event (let's call it a "failure" - though + we could equally well call it a success if we felt it was a 'good' + event) that you know will occur in 1 in N trials. You may want to know + how many trials you need to conduct to be P% sure of observing at least + k such failures. If the failure events follow a negative binomial distribution + (each trial either succeeds or fails) then the static member function + negative_binomial_distibution<>::find_minimum_number_of_trials + can be used to estimate the minimum number of trials required to be + P% sure of observing the desired number of failures. +

+

+ The example program neg_binomial_sample_sizes.cpp + demonstrates its usage. +

+

+

+

+ It centres around a routine that prints out a table of minimum sample + sizes for various probability thresholds: +

+

+

+

+ +

+
+void find_number_of_trials(double failures, double p);
+

+

+

+

+

+ First define a table of significance levels: these are the maximum + acceptable probability that failure or fewer + events will be observed. +

+

+

+

+ +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+

+

+

+

+

+ Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% + confidence that the desired number of failures will be observed. +

+

+

+

+ Much of the rest of the program is pretty-printing, the important + part is in the calculation of minimum number of trials required for + each value of alpha using: +

+

+ +

+
+(int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]);
+
+

+

+

+ find_minimum_number_of_trials returns a double, so ceil rounds this + up to ensure we have an integral minimum number of trials. +

+

+

+

+ +

+
+  
+void find_number_of_trials(double failures, double p)
+{
+   // trials = number of trials
+   // failures = number of failures before achieving required success(es).
+   // p        = success fraction (0 <= p <= 1.).
+   //
+   // Calculate how many trials we need to ensure the
+   // required number of failures DOES exceed "failures".
+
+  cout << "\n""Target number of failures = " << failures;
+  cout << ",   Success fraction = " << 100 * p << "%" << endl;
+   // Print table header:
+   cout << "\n\n"
+           "____________________________\n"
+           "Confidence        Min Number\n"
+           " Value (%)        Of Trials \n"
+           "____________________________\n";
+   // Now print out the data for the alpha table values.
+  for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+   { // Confidence values %:
+      cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]) << "      "
+      // find_minimum_number_of_trials
+      << setw(6) << right
+      << (int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
+      << endl;
+   }
+   cout << endl;
+} // void find_number_of_trials(double failures, double p)
+

+

+

+

+

+ finally we can produce some tables of minimum trials for the chosen + confidence levels: +

+

+

+

+ +

+
+int main()
+{
+    find_number_of_trials(5, 0.5);
+    find_number_of_trials(50, 0.5);
+    find_number_of_trials(500, 0.5);
+    find_number_of_trials(50, 0.1);
+    find_number_of_trials(500, 0.1);
+    find_number_of_trials(5, 0.9);
+
+    return 0;
+} // int main()
+
+
+

+

+

+

+
+ + + + + +
[Note]Note
+

+ Since we're calculating the minimum number of + trials required, we'll err on the safe side and take the ceiling + of the result. Had we been calculating the maximum + number of trials permitted to observe less than a certain number + of failures then we would have taken the floor + instead. We would also have called find_minimum_number_of_trials + like this: +

+
+floor(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
+
+

+ which would give us the largest number of trials we could conduct + and still be P% sure of observing failures or less + failure events, when the probability of success is p. +

+
+

+ We'll finish off by looking at some sample output, firstly suppose + we wish to observe at least 5 "failures" with a 50/50 (0.5) + chance of success or failure: +

+
Target number of failures = 5,   Success fraction = 50%
+
+____________________________
+Confidence        Min Number
+ Value (%)        Of Trials
+____________________________
+    50.000          11
+    75.000          14
+    90.000          17
+    95.000          18
+    99.000          22
+    99.900          27
+    99.990          31
+    99.999          36
+
+
+

+ So 18 trials or more would yield a 95% chance that at least our 5 required + failures would be observed. +

+

+ Compare that to what happens if the success ratio is 90%: +

+
Target number of failures = 5.000,   Success fraction = 90.000%
+
+____________________________
+Confidence        Min Number
+ Value (%)        Of Trials
+____________________________
+    50.000          57
+    75.000          73
+    90.000          91
+    95.000         103
+    99.000         127
+    99.900         159
+    99.990         189
+    99.999         217
+
+

+ So now 103 trials are required to observe at least 5 failures with + 95% certainty. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html new file mode 100644 index 000000000..3680eed66 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html @@ -0,0 +1,834 @@ + + + +Negative Binomial Sales Quota Example. + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ This example program negative_binomial_example1.cpp + (full source code) demonstrates a simple use to find the probability + of meeting a sales quota. +

+

+

+

+ Based on a + problem by Dr. Diane Evans, Professor of Mathematics at Rose-Hulman + Institute of Technology. +

+

+

+

+ Pat is required to sell candy bars to raise money for the 6th grade + field trip. There are thirty houses in the neighborhood, and Pat + is not supposed to return home until five candy bars have been sold. + So the child goes door to door, selling candy bars. At each house, + there is a 0.4 probability (40%) of selling one candy bar and a 0.6 + probability (60%) of selling nothing. +

+

+

+

+ What is the probability mass (density) function (pdf) for selling + the last (fifth) candy bar at the nth house? +

+

+

+

+ The Negative Binomial(r, p) distribution describes the probability + of k failures and r successes in k+r Bernoulli(p) trials with success + on the last trial. (A Bernoulli + trial is one with only two possible outcomes, success of + failure, and p is the probability of success). See also http://en.wikipedia.org/wiki/Bernoulli_distribution + Bernoulli distribution and Bernoulli + applications. +

+

+

+

+ In this example, we will deliberately produce a variety of calculations + and outputs to demonstrate the ways that the negative binomial distribution + can be implemented with this library: it is also deliberately over-commented. +

+

+

+

+ First we need to #define macros to control the error and discrete + handling policies. For this simple example, we want to avoid throwing + an exception (the default policy) and just return infinity. We want + to treat the distribution as if it was continuous, so we choose a + discrete_quantile policy of real, rather than the default policy + integer_round_outwards. +

+

+

+

+ +

+
+#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error
+#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+

+

+

+

+

+ After that we need some includes to provide easy access to the negative + binomial distribution, +

+

+

+
+ + + + + +
[Caution]Caution

+ It is vital to #include distributions etc after + the above #defines +

+

+

+

+ and we need some std library iostream, of course. +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+  // for negative_binomial_distribution
+  using boost::math::negative_binomial; // typedef provides default type is double.
+  using  ::boost::math::pdf; // Probability mass function.
+  using  ::boost::math::cdf; // Cumulative density function.
+  using  ::boost::math::quantile;
+
+#include <iostream>
+  using std::cout; using std::endl;
+  using std::noshowpoint; using std::fixed; using std::right; using std::left;
+#include <iomanip>
+  using std::setprecision; using std::setw; 
+
+#include <limits>
+  using std::numeric_limits;
+
+

+

+

+

+

+ It is always sensible to use try and catch blocks because defaults + policies are to throw an exception if anything goes wrong. +

+

+

+

+ A simple catch block (see below) will ensure that you get a helpful + error message instead of an abrupt program abort. +

+

+

+

+ +

+
+try
+{
+

+

+

+

+

+ Selling five candy bars means getting five successes, so successes + r = 5. The total number of trials (n, in this case, houses visited) + this takes is therefore = sucesses + failures or k + r = k + 5. +

+

+

+

+ +

+
+double sales_quota = 5; // Pat's sales quota - successes (r).
+

+

+

+

+

+ At each house, there is a 0.4 probability (40%) of selling one candy + bar and a 0.6 probability (60%) of selling nothing. +

+

+

+

+ +

+
+double success_fraction = 0.4; // success_fraction (p) - so failure_fraction is 0.6.
+

+

+

+

+

+ The Negative Binomial(r, p) distribution describes the probability + of k failures and r successes in k+r Bernoulli(p) trials with success + on the last trial. (A Bernoulli + trial is one with only two possible outcomes, success of + failure, and p is the probability of success). +

+

+

+

+ We therefore start by constructing a negative binomial distribution + with parameters sales_quota (required successes) and probability + of success. +

+

+

+

+ +

+
+negative_binomial nb(sales_quota, success_fraction); // type double by default.
+

+

+

+

+

+ To confirm, display the success_fraction & successes parameters + of the distribution. +

+

+

+

+ +

+
+cout << "Pat has a sales per house success rate of " << success_fraction
+  << ".\nTherefore he would, on average, sell " << nb.success_fraction() * 100
+  << " bars after trying 100 houses." << endl;
+
+int all_houses = 30; // The number of houses on the estate.
+
+cout << "With a success rate of " << nb.success_fraction() 
+  << ", he might expect, on average,\n"
+    "to need to visit about " << success_fraction * all_houses
+    << " houses in order to sell all " << nb.successes() << " bars. " << endl;
+

+

+

+ +

+
Pat has a sales per house success rate of 0.4.
+Therefore he would, on average, sell 40 bars after trying 100 houses.
+With a success rate of 0.4, he might expect, on average,
+to need to visit about 12 houses in order to sell all 5 bars. 
+
+

+

+

+ The random variable of interest is the number of houses that must + be visited to sell five candy bars, so we substitute k = n - 5 into + a negative_binomial(5, 0.4) and obtain the probability + mass (density) function (pdf or pmf) of the distribution of + houses visited. Obviously, the best possible case is that Pat makes + sales on all the first five houses. +

+

+

+

+ We calculate this using the pdf function: +

+

+

+

+ +

+
+cout << "Probability that Pat finishes on the " << sales_quota << "th house is "
+  << pdf(nb, 5 - sales_quota) << endl; // == pdf(nb, 0)
+

+

+

+

+

+ Of course, he could not finish on fewer than 5 houses because he + must sell 5 candy bars. So the 5th house is the first that he could + possibly finish on. +

+

+

+

+ To finish on or before the 8th house, Pat must finish at the 5th, + 6th, 7th or 8th house. The probability that he will finish on exactly ( == ) on any house is the Probability + Density Function (pdf). +

+

+

+

+ +

+
+cout << "Probability that Pat finishes on the 6th house is "
+  << pdf(nb, 6 - sales_quota) << endl;
+cout << "Probability that Pat finishes on the 7th house is "
+  << pdf(nb, 7 - sales_quota) << endl;
+cout << "Probability that Pat finishes on the 8th house is "
+  << pdf(nb, 8 - sales_quota) << endl;
+

+

+

+ +

+
Probability that Pat finishes on the 6th house is 0.03072
+Probability that Pat finishes on the 7th house is 0.055296
+Probability that Pat finishes on the 8th house is 0.077414
+
+

+

+

+ The sum of the probabilities for these houses is the Cumulative Distribution + Function (cdf). We can calculate it by adding the individual probabilities. +

+

+

+

+ +

+
+cout << "Probability that Pat finishes on or before the 8th house is sum "
+  "\n" << "pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = "
+  // Sum each of the mass/density probabilities for houses sales_quota = 5, 6, 7, & 8.
+  << pdf(nb, 5 - sales_quota) // 0 failures.
+    + pdf(nb, 6 - sales_quota) // 1 failure.
+    + pdf(nb, 7 - sales_quota) // 2 failures.
+    + pdf(nb, 8 - sales_quota) // 3 failures.
+  << endl;
+

+

+

+ +

+
pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367
+
+

+

+

+ Or, usually better, by using the negative binomial cumulative + distribution function. +

+

+

+

+ +

+
+cout << "\nProbability of selling his quota of " << sales_quota
+  << " bars\non or before the " << 8 << "th house is "
+  << cdf(nb, 8 - sales_quota) << endl;
+

+

+

+ +

+
Probability of selling his quota of 5 bars on or before the 8th house is 0.17367
+
+

+

+

+ +

+
+cout << "\nProbability that Pat finishes exactly on the 10th house is "
+  << pdf(nb, 10 - sales_quota) << endl;
+cout << "\nProbability of selling his quota of " << sales_quota
+  << " bars\non or before the " << 10 << "th house is "
+  << cdf(nb, 10 - sales_quota) << endl;
+

+

+

+ +

+
Probability that Pat finishes exactly on the 10th house is 0.10033
+Probability of selling his quota of 5 bars on or before the 10th house is 0.3669
+
+

+

+

+ +

+
+cout << "Probability that Pat finishes exactly on the 11th house is "
+  << pdf(nb, 11 - sales_quota) << endl;
+cout << "\nProbability of selling his quota of " << sales_quota
+  << " bars\non or before the " << 11 << "th house is "
+  << cdf(nb, 11 - sales_quota) << endl;
+

+

+

+ +

+
Probability that Pat finishes on the 11th house is 0.10033
+Probability of selling his quota of 5 candy bars
+on or before the 11th house is 0.46723
+
+

+

+

+ +

+
+cout << "Probability that Pat finishes exactly on the 12th house is "
+  << pdf(nb, 12 - sales_quota) << endl;
+
+cout << "\nProbability of selling his quota of " << sales_quota
+  << " bars\non or before the " << 12 << "th house is "
+  << cdf(nb, 12 - sales_quota) << endl;
+

+

+

+ +

+
Probability that Pat finishes on the 12th house is 0.094596
+Probability of selling his quota of 5 candy bars
+on or before the 12th house is 0.56182
+
+

+

+

+ Finally consider the risk of Pat not selling his quota of 5 bars + even after visiting all the houses. Calculate the probability that + he will sell on or before the last house: Calculate + the probability that he would sell all his quota on the very last + house. +

+

+

+

+ +

+
+cout << "Probability that Pat finishes on the " << all_houses
+  << " house is " << pdf(nb, all_houses - sales_quota) << endl;
+

+

+

+

+

+ Probability of selling his quota of 5 bars on the 30th house is +

+

+ +

+
Probability that Pat finishes on the 30 house is 0.00069145
+
+

+

+

+ when he'd be very unlucky indeed! +

+

+

+

+ What is the probability that Pat exhausts all 30 houses in the neighborhood, + and still doesn't sell the required + 5 candy bars? +

+

+

+

+ +

+
+cout << "\nProbability of selling his quota of " << sales_quota
+  << " bars\non or before the " << all_houses << "th house is "
+  << cdf(nb, all_houses - sales_quota) << endl;
+

+

+

+ +

+
Probability of selling his quota of 5 bars
+on or before the 30th house is 0.99849
+
+

+

+

+ /*So the + risk of + failing even + after visiting + all the + houses is + 1 - + this probability, 1 + - cdf(nb, all_houses + - sales_quota + But using + this expression + may cause + serious inaccuracy, so + it would + be much + better to + use the + complement of + the cdf: So + the risk + of failing + even at, or after, + the 31th (non-existent) + houses is + 1 - + this probability, 1 + - cdf(nb, all_houses + - sales_quota)` But using this expression may cause + serious inaccuracy. So it would be much better to use the complement + of the cdf. Why complements? +

+

+

+

+ +

+
+cout << "\nProbability of failing to sell his quota of " << sales_quota
+  << " bars\neven after visiting all " << all_houses << " houses is "
+  << cdf(complement(nb, all_houses - sales_quota)) << endl;
+

+

+

+ +

+
Probability of failing to sell his quota of 5 bars
+even after visiting all 30 houses is 0.0015101
+
+

+

+

+ We can also use the quantile (percentile), the inverse of the cdf, + to predict which house Pat will finish on. So for the 8th house: +

+

+

+

+ +

+
+double p = cdf(nb, (8 - sales_quota)); 
+cout << "Probability of meeting sales quota on or before 8th house is "<< p << endl;
+

+

+

+ +

+
Probability of meeting sales quota on or before 8th house is 0.174
+
+

+

+

+ +

+
+cout << "If the confidence of meeting sales quota is " << p
+    << ", then the finishing house is " << quantile(nb, p) + sales_quota << endl;
+
+cout<< " quantile(nb, p) = " << quantile(nb, p) << endl;
+

+

+

+ +

+
If the confidence of meeting sales quota is 0.17367, then the finishing house is 8
+
+

+

+

+ Demanding absolute certainty that all 5 will be sold, implies an + infinite number of trials. (Of course, there are only 30 houses on + the estate, so he can't ever be certain + of selling his quota). +

+

+

+

+ +

+
+cout << "If the confidence of meeting sales quota is " << 1.
+    << ", then the finishing house is " << quantile(nb, 1) + sales_quota << endl;
+//  1.#INF == infinity.
+

+

+

+ +

+
If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF
+
+

+

+

+ And similarly for a few other probabilities: +

+

+

+

+ +

+
+cout << "If the confidence of meeting sales quota is " << 0.
+    << ", then the finishing house is " << quantile(nb, 0.) + sales_quota << endl;
+
+cout << "If the confidence of meeting sales quota is " << 0.5
+    << ", then the finishing house is " << quantile(nb, 0.5) + sales_quota << endl;
+
+cout << "If the confidence of meeting sales quota is " << 1 - 0.00151 // 30 th
+    << ", then the finishing house is " << quantile(nb, 1 - 0.00151) + sales_quota << endl;
+

+

+

+ +

+
If the confidence of meeting sales quota is 0, then the finishing house is 5
+If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337
+If the confidence of meeting sales quota is 0.99849, then the finishing house is 30
+
+

+

+

+ Notice that because we chose a discrete quantile policy of real, + the result can be an 'unreal' fractional house. +

+

+

+

+ If the opposite is true, we don't want to assume any confidence, + then this is tantamount to assuming that all the first sales_quota + trials will be successful sales. +

+

+

+

+ +

+
+cout << "If confidence of meeting quota is zero\n(we assume all houses are successful sales)" 
+  ", then finishing house is " << sales_quota << endl;
+

+

+

+ +

+
If confidence of meeting quota is zero (we assume all houses are successful sales), then finishing house is 5
+If confidence of meeting quota is 0, then finishing house is 5
+
+

+

+

+ We can list quantiles for a few probabilities: +

+

+

+

+ +

+
+ double ps[] = {0., 0.001, 0.01, 0.05, 0.1, 0.5, 0.9, 0.95, 0.99, 0.999, 1.};
+ // Confidence as fraction = 1-alpha, as percent =  100 * (1-alpha[i]) %
+ cout.precision(3);
+ for (int i = 0; i < sizeof(ps)/sizeof(ps[0]); i++)
+ {
+   cout << "If confidence of meeting quota is " << ps[i]
+     << ", then finishing house is " << quantile(nb, ps[i]) + sales_quota
+     << endl;
+}
+

+

+

+ +

+
If confidence of meeting quota is 0, then finishing house is 5
+If confidence of meeting quota is 0.001, then finishing house is 5
+If confidence of meeting quota is 0.01, then finishing house is 5
+If confidence of meeting quota is 0.05, then finishing house is 6.2
+If confidence of meeting quota is 0.1, then finishing house is 7.06
+If confidence of meeting quota is 0.5, then finishing house is 11.3
+If confidence of meeting quota is 0.9, then finishing house is 17.8
+If confidence of meeting quota is 0.95, then finishing house is 20.1
+If confidence of meeting quota is 0.99, then finishing house is 24.8
+If confidence of meeting quota is 0.999, then finishing house is 31.1
+If confidence of meeting quota is 1, then finishing house is 1.#INF
+
+

+

+

+ We could have applied a ceil function to obtain a 'worst case' integer + value for house. +

+
+ceil(quantile(nb, ps[i]))
+

+

+

+

+

+ Or, if we had used the default discrete quantile policy, integer_outside, + by omitting +

+
+#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+

+ we would have achieved the same effect. +

+

+

+

+ The real result gives some suggestion which house is most likely. + For example, compare the real and integer_outside for 95% confidence. +

+

+ +

+
If confidence of meeting quota is 0.95, then finishing house is 20.1
+If confidence of meeting quota is 0.95, then finishing house is 21
+
+

+

+

+ The real value 20.1 is much closer to 20 than 21, so integer_outside + is pessimistic. We could also use integer_round_nearest policy to + suggest that 20 is more likely. +

+

+

+

+ Finally, we can tabulate the probability for the last sale being + exactly on each house. +

+

+

+

+ +

+
+cout << "\nHouse for " << sales_quota << "th (last) sale.  Probability (%)" << endl;
+cout.precision(5);
+for (int i = (int)sales_quota; i < all_houses+1; i++)
+{
+  cout << left << setw(3) << i << "                             " << setw(8) << cdf(nb, i - sales_quota)  << endl;
+}
+cout << endl;
+

+

+

+ +

+
House for 5 th (last) sale.  Probability (%)
+5                               0.01024 
+6                               0.04096 
+7                               0.096256
+8                               0.17367 
+9                               0.26657 
+10                              0.3669  
+11                              0.46723 
+12                              0.56182 
+13                              0.64696 
+14                              0.72074 
+15                              0.78272 
+16                              0.83343 
+17                              0.874   
+18                              0.90583 
+19                              0.93039 
+20                              0.94905 
+21                              0.96304 
+22                              0.97342 
+23                              0.98103 
+24                              0.98655 
+25                              0.99053 
+26                              0.99337 
+27                              0.99539 
+28                              0.99681 
+29                              0.9978  
+30                              0.99849
+
+

+

+

+ As noted above, using a catch block is always a good idea, even if + you do not expect to use it. +

+

+

+

+ +

+
+}
+catch(const std::exception& e)
+{ // Since we have set an overflow policy of ignore_error,
+  // an overflow exception should never be thrown.
+   std::cout << "\nMessage from thrown exception was:\n " << e.what() << std::endl;
+

+

+

+

+

+ For example, without a ignore domain error policy, if we asked for + +

+
+pdf(nb, -1)
+

+ for example, we would get: +

+

+ +

+
Message from thrown exception was:
+ Error in function boost::math::pdf(const negative_binomial_distribution<double>&, double):
+ Number of failures argument is -1, but must be >= 0 !
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html new file mode 100644 index 000000000..b0c7d6dcf --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html @@ -0,0 +1,152 @@ + + + +Negative Binomial Table Printing Example. + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Example program showing output of a table of values of cdf and pdf + for various k failures. +

+

+

+

+ +

+
+// Print a table of values that can be used to plot
+// using Excel, or some other superior graphical display tool.
+
+cout.precision(17); // Use max_digits10 precision, the maximum available for a reference table.
+cout << showpoint << endl; // include trailing zeros.
+// This is a maximum possible precision for the type (here double) to suit a reference table.
+int maxk = static_cast<int>(2. * mynbdist.successes() /  mynbdist.success_fraction());
+// This maxk shows most of the range of interest, probability about 0.0001 to 0.999.
+cout << "\n"" k            pdf                      cdf""\n" << endl;
+for (int k = 0; k < maxk; k++)
+{
+  cout << right << setprecision(17) << showpoint
+    << right << setw(3) << k  << ", "
+    << left << setw(25) << pdf(mynbdist, static_cast<double>(k))
+    << left << setw(25) << cdf(mynbdist, static_cast<double>(k))
+    << endl;
+}
+cout << endl;
+
+

+

+

+

+

+ +

+
+k            pdf                      cdf
+ 0, 1.5258789062500000e-005  1.5258789062500003e-005  
+ 1, 9.1552734375000000e-005  0.00010681152343750000   
+ 2, 0.00030899047851562522   0.00041580200195312500   
+ 3, 0.00077247619628906272   0.0011882781982421875    
+ 4, 0.0015932321548461918    0.0027815103530883789    
+ 5, 0.0028678178787231476    0.0056493282318115234    
+ 6, 0.0046602040529251142    0.010309532284736633     
+ 7, 0.0069903060793876605    0.017299838364124298     
+ 8, 0.0098301179241389001    0.027129956288263202     
+ 9, 0.013106823898851871     0.040236780187115073     
+10, 0.016711200471036140     0.056947980658151209     
+11, 0.020509200578089786     0.077457181236241013     
+12, 0.024354675686481652     0.10181185692272265      
+13, 0.028101548869017230     0.12991340579173993      
+14, 0.031614242477644432     0.16152764826938440      
+15, 0.034775666725408917     0.19630331499479325      
+16, 0.037492515688331451     0.23379583068312471      
+17, 0.039697957787645101     0.27349378847076977      
+18, 0.041352039362130305     0.31484582783290005      
+19, 0.042440250924291580     0.35728607875719176      
+20, 0.042970754060845245     0.40025683281803687      
+21, 0.042970754060845225     0.44322758687888220      
+22, 0.042482450037426581     0.48571003691630876      
+23, 0.041558918514873783     0.52726895543118257      
+24, 0.040260202311284021     0.56752915774246648      
+25, 0.038649794218832620     0.60617895196129912      
+26, 0.036791631035234917     0.64297058299653398      
+27, 0.034747651533277427     0.67771823452981139      
+28, 0.032575923312447595     0.71029415784225891      
+29, 0.030329307911589130     0.74062346575384819      
+30, 0.028054609818219924     0.76867807557206813      
+31, 0.025792141284492545     0.79447021685656061      
+32, 0.023575629142856460     0.81804584599941710      
+33, 0.021432390129869489     0.83947823612928651      
+34, 0.019383705779220189     0.85886194190850684      
+35, 0.017445335201298231     0.87630727710980494      
+36, 0.015628112784496322     0.89193538989430121      
+37, 0.013938587078064250     0.90587397697236549      
+38, 0.012379666154859701     0.91825364312722524      
+39, 0.010951243136991251     0.92920488626421649      
+40, 0.0096507830144735539    0.93885566927869002      
+41, 0.0084738582566109364    0.94732952753530097      
+42, 0.0074146259745345548    0.95474415350983555      
+43, 0.0064662435824429246    0.96121039709227851      
+44, 0.0056212231142827853    0.96683162020656122      
+45, 0.0048717266990450708    0.97170334690560634      
+46, 0.0042098073105878630    0.97591315421619418      
+47, 0.0036275999165703964    0.97954075413276465      
+48, 0.0031174686783026818    0.98265822281106729      
+49, 0.0026721160099737302    0.98533033882104104      
+50, 0.0022846591885275322    0.98761499800956853      
+51, 0.0019486798960970148    0.98956367790566557      
+52, 0.0016582516423517923    0.99122192954801736      
+53, 0.0014079495076571762    0.99262987905567457      
+54, 0.0011928461106539983    0.99382272516632852      
+55, 0.0010084971662802015    0.99483122233260868      
+56, 0.00085091948404891532   0.99568214181665760      
+57, 0.00071656377604119542   0.99639870559269883      
+58, 0.00060228420831048650   0.99700098980100937      
+59, 0.00050530624256557675   0.99750629604357488      
+60, 0.00042319397814867202   0.99792949002172360      
+61, 0.00035381791615708398   0.99828330793788067      
+62, 0.00029532382517950324   0.99857863176306016      
+63, 0.00024610318764958566   0.99882473495070978      
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example.html new file mode 100644 index 000000000..4d62e8b06 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example.html @@ -0,0 +1,51 @@ + + + +Normal Distribution Examples + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ Some Miscellaneous Examples of the Normal (Gaussian) Distribution
+

+ (See also the reference documentation for the Normal + Distribution.) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html new file mode 100644 index 000000000..b72102993 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html @@ -0,0 +1,782 @@ + + + +Some Miscellaneous Examples of the Normal (Gaussian) Distribution + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ The sample program normal_misc_examples.cpp + illustrates their use. +

+
+ + Traditional + Tables +
+

+

+

+ First we need some includes to access the normal distribution (and + some std output of course). +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp> // for normal_distribution
+  using boost::math::normal; // typedef provides default type is double.
+
+#include <iostream>
+  using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
+#include <iomanip>
+  using std::setw; using std::setprecision;
+#include <limits>
+  using std::numeric_limits;
+
+int main()
+{
+  cout << "Example: Normal distribution, Miscellaneous Applications.";
+
+  try
+  {
+    { // Traditional tables and values.
+

+

+

+

+

+ Let's start by printing some traditional tables. +

+

+

+

+ +

+
+double step = 1.; // in z 
+double range = 4; // min and max z = -range to +range.
+int precision = 17; // traditional tables are only computed to much lower precision.
+
+// Construct a standard normal distribution s
+  normal s; // (default mean = zero, and standard deviation = unity)
+  cout << "Standard normal distribution, mean = "<< s.mean()
+    << ", standard deviation = " << s.standard_deviation() << endl;
+

+

+

+

+

+ First the probability distribution function (pdf). +

+

+

+

+ +

+
+cout << "Probability distribution function values" << endl;
+cout << "  z " "      pdf " << endl;
+cout.precision(5);
+for (double z = -range; z < range + step; z += step)
+{
+  cout << left << setprecision(3) << setw(6) << z << " " 
+    << setprecision(precision) << setw(12) << pdf(s, z) << endl;
+}
+cout.precision(6); // default
+

+

+

+

+

+ And the area under the normal curve from -∞ up to z, the cumulative + distribution function (cdf). +

+

+

+

+ +

+
+// For a standard normal distribution 
+cout << "Standard normal mean = "<< s.mean()
+  << ", standard deviation = " << s.standard_deviation() << endl;
+cout << "Integral (area under the curve) from - infinity up to z " << endl;
+cout << "  z " "      cdf " << endl;
+for (double z = -range; z < range + step; z += step)
+{
+  cout << left << setprecision(3) << setw(6) << z << " " 
+    << setprecision(precision) << setw(12) << cdf(s, z) << endl;
+}
+cout.precision(6); // default
+

+

+

+

+

+ And all this you can do with a nanoscopic amount of work compared + to the team of human computers toiling + with Milton Abramovitz and Irene Stegen at the US National Bureau + of Standards (now NIST). + Starting in 1938, their "Handbook of Mathematical Functions + with Formulas, Graphs and Mathematical Tables", was eventually + published in 1964, and has been reprinted numerous times since. (A + major replacement is planned at Digital + Library of Mathematical Functions). +

+

+

+

+ Pretty-printing a traditional 2-dimensional table is left as an exercise + for the student, but why bother now that the Math Toolkit lets you + write +

+

+

+

+ +

+
+double z = 2.; 
+cout << "Area for z = " << z << " is " << cdf(s, z) << endl; // to get the area for z.
+

+

+

+

+

+ Correspondingly, we can obtain the traditional 'critical' values + for significance levels. For the 95% confidence level, the significance + level usually called alpha, is 0.05 = 1 - 0.95 (for a one-sided test), + so we can write +

+

+

+

+ +

+
+  cout << "95% of area has a z below " << quantile(s, 0.95) << endl;
+// 95% of area has a z below 1.64485
+

+

+

+

+

+ and a two-sided test (a comparison between two levels, rather than + a one-sided test) +

+

+

+

+ +

+
+  cout << "95% of area has a z between " << quantile(s, 0.975)
+    << " and " << -quantile(s, 0.975) << endl;
+// 95% of area has a z between 1.95996 and -1.95996
+

+

+

+

+

+ First, define a table of significance levels: these are the probabilities + that the true occurrence frequency lies outside the calculated interval. +

+

+

+

+ It is convenient to have an alpha level for the probability that + z lies outside just one standard deviation. This will not be some + nice neat number like 0.05, but we can easily calculate it, +

+

+

+

+ +

+
+double alpha1 = cdf(s, -1) * 2; // 0.3173105078629142
+cout << setprecision(17) << "Significance level for z == 1 is " << alpha1 << endl;
+

+

+

+

+

+ and place in our array of favorite alpha values. +

+

+

+

+ +

+
+double alpha[] = {0.3173105078629142, // z for 1 standard deviation.
+  0.20, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+

+

+

+

+

+ Confidence value as % is (1 - alpha) * 100 (so alpha 0.05 == 95% + confidence) that the true occurrence frequency lies inside + the calculated interval. +

+

+

+

+ +

+
+cout << "level of significance (alpha)" << setprecision(4) << endl;
+cout << "2-sided       1 -sided          z(alpha) " << endl;
+for (int i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+{
+  cout << setw(15) << alpha[i] << setw(15) << alpha[i] /2 << setw(10) << quantile(complement(s,  alpha[i]/2)) << endl;
+  // Use quantile(complement(s, alpha[i]/2)) to avoid potential loss of accuracy from quantile(s,  1 - alpha[i]/2) 
+}
+cout << endl;
+

+

+

+

+

+ Notice the distinction between one-sided (also called one-tailed) + where we are using a > or < + test (and not both) and considering the area of the tail (integral) + from z up to +∞, and a two-sided test where we are using two > + and < tests, and thus considering + two tails, from -∞ up to z low and z high up to +∞. +

+

+

+

+ So the 2-sided values alpha[i] are calculated using alpha[i]/2. +

+

+

+

+ If we consider a simple example of alpha = 0.05, then for a two-sided + test, the lower tail area from -∞ up to -1.96 is 0.025 (alpha/2) and + the upper tail area from +z up to +1.96 is also 0.025 (alpha/2), + and the area between -1.96 up to 12.96 is alpha = 0.95. and the sum + of the two tails is 0.025 + 0.025 = 0.05, +

+

+

+
+ + Standard + deviations either side of the Mean +
+

+

+

+ Armed with the cumulative distribution function, we can easily calculate + the easy to remember proportion of values that lie within 1, 2 and + 3 standard deviations from the mean. +

+

+

+

+ +

+
+cout.precision(3);
+cout << showpoint << "cdf(s, s.standard_deviation()) = "
+  << cdf(s, s.standard_deviation()) << endl;  // from -infinity to 1 sd
+cout << "cdf(complement(s, s.standard_deviation())) = "
+  << cdf(complement(s, s.standard_deviation())) << endl;
+cout << "Fraction 1 standard deviation within either side of mean is "
+  << 1 -  cdf(complement(s, s.standard_deviation())) * 2 << endl;
+cout << "Fraction 2 standard deviations within either side of mean is "
+  << 1 -  cdf(complement(s, 2 * s.standard_deviation())) * 2 << endl;
+cout << "Fraction 3 standard deviations within either side of mean is "
+  << 1 -  cdf(complement(s, 3 * s.standard_deviation())) * 2 << endl;
+

+

+

+

+

+ To a useful precision, the 1, 2 & 3 percentages are 68, 95 and + 99.7, and these are worth memorising as useful 'rules of thumb', + as, for example, in standard + deviation: +

+

+ +

+
Fraction 1 standard deviation within either side of mean is 0.683
+Fraction 2 standard deviations within either side of mean is 0.954
+Fraction 3 standard deviations within either side of mean is 0.997
+
+

+

+

+ We could of course get some really accurate values for these confidence + intervals by using cout.precision(15); +

+

+ +

+
Fraction 1 standard deviation within either side of mean is 0.682689492137086
+Fraction 2 standard deviations within either side of mean is 0.954499736103642
+Fraction 3 standard deviations within either side of mean is 0.997300203936740
+
+

+

+

+ But before you get too excited about this impressive precision, don't + forget that the confidence intervals of the + standard deviation are surprisingly wide, especially if + you have estimated the standard deviation from only a few measurements. +

+

+

+
+ + Some + simple examples +
+
+ + Life + of light bulbs +
+

+

+

+ Examples from K. Krishnamoorthy, Handbook of Statistical Distributions + with Applications, ISBN 1 58488 635 8, page 125... implemented using + the Math Toolkit library. +

+

+

+

+ A few very simple examples are shown here: +

+

+

+

+ +

+
+// K. Krishnamoorthy, Handbook of Statistical Distributions with Applications,
+ // ISBN 1 58488 635 8, page 125, example 10.3.5
+

+

+

+

+

+ Mean lifespan of 100 W bulbs is 1100 h with standard deviation of + 100 h. Assuming, perhaps with little evidence and much faith, that + the distribution is normal, we construct a normal distribution called + bulbs with these values: +

+

+

+

+ +

+
+double mean_life = 1100.;
+double life_standard_deviation = 100.;
+normal bulbs(mean_life, life_standard_deviation); 
+double expected_life = 1000.;
+

+

+

+

+

+ The we can use the Cumulative distribution function to predict fractions + (or percentages, if * 100) that will last various lifetimes. +

+

+

+

+ +

+
+cout << "Fraction of bulbs that will last at best (<=) " // P(X <= 1000)
+  << expected_life << " is "<< cdf(bulbs, expected_life) << endl;
+cout << "Fraction of bulbs that will last at least (>) " // P(X > 1000)
+  << expected_life << " is "<< cdf(complement(bulbs, expected_life)) << endl;
+double min_life = 900;
+double max_life = 1200;
+cout << "Fraction of bulbs that will last between "
+  << min_life << " and " << max_life << " is "
+  << cdf(bulbs, max_life)  // P(X <= 1200)
+   - cdf(bulbs, min_life) << endl; // P(X <= 900)
+

+

+

+

+
+ + + + + +
[Note]Note

+ Real-life failures are often very ab-normal, with a significant + number that 'dead-on-arrival' or suffer failure very early in their + life: the lifetime of the survivors of 'early mortality' may be + well described by the normal distribution. +

+

+

+
+ + How + many onions? +
+

+

+

+ Weekly demand for 5 lb sacks of onions at a store is normally distributed + with mean 140 sacks and standard deviation 10. +

+

+

+

+ +

+
+double mean = 140.; // sacks per week.
+double standard_deviation = 10; 
+normal sacks(mean, standard_deviation);
+
+double stock = 160.; // per week.
+cout << "Percentage of weeks overstocked "
+  << cdf(sacks, stock) * 100. << endl; // P(X <=160)
+// Percentage of weeks overstocked 97.7
+

+

+

+

+

+ So there will be lots of mouldy onions! So we should be able to say + what stock level will meet demand 95% of the weeks. +

+

+

+

+ +

+
+double stock_95 = quantile(sacks, 0.95);
+cout << "Store should stock " << int(stock_95) << " sacks to meet 95% of demands." << endl;
+

+

+

+

+

+ And it is easy to estimate how to meet 80% of demand, and waste even + less. +

+

+

+

+ +

+
+double stock_80 = quantile(sacks, 0.80);
+cout << "Store should stock " << int(stock_80) << " sacks to meet 8 out of 10 demands." << endl;
+
+

+

+

+

+
+ + Packing + beef +
+

+

+

+ A machine is set to pack 3 kg of ground beef per pack. Over a long + period of time it is found that the average packed was 3 kg with + a standard deviation of 0.1 kg. Assuming the packing is normally + distributed, we can find the fraction (or %) of packages that weigh + more than 3.1 kg. +

+

+

+

+ +

+
+double mean = 3.; // kg
+double standard_deviation = 0.1; // kg
+normal packs(mean, standard_deviation);
+
+double max_weight = 3.1; // kg
+cout << "Percentage of packs > " << max_weight << " is "
+<< cdf(complement(packs, max_weight)) << endl; // P(X > 3.1)
+
+double under_weight = 2.9;
+cout <<"fraction of packs <= " << under_weight << " with a mean of " << mean 
+  << " is " << cdf(complement(packs, under_weight)) << endl;
+// fraction of packs <= 2.9 with a mean of 3 is 0.841345
+// This is 0.84 - more than the target 0.95
+// Want 95% to be over this weight, so what should we set the mean weight to be?
+// KK StatCalc says:
+double over_mean = 3.0664;
+normal xpacks(over_mean, standard_deviation);
+cout << "fraction of packs >= " << under_weight
+<< " with a mean of " << xpacks.mean() 
+  << " is " << cdf(complement(xpacks, under_weight)) << endl;
+// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005
+double under_fraction = 0.05;  // so 95% are above the minimum weight mean - sd = 2.9
+double low_limit = standard_deviation;
+double offset = mean - low_limit - quantile(packs, under_fraction);
+double nominal_mean = mean + offset;
+
+normal nominal_packs(nominal_mean, standard_deviation);
+cout << "Setting the packer to " << nominal_mean << " will mean that "
+  << "fraction of packs >= " << under_weight 
+  << " is " << cdf(complement(nominal_packs, under_weight)) << endl;
+

+

+

+

+

+ Setting the packer to 3.06449 will mean that fraction of packs >= + 2.9 is 0.95. +

+

+

+

+ Setting the packer to 3.13263 will mean that fraction of packs >= + 2.9 is 0.99, but will more than double the mean loss from 0.0644 + to 0.133. +

+

+

+

+ Alternatively, we could invest in a better (more precise) packer + with a lower standard deviation. +

+

+

+

+ To estimate how much better (how much smaller standard deviation) + it would have to be, we need to get the 5% quantile to be located + at the under_weight limit, 2.9 +

+

+

+

+ +

+
+double p = 0.05; // wanted p th quantile.
+cout << "Quantile of " << p << " = " << quantile(packs, p)
+  << ", mean = " << packs.mean() << ", sd = " << packs.standard_deviation() << endl; //
+

+

+

+

+

+ Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 +

+

+

+

+ With the current packer (mean = 3, sd = 0.1), the 5% quantile is + at 2.8551 kg, a little below our target of 2.9 kg. So we know that + the standard deviation is going to have to be smaller. +

+

+

+

+ Let's start by guessing that it (now 0.1) needs to be halved, to + a standard deviation of 0.05 +

+

+

+

+ +

+
+normal pack05(mean, 0.05); 
+cout << "Quantile of " << p << " = " << quantile(pack05, p) 
+  << ", mean = " << pack05.mean() << ", sd = " << pack05.standard_deviation() << endl;
+
+cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean 
+  << " and standard deviation of " << pack05.standard_deviation()
+  << " is " << cdf(complement(pack05, under_weight)) << endl;
+//
+

+

+

+

+

+ Fraction of packs >= 2.9 with a mean of 3 and standard deviation + of 0.05 is 0.9772 +

+

+

+

+ So 0.05 was quite a good guess, but we are a little over the 2.9 + target, so the standard deviation could be a tiny bit more. So we + could do some more guessing to get closer, say by increasing to 0.06 +

+

+

+

+ +

+
+normal pack06(mean, 0.06); 
+cout << "Quantile of " << p << " = " << quantile(pack06, p) 
+  << ", mean = " << pack06.mean() << ", sd = " << pack06.standard_deviation() << endl;
+
+cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean 
+  << " and standard deviation of " << pack06.standard_deviation()
+  << " is " << cdf(complement(pack06, under_weight)) << endl;
+

+

+

+

+

+ Fraction of packs >= 2.9 with a mean of 3 and standard deviation + of 0.06 is 0.9522 +

+

+

+

+ Now we are getting really close, but to do the job properly, we could + use root finding method, for example the tools provided, and used + elsewhere, in the Math Toolkit, see Root + Finding Without Derivatives. +

+

+

+

+ But in this normal distribution case, we could be even smarter and + make a direct calculation. +

+

+

+

+ +

+
+normal s; // For standard normal distribution, 
+double sd = 0.1;
+double x = 2.9; // Our required limit.
+// then probability p = N((x - mean) / sd)
+// So if we want to find the standard deviation that would be required to meet this limit,
+// so that the p th quantile is located at x,
+// in this case the 0.95 (95%) quantile at 2.9 kg pack weight, when the mean is 3 kg.
+
+double prob =  pdf(s, (x - mean) / sd);
+double qp = quantile(s, 0.95);
+cout << "prob = " << prob << ", quantile(p) " << qp << endl; // p = 0.241971, quantile(p) 1.64485
+// Rearranging, we can directly calculate the required standard deviation:
+double sd95 = abs((x - mean)) / qp;
+
+cout << "If we want the "<< p << " th quantile to be located at "  
+  << x << ", would need a standard deviation of " << sd95 << endl;
+
+normal pack95(mean, sd95);  // Distribution of the 'ideal better' packer.
+cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean 
+  << " and standard deviation of " << pack95.standard_deviation()
+  << " is " << cdf(complement(pack95, under_weight)) << endl;
+
+// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0608 is 0.95
+

+

+

+

+

+ Notice that these two deceptively simple questions (do we over-fill + or measure better) are actually very common. The weight of beef might + be replaced by a measurement of more or less anything. But the calculations + rely on the accuracy of the standard deviation - something that is + almost always less good than we might wish, especially if based on + a few measurements. +

+

+

+
+ + Length + of bolts +
+

+

+

+ A bolt is usable if between 3.9 and 4.1 long. From a large batch + of bolts, a sample of 50 show a mean length of 3.95 with standard + deviation 0.1. Assuming a normal distribution, what proportion is + usable? The true sample mean is unknown, but we can use the sample + mean and standard deviation to find approximate solutions. +

+

+

+

+ +

+
+    normal bolts(3.95, 0.1);
+    double top = 4.1;
+    double bottom = 3.9; 
+
+cout << "Fraction long enough [ P(X <= " << top << ") ] is " << cdf(bolts, top) << endl;
+cout << "Fraction too short [ P(X <= " << bottom << ") ] is " << cdf(bolts, bottom) << endl;
+cout << "Fraction OK  -between " << bottom << " and " << top
+  << "[ P(X <= " << top  << ") - P(X<= " << bottom << " ) ] is "
+  << cdf(bolts, top) - cdf(bolts, bottom) << endl;
+
+cout << "Fraction too long [ P(X > " << top << ") ] is "
+  << cdf(complement(bolts, top)) << endl;
+
+cout << "95% of bolts are shorter than " << quantile(bolts, 0.95) << endl;
+ 
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg.html new file mode 100644 index 000000000..7d4359c42 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg.html @@ -0,0 +1,58 @@ + + + +Student's t Distribution Examples + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ Calculating confidence intervals on the mean with the Students-t distribution
+
+ Testing a sample mean for difference from a "true" mean
+
+ Estimating how large a sample size would have to become in order to give + a significant Students-t test result with a single sample test
+
+ Comparing the means of two samples with the Students-t test
+
+ Comparing two paired samples with the Student's t distribution
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html new file mode 100644 index 000000000..4fc5c4a19 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html @@ -0,0 +1,79 @@ + + + +Comparing two paired samples with the Student's t distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Imagine that we have a before and after reading for each item in the + sample: for example we might have measured blood pressure before and + after administration of a new drug. We can't pool the results and compare + the means before and after the change, because each patient will have + a different baseline reading. Instead we calculate the difference between + before and after measurements in each patient, and calculate the mean + and standard deviation of the differences. To test whether a significant + change has taken place, we can then test the null-hypothesis that the + true mean is zero using the same procedure we used in the single sample + cases previously discussed. +

+

+ That means we can: +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html new file mode 100644 index 000000000..2a1f81b5e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html @@ -0,0 +1,273 @@ + + + +Calculating confidence intervals on the mean with the Students-t distribution + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Let's say you have a sample mean, you may wish to know what confidence + intervals you can place on that mean. Colloquially: "I want an + interval that I can be P% sure contains the true mean". (On a + technical point, note that the interval either contains the true mean + or it does not: the meaning of the confidence level is subtly different + from this colloquialism. More background information can be found on + the NIST + site). +

+

+ The formula for the interval can be expressed as: +

+

+ +

+

+ Where, Ys is the sample mean, s + is the sample standard deviation, N is the sample + size, [alpha] is the desired significance level + and t(α/2,N-1) is the upper critical value of the + Students-t distribution with N-1 degrees of freedom. +

+
+ + + + + +
[Note]Note
+

+ The quantity α is the maximum acceptable risk of falsely rejecting + the null-hypothesis. The smaller the value of α the greater the strength + of the test. +

+

+ The confidence level of the test is defined as 1 - α, and often expressed + as a percentage. So for example a significance level of 0.05, is + equivalent to a 95% confidence level. Refer to "What + are confidence intervals?" in NIST/SEMATECH + e-Handbook of Statistical Methods. for more information. +

+
+
+ + + + + +
[Note]Note

+ The usual assumptions of independent + and identically distributed (i.i.d.) variables and normal distribution + of course apply here, as they do in other examples. +

+

+ From the formula, it should be clear that: +

+
    +
  • + The width of the confidence interval decreases as the sample size + increases. +
    • +
  • + The width increases as the standard deviation increases. +
    • +
  • + The width increases as the confidence level increases + (0.5 towards 0.99999 - stronger). +
    • +
  • + The width increases as the significance level decreases + (0.5 towards 0.00000...01 - stronger). +
    • +
+

+ The following example code is taken from the example program students_t_single_sample.cpp. +

+

+ We'll begin by defining a procedure to calculate intervals for various + confidence levels; the procedure will print these out as a table: +

+
+// Needed includes:
+#include <boost/math/distributions/students_t.hpp>
+#include <iostream>
+#include <iomanip>
+// Bring everything into global namespace for ease of use:
+using namespace boost::math;
+using namespace std;   
+
+void confidence_limits_on_mean(
+   double Sm,           // Sm = Sample Mean.
+   double Sd,           // Sd = Sample Standard Deviation.
+   unsigned Sn)         // Sn = Sample Size.
+{
+   using namespace std;
+   using namespace boost::math;
+
+   // Print out general info:
+   cout << 
+      "__________________________________\n"
+      "2-Sided Confidence Limits For Mean\n"
+      "__________________________________\n\n";
+   cout << setprecision(7);
+   cout << setw(40) << left << "Number of Observations" << "=  " << Sn << "\n";
+   cout << setw(40) << left << "Mean" << "=  " << Sm << "\n";
+   cout << setw(40) << left << "Standard Deviation" << "=  " << Sd << "\n";
+
+

+ We'll define a table of significance/risk levels for which we'll compute + intervals: +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ Note that these are the complements of the confidence/probability levels: + 0.5, 0.75, 0.9 .. 0.99999). +

+

+ Next we'll declare the distribution object we'll need, note that the + degrees of freedom parameter is the sample size + less one: +

+
+students_t dist(Sn - 1);
+
+

+ Most of what follows in the program is pretty printing, so let's focus + on the calculation of the interval. First we need the t-statistic, + computed using the quantile function and our significance + level. Note that since the significance levels are the complement of + the probability, we have to wrap the arguments in a call to complement(...): +

+
+double T = quantile(complement(dist, alpha[i] / 2));
+
+

+ Note that alpha was divided by two, since we'll be calculating both + the upper and lower bounds: had we been interested in a single sided + interval then we would have omitted this step. +

+

+ Now to complete the picture, we'll get the (one-sided) width of the + interval from the t-statistic by multiplying by the standard deviation, + and dividing by the square root of the sample size: +

+
+double w = T * Sd / sqrt(double(Sn));
+
+

+ The two-sided interval is then the sample mean plus and minus this + width. +

+

+ And apart from some more pretty-printing that completes the procedure. +

+

+ Let's take a look at some sample output, first using the Heat + flow data from the NIST site. The data set was collected by + Bob Zarr of NIST in January, 1990 from a heat flow meter calibration + and stability analysis. The corresponding dataplot output for this + test can be found in section + 3.5.2 of the NIST/SEMATECH + e-Handbook of Statistical Methods.. +

+
   __________________________________
+   2-Sided Confidence Limits For Mean
+   __________________________________
+
+   Number of Observations                  =  195
+   Mean                                    =  9.26146
+   Standard Deviation                      =  0.02278881
+
+
+   ___________________________________________________________________
+   Confidence       T           Interval          Lower          Upper
+    Value (%)     Value          Width            Limit          Limit
+   ___________________________________________________________________
+       50.000     0.676       1.103e-003        9.26036        9.26256
+       75.000     1.154       1.883e-003        9.25958        9.26334
+       90.000     1.653       2.697e-003        9.25876        9.26416
+       95.000     1.972       3.219e-003        9.25824        9.26468
+       99.000     2.601       4.245e-003        9.25721        9.26571
+       99.900     3.341       5.453e-003        9.25601        9.26691
+       99.990     3.973       6.484e-003        9.25498        9.26794
+       99.999     4.537       7.404e-003        9.25406        9.26886
+
+

+ As you can see the large sample size (195) and small standard deviation + (0.023) have combined to give very small intervals, indeed we can be + very confident that the true mean is 9.2. +

+

+ For comparison the next example data output is taken from P.K.Hou, + O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from + Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. + Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. + The values result from the determination of mercury by cold-vapour + atomic absorption. +

+
   __________________________________
+   2-Sided Confidence Limits For Mean
+   __________________________________
+
+   Number of Observations                  =  3
+   Mean                                    =  37.8000000
+   Standard Deviation                      =  0.9643650
+
+
+   ___________________________________________________________________
+   Confidence       T           Interval          Lower          Upper
+    Value (%)     Value          Width            Limit          Limit
+   ___________________________________________________________________
+       50.000     0.816            0.455       37.34539       38.25461
+       75.000     1.604            0.893       36.90717       38.69283
+       90.000     2.920            1.626       36.17422       39.42578
+       95.000     4.303            2.396       35.40438       40.19562
+       99.000     9.925            5.526       32.27408       43.32592
+       99.900    31.599           17.594       20.20639       55.39361
+       99.990    99.992           55.673      -17.87346       93.47346
+       99.999   316.225          176.067     -138.26683      213.86683
+
+

+ This time the fact that there are only three measurements leads to + much wider intervals, indeed such large intervals that it's hard to + be very confident in the location of the mean. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html new file mode 100644 index 000000000..51a99f352 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html @@ -0,0 +1,184 @@ + + + +Estimating how large a sample size would have to become in order to give a significant Students-t test result with a single sample test + + + + + + + + + + + + + + + +
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+

+ Imagine you have conducted a Students-t test on a single sample in + order to check for systematic errors in your measurements. Imagine + that the result is borderline. At this point one might go off and collect + more data, but it might be prudent to first ask the question "How + much more?". The parameter estimators of the students_t_distribution + class can provide this information. +

+

+ This section is based on the example code in students_t_single_sample.cpp + and we begin by defining a procedure that will print out a table of + estimated sample sizes for various confidence levels: +

+
+// Needed includes:
+#include <boost/math/distributions/students_t.hpp>
+#include <iostream>
+#include <iomanip>
+// Bring everything into global namespace for ease of use:
+using namespace boost::math;
+using namespace std;   
+
+void single_sample_find_df(
+   double M,          // M = true mean.
+   double Sm,         // Sm = Sample Mean.
+   double Sd)         // Sd = Sample Standard Deviation.
+{
+
+

+ Next we define a table of significance levels: +

+
+double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+
+

+ Printing out the table of sample sizes required for various confidence + levels begins with the table header: +

+
+cout << "\n\n"
+        "_______________________________________________________________\n"
+        "Confidence       Estimated          Estimated\n"
+        " Value (%)      Sample Size        Sample Size\n"
+        "              (one sided test)    (two sided test)\n"
+        "_______________________________________________________________\n";
+
+

+ And now the important part: the sample sizes required. Class students_t_distribution has a static + member function find_degrees_of_freedom + that will calculate how large a sample size needs to be in order to + give a definitive result. +

+

+ The first argument is the difference between the means that you wish + to be able to detect, here it's the absolute value of the difference + between the sample mean, and the true mean. +

+

+ Then come two probability values: alpha and beta. Alpha is the maximum + acceptable risk of rejecting the null-hypothesis when it is in fact + true. Beta is the maximum acceptable risk of failing to reject the + null-hypothesis when in fact it is false. Also note that for a two-sided + test, alpha must be divided by 2. +

+

+ The final parameter of the function is the standard deviation of the + sample. +

+

+ In this example, we assume that alpha and beta are the same, and call + find_degrees_of_freedom + twice: once with alpha for a one-sided test, and once with alpha/2 + for a two-sided test. +

+
+   for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+   {
+      // Confidence value:
+      cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+      // calculate df for single sided test:
+      double df = students_t::find_degrees_of_freedom(
+         fabs(M - Sm), alpha[i], alpha[i], Sd);
+      // convert to sample size:
+      double size = ceil(df) + 1;
+      // Print size:
+      cout << fixed << setprecision(0) << setw(16) << right << size;
+      // calculate df for two sided test:
+      df = students_t::find_degrees_of_freedom(
+         fabs(M - Sm), alpha[i]/2, alpha[i], Sd);
+      // convert to sample size:
+      size = ceil(df) + 1;
+      // Print size:
+      cout << fixed << setprecision(0) << setw(16) << right << size << endl;
+   }
+   cout << endl;
+}
+
+

+ Let's now look at some sample output using data taken from P.K.Hou, + O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from + Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. + Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. + The values result from the determination of mercury by cold-vapour + atomic absorption. +

+

+ Only three measurements were made, and the Students-t test above gave + a borderline result, so this example will show us how many samples + would need to be collected: +

+
_____________________________________________________________
+Estimated sample sizes required for various confidence levels
+_____________________________________________________________
+
+True Mean                               =  38.90000
+Sample Mean                             =  37.80000
+Sample Standard Deviation               =  0.96437
+
+
+_______________________________________________________________
+Confidence       Estimated          Estimated
+ Value (%)      Sample Size        Sample Size
+              (one sided test)    (two sided test)
+_______________________________________________________________
+    50.000               2               3
+    75.000               4               5
+    90.000               8              10
+    95.000              12              14
+    99.000              21              23
+    99.900              36              38
+    99.990              51              54
+    99.999              67              69
+
+

+ So in this case, many more measurements would have had to be made, + for example at the 95% level, 14 measurements in total for a two-sided + test. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html new file mode 100644 index 000000000..08f166c8a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html @@ -0,0 +1,337 @@ + + + +Testing a sample mean for difference from a "true" mean + + + + + + + + + + + + + + + +
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+
+

+ When calibrating or comparing a scientific instrument or measurement + method of some kind, we want to be answer the question "Does an + observed sample mean differ from the "true" mean in any significant + way?". If it does, then we have evidence of a systematic difference. + This question can be answered with a Students-t test: more information + can be found on + the NIST site. +

+

+ Of course, the assignment of "true" to one mean may be quite + arbitrary, often this is simply a "traditional" method of + measurement. +

+

+ The following example code is taken from the example program students_t_single_sample.cpp. +

+

+ We'll begin by defining a procedure to determine which of the possible + hypothesis are rejected or not-rejected at a given significance level: +

+
+ + + + + +
[Note]Note

+ Non-statisticians might say 'not-rejected' means 'accepted', (often + of the null-hypothesis) implying, wrongly, that there really IS no difference, but statisticans eschew + this to avoid implying that there is positive evidence of 'no difference'. + 'Not-rejected' here means there is no evidence + of difference, but there still might well be a difference. For example, + see argument + from ignorance and Absence + of evidence does not constitute evidence of absence. +

+
+// Needed includes:
+#include <boost/math/distributions/students_t.hpp>
+#include <iostream>
+#include <iomanip>
+// Bring everything into global namespace for ease of use:
+using namespace boost::math;
+using namespace std;   
+
+void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha)
+{
+   //
+   // M = true mean.
+   // Sm = Sample Mean.
+   // Sd = Sample Standard Deviation.
+   // Sn = Sample Size.
+   // alpha = Significance Level.
+
+

+ Most of the procedure is pretty-printing, so let's just focus on the + calculation, we begin by calculating the t-statistic: +

+
+// Difference in means:
+double diff = Sm - M;
+// Degrees of freedom:
+unsigned v = Sn - 1;
+// t-statistic:
+double t_stat = diff * sqrt(double(Sn)) / Sd;
+
+

+ Finally calculate the probability from the t-statistic. If we're interested + in simply whether there is a difference (either less or greater) or + not, we don't care about the sign of the t-statistic, and we take the + complement of the probability for comparison to the significance level: +

+
+students_t dist(v);
+double q = cdf(complement(dist, fabs(t_stat)));
+
+

+ The procedure then prints out the results of the various tests that + can be done, these can be summarised in the following table: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Hypothesis +

+ 		+

+ Test +

+
+

+ The Null-hypothesis: there is no difference + in means +

+ 		+

+ Reject if complement of CDF for |t| < significance level + / 2: +

+

+ cdf(complement(dist, + fabs(t))) + < alpha + / 2 +

+
+

+ The Alternative-hypothesis: there is + difference in means +

+ 		+

+ Reject if complement of CDF for |t| > significance level + / 2: +

+

+ cdf(complement(dist, + fabs(t))) + > alpha + / 2 +

+
+

+ The Alternative-hypothesis: the sample mean is + less than the true mean. +

+ 		+

+ Reject if CDF of t > significance level: +

+

+ cdf(dist, + t) + > alpha +

+
+

+ The Alternative-hypothesis: the sample mean is + greater than the true mean. +

+ 		+

+ Reject if complement of CDF of t > significance level: +

+

+ cdf(complement(dist, + t)) + > alpha +

+
+
+ + + + + +
[Note]Note

+ Notice that the comparisons are against alpha + / 2 + for a two-sided test and against alpha + for a one-sided test +

+

+ Now that we have all the parts in place, let's take a look at some + sample output, first using the Heat + flow data from the NIST site. The data set was collected by + Bob Zarr of NIST in January, 1990 from a heat flow meter calibration + and stability analysis. The corresponding dataplot output for this + test can be found in section + 3.5.2 of the NIST/SEMATECH + e-Handbook of Statistical Methods.. +

+
   __________________________________
+   Student t test for a single sample
+   __________________________________
+
+   Number of Observations                                 =  195
+   Sample Mean                                            =  9.26146
+   Sample Standard Deviation                              =  0.02279
+   Expected True Mean                                     =  5.00000
+
+   Sample Mean - Expected Test Mean                       =  4.26146
+   Degrees of Freedom                                     =  194
+   T Statistic                                            =  2611.28380
+   Probability that difference is due to chance           =  0.000e+000
+
+   Results for Alternative Hypothesis and alpha           =  0.0500
+
+   Alternative Hypothesis     Conclusion
+   Mean != 5.000              NOT REJECTED
+   Mean  < 5.000              REJECTED
+   Mean  > 5.000              NOT REJECTED
+
+

+ You will note the line that says the probability that the difference + is due to chance is zero. From a philosophical point of view, of course, + the probability can never reach zero. However, in this case the calculated + probability is smaller than the smallest representable double precision + number, hence the appearance of a zero here. Whatever its "true" + value is, we know it must be extraordinarily small, so the alternative + hypothesis - that there is a difference in means - is not rejected. +

+

+ For comparison the next example data output is taken from P.K.Hou, + O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from + Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. + Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. + The values result from the determination of mercury by cold-vapour + atomic absorption. +

+
   __________________________________
+   Student t test for a single sample
+   __________________________________
+
+   Number of Observations                                 =  3
+   Sample Mean                                            =  37.80000
+   Sample Standard Deviation                              =  0.96437
+   Expected True Mean                                     =  38.90000
+
+   Sample Mean - Expected Test Mean                       =  -1.10000
+   Degrees of Freedom                                     =  2
+   T Statistic                                            =  -1.97566
+   Probability that difference is due to chance           =  1.869e-001
+
+   Results for Alternative Hypothesis and alpha           =  0.0500
+
+   Alternative Hypothesis     Conclusion
+   Mean != 38.900             REJECTED
+   Mean  < 38.900             REJECTED
+   Mean  > 38.900             REJECTED
+
+

+ As you can see the small number of measurements (3) has led to a large + uncertainty in the location of the true mean. So even though there + appears to be a difference between the sample mean and the expected + true mean, we conclude that there is no significant difference, and + are unable to reject the null hypothesis. However, if we were to lower + the bar for acceptance down to alpha = 0.1 (a 90% confidence level) + we see a different output: +

+
__________________________________
+Student t test for a single sample
+__________________________________
+
+Number of Observations                                 =  3
+Sample Mean                                            =  37.80000
+Sample Standard Deviation                              =  0.96437
+Expected True Mean                                     =  38.90000
+
+Sample Mean - Expected Test Mean                       =  -1.10000
+Degrees of Freedom                                     =  2
+T Statistic                                            =  -1.97566
+Probability that difference is due to chance           =  1.869e-001
+
+Results for Alternative Hypothesis and alpha           =  0.1000
+
+Alternative Hypothesis     Conclusion
+Mean != 38.900            REJECTED
+Mean  < 38.900            NOT REJECTED
+Mean  > 38.900            REJECTED
+
+

+ In this case, we really have a borderline result, and more data (and/or + more accurate data), is needed for a more convincing conclusion. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html new file mode 100644 index 000000000..0840674d1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html @@ -0,0 +1,362 @@ + + + +Comparing the means of two samples with the Students-t test + + + + + + + + + + + + + + + +
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+
+

+ Imagine that we have two samples, and we wish to determine whether + their means are different or not. This situation often arises when + determining whether a new process or treatment is better than an old + one. +

+

+ In this example, we'll be using the Car + Mileage sample data from the NIST + website. The data compares miles per gallon of US cars with + miles per gallon of Japanese cars. +

+

+ The sample code is in students_t_two_samples.cpp. +

+

+ There are two ways in which this test can be conducted: we can assume + that the true standard deviations of the two samples are equal or not. + If the standard deviations are assumed to be equal, then the calculation + of the t-statistic is greatly simplified, so we'll examine that case + first. In real life we should verify whether this assumption is valid + with a Chi-Squared test for equal variances. +

+

+ We begin by defining a procedure that will conduct our test assuming + equal variances: +

+
+// Needed headers:
+#include <boost/math/distributions/students_t.hpp>
+#include <iostream>
+#include <iomanip>
+// Simplify usage:
+using namespace boost::math;
+using namespace std;
+
+void two_samples_t_test_equal_sd(
+        double Sm1,       // Sm1 = Sample 1 Mean.
+        double Sd1,       // Sd1 = Sample 1 Standard Deviation.
+        unsigned Sn1,     // Sn1 = Sample 1 Size.
+        double Sm2,       // Sm2 = Sample 2 Mean.
+        double Sd2,       // Sd2 = Sample 2 Standard Deviation.
+        unsigned Sn2,     // Sn2 = Sample 2 Size.
+        double alpha)     // alpha = Significance Level.
+{
+
+

+ Our procedure will begin by calculating the t-statistic, assuming equal + variances the needed formulae are: +

+

+ +

+

+ where Sp is the "pooled" standard deviation of the two samples, + and v is the number of degrees of freedom of the + two combined samples. We can now write the code to calculate the t-statistic: +

+
+// Degrees of freedom:
+double v = Sn1 + Sn2 - 2;
+cout << setw(55) << left << "Degrees of Freedom" << "=  " << v << "\n";
+// Pooled variance:
+double sp = sqrt(((Sn1-1) * Sd1 * Sd1 + (Sn2-1) * Sd2 * Sd2) / v);
+cout << setw(55) << left << "Pooled Standard Deviation" << "=  " << v << "\n";
+// t-statistic:
+double t_stat = (Sm1 - Sm2) / (sp * sqrt(1.0 / Sn1 + 1.0 / Sn2));
+cout << setw(55) << left << "T Statistic" << "=  " << t_stat << "\n";
+
+

+ The next step is to define our distribution object, and calculate the + complement of the probability: +

+
+students_t dist(v);
+double q = cdf(complement(dist, fabs(t_stat)));
+cout << setw(55) << left << "Probability that difference is due to chance" << "=  " 
+   << setprecision(3) << scientific << 2 * q << "\n\n";
+
+

+ Here we've used the absolute value of the t-statistic, because we initially + want to know simply whether there is a difference or not (a two-sided + test). However, we can also test whether the mean of the second sample + is greater or is less (one-sided test) than that of the first: all + the possible tests are summed up in the following table: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Hypothesis +

+ 		+

+ Test +

+
+

+ The Null-hypothesis: there is no difference + in means +

+ 		+

+ Reject if complement of CDF for |t| < significance level + / 2: +

+

+ cdf(complement(dist, + fabs(t))) + < alpha + / 2 +

+
+

+ The Alternative-hypothesis: there is a difference + in means +

+ 		+

+ Reject if complement of CDF for |t| > significance level + / 2: +

+

+ cdf(complement(dist, + fabs(t))) + < alpha + / 2 +

+
+

+ The Alternative-hypothesis: Sample 1 Mean is less + than Sample 2 Mean. +

+ 		+

+ Reject if CDF of t > significance level: +

+

+ cdf(dist, + t) + > alpha +

+
+

+ The Alternative-hypothesis: Sample 1 Mean is greater + than Sample 2 Mean. +

+ 		+

+ Reject if complement of CDF of t > significance level: +

+

+ cdf(complement(dist, + t)) + > alpha +

+
+
+ + + + + +
[Note]Note

+ For a two-sided test we must compare against alpha / 2 and not alpha. +

+

+ Most of the rest of the sample program is pretty-printing, so we'll + skip over that, and take a look at the sample output for alpha=0.05 + (a 95% probability level). For comparison the dataplot output for the + same data is in section + 1.3.5.3 of the NIST/SEMATECH + e-Handbook of Statistical Methods.. +

+
   ________________________________________________
+   Student t test for two samples (equal variances)
+   ________________________________________________
+
+   Number of Observations (Sample 1)                      =  249
+   Sample 1 Mean                                          =  20.14458
+   Sample 1 Standard Deviation                            =  6.41470
+   Number of Observations (Sample 2)                      =  79
+   Sample 2 Mean                                          =  30.48101
+   Sample 2 Standard Deviation                            =  6.10771
+   Degrees of Freedom                                     =  326.00000
+   Pooled Standard Deviation                              =  326.00000
+   T Statistic                                            =  -12.62059
+   Probability that difference is due to chance           =  5.273e-030
+
+   Results for Alternative Hypothesis and alpha           =  0.0500
+
+   Alternative Hypothesis              Conclusion
+   Sample 1 Mean != Sample 2 Mean       NOT REJECTED
+   Sample 1 Mean <  Sample 2 Mean       NOT REJECTED
+   Sample 1 Mean >  Sample 2 Mean       REJECTED
+
+

+ So with a probability that the difference is due to chance of just + 5.273e-030, we can safely conclude that there is indeed a difference. +

+

+ The tests on the alternative hypothesis show that we must also reject + the hypothesis that Sample 1 Mean is greater than that for Sample 2: + in this case Sample 1 represents the miles per gallon for Japanese + cars, and Sample 2 the miles per gallon for US cars, so we conclude + that Japanese cars are on average more fuel efficient. +

+

+ Now that we have the simple case out of the way, let's look for a moment + at the more complex one: that the standard deviations of the two samples + are not equal. In this case the formula for the t-statistic becomes: +

+

+ +

+

+ And for the combined degrees of freedom we use the Welch-Satterthwaite + approximation: +

+

+ +

+

+ Note that this is one of the rare situations where the degrees-of-freedom + parameter to the Student's t distribution is a real number, and not + an integer value. +

+
+ + + + + +
[Note]Note

+ Some statistical packages truncate the effective degrees of freedom + to an integer value: this may be necessary if you are relying on + lookup tables, but since our code fully supports non-integer degrees + of freedom there is no need to truncate in this case. Also note that + when the degrees of freedom is small then the Welch-Satterthwaite + approximation may be a significant source of error. +

+

+ Putting these formulae into code we get: +

+
+// Degrees of freedom:
+double v = Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2;
+v *= v;
+double t1 = Sd1 * Sd1 / Sn1;
+t1 *= t1;
+t1 /=  (Sn1 - 1);
+double t2 = Sd2 * Sd2 / Sn2;
+t2 *= t2;
+t2 /= (Sn2 - 1);
+v /= (t1 + t2);
+cout << setw(55) << left << "Degrees of Freedom" << "=  " << v << "\n";
+// t-statistic:
+double t_stat = (Sm1 - Sm2) / sqrt(Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2);
+cout << setw(55) << left << "T Statistic" << "=  " << t_stat << "\n";
+
+

+ Thereafter the code and the tests are performed the same as before. + Using are car mileage data again, here's what the output looks like: +

+
   __________________________________________________
+   Student t test for two samples (unequal variances)
+   __________________________________________________
+
+   Number of Observations (Sample 1)                      =  249
+   Sample 1 Mean                                          =  20.145
+   Sample 1 Standard Deviation                            =  6.4147
+   Number of Observations (Sample 2)                      =  79
+   Sample 2 Mean                                          =  30.481
+   Sample 2 Standard Deviation                            =  6.1077
+   Degrees of Freedom                                     =  136.87
+   T Statistic                                            =  -12.946
+   Probability that difference is due to chance           =  1.571e-025
+
+   Results for Alternative Hypothesis and alpha           =  0.0500
+
+   Alternative Hypothesis              Conclusion
+   Sample 1 Mean != Sample 2 Mean       NOT REJECTED
+   Sample 1 Mean <  Sample 2 Mean       NOT REJECTED
+   Sample 1 Mean >  Sample 2 Mean       REJECTED
+
+

+ This time allowing the variances in the two samples to differ has yielded + a higher likelihood that the observed difference is down to chance + alone (1.571e-025 compared to 5.273e-030 when equal variances were + assumed). However, the conclusion remains the same: US cars are less + fuel efficient than Japanese models. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview.html b/doc/sf_and_dist/html/math_toolkit/main_overview.html new file mode 100644 index 000000000..cc999d7bd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview.html @@ -0,0 +1,67 @@ + + + +Overview + + + + + + + + + + + + + + + +
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About the Math Toolkit
+
Navigation
+
Directory and + File Structure
+
Namespaces
+
Calculation + of the Type of the Result
+
Error Handling
+
Compilers
+
Configuration + and Policies
+
Thread Safety
+
Performance
+
History and What's + New
+
Contact Info and + Support
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/contact.html b/doc/sf_and_dist/html/math_toolkit/main_overview/contact.html new file mode 100644 index 000000000..0467b0ad8 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/contact.html @@ -0,0 +1,62 @@ + + + +Contact Info and Support + + + + + + + + + + + + + + + +
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+ The main support for this library is via the Boost mailing lists: +

+
+

+ You can also find JM at john - at - johnmaddock.co.uk and PAB at pbristow + - at - hetp.u-net.com. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
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+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/directories.html b/doc/sf_and_dist/html/math_toolkit/main_overview/directories.html new file mode 100644 index 000000000..8c0318b2b --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/directories.html @@ -0,0 +1,124 @@ + + + +Directory and File Structure + + + + + + + + + + + + + + + +
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+ + boost/math +
+
+

+
+
/concepts/
+

+ Prototype defining the essential features + of a RealType class (see real_concept.hpp). Most applications will use + double as the RealType (and + short typedef names of distributions + are reserved for this type where possible), a few will use float or long + double, but it is also possible + to use higher precision types like NTL::RR + that conform to the requirements specified by real_concept. +

+
/constants/
+

+ Templated definition of some highly accurate math constants (in constants.hpp). +

+
/distributions/
+

+ Distributions used in mathematics and, especially, statistics: Gaussian, + Students-t, Fisher, Binomial etc +

+
/policies/
+

+ Policy framework, for handling user requested behaviour modifications. +

+
/special_functions/
+

+ Math functions generally regarded as 'special', like beta, cbrt, erf, + gamma, lgamma, tgamma ... (Some of these are specified in C++, and C99/TR1, + and perhaps TR2). +

+
/tools/
+

+ Tools used by functions, like evaluating polynomials, continued fractions, + root finding, precision and limits, and by tests. Some will find application + outside this package. +

+
+
+
+ + boost/libs +
+
+

+
+
/doc/
+

+ Documentation source files in Quickbook format processed into html and + pdf formats. +

+
/examples/
+

+ Examples and demos of using math functions and distributions. +

+
/performance/
+

+ Performance testing and tuning program. +

+
/test/
+

+ Test files, in various .cpp files, most using Boost.Test (some with test + data as .ipp files, usually generated using NTL RR type with ample precision + for the type, often for precisions suitable for up to 256-bit significand + real types). +

+
/tools/
+

+ Programs used to generate test data. Also changes to the NTL + released package to provide a few additional (and vital) extra features. +

+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/error_handling.html b/doc/sf_and_dist/html/math_toolkit/main_overview/error_handling.html new file mode 100644 index 000000000..2f01ce2bb --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/error_handling.html @@ -0,0 +1,899 @@ + + + +Error Handling + + + + + + + + + + + + + + + +
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+ + Quick + Reference +
+

+ Handling of errors by this library is split into two orthogonal parts: +

+
    +
  • + What kind of error has been raised? +
    • +
  • + What should be done when the error is raised? +
    • +
+

+ The kinds of errors that can be raised are: +

+
+

+
+
Domain Error
+

+ Occurs when one or more arguments to a function are out of range. +

+
Pole Error
+

+ Occurs when the particular arguments cause the function to be evaluated + at a pole with no well defined residual value. For example if tgamma + is evaluated at exactly -2, the function approaches different limiting + values depending upon whether you approach from just above or just below + -2. Hence the function has no well defined value at this point and a + Pole Error will be raised. +

+
Overflow Error
+

+ Occurs when the result is either infinite, or too large to represent + in the numeric type being returned by the function. +

+
Underflow Error
+

+ Occurs when the result is not zero, but is too small to be represented + by any other value in the type being returned by the function. +

+
Denormalisation Error
+

+ Occurs when the returned result would be a denormalised value. +

+
Evaluation Error
+

+ Occurs when an internal error occured that prevented the result from + being evaluated: this should never occur, but if it does, then it's likely + to be due to an iterative method not converging fast enough. +

+
+
+

+ The action undertaken by each error condition is determined by the current + Policy in effect. This can be + changed program-wide by setting some configuration macros, or at namespace + scope, or at the call site (by specifying a specific policy in the function + call). +

+

+ The available actions are: +

+
+

+
+
throw_on_error
+

+ Throws the exception most appropriate to the error condition. +

+
errno_on_error
+

+ Sets ::errno to an appropriate value, and then returns the most appropriate + result +

+
ignore_error
+

+ Ignores the error and simply the returns the most appropriate result. +

+
user_error
+

+ Calls a user-supplied + error handler. +

+
+
+

+ The following tables show all the permutations of errors and actions, with + the default action for each error shown in bold: +

+
+

Table 1. Possible Actions for Domain Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws std::domain_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to EDOM and returns + std::numeric_limits<T>::quiet_NaN() +

+
+

+ ignore_error +

+ 		+

+ Returns std::numeric_limits<T>::quiet_NaN() +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_domain_error: + this + function must be defined by the user. +

+
+
+
+

Table 2. Possible Actions for Pole Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws std::domain_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to EDOM and returns + std::numeric_limits<T>::quiet_NaN() +

+
+

+ ignore_error +

+ 		+

+ Returns std::numeric_limits<T>::quiet_NaN() +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_pole_error: + this + function must be defined by the user. +

+
+
+
+

Table 3. Possible Actions for Overflow Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws std::overflow_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to ERANGE and returns + std::numeric_limits<T>::infinity() +

+
+

+ ignore_error +

+ 		+

+ Returns std::numeric_limits<T>::infinity() +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_overflow_error: + this + function must be defined by the user. +

+
+
+
+

Table 4. Possible Actions for Underflow Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws std::underflow_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to ERANGE and returns + 0. +

+
+

+ ignore_error +

+ 		+

+ Returns 0 +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_underflow_error: + this + function must be defined by the user. +

+
+
+
+

Table 5. Possible Actions for Denorm Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws std::underflow_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to ERANGE and returns + the denormalised value. +

+
+

+ ignore_error +

+ 		+

+ Returns the denormalised value. +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_denorm_error: + this + function must be defined by the user. +

+
+
+
+

Table 6. Possible Actions for Internal Evaluation + Errors

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Action +

+ 		+

+ Behaviour +

+
+

+ throw_on_error +

+ 		+

+ Throws boost::math::evaluation_error +

+
+

+ errno_on_error +

+ 		+

+ Sets ::errno + to EDOM and returns + std::numeric_limits<T>::infinity(). +

+
+

+ ignore_error +

+ 		+

+ Returns std::numeric_limits<T>::infinity(). +

+
+

+ user_error +

+ 		+

+ Returns the result of boost::math::policies::user_evaluation_error: + this + function must be defined by the user. +

+
+
+
+ + Rationale +
+

+ The flexibility of the current implementation should be reasonably obvious, + the default behaviours were chosen based on feedback during the formal review + of this library. It was felt that: +

+
    +
  • + Genuine errors should be flagged with exceptions rather than following + C-compatible behaviour and setting ::errno. +
    • +
  • + Numeric underflow and denormalised results were not considered to be fatal + errors in most cases, so it was felt that these should be ignored. +
    • +
+
+ + Finding + More Information +
+

+ There are some pre-processor macro defines that can be used to change + the policy defaults. See also the policy + section. +

+

+ An example is at the Policy tutorial in Changing + the Policy Defaults. +

+

+ Full source code of this typical example of passing a 'bad' argument (negative + degrees of freedom) to Student's t distribution is in + the error handling example. +

+

+ The various kind of errors are described in more detail below. +

+
+ + Domain + Errors +
+

+ When a special function is passed an argument that is outside the range of + values for which that function is defined, then the function returns the + result of: +

+
+boost::math::policies::raise_domain_error<T>(FunctionName, Message, Val, Policy);
+
+

+ Where T is the floating-point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, Val is the value that was out + of range, and Policy is the current + policy in use for the function that was called. +

+

+ The default policy behaviour of this function is to throw a std::domain_error + C++ exception. But if the Policy + is to ignore the error, or set global ::errno, then a NaN will be returned. +

+

+ This behaviour is chosen to assist compatibility with the behaviour of ISO/IEC + 9899:1999 Programming languages - C and with the Draft + Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph + 6: +

+
+

+

+

+ "Each of the functions declared above shall return a NaN + (Not a Number) if any argument value is a NaN, but it shall not report + a domain error. Otherwise, each of the functions declared above shall + report a domain error for just those argument values for which: +

+

+

+
+
+

+

+

+ "the function description's Returns clause explicitly + specifies a domain, and those arguments fall outside the specified domain; + or +

+

+

+

+ "the corresponding mathematical function value has a non-zero + imaginary component; or +

+

+

+

+ "the corresponding mathematical function is not mathematically + defined. +

+

+

+
+
+

+

+

+ "Note 2: A mathematical function is mathematically defined + for a given set of argument values if it is explicitly defined for that + set of argument values or if its limiting value exists and does not depend + on the direction of approach." +

+

+

+
+

+ Note that in order to support information-rich error messages when throwing + exceptions, Message must + contain a Boost.Format + recognised format specifier: the argument Val + is inserted into the error message according to the specifier used. +

+

+ For example if Message contains + a "%1%" then it is replaced by the value of Val + to the full precision of T, where as "%.3g" would contain the value + of Val to 3 digits. See the + Boost.Format documentation + for more details. +

+
+ + Evaluation + at a pole +
+

+ When a special function is passed an argument that is at a pole without a + well defined residual value, then the function returns the result of: +

+
+boost::math::policies::raise_pole_error<T>(FunctionName, Message, Val, Policy);
+
+

+ Where T is the floating point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, Val + is the value of the argument that is at a pole, and Policy + is the current policy in use for the function that was called. +

+

+ The default behaviour of this function is to throw a std::domain_error exception. + But error + handling policies can be used to change this, for example to ignore_error and return NaN. +

+

+ Note that in order to support information-rich error messages when throwing + exceptions, Message must + contain a Boost.Format + recognised format specifier: the argument val + is inserted into the error message according to the specifier used. +

+

+ For example if Message contains + a "%1%" then it is replaced by the value of val + to the full precision of T, where as "%.3g" would contain the value + of val to 3 digits. See the + Boost.Format documentation + for more details. +

+
+ + Numeric + Overflow +
+

+ When the result of a special function is too large to fit in the argument + floating-point type, then the function returns the result of: +

+
+boost::math::policies::raise_overflow_error<T>(FunctionName, Message, Policy);
+
+

+ Where T is the floating-point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, and Policy + is the current policy in use for the function that was called. +

+

+ The default policy for this function is that std::overflow_error + C++ exception is thrown. But if, for example, an ignore_error + policy is used, then returns std::numeric_limits<T>::infinity(). + In this situation if the type T + doesn't support infinities, the maximum value for the type is returned. +

+
+ + Numeric + Underflow +
+

+ If the result of a special function is known to be non-zero, but the calculated + result underflows to zero, then the function returns the result of: +

+
+boost::math::policies::raise_underflow_error<T>(FunctionName, Message, Policy);
+
+

+ Where T is the floating point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, and Policy + is the current policy in use for the called function. +

+

+ The default version of this function returns zero. But with another policy, + like throw_on_error, throws + an std::underflow_error C++ exception. +

+
+ + Denormalisation + Errors +
+

+ If the result of a special function is a denormalised value z + then the function returns the result of: +

+
+boost::math::policies::raise_denorm_error<T>(z, FunctionName, Message, Policy);
+
+

+ Where T is the floating point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, and Policy + is the current policy in use for the called function. +

+

+ The default version of this function returns z. But + with another policy, like throw_on_error + throws an std::underflow_error C++ exception. +

+
+ + Evaluation + Errors +
+

+ When a special function calculates a result that is known to be erroneous, + or where the result is incalculable then it calls: +

+
+boost::math::policies::raise_evaluation_error<T>(FunctionName, Message, Val, Policy);
+
+

+ Where T is the floating point + type passed to the function, FunctionName + is the name of the function, Message + is an error message describing the problem, Val + is the erroneous value, and Policy + is the current policy in use for the called function. +

+

+ The default behaviour of this function is to throw a boost::math::evaluation_error. +

+

+ Note that in order to support information rich error messages when throwing + exceptions, Message must + contain a Boost.Format + recognised format specifier: the argument val + is inserted into the error message according to the specifier used. +

+

+ For example if Message contains + a "%1%" then it is replaced by the value of val + to the full precision of T, where as "%.3g" would contain the value + of val to 3 digits. See the + Boost.Format documentation + for more details. +

+
+ + Errors + from typecasts +
+

+ Many special functions evaluate their results at a higher precision than + their arguments in order to ensure full machine precision in the result: + for example, a function passed a float argument may evaluate its result using + double precision internally. Many of the errors listed above may therefore + occur not during evaluation, but when converting the result to the narrower + result type. The function: +

+
+template <class T, class Policy, class U>
+T checked_narrowing_cast(U const& val, const char* function);
+
+

+ Is used to perform these conversions, and will call the error handlers listed + above on overflow, underflow + or denormalisation. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/history1.html b/doc/sf_and_dist/html/math_toolkit/main_overview/history1.html new file mode 100644 index 000000000..6ab8de5c7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/history1.html @@ -0,0 +1,218 @@ + + + +History and What's New + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+ + Milestone + 5: Post Review First Official Release +
+

+

+
    +
  • + Added Policy based framework that allows fine grained control over function + behaviour. +
    • +
  • +Breaking change: Changed default behaviour + for domain, pole and overflow errors to throw an exception (based on + review feedback), this behaviour can be customised using Policy's. +
    • +
  • +Breaking change: Changed exception thrown + when an internal evaluation error occurs to boost::math::evaluation_error. +
    • +
  • +Breaking change: Changed discrete quantiles + to return an integer result: this is anything up to 20 times faster than + finding the true root, this behaviour can be customised using Policy's. +
    • +
  • + Polynomial/rational function evaluation is now customisable and hopefully + faster than before. +
    • +
  • + Added performance test program. +
    • +
+

+ +

+
+ + Milestone + 4: Second Review Candidate (1st March 2007) +
+

+

+
    +
  • + Moved Xiaogang Zhang's Bessel Functions code into the library, and brought + them into line with the rest of the code. +
    • +
  • + Added C# "Distribution Explorer" demo application. +
    • +
+

+ +

+
+ + Milestone + 3: First Review Candidate (31st Dec 2006) +
+

+

+
    +
  • + Implemented the main probability distribution and density functions. +
    • +
  • + Implemented digamma. +
    • +
  • + Added more factorial functions. +
    • +
  • + Implemented the Hermite, Legendre and Laguerre polynomials plus the spherical + harmonic functions from TR1. +
    • +
  • + Moved Xiaogang Zhang's elliptic integral code into the library, and brought + them into line with the rest of the code. +
    • +
  • + Moved Hubert Holin's existing Boost.Math special functions into this + library and brought them into line with the rest of the code. +
    • +
+

+ +

+
+ + Milestone + 2: Released September 10th 2006 +
+

+

+
    +
  • + Implement preview release of the statistical distributions. +
    • +
  • + Added statistical distributions tutorial. +
    • +
  • + Implemented root finding algorithms. +
    • +
  • + Implemented the inverses of the incomplete gamma and beta functions. +
    • +
  • + Rewrite erf/erfc as rational approximations (valid to 128-bit precision). +
    • +
  • + Integrated the statistical results generated from the test data with + Boost.Test: uses a database of expected results, indexed by test, floating + point type, platform, and compiler. +
    • +
  • + Improved lgamma near 1 and 2 (rational approximations). +
    • +
  • + Improved erf/erfc inverses (rational approximations). +
    • +
  • + Implemented Rational function generation (the Remez method). +
    • +
+

+ +

+
+ + Milestone + 1: Released March 31st 2006 +
+

+

+
    +
  • + Implement gamma/beta/erf functions along with their incomplete counterparts. +
    • +
  • + Generate high quality test data, against which future improvements can + be judged. +
    • +
  • + Provide tools for the evaluation of infinite series, continued fractions, + and rational functions. +
    • +
  • + Provide tools for testing against tabulated test data, and collecting + statistics on error rates. +
    • +
  • + Provide sufficient docs for people to be able to find their way around + the library. +
    • +
+

+

+

+ SVN Revisions: +

+

+

+

+ Sandbox and trunk last synchonised at revision: 40161. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/intro.html b/doc/sf_and_dist/html/math_toolkit/main_overview/intro.html new file mode 100644 index 000000000..8377d8275 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/intro.html @@ -0,0 +1,134 @@ + + + +About the Math Toolkit + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ This library is divided into three interconnected parts: +

+
+ + Statistical + Distributions +
+

+ Provides a reasonably comprehensive set of statistical + distributions, upon which higher level statistical tests can be built. +

+

+ The initial focus is on the central univariate + distributions. + Both continuous + (like normal + & Fisher) + and discrete + (like binomial + & Poisson) + distributions are provided. +

+

+ A comprehensive tutorial is provided, + along with a series of worked + examples illustrating how the library is used to conduct statistical + tests. +

+
+ + Mathematical + Special Functions +
+

+ Provides a small number of high quality special + functions, initially these were concentrated on functions used in + statistical applications along with those in the Technical + Report on C++ Library Extensions. +

+

+ The function families currently implemented are the gamma, beta & erf + functions along with the incomplete gamma and beta functions (four variants + of each) and all the possible inverses of these, plus digamma, various factorial + functions, Bessel functions, elliptic integrals, sinus cardinals (along with + their hyperbolic variants), inverse hyperbolic functions, Legrendre/Laguerre/Hermite + polynomials and various special power and logarithmic functions. +

+

+ All the implementations are fully generic and support the use of arbitrary + "real-number" types, although they are optimised for use with types + with known-about significand + (or mantissa) sizes: typically float, + double or long + double. +

+
+ + Implementation + Toolkit +
+

+ Provides many of the tools required + to implement mathematical special functions: hopefully the presence of these + will encourage other authors to contribute more special function implementations + in the future. These tools are currently considered experimental: they are + "exposed implementation details" whose interfaces and/or implementations + may change. +

+

+ There are helpers for the evaluation + of infinite series, continued + fractions and rational + approximations. +

+

+ There is a fairly comprehensive set of root finding and function + minimisation algorithms: the root finding algorithms are both with and without + derivative support. +

+

+ A Remez algorithm + implementation allows for the locating of minimax rational approximations. +

+

+ There are also (experimental) classes for the manipulation + of polynomials, for testing + a special function against tabulated test data, and for the rapid generation of test + data and/or data for output to an external graphing application. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/namespaces.html b/doc/sf_and_dist/html/math_toolkit/main_overview/namespaces.html new file mode 100644 index 000000000..ece07a4fd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/namespaces.html @@ -0,0 +1,76 @@ + + + +Namespaces + + + + + + + + + + + + + + + +
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+
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+
+

+ All math functions and distributions are in namespace + boost::math +

+

+ So, for example, the Students-t distribution template in namespace + boost::math is +

+
+template <class RealType> class students_t_distribution
+
+

+ and can be instantiated with the help of the reserved name students_t(for RealType + double) +

+
+typedef students_t_distribution<double> students_t;
+
+student_t mydist(10);
+
+

+ Functions not intended for use by applications are in boost::math::detail. +

+

+ Functions that may have more general use, like digits + (significand), max_value, + min_value and epsilon are in boost::math::tools. +

+

+ Policy and configuration information + is in namespace boost::math::policies. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/navigation.html b/doc/sf_and_dist/html/math_toolkit/main_overview/navigation.html new file mode 100644 index 000000000..576cac5da --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/navigation.html @@ -0,0 +1,154 @@ + + + +Navigation + + + + + + + + + + + + + + + +
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+ Used in combination with the configured browser key, the following keys act + as handy shortcuts for common navigation tasks. +

+
+ + Shortcuts +
+
+

+

+

+ p - Previous page +

+

+

+

+ n - Next page +

+

+

+

+ h - home +

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+

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+ u - Up +

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+

+
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+ The following table shows how to access these from common browsers: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Browser +

+ 		+

+ Access Method +

+
+

+ Internet Explorer +

+ 		+

+ Alt+Key highlights the link only, so for example to move to the next + topic you would need "Alt+n" followed by "Enter". +

+
+

+ Firefox 2.0 and later +

+ 		+

+ Alt+Shift+Key follows the link, so for example "Alt+Shift+n" + will take you to the next topic. +

+
+

+ Opera +

+ 		+

+ Press Shift+Esc followed by the access key. +

+
+

+ Konqueror +

+ 		+

+ Press and release the Ctrl key, followed by the access key +

+
+

+ Some browsers also make these links available in their site-navigation toolbars: + in Opera for example you can use Ctrl plus the left and right arrow keys + to move between "next" and "previous" topics. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/perf_over.html b/doc/sf_and_dist/html/math_toolkit/main_overview/perf_over.html new file mode 100644 index 000000000..19d6aafb5 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/perf_over.html @@ -0,0 +1,93 @@ + + + +Performance + + + + + + + + + + + + + + + +
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+

+

+ By and large the performance of this library should be acceptable for most + needs. However, you should note that the library's primary emphasis is + on accuracy and numerical stability, and not speed. +

+

+

+

+ In terms of the algorithms used, this library aims to use the same "best + of breed" algorithms as many other libraries: the principle difference + is that this library is implemented in C++ - taking advantage of all the + abstraction mechanisms that C++ offers - where as most traditional numeric + libraries are implemented in C or FORTRAN. Traditionally languages such + as C or FORTAN are perceived as easier to optimise than more complex languages + like C++, so in a sense this library provides a good test of current compiler + technology, and the "abstraction penalty" - if any - of C++ compared + to other languages. +

+

+

+

+ The two most important things you can do to ensure the best performance + from this library are: +

+

+

+
    +
  1. + Turn on your compilers optimisations: the difference between "release" + and "debug" builds can easily be a factor + of 20. +
    2. +
  2. + Pick your compiler carefully: performance + differences of up to 8 fold have been found between some windows + compilers for example. +
    3. +
+

+

+

+ The performance section contains + more information on the performance of this library, what you can do to + fine tune it, and how this library compares to some other open source alternatives. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/pol_overview.html b/doc/sf_and_dist/html/math_toolkit/main_overview/pol_overview.html new file mode 100644 index 000000000..dbbeb45a2 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/pol_overview.html @@ -0,0 +1,136 @@ + + + +Configuration and Policies + + + + + + + + + + + + + + + +
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+
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+
+
+

+

+

+ Policies are a powerful fine-grain mechanism that allow you to customise + the behaviour of this library according to your needs. There is more information + available in the policy + tutorial and the policy + reference. +

+

+

+

+ Generally speaking unless you find that the default + policy behaviour when encountering 'bad' argument values does not + meet your needs, you should not need to worry about policies. +

+

+

+

+ Policies are a compile-time mechanism that allow you to change error-handling + or calculation precision either program wide, or at the call site. +

+

+

+

+ Although the policy mechanism itself is rather complicated, in practice + it is easy to use, and very flexible. +

+

+

+

+ Using policies you can control: +

+

+

+
+

+

+

+ You can control policies: +

+

+

+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/result_type.html b/doc/sf_and_dist/html/math_toolkit/main_overview/result_type.html new file mode 100644 index 000000000..ffed2febe --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/result_type.html @@ -0,0 +1,154 @@ + + + +Calculation of the Type of the Result + + + + + + + + + + + + + + + +
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+
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+
+
+

+ The functions in this library are all overloaded to accept mixed floating + point (or mixed integer and floating point type) arguments. So for example: +

+
+foo(1.0, 2.0);
+foo(1.0f, 2);
+foo(1.0, 2L);
+
+

+ etc, are all valid calls, as long as "foo" is a function taking + two floating-point arguments. But that leaves the question: +

+
+

+

+ "Given a special function with N arguments of types T1, T2, + T3 ... TN, then what type is the result?" +

+
+

+ If all the arguments are of the same (floating point) + type then the result is the same type as the arguments. +

+

+ Otherwise, the type of the result is computed using the following logic: +

+
    +
  1. + Any arguments that are not template arguments are disregarded from further + analysis. +
    2. +
  2. + For each type in the argument list, if that type is an integer type then + it is treated as if it were of type double for the purposes of further + analysis. +
    3. +
  3. + If any of the arguments is a user-defined class type, then the result type + is the first such class type that is constructible from all of the other + argument types. +
    4. +
  4. + If any of the arguments is of type long + double, then the result is of type + long double. +
    5. +
  5. + If any of the arguments is of type double, + then the result is of type double. +
    6. +
  6. + Otherwise the result is of type float. +
    7. +
+

+ For example: +

+
+cyl_bessel(2, 3.0);
+
+

+ Returns a double result, as + does: +

+
+cyl_bessel(2, 3.0f);
+
+

+ as in this case the integer first argument is treated as a double and takes precedence over the float second argument. To get a float result we would need all the arguments + to be of type float: +

+
+cyl_bessel_j(2.0f, 3.0f);
+
+

+ When one or more of the arguments is not a template argument then it doesn't + effect the return type at all, for example: +

+
+sph_bessel(2, 3.0f);
+
+

+ returns a float, since the first + argument is not a template argument and so doesn't effect the result: without + this rule functions that take explicitly integer arguments could never return + float. +

+

+ And for user defined types, all of the following return an NTL::RR result: +

+
+cyl_bessel_j(0, NTL::RR(2));
+
+cyl_bessel_j(NTL::RR(2), 3);
+
+cyl_bessel_j(NTL::quad_float(2), NTL::RR(3));
+
+

+ In the last case, quad_float is convertible to RR, but not vice-versa, so + the result will be an NTL::RR. Note that this assumes that you are using + a patched NTL library. +

+

+ These rules are chosen to be compatible with the behaviour of ISO/IEC + 9899:1999 Programming languages - C and with the Draft + Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph + 5. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/main_overview/threads.html b/doc/sf_and_dist/html/math_toolkit/main_overview/threads.html new file mode 100644 index 000000000..6aba11951 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/main_overview/threads.html @@ -0,0 +1,70 @@ + + + +Thread Safety + + + + + + + + + + + + + + + +
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+
+

+ The library is fully thread safe and re-entrant provided the function and + class templates in the library are instantiated with built-in floating point + types: i.e. the types float, + double and long + double. +

+

+ However, the library is not thread safe + when used with user-defined (i.e. class type) numeric types. +

+

+ The reason for the latter limitation is the need to initialise symbolic constants + using constructs such as: +

+
+static const T coefficient_array = { ... list of values ... };
+
+

+ Which is always thread safe when T is a built-in floating point type, but + not when T is a user defined type: as in this case there is a need for T's + constructors to be run, leading to potential race conditions. +

+

+ This limitation may be addressed in a future release. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf.html b/doc/sf_and_dist/html/math_toolkit/perf.html new file mode 100644 index 000000000..440960f3a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf.html @@ -0,0 +1,56 @@ + + + +Performance + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Performance Overview
+
Interpreting these Results
+
Getting the Best Performance + from this Library
+
Comparing Compilers
+
Performance Tuning Macros
+
Comparisons to Other + Open Source Libraries
+
The Performance Test + Application
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/comp_compilers.html b/doc/sf_and_dist/html/math_toolkit/perf/comp_compilers.html new file mode 100644 index 000000000..8404313e5 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/comp_compilers.html @@ -0,0 +1,377 @@ + + + +Comparing Compilers + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ After a good choice of build settings the next most important thing you can + do, is choose your compiler - and the standard C library it sits on top of + - very carefully. GCC-3.x in particular has been found to be particularly + bad at inlining code, and performing the kinds of high level transformations + that good C++ performance demands (thankfully GCC-4.x is somewhat better + in this respect). +

+
+

Table 40. Performance Comparison of Various Windows Compilers

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Intel C++ 10.0 +

+

+ ( /Ox /Qipo /QxN ) +

+ 		+

+ Microsoft Visual C++ 8.0 +

+

+ ( /Ox /arch:SSE2 ) +

+ 		+

+ Cygwin G++ 3.4 +

+

+ ( /O3 ) +

+
+

+ erf +

+ 		+

+

+

1.00

+

+

(4.118e-008s)

+

+

+ 		+

+

+

1.50

+

+

(6.173e-008s)

+

+

+ 		+

+

+

3.24

+

+

(1.336e-007s)

+

+

+
+

+ erf_inv +

+ 		+

+

+

1.00

+

+

(4.439e-008s)

+

+

+ 		+

+

+

1.42

+

+

(6.302e-008s)

+

+

+ 		+

+

+

7.88

+

+

(3.500e-007s)

+

+

+
+

+ ibeta + and ibetac +

+ 		+

+

+

1.00

+

+

(1.631e-006s)

+

+

+ 		+

+

+

1.14

+

+

(1.852e-006s)

+

+

+ 		+

+

+

3.05

+

+

(4.975e-006s)

+

+

+
+

+ ibeta_inv + and ibetac_inv +

+ 		+

+

+

1.00

+

+

(6.133e-006s)

+

+

+ 		+

+

+

1.19

+

+

(7.311e-006s)

+

+

+ 		+

+

+

2.60

+

+

(1.597e-005s)

+

+

+
+

+ ibeta_inva, + ibetac_inva, + ibeta_invb + and ibetac_invb +

+ 		+

+

+

1.00

+

+

(2.453e-005s)

+

+

+ 		+

+

+

1.16

+

+

(2.847e-005s)

+

+

+ 		+

+

+

2.83

+

+

(6.947e-005s)

+

+

+
+

+ gamma_p + and gamma_q +

+ 		+

+

+

1.00

+

+

(6.735e-007s)

+

+

+ 		+

+

+

1.41

+

+

(9.504e-007s)

+

+

+ 		+

+

+

2.78

+

+

(1.872e-006s)

+

+

+
+

+ gamma_p_inv + and gamma_q_inv +

+ 		+

+

+

1.00

+

+

(2.637e-006s)

+

+

+ 		+

+

+

1.38

+

+

(3.631e-006s)

+

+

+ 		+

+

+

3.31

+

+

(8.736e-006s)

+

+

+
+

+ gamma_p_inva + and gamma_q_inva +

+ 		+

+

+

1.00

+

+

(7.716e-006s)

+

+

+ 		+

+

+

1.29

+

+

(9.982e-006s)

+

+

+ 		+

+

+

2.56

+

+

(1.974e-005s)

+

+

+
+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/comparisons.html b/doc/sf_and_dist/html/math_toolkit/perf/comparisons.html new file mode 100644 index 000000000..c3b7d544e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/comparisons.html @@ -0,0 +1,1772 @@ + + + +Comparisons to Other Open Source Libraries + + + + + + + + + + + + + + + +
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+
+
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+
+
+

+ We've run our performance tests both for our own code, and against other + open source implementations of the same functions. The results are presented + below to give you a rough idea of how they all compare. +

+
+ + + + + +
[Caution]Caution

+ You should exercise extreme caution when interpreting these results, relative + performance may vary by platform, the tests use data that gives good code + coverage of our code, but which may skew the results + towards the corner cases. Finally, remember that different libraries make + different choices with regard to performance verses numerical stability. +

+
+ + Comparison + to GSL-1.9 and Cephes +
+

+ All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows + XP machine, with all the libraries compiled with Microsoft Visual C++ 2005 + using the /Ox + /arch:SSE2 options. +

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Boost +

+ 		+

+ GSL-1.9 +

+ 		+

+ Cephes +

+
+

+ tgamma +

+ 		+

+

+

1.50

+

+

(2.566e-007s)

+

+

+ 		+

+

+

1.54

+

+

(2.627e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.709e-007s)

+

+

+
+

+ lgamma +

+ 		+

+

+

1.73

+

+

(2.688e-007s)

+

+

+ 		+

+

+

3.61

+

+

(5.621e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.556e-007s)

+

+

+
+

+ gamma_p + and gamma_q +

+ 		+

+

+

1.00

+

+

(9.504e-007s)

+

+

+ 		+

+

+

2.15

+

+

(2.042e-006s)

+

+

+ 		+

+

+

2.57

+

+

(2.439e-006s)

+

+

+
+

+ gamma_p_inv + and gamma_q_inv +

+ 		+

+

+

1.00

+

+

(3.631e-006s)

+

+

+ 		+

+ N/A +

+ 		+

+ +INF + [1] +

+
+

+ ibeta + and ibetac +

+ 		+

+

+

1.00

+

+

(1.852e-006s)

+

+

+ 		+

+

+

1.07

+

+

(1.974e-006s)

+

+

+ 		+

+

+

1.07

+

+

(1.974e-006s)

+

+

+
+

+ ibeta_inv + and ibetac_inv +

+ 		+

+

+

1.00

+

+

(7.311e-006s)

+

+

+ 		+

+ N/A +

+ 		+

+

+

2.24

+

+

(1.637e-005s)

+

+

+

[1] + Cephes gets stuck in an infinite loop while trying to execute + our test cases. +

+
+ + Comparison + to the R Statistical Library on Windows +
+

+ All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows + XP machine, with the test program compiled with Microsoft Visual C++ 2005, + and R-2.5.0 compiled in "standalone mode" with MinGW-3.4 (R-2.5.0 + appears not to be buildable with Visual C++). +

+
+

Table 43. A Comparison to the R Statistical Library on Windows + XP

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Statistical Function +

+ 		+

+ Boost +

+ 		+

+ R +

+
+

+ Beta Distribution + CDF +

+ 		+

+

+

1.20

+

+

(1.916e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.597e-006s)

+

+

+
+

+ Beta Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(6.570e-006s)

+

+

+ 		+

+

+

74.66 + [1] +

+

+

(4.905e-004s)

+

+

+
+

+ Binomial + Distribution CDF +

+ 		+

+

+

1.00

+

+

(5.276e-007s)

+

+

+ 		+

+

+

2.45

+

+

(1.293e-006s)

+

+

+
+

+ Binomial + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(4.013e-006s)

+

+

+ 		+

+

+

1.32

+

+

(5.280e-006s)

+

+

+
+

+ Cauchy + Distribution CDF +

+ 		+

+

+

1.00

+

+

(1.231e-007s)

+

+

+ 		+

+

+

1.28

+

+

(1.576e-007s)

+

+

+
+

+ Cauchy + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.498e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.498e-007s)

+

+

+
+

+ Chi + Squared Distribution CDF +

+ 		+

+

+

1.00

+

+

(7.889e-007s)

+

+

+ 		+

+

+

2.48

+

+

(1.955e-006s)

+

+

+
+

+ Chi + Squared Distribution Quantile +

+ 		+

+

+

1.00

+

+

(4.303e-006s)

+

+

+ 		+

+

+

1.61

+

+

(6.925e-006s)

+

+

+
+

+ Exponential + Distribution CDF +

+ 		+

+

+

1.00

+

+

(1.955e-007s)

+

+

+ 		+

+

+

1.97

+

+

(3.844e-007s)

+

+

+
+

+ Exponential + Distribution Quantile +

+ 		+

+

+

1.07

+

+

(1.206e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.126e-007s)

+

+

+
+

+ Fisher F Distribution + CDF +

+ 		+

+

+

1.00

+

+

(1.309e-006s)

+

+

+ 		+

+

+

2.12

+

+

(2.780e-006s)

+

+

+
+

+ Fisher F Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(7.204e-006s)

+

+

+ 		+

+

+

1.78

+

+

(1.280e-005s)

+

+

+
+

+ Gamma Distribution + CDF +

+ 		+

+

+

1.00

+

+

(1.076e-006s)

+

+

+ 		+

+

+

2.07

+

+

(2.227e-006s)

+

+

+
+

+ Gamma Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(5.189e-006s)

+

+

+ 		+

+

+

1.14

+

+

(5.937e-006s)

+

+

+
+

+ Log-normal + Distribution CDF +

+ 		+

+

+

1.00

+

+

(2.078e-007s)

+

+

+ 		+

+

+

1.41

+

+

(2.930e-007s)

+

+

+
+

+ Log-normal + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(6.692e-007s)

+

+

+ 		+

+

+

1.63

+

+

(1.090e-006s)

+

+

+
+

+ Negative + Binomial Distribution CDF +

+ 		+

+

+

1.00

+

+

(9.005e-007s)

+

+

+ 		+

+

+

2.42

+

+

(2.178e-006s)

+

+

+
+

+ Negative + Binomial Distribution Quantile +

+ 		+

+

+

1.00

+

+

(9.601e-006s)

+

+

+ 		+

+

+

53.59 + [2] +

+

+

(5.145e-004s)

+

+

+
+

+ Normal + Distribution CDF +

+ 		+

+

+

1.00

+

+

(5.926e-008s)

+

+

+ 		+

+

+

3.01

+

+

(1.785e-007s)

+

+

+
+

+ Normal + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.248e-007s)

+

+

+ 		+

+

+

1.05

+

+

(1.311e-007s)

+

+

+
+

+ Poisson + Distribution CDF +

+ 		+

+

+

1.00

+

+

(8.999e-007s)

+

+

+ 		+

+

+

2.42

+

+

(2.175e-006s)

+

+

+
+

+ Poisson + Distribution +

+ 		+

+

+

1.00

+

+

(1.853e-006s)

+

+

+ 		+

+

+

2.17

+

+

(4.014e-006s)

+

+

+
+

+ Students + t Distribution CDF +

+ 		+

+

+

1.00

+

+

(1.223e-006s)

+

+

+ 		+

+

+

1.13

+

+

(1.376e-006s)

+

+

+
+

+ Students + t Distribution Quantile +

+ 		+

+

+

1.00

+

+

(2.570e-006s)

+

+

+ 		+

+

+

1.04

+

+

(2.668e-006s)

+

+

+
+

+ Weibull Distribution + CDF +

+ 		+

+

+

1.00

+

+

(4.741e-007s)

+

+

+ 		+

+

+

1.46

+

+

(6.943e-007s)

+

+

+
+

+ Weibull Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(7.926e-007s)

+

+

+ 		+

+

+

1.08

+

+

(8.542e-007s)

+

+

+
+

[1] + There are a small number of our test cases where the R library + fails to converge on a result: these tend to dominate the performance + result. +

+

[2] + The R library appears to use a linear-search strategy, that can + perform very badly in a small number of pathological cases, but + may or may not be more efficient in "typical" cases +

+
+
+
+ + Comparison + to the R Statistical Library on Linux +
+

+ All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Mandriva + Linux machine, with the test program and R-2.5.0 compiled with GNU G++ 4.2.0. +

+
+

Table 44. A Comparison to the R Statistical Library on Linux

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Statistical Function +

+ 		+

+ Boost +

+ 		+

+ R +

+
+

+ Beta Distribution + CDF +

+ 		+

+

+

1.71

+

+

(3.508e-006s)

+

+

+ 		+

+

+

1.00

+

+

(2.050e-006s)

+

+

+
+

+ Beta Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(1.294e-005s)

+

+

+ 		+

+

+

44.06 + [1] +

+

+

(5.701e-004s)

+

+

+
+

+ Binomial + Distribution CDF +

+ 		+

+

+

1.22

+

+

(1.342e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.104e-006s)

+

+

+
+

+ Binomial + Distribution Quantile +

+ 		+

+

+

1.36

+

+

(7.083e-006s)

+

+

+ 		+

+

+

1.00

+

+

(5.194e-006s)

+

+

+
+

+ Cauchy + Distribution CDF +

+ 		+

+

+

1.00

+

+

(1.372e-007s)

+

+

+ 		+

+

+

1.47

+

+

(2.017e-007s)

+

+

+
+

+ Cauchy + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.542e-007s)

+

+

+ 		+

+

+

1.14

+

+

(1.752e-007s)

+

+

+
+

+ Chi + Squared Distribution CDF +

+ 		+

+

+

1.04

+

+

(1.820e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.753e-006s)

+

+

+
+

+ Chi + Squared Distribution Quantile +

+ 		+

+

+

1.39

+

+

(9.345e-006s)

+

+

+ 		+

+

+

1.00

+

+

(6.728e-006s)

+

+

+
+

+ Exponential + Distribution CDF +

+ 		+

+

+

1.00

+

+

(2.195e-007s)

+

+

+ 		+

+

+

1.17

+

+

(2.561e-007s)

+

+

+
+

+ Exponential + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.123e-007s)

+

+

+ 		+

+

+

1.03

+

+

(1.155e-007s)

+

+

+
+

+ Fisher F Distribution + CDF +

+ 		+

+

+

1.00

+

+

(2.744e-006s)

+

+

+ 		+

+

+

1.08

+

+

(2.970e-006s)

+

+

+
+

+ Fisher F Distribution + Quantile +

+ 		+

+

+

1.14

+

+

(1.550e-005s)

+

+

+ 		+

+

+

1.00

+

+

(1.359e-005s)

+

+

+
+

+ Gamma Distribution + CDF +

+ 		+

+

+

1.29

+

+

(2.578e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.992e-006s)

+

+

+
+

+ Gamma Distribution + Quantile +

+ 		+

+

+

1.77

+

+

(1.020e-005s)

+

+

+ 		+

+

+

1.00

+

+

(5.757e-006s)

+

+

+
+

+ Log-normal + Distribution CDF +

+ 		+

+

+

1.00

+

+

(1.782e-007s)

+

+

+ 		+

+

+

2.00

+

+

(3.564e-007s)

+

+

+
+

+ Log-normal + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(7.093e-007s)

+

+

+ 		+

+

+

1.07

+

+

(7.607e-007s)

+

+

+
+

+ Negative + Binomial Distribution CDF +

+ 		+

+

+

1.03

+

+

(2.209e-006s)

+

+

+ 		+

+

+

1.00

+

+

(2.141e-006s)

+

+

+
+

+ Negative + Binomial Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.826e-005s)

+

+

+ 		+

+

+

30.07 + [2] +

+

+

(5.490e-004s)

+

+

+
+

+ Normal + Distribution CDF +

+ 		+

+

+

1.00

+

+

(8.542e-008s)

+

+

+ 		+

+

+

2.09

+

+

(1.782e-007s)

+

+

+
+

+ Normal + Distribution Quantile +

+ 		+

+

+

1.00

+

+

(1.362e-007s)

+

+

+ 		+

+

+

1.26

+

+

(1.722e-007s)

+

+

+
+

+ Poisson + Distribution CDF +

+ 		+

+

+

1.10

+

+

(1.953e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.775e-006s)

+

+

+
+

+ Poisson + Distribution +

+ 		+

+

+

1.12

+

+

(4.214e-006s)

+

+

+ 		+

+

+

1.00

+

+

(3.752e-006s)

+

+

+
+

+ Students + t Distribution CDF +

+ 		+

+

+

1.55

+

+

(2.441e-006s)

+

+

+ 		+

+

+

1.00

+

+

(1.576e-006s)

+

+

+
+

+ Students + t Distribution Quantile +

+ 		+

+

+

1.33

+

+

(3.972e-006s)

+

+

+ 		+

+

+

1.00

+

+

(2.990e-006s)

+

+

+
+

+ Weibull Distribution + CDF +

+ 		+

+

+

1.00

+

+

(6.640e-007s)

+

+

+ 		+

+

+

1.06

+

+

(7.031e-007s)

+

+

+
+

+ Weibull Distribution + Quantile +

+ 		+

+

+

1.00

+

+

(7.504e-007s)

+

+

+ 		+

+

+

1.03

+

+

(7.710e-007s)

+

+

+
+

[1] + There are a small number of our test cases where the R library + fails to converge on a result: these tend to dominate the performance + result. +

+

[2] + The R library appears to use a linear-search strategy, that can + perform very badly in a small number of pathological cases, but + may or may not be more efficient in "typical" cases +

+
+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/getting_best.html b/doc/sf_and_dist/html/math_toolkit/perf/getting_best.html new file mode 100644 index 000000000..80e1f9978 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/getting_best.html @@ -0,0 +1,294 @@ + + + +Getting the Best Performance from this Library + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ By far the most important thing you can do when using this library is turn + on your compiler's optimisation options. As the following table shows the + penalty for using the library in debug mode can be quite large. +

+
+

Table 39. Performance Comparison of Release and Debug Settings

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ Microsoft Visual C++ 8.0 +

+

+ Debug Settings: /Od /ZI +

+ 		+

+ Microsoft Visual C++ 8.0 +

+

+ Release settings: /Ox /arch:SSE2 +

+
+

+ erf +

+ 		+

+

+

16.65

+

+

(1.028e-006s)

+

+

+ 		+

+

+

1.00

+

+

(6.173e-008s)

+

+

+
+

+ erf_inv +

+ 		+

+

+

19.28

+

+

(1.215e-006s)

+

+

+ 		+

+

+

1.00

+

+

(6.302e-008s)

+

+

+
+

+ ibeta + and ibetac +

+ 		+

+

+

8.32

+

+

(1.540e-005s)

+

+

+ 		+

+

+

1.00

+

+

(1.852e-006s)

+

+

+
+

+ ibeta_inv + and ibetac_inv +

+ 		+

+

+

10.25

+

+

(7.492e-005s)

+

+

+ 		+

+

+

1.00

+

+

(7.311e-006s)

+

+

+
+

+ ibeta_inva, + ibetac_inva, + ibeta_invb + and ibetac_invb +

+ 		+

+

+

8.57

+

+

(2.441e-004s)

+

+

+ 		+

+

+

1.00

+

+

(2.847e-005s)

+

+

+
+

+ gamma_p + and gamma_q +

+ 		+

+

+

10.98

+

+

(1.044e-005s)

+

+

+ 		+

+

+

1.00

+

+

(9.504e-007s)

+

+

+
+

+ gamma_p_inv + and gamma_q_inv +

+ 		+

+

+

10.25

+

+

(3.721e-005s)

+

+

+ 		+

+

+

1.00

+

+

(3.631e-006s)

+

+

+
+

+ gamma_p_inva + and gamma_q_inva +

+ 		+

+

+

11.26

+

+

(1.124e-004s)

+

+

+ 		+

+

+

1.00

+

+

(9.982e-006s)

+

+

+
+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/interp.html b/doc/sf_and_dist/html/math_toolkit/perf/interp.html new file mode 100644 index 000000000..dc8debb71 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/interp.html @@ -0,0 +1,70 @@ + + + +Interpreting these Results + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ In all of the following tables, the best performing result in each row, is + assigned a relative value of "1" and shown in bold, so a score + of "2" means "twice as slow as the best performing + result". Actual timings in seconds per function call are + also shown in parenthesis. +

+

+ Result were obtained on a system with an Intel 2.8GHz Pentium 4 processor + with 2Gb of RAM and running either Windows XP or Mandriva Linux. +

+
+ + + + + +
[Caution]Caution

+ As usual with performance results these should be taken with a large pinch + of salt: relative performance is known to shift quite a bit depending upon + the architecture of the particular test system used. Further more, our + performance results were obtained using our own test data: these test values + are designed to provide good coverage of our code and test all the appropriate + corner cases. They do not necessarily represent "typical" usage: + whatever that may be! +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/perf_over.html b/doc/sf_and_dist/html/math_toolkit/perf/perf_over.html new file mode 100644 index 000000000..7d7046fa4 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/perf_over.html @@ -0,0 +1,93 @@ + + + +Performance Overview + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+

+

+ By and large the performance of this library should be acceptable for most + needs. However, you should note that the library's primary emphasis is + on accuracy and numerical stability, and not speed. +

+

+

+

+ In terms of the algorithms used, this library aims to use the same "best + of breed" algorithms as many other libraries: the principle difference + is that this library is implemented in C++ - taking advantage of all the + abstraction mechanisms that C++ offers - where as most traditional numeric + libraries are implemented in C or FORTRAN. Traditionally languages such + as C or FORTAN are perceived as easier to optimise than more complex languages + like C++, so in a sense this library provides a good test of current compiler + technology, and the "abstraction penalty" - if any - of C++ compared + to other languages. +

+

+

+

+ The two most important things you can do to ensure the best performance + from this library are: +

+

+

+
    +
  1. + Turn on your compilers optimisations: the difference between "release" + and "debug" builds can easily be a factor + of 20. +
    2. +
  2. + Pick your compiler carefully: performance + differences of up to 8 fold have been found between some windows + compilers for example. +
    3. +
+

+

+

+ The performance section contains + more information on the performance of this library, what you can do to + fine tune it, and how this library compares to some other open source alternatives. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/perf_test_app.html b/doc/sf_and_dist/html/math_toolkit/perf/perf_test_app.html new file mode 100644 index 000000000..16d0b8466 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/perf_test_app.html @@ -0,0 +1,63 @@ + + + +The Performance Test Application + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Under boost-path/libs/math/performance you will find + a (fairly rudimentary) performance test application for this library. +

+

+ To run this application yourself, build the all the .cpp files in boost-path/libs/math/performance + into an application using your usual release-build settings. Run the application + with --help to see a full list of options, or with --all to test everything + (which takes quite a while), or with --tune to test the available + performance tuning options. +

+

+ If you want to use this application to test the effect of changing any of + the Policies, then you will need + to build and run it twice: once with the default Policies, + and then a second time with the Policies + you want to test set as the default. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/perf/tuning.html b/doc/sf_and_dist/html/math_toolkit/perf/tuning.html new file mode 100644 index 000000000..5b23d4f81 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/perf/tuning.html @@ -0,0 +1,897 @@ + + + +Performance Tuning Macros + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ There are a small number of performance tuning options that are determined + by configuration macros. These should be set in boost/math/tools/user.hpp; + or else reported to the Boost-development mailing list so that the appropriate + option for a given compiler and OS platform can be set automatically in our + configuration setup. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Macro +

+ 		+

+ Meaning +

+
+

+ BOOST_MATH_POLY_METHOD +

+ 		+

+ Determines how polynomials and most rational functions are evaluated. + Define to one of the values 0, 1, 2 or 3: see below for the meaning + of these values. +

+
+

+ BOOST_MATH_RATIONAL_METHOD +

+ 		+

+ Determines how symmetrical rational functions are evaluated: mostly + this only effects how the Lanczos approximation is evaluated, and + how the evaluate_rational + function behaves. Define to one of the values 0, 1, 2 or 3: see below + for the meaning of these values. +

+
+

+ BOOST_MATH_MAX_POLY_ORDER +

+ 		+

+ The maximum order of polynomial or rational function that will be + evaluated by a method other than 0 (a simple "for" loop). +

+
+

+ BOOST_MATH_INT_TABLE_TYPE(RT, IT) +

+ 		+

+ Many of the coefficients to the polynomials and rational functions + used by this library are integers. Normally these are stored as tables + as integers, but if mixed integer / floating point arithmetic is + much slower than regular floating point arithmetic then they can + be stored as tables of floating point values instead. If mixed arithmetic + is slow then add: +

+

+ #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT +

+

+ to boost/math/tools/user.hpp, otherwise the default of: +

+

+ #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) IT +

+

+ Set in boost/math/config.hpp is fine, and may well result in smaller + code. +

+
+

+ The values to which BOOST_MATH_POLY_METHOD + and BOOST_MATH_RATIONAL_METHOD + may be set are as follows: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + +
+

+ Value +

+ 		+

+ Effect +

+
+

+ 0 +

+ 		+

+ The polynomial or rational function is evaluated using Horner's method, + and a simple for-loop. +

+

+ Note that if the order of the polynomial or rational function is + a runtime parameter, or the order is greater than the value of BOOST_MATH_MAX_POLY_ORDER, then + this method is always used, irrespective of the value of BOOST_MATH_POLY_METHOD or BOOST_MATH_RATIONAL_METHOD. +

+
+

+ 1 +

+ 		+

+ The polynomial or rational function is evaluated without the use + of a loop, and using Horner's method. This only occurs if the order + of the polynomial is known at compile time and is less than or equal + to BOOST_MATH_MAX_POLY_ORDER. +

+
+

+ 2 +

+ 		+

+ The polynomial or rational function is evaluated without the use + of a loop, and using a second order Horner's method. In theory this + permits two operations to occur in parallel for polynomials, and + four in parallel for rational functions. This only occurs if the + order of the polynomial is known at compile time and is less than + or equal to BOOST_MATH_MAX_POLY_ORDER. +

+
+

+ 3 +

+ 		+

+ The polynomial or rational function is evaluated without the use + of a loop, and using a second order Horner's method. In theory this + permits two operations to occur in parallel for polynomials, and + four in parallel for rational functions. This differs from method + "2" in that the code is carefully ordered to make the parallelisation + more obvious to the compiler: rather than relying on the compiler's + optimiser to spot the parallelisation opportunities. This only occurs + if the order of the polynomial is known at compile time and is less + than or equal to BOOST_MATH_MAX_POLY_ORDER. +

+
+

+ To determine which of these options is best for your particular compiler/platform + build the performance test application with your usual release settings, + and run the program with the --tune command line option. +

+

+ In practice the difference between methods is rather small at present, as + the following table shows. However, parallelisation /vectorisation is likely + to become more important in the future: quite likely the methods currently + supported will need to be supplemented or replaced by ones more suited to + highly vectorisable processors in the future. +

+
+

Table 41. A Comparison of Polynomial Evaluation Methods

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Compiler/platform +

+ 		+

+ Method 0 +

+ 		+

+ Method 1 +

+ 		+

+ Method 2 +

+ 		+

+ Method 3 +

+
+

+ Microsoft C++ 8.0, Polynomial evaluation +

+ 		+

+

+

1.34

+

+

(1.161e-007s)

+

+

+ 		+

+

+

1.13

+

+

(9.777e-008s)

+

+

+ 		+

+

+

1.07

+

+

(9.289e-008s)

+

+

+ 		+

+

+

1.00

+

+

(8.678e-008s)

+

+

+
+

+ Microsoft C++ 8.0, Rational evaluation +

+ 		+

+

+

1.00

+

+

(1.443e-007s)

+

+

+ 		+

+

+

1.03

+

+

(1.492e-007s)

+

+

+ 		+

+

+

1.20

+

+

(1.736e-007s)

+

+

+ 		+

+

+

1.07

+

+

(1.540e-007s)

+

+

+
+

+ Intel C++ 10.0 (Windows), Polynomial evaluation +

+ 		+

+

+

1.03

+

+

(7.702e-008s)

+

+

+ 		+

+

+

1.03

+

+

(7.702e-008s)

+

+

+ 		+

+

+

1.00

+

+

(7.446e-008s)

+

+

+ 		+

+

+

1.03

+

+

(7.690e-008s)

+

+

+
+

+ Intel C++ 10.0 (Windows), Rational evaluation +

+ 		+

+

+

1.00

+

+

(1.245e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.245e-007s)

+

+

+ 		+

+

+

1.18

+

+

(1.465e-007s)

+

+

+ 		+

+

+

1.06

+

+

(1.318e-007s)

+

+

+
+

+ GNU G++ 4.2 (Linux), Polynomial evaluation +

+ 		+

+

+

1.61

+

+

(1.220e-007s)

+

+

+ 		+

+

+

1.68

+

+

(1.269e-007s)

+

+

+ 		+

+

+

1.23

+

+

(9.275e-008s)

+

+

+ 		+

+

+

1.00

+

+

(7.566e-008s)

+

+

+
+

+ GNU G++ 4.2 (Linux), Rational evaluation +

+ 		+

+

+

1.26

+

+

(1.660e-007s)

+

+

+ 		+

+

+

1.33

+

+

(1.758e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.318e-007s)

+

+

+ 		+

+

+

1.15

+

+

(1.513e-007s)

+

+

+
+

+ Intel C++ 10.0 (Linux), Polynomial evaluation +

+ 		+

+

+

1.15

+

+

(9.154e-008s)

+

+

+ 		+

+

+

1.15

+

+

(9.154e-008s)

+

+

+ 		+

+

+

1.00

+

+

(7.934e-008s)

+

+

+ 		+

+

+

1.00

+

+

(7.934e-008s)

+

+

+
+

+ Intel C++ 10.0 (Linux), Rational evaluation +

+ 		+

+

+

1.00

+

+

(1.245e-007s)

+

+

+ 		+

+

+

1.00

+

+

(1.245e-007s)

+

+

+ 		+

+

+

1.35

+

+

(1.684e-007s)

+

+

+ 		+

+

+

1.04

+

+

(1.294e-007s)

+

+

+
+
+

+ There is one final performance tuning option that is available as a compile + time policy. Normally when evaluating + functions at double precision, + these are actually evaluated at long + double precision internally: this + helps to ensure that as close to full double + precision as possible is achieved, but may slow down execution in some environments. + The defaults for this policy can be changed by defining + the macro BOOST_MATH_PROMOTE_DOUBLE_POLICY + to false, or by + specifying a specific policy when calling the special functions or + distributions. See also the policy + tutorial. +

+
+

Table 42. Performance Comparison with and Without Internal + Promotion to long double

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Function +

+ 		+

+ GCC 4.2 , Linux +

+

+ (with internal promotion of double to long double). +

+ 		+

+ GCC 4.2, Linux +

+

+ (without promotion of double). +

+
+

+ erf +

+ 		+

+

+

1.48

+

+

(1.387e-007s)

+

+

+ 		+

+

+

1.00

+

+

(9.377e-008s)

+

+

+
+

+ erf_inv +

+ 		+

+

+

1.11

+

+

(4.009e-007s)

+

+

+ 		+

+

+

1.00

+

+

(3.598e-007s)

+

+

+
+

+ ibeta + and ibetac +

+ 		+

+

+

1.29

+

+

(5.354e-006s)

+

+

+ 		+

+

+

1.00

+

+

(4.137e-006s)

+

+

+
+

+ ibeta_inv + and ibetac_inv +

+ 		+

+

+

1.44

+

+

(2.220e-005s)

+

+

+ 		+

+

+

1.00

+

+

(1.538e-005s)

+

+

+
+

+ ibeta_inva, + ibetac_inva, + ibeta_invb + and ibetac_invb +

+ 		+

+

+

1.25

+

+

(7.009e-005s)

+

+

+ 		+

+

+

1.00

+

+

(5.607e-005s)

+

+

+
+

+ gamma_p + and gamma_q +

+ 		+

+

+

1.26

+

+

(3.116e-006s)

+

+

+ 		+

+

+

1.00

+

+

(2.464e-006s)

+

+

+
+

+ gamma_p_inv + and gamma_q_inv +

+ 		+

+

+

1.27

+

+

(1.178e-005s)

+

+

+ 		+

+

+

1.00

+

+

(9.291e-006s)

+

+

+
+

+ gamma_p_inva + and gamma_q_inva +

+ 		+

+

+

1.20

+

+

(2.765e-005s)

+

+

+ 		+

+

+

1.00

+

+

(2.311e-005s)

+

+

+
+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy.html b/doc/sf_and_dist/html/math_toolkit/policy.html new file mode 100644 index 000000000..06246e462 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy.html @@ -0,0 +1,91 @@ + + + +Policies + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Policy Overview
+
Policy Tutorial
+
+
+ So Just What is a Policy Anyway?
+
+ Policies Have Sensible Defaults
+
So + How are Policies Used Anyway?
+
+ Changing the Policy Defaults
+
+ Setting Policies for Distributions on an Ad Hoc Basis
+
+ Changing the Policy on an Ad Hoc Basis for the Special Functions
+
+ Setting Policies at Namespace or Translation Unit Scope
+
+ Calling User Defined Error Handlers
+
+ Understanding Quantiles of Discrete Distributions
+
+
Policy Reference
+
+
+ Error Handling Policies
+
Internal + Promotion Policies
+
Mathematically + Undefined Function Policies
+
Discrete + Quantile Policies
+
Precision + Policies
+
Iteration + Limits Policies
+
Using + macros to Change the Policy Defaults
+
Setting + Polices at Namespace Scope
+
Policy Class + Reference
+
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_overview.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_overview.html new file mode 100644 index 000000000..1a579b095 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_overview.html @@ -0,0 +1,135 @@ + + + +Policy Overview + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+

+

+ Policies are a powerful fine-grain mechanism that allow you to customise + the behaviour of this library according to your needs. There is more information + available in the policy + tutorial and the policy + reference. +

+

+

+

+ Generally speaking unless you find that the default + policy behaviour when encountering 'bad' argument values does not + meet your needs, you should not need to worry about policies. +

+

+

+

+ Policies are a compile-time mechanism that allow you to change error-handling + or calculation precision either program wide, or at the call site. +

+

+

+

+ Although the policy mechanism itself is rather complicated, in practice + it is easy to use, and very flexible. +

+

+

+

+ Using policies you can control: +

+

+

+
+

+

+

+ You can control policies: +

+

+

+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref.html new file mode 100644 index 000000000..92249cc9d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref.html @@ -0,0 +1,64 @@ + + + +Policy Reference + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
+ Error Handling Policies
+
Internal + Promotion Policies
+
Mathematically + Undefined Function Policies
+
Discrete + Quantile Policies
+
Precision + Policies
+
Iteration + Limits Policies
+
Using + macros to Change the Policy Defaults
+
Setting + Polices at Namespace Scope
+
Policy Class + Reference
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/assert_undefined.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/assert_undefined.html new file mode 100644 index 000000000..918ff03ae --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/assert_undefined.html @@ -0,0 +1,92 @@ + + + +Mathematically Undefined Function Policies + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ There are some functions that are generic (they are present for all the + statistical distributions supported) but which may be mathematically undefined + for certain distributions, but defined for others. +

+

+ For example, the Cauchy distribution does not have a mean, so what should +

+
+mean(cauchy<>());
+
+

+ return, and should such an expression even compile at all? +

+

+ The default behaviour is for all such functions to not compile at all - + in fact they will raise a static + assertion - but by changing the policy we can have them return + the result of a domain error instead (which may well throw an exception, + depending on the error handling policy). +

+

+ This behaviour is controlled by the assert_undefined<> policy: +

+
+namespace boost{ namespace math{ namespace policies {
+
+template <bool b>
+class assert_undefined;
+
+}}} //namespaces
+
+

+ For example: +

+
+#include <boost/math/distributions/cauchy.hpp>
+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+// This will not compile, cauchy has no mean!
+double m1 = mean(cauchy()); 
+
+// This will compile, but raises a domain error!
+double m2 = mean(cauchy_distribution<double, policy<assert_undefined<false> > >());
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html new file mode 100644 index 000000000..a7df2779a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html @@ -0,0 +1,297 @@ + + + +Discrete Quantile Policies + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ If a statistical distribution is discrete then the + random variable can only have integer values - this leaves us with a problem + when calculating quantiles - we can either ignore the discreteness of the + distribution and return a real value, or we can round to an integer. As + it happens, computing integer values can be substantially faster than calculating + a real value, so there are definite advantages to returning an integer, + but we do then need to decide how best to round the result. The discrete_quantile policy defines how + discrete quantiles work, and how integer results are rounded: +

+
+enum discrete_quantile_policy_type
+{
+   real,
+   integer_round_outwards, // default
+   integer_round_inwards,
+   integer_round_down,
+   integer_round_up,
+   integer_round_nearest
+};
+
+template <discrete_quantile_policy_type>
+struct discrete_quantile;
+
+

+ The values that discrete_quantile + can take have the following meanings: +

+
+ + real +
+

+ Ignores the discreteness of the distribution, and returns a real-valued + result. For example: +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+typedef negative_binomial_distribution<
+      double, 
+      policy<discrete_quantile<integer_round_inwards> > 
+   > dist_type;
+   
+// Lower quantile:
+double x = quantile(dist_type(20, 0.3), 0.05);
+// Upper quantile:
+double y = quantile(complement(dist_type(20, 0.3), 0.05));
+
+
+

+

+

+

+

+ Results in x = + 27.3898 and y + = 68.1584. +

+
+ + integer_round_outwards +
+

+ This is the default policy: an integer value is returned so that: +

+
    +
  • + Lower quantiles (where the probability is less than 0.5) are rounded + down. +
    • +
  • + Upper quantiles (where the probability is greater than 0.5) are rounded + up. +
    • +
+

+ This is normally the safest rounding policy, since it ensures that both + one and two sided intervals are guaranteed to have at least + the requested coverage. For example: +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+
+using namespace boost::math;
+
+// Lower quantile rounded down:
+double x = quantile(negative_binomial(20, 0.3), 0.05);
+// Upper quantile rounded up:
+double y = quantile(complement(negative_binomial(20, 0.3), 0.05));
+
+
+

+

+

+

+

+ Results in x = + 27 (rounded down from 27.3898) and + y = + 69 (rounded up from 68.1584). +

+

+ The variables x and y are now defined so that: +

+
+cdf(negative_binomial(20), x) <= 0.05
+cdf(negative_binomial(20), y) >= 0.95
+
+

+ In other words we guarantee at least 90% coverage in the central + region overall, and also no more than 5% coverage + in each tail. +

+
+ + integer_round_inwards +
+

+ This is the opposite of integer_round_outwards: an + integer value is returned so that: +

+
    +
  • + Lower quantiles (where the probability is less than 0.5) are rounded + up. +
    • +
  • + Upper quantiles (where the probability is greater than 0.5) are rounded + down. +
    • +
+

+ For example: +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+typedef negative_binomial_distribution<
+      double, 
+      policy<discrete_quantile<integer_round_inwards> > 
+   > dist_type;
+   
+// Lower quantile rounded up:
+double x = quantile(dist_type(20, 0.3), 0.05);
+// Upper quantile rounded down:
+double y = quantile(complement(dist_type(20, 0.3), 0.05));
+
+
+

+

+

+

+

+ Results in x = + 28 (rounded up from 27.3898) and + y = + 68 (rounded down from 68.1584). +

+

+ The variables x and y are now defined so that: +

+
+cdf(negative_binomial(20), x) >= 0.05
+cdf(negative_binomial(20), y) <= 0.95
+
+

+ In other words we guarantee at no more than 90% coverage in the + central region overall, and also at least 5% coverage + in each tail. +

+
+ + integer_round_down +
+

+ Always rounds down to an integer value, no matter whether it's an upper + or a lower quantile. +

+
+ + integer_round_up +
+

+ Always rounds up to an integer value, no matter whether it's an upper or + a lower quantile. +

+
+ + integer_round_nearest +
+

+ Always rounds to the nearest integer value, no matter whether it's an upper + or a lower quantile. This will produce the requested coverage in + the average case, but for any specific example may results in + either significantly more or less coverage than the requested amount. For + example: +

+

+ For example: +

+

+

+

+ +

+
+#include <boost/math/distributions/negative_binomial.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+typedef negative_binomial_distribution<
+      double, 
+      policy<discrete_quantile<integer_round_nearest> > 
+   > dist_type;
+   
+// Lower quantile rounded up:
+double x = quantile(dist_type(20, 0.3), 0.05);
+// Upper quantile rounded down:
+double y = quantile(complement(dist_type(20, 0.3), 0.05));
+
+
+

+

+

+

+

+ Results in x = + 27 (rounded from 27.3898) and y = 68 (rounded from 68.1584). +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/error_handling_policies.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/error_handling_policies.html new file mode 100644 index 000000000..493685374 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/error_handling_policies.html @@ -0,0 +1,687 @@ + + + +Error Handling Policies + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ There are two orthogonal aspects to error handling: +

+
    +
  • + What to do (if anything) with the error. +
    • +
  • + What kind of error is being raised. +
    • +
+
+ + Available + Actions When an Error is Raised +
+

+ What to do with the error is encapsulated by an enumerated type: +

+
+namespace boost { namespace math { namespace policies {
+
+enum error_policy_type
+{
+   throw_on_error = 0, // throw an exception.
+   errno_on_error = 1, // set ::errno & return 0, NaN, infinity or best guess.
+   ignore_error = 2, // return 0, NaN, infinity or best guess.
+   user_error = 3  // call a user-defined error handler.
+};
+
+}}} // namespaces
+
+

+ The various enumerated values have the following meanings: +

+
+ + throw_on_error +
+

+ Will throw one of the following exceptions, depending upon the type of + the error: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Error Type +

+ 		+

+ Exception +

+
+

+ Domain Error +

+ 		+

+ std::domain_error +

+
+

+ Pole Error +

+ 		+

+ std::domain_error +

+
+

+ Overflow Error +

+ 		+

+ std::overflow_error +

+
+

+ Underflow Error +

+ 		+

+ std::underflow_error +

+
+

+ Denorm Error +

+ 		+

+ std::underflow_error +

+
+

+ Evaluation Error +

+ 		+

+ boost::math::evaluation_error +

+
+
+ + errno_on_error +
+

+ Will set global ::errno + to one of the following values depending upon the error type, and then + return the same value as if the error had been ignored: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Error Type +

+ 		+

+ errno value +

+
+

+ Domain Error +

+ 		+

+ EDOM +

+
+

+ Pole Error +

+ 		+

+ EDOM +

+
+

+ Overflow Error +

+ 		+

+ ERANGE +

+
+

+ Underflow Error +

+ 		+

+ ERANGE +

+
+

+ Denorm Error +

+ 		+

+ ERANGE +

+
+

+ Evaluation Error +

+ 		+

+ EDOM +

+
+
+ + ignore_error +
+

+ Will return a one of the values below depending on the error type (::errno + is NOT changed):: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Error Type +

+ 		+

+ Returned Value +

+
+

+ Domain Error +

+ 		+

+ std::numeric_limits<T>::quiet_NaN() +

+
+

+ Pole Error +

+ 		+

+ std::numeric_limits<T>::quiet_NaN() +

+
+

+ Overflow Error +

+ 		+

+ std::numeric_limits<T>::infinity() +

+
+

+ Underflow Error +

+ 		+

+ 0 +

+
+

+ Denorm Error +

+ 		+

+ The denormalised value. +

+
+

+ Evaluation Error +

+ 		+

+ The best guess as to the result: which may be significantly in + error. +

+
+
+ + user_error +
+

+ Will call a user defined error handler: these are forward declared in boost/math/policies/error_handling.hpp, + but the actual definitions must be provided by the user: +

+
+namespace boost{ namespace math{ namespace policies{
+
+template <class T>
+T user_domain_error(const char* function, const char* message, const T& val);
+
+template <class T>
+T user_pole_error(const char* function, const char* message, const T& val);
+
+template <class T>
+T user_overflow_error(const char* function, const char* message, const T& val);
+
+template <class T>
+T user_underflow_error(const char* function, const char* message, const T& val);
+
+template <class T>
+T user_denorm_error(const char* function, const char* message, const T& val);
+
+template <class T>
+T user_evaluation_error(const char* function, const char* message, const T& val);
+
+}}} // namespaces
+
+

+ Note that the strings function and message + may contain "%1%" format specifiers designed to be used in conjunction + with Boost.Format. If these strings are to be presented to the program's + end-user then the "%1%" format specifier should be replaced with + the name of type T in the function string, and if + there is a %1% specifier in the message string then + it should be replaced with the value of val. +

+

+ There is more information on user-defined error handlers in the tutorial + here. +

+
+ + Kinds + of Error Raised +
+

+ There are five kinds of error reported by this library, which are summarised + in the following table: +

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Error Type +

+ 		+

+ Policy Class +

+ 		+

+ Description +

+
+

+ Domain Error +

+ 		+

+ boost::math::policies::domain_error<action> +

+ 		+

+ Raised when more or more arguments are outside the defined range + of the function. +

+

+ Defaults to boost::math::policies::domain_error<throw_on_error> +

+

+ When the action is set to throw_on_error then + throws std::domain_error +

+
+

+ Pole Error +

+ 		+

+ boost::math::policies::pole_error<action> +

+ 		+

+ Raised when more or more arguments would cause the function to + be evaluated at a pole. +

+

+ Defaults to boost::math::policies::pole_error<throw_on_error> +

+

+ When the action is throw_on_error then throw + a std::domain_error +

+
+

+ Overflow Error +

+ 		+

+ boost::math::policies::overflow_error<action> +

+ 		+

+ Raised when the result of the function is outside the representable + range of the floating point type used. +

+

+ Defaults to boost::math::policies::overflow_error<throw_on_error>. +

+

+ When the action is throw_on_error then throws + a std::overflow_error. +

+
+

+ Underflow Error +

+ 		+

+ boost::math::policies::underflow_error<action> +

+ 		+

+ Raised when the result of the function is too small to be represented + in the floating point type used. +

+

+ Defaults to boost::math::policies::underflow_error<ignore_error> +

+

+ When the specified action is throw_on_error + then throws a std::underflow_error +

+
+

+ Denorm Error +

+ 		+

+ boost::math::policies::denorm_error<action> +

+ 		+

+ Raised when the result of the function is a denormalised value. +

+

+ Defaults to boost::math::policies::denorm_error<ignore_error> +

+

+ When the action is throw_on_error then throws + a std::underflow_error +

+
+

+ Evaluation Error +

+ 		+

+ boost::math::policies::evaluation_error<action> +

+ 		+

+ Raised when the result of the function is well defined and finite, + but we were unable to compute it. Typically this occurs when an + iterative method fails to converge. Of course ideally this error + should never be raised: feel free to report it as a bug if it is! +

+

+ Defaults to boost::math::policies::evaluation_error<throw_on_error> +

+

+ When the action is throw_on_error then throws + boost::math::evaluation_error +

+
+
+ + Examples +
+

+ Suppose we want a call to tgamma + to behave in a C-compatible way and set global ::errno rather than throw an exception, + we can achieve this at the call site using: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+// Define a policy:
+typedef policy<
+      domain_error<errno_on_error>, 
+      pole_error<errno_on_error>,
+      overflow_error<errno_on_error>,
+      policies::evaluation_error<errno_on_error> 
+      > my_policy;
+      
+// call the function:
+double t1 = tgamma(some_value, my_policy());
+
+// Alternatively we could use make_policy and define everything at the call site:
+double t2 = tgamma(some_value, make_policy(
+         domain_error<errno_on_error>(), 
+         pole_error<errno_on_error>(),
+         overflow_error<errno_on_error>(),
+         policies::evaluation_error<errno_on_error>() 
+      ));
+
+
+

+

+

+

+

+ Suppose we want a statistical distribution to return infinities, rather + than throw exceptions, then we can use: +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp>
+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+// Define a policy:
+typedef policy<
+      overflow_error<ignore_error>
+      > my_policy;
+      
+// Define the distribution:
+typedef normal_distribution<double, my_policy> my_norm;
+
+// Get a quantile:
+double q = quantile(my_norm(), 0.05);
+
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/internal_promotion.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/internal_promotion.html new file mode 100644 index 000000000..f0f536daf --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/internal_promotion.html @@ -0,0 +1,168 @@ + + + +Internal Promotion Policies + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Normally when evaluating a function at say float + precision, maximal accuracy is assured by conducting the calculation at + double precision internally, + and then rounding the result. There are two policies that effect whether + internal promotion takes place or not: +

+
++++ + + + + + + + + + + + + + + +
+

+ Policy +

+ 		+

+ Meaning +

+
+

+ boost::math::policies::promote_float<B> +

+ 		+

+ Indicates whether float + arguments should be promoted to double + precision internally: defaults to boost::math::policies::promote_float<true> +

+
+

+ boost::math::policies::promote_double<B> +

+ 		+

+ Indicates whether double + arguments should be promoted to long + double precision internally: + defaults to boost::math::policies::promote_double<true> +

+
+
+ + Examples +
+

+ Suppose we want tgamma + to be evaluated without internal promotion to long + double, then we could use: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+// Define a policy:
+typedef policy<
+      promote_double<false> 
+      > my_policy;
+      
+// Call the function:
+double t1 = tgamma(some_value, my_policy());
+
+// Alternatively we could use make_policy and define everything at the call site:
+double t2 = tgamma(some_value, make_policy(promote_double<false>()));
+
+
+

+

+

+

+

+ Alternatively, suppose we want a distribution to perform calculations without + promoting float to double, then we could use: +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp>
+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+// Define a policy:
+typedef policy<
+      promote_float<false>
+      > my_policy;
+
+// Define the distribution:
+typedef normal_distribution<float, my_policy> my_norm;
+
+// Get a quantile:
+float q = quantile(my_norm(), 0.05f);
+
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/iteration_pol.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/iteration_pol.html new file mode 100644 index 000000000..5766917ca --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/iteration_pol.html @@ -0,0 +1,67 @@ + + + +Iteration Limits Policies + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ There are two policies that effect the iterative algorithms used to implement + the special functions in this library: +

+
+template <unsigned long limit = BOOST_MATH_MAX_SERIES_ITERATION_POLICY>
+class max_series_iterations;
+
+template <unsigned long limit = BOOST_MATH_MAX_ROOT_ITERATION_POLICY>
+class max_root_iterations;
+
+

+ The class max_series_iterations + determines the maximum number of iterations permitted in a series evaluation, + before the special function gives up and returns the result of evaluation_error. +

+

+ The class max_root_iterations + determines the maximum number of iterations permitted in a root-finding + algorithm before the special function gives up and returns the result of + evaluation_error. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/namespace_pol.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/namespace_pol.html new file mode 100644 index 000000000..126f29020 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/namespace_pol.html @@ -0,0 +1,172 @@ + + + +Setting Polices at Namespace Scope + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ Sometimes what you really want to do is bring all the special functions, + or all the distributions into a specific namespace-scope, along with a + specific policy to use with them. There are two macros defined to assist + with that: +

+
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(Policy)
+
+

+ and: +

+
+BOOST_MATH_DECLARE_DISTRIBUTIONS(Type, Policy)
+
+

+ You can use either of these macros after including any special function + or distribution header. For example: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+namespace myspace{
+
+using namespace boost::math::policies;
+
+// Define a policy that does not throw on overflow:
+typedef policy<overflow_error<errno_on_error> > my_policy;
+
+// Define the special functions in this scope to use the policy:   
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(my_policy)
+
+}
+
+//
+// Now we can use myspace::tgamma etc.
+// They will automatically use "my_policy":
+//
+double t = myspace::tgamma(30.0); // will not throw on overflow
+
+
+

+

+

+

+

+ In this example, using BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS results in + a set of thin inline forwarding functions being defined: +

+
+template <class T>
+inline T tgamma(T a){ return ::boost::math::tgamma(a, mypolicy()); }
+
+template <class T>
+inline T lgamma(T a) ( return ::boost::math::lgamma(a, mypolicy()); }
+
+

+ and so on. Note that while a forwarding function is defined for all the + special functions, however, unless you include the specific header for + the special function you use (or boost/math/special_functions.hpp to include + everything), you will get linker errors from functions that are forward + declared, but not defined. +

+

+ We can do the same thing with the distributions, but this time we need + to specify the floating-point type to use: +

+

+

+

+ +

+
+#include <boost/math/distributions/cauchy.hpp>
+
+namespace myspace{
+
+using namespace boost::math::policies;
+
+// Define a policy to use, in this case we want all the distribution
+// accessor functions to compile, even if they are mathematically
+// undefined:
+typedef policy<assert_undefined<false> > my_policy;
+
+BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy)
+
+}
+
+// Now we can use myspace::cauchy etc, which will use policy
+// myspace::mypolicy:
+//
+// This compiles but raises a domain error at runtime:
+//
+void test_cauchy()
+{
+   try
+   {
+      double d = mean(myspace::cauchy());
+   }
+   catch(const std::domain_error& e)
+   {
+      std::cout << e.what() << std::endl;
+   }
+}
+
+
+

+

+

+

+

+ In this example the result of BOOST_MATH_DECLARE_DISTRIBUTIONS is to declare + a typedef for each distribution like this: +

+
+typedef boost::math::cauchy_distribution<double, my_policy> cauchy;
+tyepdef boost::math::gamma_distribution<double, my_policy> gamma;
+
+

+ and so on. The name given to each typedef is the name of the distribution + with the "_distribution" suffix removed. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/pol_ref_ref.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/pol_ref_ref.html new file mode 100644 index 000000000..62bd058b0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/pol_ref_ref.html @@ -0,0 +1,250 @@ + + + +Policy Class Reference + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ There's very little to say here, the policy + class is just a rag-bag compile-time container for a collection of policies: +

+

+ +

+
+#include <boost/math/policies/policy.hpp>
+

+

+
+namespace boost{
+namespace math{
+namespace policies
+
+template <class A1 = default_policy, 
+          class A2 = default_policy, 
+          class A3 = default_policy,
+          class A4 = default_policy,
+          class A5 = default_policy,
+          class A6 = default_policy,
+          class A7 = default_policy,
+          class A8 = default_policy,
+          class A9 = default_policy,
+          class A10 = default_policy,
+          class A11 = default_policy>
+struct policy
+{
+public:
+   typedef computed-from-template-arguments domain_error_type;
+   typedef computed-from-template-arguments pole_error_type;
+   typedef computed-from-template-arguments overflow_error_type;
+   typedef computed-from-template-arguments underflow_error_type;
+   typedef computed-from-template-arguments denorm_error_type;
+   typedef computed-from-template-arguments evaluation_error_type;
+   typedef computed-from-template-arguments precision_type;
+   typedef computed-from-template-arguments promote_float_type;
+   typedef computed-from-template-arguments promote_double_type;
+   typedef computed-from-template-arguments discrete_quantile_type;
+   typedef computed-from-template-arguments assert_undefined_type;
+};
+
+template <...argument list...>
+typename normalise<policy<>, A1>::type make_policy(...argument list..);
+
+template <class Policy, 
+          class A1 = default_policy, 
+          class A2 = default_policy, 
+          class A3 = default_policy,
+          class A4 = default_policy,
+          class A5 = default_policy,
+          class A6 = default_policy,
+          class A7 = default_policy,
+          class A8 = default_policy,
+          class A9 = default_policy,
+          class A10 = default_policy,
+          class A11 = default_policy>
+struct normalise
+{
+   typedef computed-from-template-arguments type;
+};
+
+

+ The member typedefs of class policy + are intended for internal use but are documented briefly here for the sake + of completeness. +

+
+policy<...>::domain_error_type
+
+

+ Specifies how domain errors are handled, will be an instance of boost::math::policies::domain_error<> + with the template argument to domain_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::pole_error_type
+
+

+ Specifies how pole-errors are handled, will be an instance of boost::math::policies::pole_error<> + with the template argument to pole_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::overflow_error_type
+
+

+ Specifies how overflow errors are handled, will be an instance of boost::math::policies::overflow_error<> + with the template argument to overflow_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::underflow_error_type
+
+

+ Specifies how underflow errors are handled, will be an instance of boost::math::policies::underflow_error<> + with the template argument to underflow_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::denorm_error_type
+
+

+ Specifies how denorm errors are handled, will be an instance of boost::math::policies::denorm_error<> + with the template argument to denorm_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::evaluation_error_type
+
+

+ Specifies how evaluation errors are handled, will be an instance of boost::math::policies::evaluation_error<> + with the template argument to evaluation_error + one of the error_policy_type + enumerated values. +

+
+policy<...>::precision_type
+
+

+ Specifies the internal precision to use in binary digits (uses zero to + represent whatever the default precision is). Will be an instance of boost::math::policies::digits2<N> + which in turn inherits from boost::mpl::int_<N>. +

+
+policy<...>::promote_float_type
+
+

+ Specifies whether or not to promote float + arguments to double precision + internally. Will be an instance of boost::math::policies::promote_float<B> which in turn inherits from boost::mpl::bool_<B>. +

+
+policy<...>::promote_double_type
+
+

+ Specifies whether or not to promote double + arguments to long double + precision internally. Will be an instance of boost::math::policies::promote_float<B> which in turn inherits from boost::mpl::bool_<B>. +

+
+policy<...>::discrete_quantile_type
+
+

+ Specifies how discrete quantiles are evaluated, will be an instance of + boost::math::policies::discrete_quantile<> + instantiated with one of the discrete_quantile_policy_type + enumerated type. +

+
+policy<...>::assert_undefined_type
+
+

+ Specifies whether mathematically-undefined properties are asserted as compile-time + errors, or treated as runtime errors instead. Will be an instance of boost::math::policies::assert_undefined<B> + which in turn inherits from boost::math::mpl::bool_<B>. +

+
+template <...argument list...>
+typename normalise<policy<>, A1>::type make_policy(...argument list..);
+
+

+ make_policy is a helper + function that converts a list of policies into a normalised policy class. +

+
+template <class Policy, 
+          class A1 = default_policy, 
+          class A2 = default_policy, 
+          class A3 = default_policy,
+          class A4 = default_policy,
+          class A5 = default_policy,
+          class A6 = default_policy,
+          class A7 = default_policy,
+          class A8 = default_policy,
+          class A9 = default_policy,
+          class A10 = default_policy,
+          class A11 = default_policy>
+struct normalise
+{
+   typedef computed-from-template-arguments type;
+};
+
+

+ The normalise class template + converts one instantiation of the policy + class into a normalised form. This is used internally to reduce code bloat: + so that instantiating a special function on policy<A,B> + or policy<B,A> actually both generate the same code + internally. +

+

+ Further more, normalise + can be used to combine a policy with one or more policies: for example + many of the special functions will use this to set policies which they + don't make use of to their default values, before forwarding to the actual + implementation. In this way code bloat is reduced, since the actual implementation + depends only on the policy types that they actually use. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/policy_defaults.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/policy_defaults.html new file mode 100644 index 000000000..9907638dd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/policy_defaults.html @@ -0,0 +1,240 @@ + + + +Using macros to Change the Policy Defaults + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ You can use the various macros below to change any (or all) of the policies. +

+

+ You can make a local change by placing a macro definition before + a function or distribution #include. +

+
+ + + + + +
[Caution]Caution

+ There is a danger of One-Definition-Rule violations if you add ad-hock + macros to more than one source files: these must be set the same in + every translation unit. +

+
+ + + + + +
[Caution]Caution

+ If you place it after the #include it will have no effect, (and it will + affect only any other following #includes). This is probably not what + you intend! +

+

+ If you want to alter the defaults for any or all of the policies for all functions and distributions, installation-wide, + then you can do so by defining various macros in boost/math/tools/user.hpp. +

+
+ + BOOST_MATH_DOMAIN_ERROR_POLICY +
+

+ Defines what happens when a domain error occurs, if not defined then defaults + to throw_on_error, but + can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_POLE_ERROR_POLICY +
+

+ Defines what happens when a pole error occurs, if not defined then defaults + to throw_on_error, but + can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_OVERFLOW_ERROR_POLICY +
+

+ Defines what happens when an overflow error occurs, if not defined then + defaults to throw_on_error, + but can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_EVALUATION_ERROR_POLICY +
+

+ Defines what happens when an internal evaluation error occurs, if not defined + then defaults to throw_on_error, + but can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_UNDERFLOW_ERROR_POLICY +
+

+ Defines what happens when an overflow error occurs, if not defined then + defaults to ignore_error, + but can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_DENORM_ERROR_POLICY +
+

+ Defines what happens when a denormalisation error occurs, if not defined + then defaults to ignore_error, + but can be set to any of the enumerated actions for error handing: throw_on_error, errno_on_error, + ignore_error or user_error. +

+
+ + BOOST_MATH_DIGITS10_POLICY +
+

+ Defines how many decimal digits to use in internal computations: defaults + to 0 - meaning use all available + digits - but can be set to some other decimal value. Since setting this + is likely to have a substantial impact on accuracy, it's not generally + recommended that you change this from the default. +

+
+ + BOOST_MATH_PROMOTE_FLOAT_POLICY +
+

+ Determines whether float types + get promoted to double internally + to ensure maximum precision in the result, defaults to true, + but can be set to false to + turn promotion of float's + off. +

+
+ + BOOST_MATH_PROMOTE_DOUBLE_POLICY +
+

+ Determines whether double + types get promoted to long double internally to ensure maximum precision + in the result, defaults to true, + but can be set to false to + turn promotion of double's + off. +

+
+ + BOOST_MATH_DISCRETE_QUANTILE_POLICY +
+

+ Determines how discrete quantiles return their results: either as an integer, + or as a real value, can be set to one of the enumerated values: real, integer_round_outwards, + integer_round_inwards, + integer_round_down, integer_round_up, integer_round_nearest. + Defaults to integer_round_outwards. +

+
+ + BOOST_MATH_ASSERT_UNDEFINED_POLICY +
+

+ Determines whether functions that are mathematically undefined for a specific + distribution compile or raise a static (i.e. compile-time) assertion. Defaults + to true: meaning that any + mathematically undefined function will not compile. When set to false then the function will compile but + return the result of a domain error: this can be useful for some generic + code, that needs to work with all distributions and determine at runtime + whether or not a particular property is well defined. +

+
+ + BOOST_MATH_MAX_SERIES_ITERATION_POLICY +
+

+ Determines how many series iterations a special function is permitted to + perform before it gives up and returns an evaluation_error: + Defaults to 1000000. +

+
+ + BOOST_MATH_MAX_ROOT_ITERATION_POLICY +
+

+ Determines how many root-finding iterations a special function is permitted + to perform before it gives up and returns an evaluation_error: + Defaults to 200. +

+
+ + Example +
+

+ Suppose we want overflow errors to set ::errno and return an infinity, discrete + quantiles to return a real-valued result (rather than round to integer), + and for mathematically undefined functions to compile, but return a domain + error. Then we could add the following to boost/math/tools/user.hpp: +

+
+#define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error
+#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false
+
+

+ or we could place these definitions before +

+
+#include <boost/math/distributions/normal.hpp>
+  using boost::math::normal_distribution;
+
+

+ in a source .cpp file. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/precision_pol.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/precision_pol.html new file mode 100644 index 000000000..280200dc4 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_ref/precision_pol.html @@ -0,0 +1,141 @@ + + + +Precision Policies + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ There are two equivalent policies that effect the working precision + used to calculate results, these policies both default to 0 - meaning calculate + to the maximum precision available in the type being used - but can be + set to other values to cause lower levels of precision to be used. +

+
+namespace boost{ namespace math{ namespace policies{
+
+template <int N>
+digits10;
+
+template <int N>
+digits2;
+
+}}} // namespaces
+
+

+ As you would expect, digits10 specifies the number + of decimal digits to use, and digits2 the number of + binary digits. Internally, whichever is used, the precision is always converted + to binary digits. +

+

+ These policies are specified at compile-time, because many of the special + functions use compile-time-dispatch to select which approximation to use + based on the precision requested and the numeric type being used. +

+

+ For example we could calculate tgamma + to approximately 5 decimal digits using: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+typedef policy<digits10<5> > pol;
+
+double t = tgamma(12, pol());
+
+
+

+

+

+

+

+ Or again using make_policy: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+double t = tgamma(12, policy<digits10<5> >());
+
+
+

+

+

+

+

+ And for a quantile of a distribution to approximately 25-bit precision: +

+

+

+

+ +

+
+#include <boost/math/distributions/normal.hpp>
+
+using namespace boost::math;
+using namespace boost::math::policies;
+
+double q = quantile(
+      normal_distribution<double, policy<digits2<25> > >(), 
+      0.05);
+
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial.html new file mode 100644 index 000000000..f7593b3cb --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial.html @@ -0,0 +1,64 @@ + + + +Policy Tutorial + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
+ So Just What is a Policy Anyway?
+
+ Policies Have Sensible Defaults
+
So + How are Policies Used Anyway?
+
+ Changing the Policy Defaults
+
+ Setting Policies for Distributions on an Ad Hoc Basis
+
+ Changing the Policy on an Ad Hoc Basis for the Special Functions
+
+ Setting Policies at Namespace or Translation Unit Scope
+
+ Calling User Defined Error Handlers
+
+ Understanding Quantiles of Discrete Distributions
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html new file mode 100644 index 000000000..27b1b0784 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html @@ -0,0 +1,110 @@ + + + +Setting Policies for Distributions on an Ad Hoc Basis + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ All of the statistical distributions in this library are class templates + that accept two template parameters, both with sensible defaults, for example: +

+
+namespace boost{ namespace math{
+
+template <class RealType = double, class Policy = policies::policy<> >
+class fisher_f_distribution;
+
+typedef fisher_f_distribution<> fisher_f;
+
+}}
+
+

+ This policy gets used by all the accessor functions that accept a distribution + as an argument, and forwarded to all the functions called by these. So + if you use the shorthand-typedef for the distribution, then you get double precision arithmetic and all the + default policies. +

+

+ However, say for example we wanted to evaluate the quantile of the binomial + distribution at float precision, without internal promotion to double, + and with the result rounded to the nearest integer, + then here's how it can be done: +

+

+

+

+ +

+
+#include <boost/math/distributions/binomial.hpp>
+
+//
+// Begin by defining a policy type, that gives the
+// behaviour we want:
+//
+using namespace boost::math::policies;
+typedef policy<
+   promote_float<false>,
+   discrete_quantile<integer_round_nearest>
+> mypolicy;
+//
+// Then define a distribution that uses it:
+//
+typedef boost::math::binomial_distribution<float, mypolicy> mybinom;
+//
+//  And now use it to get the quantile:
+//
+int main()
+{
+   std::cout << "quantile is: " <<
+      quantile(mybinom(200, 0.25), 0.05) << std::endl;
+}
+
+
+

+

+

+

+

+ Which outputs: +

+
quantile is: 40
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html new file mode 100644 index 000000000..2af712b53 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html @@ -0,0 +1,171 @@ + + + +Changing the Policy on an Ad Hoc Basis for the Special Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ All of the special functions in this library come in two overloaded forms, + one with a final "policy" parameter, and one without. For example: +

+
+namespace boost{ namespace math{
+
+template <class RealType, class Policy>
+RealType tgamma(RealType, const Policy&);
+
+template <class RealType>
+RealType tgamma(RealType);
+
+}} // namespaces
+
+

+ Normally, the second version is just a forwarding wrapper to the first + like this: +

+
+template <class RealType>
+inline RealType tgamma(RealType x)
+{
+   return tgamma(x, policies::policy<>());
+}
+
+

+ So calling a special function with a specific policy is just a matter of + defining the policy type to use and passing it as the final parameter. + For example, suppose we want tgamma + to behave in a C-compatible fashion and set ::errno when an error occurs, and never + throw an exception: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+//
+// Define the policy to use:
+//
+using namespace boost::math::policies;
+typedef policy<
+   domain_error<errno_on_error>,
+   pole_error<errno_on_error>,
+   overflow_error<errno_on_error>,
+   evaluation_error<errno_on_error> 
+> c_policy;
+//
+// Now use the policy when calling tgamma:
+//
+int main()
+{
+   errno = 0;
+   std::cout << "Result of tgamma(30000) is: " 
+      << boost::math::tgamma(30000, c_policy()) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   std::cout << "Result of tgamma(-10) is: " 
+      << boost::math::tgamma(-10, c_policy()) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+}
+
+
+

+

+

+

+

+ which outputs: +

+
Result of tgamma(30000) is: 1.#INF
+errno = 34
+Result of tgamma(-10) is: 1.#QNAN
+errno = 33
+
+

+ Alternatively, for ad hoc use, we can use the make_policy + helper function to create a policy for us: this usage is more verbose, + so is probably only preferred when a policy is going to be used once only: +

+

+

+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+int main()
+{
+   using namespace boost::math::policies;
+   errno = 0;
+   std::cout << "Result of tgamma(30000) is: " 
+      << boost::math::tgamma(
+         30000, 
+         make_policy(
+            domain_error<errno_on_error>(),
+            pole_error<errno_on_error>(),
+            overflow_error<errno_on_error>(),
+            evaluation_error<errno_on_error>() 
+         )
+      ) << std::endl;
+   // Check errno was set:
+   std::cout << "errno = " << errno << std::endl;
+   // and again with evaluation at a pole:
+   std::cout << "Result of tgamma(-10) is: " 
+      << boost::math::tgamma(
+         -10, 
+         make_policy(
+            domain_error<errno_on_error>(),
+            pole_error<errno_on_error>(),
+            overflow_error<errno_on_error>(),
+            evaluation_error<errno_on_error>() 
+         )
+      ) << std::endl;
+   // Check errno was set:
+   std::cout << "errno = " << errno << std::endl;
+}
+
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html new file mode 100644 index 000000000..3a5270d59 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html @@ -0,0 +1,340 @@ + + + +Changing the Policy Defaults + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ The default policies used by the library are changed by the usual configuration + macro method. +

+

+ For example, passing -DBOOST_MATH_DOMAIN_ERROR_POLICY=errno_on_error + to your compiler will cause domain errors to set ::errno and return a NaN rather than the + usual default behaviour of throwing a std::domain_error + exception. +

+
+ + + + + +
[Tip]Tip
+

+ For Microsoft Visual Studio,you can add to the Project Property Page, + C/C++, Preprocessor, Preprocessor definitions like: +

+

+ +

+
+BOOST_MATH_ASSERT_UNDEFINED_POLICY=0
+BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error
+

+

+

+ This may be helpful to avoid complications with pre-compiled headers + that may mean that the equivalent definitions in source code: +

+

+ +

+
+#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false
+#define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error
+

+

+

+ may be ignored. +

+

+ The compiler command line shows: +

+

+ +

+
+/D "BOOST_MATH_ASSERT_UNDEFINED_POLICY=0"
+/D "BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error"
+

+

+
+

+ There is however a very important caveat to this: +

+
+ + + + + +
[Important]Important
+

+ Default policies changed by setting configuration + macros must be changed uniformly in every translation unit in the program. +

+

+ Failure to follow this rule may result in violations of the "One + Definition Rule (ODR)" and result in unpredictable program behaviour. +

+
+

+ That means there are only two safe ways to use these macros: +

+
    +
  • + Edit them in boost/math/tools/user.hpp, + so that the defaults are set on an installation-wide basis. Unfortunately + this may not be convenient if you are using a pre-installed Boost distribution + (on Linux for example). +
    • +
  • + Set the defines in your project's Makefile or build environment, so that + they are set uniformly across all translation units. +
    • +
+

+ What you should not do is: +

+
  • + Set the defines in the source file using #define + as doing so almost certainly will break your program, unless you're absolutely + certain that the program is restricted to a single translation unit. +
+

+ And, yes, you will find examples in our test programs where we break this + rule: but only because we know there will always be a single translation + unit only: don't say that you weren't warned! +

+

+

+

+ The following example demonstrates the effect of setting the macro BOOST_MATH_DOMAIN_ERROR_POLICY + when an invalid argument is encountered. For the purposes of this example, + we'll pass a negative degrees of freedom parameter to the student's t + distribution. +

+

+

+

+ Since we know that this is a single file program we could just add: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
+
+

+

+

+ to the top of the source file to change the default policy to one that + simply returns a NaN when a domain error occurs. Alternatively we could + use: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
+
+

+

+

+ To ensure the ::errno + is set when a domain error occurs as well as returning a NaN. +

+

+

+

+ This is safe provided the program consists of a single translation unit + and we place the define before + any #includes. Note that should we add the define after the includes + then it will have no effect! A warning such as: +

+

+ +

+
warning C4005: 'BOOST_MATH_OVERFLOW_ERROR_POLICY' : macro redefinition
+

+

+

+ is a certain sign that it will not have the desired + effect. +

+

+

+

+ We'll begin our sample program with the needed includes: +

+

+

+

+ +

+
+// Boost
+#include <boost/math/distributions/students_t.hpp>
+   using boost::math::students_t;  // Probability of students_t(df, t).
+
+// std
+#include <iostream>
+   using std::cout;
+   using std::endl;
+
+#include <stdexcept>
+   using std::exception;
+

+

+

+

+

+ Next we'll define the program's main() to call the student's t distribution + with an invalid degrees of freedom parameter, the program is set up to + handle either an exception or a NaN: +

+

+

+

+ +

+
+int main()
+{
+   cout << "Example error handling using Student's t function. " << endl;
+   cout << "BOOST_MATH_DOMAIN_ERROR_POLICY is set to: "
+      << BOOST_STRINGIZE(BOOST_MATH_DOMAIN_ERROR_POLICY) << endl;
+
+   double degrees_of_freedom = -1; // A bad argument!
+   double t = 10;
+
+   try
+   {
+      errno = 0;
+      students_t dist(degrees_of_freedom); // exception is thrown here if enabled
+      double p = cdf(dist, t);
+      // test for error reported by other means:
+      if((boost::math::isnan)(p))
+      {
+         cout << "cdf returned a NaN!" << endl;
+         cout << "errno is set to: " << errno << endl;
+      }
+      else
+         cout << "Probability of Student's t is " << p << endl;
+   }
+   catch(const std::exception& e)
+   {
+      std::cout <<
+         "\n""Message from thrown exception was:\n   " << e.what() << std::endl;
+   }
+
+   return 0;
+} // int main()
+

+

+

+

+

+ Here's what the program output looks like with a default build (one that + does throw exceptions): +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: throw_on_error
+
+Message from thrown exception was:
+   Error in function boost::math::students_t_distribution<double>::students_t_distribution:
+   Degrees of freedom argument is -1, but must be > 0 !
+
+

+

+

+ Alternatively let's build with: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
+
+

+

+

+ Now the program output is: +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: ignore_error
+cdf returned a NaN!
+errno is set to: 0
+
+

+

+

+ And finally let's build with: +

+

+ +

+
+#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
+
+

+

+

+ Which gives the output: +

+

+ +

+
Example error handling using Student's t function.
+BOOST_MATH_DOMAIN_ERROR_POLICY is set to: errno_on_error
+cdf returned a NaN!
+errno is set to: 33
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/namespace_policies.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/namespace_policies.html new file mode 100644 index 000000000..bc96296d5 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/namespace_policies.html @@ -0,0 +1,477 @@ + + + +Setting Policies at Namespace or Translation Unit Scope + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Sometimes what you want to do is just change a set of policies within the + current scope: the one thing you should not do in this situation is use + the configuration macros, as this can lead to "One Definition Rule" + violations. Instead this library provides a pair of macros especially for + this purpose. +

+

+ Let's consider the special functions first: we can declare a set of forwarding + functions that all use a specific policy using the macro BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(Policy). + This macro should be used either inside a unique namespace set aside for + the purpose, or an unnamed namespace if you just want the functions visible + in global scope for the current file only. +

+

+

+

+ Suppose we want C::foo() + to behave in a C-compatible way and set ::errno on error rather than throwing + any exceptions. +

+

+

+

+ We'll begin by including the needed header: +

+

+

+

+ +

+
+#include <boost/math/special_functions.hpp>
+

+

+

+

+

+ Open up the "C" namespace that we'll use for our functions, + and define the policy type we want: in this case one that sets ::errno + rather than throwing exceptions. Any policies we don't specify here will + inherit the defaults: +

+

+

+

+ +

+
+namespace C{
+
+using namespace boost::math::policies;
+
+typedef policy<
+   domain_error<errno_on_error>,
+   pole_error<errno_on_error>,
+   overflow_error<errno_on_error>,
+   evaluation_error<errno_on_error>
+> c_policy;
+

+

+

+

+

+ All we need do now is invoke the BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS + macro passing our policy type as the single argument: +

+

+

+

+ +

+
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(c_policy)
+
+} // close namespace C
+

+

+

+

+

+ We now have a set of forwarding functions defined in namespace C that + all look something like this: +

+

+

+

+ +

+
+template <class RealType>
+inline typename boost::math::tools::promote_args<RT>::type
+   tgamma(RT z)
+{
+   return boost::math::tgamma(z, c_policy());
+}
+
+

+

+

+

+

+ So that when we call C::tgamma(z) we really end up calling boost::math::tgamma(z, C::c_policy()): +

+

+

+

+ +

+
+int main()
+{
+   errno = 0;
+   std::cout << "Result of tgamma(30000) is: "
+      << C::tgamma(30000) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   std::cout << "Result of tgamma(-10) is: "
+      << C::tgamma(-10) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+}
+

+

+

+

+

+ Which outputs: +

+

+ +

+
Result of C::tgamma(30000) is: 1.#INF
+errno = 34
+Result of C::tgamma(-10) is: 1.#QNAN
+errno = 33
+
+

+

+

+ This mechanism is particularly useful when we want to define a project-wide + policy, and don't want to modify the Boost source or set - possibly fragile + and easy to forget - project wide build macros. +

+

+

+

+ The same mechanism works well at file scope as well, by using an unnamed + namespace, we can ensure that these declarations don't conflict with any + alternate policies present in other translation units: +

+

+

+

+ +

+
+#include <boost/math/special_functions.hpp>
+
+namespace {
+
+using namespace boost::math::policies;
+
+typedef policy<
+   domain_error<errno_on_error>,
+   pole_error<errno_on_error>,
+   overflow_error<errno_on_error>,
+   evaluation_error<errno_on_error> 
+> c_policy;
+
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(c_policy)
+
+} // close unnamed namespace
+
+int main()
+{
+   errno = 0;
+   std::cout << "Result of tgamma(30000) is: " 
+      << tgamma(30000) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   std::cout << "Result of tgamma(-10) is: " 
+      << tgamma(-10) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+}
+
+
+

+

+

+

+

+ Handling the statistical distributions is very similar except that now + the macro BOOST_MATH_DECLARE_DISTRIBUTIONS accepts two parameters: the + floating point type to use, and the policy type to apply. For example: +

+
+BOOST_MATH_DECLARE_DISTRIBUTIONS(double, mypolicy)
+
+

+ Results a set of typedefs being defined like this: +

+
+typedef boost::math::normal_distribution<double, mypolicy> normal;
+
+

+ The name of each typedef is the same as the name of the distribution class + template, but without the "_distribution" suffix. +

+

+

+

+ Suppose we want a set of distributions to behave as follows: +

+

+

+
    +
  • + Return infinity on overflow, rather than throwing an exception. +
    • +
  • + Don't perform any promotion from double to long double internally. +
    • +
  • + Return the closest integer result from the quantiles of discrete distributions. +
    • +
+

+

+

+ We'll begin by including the needed header: +

+

+

+

+ +

+
+#include <boost/math/distributions.hpp>
+

+

+

+

+

+ Open up an appropriate namespace for our distributions, and define the + policy type we want. Any policies we don't specify here will inherit + the defaults: +

+

+

+

+ +

+
+namespace my_distributions{
+
+using namespace boost::math::policies;
+
+typedef policy<
+   // return infinity and set errno rather than throw:
+   overflow_error<errno_on_error>,
+   // Don't promote double -> long double internally:
+   promote_double<false>,
+   // Return the closest integer result for discrete quantiles:
+   discrete_quantile<integer_round_nearest>
+> my_policy;
+

+

+

+

+

+ All we need do now is invoke the BOOST_MATH_DECLARE_DISTRIBUTIONS macro + passing the floating point and policy types as arguments: +

+

+

+

+ +

+
+BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy)
+
+} // close namespace my_namespace
+

+

+

+

+

+ We now have a set of typedefs defined in namespace my_namespace that + all look something like this: +

+

+

+

+ +

+
+typedef boost::math::normal_distribution<double, my_policy> normal;
+typedef boost::math::cauchy_distribution<double, my_policy> cauchy;
+typedef boost::math::gamma_distribution<double, my_policy> gamma;
+// etc
+
+

+

+

+

+

+ So that when we use my_namespace::normal we really end up using boost::math::normal_distribution<double, my_policy>: +

+

+

+

+ +

+
+int main()
+{
+   //
+   // Start with something we know will overflow:
+   //
+   my_distributions::normal norm(10, 2);
+   errno = 0;
+   std::cout << "Result of quantile(norm, 0) is: " 
+      << quantile(norm, 0) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   errno = 0;
+   std::cout << "Result of quantile(norm, 1) is: " 
+      << quantile(norm, 1) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   //
+   // Now try a discrete distribution:
+   //
+   my_distributions::binomial binom(20, 0.25);
+   std::cout << "Result of quantile(binom, 0.05) is: " 
+      << quantile(binom, 0.05) << std::endl;
+   std::cout << "Result of quantile(complement(binom, 0.05)) is: " 
+      << quantile(complement(binom, 0.05)) << std::endl;
+}
+

+

+

+

+

+ Which outputs: +

+

+ +

+
Result of quantile(norm, 0) is: -1.#INF
+errno = 34
+Result of quantile(norm, 1) is: 1.#INF
+errno = 34
+Result of quantile(binom, 0.05) is: 1
+Result of quantile(complement(binom, 0.05)) is: 8
+
+

+

+

+ This mechanism is particularly useful when we want to define a project-wide + policy, and don't want to modify the Boost source or set - possibly fragile + and easy to forget - project wide build macros. +

+

+

+
+ + + + + +
[Note]Note

+ There is an important limitation to note: you can not use the macros + BOOST_MATH_DECLARE_DISTRIBUTIONS and BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS + in the same namespace, as doing so creates ambiguities + between functions and distributions of the same name. +

+

+ As before, the same mechanism works well at file scope as well: by using + an unnamed namespace, we can ensure that these declarations don't conflict + with any alternate policies present in other translation units: +

+

+

+

+ +

+
+#include <boost/math/distributions.hpp>
+
+namespace {
+
+using namespace boost::math::policies;
+
+typedef policy<
+   // return infinity and set errno rather than throw:
+   overflow_error<errno_on_error>,
+   // Don't promote double -> long double internally:
+   promote_double<false>,
+   // Return the closest integer result for discrete quantiles:
+   discrete_quantile<integer_round_nearest>
+> my_policy;
+
+BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy)
+
+} // close namespace my_namespace
+
+int main()
+{
+   //
+   // Start with something we know will overflow:
+   //
+   normal norm(10, 2);
+   errno = 0;
+   std::cout << "Result of quantile(norm, 0) is: " 
+      << quantile(norm, 0) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   errno = 0;
+   std::cout << "Result of quantile(norm, 1) is: " 
+      << quantile(norm, 1) << std::endl;
+   std::cout << "errno = " << errno << std::endl;
+   //
+   // Now try a discrete distribution:
+   //
+   binomial binom(20, 0.25);
+   std::cout << "Result of quantile(binom, 0.05) is: " 
+      << quantile(binom, 0.05) << std::endl;
+   std::cout << "Result of quantile(complement(binom, 0.05)) is: " 
+      << quantile(complement(binom, 0.05)) << std::endl;
+}
+
+
+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html new file mode 100644 index 000000000..0f1eb1a46 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html @@ -0,0 +1,130 @@ + + + +Policies Have Sensible Defaults + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Most of the time you can just ignore the policy framework, the defaults + for the various policies are as follows, if these work OK for you then + you can stop reading now! +

+
+

+
+
Domain Error
+

+ Throws a std::domain_error exception. +

+
Pole Error
+

+ Occurs when a function is evaluated at a pole: throws a std::domain_error exception. +

+
Overflow Error
+

+ Throws a std::overflow_error exception. +

+
Underflow
+

+ Ignores the underflow, and returns zero. +

+
Denormalised Result
+

+ Ignores the fact that the result is denormalised, and returns it. +

+
Internal Evaluation Error
+

+ Throws a boost::math::evaluation_error exception. +

+
Promotion of float to double
+

+ Does occur by default - gives full float precision results. +

+
Promotion of double to long double
+

+ Does occur by default if long double offers more precision than double. +

+
Precision of Approximation Used
+

+ By default uses an approximation that will result in the lowest level + of error for the type of the result. +

+
Behaviour of Discrete Quantiles
+
+

+ The quantile function will by default return an integer result that + has been rounded outwards. That is to say lower + quantiles (where the probability is less than 0.5) are rounded downward, + and upper quantiles (where the probability is greater than 0.5) are + rounded upwards. This behaviour ensures that if an X% quantile is requested, + then at least the requested coverage will be present + in the central region, and no more than the requested + coverage will be present in the tails. +

+

+ This behaviour can be changed so that the quantile functions are rounded + differently, or even return a real-valued result using Policies. + It is strongly recommended that you read the tutorial Understanding + Quantiles of Discrete Distributions before using the quantile + function on a discrete distribution. The reference + docs describe how to change the rounding policy for these distributions. +

+
+
+
+

+ What's more, if you define your own policy type, then it automatically + inherits the defaults for any policies not explicitly set, so given: +

+
+using namespace boost::math::policies;
+//
+// Define a policy that sets ::errno on overflow, and does
+// not promote double to long double internally:
+//
+typedef policy<domain_error<errno_on_error>, promote_double<false> > mypolicy;
+
+

+ then mypolicy defines a + policy where only the overflow error handling and double-promotion + policies differ from the defaults. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_usage.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_usage.html new file mode 100644 index 000000000..afc754361 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/policy_usage.html @@ -0,0 +1,71 @@ + + + +So How are Policies Used Anyway? + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ The details follow later, but basically policies can be set by either: +

+
    +
  • + Defining some macros that change the default behaviour: this + is the recommended method for setting installation-wide policies. +
    • +
  • + By instantiating a distribution object with an explicit policy: this + is mainly reserved for ad hoc policy changes. +
    • +
  • + By passing a policy to a special function as an optional final argument: + this is mainly reserved for ad hoc policy changes. +
    • +
  • + By using some helper macros to define a set of functions or distributions + in the current namespace that use a specific policy: this + is the recommended method for setting policies on a project- or translation-unit-wide + basis. +
    • +
+

+ The following sections introduce these methods in more detail. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html new file mode 100644 index 000000000..1a05c9788 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html @@ -0,0 +1,442 @@ + + + +Understanding Quantiles of Discrete Distributions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ Discrete distributions present us with a problem when calculating the quantile: + we are starting from a continuous real-valued variable - the probability + - but the result (the value of the random variable) should really be discrete. +

+

+ Consider for example a Binomial distribution, with a sample size of 50, + and a success fraction of 0.5. There are a variety of ways we can plot + a discrete distribution, but if we plot the PDF as a step-function then + it looks something like this: +

+

+ binomial_pdf +

+

+ Now lets suppose that the user asks for a the quantile that corresponds + to a probability of 0.05, if we zoom in on the CDF for that region here's + what we see: +

+

+ binomial_quantile_1 +

+

+ As can be seen there is no random variable that corresponds to a probability + of exactly 0.05, so we're left with two choices as shown in the figure: +

+
    +
  • + We could round the result down to 18. +
    • +
  • + We could round the result up to 19. +
    • +
+

+ In fact there's actually a third choice as well: we could "pretend" + that the distribution was continuous and return a real valued result: in + this case we would calculate a result of approximately 18.701 (this accurately + reflects the fact that the result is nearer to 19 than 18). +

+

+ By using policies we can offer any of the above as options, but that still + leaves the question: What is actually the right thing to do? +

+

+ And in particular: What policy should we use by default? +

+

+ In coming to an answer we should realise that: +

+
    +
  • + Calculating an integer result is often much faster than calculating a + real-valued result: in fact in our tests it was up to 20 times faster. +
    • +
  • + Normally people calculate quantiles so that they can perform a test of + some kind: "If the random variable is less than N then + we can reject our null-hypothesis with 90% confidence." +
    • +
+

+ So there is a genuine benefit to calculating an integer result as well + as it being "the right thing to do" from a philosophical point + of view. What's more if someone asks for a quantile at 0.05, then we can + normally assume that they are asking for at + least 95% of the probability to the right of the value chosen, + and no more than 5% of the probability + to the left of the value chosen. +

+

+ In the above binomial example we would therefore round the result down + to 18. +

+

+ The converse applies to upper-quantiles: If the probability is greater + than 0.5 we would want to round the quantile up, so that at least the requested probability is to the left + of the value returned, and no more than + 1 - the requested probability is to the right of the value returned. +

+

+ Likewise for two-sided intervals, we would round lower quantiles down, + and upper quantiles up. This ensures that we have at least the + requested probability in the central region and no + more than 1 minus the requested probability in the tail areas. +

+

+ For example, taking our 50 sample binomial distribution with a success + fraction of 0.5, if we wanted a two sided 90% confidence interval, then + we would ask for the 0.05 and 0.95 quantiles with the results rounded + outwards so that at least 90% of the probability + is in the central area: +

+

+ binomial_pdf_3 +

+

+ So far so good, but there is in fact a trap waiting for the unwary here: +

+
+quantile(binomial(50, 0.5), 0.05);
+
+

+ returns 18 as the result, which is what we would expect from the graph + above, and indeed there is no x greater than 18 for which: +

+
+cdf(binomial(50, 0.5), x) <= 0.05;
+
+

+ However: +

+
+quantile(binomial(50, 0.5), 0.95);
+
+

+ returns 31, and indeed while there is no x less than 31 for which: +

+
+cdf(binomial(50, 0.5), x) >= 0.95;
+
+

+ We might naively expect that for this symmetrical distribution the result + would be 32 (since 32 = 50 - 18), but we need to remember that the cdf + of the binomial is inclusive of the random variable. + So while the left tail area includes the quantile + returned, the right tail area always excludes an upper quantile value: + since that "belongs" to the central area. +

+

+ Look at the graph above to see what's going on here: the lower quantile + of 18 belongs to the left tail, so any value <= 18 is in the left tail. + The upper quantile of 31 on the other hand belongs to the central area, + so the tail area actually starts at 32, so any value > 31 is in the + right tail. +

+

+ Therefore if U and L are the upper and lower quantiles respectively, then + a random variable X is in the tail area - where we would reject the null + hypothesis if: +

+
+X <= L || X > U
+
+

+ And the a variable X is inside the central region if: +

+
+L < X <= U
+
+

+ The moral here is to always be very careful with your comparisons + when dealing with a discrete distribution, and if in doubt, + base your comparisons on CDF's instead. +

+
+ + Other + Rounding Policies are Available +
+

+ As you would expect from a section on policies, you won't be surprised + to know that other rounding options are available: +

+
+

+
+
integer_round_outwards
+
+

+ This is the default policy as described above: lower quantiles are + rounded down (probability < 0.5), and upper quantiles (probability + > 0.5) are rounded up. +

+

+ This gives no more than the requested probability + in the tails, and at least the requested probability + in the central area. +

+
+
integer_round_inwards
+
+

+ This is the exact opposite of the default policy: lower quantiles are + rounded up (probability < 0.5), and upper quantiles (probability + > 0.5) are rounded down. +

+

+ This gives at least the requested probability + in the tails, and no more than the requested probability + in the central area. +

+
+
integer_round_down
+

+ This policy will always round the result down no matter whether it + is an upper or lower quantile +

+
integer_round_up
+

+ This policy will always round the result up no matter whether it is + an upper or lower quantile +

+
integer_round_nearest
+

+ This policy will always round the result to the nearest integer no + matter whether it is an upper or lower quantile +

+
real
+

+ This policy will return a real valued result for the quantile of a + discrete distribution: this is generally much slower than finding an + integer result but does allow for more sophisticated rounding policies. +

+
+
+

+

+

+ To understand how the rounding policies for the discrete distributions + can be used, we'll use the 50-sample binomial distribution with a success + fraction of 0.5 once again, and calculate all the possible quantiles + at 0.05 and 0.95. +

+

+

+

+ Begin by including the needed headers: +

+

+

+

+ +

+
+#include <iostream>
+#include <boost/math/distributions/binomial.hpp>
+

+

+

+

+

+ Next we'll bring the needed declarations into scope, and define distribution + types for all the available rounding policies: +

+

+

+

+ +

+
+using namespace boost::math::policies;
+using namespace boost::math;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<integer_round_outwards> > > 
+        binom_round_outwards;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<integer_round_inwards> > > 
+        binom_round_inwards;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<integer_round_down> > > 
+        binom_round_down;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<integer_round_up> > > 
+        binom_round_up;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<integer_round_nearest> > > 
+        binom_round_nearest;
+
+typedef binomial_distribution<
+            double, 
+            policy<discrete_quantile<real> > > 
+        binom_real_quantile;
+

+

+

+

+

+ Now let's set to work calling those quantiles: +

+

+

+

+ +

+
+int main()
+{
+   std::cout << 
+      "Testing rounding policies for a 50 sample binomial distribution,\n"
+      "with a success fraction of 0.5.\n\n"
+      "Lower quantiles are calculated at p = 0.05\n\n"
+      "Upper quantiles at p = 0.95.\n\n";
+
+   std::cout << std::setw(25) << std::right
+      << "Policy"<< std::setw(18) << std::right 
+      << "Lower Quantile" << std::setw(18) << std::right 
+      << "Upper Quantile" << std::endl;
+   
+   // Test integer_round_outwards:
+   std::cout << std::setw(25) << std::right
+      << "integer_round_outwards"
+      << std::setw(18) << std::right
+      << quantile(binom_round_outwards(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_round_outwards(50, 0.5), 0.95) 
+      << std::endl;
+   
+   // Test integer_round_inwards:
+   std::cout << std::setw(25) << std::right
+      << "integer_round_inwards"
+      << std::setw(18) << std::right
+      << quantile(binom_round_inwards(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_round_inwards(50, 0.5), 0.95) 
+      << std::endl;
+   
+   // Test integer_round_down:
+   std::cout << std::setw(25) << std::right
+      << "integer_round_down"
+      << std::setw(18) << std::right
+      << quantile(binom_round_down(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_round_down(50, 0.5), 0.95) 
+      << std::endl;
+   
+   // Test integer_round_up:
+   std::cout << std::setw(25) << std::right
+      << "integer_round_up"
+      << std::setw(18) << std::right
+      << quantile(binom_round_up(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_round_up(50, 0.5), 0.95) 
+      << std::endl;
+   
+   // Test integer_round_nearest:
+   std::cout << std::setw(25) << std::right
+      << "integer_round_nearest"
+      << std::setw(18) << std::right
+      << quantile(binom_round_nearest(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_round_nearest(50, 0.5), 0.95) 
+      << std::endl;
+   
+   // Test real:
+   std::cout << std::setw(25) << std::right
+      << "real"
+      << std::setw(18) << std::right
+      << quantile(binom_real_quantile(50, 0.5), 0.05)
+      << std::setw(18) << std::right
+      << quantile(binom_real_quantile(50, 0.5), 0.95) 
+      << std::endl;
+}
+

+

+

+

+

+ Which produces the program output: +

+

+ +

+
Testing rounding policies for a 50 sample binomial distribution,
+with a success fraction of 0.5.
+
+Lower quantiles are calculated at p = 0.05
+
+Upper quantiles at p = 0.95.
+
+Testing rounding policies for a 50 sample binomial distribution,
+with a success fraction of 0.5.
+
+Lower quantiles are calculated at p = 0.05
+
+Upper quantiles at p = 0.95.
+
+                   Policy    Lower Quantile    Upper Quantile
+   integer_round_outwards                18                31
+    integer_round_inwards                19                30
+       integer_round_down                18                30
+         integer_round_up                19                31
+    integer_round_nearest                19                30
+                     real            18.701            30.299
+
+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html new file mode 100644 index 000000000..aca1a8dd7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html @@ -0,0 +1,647 @@ + + + +Calling User Defined Error Handlers + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+

+

+ Suppose we want our own user-defined error handlers rather than the any + of the default ones supplied by the library to be used. If we set the + policy for a specific type of error to user_error + then the library will call a user-supplied error handler. These are forward + declared, but not defined in boost/math/policies/error_handling.hpp like + this: +

+

+ +

+
+namespace boost{ namespace math{ namespace policy{
+
+template <class T>
+T user_domain_error(const char* function, const char* message, const T& val);
+template <class T>
+T user_pole_error(const char* function, const char* message, const T& val);
+template <class T>
+T user_overflow_error(const char* function, const char* message, const T& val);
+template <class T>
+T user_underflow_error(const char* function, const char* message, const T& val);
+template <class T>
+T user_denorm_error(const char* function, const char* message, const T& val);
+template <class T>
+T user_evaluation_error(const char* function, const char* message, const T& val);
+
+}}} // namespaces
+
+

+

+

+ So out first job is to include the header we want to use, and then provide + definitions for the user-defined error handlers we want to use: +

+

+

+

+ +

+
+#include <iostream>
+#include <boost/math/special_functions.hpp>
+
+namespace boost{ namespace math{ namespace policies{
+
+template <class T>
+T user_domain_error(const char* function, const char* message, const T& val)
+{
+   std::cerr << "Domain Error." << std::endl;
+   return std::numeric_limits<T>::quiet_NaN();
+}
+
+template <class T>
+T user_pole_error(const char* function, const char* message, const T& val)
+{
+   std::cerr << "Pole Error." << std::endl;
+   return std::numeric_limits<T>::quiet_NaN();
+}
+
+
+}}} // namespaces
+

+

+

+

+

+ Now we'll need to define a suitable policy that will call these handlers, + and define some forwarding functions that make use of the policy: +

+

+

+

+ +

+
+namespace{
+
+using namespace boost::math::policies;
+
+typedef policy<
+   domain_error<user_error>,
+   pole_error<user_error>
+> user_error_policy;
+
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(user_error_policy)
+
+} // close unnamed namespace
+

+

+

+

+

+ We now have a set of forwarding functions defined in an unnamed namespace + that all look something like this: +

+

+

+

+ +

+
+template <class RealType>
+inline typename boost::math::tools::promote_args<RT>::type
+   tgamma(RT z)
+{
+   return boost::math::tgamma(z, user_error_policy());
+}
+
+

+

+

+

+

+ So that when we call tgamma(z) we really end up calling boost::math::tgamma(z, user_error_policy()), + and any errors will get directed to our own error handlers: +

+

+

+

+ +

+
+int main()
+{
+   std::cout << "Result of erf_inv(-10) is: "
+      << erf_inv(-10) << std::endl;
+   std::cout << "Result of tgamma(-10) is: "
+      << tgamma(-10) << std::endl;
+}
+

+

+

+

+

+ Which outputs: +

+

+ +

+
Domain Error.
+Result of erf_inv(-10) is: 1.#QNAN
+Pole Error.
+Result of tgamma(-10) is: 1.#QNAN
+
+

+

+

+

+

+ The previous example was all well and good, but the custom error handlers + didn't really do much of any use. In this example we'll implement all + the custom handlers and show how the information provided to them can + be used to generate nice formatted error messages. +

+

+

+

+ Each error handler has the general form: +

+

+ +

+
+template <class T>
+T user_error_type(
+   const char* function, 
+   const char* message, 
+   const T& val);
+
+

+

+

+ and accepts three arguments: +

+

+

+
+

+
+
const char* function
+

+ The name of the function that raised the error, this string contains + one or more %1% format specifiers that should be replaced by the + name of type T. +

+
const char* message
+

+ A message associated with the error, normally this contains a %1% + format specifier that should be replaced with the value of value: + however note that overflow and underflow messages do not contain + this %1% specifier (since the value of value + is immaterial in these cases). +

+
const T& value
+

+ The value that caused the error: either an argument to the function + if this is a domain or pole error, the tentative result if this is + a denorm or evaluation error, or zero or infinity for underflow or + overflow errors. +

+
+
+

+

+

+ As before we'll include the headers we need first: +

+

+

+

+ +

+
+#include <iostream>
+#include <boost/math/special_functions.hpp>
+

+

+

+

+

+ Next we'll implement the error handlers for each type of error, starting + with domain errors: +

+

+

+

+ +

+
+namespace boost{ namespace math{ namespace policies{
+
+template <class T>
+T user_domain_error(const char* function, const char* message, const T& val)
+{
+

+

+

+

+

+ We'll begin with a bit of defensive programming: +

+

+

+

+ +

+
+if(function == 0)
+    function = "Unknown function with arguments of type %1%";
+if(message == 0)
+    message = "Cause unknown with bad argument %1%";
+

+

+

+

+

+ Next we'll format the name of the function with the name of type T: +

+

+

+

+ +

+
+std::string msg("Error in function ");
+msg += (boost::format(function) % typeid(T).name()).str();
+

+

+

+

+

+ Then likewise format the error message with the value of parameter val, + making sure we output all the digits of val: +

+

+

+

+ +

+
+msg += ": \n";
+int prec = 2 + (std::numeric_limits<T>::digits * 30103UL) / 100000UL;
+msg += (boost::format(message) % boost::io::group(std::setprecision(prec), val)).str();
+

+

+

+

+

+ Now we just have to do something with the message, we could throw an + exception, but for the purposes of this example we'll just dump the message + to std::cerr: +

+

+

+

+ +

+
+std::cerr << msg << std::endl;
+

+

+

+

+

+ Finally the only sensible value we can return from a domain error is + a NaN: +

+

+

+

+ +

+
+   return std::numeric_limits<T>::quiet_NaN();
+}
+

+

+

+

+

+ Pole errors are essentially a special case of domain errors, so in this + example we'll just return the result of a domain error: +

+

+

+

+ +

+
+template <class T>
+T user_pole_error(const char* function, const char* message, const T& val)
+{
+   return user_domain_error(function, message, val);
+}
+

+

+

+

+

+ Overflow errors are very similar to domain errors, except that there's + no %1% format specifier in the message parameter: +

+

+

+

+ +

+
+template <class T>
+T user_overflow_error(const char* function, const char* message, const T& val)
+{
+   if(function == 0)
+       function = "Unknown function with arguments of type %1%";
+   if(message == 0)
+       message = "Result of function is too large to represent";
+
+   std::string msg("Error in function ");
+   msg += (boost::format(function) % typeid(T).name()).str();
+
+   msg += ": \n";
+   msg += message;
+
+   std::cerr << msg << std::endl;
+   
+   // Value passed to the function is an infinity, just return it:
+   return val; 
+}
+

+

+

+

+

+ Underflow errors are much the same as overflow: +

+

+

+

+ +

+
+template <class T>
+T user_underflow_error(const char* function, const char* message, const T& val)
+{
+   if(function == 0)
+       function = "Unknown function with arguments of type %1%";
+   if(message == 0)
+       message = "Result of function is too small to represent";
+
+   std::string msg("Error in function ");
+   msg += (boost::format(function) % typeid(T).name()).str();
+
+   msg += ": \n";
+   msg += message;
+
+   std::cerr << msg << std::endl;
+   
+   // Value passed to the function is zero, just return it:
+   return val; 
+}
+

+

+

+

+

+ Denormalised results are much the same as underflow: +

+

+

+

+ +

+
+template <class T>
+T user_denorm_error(const char* function, const char* message, const T& val)
+{
+   if(function == 0)
+       function = "Unknown function with arguments of type %1%";
+   if(message == 0)
+       message = "Result of function is denormalised";
+
+   std::string msg("Error in function ");
+   msg += (boost::format(function) % typeid(T).name()).str();
+
+   msg += ": \n";
+   msg += message;
+
+   std::cerr << msg << std::endl;
+   
+   // Value passed to the function is denormalised, just return it:
+   return val; 
+}
+

+

+

+

+

+ Which leaves us with evaluation errors, these occur when an internal + error occurs that prevents the function being fully evaluated. The parameter + val contains the closest approximation to the result + found so far: +

+

+

+

+ +

+
+template <class T>
+T user_evaluation_error(const char* function, const char* message, const T& val)
+{
+   if(function == 0)
+       function = "Unknown function with arguments of type %1%";
+   if(message == 0)
+       message = "An internal evaluation error occured with "
+                  "the best value calculated so far of %1%";
+
+   std::string msg("Error in function ");
+   msg += (boost::format(function) % typeid(T).name()).str();
+
+   msg += ": \n";
+   int prec = 2 + (std::numeric_limits<T>::digits * 30103UL) / 100000UL;
+   msg += (boost::format(message) % boost::io::group(std::setprecision(prec), val)).str();
+
+   std::cerr << msg << std::endl;
+
+   // What do we return here?  This is generally a fatal error,
+   // that should never occur, just return a NaN for the purposes
+   // of the example:
+   return std::numeric_limits<T>::quiet_NaN();
+}
+
+}}} // namespaces
+

+

+

+

+

+ Now we'll need to define a suitable policy that will call these handlers, + and define some forwarding functions that make use of the policy: +

+

+

+

+ +

+
+namespace{
+
+using namespace boost::math::policies;
+
+typedef policy<
+   domain_error<user_error>,
+   pole_error<user_error>,
+   overflow_error<user_error>,
+   underflow_error<user_error>,
+   denorm_error<user_error>,
+   evaluation_error<user_error>
+> user_error_policy;
+
+BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(user_error_policy)
+
+} // close unnamed namespace
+

+

+

+

+

+ We now have a set of forwarding functions defined in an unnamed namespace + that all look something like this: +

+

+

+

+ +

+
+template <class RealType>
+inline typename boost::math::tools::promote_args<RT>::type
+   tgamma(RT z)
+{
+   return boost::math::tgamma(z, user_error_policy());
+}
+
+

+

+

+

+

+ So that when we call tgamma(z) we really end up calling boost::math::tgamma(z, user_error_policy()), + and any errors will get directed to our own error handlers: +

+

+

+

+ +

+
+int main()
+{
+   // Raise a domain error:
+   std::cout << "Result of erf_inv(-10) is: "
+      << erf_inv(-10) << std::endl << std::endl;
+   // Raise a pole error:
+   std::cout << "Result of tgamma(-10) is: "
+      << tgamma(-10) << std::endl << std::endl;
+   // Raise an overflow error:
+   std::cout << "Result of tgamma(3000) is: "
+      << tgamma(3000) << std::endl << std::endl;
+   // Raise an underflow error:
+   std::cout << "Result of tgamma(-190.5) is: "
+      << tgamma(-190.5) << std::endl << std::endl;
+   // Unfortunately we can't predicably raise a denormalised
+   // result, nor can we raise an evaluation error in this example
+   // since these should never really occur!
+}
+

+

+

+

+

+ Which outputs: +

+

+ +

+
Error in function boost::math::erf_inv<double>(double, double):
+Argument outside range [-1, 1] in inverse erf function (got p=-10).
+Result of erf_inv(-10) is: 1.#QNAN
+
+Error in function boost::math::tgamma<long double>(long double):
+Evaluation of tgamma at a negative integer -10.
+Result of tgamma(-10) is: 1.#QNAN
+
+Error in function boost::math::tgamma<long double>(long double):
+Result of tgamma is too large to represent.
+Error in function boost::math::tgamma<double>(double):
+Result of function is too large to represent
+Result of tgamma(3000) is: 1.#INF
+
+Error in function boost::math::tgamma<long double>(long double):
+Result of tgamma is too large to represent.
+Error in function boost::math::tgamma<long double>(long double):
+Result of tgamma is too small to represent.
+Result of tgamma(-190.5) is: 0
+
+

+

+

+ Notice how some of the calls result in an error handler being called + more than once, or for more than one handler to be called: this is an + artefact of the fact that many functions are implemented in terms of + one or more sub-routines each of which may have it's own error handling. + For example tgamma(-190.5) + is implemented in terms of tgamma(190.5) - which overflows - the reflection formula + for tgamma then notices + that it's dividing by infinity and underflows. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html new file mode 100644 index 000000000..aa0461c77 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html @@ -0,0 +1,88 @@ + + + +So Just What is a Policy Anyway? + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ A policy is a compile-time mechanism for customising the behaviour of a + special function, or a statistical distribution. With Policies you can + control: +

+
    +
  • + What action to take when an error occurs. +
    • +
  • + What happens when you call a function that is mathematically undefined + (for example if you ask for the mean of a Cauchy distribution). +
    • +
  • + What happens when you ask for a quantile of a discrete distribution. +
    • +
  • + Whether the library is allowed to internally promote float + to double and double to long + double in order to improve precision. +
    • +
  • + What precision to use when calculating the result. +
    • +
+

+ Some of these policies could arguably be runtime variables, but then we + couldn't use compile-time dispatch internally to select the best evaluation + method for the given policies. +

+

+ For this reason a Policy is a type: in fact it's an + instance of the class template boost::math::policies::policy<>. This class is just a compile-time-container + of user-selected policies (sometimes called a type-list): +

+
+using namespace boost::math::policies;
+//
+// Define a policy that sets ::errno on overflow, and does
+// not promote double to long double internally:
+//
+typedef policy<domain_error<errno_on_error>, promote_double<false> > mypolicy;
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special.html b/doc/sf_and_dist/html/math_toolkit/special.html new file mode 100644 index 000000000..996556103 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special.html @@ -0,0 +1,151 @@ + + + +Special Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Gamma Functions
+
+
Gamma
+
Log Gamma
+
Digamma
+
Ratios + of Gamma Functions
+
Incomplete Gamma + Functions
+
Incomplete + Gamma Function Inverses
+
Derivative + of the Incomplete Gamma Function
+
+
Factorials and Binomial + Coefficients
+
+
Factorial
+
+ Double Factorial
+
+ Rising Factorial
+
+ Falling Factorial
+
Binomial + Coefficients
+
+
Beta Functions
+
+
Beta
+
Incomplete + Beta Functions
+
The + Incomplete Beta Function Inverses
+
Derivative + of the Incomplete Beta Function
+
+
Error Functions
+
+
Error + Functions
+
Error Function + Inverses
+
+
Polynomials
+
+
Legendre (and + Associated) Polynomials
+
Laguerre (and + Associated) Polynomials
+
Hermite Polynomials
+
Spherical Harmonics
+
+
Bessel Functions
+
+
Bessel Function + Overview
+
Bessel Functions + of the First and Second Kinds
+
Modified Bessel + Functions of the First and Second Kinds
+
Spherical + Bessel Functions of the First and Second Kinds
+
+
Elliptic Integrals
+
+
Elliptic + Integral Overview
+
Elliptic + Integrals - Carlson Form
+
Elliptic Integrals + of the First Kind - Legendre Form
+
Elliptic Integrals + of the Second Kind - Legendre Form
+
Elliptic Integrals + of the Third Kind - Legendre Form
+
+
Logs, Powers, Roots and + Exponentials
+
+
log1p
+
expm1
+
cbrt
+
sqrt1pm1
+
powm1
+
hypot
+
+
Sinus Cardinal and Hyperbolic + Sinus Cardinal Functions
+
+
Sinus Cardinal + and Hyperbolic Sinus Cardinal Functions Overview
+
sinc_pi
+
sinhc_pi
+
+
Inverse Hyperbolic Functions
+
+
Inverse + Hyperbolic Functions Overview
+
acosh
+
asinh
+
atanh
+
+
Floating Point Classification: + Infinities and NaN's
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/bessel.html b/doc/sf_and_dist/html/math_toolkit/special/bessel.html new file mode 100644 index 000000000..161e474ce --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/bessel.html @@ -0,0 +1,62 @@ + + + +Bessel Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Bessel Function + Overview
+
Bessel Functions + of the First and Second Kinds
+
Modified Bessel + Functions of the First and Second Kinds
+
Spherical + Bessel Functions of the First and Second Kinds
+
+

+

+

+

+

+

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html b/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html new file mode 100644 index 000000000..0a743918d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html @@ -0,0 +1,673 @@ + + + +Bessel Functions of the First and Second Kinds + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+
+template <class T1, class T2>
+calculated-result-type cyl_bessel_j(T1 v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type cyl_bessel_j(T1 v, T2 x, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type cyl_neumann(T1 v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type cyl_neumann(T1 v, T2 x, const Policy&);
+
+
+ + Description +
+

+ The functions cyl_bessel_j + and cyl_neumann + return the result of the Bessel functions of the first and second kinds + respectively: +

+

+ cyl_bessel_j(v, x) = Jv(x) +

+

+ cyl_neumann(v, x) = Yv(x) = Nv(x) +

+

+ where: +

+

+ +

+

+ +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types. + The functions are also optimised for the relatively common case that T1 + is an integer. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ The functions return the result of domain_error + whenever the result is undefined or complex. For cyl_bessel_j + this occurs when x < + 0 and v is not an integer, or when + x == + 0 and v + != 0. + For cyl_neumann + this occurs when x <= + 0. +

+

+ The following graph illustrates the cyclic nature of Jv: +

+

+ bessel_jn +

+

+ The following graph shows the behaviour of Yv: this is also cyclic for + large x, but tends to -∞ for small x: +

+

+ bessel_yv +

+
+ + Testing +
+

+ There are two sets of test values: spot values calculated using functions.wolfram.com, and a + much larger set of tests computed using a simplified version of this implementation + (with all the special case handling removed). +

+
+ + Accuracy +
+

+ The following tables show how the accuracy of these functions varies on + various platforms, along with comparisons to the GSL-1.9 + and Cephes libraries. + Note that the cyclic nature of these functions means that they have an + infinite number of irrational roots: in general these functions have arbitrarily + large relative errors when the arguments are sufficiently + close to a root. Of course the absolute error in such cases is always small. + Note that only results for the widest floating-point type on the system + are given as narrower types have effectively + zero error. All values are relative errors in units of epsilon. +

+
+

Table 31. Errors Rates in cyl_bessel_j

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ J0 and J1 +

+ 		+

+ Jv +

+ 		+

+ Jv (large values of x > 1000) +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=2.5 Mean=1.1 +

+

+ GSL Peak=6.6 +

+

+ Cephes Peak=2.5 + Mean=1.1 +

+ 		+

+ Peak=11 Mean=2.2 +

+

+ GSL Peak=11 +

+

+ Cephes Peak=17 + Mean=2.5 +

+ 		+

+ Peak=413 Mean=110 +

+

+ GSL Peak=6x1011 +

+

+ Cephes Peak=2x105 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64 / G++ 3.4 +

+ 		+

+ Peak=7 Mean=3 +

+ 		+

+ Peak=117 Mean=10 +

+ 		+

+ Peak=2x104 Mean=6x103 +

+
+

+ 64 +

+ 		+

+ SUSE Linux AMD64 / G++ 4.1 +

+ 		+

+ Peak=7 Mean=3 +

+ 		+

+ Peak=400 Mean=40 +

+ 		+

+ Peak=2x104 Mean=1x104 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=14 Mean=6 +

+ 		+

+ Peak=29 Mean=3 +

+ 		+

+ Peak=2700 Mean=450 +

+
+
+
+

Table 32. Errors Rates in cyl_neumann

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ J0 and J1 +

+ 		+

+ Jn (integer orders) +

+ 		+

+ Jv (fractional orders) +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=330 Mean=54 +

+

+ GSL Peak=34 Mean=9 +

+

+ Cephes Peak=330 + Mean=54 +

+ 		+

+ Peak=923 Mean=83 +

+

+ GSL Peak=500 Mean=54 +

+

+ Cephes Peak=923 + Mean=83 +

+ 		+

+ Peak=561 Mean=36 +

+

+ GSL Peak=1.4x106 Mean=7x104 +

+

+ Cephes Peak=+INF +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64 / G++ 3.4 +

+ 		+

+ Peak=470 Mean=56 +

+ 		+

+ Peak=843 Mean=51 +

+ 		+

+ Peak=741 Mean=51 +

+
+

+ 64 +

+ 		+

+ SUSE Linux AMD64 / G++ 4.1 +

+ 		+

+ Peak=1300 Mean=424 +

+ 		+

+ Peak=2x104 Mean=8x103 +

+ 		+

+ Peak=1x105 Mean=6x103 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=180 Mean=63 +

+ 		+

+ Peak=340 Mean=150 +

+ 		+

+ Peak=2x104 Mean=1200 +

+
+
+

+ Note that for large x these functions are largely + dependent on the accuracy of the std::sin + and std::cos functions. +

+

+ Comparison to GSL and Cephes + is interesting: both Cephes + and this library optimise the integer order case - leading to identical + results - simply using the general case is for the most part slightly more + accurate though, as noted by the better accuracy of GSL in the integer + argument cases. This implementation tends to perform much better when the + arguments become large, Cephes + in particular produces some remarkably inaccurate results with some of + the test data (no significant figures correct), and even GSL performs badly + with some inputs to Jv. Note that by way of double-checking these results, + the worst performing Cephes + and GSL cases were recomputed using functions.wolfram.com, + and the result checked against our test data: no errors in the test data + were found. +

+
+ + Implementation +
+

+ The implementation is mostly about filtering off various special cases: +

+

+ When x is negative, then the order v + must be an integer or the result is a domain error. If the order is an + integer then the function is odd for odd orders and even for even orders, + so we reflect to x > 0. +

+

+ When the order v is negative then the reflection formulae + can be used to move to v > 0: +

+

+ +

+

+ +

+

+ Note that if the order is an integer, then these formulae reduce to: +

+

+ J-n = (-1)nJn +

+

+ Y-n = (-1)nYn +

+

+ However, in general, a negative order implies that we will need to compute + both J and Y. +

+

+ When x is large compared to the order v + then the asymptotic expansions for large x in M. Abramowitz + and I.A. Stegun, Handbook of Mathematical Functions + 9.2.19 are used (these were found to be more reliable than those in A&S + 9.2.5). +

+

+ When the order v is an integer the method first relates + the result to J0, J1, Y0 and Y1 using either forwards or backwards recurrence + (Miller's algorithm) depending upon which is stable. The values for J0, + J1, Y0 and Y1 are calculated using the rational minimax approximations on + root-bracketing intervals for small |x| and Hankel + asymptotic expansion for large |x|. The coefficients + are from: +

+

+ W.J. Cody, ALGORITHM 715: SPECFUN - A Portable FORTRAN Package + of Special Function Routines and Test Drivers, ACM Transactions + on Mathematical Software, vol 19, 22 (1993). +

+

+ and +

+

+ J.F. Hart et al, Computer Approximations, John Wiley + & Sons, New York, 1968. +

+

+ These approximations are accurate to around 19 decimal digits: therefore + these methods are not used when type T has more than 64 binary digits. +

+

+ When x is small, Jx is best computed directly from + the series: +

+

+ +

+

+ In the general case we compute Jv and Yv simultaneously. +

+

+ To get the initial values, let μ = ν - floor(ν + 1/2), then μ is the fractional + part of ν such that |μ| <= 1/2 (we need this for convergence later). The + idea is to calculate Jμ(x), Jμ+1(x), Yμ(x), Yμ+1(x) and use them to obtain + Jν(x), Yν(x). +

+

+ The algorithm is called Steed's method, which needs two continued fractions + as well as the Wronskian: +

+

+ +

+

+ +

+

+ +

+

+ See: F.S. Acton, Numerical Methods that Work, The + Mathematical Association of America, Washington, 1997. +

+

+ The continued fractions are computed using the modified Lentz's method + (W.J. Lentz, Generating Bessel functions in Mie scattering calculations + using continued fractions, Applied Optics, vol 15, 668 (1976)). + Their convergence rates depend on x, therefore we + need different strategies for large x and small x. +

+

+ x > v, CF1 needs O(x) iterations + to converge, CF2 converges rapidly +

+

+ x <= v, CF1 converges rapidly, CF2 fails to converge + when x -> 0 +

+

+ When x is large (x > 2), both + continued fractions converge (CF1 may be slow for really large x). + Jμ, Jμ+1, Yμ, Yμ+1 can be calculated by +

+

+ +

+

+ where +

+

+ +

+

+ Jν and Yμ are then calculated using backward (Miller's algorithm) and forward + recurrence respectively. +

+

+ When x is small (x <= 2), + CF2 convergence may fail (but CF1 works very well). The solution here is + Temme's series: +

+

+ +

+

+ where +

+

+ +

+

+ gk and hk +are also computed by recursions (involving gamma functions), but + the formulas are a little complicated, readers are refered to N.M. Temme, + On the numerical evaluation of the ordinary Bessel function of + the second kind, Journal of Computational Physics, vol 21, 343 + (1976). Note Temme's series converge only for |μ| <= 1/2. +

+

+ As the previous case, Yν is calculated from the forward recurrence, so is + Yν+1. With these two values and fν, the Wronskian yields Jν(x) directly without + backward recurrence. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel_over.html b/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel_over.html new file mode 100644 index 000000000..d0261e604 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel_over.html @@ -0,0 +1,213 @@ + + + +Bessel Function Overview + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Ordinary + Bessel Functions +
+

+ Bessel Functions are solutions to Bessel's ordinary differential equation: +

+

+ +

+

+ where ν is the order of the equation, and may be an + arbitrary real or complex number, although integer orders are the most + common occurrence. +

+

+ This library supports either integer or real orders. +

+

+ Since this is a second order differential equation, there must be two linearly + independent solutions, the first of these is denoted Jv +and known as a Bessel + function of the first kind: +

+

+ +

+

+ This function is implemented in this library as cyl_bessel_j. +

+

+ The second solution is denoted either Yv or Nv +and is known as either a Bessel + Function of the second kind, or as a Neumann function: +

+

+ +

+

+ This function is implemented in this library as cyl_neumann. +

+

+ The Bessel functions satisfy the recurrence relations: +

+

+ +

+

+ +

+

+ Have the derivatives: +

+

+ +

+

+ +

+

+ Have the Wronskian relation: +

+

+ +

+

+ and the reflection formulae: +

+

+ +

+

+ +

+
+ + Modified + Bessel Functions +
+

+ The Bessel functions are valid for complex argument x, + and an important special case is the situation where x + is purely imaginary: giving a real valued result. In this case the functions + are the two linearly independent solutions to the modified Bessel equation: +

+

+ +

+

+ The solutions are known as the modified Bessel functions of the first and + second kind (or occasionally as the hyperbolic Bessel functions of the + first and second kind). They are denoted Iv and Kv +respectively: +

+

+ +

+

+ +

+

+ These functions are implemented in this library as cyl_bessel_i + and cyl_bessel_k + respectively. +

+

+ The modified Bessel functions satisfy the recurrence relations: +

+

+ +

+

+ +

+

+ Have the derivatives: +

+

+ +

+

+ +

+

+ Have the Wronskian relation: +

+

+ +

+

+ and the reflection formulae: +

+

+ +

+

+ +

+
+ + Spherical + Bessel Functions +
+

+ When solving the Helmholtz equation in spherical coordinates by separation + of variables, the radial equation has the form: +

+

+ +

+

+ The two linearly independent solutions to this equation are called the + spherical Bessel functions jn and yn, and are related to the ordinary Bessel + functions Jn and Yn by: +

+

+ +

+

+ The spherical Bessel function of the second kind yn +is also known as the + spherical Neumann function nn. +

+

+ These functions are implemented in this library as sph_bessel + and sph_neumann. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html b/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html new file mode 100644 index 000000000..b8bb64edd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html @@ -0,0 +1,485 @@ + + + +Modified Bessel Functions of the First and Second Kinds + + + + + + + + + + + + + + + +
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+
+
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+
+
+
+ + Synopsis +
+
+template <class T1, class T2>
+calculated-result-type cyl_bessel_i(T1 v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type cyl_bessel_i(T1 v, T2 x, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type cyl_bessel_k(T1 v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type cyl_bessel_k(T1 v, T2 x, const Policy&);
+
+
+ + Description +
+

+ The functions cyl_bessel_i + and cyl_bessel_k + return the result of the modified Bessel functions of the first and second + kind respectively: +

+

+ cyl_bessel_i(v, x) = Iv(x) +

+

+ cyl_bessel_k(v, x) = Kv(x) +

+

+ where: +

+

+ +

+

+ +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types. + The functions are also optimised for the relatively common case that T1 + is an integer. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ The functions return the result of domain_error + whenever the result is undefined or complex. For cyl_bessel_j + this occurs when x < + 0 and v is not an integer, or when + x == + 0 and v + != 0. + For cyl_neumann + this occurs when x <= + 0. +

+

+ The following graph illustrates the exponential behaviour of Iv. +

+

+ bessel_i +

+

+ The following graph illustrates the exponential decay of Kv. +

+

+ bessel_k +

+
+ + Testing +
+

+ There are two sets of test values: spot values calculated using functions.wolfram.com, and a + much larger set of tests computed using a simplified version of this implementation + (with all the special case handling removed). +

+
+ + Accuracy +
+

+ The following tables show how the accuracy of these functions varies on + various platforms, along with a comparison to the GSL-1.9 + library. Note that only results for the widest floating-point type on the + system are given, as narrower types have effectively + zero error. All values are relative errors in units of epsilon. +

+
+

Table 33. Errors Rates in cyl_bessel_i

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Iv +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=10 Mean=3.4 GSL Peak=6000 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64 / G++ 3.4 +

+ 		+

+ Peak=11 Mean=3 +

+
+

+ 64 +

+ 		+

+ SUSE Linux AMD64 / G++ 4.1 +

+ 		+

+ Peak=11 Mean=4 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=15 Mean=4 +

+
+
+
+

Table 34. Errors Rates in cyl_bessel_k

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Kv +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=9 Mean=2 +

+

+ GSL Peak=9 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64 / G++ 3.4 +

+ 		+

+ Peak=10 Mean=2 +

+
+

+ 64 +

+ 		+

+ SUSE Linux AMD64 / G++ 4.1 +

+ 		+

+ Peak=10 Mean=2 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=12 Mean=5 +

+
+
+
+ + Implementation +
+

+ The following are handled as special cases first: +

+

+ When computing Iv for x < 0, then ν must be an integer + or a domain error occurs. If ν is an integer, then the function is odd if + ν is odd and even if ν is even, and we can reflect to x > 0. +

+

+ For Iv with v equal to 0, 1 or 0.5 are handled as special cases. +

+

+ The 0 and 1 cases use minimax rational approximations on finite and infinite + intervals. The coefficients are from: +

+
    +
  • + J.M. Blair and C.A. Edwards, Stable rational minimax approximations + to the modified Bessel functions I_0(x) and I_1(x), Atomic + Energy of Canada Limited Report 4928, Chalk River, 1974. +
    • +
  • + S. Moshier, Methods and Programs for Mathematical Functions, + Ellis Horwood Ltd, Chichester, 1989. +
    • +
+

+ While the 0.5 case is a simple trigonometric function: +

+

+ I0.5(x) = sqrt(2 / πx) * sinh(x) +

+

+ For Kv with v an integer, the result is calculated + using the recurrence relation: +

+

+ +

+

+ starting from K0 and K1 which are calculated using rational the approximations + above. These rational approximations are accurate to around 19 digits, + and are therefore only used when T has no more than 64 binary digits of + precision. +

+

+ In the general case, we first normalize ν to [0, [inf]) + with the help of the reflection formulae: +

+

+ +

+

+ +

+

+ Let μ = ν - floor(ν + 1/2), then μ is the fractional part of ν such that |μ| <= + 1/2 (we need this for convergence later). The idea is to calculate Kμ(x) + and Kμ+1(x), and use them to obtain Iν(x) and Kν(x). +

+

+ The algorithm is proposed by Temme in N.M. Temme, On the numerical + evaluation of the modified bessel function of the third kind, + Journal of Computational Physics, vol 19, 324 (1975), which needs two continued + fractions as well as the Wronskian: +

+

+ +

+

+ +

+

+ +

+

+ The continued fractions are computed using the modified Lentz's method + (W.J. Lentz, Generating Bessel functions in Mie scattering calculations + using continued fractions, Applied Optics, vol 15, 668 (1976)). + Their convergence rates depend on x, therefore we + need different strategies for large x and small x. +

+

+ x > v, CF1 needs O(x) iterations + to converge, CF2 converges rapidly. +

+

+ x <= v, CF1 converges rapidly, CF2 fails to converge + when x -> 0. +

+

+ When x is large (x > 2), both + continued fractions converge (CF1 may be slow for really large x). + Kμ and Kμ+1 +can be calculated by +

+

+ +

+

+ where +

+

+ +

+

+ S is also a series that is summed along with CF2, + see I.J. Thompson and A.R. Barnett, Modified Bessel functions + I_v and K_v of real order and complex argument to selected accuracy, + Computer Physics Communications, vol 47, 245 (1987). +

+

+ When x is small (x <= 2), + CF2 convergence may fail (but CF1 works very well). The solution here is + Temme's series: +

+

+ +

+

+ where +

+

+ +

+

+ fk and hk +are also computed by recursions (involving gamma functions), but + the formulas are a little complicated, readers are referred to N.M. Temme, + On the numerical evaluation of the modified Bessel function of + the third kind, Journal of Computational Physics, vol 19, 324 + (1975). Note: Temme's series converge only for |μ| <= 1/2. +

+

+ Kν(x) is then calculated from the forward recurrence, as is Kν+1(x). With + these two values and fν, the Wronskian yields Iν(x) directly. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html b/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html new file mode 100644 index 000000000..cef8e3bbb --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/bessel/sph_bessel.html @@ -0,0 +1,162 @@ + + + +Spherical Bessel Functions of the First and Second Kinds + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+
+template <class T1, class T2>
+calculated-result-type sph_bessel(unsigned v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type sph_bessel(unsigned v, T2 x, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type sph_neumann(unsigned v, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type sph_neumann(unsigned v, T2 x, const Policy&);
+
+
+ + Description +
+

+ The functions sph_bessel + and sph_neumann + return the result of the Spherical Bessel functions of the first and second + kinds respectively: +

+

+ sph_bessel(v, x) = jv(x) +

+

+ sph_neumann(v, x) = yv(x) = nv(x) +

+

+ where: +

+

+ +

+

+ The return type of these functions is computed using the result + type calculation rules for the single argument type T. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ The functions return the result of domain_error + whenever the result is undefined or complex: this occurs when x < 0. +

+

+ The jv function is cyclic like Jv but differs in its behaviour at the origin: +

+

+ sph_bessel_j +

+

+ Likewise yv is also cyclic for large x, but tends to -∞ +for small x: +

+

+ sph_bessel_y +

+
+ + Testing +
+

+ There are two sets of test values: spot values calculated using functions.wolfram.com, and + a much larger set of tests computed using a simplified version of this + implementation (with all the special case handling removed). +

+
+ + Accuracy +
+

+ Other than for some special cases, these functions are computed in terms + of cyl_bessel_j + and cyl_neumann: + refer to these functions for accuracy data. +

+
+ + Implementation +
+

+ Other than error handling and a couple of special cases these functions + are implemented directly in terms of their definitions: +

+

+ +

+

+ The special cases occur for: +

+

+ j0= sinc_pi(x) + = sin(x) / x +

+

+ and for small x < 1, we can use the series: +

+

+ +

+

+ which neatly avoids the problem of calculating 0/0 that can occur with + the main definition as x → 0. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint.html b/doc/sf_and_dist/html/math_toolkit/special/ellint.html new file mode 100644 index 000000000..c4feb9705 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint.html @@ -0,0 +1,56 @@ + + + +Elliptic Integrals + + + + + + + + + + + + + + + +
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+
+
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+
+
Elliptic + Integral Overview
+
Elliptic + Integrals - Carlson Form
+
Elliptic Integrals + of the First Kind - Legendre Form
+
Elliptic Integrals + of the Second Kind - Legendre Form
+
Elliptic Integrals + of the Third Kind - Legendre Form
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_1.html b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_1.html new file mode 100644 index 000000000..d2beb00e1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_1.html @@ -0,0 +1,285 @@ + + + +Elliptic Integrals of the First Kind - Legendre Form + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/ellint_1.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2>
+calculated-result-type ellint_1(T1 k, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_1(T1 k, T2 phi, const Policy&);
+
+template <class T>
+calculated-result-type ellint_1(T k);
+
+template <class T, class Policy>
+calculated-result-type ellint_1(T k, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ These two functions evaluate the incomplete elliptic integral of the first + kind F(φ, k) and its complete counterpart K(k) + = F(π/2, k). +

+

+ ellint_1 +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types: + when they are the same type then the result is the same type as the arguments. +

+
+template <class T1, class T2>
+calculated-result-type ellint_1(T1 k, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_1(T1 k, T2 phi, const Policy&);
+
+

+ Returns the incomplete elliptic integral of the first kind F(φ, + k): +

+

+ +

+

+ Requires -1 <= k <= 1, otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T>
+calculated-result-type ellint_1(T k);
+
+template <class T>
+calculated-result-type ellint_1(T k, const Policy&);
+
+

+ Returns the complete elliptic integral of the first kind K(k): +

+

+ +

+

+ Requires -1 <= k <= 1, otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Accuracy +
+

+ These functions are computed using only basic arithmetic operations, so + there isn't much variation in accuracy over differing platforms. Note that + only results for the widest floating point type on the system are given + as narrower types have effectively zero error. + All values are relative errors in units of epsilon. +

+
+

Table 36. Errors Rates in the Elliptic Integrals of the + First Kind

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ F(φ, k) +

+ 		+

+ K(k) +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=3 Mean=0.8 +

+ 		+

+ Peak=1.8 Mean=0.7 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux / G++ 3.4 +

+ 		+

+ Peak=2.6 Mean=1.7 +

+ 		+

+ Peak=2.2 Mean=1.8 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=4.6 Mean=1.5 +

+ 		+

+ Peak=3.7 Mean=1.5 +

+
+
+
+ + Testing +
+

+ The tests use a mixture of spot test values calculated using the online + calculator at functions.wolfram.com, + and random test data generated using NTL::RR at 1000-bit precision and + this implementation. +

+
+ + Implementation +
+

+ These functions are implemented in terms of Carlson's integrals using the + relations: +

+

+ +

+

+ and +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_2.html b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_2.html new file mode 100644 index 000000000..b9dcb93f9 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_2.html @@ -0,0 +1,285 @@ + + + +Elliptic Integrals of the Second Kind - Legendre Form + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/ellint_2.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2>
+calculated-result-type ellint_2(T1 k, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_2(T1 k, T2 phi, const Policy&);
+
+template <class T>
+calculated-result-type ellint_2(T k);
+
+template <class T, class Policy>
+calculated-result-type ellint_2(T k, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ These two functions evaluate the incomplete elliptic integral of the second + kind E(φ, k) and its complete counterpart E(k) + = E(π/2, k). +

+

+ ellint_2 +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types: + when they are the same type then the result is the same type as the arguments. +

+
+template <class T1, class T2>
+calculated-result-type ellint_2(T1 k, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_2(T1 k, T2 phi, const Policy&);
+
+

+ Returns the incomplete elliptic integral of the second kind E(φ, + k): +

+

+ +

+

+ Requires -1 <= k <= 1, otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T>
+calculated-result-type ellint_2(T k);
+
+template <class T>
+calculated-result-type ellint_2(T k, const Policy&);
+
+

+ Returns the complete elliptic integral of the first kind E(k): +

+

+ +

+

+ Requires -1 <= k <= 1, otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Accuracy +
+

+ These functions are computed using only basic arithmetic operations, so + there isn't much variation in accuracy over differing platforms. Note that + only results for the widest floating point type on the system are given + as narrower types have effectively zero error. + All values are relative errors in units of epsilon. +

+
+

Table 37. Errors Rates in the Elliptic Integrals of the + Second Kind

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ F(φ, k) +

+ 		+

+ K(k) +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=4.6 Mean=1.2 +

+ 		+

+ Peak=3.5 Mean=1.0 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux / G++ 3.4 +

+ 		+

+ Peak=4.3 Mean=1.1 +

+ 		+

+ Peak=4.6 Mean=1.2 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=5.8 Mean=2.2 +

+ 		+

+ Peak=10.8 Mean=2.3 +

+
+
+
+ + Testing +
+

+ The tests use a mixture of spot test values calculated using the online + calculator at [http://@functions.wolfram.com functions.wolfram.com], and + random test data generated using NTL::RR at 1000-bit precision and this + implementation. +

+
+ + Implementation +
+

+ These functions are implemented in terms of Carlson's integrals using the + relations: +

+

+ +

+

+ and +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_3.html b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_3.html new file mode 100644 index 000000000..fb333e004 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_3.html @@ -0,0 +1,339 @@ + + + +Elliptic Integrals of the Third Kind - Legendre Form + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/ellint_3.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_3(T1 k, T2 n, T3 phi);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_3(T1 k, T2 n, T3 phi, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type ellint_3(T1 k, T2 n);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_3(T1 k, T2 n, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ These two functions evaluate the incomplete elliptic integral of the third + kind Π(n, φ, k) and its complete counterpart Π(n, + k) = E(n, π/2, k). +

+

+ ellint_3 +

+

+ The return type of these functions is computed using the result + type calculation rules when the arguments are of different + types: when they are the same type then the result is the same type as + the arguments. +

+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_3(T1 k, T2 n, T3 phi);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_3(T1 k, T2 n, T3 phi, const Policy&);
+
+

+ Returns the incomplete elliptic integral of the third kind Π(n, + φ, k): +

+

+ +

+

+ Requires -1 <= k <= 1 and n < 1/sin2(φ), + otherwise returns the result of domain_error + (outside this range the result would be complex). +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + + + +
[Caution]Caution

+ In addition, the region where n > 1 and φ is + not in the range [0, π/2] is currently unsupported and returns + the result of domain_error. For this + reason it is recomended that you keep φ inside its "natural" + range of [0, π/2]. +

+
+template <class T1, class T2>
+calculated-result-type ellint_3(T1 k, T2 n);
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_3(T1 k, T2 n, const Policy&);
+
+

+ Returns the complete elliptic integral of the first kind Π(n, + k): +

+

+ +

+

+ Requires -1 <= k <= 1 and n < 1, + otherwise returns the result of domain_error + (outside this range the result would be complex). +

+

+ [opitonal_policy] +

+
+ + Accuracy +
+

+ These functions are computed using only basic arithmetic operations, so + there isn't much variation in accuracy over differing platforms. Note that + only results for the widest floating point type on the system are given + as narrower types have effectively zero error. + All values are relative errors in units of epsilon. +

+
+

Table 38. Errors Rates in the Elliptic Integrals of the + Third Kind

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Π(n, φ, k) +

+ 		+

+ Π(n, k) +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=29 Mean=2.2 +

+ 		+

+ Peak=3 Mean=0.8 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux / G++ 3.4 +

+ 		+

+ Peak=14 Mean=1.3 +

+ 		+

+ Peak=2.3 Mean=0.8 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=10 Mean=1.4 +

+ 		+

+ Peak=4.2 Mean=1.1 +

+
+
+
+ + Testing +
+

+ The tests use a mixture of spot test values calculated using the online + calculator at functions.wolfram.com, + and random test data generated using NTL::RR at 1000-bit precision and + this implementation. +

+
+ + Implementation +
+

+ The implementation for Π(n, φ, k) first siphons off the special cases: +

+

+ Π(0, φ, k) = F(φ, k) +

+

+ Π(n, π/2, k) = Π(n, k) +

+

+ and +

+

+ +

+

+ Then if n < 0 the relations (A&S 17.7.15/16): +

+

+ +

+

+ are used to shift n to the range [0, 1]. +

+

+ Then the relations: +

+

+ Π(n, -φ, k) = -Π(n, φ, k) +

+

+ Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k) +

+

+ are used to move φ to the range [0, π/2]. +

+

+ The functions are then implemented in terms of Carlson's integrals using + the relations: +

+

+ +

+

+ and +

+

+ +

+

+ The remaining problem area occurs when n > 1 and φ is outside the range + [0, π/2]. In this range the reduction formula for large φ can no longer be + applied. Likewise the identities 17.7.7/8 in A&S for reducing n to + the range [0,1] appear to be no longer applicable. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_carlson.html b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_carlson.html new file mode 100644 index 000000000..b9b95118f --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_carlson.html @@ -0,0 +1,484 @@ + + + +Elliptic Integrals - Carlson Form + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/ellint_rf.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_rf(T1 x, T2 y, T3 z)
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_rf(T1 x, T2 y, T3 z, const Policy&)
+
+}} // namespaces
+
+

+ +

+
+#include <boost/math/special_functions/ellint_rd.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_rd(T1 x, T2 y, T3 z)
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_rd(T1 x, T2 y, T3 z, const Policy&)
+
+}} // namespaces
+
+

+ +

+
+#include <boost/math/special_functions/ellint_rj.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ellint_rj(T1 x, T2 y, T3 z, T4 p)
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy&)
+
+}} // namespaces
+
+

+ +

+
+#include <boost/math/special_functions/ellint_rc.hpp>
+
+

+

+
+namespace boost { namespace math {
+
+template <class T1, class T2>
+calculated-result-type ellint_rc(T1 x, T2 y)
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_rc(T1 x, T2 y, const Policy&)
+
+}} // namespaces
+
+
+ + Description +
+

+ These functions return Carlson's symmetrical elliptic integrals, the functions + have complicated behavior over all their possible domains, but the following + graph gives an idea of their behavior: +

+

+ ellint_c +

+

+ The return type of these functions is computed using the result + type calculation rules when the arguments are of different + types: otherwise the return is the same type as the arguments. +

+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_rf(T1 x, T2 y, T3 z)
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_rf(T1 x, T2 y, T3 z, const Policy&)
+
+

+ Returns Carlson's Elliptic Integral RF: +

+

+ +

+

+ Requires that all of the arguments are non-negative, and at most one may + be zero. Otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ellint_rd(T1 x, T2 y, T3 z)
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ellint_rd(T1 x, T2 y, T3 z, const Policy&)
+
+

+ Returns Carlson's elliptic integral RD: +

+

+ +

+

+ Requires that x and y are non-negative, with at most one of them zero, + and that z >= 0. Otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ellint_rj(T1 x, T2 y, T3 z, T4 p)
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy&)
+
+

+ Returns Carlson's elliptic integral RJ: +

+

+ +

+

+ Requires that x, y and z are non-negative, with at most one of them zero, + and that p != 0. Otherwise returns the result of + domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ When p < 0 the function returns the Cauchy + principal value using the relation: +

+

+ +

+
+template <class T1, class T2>
+calculated-result-type ellint_rc(T1 x, T2 y)
+
+template <class T1, class T2, class Policy>
+calculated-result-type ellint_rc(T1 x, T2 y, const Policy&)
+
+

+ Returns Carlson's elliptic integral RC: +

+

+ +

+

+ Requires that x > 0 and that y != 0. + Otherwise returns the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ When y < 0 the function returns the Cauchy + principal value using the relation: +

+

+ +

+
+ + Testing +
+

+ There are two sets of tests. +

+

+ Spot tests compare selected values with test data given in: +

+
+

+

+

+ B. C. Carlson, Numerical + computation of real or complex elliptic integrals. + Numerical Algorithms, Volume 10, Number 1 / March, 1995, pp 13-26. +

+

+

+
+

+ Random test data generated using NTL::RR at 1000-bit precision and our + implementation checks for rounding-errors and/or regressions. +

+

+ There are also sanity checks that use the inter-relations between the integrals + to verify their correctness: see the above Carlson paper for details. +

+
+ + Accuracy +
+

+ These functions are computed using only basic arithmetic operations, so + there isn't much variation in accuracy over differing platforms. Note that + only results for the widest floating-point type on the system are given + as narrower types have effectively zero error. + All values are relative errors in units of epsilon. +

+
+

Table 35. Errors Rates in the Carlson Elliptic Integrals

+
++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ RF +

+ 		+

+ RD +

+ 		+

+ RJ +

+ 		+

+ RC +

+
+

+ 53 +

+ 		+

+ Win32 / Visual C++ 8.0 +

+ 		+

+ Peak=2.9 Mean=0.75 +

+ 		+

+ Peak=2.6 Mean=0.9 +

+ 		+

+ Peak=108 Mean=6.9 +

+ 		+

+ Peak=2.4 Mean=0.6 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux / G++ 3.4 +

+ 		+

+ Peak=2.5 Mean=0.75 +

+ 		+

+ Peak=2.7 Mean=0.9 +

+ 		+

+ Peak=105 Mean=8 +

+ 		+

+ Peak=1.9 Mean=0.7 +

+
+

+ 113 +

+ 		+

+ HP-UX / HP aCC 6 +

+ 		+

+ Peak=5.3 Mean=1.6 +

+ 		+

+ Peak=2.9 Mean=0.99 +

+ 		+

+ Peak=180 Mean=12 +

+ 		+

+ Peak=1.8 Mean=0.7 +

+
+
+
+ + Implementation +
+

+ The key of Carlson's algorithm [Carlson79] + is the duplication theorem: +

+

+ +

+

+ By applying it repeatedly, x, y, + z get closer and closer. When they are nearly equal, + the special case equation +

+

+ +

+

+ is used. More specifically, [R F] is evaluated from + a Taylor series expansion to the fifth order. The calculations of the other + three integrals are analogous. +

+

+ For p < 0 in RJ(x, y, z, p) + and y < 0 in RC(x, y), the + integrals are singular and their Cauchy + principal values are returned via the relations: +

+

+ +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html new file mode 100644 index 000000000..4e04eddb0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/ellint/ellint_intro.html @@ -0,0 +1,423 @@ + + + +Elliptic Integral Overview + + + + + + + + + + + + + + + +
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+
+
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+
+
+

+ The main reference for the elliptic integrals is: +

+
+

+

+

+ M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical + Functions with Formulas, Graphs, and Mathematical Tables, National + Bureau of Standards Applied Mathematics Series, U.S. Government Printing + Office, Washington, D.C. +

+

+

+
+

+ Mathworld also contain a lot of useful background information: +

+
+

+

+

+ Weisstein, + Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web + Resource. +

+

+

+
+

+ As does Wikipedia + Elliptic integral. +

+
+ + Notation +
+

+ All variables are real numbers unless otherwise noted. +

+
+ + Definition +
+

+ +

+

+ is called elliptic integral if R(t, s) is a rational + function of t and s, and s2 + is a cubic or quartic polynomial in t. +

+

+ Elliptic integrals generally can not be expressed in terms of elementary + functions. However, Legendre showed that all elliptic integrals can be + reduced to the following three canonical forms: +

+

+ Elliptic Integral of the First Kind (Legendre form) +

+

+ +

+

+ Elliptic Integral of the Second Kind (Legendre form) +

+

+ +

+

+ Elliptic Integral of the Third Kind (Legendre form) +

+

+ +

+

+ where +

+

+ +

+
+ + + + + +
[Note]Note
+

+ φ is called the amplitude. +

+

+ k is called the modulus. +

+

+ α is called the modular angle. +

+

+ n is called the characteristic. +

+
+
+ + + + + +
[Caution]Caution
+

+ Perhaps more than any other special functions the elliptic integrals + are expressed in a variety of different ways. In particular, the final + parameter k (the modulus) may be expressed using + a modular angle α, or a parameter m. These are related + by: +

+

+ k = sinα +

+

+ m = k2 = sin2α +

+

+ So that the integral of the third kind (for example) may be expressed + as either: +

+

+ Π(n, φ, k) +

+

+ Π(n, φ \ α) +

+

+ Π(n, φ| m) +

+

+ To further complicate matters, some texts refer to the complement + of the parameter m, or 1 - m, where: +

+

+ 1 - m = 1 - k2 = cos2α +

+

+ This implementation uses k throughout: this matches + the requirements of the Technical + Report on C++ Library Extensions. However, you should be extra + careful when using these functions! +

+
+

+ When φ = π / 2, the elliptic integrals + are called complete. +

+

+ Complete Elliptic Integral of the First Kind (Legendre form) +

+

+ +

+

+ Complete Elliptic Integral of the Second Kind (Legendre form) +

+

+ +

+

+ Complete Elliptic Integral of the Third Kind (Legendre form) +

+

+ +

+

+ Carlson [Carlson77] [Carlson78] gives an alternative definition + of elliptic integral's canonical forms: +

+

+ Carlson's Elliptic Integral of the First Kind +

+

+ +

+

+ where x, y, z + are nonnegative and at most one of them may be zero. +

+

+ Carlson's Elliptic Integral of the Second Kind +

+

+ +

+

+ where x, y are nonnegative, at + most one of them may be zero, and z must be positive. +

+

+ Carlson's Elliptic Integral of the Third Kind +

+

+ +

+

+ where x, y, z + are nonnegative, at most one of them may be zero, and p + must be nonzero. +

+

+ Carlson's Degenerate Elliptic Integral +

+

+ +

+

+ where x is nonnegative and y + is nonzero. +

+
+ + + + + +
[Note]Note
+

+ RC(x, y) = RF(x, y, y) +

+

+ RD(x, y, z) = RJ(x, y, z, z) +

+
+
+ + Duplication + Theorem +
+

+ Carlson proved in [Carlson78] + that +

+

+ +

+
+ + Carlson's + Formulas +
+

+ The Legendre form and Carlson form of elliptic integrals are related by + equations: +

+

+ +

+

+ In particular, +

+

+ +

+
+ + Numerical + Algorithms +
+

+ The conventional methods for computing elliptic integrals are Gauss and + Landen transformations, which converge quadratically and work well for + elliptic integrals of the first and second kinds. Unfortunately they suffer + from loss of significant digits for the third kind. Carlson's algorithm + [Carlson79] [Carlson78], + by contrast, provides a unified method for all three kinds of elliptic + integrals with satisfactory precisions. +

+
+ + References +
+

+ Special mention goes to: +

+
+

+

+

+ A. M. Legendre, Traitd des Fonctions Elliptiques et des Integrales + Euleriennes, Vol. 1. Paris (1825). +

+

+

+
+

+ However the main references are: +

+
    +
  1. + M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical + Functions with Formulas, Graphs, and Mathematical Tables, National Bureau + of Standards Applied Mathematics Series, U.S. Government Printing Office, + Washington, D.C. +
    2. +
  2. + B.C. Carlson, Computing elliptic integrals by duplication, + Numerische Mathematik, vol 33, 1 (1979). +
    3. +
  3. + B.C. Carlson, Elliptic Integrals of the First Kind, + SIAM Journal on Mathematical Analysis, vol 8, 231 (1977). +
    4. +
  4. + B.C. Carlson, Short Proofs of Three Theorems on Elliptic Integrals, + SIAM Journal on Mathematical Analysis, vol 9, 524 (1978). +
    5. +
  5. + B.C. Carlson and E.M. Notis, ALGORITHM 577: Algorithms for + Incomplete Elliptic Integrals, ACM Transactions on Mathematmal + Software, vol 7, 398 (1981). +
    6. +
  6. + B. C. Carlson, On computing elliptic integrals and functions. + J. Math. and Phys., 44 (1965), pp. 36-51. +
    7. +
  7. + B. C. Carlson, A table of elliptic integrals of the second + kind. Math. Comp., 49 (1987), pp. 595-606. (Supplement, ibid., + pp. S13-S17.) +
    8. +
  8. + B. C. Carlson, A table of elliptic integrals of the third kind. + Math. Comp., 51 (1988), pp. 267-280. (Supplement, ibid., pp. S1-S5.) +
    9. +
  9. + B. C. Carlson, A table of elliptic integrals: cubic cases. + Math. Comp., 53 (1989), pp. 327-333. +
    10. +
  10. + B. C. Carlson, A table of elliptic integrals: one quadratic + factor. Math. Comp., 56 (1991), pp. 267-280. +
    11. +
  11. + B. C. Carlson, A table of elliptic integrals: two quadratic + factors. Math. Comp., 59 (1992), pp. 165-180. +
    12. +
  12. + B. C. Carlson, Numerical + computation of real or complex elliptic integrals. + Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26. +
    13. +
  13. + B. C. Carlson and John L. Gustafson, Asymptotic + Approximations for Symmetric Elliptic Integrals, SIAM + Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303. +
    14. +
+

+ The following references, while not directly relevent to our implementation, + may also be of interest: +

+
    +
  1. + R. Burlisch, Numerical Compuation of Elliptic Integrals and + Elliptic Functions. Numerical Mathematik 7, 78-90. +
    2. +
  2. + R. Burlisch, An extension of the Bartky Transformation to Incomplete + Elliptic Integrals of the Third Kind. Numerical Mathematik + 13, 266-284. +
    3. +
  3. + R. Burlisch, Numerical Compuation of Elliptic Integrals and + Elliptic Functions. III. Numerical Mathematik 13, 305-315. +
    4. +
  4. + T. Fukushima and H. Ishizaki, Numerical + Computation of Incomplete Elliptic Integrals of a General Form. + Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, + 1994, 237-251. +
    5. +
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials.html b/doc/sf_and_dist/html/math_toolkit/special/factorials.html new file mode 100644 index 000000000..8c2c0697c --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials.html @@ -0,0 +1,56 @@ + + + +Factorials and Binomial Coefficients + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Factorial
+
+ Double Factorial
+
+ Rising Factorial
+
+ Falling Factorial
+
Binomial + Coefficients
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_binomial.html b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_binomial.html new file mode 100644 index 000000000..35522c7a6 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_binomial.html @@ -0,0 +1,125 @@ + + + +Binomial Coefficients + + + + + + + + + + + + + + + +
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+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/binomial.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+T binomial_coefficient(unsigned n, unsigned k);
+
+template <class T, class Policy>
+T binomial_coefficient(unsigned n, unsigned k, const Policy&);
+
+}} // namespaces
+
+

+ Returns the binomial coefficient: nCk. +

+

+ Requires k <= n. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ May return the result of overflow_error + if the result is too large to represent in type T. +

+
+ + Accuracy +
+

+ The accuracy will be the same as for the factorials for small arguments + (i.e. no more than one or two epsilon), and the beta + function for larger arguments. +

+
+ + Testing +
+

+ The spot tests for the binomial coefficients use data generated by functions.wolfram.com. +

+
+ + Implementation +
+

+ Binomial coefficients are calculated using table lookup of factorials where + possible using: +

+

+ nCk = n! / (k!(n-k)!) +

+

+ Otherwise it is implemented in terms of the beta function using the relations: +

+

+ nCk = 1 / (k * beta(k, + n-k+1)) +

+

+ and +

+

+ nCk = 1 / ((n-k) * beta(k+1, + n-k)) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_double_factorial.html b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_double_factorial.html new file mode 100644 index 000000000..3cf95f1c6 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_double_factorial.html @@ -0,0 +1,118 @@ + + + +Double Factorial + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/factorials.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+T double_factorial(unsigned i);
+
+template <class T, class Policy>
+T double_factorial(unsigned i, const Policy&);
+
+}} // namespaces
+
+

+ Returns i!!. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ May return the result of overflow_error + if the result is too large to represent in type T. The implementation is + designed to be optimised for small i where table lookup + of i! is possible. +

+
+ + Accuracy +
+

+ The implementation uses a trivial adaptation of the factorial function, + so error rates should be no more than a couple of epsilon higher. +

+
+ + Testing +
+

+ The spot tests for the double factorial use data generated by functions.wolfram.com. +

+
+ + Implementation +
+

+ The double factorial is implemented in terms of the factorial and gamma + functions using the relations: +

+

+ (2n)!! = 2n * n! +

+

+ (2n+1)!! = (2n+1)! / (2n n!) +

+

+ and +

+

+ (2n-1)!! = Γ((2n+1)/2) * 2n / sqrt(pi) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_factorial.html b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_factorial.html new file mode 100644 index 000000000..50bb6676c --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_factorial.html @@ -0,0 +1,161 @@ + + + +Factorial + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/factorials.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+T factorial(unsigned i);
+
+template <class T, class Policy>
+T factorial(unsigned i, const Policy&);
+
+template <class T>
+T unchecked_factorial(unsigned i);
+
+template <class T>
+struct max_factorial;
+
+}} // namespaces
+
+
+ + Description +
+
+template <class T>
+T factorial(unsigned i);
+
+template <class T, class Policy>
+T factorial(unsigned i, const Policy&);
+
+

+ Returns i!. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ For i <= max_factorial<T>::value this is implemented + by table lookup, for larger values of i, this function + is implemented in terms of tgamma. +

+

+ If i is so large that the result can not be represented + in type T, then calls overflow_error. +

+
+template <class T>
+T unchecked_factorial(unsigned i);
+
+

+ Returns i!. +

+

+ Internally this function performs table lookup of the result. Further it + performs no range checking on the value of i: it is up to the caller to + ensure that i <= max_factorial<T>::value. This + function is intended to be used inside inner loops that require fast table + lookup of factorials, but requires care to ensure that argument i + never grows too large. +

+
+template <class T>
+struct max_factorial
+{
+   static const unsigned value = X;
+};
+
+

+ This traits class defines the largest value that can be passed to unchecked_factorial. + The member value can be + used where integral constant expressions are required: for example to define + the size of further tables that depend on the factorials. +

+
+ + Accuracy +
+

+ For arguments smaller than max_factorial<T>::value + the result should be correctly rounded. For larger arguments the accuracy + will be the same as for tgamma. +

+
+ + Testing +
+

+ Basic sanity checks and spot values to verify the data tables: the main + tests for the tgamma + function handle those cases already. +

+
+ + Implementation +
+

+ The factorial function is table driven for small arguments, and is implemented + in terms of tgamma + for larger arguments. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_falling_factorial.html b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_falling_factorial.html new file mode 100644 index 000000000..806197054 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_falling_factorial.html @@ -0,0 +1,120 @@ + + + +Falling Factorial + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/factorials.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type falling_factorial(T x, unsigned i);
+
+template <class T, class Policy>
+calculated-result-type falling_factorial(T x, unsigned i, const Policy&);
+
+}} // namespaces
+
+

+ Returns the falling factorial of x and i: +

+

+ falling_factorial(x, i) = x(x-1)(x-2)(x-3)...(x-i+1) +

+

+ Note that this function is only defined for positive i, + hence the unsigned second + argument. Argument x can be either positive or negative + however. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ May return the result of overflow_error + if the result is too large to represent in type T. +

+

+ The return type of these functions is computed using the result + type calculation rules: the type of the result is double if T is an integer type, otherwise + the type of the result is T. +

+
+ + Accuracy +
+

+ The accuracy will be the same as the tgamma_delta_ratio + function. +

+
+ + Testing +
+

+ The spot tests for the falling factorials use data generated by functions.wolfram.com. +

+
+ + Implementation +
+

+ Rising and falling factorials are implemented as ratios of gamma functions + using tgamma_delta_ratio. + Optimisations for small integer arguments are handled internally by that + function. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_rising_factorial.html b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_rising_factorial.html new file mode 100644 index 000000000..5e456f262 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/factorials/sf_rising_factorial.html @@ -0,0 +1,124 @@ + + + +Rising Factorial + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/factorials.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type rising_factorial(T x, int i);
+
+template <class T, class Policy>
+calculated-result-type rising_factorial(T x, int i, const Policy&);
+
+}} // namespaces
+
+

+ Returns the rising factorial of x and i: +

+

+ rising_factorial(x, i) = Γ(x + i) / Γ(x); +

+

+ or +

+

+ rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i) +

+

+ Note that both x and i can be + negative as well as positive. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ May return the result of overflow_error + if the result is too large to represent in type T. +

+

+ The return type of these functions is computed using the result + type calculation rules: the type of the result is double if T is an integer type, otherwise + the type of the result is T. +

+
+ + Accuracy +
+

+ The accuracy will be the same as the tgamma_delta_ratio + function. +

+
+ + Testing +
+

+ The spot tests for the rising factorials use data generated by functions.wolfram.com. +

+
+ + Implementation +
+

+ Rising and falling factorials are implemented as ratios of gamma functions + using tgamma_delta_ratio. + Optimisations for small integer arguments are handled internally by that + function. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/fpclass.html b/doc/sf_and_dist/html/math_toolkit/special/fpclass.html new file mode 100644 index 000000000..a57f05a60 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/fpclass.html @@ -0,0 +1,227 @@ + + + +Floating Point Classification: Infinities and NaN's + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+
+#define FP_ZERO        /* implementation specific value */
+#define FP_NORMAL      /* implementation specific value */
+#define FP_INFINITE    /* implementation specific value */
+#define FP_NAN         /* implementation specific value */
+#define FP_SUBNORMAL   /* implementation specific value */
+
+template <class T>
+int fpclassify(T t);
+
+template <class T>
+bool isfinite(T z);
+
+template <class T>
+bool isinf(T t);
+
+template <class T>
+bool isnan(T t);
+
+template <class T>
+bool isnormal(T t);
+
+
+ + Description +
+

+ These functions provide the same functionality as the macros with the same + name in C99, indeed if the C99 macros are available, then these functions + are implemented in terms of them, otherwise they rely on std::numeric_limits<> + to function. +

+

+ Note that the definition of these functions does not suppress the + definition of these names as macros by math.h on those platforms + that already provide these as macros. That mean that the following have differing + meanings: +

+
+using namespace boost::math;
+
+// This might call a global macro if defined,
+// but might not work if the type of z is unsupported 
+// by the std lib macro:
+isnan(z);
+//
+// This calls the Boost version
+// (found via the "using namespace boost::math" declaration)
+// it works for any type that has numeric_limits support for type z:
+(isnan)(z); 
+//
+// As above but with namespace qualification.
+(boost::math::isnan)(z); 
+//
+// This will cause a compiler error is isnan is a native macro:
+boost::math::isnan(z);
+// So always use (boost::math::isnan)(z); instead. 
+
+

+ Detailed descriptions for each of these functions follows: +

+
+template <class T>
+int fpclassify(T t);
+
+

+ Returns an integer value that classifies the value t: +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ fpclassify value +

+ 		+

+ class of t. +

+
+

+ FP_ZERO +

+ 		+

+ If t is zero. +

+
+

+ FP_NORMAL +

+ 		+

+ If t is a non-zero, non-denormalised finite + value. +

+
+

+ FP_INFINITE +

+ 		+

+ If t is plus or minus infinity. +

+
+

+ FP_NAN +

+ 		+

+ If t is a NaN. +

+
+

+ FP_SUBNORMAL +

+ 		+

+ If t is a denormalised number. +

+
+
+template <class T>
+bool isfinite(T z);
+
+

+ Returns true only if z is not an infinity or a NaN. +

+
+template <class T>
+bool isinf(T t);
+
+

+ Returns true only if z is plus or minus infinity. +

+
+template <class T>
+bool isnan(T t);
+
+

+ Returns true only if z is a NaN. +

+
+template <class T>
+bool isnormal(T t);
+
+

+ Returns true only if z is a normal number (not zero, + infinite, NaN, or denormalised). +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/inv_hyper.html b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper.html new file mode 100644 index 000000000..a115aa188 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper.html @@ -0,0 +1,51 @@ + + + +Inverse Hyperbolic Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Inverse + Hyperbolic Functions Overview
+
acosh
+
asinh
+
atanh
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/acosh.html b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/acosh.html new file mode 100644 index 000000000..89a0295f8 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/acosh.html @@ -0,0 +1,85 @@ + + + +acosh + + + + + + + + + + + + + + + +
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+
+PrevUpHomeNext +
+
+

+ acosh +

+

+ +

+
+#include <boost/math/special_functions/acosh.hpp>
+
+

+

+
+template<class T> 
+calculated-result-type acosh(const T x);
+
+template<class T, class Policy> 
+calculated-result-type acosh(const T x, const Policy&);
+
+

+ Computes the reciprocal of (the restriction to the range of [0;+∞[) + the hyperbolic + cosine function, at x. Values returned are positive. Generalised + Taylor series are used near 1 and Laurent series are used near the infinity + to ensure accuracy. +

+

+ If x is in the range ]-∞;+1[ then returns the result + of domain_error. +

+

+ The return type of this function is computed using the result + type calculation rules: the return type is double when T is an integer type, and T + otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/asinh.html b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/asinh.html new file mode 100644 index 000000000..f4700a148 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/asinh.html @@ -0,0 +1,79 @@ + + + +asinh + + + + + + + + + + + + + + + +
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+
+PrevUpHomeNext +
+
+

+ asinh +

+

+ +

+
+#include <boost/math/special_functions/asinh.hpp>
+
+

+

+
+template<class T> 
+calculated-result-type asinh(const T x);
+
+template<class T, class Policy> 
+calculated-result-type asinh(const T x, const Policy&);
+
+

+ Computes the reciprocal of the + hyperbolic sine function. Taylor series are used at the origin and + Laurent series are used near the infinity to ensure accuracy. +

+

+ The return type of this function is computed using the result + type calculation rules: the return type is double when T is an integer type, and T + otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/atanh.html b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/atanh.html new file mode 100644 index 000000000..c5829c5b0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/atanh.html @@ -0,0 +1,93 @@ + + + +atanh + + + + + + + + + + + + + + + +
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+
+PrevUpHomeNext +
+
+

+ atanh +

+

+ +

+
+#include <boost/math/special_functions/atanh.hpp>
+
+

+

+
+template<class T> 
+calculated-result-type atanh(const T x);
+
+template<class T, class Policy> 
+calculated-result-type atanh(const T x, const Policy&);
+
+

+ Computes the reciprocal of the + hyperbolic tangent function, at x. Taylor series are used at the + origin to ensure accuracy. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ If x is in the range ]-∞;-1[ or in the range ]+1;+∞[ + then returns the result of domain_error. +

+

+ If x is in the range [-1;-1+ε[, then the result of -overflow_error is returned, with ε +denoting + numeric_limits<T>::epsilon(). +

+

+ If x is in the range ]+1-ε;+1], then the result of overflow_error is returned, with ε +denoting + numeric_limits<T>::epsilon(). +

+

+ The return type of this function is computed using the result + type calculation rules: the return type is double when T is an integer type, and T + otherwise. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/inv_hyper_over.html b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/inv_hyper_over.html new file mode 100644 index 000000000..7c7c6dfb1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/inv_hyper/inv_hyper_over.html @@ -0,0 +1,163 @@ + + + +Inverse Hyperbolic Functions Overview + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ The exponential funtion is defined, for all objects for which this makes + sense, as the power series , + with n! = 1x2x3x4x5...xn (and + 0! = 1 by definition) being the + factorial of n. In particular, + the exponential function is well defined for real numbers, complex number, + quaternions, octonions, and matrices of complex numbers, among others. +

+
+

+

+

+ Graph of exp on R +

+

+

+
+
+

+

+

+ exp_on_r +

+

+

+
+
+

+

+

+ Real and Imaginary parts of exp on + C +

+

+

+
+
+

+

+

+ im_exp_on_c +

+

+

+
+

+ The hyperbolic functions are defined as power series which can be computed + (for reals, complex, quaternions and octonions) as: +

+

+ Hyperbolic cosine: +

+

+ Hyperbolic sine: +

+

+ Hyperbolic tangent: +

+
+

+

+

+ Trigonometric functions on R (cos: + purple; sin: red; tan: blue) +

+

+

+
+
+

+

+

+ trigonometric +

+

+

+
+
+

+

+

+ Hyperbolic functions on r (cosh: purple; + sinh: red; tanh: blue) +

+

+

+
+
+

+

+

+ hyperbolic +

+

+

+
+

+ The hyperbolic sine is one to one on the set of real numbers, with range + the full set of reals, while the hyperbolic tangent is also one to one + on the set of real numbers but with range [0;+∞[, and + therefore both have inverses. The hyperbolic cosine is one to one from + ]-∞;+1[ onto ]-∞;-1[ (and from ]+1;+∞[ + onto ]-∞;-1[); the inverse function we use here is defined + on ]-∞;-1[ with range ]-∞;+1[. +

+

+ The inverse of the hyperbolic tangent is called the Argument hyperbolic + tangent, and can be computed as . +

+

+ The inverse of the hyperbolic sine is called the Argument hyperbolic sine, + and can be computed (for [-1;-1+ε[) as . +

+

+ The inverse of the hyperbolic cosine is called the Argument hyperbolic + cosine, and can be computed as . +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers.html b/doc/sf_and_dist/html/math_toolkit/special/powers.html new file mode 100644 index 000000000..082245155 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers.html @@ -0,0 +1,53 @@ + + + +Logs, Powers, Roots and Exponentials + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
log1p
+
expm1
+
cbrt
+
sqrt1pm1
+
powm1
+
hypot
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/cbrt.html b/doc/sf_and_dist/html/math_toolkit/special/powers/cbrt.html new file mode 100644 index 000000000..071714ff9 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/cbrt.html @@ -0,0 +1,100 @@ + + + +cbrt + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ cbrt +

+

+ +

+
+#include <boost/math/special_functions/cbrt.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type cbrt(T x);
+
+template <class T, class Policy>
+calculated-result-type cbrt(T x, const Policy&);
+
+}} // namespaces
+
+

+ Returns the cubed root of x: x1/3. +

+

+ The return type of this function is computed using the result + type calculation rules: the return is double + when x is an integer type and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Implemented using Halley iteration. +

+
+ + Accuracy +
+

+ For built in floating-point types cbrt + should have approximately 2 epsilon accuracy. +

+
+ + Testing +
+

+ A mixture of spot test sanity checks, and random high precision test values + calculated using NTL::RR at 1000-bit precision. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/expm1.html b/doc/sf_and_dist/html/math_toolkit/special/powers/expm1.html new file mode 100644 index 000000000..95254aa9d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/expm1.html @@ -0,0 +1,109 @@ + + + +expm1 + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ expm1 +

+

+ +

+
+#include <boost/math/special_functions/expm1.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type expm1(T x);
+
+template <class T, class Policy>
+calculated-result-type expm1(T x, const Policy&);
+
+}} // namespaces
+
+

+ Returns ex - 1. +

+

+ The return type of this function is computed using the result + type calculation rules: the return is double + when x is an integer type and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ For small x, then ex is very close to 1, as a result calculating ex - 1 results in + catastrophic cancellation errors when x is small. expm1 + calculates ex - 1 using rational approximations (for up to 128-bit long doubles), + otherwise via a series expansion when x is small (giving an accuracy of + less than 2ɛ). +

+

+ Finally when BOOST_HAS_EXPM1 is defined then the float/double/long double + specializations of this template simply forward to the platform's native + (POSIX) implementation of this function. +

+
+ + Accuracy +
+

+ For built in floating point types expm1 + should have approximately 1 epsilon accuracy. +

+
+ + Testing +
+

+ A mixture of spot test sanity checks, and random high precision test values + calculated using NTL::RR at 1000-bit precision. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/hypot.html b/doc/sf_and_dist/html/math_toolkit/special/powers/hypot.html new file mode 100644 index 000000000..8062456d8 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/hypot.html @@ -0,0 +1,94 @@ + + + +hypot + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ hypot +

+
+template <class T1, class T2>
+calculated-result-type hypot(T1 x, T2 y);
+
+template <class T1, class T2, class Policy>
+calculated-result-type hypot(T1 x, T2 y, const Policy&);
+
+

+ Effects: computes +in such a + way as to avoid undue underflow and overflow. +

+

+ The return type of this function is computed using the result + type calculation rules when T1 and T2 are of different + types. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ When calculating it's quite easy for the intermediate terms to + either overflow or underflow, even though the result is in fact perfectly + representable. +

+
+ + Implementation +
+

+ The function is even and symmetric in x and y, so first take assume x,y + > 0 and x > y (we can permute the + arguments if this is not the case). +

+

+ Then if x * ε >= y we can simply return x. +

+

+ Otherwise the result is given by: +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/log1p.html b/doc/sf_and_dist/html/math_toolkit/special/powers/log1p.html new file mode 100644 index 000000000..77d8ff6b3 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/log1p.html @@ -0,0 +1,123 @@ + + + +log1p + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ log1p +

+

+ +

+
+#include <boost/math/special_functions/log1p.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type log1p(T x);
+
+template <class T, class Policy>
+calculated-result-type log1p(T x, const Policy&);
+
+}} // namespaces
+
+

+ Returns the natural logarithm of x+1. +

+

+ The return type of this function is computed using the result + type calculation rules: the return is double + when x is an integer type and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ There are many situations where it is desirable to compute log(x+1). However, for small x + then x+1 suffers from catastrophic cancellation + errors so that x+1 == 1 + and log(x+1) == 0, + when in fact for very small x, the best approximation to log(x+1) would be x. + log1p calculates the best + approximation to log(1+x) using a Taylor series expansion for accuracy + (less than 2ɛ). Alternatively note that there are faster methods available, + for example using the equivalence: +

+
+log(1+x) == (log(1+x) * x) / ((1-x) - 1)
+
+

+ However, experience has shown that these methods tend to fail quite spectacularly + once the compiler's optimizations are turned on, consequently they are + used only when known not to break with a particular compiler. In contrast, + the series expansion method seems to be reasonably immune to optimizer-induced + errors. +

+

+ Finally when BOOST_HAS_LOG1P is defined then the float/double/long double + specializations of this template simply forward to the platform's native + (POSIX) implementation of this function. +

+
+ + Accuracy +
+

+ For built in floating point types log1p + should have approximately 1 epsilon accuracy. +

+
+ + Testing +
+

+ A mixture of spot test sanity checks, and random high precision test values + calculated using NTL::RR at 1000-bit precision. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/powm1.html b/doc/sf_and_dist/html/math_toolkit/special/powers/powm1.html new file mode 100644 index 000000000..34812b656 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/powm1.html @@ -0,0 +1,102 @@ + + + +powm1 + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ powm1 +

+

+ +

+
+#include <boost/math/special_functions/powm1.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+calculated-result-type powm1(T1 x, T2 y);
+
+template <class T1, class T2, class Policy>
+calculated-result-type powm1(T1 x, T2 y, const Policy&);
+
+}} // namespaces
+
+

+ Returns xy - 1. +

+

+ The return type of this function is computed using the result + type calculation rules when T1 and T2 are dufferent types. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ There are two domains where this is useful: when y is very small, or when + x is close to 1. +

+

+ Implemented in terms of expm1. +

+
+ + Accuracy +
+

+ Should have approximately 2-3 epsilon accuracy. +

+
+ + Testing +
+

+ A selection of random high precision test values calculated using NTL::RR + at 1000-bit precision. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/powers/sqrt1pm1.html b/doc/sf_and_dist/html/math_toolkit/special/powers/sqrt1pm1.html new file mode 100644 index 000000000..a5463112f --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/powers/sqrt1pm1.html @@ -0,0 +1,105 @@ + + + +sqrt1pm1 + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/sqrt1pm1.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type sqrt1pm1(T x);
+
+template <class T, class Policy>
+calculated-result-type sqrt1pm1(T x, const Policy&);
+
+}} // namespaces
+
+

+ Returns sqrt(1+x) - 1. +

+

+ The return type of this function is computed using the result + type calculation rules: the return is double + when x is an integer type and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ This function is useful when you need the difference between sqrt(x) and + 1, when x is itself close to 1. +

+

+ Implemented in terms of log1p + and expm1. +

+
+ + Accuracy +
+

+ For built in floating-point types sqrt1pm1 + should have approximately 3 epsilon accuracy. +

+
+ + Testing +
+

+ A selection of random high precision test values calculated using NTL::RR + at 1000-bit precision. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_beta.html b/doc/sf_and_dist/html/math_toolkit/special/sf_beta.html new file mode 100644 index 000000000..c4111370a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_beta.html @@ -0,0 +1,53 @@ + + + +Beta Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Beta
+
Incomplete + Beta Functions
+
The + Incomplete Beta Function Inverses
+
Derivative + of the Incomplete Beta Function
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_derivative.html b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_derivative.html new file mode 100644 index 000000000..b367aea74 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_derivative.html @@ -0,0 +1,111 @@ + + + +Derivative of the Incomplete Beta Function + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/beta.hpp>
+
+

+

+
+namespace boost{ namespace math{ 
+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_derivative(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_derivative(T1 a, T2 b, T3 x, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ This function finds some uses in statistical distributions: it computes + the partial derivative with respect to x of the incomplete + beta function ibeta. +

+

+ +

+

+ The return type of this function is computed using the result + type calculation rules when T1, T2 and T3 are different + types. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Accuracy +
+

+ Almost identical to the incomplete beta function ibeta. +

+
+ + Implementation +
+

+ This function just expose some of the internals of the incomplete beta + function ibeta: + refer to the documentation for that function for more information. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_function.html b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_function.html new file mode 100644 index 000000000..99dbb77bd --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/beta_function.html @@ -0,0 +1,336 @@ + + + +Beta + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+

+ Beta +

+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/beta.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+calculated-result-type beta(T1 a, T2 b);
+
+template <class T1, class T2, class Policy>
+calculated-result-type beta(T1 a, T2 b, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ The beta function is defined by: +

+

+ +

+

+ beta +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ There are effectively two versions of this function internally: a fully + generic version that is slow, but reasonably accurate, and a much more + efficient approximation that is used where the number of digits in the + significand of T correspond to a certain Lanczos + approximation. In practice any built-in floating-point type you + will encounter has an appropriate Lanczos + approximation defined for it. It is also possible, given enough + machine time, to generate further Lanczos + approximation's using the program libs/math/tools/lanczos_generator.cpp. +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types. +

+
+ + Accuracy +
+

+ The following table shows peak errors for various domains of input arguments, + along with comparisons to the GSL-1.9 + and Cephes libraries. + Note that only results for the widest floating point type on the system + are given as narrower types have effectively + zero error. +

+
+

Table 17. Peak Errors In the Beta Function

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0.4 < a,b < 100 +

+ 		+

+ Errors in range +

+

+ 1e-6 < a,b < 36 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=99 Mean=22 +

+

+ (GSL Peak=1178 Mean=238) +

+

+ (Cephes=1612) +

+ 		+

+ Peak=10.7 Mean=2.6 +

+

+ (GSL Peak=12 Mean=2.0) +

+

+ (Cephes=174) +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA32, g++ 3.4.4 +

+ 		+

+ Peak=112.1 Mean=26.9 +

+ 		+

+ Peak=15.8 Mean=3.6 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=61.4 Mean=19.5 +

+ 		+

+ Peak=12.2 Mean=3.6 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=42.03 Mean=13.94 +

+ 		+

+ Peak=9.8 Mean=3.1 +

+
+
+

+ Note that the worst errors occur when a or b are large, and that when this + is the case the result is very close to zero, so absolute errors will be + very small. +

+
+ + Testing +
+

+ A mixture of spot tests of exact values, and randomly generated test data + are used: the test data was computed using NTL::RR + at 1000-bit precision. +

+
+ + Implementation +
+

+ Traditional methods of evaluating the beta function either involve evaluating + the gamma functions directly, or taking logarithms and then exponentiating + the result. However, the former is prone to overflows for even very modest + arguments, while the latter is prone to cancellation errors. As an alternative, + if we regard the gamma function as a white-box containing the Lanczos + approximation, then we can combine the power terms: +

+

+ +

+

+ which is almost the ideal solution, however almost all of the error occurs + in evaluating the power terms when a or b + are large. If we assume that a > b then the larger + of the two power terms can be reduced by a factor of b, + which immediately cuts the maximum error in half: +

+

+ +

+

+ This may not be the final solution, but it is very competitive compared + to other implementation methods. +

+

+ The generic implementation - where no Lanczos + approximation approximation is available - is implemented in a very + similar way to the generic version of the gamma function. Again in order + to avoid numerical overflow the power terms that prefix the series and + continued fraction parts are collected together into: +

+

+ +

+

+ where la, lb and lc are the integration limits used for a, b, and a+b. +

+

+ There are a few special cases worth mentioning: +

+

+ When a or b are less than one, + we can use the recurrence relations: +

+

+ +

+

+ +

+

+ to move to a more favorable region where they are both greater than 1. +

+

+ In addition: +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html new file mode 100644 index 000000000..e5b847e6d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html @@ -0,0 +1,982 @@ + + + +Incomplete Beta Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta(T1 a, T2 b, T3 x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac(T1 a, T2 b, T3 x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type beta(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type beta(T1 a, T2 b, T3 x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type betac(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type betac(T1 a, T2 b, T3 x, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ There are four incomplete + beta functions : two are normalised versions (also known as regularized + beta functions) that return values in the range [0, 1], and two are non-normalised + and return values in the range [0, beta(a, + b)]. Users interested in statistical applications should use the normalised + (or regularized + ) versions (ibeta and ibetac). +

+

+ All of these functions require a > 0, b + > 0 and 0 <= x <= 1. +

+

+ The return type of these functions is computed using the result + type calculation rules when T1, T2 and T3 are different + types. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta(T1 a, T2 b, T3 x, const Policy&);
+
+

+ Returns the normalised incomplete beta function of a, b and x: +

+

+ +

+

+ ibeta +

+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac(T1 a, T2 b, T3 x, const Policy&);
+
+

+ Returns the normalised complement of the incomplete beta function of a, + b and x: +

+

+ +

+
+template <class T1, class T2, class T3>
+calculated-result-type beta(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type beta(T1 a, T2 b, T3 x, const Policy&);
+
+

+ Returns the full (non-normalised) incomplete beta function of a, b and + x: +

+

+ +

+
+template <class T1, class T2, class T3>
+calculated-result-type betac(T1 a, T2 b, T3 x);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type betac(T1 a, T2 b, T3 x, const Policy&);
+
+

+ Returns the full (non-normalised) complement of the incomplete beta function + of a, b and x: +

+

+ +

+
+ + Accuracy +
+

+ The following tables give peak and mean relative errors in over various + domains of a, b and x, along with comparisons to the GSL-1.9 + and Cephes libraries. + Note that only results for the widest floating-point type on the system + are given as narrower types have effectively + zero error. +

+

+ Note that the results for 80 and 128-bit long doubles are noticeably higher + than for doubles: this is because the wider exponent range of these types + allow more extreme test cases to be tested. For example expected results + that are zero at double precision, may be finite but exceptionally small + with the wider exponent range of the long double types. +

+
+

Table 18. Errors In the Function ibeta(a,b,x)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0 < a,b < 10 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 0 < a,b < 100 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 1x10-5 < a,b < 1x105 +

+

+ and +

+

+ 0 < x < 1 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=42.3 Mean=2.9 +

+

+ (GSL Peak=682 Mean=32.5) +

+

+ (Cephes Peak=42.7 + Mean=7.0) +

+ 		+

+ Peak=108 Mean=16.6 +

+

+ (GSL Peak=690 Mean=151) +

+

+ (Cephes Peak=1545 + Mean=218) +

+ 		+

+ Peak=4x103 Mean=203 +

+

+ (GSL Peak~3x105 Mean~2x104) +

+

+ (Cephes Peak~5x105 Mean~2x104) +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=21.9 Mean=3.1 +

+ 		+

+ Peak=270.7 Mean=26.8 +

+ 		+

+ Peak~5x104 Mean=3x103 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=15.4 Mean=3.0 +

+ 		+

+ Peak=112.9 Mean=14.3 +

+ 		+

+ Peak~5x104 Mean=3x103 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=20.9 Mean=2.6 +

+ 		+

+ Peak=88.1 Mean=14.3 +

+ 		+

+ Peak~2x104 Mean=1x103 +

+
+
+
+

Table 19. Errors In the Function ibetac(a,b,x)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0 < a,b < 10 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 0 < a,b < 100 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 1x10-5 < a,b < 1x105 +

+

+ and +

+

+ 0 < x < 1 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=13.9 Mean=2.0 +

+ 		+

+ Peak=56.2 Mean=14 +

+ 		+

+ Peak=3x103 Mean=159 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=21.1 Mean=3.6 +

+ 		+

+ Peak=221.7 Mean=25.8 +

+ 		+

+ Peak~9x104 Mean=3x103 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=10.6 Mean=2.2 +

+ 		+

+ Peak=73.9 Mean=11.9 +

+ 		+

+ Peak~9x104 Mean=3x103 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=9.9 Mean=2.6 +

+ 		+

+ Peak=117.7 Mean=15.1 +

+ 		+

+ Peak~3x104 Mean=1x103 +

+
+
+
+

Table 20. Errors In the Function beta(a, b, x)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0 < a,b < 10 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 0 < a,b < 100 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 1x10-5 < a,b < 1x105 +

+

+ and +

+

+ 0 < x < 1 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=39 Mean=2.9 +

+ 		+

+ Peak=91 Mean=12.7 +

+ 		+

+ Peak=635 Mean=25 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=26 Mean=3.6 +

+ 		+

+ Peak=180.7 Mean=30.1 +

+ 		+

+ Peak~7x104 Mean=3x103 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=13 Mean=2.4 +

+ 		+

+ Peak=67.1 Mean=13.4 +

+ 		+

+ Peak~7x104 Mean=3x103 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=27.3 Mean=3.6 +

+ 		+

+ Peak=49.8 Mean=9.1 +

+ 		+

+ Peak~6x104 Mean=3x103 +

+
+
+
+

Table 21. Errors In the Function betac(a,b,x)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0 < a,b < 10 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 0 < a,b < 100 +

+

+ and +

+

+ 0 < x < 1 +

+ 		+

+ 1x10-5 < a,b < 1x105 +

+

+ and +

+

+ 0 < x < 1 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=12.0 Mean=2.4 +

+ 		+

+ Peak=91 Mean=15 +

+ 		+

+ Peak=4x103 Mean=113 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=19.8 Mean=3.8 +

+ 		+

+ Peak=295.1 Mean=33.9 +

+ 		+

+ Peak~1x105 Mean=5x103 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=11.2 Mean=2.4 +

+ 		+

+ Peak=63.5 Mean=13.6 +

+ 		+

+ Peak~1x105 Mean=5x103 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=15.6 Mean=3.5 +

+ 		+

+ Peak=39.8 Mean=8.9 +

+ 		+

+ Peak~9x104 Mean=5x103 +

+
+
+
+ + Testing +
+

+ There are two sets of tests: spot tests compare values taken from Mathworld's + online function evaluator with this implementation: they provide + a basic "sanity check" for the implementation, with one spot-test + in each implementation-domain (see implementation notes below). +

+

+ Accuracy tests use data generated at very high precision (with NTL + RR class set at 1000-bit precision), using the "textbook" + continued fraction representation (refer to the first continued fraction + in the implementation discussion below). Note that this continued fraction + is not used in the implementation, and therefore we + have test data that is fully independent of the code. +

+
+ + Implementation +
+

+ This implementation is closely based upon "Algorithm + 708; Significant digit computation of the incomplete beta function ratios", + DiDonato and Morris, ACM, 1992. +

+

+ All four of these functions share a common implementation: this is passed + both x and y, and can return either p or q where these are related by: +

+

+ +

+

+ so at any point we can swap a for b, x for y and p for q if this results + in a more favourable position. Generally such swaps are performed so that + we always compute a value less than 0.9: when required this can then be + subtracted from 1 without undue cancellation error. +

+

+ The following continued fraction representation is found in many textbooks + but is not used in this implementation - it's both slower and less accurate + than the alternatives - however it is used to generate test data: +

+

+ +

+

+ The following continued fraction is due to Didonato + and Morris, and is used in this implementation when a and b are + both greater than 1: +

+

+ +

+

+ For smallish b and x then a series representation can be used: +

+

+ +

+

+ When b << a then the transition from 0 to 1 occurs very close to + x = 1 and some care has to be taken over the method of computation, in + that case the following series representation is used: +

+

+ +

+

+ Where Q(a,x) is an incomplete + gamma function. Note that this method relies on keeping a table + of all the pn previously computed, which does limit the precision of the + method, depending upon the size of the table used. +

+

+ When a and b are both small integers, + then we can relate the incomplete beta to the binomial distribution and + use the following finite sum: +

+

+ +

+

+ Finally we can sidestep difficult areas, or move to an area with a more + efficient means of computation, by using the duplication formulae: +

+

+ +

+

+ +

+

+ The domains of a, b and x for which the various methods are used are identical + to those described in the Didonato + and Morris TOMS 708 paper. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_inv_function.html b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_inv_function.html new file mode 100644 index 000000000..3cca643e4 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_inv_function.html @@ -0,0 +1,577 @@ + + + +The Incomplete Beta Function Inverses + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/beta.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, const Policy&);
+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, T4* py);
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, const Policy&);
+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, T4* py);
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_inva(T1 b, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_inva(T1 b, T2 x, T3 p, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_inva(T1 b, T2 x, T3 q);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_inva(T1 b, T2 x, T3 q, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_invb(T1 a, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_invb(T1 a, T2 x, T3 p, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_invb(T1 a, T2 x, T3 q);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_invb(T1 a, T2 x, T3 q, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ There are six incomplete + beta function inverses which allow you solve for any of the three + parameters to the incomplete beta, starting from either the result of the + incomplete beta (p) or its complement (q). +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + + + +
[Tip]Tip
+

+ When people normally talk about the inverse of the incomplete beta function, + they are talking about inverting on parameter x. + These are implemented here as ibeta_inv and ibeta_inv, and are by far + the most efficient of the inverses presented here. +

+

+ The inverses on the a and b + parameters find use in some statistical applications, but have to be + computed by rather brute force numerical techniques and are consequently + several times slower. These are implemented here as ibeta_inva and ibeta_invb, + and complement versions ibetac_inva and ibetac_invb. +

+
+

+ The return type of these functions is computed using the result + type calculation rules when called with arguments T1...TN + of different types. +

+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, const Policy&);
+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, T4* py);
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy&);
+
+

+ Returns a value x such that: p + = ibeta(a, + b, + x); + and sets *py + = 1 - x when + the py parameter is provided + and is non-null. Note that internally this function computes whichever + is the smaller of x and + 1-x, and therefore the value assigned to + *py + is free from cancellation errors. That means that even if the function + returns 1, the value stored + in *py + may be non-zero, albeit very small. +

+

+ Requires: a,b > 0 and 0 <= p <= + 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, const Policy&);
+
+template <class T1, class T2, class T3, class T4>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, T4* py);
+
+template <class T1, class T2, class T3, class T4, class Policy>
+calculated-result-type ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy&);
+
+

+ Returns a value x such that: q + = ibetac(a, + b, + x); + and sets *py + = 1 - x when + the py parameter is provided + and is non-null. Note that internally this function computes whichever + is the smaller of x and + 1-x, and therefore the value assigned to + *py + is free from cancellation errors. That means that even if the function + returns 1, the value stored + in *py + may be non-zero, albeit very small. +

+

+ Requires: a,b > 0 and 0 <= q <= + 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_inva(T1 b, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_inva(T1 b, T2 x, T3 p, const Policy&);
+
+

+ Returns a value a such that: p + = ibeta(a, + b, + x); +

+

+ Requires: b > 0, 0 < x < 1 + and 0 <= p <= 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_inva(T1 b, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_inva(T1 b, T2 x, T3 p, const Policy&);
+
+

+ Returns a value a such that: q + = ibetac(a, + b, + x); +

+

+ Requires: b > 0, 0 < x < 1 + and 0 <= q <= 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibeta_invb(T1 b, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibeta_invb(T1 b, T2 x, T3 p, const Policy&);
+
+

+ Returns a value b such that: p + = ibeta(a, + b, + x); +

+

+ Requires: a > 0, 0 < x < 1 + and 0 <= p <= 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2, class T3>
+calculated-result-type ibetac_invb(T1 b, T2 x, T3 p);
+
+template <class T1, class T2, class T3, class Policy>
+calculated-result-type ibetac_invb(T1 b, T2 x, T3 p, const Policy&);
+
+

+ Returns a value b such that: q + = ibetac(a, + b, + x); +

+

+ Requires: a > 0, 0 < x < 1 + and 0 <= q <= 1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Accuracy +
+

+ The accuracy of these functions should closely follow that of the regular + forward incomplete beta functions. However, note that in some parts of + their domain, these functions can be extremely sensitive to changes in + input, particularly when the argument p (or it's complement + q) is very close to 0 + or 1. +

+
+ + Testing +
+

+ There are two sets of tests: +

+
    +
  • + Basic sanity checks attempt to "round-trip" from a, + b and x to p or + q and back again. These tests have quite generous + tolerances: in general both the incomplete beta and its inverses change + so rapidly, that round tripping to more than a couple of significant + digits isn't possible. This is especially true when p + or q is very near one: in this case there isn't + enough "information content" in the input to the inverse function + to get back where you started. +
    • +
  • + Accuracy checks using high precision test values. These measure the accuracy + of the result, given exact input values. +
    • +
+
+ + Implementation + of ibeta_inv and ibetac_inv +
+

+ These two functions share a common implementation. +

+

+ First an initial approximation to x is computed then the last few bits + are cleaned up using Halley + iteration. The iteration limit is set to 12 of the number + of bits in T, which by experiment is sufficient to ensure that the inverses + are at least as accurate as the normal incomplete beta functions. Up to + 5 iterations may be required in extreme cases, although normally only one + or two are required. Further, the number of iterations required decreases + with increasing /a and b (which generally + form the more important use cases). +

+

+ The initial guesses used for iteration are obtained as follows: +

+

+ Firstly recall that: +

+

+ +

+

+ We may wish to start from either p or q, and to calculate either x or y. + In addition at any stage we can exchange a for b, p for q, and x for y + if it results in a more manageable problem. +

+

+ For a+b >= 5 the initial guess is computed using the + methods described in: +

+

+ Asymptotic Inversion of the Incomplete Beta Function, by N. M. Temme. + Journal of Computational and Applied Mathematics 41 (1992) 145-157. +

+

+ The nearly symmetrical case (section 2 of the paper) is used for +

+

+ +

+

+ and involves solving the inverse error function first. The method is accurate + to at least 2 decimal digits when a = 5 rising to at + least 8 digits when a = 105. +

+

+ The general error function case (section 3 of the paper) is used for +

+

+ +

+

+ and again expresses the inverse incomplete beta in terms of the inverse + of the error function. The method is accurate to at least 2 decimal digits + when a+b = 5 rising to 11 digits when a+b = + 105. However, when the result is expected to be very small, and + when a+b is also small, then its accuracy tails off, in this case when + p1/a < 0.0025 then it is better to use the following as an initial estimate: +

+

+ +

+

+ Finally the for all other cases where a+b > + 5 the method of section 4 of the + paper is used. This expresses the inverse incomplete beta in terms of the + inverse of the incomplete gamma function, and is therefore significantly + more expensive to compute than the other cases. However the method is accurate + to at least 3 decimal digits when a = 5 rising to at + least 10 digits when a = 105. This method is limited + to a > b, and therefore we need to perform an exchange a for b, p for + q and x for y when this is not the case. In addition when p is close to + 1 the method is inaccurate should we actually want y rather than x as output. + Therefore when q is small (q1/p < 10-3) we use: +

+

+ +

+

+ which is both cheaper to compute than the full method, and a more accurate + estimate on q. +

+

+ When a and b are both small there is a distinct lack of information in + the literature on how to proceed. I am extremely grateful to Prof Nico + Temme who provided the following information with a great deal of patience + and explanation on his part. Any errors that follow are entirely my own, + and not Prof Temme's. +

+

+ When a and b are both less than 1, then there is a point of inflection + in the incomplete beta at point xs + = (1 - a) / (2 - a + - b). Therefore if p > Ix(a,b) + we swap a for b, p for q and x for y, so that now we always look for a + point x below the point of inflection xs, + and on a convex curve. An initial estimate for x is made with: +

+

+ +

+

+ which is provably below the true value for x: Newton + iteration will therefore smoothly converge on x without problems + caused by overshooting etc. +

+

+ When a and b are both greater than 1, but a+b is too small to use the other + methods mentioned above, we proceed as follows. Observe that there is a + point of inflection in the incomplete beta at xs + = (1 - a) / (2 - a + - b). Therefore if p > Ix(a,b) + we swap a for b, p for q and x for y, so that now we always look for a + point x below the point of inflection xs, + and on a concave curve. An initial estimate for x is made with: +

+

+ +

+

+ which can be improved somewhat to: +

+

+ +

+

+ when b and x are both small (I've used b < a and x < 0.2). This actually + under-estimates x, which drops us on the wrong side of x for Newton iteration + to converge monotonically. However, use of higher derivatives and Halley + iteration keeps everything under control. +

+

+ The final case to be considered if when one of a and b is less than or + equal to 1, and the other greater that 1. Here, if b < a we swap a for + b, p for q and x for y. Now the curve of the incomplete beta is convex + with no points of inflection in [0,1]. For small p, x can be estimated + using +

+

+ +

+

+ which under-estimates x, and drops us on the right side of the true value + for Newton iteration to converge monotonically. However, when p is large + this can quite badly underestimate x. This is especially an issue when + we really want to find y, in which case this method can be an arbitrary + number of order of magnitudes out, leading to very poor convergence during + iteration. +

+

+ Things can be improved by considering the incomplete beta as a distorted + quarter circle, and estimating y from: +

+

+ +

+

+ This doesn't guarantee that we will drop in on the right side of x for + monotonic convergence, but it does get us close enough that Halley iteration + rapidly converges on the true value. +

+
+ + Implementation + of inverses on the a and b parameters +
+

+ These four functions share a common implementation. +

+

+ First an initial approximation is computed for a or + b: where possible this uses a Cornish-Fisher expansion + for the negative binomial distribution to get within around 1 of the result. + However, when a or b are very + small the Cornish Fisher expansion is not usable, in this case the initial + approximation is chosen so that Ix(a, b) is near the middle of the range + [0,1]. +

+

+ This initial guess is then used as a starting value for a generic root + finding algorithm. The algorithm converges rapidly on the root once it + has been bracketed, but bracketing the root may take several iterations. + A better initial approximation for a or b + would improve these functions quite substantially: currently 10-20 incomplete + beta function invocations are required to find the root. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_erf.html b/doc/sf_and_dist/html/math_toolkit/special/sf_erf.html new file mode 100644 index 000000000..3e670b6e1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_erf.html @@ -0,0 +1,50 @@ + + + +Error Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Error + Functions
+
Error Function + Inverses
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_function.html b/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_function.html new file mode 100644 index 000000000..3b74e3568 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_function.html @@ -0,0 +1,642 @@ + + + +Error Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/erf.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type erf(T z);
+
+template <class T, class Policy>
+calculated-result-type erf(T z, const Policy&);
+
+template <class T>
+calculated-result-type erfc(T z);
+
+template <class T, class Policy>
+calculated-result-type erfc(T z, const Policy&);
+
+}} // namespaces
+
+

+ The return type of these functions is computed using the result + type calculation rules: the return type is double if T is an integer type, and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Description +
+
+template <class T>
+calculated-result-type erf(T z);
+
+template <class T, class Policy>
+calculated-result-type erf(T z, const Policy&);
+
+

+ Returns the error + function erf + of z: +

+

+ +

+

+ erf1 +

+
+template <class T>
+calculated-result-type erfc(T z);
+
+template <class T, class Policy>
+calculated-result-type erfc(T z, const Policy&);
+
+

+ Returns the complement of the error + function of z: +

+

+ +

+

+ erf2 +

+
+ + Accuracy +
+

+ The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types, along with comparisons + to the GSL-1.9, + GNU C Lib, HP-UX C Library + and Cephes libraries. + Unless otherwise specified any floating point type that is narrower than + the one shown will have effectively zero error. +

+
+

Table 22. Errors In the Function erf(z)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ z < 0.5 +

+ 		+

+ 0.5 < z < 8 +

+ 		+

+ z > 8 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GSL Peak=2.0 Mean=0.3 +

+

+ Cephes Peak=1.1 + Mean=0.7 +

+ 		+

+ Peak=0.9 Mean=0.09 +

+

+ GSL Peak=2.3 Mean=0.3 +

+

+ Cephes Peak=1.3 + Mean=0.2 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GSL Peak=0 Mean=0 +

+

+ Cephes Peak=0 + Mean=0 +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=0.7 Mean=0.07 +

+

+ GNU C Lib + Peak=0.9 Mean=0.2 +

+ 		+

+ Peak=0.9 Mean=0.2 +

+

+ GNU C Lib + Peak=0.9 Mean=0.07 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=0.7 Mean=0.07 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+ 		+

+ Peak=0.9 Mean=0.1 +

+

+ GNU C Lib + Peak=0.5 Mean=0.03 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=0.8 Mean=0.1 +

+

+ HP-UX C + Library Lib Peak=0.9 Mean=0.2 +

+ 		+

+ Peak=0.9 Mean=0.1 +

+

+ HP-UX C + Library Lib Peak=0.5 Mean=0.02 +

+ 		+

+ Peak=0 Mean=0 +

+

+ HP-UX C + Library Lib Peak=0 Mean=0 +

+
+
+
+

Table 23. Errors In the Function erfc(z)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ z < 0.5 +

+ 		+

+ 0.5 < z < 8 +

+ 		+

+ z > 8 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=0.7 Mean=0.06 +

+

+ GSL Peak=1.0 Mean=0.4 +

+

+ Cephes Peak=0.7 + Mean=0.06 +

+ 		+

+ Peak=0.99 Mean=0.3 +

+

+ GSL Peak=2.6 Mean=0.6 +

+

+ Cephes Peak=3.6 + Mean=0.7 +

+ 		+

+ Peak=1.0 Mean=0.2 +

+

+ GSL Peak=3.9 Mean=0.4 +

+

+ Cephes Peak=2.7 + Mean=0.4 +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+ 		+

+ Peak=1.4 Mean=0.3 +

+

+ GNU C Lib + Peak=1.3 Mean=0.3 +

+ 		+

+ Peak=1.6 Mean=0.4 +

+

+ GNU C Lib + Peak=1.3 Mean=0.4 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=0 Mean=0 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+ 		+

+ Peak=1.4 Mean=0.3 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+ 		+

+ Peak=1.5 Mean=0.4 +

+

+ GNU C Lib + Peak=0 Mean=0 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=0 Mean=0 +

+

+ HP-UX C + Library Peak=0 Mean=0 +

+ 		+

+ Peak=1.5 Mean=0.3 +

+

+ HP-UX C + Library Peak=0.9 Mean=0.08 +

+ 		+

+ Peak=1.6 Mean=0.4 +

+

+ HP-UX C + Library Peak=0.9 Mean=0.1 +

+
+
+
+ + Testing +
+

+ The tests for these functions come in two parts: basic sanity checks use + spot values calculated using Mathworld's + online evaluator, while accuracy checks use high-precision test + values calculated at 1000-bit precision with NTL::RR + and this implementation. Note that the generic and type-specific versions + of these functions use differing implementations internally, so this gives + us reasonably independent test data. Using our test data to test other + "known good" implementations also provides an additional sanity + check. +

+
+ + Implementation +
+

+ All versions of these functions first use the usual reflection formulas + to make their arguments positive: +

+
+erf(-z) = 1 - erf(z);
+
+erfc(-z) = 2 - erfc(z);  // preferred when -z < -0.5
+
+erfc(-z) = 1 + erf(z);   // preferred when -0.5 <= -z < 0
+
+

+ The generic versions of these functions are implemented in terms of the + incomplete gamma function. +

+

+ When the significand (mantissa) size is recognised (currently for 53, 64 + and 113-bit reals, plus single-precision 24-bit handled via promotion to + double) then a series of rational approximations devised + by JM are used. +

+

+ For z <= + 0.5 then a rational approximation + to erf is used, based on the observation that: +

+
+erf(z)/z ~ 1.12....
+
+

+ Therefore erf is calculated using: +

+
+erf(z) = z * (1.125F + R(z));
+
+

+ where the rational approximation R(z) is optimised for absolute error: + as long as its absolute error is small enough compared to 1.125, then any + round-off error incurred during the computation of R(z) will effectively + disappear from the result. As a result the error for erf and erfc in this + region is very low: the last bit is incorrect in only a very small number + of cases. +

+

+ For z > + 0.5 we observe that over a small + interval [a, b) then: +

+
+erfc(z) * exp(z*z) * z ~ c
+
+

+ for some constant c. +

+

+ Therefore for z > + 0.5 we calculate erfc using: +

+
+erfc(z) = exp(-z*z) * (c + R(z)) / z;
+
+

+ Again R(z) is optimised for absolute error, and the constant c is the average of erfc(z) + * exp(z*z) * + z taken at the endpoints of the + range. Once again, as long as the absolute error in R(z) is small compared + to c then c + R(z) will be correctly rounded, and the error + in the result will depend only on the accuracy of the exp function. In + practice, in all but a very small number of cases, the error is confined + to the last bit of the result. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_inv.html b/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_inv.html new file mode 100644 index 000000000..b333b9c87 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_erf/error_inv.html @@ -0,0 +1,220 @@ + + + +Error Function Inverses + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/erf.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type erf_inv(T p);
+
+template <class T, class Policy>
+calculated-result-type erf_inv(T p, const Policy&);
+
+template <class T>
+calculated-result-type erfc_inv(T p);
+
+template <class T, class Policy>
+calculated-result-type erfc_inv(T p, const Policy&);
+
+}} // namespaces
+
+

+ The return type of these functions is computed using the result + type calculation rules: the return type is double if T is an integer type, and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Description +
+
+template <class T>
+calculated-result-type erf_inv(T z);
+
+template <class T, class Policy>
+calculated-result-type erf_inv(T z, const Policy&);
+
+

+ Returns the inverse + error function of z, that is a value x such that: +

+
+p = erf(x);
+
+

+ erf_inv +

+
+template <class T>
+calculated-result-type erfc_inv(T z);
+
+template <class T, class Policy>
+calculated-result-type erfc_inv(T z, const Policy&);
+
+

+ Returns the inverse of the complement of the error function of z, that + is a value x such that: +

+
+p = erfc(x);
+
+

+ erfc_inv +

+
+ + Accuracy +
+

+ For types up to and including 80-bit long doubles the approximations used + are accurate to less than ~ 2 epsilon. For higher precision types these + functions have the same accuracy as the forward + error functions. +

+
+ + Testing +
+

+ There are two sets of tests: +

+
    +
  • + Basic sanity checks attempt to "round-trip" from x + to p and back again. These tests have quite generous + tolerances: in general both the error functions and their inverses change + so rapidly in some places that round tripping to more than a couple of + significant digits isn't possible. This is especially true when p + is very near one: in this case there isn't enough "information content" + in the input to the inverse function to get back where you started. +
    • +
  • + Accuracy checks using high-precision test values. These measure the accuracy + of the result, given exact input values. +
    • +
+
+ + Implementation +
+

+ These functions use a rational approximation devised + by JM to calculate an initial approximation to the result that is + accurate to ~10-19, then only if that has insufficient accuracy compared + to the epsilon for T, do we clean up the result using Halley + iteration. +

+

+ Constructing rational approximations to the erf/erfc functions is actually + surprisingly hard, especially at high precision. For this reason no attempt + has been made to achieve 10-34 accuracy suitable for use with 128-bit reals. +

+

+ In the following discussion, p is the value passed + to erf_inv, and q is the value passed to erfc_inv, + so that p = 1 - q and q = 1 - p + and in both cases we want to solve for the same result x. +

+

+ For p < 0.5 the inverse erf function is reasonably + smooth and the approximation: +

+
+x = p(p + 10)(Y + R(p))
+
+

+ Gives a good result for a constant Y, and R(p) optimised for low absolute + error compared to |Y|. +

+

+ For q < 0.5 things get trickier, over the interval 0.5 > + q > 0.25 the following approximation works well: +

+
+x = sqrt(-2log(q)) / (Y + R(q))
+
+

+ While for q < 0.25, let +

+
+z = sqrt(-log(q))
+
+

+ Then the result is given by: +

+
+x = z(Y + R(z - B))
+
+

+ As before Y is a constant and the rational function R is optimised for + low absolute error compared to |Y|. B is also a constant: it is the smallest + value of z for which each approximation is valid. + There are several approximations of this form each of which reaches a little + further into the tail of the erfc function (at long + double precision the extended exponent + range compared to double means + that the tail goes on for a very long way indeed). +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma.html new file mode 100644 index 000000000..760b3a1f6 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma.html @@ -0,0 +1,57 @@ + + + +Gamma Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Gamma
+
Log Gamma
+
Digamma
+
Ratios + of Gamma Functions
+
Incomplete Gamma + Functions
+
Incomplete + Gamma Function Inverses
+
Derivative + of the Incomplete Gamma Function
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/digamma.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/digamma.html new file mode 100644 index 000000000..f3a614dc7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/digamma.html @@ -0,0 +1,400 @@ + + + +Digamma + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/digamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type digamma(T z);
+
+template <class T, class Policy>
+calculated-result-type digamma(T z, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ Returns the digamma or psi function of x. Digamma + is defined as the logarithmic derivative of the gamma function: +

+

+ +

+

+ digamma +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ There is no fully generic version of this function: all the implementations + are tuned to specific accuracy levels, the most precise of which delivers + 34-digits of precision. +

+

+ The return type of this function is computed using the result + type calculation rules: the result is of type double when T is an integer type, and type + T otherwise. +

+
+ + Accuracy +
+

+ The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types. Unless otherwise specified + any floating point type that is narrower than the one shown will have + effectively zero error. +

+
++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Random Positive Values +

+ 		+

+ Values Near The Positive Root +

+ 		+

+ Values Near Zero +

+ 		+

+ Negative Values +

+
+

+ 53 +

+ 		+

+ Win32 Visual C++ 8 +

+ 		+

+ Peak=0.98 Mean=0.36 +

+ 		+

+ Peak=0.99 Mean=0.5 +

+ 		+

+ Peak=0.95 Mean=0.5 +

+ 		+

+ Peak=214 Mean=16 +

+
+

+ 64 +

+ 		+

+ Linux IA32 / GCC +

+ 		+

+ Peak=1.4 Mean=0.4 +

+ 		+

+ Peak=1.3 Mean=0.45 +

+ 		+

+ Peak=0.98 Mean=0.35 +

+ 		+

+ Peak=180 Mean=13 +

+
+

+ 64 +

+ 		+

+ Linux IA64 / GCC +

+ 		+

+ Peak=0.92 Mean=0.4 +

+ 		+

+ Peak=1.3 Mean=0.45 +

+ 		+

+ Peak=0.98 Mean=0.4 +

+ 		+

+ Peak=180 Mean=13 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=0.9 Mean=0.4 +

+ 		+

+ Peak=1.1 Mean=0.5 +

+ 		+

+ Peak=0.99 Mean=0.4 +

+ 		+

+ Peak=64 Mean=6 +

+
+

+ As shown above, error rates for positive arguments are generally very low. + For negative arguments there are an infinite number of irrational roots: + relative errors very close to these can be arbitrarily large, although + absolute error will remain very low. +

+
+ + Testing +
+

+ There are two sets of tests: spot values are computed using the online + calculator at functions.wolfram.com, while random test values are generated + using the high-precision reference implementation (a differentiated Lanczos approximation + see below). +

+
+ + Implementation +
+

+ The implementation is divided up into the following domains: +

+

+ For Negative arguments the reflection formula: +

+
+digamma(1-x) = digamma(x) + pi/tan(pi*x);
+
+

+ is used to make x positive. +

+

+ For arguments in the range [0,1] the recurrence relation: +

+
+digamma(x) = digamma(x+1) - 1/x
+
+

+ is used to shift the evaluation to [1,2]. +

+

+ For arguments in the range [1,2] a rational approximation devised + by JM is used (see below). +

+

+ For arguments in the range [2,BIG] the recurrence relation: +

+
+digamma(x+1) = digamma(x) + 1/x;
+
+

+ is used to shift the evaluation to the range [1,2]. +

+

+ For arguments > BIG the asymptotic expansion: +

+

+ +

+

+ can be used. However, this expansion is divergent after a few terms: exactly + how many terms depends on the size of x. Therefore + the value of BIG must be chosen so that the series + can be truncated at a term that is too small to have any effect on the + result when evaluated at BIG. Choosing BIG=10 for + up to 80-bit reals, and BIG=20 for 128-bit reals allows the series to truncated + after a suitably small number of terms and evaluated as a polynomial in + 1/(x*x). +

+

+ The rational approximation devised + by JM in the range [1,2] is derived as follows. +

+

+ First a high precision approximation to digamma was constructed using a + 60-term differentiated Lanczos + approximation, the form used is: +

+

+ +

+

+ Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos + sum, and P'(x) and Q'(x) are their first derivatives. The Lanzos part of + this approximation has a theoretical precision of ~100 decimal digits. + However, cancellation in the above sum will reduce that to around 99-(1/y) + digits if y is the result. This approximation was + used to calculate the positive root of digamma, and was found to agree + with the value used by Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) + by Cody, Strecok and Thacher) and with the value used by Morris to 35 digits + (See TOMS Algorithm 708). +

+

+ Likewise a few spot tests agreed with values calculated using functions.wolfram.com + to >40 digits. That's sufficiently precise to insure that the approximation + below is accurate to double precision. Achieving 128-bit long double precision + requires that the location of the root is known to ~70 digits, and it's + not clear whether the value calculated by this method meets that requirement: + the difficulty lies in independently verifying the value obtained. +

+

+ The rational approximation devised + by JM was optimised for absolute error using the form: +

+
+digamma(x) = (x - X0)(Y + R(x - 1));
+
+

+ Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) + is the rational approximation. Note that since X0 is irrational, we need + twice as many digits in X0 as in x in order to avoid cancellation error + during the subtraction (this assumes that x is an + exact value, if it's not then all bets are off). That means that even when + x is the value of the root rounded to the nearest representable value, + the result of digamma(x) will not be zero. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_derivatives.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_derivatives.html new file mode 100644 index 000000000..50d215d59 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_derivatives.html @@ -0,0 +1,116 @@ + + + +Derivative of the Incomplete Gamma Function + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{ 
+
+template <class T1, class T2>
+calculated-result-type gamma_p_derivative(T1 a, T2 x);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p_derivative(T1 a, T2 x, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ This function find some uses in statistical distributions: it implements + the partial derivative with respect to x of the incomplete + gamma function. +

+

+ +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Note that the derivative of the function gamma_q + can be obtained by negating the result of this function. +

+

+ The return type of this function is computed using the result + type calculation rules when T1 and T2 are different types, + otherwise the return type is simply T1. +

+
+ + Accuracy +
+

+ Almost identical to the incomplete gamma function gamma_p: + refer to the documentation for that function for more information. +

+
+ + Implementation +
+

+ This function just expose some of the internals of the incomplete gamma + function gamma_p: + refer to the documentation for that function for more information. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_ratios.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_ratios.html new file mode 100644 index 000000000..638e45784 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/gamma_ratios.html @@ -0,0 +1,371 @@ + + + +Ratios of Gamma Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+calculated-result-type tgamma_ratio(T1 a, T2 b);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma_ratio(T1 a, T2 b, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type tgamma_delta_ratio(T1 a, T2 delta);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma_delta_ratio(T1 a, T2 delta, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+
+template <class T1, class T2> 
+calculated-result-type tgamma_ratio(T1 a, T2 b);
+
+template <class T1, class T2, class Policy> 
+calculated-result-type tgamma_ratio(T1 a, T2 b, const Policy&);
+
+

+ Returns the ratio of gamma functions: +

+

+ +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Internally this just calls tgamma_delta_ratio(a, + b-a). +

+
+template <class T1, class T2>
+calculated-result-type tgamma_delta_ratio(T1 a, T2 delta);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma_delta_ratio(T1 a, T2 delta, const Policy&);
+
+

+ Returns the ratio of gamma functions: +

+

+ +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Note that the result is calculated accurately even when delta + is small compared to a: indeed even if a+delta + ~ a. The function is typically used when a + is large and delta is very small. +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types, + otherwise the result type is simple T1. +

+
+ + Accuracy +
+

+ The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types. Unless otherwise specified + any floating point type that is narrower than the one shown will have + effectively zero error. +

+
+

Table 11. Errors In the Function tgamma_delta_ratio(a, delta)

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 20 < a < 80 +

+

+ and +

+

+ delta < 1 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=16.9 Mean=1.7 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=24 Mean=2.7 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=12.8 Mean=1.8 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=21.4 Mean=2.3 +

+
+
+
+

Table 12. Errors In the Function tgamma_ratio(a, + b)

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 6 < a,b < 50 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=34 Mean=9 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA32, gcc-3.4.4 +

+ 		+

+ Peak=91 Mean=23 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=35.6 Mean=9.3 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=43.9 Mean=13.2 +

+
+
+
+ + Testing +
+

+ Accuracy tests use data generated at very high precision (with NTL + RR class set at 1000-bit precision: about 300 decimal digits) and + a deliberately naive calculation of Γ(x)/Γ(y). +

+
+ + Implementation +
+

+ The implementation of these functions is very similar to that of beta, and is + based on combining similar power terms to improve accuracy and avoid spurious + overflow/underflow. +

+

+ In addition there are optimisations for the situation where delta + is a small integer: in which case this function is basically the reciprocal + of a rising factorial, or where both arguments are smallish integers: in + which case table lookup of factorials can be used to calculate the ratio. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html new file mode 100644 index 000000000..c61161737 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html @@ -0,0 +1,1062 @@ + + + +Incomplete Gamma Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+calculated-result-type gamma_p(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p(T1 a, T2 z, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type gamma_q(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q(T1 a, T2 z, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type tgamma_lower(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma_lower(T1 a, T2 z, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type tgamma(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma(T1 a, T2 z, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ There are four incomplete + gamma functions: two are normalised versions (also known as regularized + incomplete gamma functions) that return values in the range [0, 1], and + two are non-normalised and return values in the range [0, Γ(a)]. Users interested + in statistical applications should use the normalised + versions (gamma_p and gamma_q). +

+

+ All of these functions require a > 0 and z + >= 0, otherwise they return the result of domain_error. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types, + otherwise the return type is simply T1. +

+
+template <class T1, class T2>
+calculated-result-type gamma_p(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p(T1 a, T2 z, const Policy&);
+
+

+ Returns the normalised lower incomplete gamma function of a and z: +

+

+ +

+

+ This function changes rapidly from 0 to 1 around the point z == a: +

+

+ gamma_p +

+
+template <class T1, class T2>
+calculated-result-type gamma_q(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q(T1 a, T2 z, const Policy&);
+
+

+ Returns the normalised upper incomplete gamma function of a and z: +

+

+ +

+

+ This function changes rapidly from 1 to 0 around the point z == a: +

+

+ gamma_q +

+
+template <class T1, class T2>
+calculated-result-type tgamma_lower(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma_lower(T1 a, T2 z, const Policy&);
+
+

+ Returns the full (non-normalised) lower incomplete gamma function of a + and z: +

+

+ +

+
+template <class T1, class T2>
+calculated-result-type tgamma(T1 a, T2 z);
+
+template <class T1, class T2, class Policy>
+calculated-result-type tgamma(T1 a, T2 z, const Policy&);
+
+

+ Returns the full (non-normalised) upper incomplete gamma function of a + and z: +

+

+ +

+
+ + Accuracy +
+

+ The following tables give peak and mean relative errors in over various + domains of a and z, along with comparisons to the GSL-1.9 + and Cephes libraries. + Note that only results for the widest floating point type on the system + are given as narrower types have effectively + zero error. +

+

+ Note that errors grow as a grows larger. +

+

+ Note also that the higher error rates for the 80 and 128 bit long double + results are somewhat misleading: expected results that are zero at 64-bit + double precision may be non-zero - but exceptionally small - with the larger + exponent range of a long double. These results therefore reflect the more + extreme nature of the tests conducted for these types. +

+

+ All values are in units of epsilon. +

+
+

Table 13. Errors In the Function gamma_p(a,z)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0.5 < a < 100 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1x10-12 < a < 5x10-2 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1e-6 < a < 1.7x106 +

+

+ and +

+

+ 1 < z < 100*a +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=36 Mean=9.1 +

+

+ (GSL Peak=342 Mean=46) +

+

+ (Cephes Peak=491 + Mean=102) +

+ 		+

+ Peak=4.5 Mean=1.4 +

+

+ (GSL Peak=4.8 Mean=0.76) +

+

+ (Cephes Peak=21 + Mean=5.6) +

+ 		+

+ Peak=244 Mean=21 +

+

+ (GSL Peak=1022 Mean=1054) +

+

+ (Cephes Peak~8x106 Mean~7x104) +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=241 Mean=36 +

+ 		+

+ Peak=4.7 Mean=1.5 +

+ 		+

+ Peak~30,220 Mean=1929 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4 +

+ 		+

+ Peak=41 Mean=10 +

+ 		+

+ Peak=4.7 Mean=1.4 +

+ 		+

+ Peak~30,790 Mean=1864 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=40.2 Mean=10.2 +

+ 		+

+ Peak=5 Mean=1.6 +

+ 		+

+ Peak=5,476 Mean=440 +

+
+
+
+

Table 14. Errors In the Function gamma_q(a,z)

+
+++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0.5 < a < 100 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1x10-12 < a < 5x10-2 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1x10-6 < a < 1.7x106 +

+

+ and +

+

+ 1 < z < 100*a +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=28.3 Mean=7.2 +

+

+ (GSL Peak=201 Mean=13) +

+

+ (Cephes Peak=556 + Mean=97) +

+ 		+

+ Peak=4.8 Mean=1.6 +

+

+ (GSL Peak~1.3x1010 Mean=1x10+9) +

+

+ (Cephes Peak~3x1011 Mean=4x1010) +

+ 		+

+ Peak=469 Mean=33 +

+

+ (GSL Peak=27,050 Mean=2159) +

+

+ (Cephes Peak~8x106 Mean~7x105) +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=280 Mean=33 +

+ 		+

+ Peak=4.1 Mean=1.6 +

+ 		+

+ Peak=11,490 Mean=732 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4 +

+ 		+

+ Peak=32 Mean=9.4 +

+ 		+

+ Peak=4.7 Mean=1.5 +

+ 		+

+ Peak=6815 Mean=414 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=37 Mean=10 +

+ 		+

+ Peak=11.2 Mean=2.0 +

+ 		+

+ Peak=4,999 Mean=298 +

+
+
+
+

Table 15. Errors In the Function tgamma_lower(a,z)

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0.5 < a < 100 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1x10-12 < a < 5x10-2 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=5.5 Mean=1.4 +

+ 		+

+ Peak=3.6 Mean=0.78 +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=402 Mean=79 +

+ 		+

+ Peak=3.4 Mean=0.8 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4 +

+ 		+

+ Peak=6.8 Mean=1.4 +

+ 		+

+ Peak=3.4 Mean=0.78 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=6.1 Mean=1.8 +

+ 		+

+ Peak=3.7 Mean=0.89 +

+
+
+
+

Table 16. Errors In the Function tgamma(a,z)

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ 0.5 < a < 100 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+ 		+

+ 1x10-12 < a < 5x10-2 +

+

+ and +

+

+ 0.01*a < z < 100*a +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=5.9 Mean=1.5 +

+ 		+

+ Peak=1.8 Mean=0.6 +

+
+

+ 64 +

+ 		+

+ RedHat Linux IA32, gcc-3.3 +

+ 		+

+ Peak=596 Mean=116 +

+ 		+

+ Peak=3.2 Mean=0.84 +

+
+

+ 64 +

+ 		+

+ Redhat Linux IA64, gcc-3.4.4 +

+ 		+

+ Peak=40.2 Mean=2.5 +

+ 		+

+ Peak=3.2 Mean=0.8 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=364 Mean=17.6 +

+ 		+

+ Peak=12.7 Mean=1.8 +

+
+
+
+ + Testing +
+

+ There are two sets of tests: spot tests compare values taken from Mathworld's online evaluator + with this implementation to perform a basic "sanity check". Accuracy + tests use data generated at very high precision (using NTL's RR class set + at 1000-bit precision) using this implementation with a very high precision + 60-term Lanczos approximation, + and some but not all of the special case handling disabled. This is less + than satisfactory: an independent method should really be used, but apparently + a complete lack of such methods are available. We can't even use a deliberately + naive implementation without special case handling since Legendre's continued + fraction (see below) is unstable for small a and z. +

+
+ + Implementation +
+

+ These four functions share a common implementation since they are all related + via: +

+

+ 1) +

+

+ 2) +

+

+ 3) +

+

+ The lower incomplete gamma is computed from its series representation: +

+

+ 4) +

+

+ Or by subtraction of the upper integral from either Γ(a) or 1 when x + > a and x > 1.1. +

+

+ The upper integral is computed from Legendre's continued fraction representation: +

+

+ 5) +

+

+ When x > 1.1 or by subtraction of the lower integral + from either Γ(a) or 1 when x < a. +

+

+ For x < 1.1 computation of the upper integral is + more complex as the continued fraction representation is unstable in this + area. However there is another series representation for the lower integral: +

+

+ 6) +

+

+ That lends itself to calculation of the upper integral via rearrangement + to: +

+

+ 7) +

+

+ Refer to the documentation for powm1 + and tgamma1pm1 + for details of their implementation. Note however that the precision of + tgamma1pm1 + is capped to either around 35 digits, or to that of the Lanczos + approximation associated with type T - if there is one - whichever + of the two is the greater. That therefore imposes a similar limit on the + precision of this function in this region. +

+

+ For x < 1.1 the crossover point where the result + is ~0.5 no longer occurs for x ~ y. Using x + * 1.1 < a as the crossover criterion for 0.5 < + x <= 1.1 keeps the maximum value computed (whether it's the + upper or lower interval) to around 0.6. Likewise for x <= + 0.5 then using -0.4 / log(x) < a as + the crossover criterion keeps the maximum value computed to around 0.7 + (whether it's the upper or lower interval). +

+

+ There are two special cases used when a is an integer or half integer, + and the crossover conditions listed above indicate that we should compute + the upper integral Q. If a is an integer in the range 1 <= + a < 30 then the following finite sum is used: +

+

+ 9) +

+

+ While for half integers in the range 0.5 <= a < 30 + then the following finite sum is used: +

+

+ 10) +

+

+ These are both more stable and more efficient than the continued fraction + alternative. +

+

+ When the argument a is large, and x ~ a + then the series (4) and continued fraction (5) above are very slow to converge. + In this area an expansion due to Temme is used: +

+

+ 11) +

+

+ 12) +

+

+ 13) +

+

+ 14) +

+

+ The double sum is truncated to a fixed number of terms - to give a specific + target precision - and evaluated as a polynomial-of-polynomials. There + are versions for up to 128-bit long double precision: types requiring greater + precision than that do not use these expansions. The coefficients Ckn are + computed in advance using the recurrence relations given by Temme. The + zone where these expansions are used is +

+
+(a > 20) && (a < 200) && fabs(x-a)/a < 0.4
+
+

+ And: +

+
+(a > 200) && (fabs(x-a)/a < 4.5/sqrt(a))
+
+

+ The latter range is valid for all types up to 128-bit long doubles, and + is designed to ensure that the result is larger than 10-6, the first range + is used only for types up to 80-bit long doubles. These domains are narrower + than the ones recommended by either Temme or Didonato and Morris. However, + using a wider range results in large and inexact (i.e. computed) values + being passed to the exp + and erfc functions resulting + in significantly larger error rates. In other words there is a fine trade + off here between efficiency and error. The current limits should keep the + number of terms required by (4) and (5) to no more than ~20 at double precision. +

+

+ For the normalised incomplete gamma functions, calculation of the leading + power terms is central to the accuracy of the function. For smallish a + and x combining the power terms with the Lanczos + approximation gives the greatest accuracy: +

+

+ 15) +

+

+ In the event that this causes underflowoverflow then the exponent + can be reduced by a factor of /a and brought inside the power + term. +

+

+ When a and x are large, we end up with a very large exponent with a base + near one: this will not be computed accurately via the pow function, and + taking logs simply leads to cancellation errors. The worst of the errors + can be avoided by using: +

+

+ 16) +

+

+ when a-x is small and a and x are large. There is + still a subtraction and therefore some cancellation errors - but the terms + are small so the absolute error will be small - and it is absolute rather + than relative error that counts in the argument to the exp + function. Note that for sufficiently large a and x the errors will still + get you eventually, although this does delay the inevitable much longer + than other methods. Use of log(1+x)-x here is inspired + by Temme (see references below). +

+
+ + References +
+
    +
  • + N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, + Probability in the Engineering and Informational Sciences, 8, 1994. +
    • +
  • + N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions, + Siam J. Math Anal. Vol 10 No 4, July 1979, p757. +
    • +
  • + A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma + Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986, + p377. +
    • +
  • + W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's + Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, + n. 147, Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237. http://citeseer.ist.psu.edu/gautschi98incomplete.html +
    • +
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma_inv.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma_inv.html new file mode 100644 index 000000000..c998c98c4 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma_inv.html @@ -0,0 +1,270 @@ + + + +Incomplete Gamma Function Inverses + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+calculated-result-type gamma_q_inv(T1 a, T2 q);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q_inv(T1 a, T2 q, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type gamma_p_inv(T1 a, T2 p);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p_inv(T1 a, T2 p, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type gamma_q_inva(T1 x, T2 q);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q_inva(T1 x, T2 q, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type gamma_p_inva(T1 x, T2 p);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p_inva(T1 x, T2 p, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ There are four incomplete + gamma function inverses which either compute x + given a and p or q, + or else compute a given x and + either p or q. +

+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types, + otherwise the return type is simply T1. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + + + +
[Tip]Tip
+

+ When people normally talk about the inverse of the incomplete gamma function, + they are talking about inverting on parameter x. + These are implemented here as gamma_p_inv and gamma_q_inv, and are by + far the most efficient of the inverses presented here. +

+

+ The inverse on the a parameter finds use in some + statistical applications but has to be computed by rather brute force + numerical techniques and is consequently several times slower. These + are implemented here as gamma_p_inva and gamma_q_inva. +

+
+
+template <class T1, class T2>
+calculated-result-type gamma_q_inv(T1 a, T2 q);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q_inv(T1 a, T2 q, const Policy&);
+
+

+ Returns a value x such that: q + = gamma_q(a, + x); +

+

+ Requires: a > 0 and 1 >= p,q >= + 0. +

+
+template <class T1, class T2>
+calculated-result-type gamma_p_inv(T1 a, T2 p);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p_inv(T1 a, T2 p, const Policy&);
+
+

+ Returns a value x such that: p + = gamma_p(a, + x); +

+

+ Requires: a > 0 and 1 >= p,q >= + 0. +

+
+template <class T1, class T2>
+calculated-result-type gamma_q_inva(T1 x, T2 q);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_q_inva(T1 x, T2 q, const Policy&);
+
+

+ Returns a value a such that: q + = gamma_q(a, + x); +

+

+ Requires: x > 0 and 1 >= p,q >= + 0. +

+
+template <class T1, class T2>
+calculated-result-type gamma_p_inva(T1 x, T2 p);
+
+template <class T1, class T2, class Policy>
+calculated-result-type gamma_p_inva(T1 x, T2 p, const Policy&);
+
+

+ Returns a value a such that: p + = gamma_p(a, + x); +

+

+ Requires: x > 0 and 1 >= p,q >= + 0. +

+
+ + Accuracy +
+

+ The accuracy of these functions doesn't vary much by platform or by the + type T. Given that these functions are computed by iterative methods, they + are deliberately "detuned" so as not to be too accurate: it is + in any case impossible for these function to be more accurate than the + regular forward incomplete gamma functions. In practice, the accuracy of + these functions is very similar to that of gamma_p + and gamma_q + functions. +

+
+ + Testing +
+

+ There are two sets of tests: +

+
    +
  • + Basic sanity checks attempt to "round-trip" from a + and x to p or q + and back again. These tests have quite generous tolerances: in general + both the incomplete gamma, and its inverses, change so rapidly that round + tripping to more than a couple of significant digits isn't possible. + This is especially true when p or q + is very near one: in this case there isn't enough "information content" + in the input to the inverse function to get back where you started. +
    • +
  • + Accuracy checks using high precision test values. These measure the accuracy + of the result, given exact input values. +
    • +
+
+ + Implementation +
+

+ The functions gamma_p_inv and gamma_q_inv + share a common implementation. +

+

+ First an initial approximation is computed using the methodology described + in: +

+

+ A. + R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function + Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393. +

+

+ Finally, the last few bits are cleaned up using Halley iteration, the iteration + limit is set to 2/3 of the number of bits in T, which by experiment is + sufficient to ensure that the inverses are at least as accurate as the + normal incomplete gamma functions. In testing, no more than 3 iterations + are required to produce a result as accurate as the forward incomplete + gamma function, and in many cases only one iteration is required. +

+

+ The functions gamma_p_inva and gamma_q_inva also share a common implementation + but are handled separately from gamma_p_inv and gamma_q_inv. +

+

+ An initial approximation for a is computed very crudely + so that gamma_p(a, x) ~ 0.5, this value is then used + as a starting point for a generic derivative-free root finding algorithm. + As a consequence, these two functions are rather more expensive to compute + than the gamma_p_inv or gamma_q_inv functions. Even so, the root is usually + found in fewer than 10 iterations. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/lgamma.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/lgamma.html new file mode 100644 index 000000000..9efa34ae7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/lgamma.html @@ -0,0 +1,494 @@ + + + +Log Gamma + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type lgamma(T z);
+
+template <class T, class Policy>
+calculated-result-type lgamma(T z, const Policy&);
+
+template <class T>
+calculated-result-type lgamma(T z, int* sign);
+
+template <class T, class Policy>
+calculated-result-type lgamma(T z, int* sign, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+

+ The lgamma function + is defined by: +

+

+ +

+

+ The second form of the function takes a pointer to an integer, which if + non-null is set on output to the sign of tgamma(z). +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ lgamma +

+

+ There are effectively two versions of this function internally: a fully + generic version that is slow, but reasonably accurate, and a much more + efficient approximation that is used where the number of digits in the + significand of T correspond to a certain Lanczos + approximation. In practice, any built-in floating-point type you + will encounter has an appropriate Lanczos + approximation defined for it. It is also possible, given enough + machine time, to generate further Lanczos + approximation's using the program libs/math/tools/lanczos_generator.cpp. +

+

+ The return type of these functions is computed using the result + type calculation rules: the result is of type double if T is an integer type, or type + T otherwise. +

+
+ + Accuracy +
+

+ The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types, along with comparisons + to the GSL-1.9, + GNU C Lib, HP-UX C Library + and Cephes libraries. + Unless otherwise specified any floating point type that is narrower than + the one shown will have effectively zero error. +

+

+ Note that while the relative errors near the positive roots of lgamma are + very low, the lgamma function has an infinite number of irrational roots + for negative arguments: very close to these negative roots only a low absolute + error can be guaranteed. +

+
++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Factorials and Half factorials +

+ 		+

+ Values Near Zero +

+ 		+

+ Values Near 1 or 2 +

+ 		+

+ Values Near a Negative Pole +

+
+

+ 53 +

+ 		+

+ Win32 Visual C++ 8 +

+ 		+

+ Peak=0.88 Mean=0.14 +

+

+ (GSL=33) (Cephes=1.5) +

+ 		+

+ Peak=0.96 Mean=0.46 +

+

+ (GSL=5.2) (Cephes=1.1) +

+ 		+

+ Peak=0.86 Mean=0.46 +

+

+ (GSL=1168) (Cephes~500000) +

+ 		+

+ Peak=4.2 Mean=1.3 +

+

+ (GSL=25) (Cephes=1.6) +

+
+

+ 64 +

+ 		+

+ Linux IA32 / GCC +

+ 		+

+ Peak=1.9 Mean=0.43 +

+

+ (GNU C Lib + Peak=1.7 Mean=0.49) +

+ 		+

+ Peak=1.4 Mean=0.57 +

+

+ (GNU C Lib + Peak= 0.96 Mean=0.54) +

+ 		+

+ Peak=0.86 Mean=0.35 +

+

+ (GNU C Lib + Peak=0.74 Mean=0.26) +

+ 		+

+ Peak=6.0 Mean=1.8 +

+

+ (GNU C Lib + Peak=3.0 Mean=0.86) +

+
+

+ 64 +

+ 		+

+ Linux IA64 / GCC +

+ 		+

+ Peak=0.99 Mean=0.12 +

+

+ (GNU C Lib + Peak 0) +

+ 		+

+ Pek=1.2 Mean=0.6 +

+

+ (GNU C Lib + Peak 0) +

+ 		+

+ Peak=0.86 Mean=0.16 +

+

+ (GNU C Lib + Peak 0) +

+ 		+

+ Peak=2.3 Mean=0.69 +

+

+ (GNU C Lib + Peak 0) +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=0.96 Mean=0.13 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=0.99 Mean=0.53 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=0.9 Mean=0.4 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=3.0 Mean=0.9 +

+

+ (HP-UX + C Library Peak 0) +

+
+
+ + Testing +
+

+ The main tests for this function involve comparisons against the logs of + the factorials which can be independently calculated to very high accuracy. +

+

+ Random tests in key problem areas are also used. +

+
+ + Implementation +
+

+ The generic version of this function is implemented by combining the series + and continued fraction representations for the incomplete gamma function: +

+

+ +

+

+ where l is an arbitrary integration limit: choosing + l = max(10, a) seems to work fairly well. For negative + z the logarithm version of the reflection formula + is used: +

+

+ +

+

+ For types of known precision, the Lanczos + approximation is used, a traits class boost::math::lanczos::lanczos_traits + maps type T to an appropriate approximation. The logarithmic version of + the Lanczos approximation + is: +

+

+ +

+

+ Where Le,g is the Lanczos sum, scaled by eg. +

+

+ As before the reflection formula is used for z < 0. +

+

+ When z is very near 1 or 2, then the logarithmic version of the Lanczos + approximation suffers very badly from cancellation error: indeed + for values sufficiently close to 1 or 2, arbitrarily large relative errors + can be obtained (even though the absolute error is tiny). +

+

+ For types with up to 113 bits of precision (up to and including 128-bit + long doubles), root-preserving rational approximations devised + by JM are used over the intervals [1,2] and [2,3]. Over the interval + [2,3] the approximation form used is: +

+
+lgamma(z) = (z-2)(z+1)(Y + R(z-2));
+
+

+ Where Y is a constant, and R(z-2) is the rational approximation: optimised + so that it's absolute error is tiny compared to Y. In addition small values + of z greater than 3 can handled by argument reduction using the recurrence + relation: +

+
+lgamma(z+1) = log(z) + lgamma(z);
+
+

+ Over the interval [1,2] two approximations have to be used, one for small + z uses: +

+
+lgamma(z) = (z-1)(z-2)(Y + R(z-1));
+
+

+ Once again Y is a constant, and R(z-1) is optimised for low absolute error + compared to Y. For z > 1.5 the above form wouldn't converge to a minimax + solution but this similar form does: +

+
+lgamma(z) = (2-z)(1-z)(Y + R(2-z));
+
+

+ Finally for z < 1 the recurrence relation can be used to move to z > + 1: +

+
+lgamma(z) = lgamma(z+1) - log(z);
+
+

+ Note that while this involves a subtraction, it appears not to suffer from + cancellation error: as z decreases from 1 the -log(z) term + grows positive much more rapidly than the lgamma(z+1) term becomes + negative. So in this specific case, significant digits are preserved, rather + than cancelled. +

+

+ For other types which do have a Lanczos + approximation defined for them the current solution is as follows: + imagine we balance the two terms in the Lanczos + approximation by dividing the power term by its value at z + = 1, and then multiplying the Lanczos coefficients by the same + value. Now each term will take the value 1 at z = 1 + and we can rearrange the power terms in terms of log1p. Likewise if we + subtract 1 from the Lanczos sum part (algebraically, by subtracting the + value of each term at z = 1), we obtain a new summation + that can be also be fed into log1p. Crucially, all of the terms tend to + zero, as z -> 1: +

+

+ +

+

+ The Ck terms in the above are the same as in the Lanczos + approximation. +

+

+ A similar rearrangement can be performed at z = 2: +

+

+ +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/tgamma.html b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/tgamma.html new file mode 100644 index 000000000..0b3a0eb65 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/tgamma.html @@ -0,0 +1,483 @@ + + + +Gamma + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+

+ Gamma +

+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/gamma.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type tgamma(T z);
+
+template <class T, class Policy>
+calculated-result-type tgamma(T z, const Policy&);
+
+template <class T>
+calculated-result-type tgamma1pm1(T dz);
+
+template <class T, class Policy>
+calculated-result-type tgamma1pm1(T dz, const Policy&);
+
+}} // namespaces
+
+
+ + Description +
+
+template <class T>
+calculated-result-type tgamma(T z);
+
+template <class T, class Policy>
+calculated-result-type tgamma(T z, const Policy&);
+
+

+ Returns the "true gamma" (hence name tgamma) of value z: +

+

+ +

+

+ gamma +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ There are effectively two versions of the tgamma + function internally: a fully generic version that is slow, but reasonably + accurate, and a much more efficient approximation that is used where the + number of digits in the significand of T correspond to a certain Lanczos approximation. + In practice any built in floating point type you will encounter has an + appropriate Lanczos + approximation defined for it. It is also possible, given enough + machine time, to generate further Lanczos + approximation's using the program libs/math/tools/lanczos_generator.cpp. +

+

+ The return type of this function is computed using the result + type calculation rules: the result is double + when T is an integer type, and T otherwise. +

+
+template <class T>
+calculated-result-type tgamma1pm1(T dz);
+
+template <class T, class Policy>
+calculated-result-type tgamma1pm1(T dz, const Policy&);
+
+

+ Returns tgamma(dz + 1) - + 1. Internally the implementation + does not make use of the addition and subtraction implied by the definition, + leading to accurate results even for very small dz. + However, the implementation is capped to either 35 digit accuracy, or to + the precision of the Lanczos + approximation associated with type T, whichever is more accurate. +

+

+ The return type of this function is computed using the result + type calculation rules: the result is double + when T is an integer type, and T otherwise. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Accuracy +
+

+ The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types, along with comparisons + to the GSL-1.9, + GNU C Lib, HP-UX C Library + and Cephes libraries. + Unless otherwise specified any floating point type that is narrower than + the one shown will have effectively zero error. +

+
++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Factorials and Half factorials +

+ 		+

+ Values Near Zero +

+ 		+

+ Values Near 1 or 2 +

+ 		+

+ Values Near a Negative Pole +

+
+

+ 53 +

+ 		+

+ Win32 Visual C++ 8 +

+ 		+

+ Peak=1.9 Mean=0.7 +

+

+ (GSL=3.9) +

+

+ (Cephes=3.0) +

+ 		+

+ Peak=2.0 Mean=1.1 +

+

+ (GSL=4.5) +

+

+ (Cephes=1) +

+ 		+

+ Peak=2.0 Mean=1.1 +

+

+ (GSL=7.9) +

+

+ (Cephes=1.0) +

+ 		+

+ Peak=2.6 Mean=1.3 +

+

+ (GSL=2.5) +

+

+ (Cephes=2.7) +

+
+

+ 64 +

+ 		+

+ Linux IA32 / GCC +

+ 		+

+ Peak=300 Mean=49.5 +

+

+ (GNU C Lib + Peak=395 Mean=89) +

+ 		+

+ Peak=3.0 Mean=1.4 +

+

+ (GNU C Lib + Peak=11 Mean=3.3) +

+ 		+

+ Peak=5.0 Mean=1.8 +

+

+ (GNU C Lib + Peak=0.92 Mean=0.2) +

+ 		+

+ Peak=157 Mean=65 +

+

+ (GNU C Lib + Peak=205 Mean=108) +

+
+

+ 64 +

+ 		+

+ Linux IA64 / GCC +

+ 		+

+ GNU C Lib + Peak 2.8 Mean=0.9 +

+

+ (GNU C Lib + Peak 0.7) +

+ 		+

+ Peak=4.8 Mean=1.5 +

+

+ (GNU C Lib + Peak 0) +

+ 		+

+ Peak=4.8 Mean=1.5 +

+

+ (GNU C Lib + Peak 0) +

+ 		+

+ Peak=5.0 Mean=1.7 (GNU + C Lib Peak 0) +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=2.5 Mean=1.1 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=3.5 Mean=1.7 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=3.5 Mean=1.6 +

+

+ (HP-UX + C Library Peak 0) +

+ 		+

+ Peak=5.2 Mean=1.92 +

+

+ (HP-UX + C Library Peak 0) +

+
+
+ + Testing +
+

+ The gamma is relatively easy to test: factorials and half-integer factorials + can be calculated exactly by other means and compared with the gamma function. + In addition, some accuracy tests in known tricky areas were computed at + high precision using the generic version of this function. +

+

+ The function tgamma1pm1 + is tested against values calculated very naively using the formula tgamma(1+dz)-1 with + a lanczos approximation accurate to around 100 decimal digits. +

+
+ + Implementation +
+

+ The generic version of the tgamma + function is implemented by combining the series and continued fraction + representations for the incomplete gamma function: +

+

+ +

+

+ where l is an arbitrary integration limit: choosing + l = max(10, a) seems to work fairly well. +

+

+ For types of known precision the Lanczos + approximation is used, a traits class boost::math::lanczos::lanczos_traits + maps type T to an appropriate approximation. +

+

+ For z in the range -20 < z < 1 then recursion is used to shift to + z > 1 via: +

+

+ +

+

+ For very small z, this helps to preserve the identity: +

+

+ +

+

+ For z < -20 the reflection formula: +

+

+ +

+

+ is used. Particular care has to be taken to evaluate the z * sin([pi][space] * z) + part: a special routine is used to reduce z prior to multiplying by π to + ensure that the result in is the range [0, π/2]. Without this an excessive + amount of error occurs in this region (which is hard enough already, as + the rate of change near a negative pole is exceptionally + high). +

+

+ Finally if the argument is a small integer then table lookup of the factorial + is used. +

+

+ The function tgamma1pm1 + is implemented using rational approximations devised + by JM in the region -0.5 < dz < 2. These are the same approximations (and + internal routines) that are used for lgamma, + and so aren't detailed further here. The result of the approximation is + log(tgamma(dz+1)) which can fed into expm1 + to give the desired result. Outside the range -0.5 < dz < 2 then the naive formula tgamma1pm1(dz) + = tgamma(dz+1)-1 + can be used directly. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_poly.html b/doc/sf_and_dist/html/math_toolkit/special/sf_poly.html new file mode 100644 index 000000000..5a2812038 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_poly.html @@ -0,0 +1,52 @@ + + + +Polynomials + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Legendre (and + Associated) Polynomials
+
Laguerre (and + Associated) Polynomials
+
Hermite Polynomials
+
Spherical Harmonics
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_poly/hermite.html b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/hermite.html new file mode 100644 index 000000000..295636f28 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/hermite.html @@ -0,0 +1,294 @@ + + + +Hermite Polynomials + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/hermite.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type hermite(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type hermite(unsigned n, T x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);
+   
+}} // namespaces
+
+
+ + Description +
+

+ The return type of these functions is computed using the result + type calculation rules: note than when there is a single + template argument the result is the same type as that argument or double if the template argument is an integer + type. +

+
+template <class T>
+calculated-result-type hermite(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type hermite(unsigned n, T x, const Policy&);
+
+

+ Returns the value of the Hermite Polynomial of order n + at point x: +

+

+ +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ The following graph illustrates the behaviour of the first few Hermite + Polynomials: +

+

+ hermite +

+
+template <class T1, class T2, class T3>
+calculated-result-type hermite_next(unsigned n, T1 x, T2 Hn, T3 Hnm1);
+
+

+ Implements the three term recurrence relation for the Hermite polynomials, + this function can be used to create a sequence of values evaluated at the + same x, and for rising n. +

+

+ +

+

+ For example we could produce a vector of the first 10 polynomial values + using: +

+
+double x = 0.5;  // Abscissa value
+vector<double> v;
+v.push_back(hermite(0, x)).push_back(hermite(1, x));
+for(unsigned l = 1; l < 10; ++l)
+   v.push_back(hermite_next(l, x, v[l], v[l-1]));
+
+

+ Formally the arguments are: +

+
+

+
+
n
+

+ The degree n of the last polynomial calculated. +

+
x
+

+ The abscissa value +

+
Hn
+

+ The value of the polynomial evaluated at degree n. +

+
Hnm1
+

+ The value of the polynomial evaluated at degree n-1. +

+
+
+
+ + Accuracy +
+

+ The following table shows peak errors (in units of epsilon) for various + domains of input arguments. Note that only results for the widest floating + point type on the system are given as narrower types have effectively + zero error. +

+
+

Table 29. Peak Errors In the Hermite Polynomial

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=4.5 Mean=1.5 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA32, g++ 4.1 +

+ 		+

+ Peak=6 Mean=2 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=6 Mean=2 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=6 Mean=4 +

+
+
+

+ Note that the worst errors occur when the degree increases, values greater + than ~120 are very unlikely to produce sensible results, especially in + the associated polynomial case when the order is also large. Further the + relative errors are likely to grow arbitrarily large when the function + is very close to a root. +

+
+ + Testing +
+

+ A mixture of spot tests of values calculated using functions.wolfram.com, + and randomly generated test data are used: the test data was computed using + NTL::RR at 1000-bit + precision. +

+
+ + Implementation +
+

+ These functions are implemented using the stable three term recurrence + relations. These relations guarentee low absolute error but cannot guarentee + low relative error near one of the roots of the polynomials. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_poly/laguerre.html b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/laguerre.html new file mode 100644 index 000000000..5f97426b3 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/laguerre.html @@ -0,0 +1,473 @@ + + + +Laguerre (and Associated) Polynomials + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/laguerre.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type laguerre(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type laguerre(unsigned n, T x, const Policy&);
+
+template <class T>
+calculated-result-type laguerre(unsigned n, unsigned m, T x);
+
+template <class T, class Policy>
+calculated-result-type laguerre(unsigned n, unsigned m, T x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1);
+
+template <class T1, class T2, class T3>
+calculated-result-type laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1);
+
+
+}} // namespaces
+
+
+ + Description +
+

+ The return type of these functions is computed using the result + type calculation rules: note than when there is a single + template argument the result is the same type as that argument or double if the template argument is an integer + type. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T>
+calculated-result-type laguerre(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type laguerre(unsigned n, T x, const Policy&);
+
+

+ Returns the value of the Laguerre Polynomial of order n + at point x: +

+

+ +

+

+ The following graph illustrates the behaviour of the first few Laguerre + Polynomials: +

+

+ laguerre +

+
+template <class T>
+calculated-result-type laguerre(unsigned n, unsigned m, T x);
+
+template <class T, class Policy>
+calculated-result-type laguerre(unsigned n, unsigned m, T x, const Policy&);
+
+

+ Returns the Associated Laguerre polynomial of degree n + and order m at point x: +

+

+ +

+
+template <class T1, class T2, class T3>
+calculated-result-type laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1);
+
+

+ Implements the three term recurrence relation for the Laguerre polynomials, + this function can be used to create a sequence of values evaluated at the + same x, and for rising n. +

+

+ +

+

+ For example we could produce a vector of the first 10 polynomial values + using: +

+
+double x = 0.5;  // Abscissa value
+vector<double> v;
+v.push_back(laguerre(0, x)).push_back(laguerre(1, x));
+for(unsigned l = 1; l < 10; ++l)
+   v.push_back(laguerre_next(l, x, v[l], v[l-1]));
+
+

+ Formally the arguments are: +

+
+

+
+
n
+

+ The degree n of the last polynomial calculated. +

+
x
+

+ The abscissa value +

+
Ln
+

+ The value of the polynomial evaluated at degree n. +

+
Lnm1
+

+ The value of the polynomial evaluated at degree n-1. +

+
+
+
+template <class T1, class T2, class T3>
+calculated-result-type laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1);
+
+

+ Implements the three term recurrence relation for the Associated Laguerre + polynomials, this function can be used to create a sequence of values evaluated + at the same x, and for rising degree n. +

+

+ +

+

+ For example we could produce a vector of the first 10 polynomial values + using: +

+
+double x = 0.5;  // Abscissa value
+int m = 10;      // order
+vector<double> v;
+v.push_back(laguerre(0, m, x)).push_back(laguerre(1, m, x));
+for(unsigned l = 1; l < 10; ++l)
+   v.push_back(laguerre_next(l, m, x, v[l], v[l-1]));
+
+

+ Formally the arguments are: +

+
+

+
+
n
+

+ The degree of the last polynomial calculated. +

+
m
+

+ The order of the Associated Polynomial. +

+
x
+

+ The abscissa value. +

+
Ln
+

+ The value of the polynomial evaluated at degree n. +

+
Lnm1
+

+ The value of the polynomial evaluated at degree n-1. +

+
+
+
+ + Accuracy +
+

+ The following table shows peak errors (in units of epsilon) for various + domains of input arguments. Note that only results for the widest floating + point type on the system are given as narrower types have effectively + zero error. +

+
+

Table 27. Peak Errors In the Laguerre Polynomial

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=3000 Mean=185 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=1x104 Mean=828 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=1x104 Mean=828 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=680 Mean=40 +

+
+
+
+

Table 28. Peak Errors In the Associated Laguerre + Polynomial

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=433 Mean=11 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=61.4 Mean=19.5 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=61.4 Mean=19.5 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=540 Mean=13.94 +

+
+
+

+ Note that the worst errors occur when the degree increases, values greater + than ~120 are very unlikely to produce sensible results, especially in + the associated polynomial case when the order is also large. Further the + relative errors are likely to grow arbitrarily large when the function + is very close to a root. +

+
+ + Testing +
+

+ A mixture of spot tests of values calculated using functions.wolfram.com, + and randomly generated test data are used: the test data was computed using + NTL::RR at 1000-bit + precision. +

+
+ + Implementation +
+

+ These functions are implemented using the stable three term recurrence + relations. These relations guarentee low absolute error but cannot guarentee + low relative error near one of the roots of the polynomials. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_poly/legendre.html b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/legendre.html new file mode 100644 index 000000000..e988186f6 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/legendre.html @@ -0,0 +1,720 @@ + + + +Legendre (and Associated) Polynomials + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/legendre.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type legendre_p(int n, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_p(int n, T x, const Policy&);
+
+template <class T>
+calculated-result-type legendre_p(int n, int m, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_p(int n, int m, T x, const Policy&);
+
+template <class T>
+calculated-result-type legendre_q(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_q(unsigned n, T x, const Policy&);
+
+template <class T1, class T2, class T3>
+calculated-result-type legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);
+
+template <class T1, class T2, class T3>
+calculated-result-type legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);
+
+
+}} // namespaces
+
+

+ The return type of these functions is computed using the result + type calculation rules: note than when there is a single + template argument the result is the same type as that argument or double if the template argument is an integer + type. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + Description +
+
+template <class T>
+calculated-result-type legendre_p(int l, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_p(int l, T x, const Policy&);
+
+

+ Returns the Legendre Polynomial of the first kind: +

+

+ +

+

+ Requires -1 <= x <= 1, otherwise returns the result of domain_error. +

+

+ Negative orders are handled via the reflection formula: +

+

+ P-l-1(x) = Pl(x) +

+

+ The following graph illustrates the behaviour of the first few Legendre + Polynomials: +

+

+ legendre_p1 +

+
+template <class T>
+calculated-result-type legendre_p(int l, int m, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_p(int l, int m, T x, const Policy&);
+
+

+ Returns the associated Legendre polynomial of the first kind: +

+

+ +

+

+ Requires -1 <= x <= 1, otherwise returns the result of domain_error. +

+

+ Negative values of l and m are + handled via the identity relations: +

+

+ +

+
+ + + + + +
[Caution]Caution
+

+ The definition of the associated Legendre polynomial used here includes + a leading Condon-Shortley phase term of (-1)m. This matches the definition + given by Abramowitz and Stegun (8.6.6) and that used by Mathworld + and Mathematica's + LegendreP function. However, uses in the literature do not always + include this phase term, and strangely the specification for the associated + Legendre function in the C++ TR1 (assoc_legendre) also omits it, in spite + of stating that it uses Abramowitz and Stegun as the final arbiter on + these matters. +

+

+ See: +

+

+ Weisstein, + Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web + Resource. +

+

+ Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" + and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook + of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, + 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972. +

+
+
+template <class T>
+calculated-result-type legendre_q(unsigned n, T x);
+
+template <class T, class Policy>
+calculated-result-type legendre_q(unsigned n, T x, const Policy&);
+
+

+ Returns the value of the Legendre polynomial that is the second solution + to the Legendre differential equation, for example: +

+

+ +

+

+ Requires -1 <= x <= 1, otherwise domain_error + is called. +

+

+ The following graph illustrates the first few Legendre functions of the + second kind: +

+

+ legendre_q +

+
+template <class T1, class T2, class T3>
+calculated-result-type legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);
+
+

+ Implements the three term recurrence relation for the Legendre polynomials, + this function can be used to create a sequence of values evaluated at the + same x, and for rising l. This + recurrence relation holds for Legendre Polynomials of both the first and + second kinds. +

+

+ +

+

+ For example we could produce a vector of the first 10 polynomial values + using: +

+
+double x = 0.5;  // Abscissa value
+vector<double> v;
+v.push_back(legendre_p(0, x)).push_back(legendre_p(1, x));
+for(unsigned l = 1; l < 10; ++l)
+   v.push_back(legendre_next(l, x, v[l], v[l-1]));
+
+

+ Formally the arguments are: +

+
+

+
+
l
+

+ The degree of the last polynomial calculated. +

+
x
+

+ The abscissa value +

+
Pl
+

+ The value of the polynomial evaluated at degree l. +

+
Plm1
+

+ The value of the polynomial evaluated at degree l-1. +

+
+
+
+template <class T1, class T2, class T3>
+calculated-result-type legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);
+
+

+ Implements the three term recurrence relation for the Associated Legendre + polynomials, this function can be used to create a sequence of values evaluated + at the same x, and for rising l. +

+

+ +

+

+ For example we could produce a vector of the first m+10 polynomial values + using: +

+
+double x = 0.5;  // Abscissa value
+int m = 10;      // order
+vector<double> v;
+v.push_back(legendre_p(m, m, x)).push_back(legendre_p(1 + m, m, x));
+for(unsigned l = 1 + m; l < m + 10; ++l)
+   v.push_back(legendre_next(l, m, x, v[l], v[l-1]));
+
+

+ Formally the arguments are: +

+
+

+
+
l
+

+ The degree of the last polynomial calculated. +

+
m
+

+ The order of the Associated Polynomial. +

+
x
+

+ The abscissa value +

+
Pl
+

+ The value of the polynomial evaluated at degree l. +

+
Plm1
+

+ The value of the polynomial evaluated at degree l-1. +

+
+
+
+ + Accuracy +
+

+ The following table shows peak errors (in units of epsilon) for various + domains of input arguments. Note that only results for the widest floating + point type on the system are given as narrower types have effectively + zero error. +

+
+

Table 24. Peak Errors In the Legendre P Function

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+ 		+

+ Errors in range +

+

+ 20 < l < 120 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=211 Mean=20 +

+ 		+

+ Peak=300 Mean=33 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=70 Mean=10 +

+ 		+

+ Peak=700 Mean=60 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=70 Mean=10 +

+ 		+

+ Peak=700 Mean=60 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=35 Mean=6 +

+ 		+

+ Peak=292 Mean=41 +

+
+
+
+

Table 25. Peak Errors In the Associated Legendre + P Function

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=1200 Mean=7 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=80 Mean=5 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=80 Mean=5 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=42 Mean=4 +

+
+
+
+

Table 26. Peak Errors In the Legendre Q Function

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+ 		+

+ Errors in range +

+

+ 20 < l < 120 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=50 Mean=7 +

+ 		+

+ Peak=4600 Mean=370 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=51 Mean=8 +

+ 		+

+ Peak=6000 Mean=480 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=51 Mean=8 +

+ 		+

+ Peak=6000 Mean=480 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=90 Mean=10 +

+ 		+

+ Peak=1700 Mean=140 +

+
+
+

+ Note that the worst errors occur when the order increases, values greater + than ~120 are very unlikely to produce sensible results, especially in + the associated polynomial case when the degree is also large. Further the + relative errors are likely to grow arbitrarily large when the function + is very close to a root. +

+

+ No comparisons to other libraries are shown here: there appears to be only + one viable implementation method for these functions, the comparisons to + other libraries that have been run show identical error rates to those + given here. +

+
+ + Testing +
+

+ A mixture of spot tests of values calculated using functions.wolfram.com, + and randomly generated test data are used: the test data was computed using + NTL::RR at 1000-bit + precision. +

+
+ + Implementation +
+

+ These functions are implemented using the stable three term recurrence + relations. These relations guarentee low absolute error but cannot guarentee + low relative error near one of the roots of the polynomials. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sf_poly/sph_harm.html b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/sph_harm.html new file mode 100644 index 000000000..2695a31b5 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sf_poly/sph_harm.html @@ -0,0 +1,321 @@ + + + +Spherical Harmonics + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/special_functions/spheric_harmonic.hpp>
+
+

+

+
+namespace boost{ namespace math{
+
+template <class T1, class T2>
+std::complex<calculated-result-type> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi);
+
+template <class T1, class T2, class Policy>
+std::complex<calculated-result-type> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+
+template <class T1, class T2>
+calculated-result-type spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi);
+   
+template <class T1, class T2, class Policy>
+calculated-result-type spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+   
+}} // namespaces
+
+
+ + Description +
+

+ The return type of these functions is computed using the result + type calculation rules when T1 and T2 are different types. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+template <class T1, class T2>
+std::complex<calculated-result-type> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi);
+
+template <class T1, class T2, class Policy>
+std::complex<calculated-result-type> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+
+

+ Returns the value of the Spherical Harmonic Ynm(theta, phi): +

+

+ +

+

+ The spherical harmonics Ynm(theta, phi) are the angular portion of the + solution to Laplace's equation in spherical coordinates where azimuthal + symmetry is not present. +

+
+ + + + + +
[Caution]Caution
+

+ Care must be taken in correctly identifying the arguments to this function: + θ is taken as the polar (colatitudinal) coordinate with θ in [0, π], and φ as + the azimuthal (longitudinal) coordinate with φ in [0,2π). This is the convention + used in Physics, and matches the definition used by Mathematica + in the function SpericalHarmonicY, but is opposite to the usual + mathematical conventions. +

+

+ Some other sources include an additional Condon-Shortley phase term of + (-1)m in the definition of this function: note however that our definition + of the associated Legendre polynomial already includes this term. +

+

+ This implementation returns zero for m > n +

+

+ For θ outside [0, π] and φ outside [0, 2π] this implementation follows the + convention used by Mathematica: the function is periodic with period + π in θ and 2π in φ. Please note that this is not the behaviour one would get + from a casual application of the function's definition. Cautious users + should keep θ and φ to the range [0, π] and [0, 2π] respectively. +

+

+ See: Weisstein, + Eric W. "Spherical Harmonic." From MathWorld--A Wolfram Web + Resource. +

+
+
+template <class T1, class T2>
+calculated-result-type spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+
+

+ Returns the real part of Ynm(theta, phi): +

+

+ +

+
+template <class T1, class T2>
+calculated-result-type spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi);
+
+template <class T1, class T2, class Policy>
+calculated-result-type spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const Policy&);
+
+

+ Returns the imaginary part of Ynm(theta, phi): +

+

+ +

+
+ + Accuracy +
+

+ The following table shows peak errors for various domains of input arguments. + Note that only results for the widest floating point type on the system + are given as narrower types have effectively + zero error. Peak errors are the same for both the real and imaginary + parts, as the error is dominated by calculation of the associated Legendre + polynomials: especially near the roots of the associated Legendre function. +

+

+ All values are in units of epsilon. +

+
+

Table 30. Peak Errors In the Sperical Harmonic Functions

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Significand Size +

+ 		+

+ Platform and Compiler +

+ 		+

+ Errors in range +

+

+ 0 < l < 20 +

+
+

+ 53 +

+ 		+

+ Win32, Visual C++ 8 +

+ 		+

+ Peak=2x104 Mean=700 +

+
+

+ 64 +

+ 		+

+ SUSE Linux IA32, g++ 4.1 +

+ 		+

+ Peak=2900 Mean=100 +

+
+

+ 64 +

+ 		+

+ Red Hat Linux IA64, g++ 3.4.4 +

+ 		+

+ Peak=2900 Mean=100 +

+
+

+ 113 +

+ 		+

+ HPUX IA64, aCC A.06.06 +

+ 		+

+ Peak=6700 Mean=230 +

+
+
+

+ Note that the worst errors occur when the degree increases, values greater + than ~120 are very unlikely to produce sensible results, especially when + the order is also large. Further the relative errors are likely to grow + arbitrarily large when the function is very close to a root. +

+
+ + Testing +
+

+ A mixture of spot tests of values calculated using functions.wolfram.com, + and randomly generated test data are used: the test data was computed using + NTL::RR at 1000-bit + precision. +

+
+ + Implementation +
+

+ These functions are implemented fairly naively using the formulae given + above. Some extra care is taken to prevent roundoff error when converting + from polar coordinates (so for example the 1-x2 term + used by the associated Legendre functions is calculated without roundoff + error using x = cos(theta), and 1-x2 = sin2(theta)). + The limiting factor in the error rates for these functions is the need + to calculate values near the roots of the associated Legendre functions. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sinc.html b/doc/sf_and_dist/html/math_toolkit/special/sinc.html new file mode 100644 index 000000000..812bc00d0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sinc.html @@ -0,0 +1,51 @@ + + + +Sinus Cardinal and Hyperbolic Sinus Cardinal Functions + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Sinus Cardinal + and Hyperbolic Sinus Cardinal Functions Overview
+
sinc_pi
+
sinhc_pi
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_overview.html b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_overview.html new file mode 100644 index 000000000..6dacc9ef9 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_overview.html @@ -0,0 +1,85 @@ + + + +Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ The Sinus Cardinal + family of functions (indexed by the family of indices a + > 0) is defined by ; + it sees heavy use in signal processing tasks. +

+

+ By analogy, the Hyperbolic + Sinus Cardinal family of functions (also indexed by the family + of indices a > 0) is defined by . +

+

+ These two families of functions are composed of entire functions. +

+

+ These functions (sinc_pi + and sinhc_pi) + are needed by our + implementation of quaternions + and octonions. +

+
+

+

+

+ Sinus Cardinal of index pi (purple) + and Hyperbolic Sinus Cardinal of index pi (red) on R +

+

+

+
+
+

+

+

+ sinc_pi_and_sinhc_pi_on_r +

+

+

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_pi.html b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_pi.html new file mode 100644 index 000000000..90a6a3c2c --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinc_pi.html @@ -0,0 +1,86 @@ + + + +sinc_pi + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/sinc.hpp>
+
+

+

+
+template<class T> 
+calculated-result-type sinc_pi(const T x);
+
+template<class T, class Policy> 
+calculated-result-type sinc_pi(const T x, const Policy&);
+
+template<class T, template<typename> class U> 
+U<T> sinc_pi(const U<T> x);
+
+template<class T, template<typename> class U, class Policy> 
+U<T> sinc_pi(const U<T> x, const Policy&);
+
+

+ Computes the Sinus + Cardinal of x: +

+
+sinc_pi(x) = sin(x) / x
+
+

+ The second form is for complex numbers, quaternions, octonions etc. Taylor + series are used at the origin to ensure accuracy. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/special/sinc/sinhc_pi.html b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinhc_pi.html new file mode 100644 index 000000000..5a45ead0d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/special/sinc/sinhc_pi.html @@ -0,0 +1,90 @@ + + + +sinhc_pi + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+#include <boost/math/special_functions/sinhc.hpp>
+
+

+

+
+template<class T> 
+calculated-result-type sinhc_pi(const T x);
+
+template<class T, class Policy> 
+calculated-result-type sinhc_pi(const T x, const Policy&);
+
+template<typename T, template<typename> class U> 
+U<T> sinhc_pi(const U<T> x);
+
+template<class T, template<typename> class U, class Policy> 
+U<T> sinhc_pi(const U<T> x, const Policy&);
+
+

+ Computes http://mathworld.wolfram.com/SinhcFunction.html the + Hyperbolic Sinus Cardinal of x: +

+
+sinhc_pi(x) = sinh(x) / x
+
+

+ The second form is for complex numbers, quaternions, octonions etc. Taylor + series are used at the origin to ensure accuracy. +

+

+ The return type of the first form is computed using the result + type calculation rules when T is an integer type. +

+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/status.html b/doc/sf_and_dist/html/math_toolkit/status.html new file mode 100644 index 000000000..8e13db276 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/status.html @@ -0,0 +1,50 @@ + + + +Library Status + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
History and What's New
+
Compilers
+
Known Issues, and Todo List
+
Credits and Acknowledgements
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/status/compilers.html b/doc/sf_and_dist/html/math_toolkit/status/compilers.html new file mode 100644 index 000000000..a9e167efb --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/status/compilers.html @@ -0,0 +1,629 @@ + + + +Compilers + + + + + + + + + + + + + + + +
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+
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+
+
+

+

+

+ This section contains some information about how various compilers work + with this library. It is not comprehensive and updated experiences are + always welcome. Some effort has been made to suppress unhelpful warnings + but it is difficult to achieve this on all systems. +

+

+

+
+

Table 47. Supported/Tested Compilers

+
++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Platform +

+ 		+

+ Compiler +

+ 		+

+ Has long double support +

+ 		+

+ Notes +

+
+

+ Windows +

+ 		+

+ MSVC 7.1 and later +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free at level 4 with this compiler. +

+
+

+ Windows +

+ 		+

+ Intel 8.1 and later +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free at level 4 with this compiler. + However, The tests cases tend to generate a lot of warnings relating + to numeric underflow of the test data: these are harmless. +

+
+

+ Windows +

+ 		+

+ GNU Mingw32 C++ +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. +

+
+

+ Windows +

+ 		+

+ GNU Cygwin C++ +

+ 		+

+ No +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. +

+

+ Long double support has been disabled because there are no native + long double C std library functions available. +

+
+

+ Windows +

+ 		+

+ Borland C++ 5.8.2 (Developer studio 2006) +

+ 		+

+ No +

+ 		+

+ We have only partial compatability with this compiler: +

+

+ Long double support has been disabled because the native long double + C standard library functions really only forward to the double versions. + This can result in unpredictable behaviour when using the long double + overloads: for example sqrtl + applied to a finite value, can result in an infinite result. +

+

+ Some functions still fail to compile, there are no known workarounds + at present. +

+
+

+ Linux +

+ 		+

+ GNU C++ 3.4 and later +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. +

+
+

+ Linux +

+ 		+

+ Intel C++ 10.0 and later +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. + However, The tests cases tend to generate a lot of warnings relating + to numeric underflow of the test data: these are harmless. +

+
+

+ Linux +

+ 		+

+ Intel C++ 8.1 and 9.1 +

+ 		+

+ No +

+ 		+

+ All tests OK. +

+

+ Long double support has been disabled with these compiler releases + because calling the standard library long double math functions can + result in a segfault. The issue is Linux distribution and glibc version + specific and is Intel bug report #409291. Fully up to date releases + of Intel 9.1 (post version l_cc_c_9.1.046) shouldn't have this problem. + If you need long double support with this compiler, then comment + out the define of BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS at line + 55 of boost/math/tools/config.hpp. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. + However, The tests cases tend to generate a lot of warnings relating + to numeric underflow of the test data: these are harmless. +

+
+

+ Linux +

+ 		+

+ QLogic PathScale 3.0 +

+ 		+

+ Yes +

+ 		+

+ Some tests involving conceptual checks fail to build, otherwise there + appear to be no issues. +

+
+

+ Linux +

+ 		+

+ Sun Studio 12 +

+ 		+

+ Yes +

+ 		+

+ Some tests involving function overload resolution fail to build, + these issues should be rairly encountered in practice. +

+
+

+ Solaris +

+ 		+

+ Sun Studio 12 +

+ 		+

+ Yes +

+ 		+

+ Some tests involving function overload resolution fail to build, + these issues should be rairly encountered in practice. +

+
+

+ Solaris +

+ 		+

+ GNU C++ 4.x +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ We aim to keep our headers warning free with -Wall with this compiler. +

+
+

+ HP Tru64 +

+ 		+

+ Compaq C++ 7.1 +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+
+

+ HP-UX Itanium +

+ 		+

+ HP aCC 6.x +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+

+ Unfortunately this compiler emits quite a few warnings from libraries + upon which we depend (TR1, Array etc). +

+
+

+ HP-UX PA-RISC +

+ 		+

+ GNU C++ 3.4 +

+ 		+

+ No +

+ 		+

+ All tests OK. +

+
+

+ Apple Mac OS X, Intel +

+ 		+

+ Darwin/GNU C++ 4.x +

+ 		+

+ Yes +

+ 		+

+ All tests OK. +

+
+

+ Apple Mac OS X, PowerPC +

+ 		+

+ Darwin/GNU C++ 4.x +

+ 		+

+ No +

+ 		+

+ All tests OK. +

+

+ Long double support has been disabled on this platform due to the + rather strange nature of Darwin's 106-bit long double implementation. + It should be possible to make this work if someone is prepared to + offer assistance. +

+
+

+ IMB AIX +

+ 		+

+ IBM xlc 5.3 +

+ 		+

+ Yes +

+ 		+

+ Currently our accuracy tests are unable to be run on this platform + due to compiler errors in our test code. The + IBM compiler group are aware of the problem. +

+
+
+


+
+

Table 48. Unsupported Compilers

+
++++ + + + + + + + + + + + + + + +
+

+ Platform +

+ 		+

+ Compiler +

+
+

+ Windows +

+ 		+

+ Borland C++ 5.9.2 (Borland Developer Studio 2007) +

+
+

+ Windows +

+ 		+

+ MSVC 6 and 7 +

+
+
+


+

+

+ If you're compiler or platform is not listed above, please try running + the regression tests: cd into boost-root/libs/math/test and do a: +

+

+ +

+
+bjam mytoolset
+
+

+

+

+ where "mytoolset" is the name of the Boost.Build + toolset used for your compiler. The chances are that many + of the accuracy tests will fail at this stage - don't panic + - the default acceptable error tolerances are quite tight, especially for + long double types with an extended exponent range (these cause more extreme + test cases to be executed for some functions). You will need to cast an + eye over the output from the failing tests and make a judgement as to whether + the error rates are acceptable or not. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/status/credits.html b/doc/sf_and_dist/html/math_toolkit/status/credits.html new file mode 100644 index 000000000..44b814d99 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/status/credits.html @@ -0,0 +1,84 @@ + + + +Credits and Acknowledgements + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHome +
+
+
+

+ Hubert Holin started the Boost.Math library. The inverse hyperbolic functions, + and the sinus cardinal functions are his. +

+

+ John Maddock started this library, the beta, gamma, erf, polynomial, and + factorial functions are his, as is the "Toolkit" section, and many + of the statistical distributions. +

+

+ Paul A. Bristow threw down the challenge in A + Proposal to add Mathematical Functions for Statistics to the C++ Standard + Library to add the key math functions, especially those essential + for statistics. After JM accepted and solved the difficult problems, not + only numerically, but in full C++ template style, PAB implemented a few of + the statistical distributions. PAB also tirelessly proof-read everything + that JM threw at him (so that all remaining editorial mistakes are his fault). +

+

+ Xiaogang Zhang worked on the Bessel functions and elliptic integrals for + his Google Summer of Code project 2006. +

+

+ Professor Nico Temme for advice on the inverse incomplete beta function. +

+

+ Victor Shoup for NTL, without which + it would have much difficult to produce high accuracy constants, and especially + the tables of accurate values for testing. +

+

+ We are grateful to Joel Guzman for helping us stress-test his Boost.Quickbook + program used to generate the html and pdf versions of this document, adding + several new features en route. +

+

+ We are also indebted to Matthias Schabel for managing the formal Boost-review + of this library, and to all the reviewers - including Guillaume Melquiond, + Arnaldur Gylfason, John Phillips, Stephan Tolksdorf and Jeff Garland - for + their many helpful comments. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHome +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/status/history1.html b/doc/sf_and_dist/html/math_toolkit/status/history1.html new file mode 100644 index 000000000..de34a25f3 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/status/history1.html @@ -0,0 +1,217 @@ + + + +History and What's New + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ +

+
+ + Milestone + 5: Post Review First Official Release +
+

+

+
    +
  • + Added Policy based framework that allows fine grained control over function + behaviour. +
    • +
  • +Breaking change: Changed default behaviour + for domain, pole and overflow errors to throw an exception (based on + review feedback), this behaviour can be customised using Policy's. +
    • +
  • +Breaking change: Changed exception thrown + when an internal evaluation error occurs to boost::math::evaluation_error. +
    • +
  • +Breaking change: Changed discrete quantiles + to return an integer result: this is anything up to 20 times faster than + finding the true root, this behaviour can be customised using Policy's. +
    • +
  • + Polynomial/rational function evaluation is now customisable and hopefully + faster than before. +
    • +
  • + Added performance test program. +
    • +
+

+ +

+
+ + Milestone + 4: Second Review Candidate (1st March 2007) +
+

+

+
    +
  • + Moved Xiaogang Zhang's Bessel Functions code into the library, and brought + them into line with the rest of the code. +
    • +
  • + Added C# "Distribution Explorer" demo application. +
    • +
+

+ +

+
+ + Milestone + 3: First Review Candidate (31st Dec 2006) +
+

+

+
    +
  • + Implemented the main probability distribution and density functions. +
    • +
  • + Implemented digamma. +
    • +
  • + Added more factorial functions. +
    • +
  • + Implemented the Hermite, Legendre and Laguerre polynomials plus the spherical + harmonic functions from TR1. +
    • +
  • + Moved Xiaogang Zhang's elliptic integral code into the library, and brought + them into line with the rest of the code. +
    • +
  • + Moved Hubert Holin's existing Boost.Math special functions into this + library and brought them into line with the rest of the code. +
    • +
+

+ +

+
+ + Milestone + 2: Released September 10th 2006 +
+

+

+
    +
  • + Implement preview release of the statistical distributions. +
    • +
  • + Added statistical distributions tutorial. +
    • +
  • + Implemented root finding algorithms. +
    • +
  • + Implemented the inverses of the incomplete gamma and beta functions. +
    • +
  • + Rewrite erf/erfc as rational approximations (valid to 128-bit precision). +
    • +
  • + Integrated the statistical results generated from the test data with + Boost.Test: uses a database of expected results, indexed by test, floating + point type, platform, and compiler. +
    • +
  • + Improved lgamma near 1 and 2 (rational approximations). +
    • +
  • + Improved erf/erfc inverses (rational approximations). +
    • +
  • + Implemented Rational function generation (the Remez method). +
    • +
+

+ +

+
+ + Milestone + 1: Released March 31st 2006 +
+

+

+
    +
  • + Implement gamma/beta/erf functions along with their incomplete counterparts. +
    • +
  • + Generate high quality test data, against which future improvements can + be judged. +
    • +
  • + Provide tools for the evaluation of infinite series, continued fractions, + and rational functions. +
    • +
  • + Provide tools for testing against tabulated test data, and collecting + statistics on error rates. +
    • +
  • + Provide sufficient docs for people to be able to find their way around + the library. +
    • +
+

+

+

+ SVN Revisions: +

+

+

+

+ Sandbox and trunk last synchonised at revision: 40161. +

+

+

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/status/issues.html b/doc/sf_and_dist/html/math_toolkit/status/issues.html new file mode 100644 index 000000000..c4458a215 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/status/issues.html @@ -0,0 +1,163 @@ + + + +Known Issues, and Todo List + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ This section lists those issues that are known about. +

+

+ Predominantly this is a TODO list, or a list of possible future enhancements. + Items labled "High Priority" effect the proper functioning of the + component, and should be fixed as soon as possible. Items labled "Medium + Priority" are desirable enhancements, often pertaining to the performance + of the component, but do not effect it's accuracy or functionality. Items + labled "Low Priority" should probably be investigated at some point. + Such classifications are obviously highly subjective. +

+

+ If you don't see a component listed here, then we don't have any known issues + with it. +

+
+ + tgamma +
+
+
+ + Incomplete Beta +
+
  • + Investigate Didonato and Morris' asymptotic expansion for large a and b + (medium priority). +
+
+ + Inverse Gamma +
+
  • + Investigate whether we can skip iteration altogether if the first approximation + is good enough (Medium Priority). +
+
+ + Polynomials +
+
  • + The Legendre and Laguerre Polynomials have surprisingly different error + rates on different platforms, considering they are evaluated with only + basic arithmetic operations. Maybe this is telling us something, or maybe + not (Low Priority). +
+
+ + Elliptic Integrals +
+
    +
  • + Carlson's algorithms are essentially unchanged from Xiaogang Zhang's Google + Summer of Code student project, and are based on Carlson's original papers. + However, Carlson has revised his algorithms since then (refer to the references + in the elliptic integral docs for a list), to improve performance and accuracy, + we may be able to take advantage of these improvements too (Low Priority). +
    • +
  • +

    Carlson's algorithms (mainly RJ) are somewhat prone to internal overflow/underflow + when the arguments are very large or small. The homogeneity relations:

    +

    RF(ka, + kb, kc) = k-1/2 RF(a, b, c)

    +

    and

    +

    RJ(ka, kb, kc, kr) = k-3/2 RJ(a, b, c, r)

    +

    could + be used to sidestep trouble here: provided the problem domains can be accurately + identified. (Medium Priority).

    +
    • +
  • + Carlson's RC can be reduced to elementary funtions (asin and log), would + it be more efficient evaluated this way, rather than by Carlson's algorithms? + (Low Priority). +
    • +
  • + Should we add an implementation of Carlson's RG? It's not required for + the Legendre form integrals, but some people may find it useful (Low Priority). +
    • +
  • + There are a several other integrals: D(φ, k), Z(β, k), Λ0(β, k) and Bulirsch's + el functions that could be implemented using Carlson's + integrals (Low Priority). +
    • +
  • + The integrals K(k) and E(k) could be implemented using rational approximations + (both for efficiency and accuracy), assuming we can find them. (Medium + Priority). +
    • +
  • + There is a sub-domain of ellint_3 + that is unimplemented (see the docs for details), currently it's not clear + how to solve this issue, or if it's ever likely to be an real problem in + practice - especially as most other implementations don't support this + domain either (Medium Priority). +
    • +
+
+ + Inverse + Hyperbolic Functions +
+
  • + These functions are inherited from previous Boost versions, before log1p became widely + available. Would they be better expressed in terms of this function? This + is probably only an issue for very high precision types (Low Priority). +
+
+ + Statistical + distributions +
+
  • + Student's t Perhaps switch to normal distribution as a better approximation + for very large degrees of freedom? +
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit.html b/doc/sf_and_dist/html/math_toolkit/toolkit.html new file mode 100644 index 000000000..a7f42013a --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit.html @@ -0,0 +1,72 @@ + + + +Internal Details and Tools (Experimental) + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+
Overview
+
Reused Utilities
+
+
+ Series Evaluation
+
Continued Fraction + Evaluation
+
Polynomial + and Rational Function Evaluation
+
Root Finding + With Derivatives
+
Root Finding + Without Derivatives
+
Locating Function + Minima
+
+
Testing and Development
+
+
Polynomials
+
Minimax Approximations + and the Remez Algorithm
+
Relative + Error and Testing
+
Graphing, + Profiling, and Generating Test Data for Special Functions
+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1.html new file mode 100644 index 000000000..6d36ab455 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1.html @@ -0,0 +1,58 @@ + + + +Reused Utilities + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
+ Series Evaluation
+
Continued Fraction + Evaluation
+
Polynomial + and Rational Function Evaluation
+
Root Finding + With Derivatives
+
Root Finding + Without Derivatives
+
Locating Function + Minima
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/cf.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/cf.html new file mode 100644 index 000000000..7fae8759d --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/cf.html @@ -0,0 +1,271 @@ + + + +Continued Fraction Evaluation + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/fraction.hpp>
+
+

+

+
+namespace boost{ namespace math{ namespace tools{
+
+template <class Gen>
+typename detail::fraction_traits<Gen>::result_type 
+   continued_fraction_b(Gen& g, int bits);
+
+template <class Gen>
+typename detail::fraction_traits<Gen>::result_type 
+   continued_fraction_b(Gen& g, int bits, boost::uintmax_t& max_terms);
+
+template <class Gen>
+typename detail::fraction_traits<Gen>::result_type 
+   continued_fraction_a(Gen& g, int bits);
+   
+template <class Gen>
+typename detail::fraction_traits<Gen>::result_type 
+   continued_fraction_a(Gen& g, int bits, boost::uintmax_t& max_terms);
+   
+}}} // namespaces
+
+
+ + Description +
+

+ Continued + fractions are a common method of approximation. These functions + all evaluate the continued fraction described by the generator + type argument. The functions with an "_a" suffix evaluate the + fraction: +

+

+ +

+

+ and those with a "_b" suffix evaluate the fraction: +

+

+ +

+

+ This latter form is somewhat more natural in that it corresponds with the + usual definition of a continued fraction, but note that the first a + value returned by the generator is discarded. Further, often the first + a and b values in a continued + fraction have different defining equations to the remaining terms, which + may make the "_a" suffixed form more appropriate. +

+

+ The generator type should be a function object which supports the following + operations: +

+
++++ + + + + + + + + + + + + + + +
+

+ Expression +

+ 		+

+ Description +

+
+

+ Gen::result_type +

+ 		+

+ The type that is the result of invoking operator(). This can be + either an arithmetic type, or a std::pair<> of arithmetic + types. +

+
+

+ g() +

+ 		+

+ Returns an object of type Gen::result_type. +

+

+ Each time this operator is called then the next pair of a + and b values is returned. Or, if result_type + is an arithmetic type, then the next b value + is returned and all the a values are assumed + to 1. +

+
+

+ In all the continued fraction evaluation functions the bits + parameter is the number of bits precision desired in the result, evaluation + of the fraction will continue until the last term evaluated leaves the + first bits bits in the result unchanged. +

+

+ If the optional max_terms parameter is specified then + no more than max_terms calls to the generator will + be made, and on output, max_terms will be set to actual + number of calls made. This facility is particularly useful when profiling + a continued fraction for convergence. +

+
+ + Implementation +
+

+ Internally these algorithms all use the modified Lentz algorithm: refer + to Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 + Evaluation of continued fractions, p 175 - 179) for more information, also + Lentz, W.J. 1976, Applied Optics, vol. 15, pp. 668-671. +

+
+ + Examples +
+

+ The golden ratio + phi = 1.618033989... can be computed from the simplest continued + fraction of all: +

+

+ +

+

+ We begin by defining a generator function: +

+
+template <class T>
+struct golden_ratio_fraction
+{
+   typedef T result_type;
+   
+   result_type operator()
+   {
+      return 1;
+   }
+};
+
+

+ The golden ratio can then be computed to double precision using: +

+
+continued_fraction_a(
+   golden_ratio_fraction<double>(),
+   std::numeric_limits<double>::digits);
+
+

+ It's more usual though to have to define both the a's + and the b's when evaluating special functions by continued + fractions, for example the tan function is defined by: +

+

+ +

+

+ So it's generator object would look like: +

+
+template <class T>
+struct tan_fraction
+{
+private:
+   T a, b;
+public:
+   tan_fraction(T v)
+      : a(-v*v), b(-1)
+   {}
+
+   typedef std::pair<T,T> result_type;
+
+   std::pair<T,T> operator()()
+   {
+      b += 2;
+      return std::make_pair(a, b);
+   }
+};
+
+

+ Notice that if the continuant is subtracted from the b + terms, as is the case here, then all the a terms returned + by the generator will be negative. The tangent function can now be evaluated + using: +

+
+template <class T>
+T tan(T a)
+{
+   tan_fraction<T> fract(a);
+   return a / continued_fraction_b(fract, std::numeric_limits<T>::digits);
+}
+
+

+ Notice that this time we're using the "_b" suffixed version to + evaluate the fraction: we're removing the leading a + term during fraction evaluation as it's different from all the others. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/minima.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/minima.html new file mode 100644 index 000000000..431c18742 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/minima.html @@ -0,0 +1,128 @@ + + + +Locating Function Minima + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + synopsis +
+

+ +

+
+#include <boost/math/tools/minima.hpp>
+
+

+

+
+template <class F, class T>
+std::pair<T, T> brent_find_minima(F f, T min, T max, int bits);
+
+template <class F, class T>
+std::pair<T, T> brent_find_minima(F f, T min, T max, int bits, boost::uintmax_t& max_iter);
+
+
+ + Description +
+

+ These two functions locate the minima of the continuous function f + using Brent's algorithm. Parameters are: +

+
+

+
+
f
+

+ The function to minimise. The function should be smooth over the range + [min,max], with no maxima occurring in that interval. +

+
min
+

+ The lower endpoint of the range in which to search for the minima. +

+
max
+

+ The upper endpoint of the range in which to search for the minima. +

+
bits
+

+ The number of bits precision to which the minima should be found. Note + that in principle, the minima can not be located to greater accuracy + than the square root of machine epsilon, therefore if bits + is set to a value greater than one half of the bits in type T, then + the value will be ignored. +

+
max_iter
+

+ The maximum number of iterations to use in the algorithm, if not provided + the algorithm will just keep on going until the minima is found. +

+
+
+

+ Returns: a pair containing the value of + the abscissa at the minima and the value of f(x) at the minima. +

+
+ + Implementation +
+

+ This is a reasonably faithful implementation of Brent's algorithm, refer + to: +

+

+ Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood + Cliffs, NJ: Prentice-Hall), Chapter 5. +

+

+ Numerical Recipes in C, The Art of Scientific Computing, Second Edition, + William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. + Flannery. Cambridge University Press. 1988, 1992. +

+

+ An algorithm with guaranteed convergence for finding a zero of a function, + R. P. Brent, The Computer Journal, Vol 44, 1971. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/rational.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/rational.html new file mode 100644 index 000000000..74f9638ae --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/rational.html @@ -0,0 +1,243 @@ + + + +Polynomial and Rational Function Evaluation + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + synopsis +
+

+ +

+
+#include <boost/math/tools/rational.hpp>
+
+

+

+
+// Polynomials:
+template <std::size_t N, class T, class V>
+V evaluate_polynomial(const T(&poly)[N], const V& val);
+
+template <std::size_t N, class T, class V>
+V evaluate_polynomial(const boost::array<T,N>& poly, const V& val);
+
+template <class T, class U>
+U evaluate_polynomial(const T* poly, U z, std::size_t count);
+
+// Even polynomials:
+template <std::size_t N, class T, class V>
+V evaluate_even_polynomial(const T(&poly)[N], const V& z);
+
+template <std::size_t N, class T, class V>
+V evaluate_even_polynomial(const boost::array<T,N>& poly, const V& z);
+
+template <class T, class U>
+U evaluate_even_polynomial(const T* poly, U z, std::size_t count);
+
+// Odd polynomials   
+template <std::size_t N, class T, class V>
+V evaluate_odd_polynomial(const T(&a)[N], const V& z);
+
+template <std::size_t N, class T, class V>
+V evaluate_odd_polynomial(const boost::array<T,N>& a, const V& z);
+
+template <class T, class U>
+U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);
+
+// Rational Functions:
+template <std::size_t N, class T, class V>
+V evaluate_rational(const T(&a)[N], const T(&b)[N], const V& z);
+
+template <std::size_t N, class T, class V>
+V evaluate_rational(const boost::array<T,N>& a, const boost::array<T,N>& b, const V& z);
+
+template <class T, class U, class V>
+V evaluate_rational(const T* num, const U* denom, V z, unsigned count);
+
+
+ + Description +
+

+ Each of the functions come in three variants: a pair of overloaded functions + where the order of the polynomial or rational function is evaluated at + compile time, and an overload that accepts a runtime variable for the size + of the coefficient array. Generally speaking, compile time evaluation of + the array size results in better type safety, is less prone to programmer + errors, and may result in better optimised code. The polynomial evaluation + functions in particular, are specialised for various array sizes, allowing + for loop unrolling, and one hopes, optimal inline expansion. +

+
+template <std::size_t N, class T, class V>
+V evaluate_polynomial(const T(&poly)[N], const V& val);
+
+template <std::size_t N, class T, class V>
+V evaluate_polynomial(const boost::array<T,N>& poly, const V& val);
+
+template <class T, class U>
+U evaluate_polynomial(const T* poly, U z, std::size_t count);
+
+

+ Evaluates the polynomial + described by the coefficients stored in poly. +

+

+ If the size of the array is specified at runtime, then the polynomial most + have order count-1 with count + coefficients. Otherwise it has order N-1 with N + coefficients. +

+

+ Coefficients should be stored such that the coefficients for the xi terms + are in poly[i]. +

+

+ The types of the coefficients and of variable z may + differ as long as *poly is convertible to type U. + This allows, for example, for the coefficient table to be a table of integers + if this is appropriate. +

+
+template <std::size_t N, class T, class V>
+V evaluate_even_polynomial(const T(&poly)[N], const V& z);
+
+template <std::size_t N, class T, class V>
+V evaluate_even_polynomial(const boost::array<T,N>& poly, const V& z);
+
+template <class T, class U>
+U evaluate_even_polynomial(const T* poly, U z, std::size_t count);
+
+

+ As above, but evaluates an even polynomial: one where all the powers of + z are even numbers. Equivalent to calling evaluate_polynomial(poly, z*z, count). +

+
+template <std::size_t N, class T, class V>
+V evaluate_odd_polynomial(const T(&a)[N], const V& z);
+
+template <std::size_t N, class T, class V>
+V evaluate_odd_polynomial(const boost::array<T,N>& a, const V& z);
+
+template <class T, class U>
+U evaluate_odd_polynomial(const T* poly, U z, std::size_t count);
+
+

+ As above but evaluates a polynomial where all the powers are odd numbers. + Equivalent to evaluate_polynomial(poly+1, + z*z, count-1) + * z + + poly[0]. +

+
+template <std::size_t N, class T, class U, class V>
+V evaluate_rational(const T(&num)[N], const U(&denom)[N], const V& z);
+
+template <std::size_t N, class T, class U, class V>
+V evaluate_rational(const boost::array<T,N>& num, const boost::array<U,N>& denom, const V& z);
+
+template <class T, class U, class V>
+V evaluate_rational(const T* num, const U* denom, V z, unsigned count);
+
+

+ Evaluates the rational function (the ratio of two polynomials) described + by the coefficients stored in num and demom. +

+

+ If the size of the array is specified at runtime then both polynomials + most have order count-1 with count + coefficients. Otherwise both polynomials have order N-1 + with N coefficients. +

+

+ Array num describes the numerator, and demon + the denominator. +

+

+ Coefficients should be stored such that the coefficients for the xi terms + are in num[i] and denom[i]. +

+

+ The types of the coefficients and of variable v may + differ as long as *num and *denom + are convertible to type V. This allows, for example, + for one or both of the coefficient tables to be a table of integers if + this is appropriate. +

+

+ These functions are designed to safely evaluate the result, even when the + value z is very large. As such they do not take advantage + of compile time array sizes to make any optimisations. These functions + are best reserved for situations where z may be large: + if you can be sure that numerical overflow will not occur then polynomial + evaluation with compile-time array sizes may offer slightly better performance. +

+
+ + Implementation +
+

+ Polynomials are evaluated by Horners + method. If the array size is known at compile time then the functions + dispatch to size-specific implementations that unroll the evaluation loop. +

+

+ Rational evaluation is by Horners + method: with the two polynomials being evaluated in parallel to + make the most of the processors floating-point pipeline. If v + is greater than one, then the polynomials are evaluated in reverse order + as polynomials in 1/v: this avoids unnecessary numerical + overflow when the coefficients are large. +

+

+ Both the polynomial and rational function evaluation algorithms can be + tuned using various configuration macros to provide optimal performance + for a particular combination of compiler and platform. This includes support + for second-order Horner's methods. The various options are documented + here. However, the performance benefits to be gained from these + are marginal on most current hardware, consequently it's best to run the + performance test application + before changing the default settings. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots.html new file mode 100644 index 000000000..3bce987f7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots.html @@ -0,0 +1,390 @@ + + + +Root Finding With Derivatives + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/roots.hpp>
+
+

+

+
+namespace boost{ namespace math{ namespace tools{
+
+template <class F, class T>
+T newton_raphson_iterate(F f, T guess, T min, T max, int digits);
+
+template <class F, class T>
+T newton_raphson_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter);
+
+template <class F, class T>
+T halley_iterate(F f, T guess, T min, T max, int digits);
+
+template <class F, class T>
+T halley_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter);
+
+template <class F, class T>
+T schroeder_iterate(F f, T guess, T min, T max, int digits);
+
+template <class F, class T>
+T schroeder_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter);
+
+}}} // namespaces
+
+
+ + Description +
+

+ These functions all perform iterative root finding: newton_raphson_iterate + performs second order Newton Raphson iteration, + while halley_iterate and + schroeder_iterate perform + third order Halley and Schroeder + iteration respectively. +

+

+ The functions all take the same parameters: +

+
+

Parameters of the root finding functions

+
+
F + f
+
+

+ Type F must be a callable function object that accepts one parameter + and returns a tuple: +

+

+ For the second order iterative methods (Newton Raphson) the tuple should + have two elements containing the evaluation of the function and it's + first derivative. +

+

+ For the third order methods (Halley and Schroeder) the tuple should + have three elements containing the evaluation of the function and its + first and second derivatives. +

+
+
T guess
+

+ The initial starting value. +

+
T min
+

+ The minimum possible value for the result, this is used as an initial + lower bracket. +

+
T max
+

+ The maximum possible value for the result, this is used as an initial + upper bracket. +

+
int digits
+

+ The desired number of binary digits. +

+
uintmax_t max_iter
+

+ An optional maximum number of iterations to perform. +

+
+
+

+ When using these functions you should note that: +

+
    +
  • + They may be very sensitive to the initial guess, typically they converge + very rapidly if the initial guess has two or three decimal digits correct. + However convergence can be no better than bisection, or in some rare + cases even worse than bisection if the initial guess is a long way from + the correct value and the derivatives are close to zero. +
    • +
  • + These functions include special cases to handle zero first (and second + where appropriate) derivatives, and fall back to bisection in this case. + However, it is helpful if F is defined to return an arbitrarily small + value of the correct sign rather than zero. +
    • +
  • + If the derivative at the current best guess for the result is infinite + (or very close to being infinite) then these functions may terminate + prematurely. A large first derivative leads to a very small next step, + triggering the termination condition. Derivative based iteration may + not be appropriate in such cases. +
    • +
  • + These functions fall back to bisection if the next computed step would + take the next value out of bounds. The bounds are updated after each + step to ensure this leads to convergence. However, a good initial guess + backed up by asymptotically-tight bounds will improve performance no + end rather than relying on bisection. +
    • +
  • + The value of digits is crucial to good performance + of these functions, if it is set too high then at best you will get one + extra (unnecessary) iteration, and at worst the last few steps will proceed + by bisection. Remember that the returned value can never be more accurate + than f(x) can be evaluated, and that if f(x) suffers from cancellation + errors as it tends to zero then the computed steps will be effectively + random. The value of digits should be set so that + iteration terminates before this point: remember that for second and + third order methods the number of correct digits in the result is increasing + quite substantially with each iteration, digits + should be set by experiment so that the final iteration just takes the + next value into the zone where f(x) becomes inaccurate. +
    • +
  • + Finally: you may well be able to do better than these functions by hand-coding + the heuristics used so that they are tailored to a specific function. + You may also be able to compute the ratio of derivatives used by these + methods more efficiently than computing the derivatives themselves. As + ever, algebraic simplification can be a big win. +
    • +
+

+

+
+ + Newton + Raphson Method +
+

+ Given an initial guess x0 the subsequent values are computed using: +

+

+ +

+

+ Out of bounds steps revert to bisection of the current bounds. +

+

+ Under ideal conditions, the number of correct digits doubles with each + iteration. +

+

+

+
+ + Halley's + Method +
+

+ Given an initial guess x0 the subsequent values are computed using: +

+

+ +

+

+ Over-compensation by the second derivative (one which would proceed in + the wrong direction) causes the method to revert to a Newton-Raphson step. +

+

+ Out of bounds steps revert to bisection of the current bounds. +

+

+ Under ideal conditions, the number of correct digits trebles with each + iteration. +

+

+

+
+ + Schroeder's + Method +
+

+ Given an initial guess x0 the subsequent values are computed using: +

+

+ +

+

+ Over-compensation by the second derivative (one which would proceed in + the wrong direction) causes the method to revert to a Newton-Raphson step. + Likewise a Newton step is used whenever that Newton step would change the + next value by more than 10%. +

+

+ Out of bounds steps revert to bisection of the current bounds. +

+

+ Under ideal conditions, the number of correct digits trebles with each + iteration. +

+
+ + Example +
+

+ Lets suppose we want to find the cube root of a number, the equation we + want to solve along with its derivatives are: +

+

+ +

+

+ To begin with lets solve the problem using Newton Raphson iterations, we'll + begin be defining a function object that returns the evaluation of the + function to solve, along with its first derivative: +

+
+template <class T>
+struct cbrt_functor
+{
+   cbrt_functor(T const& target) : a(target){}
+   std::tr1::tuple<T, T> operator()(T const& z)
+   {
+      T sqr = z * z;
+      return std::tr1::make_tuple(sqr * z - a, 3 * sqr);
+   }
+private:
+   T a;
+};
+
+

+ Implementing the cube root is fairly trivial now, the hardest part is finding + a good approximation to begin with: in this case we'll just divide the + exponent by three: +

+
+template <class T>
+T cbrt(T z)
+{
+   using namespace std;
+   int exp;
+   frexp(z, &exp);
+   T min = ldexp(0.5, exp/3);
+   T max = ldexp(2.0, exp/3);
+   T guess = ldexp(1.0, exp/3);
+   int digits = std::numeric_limits<T>::digits;
+   return tools::newton_raphson_iterate(detail::cbrt_functor<T>(z), guess, min, max, digits);
+}
+
+

+ Using the test data in libs/math/test/cbrt_test.cpp this found the cube + root exact to the last digit in every case, and in no more than 6 iterations + at double precision. However, you will note that a high precision was used + in this example, exactly what was warned against earlier on in these docs! + In this particular case its possible to compute f(x) exactly and without + undue cancellation error, so a high limit is not too much of an issue. + However, reducing the limit to std::numeric_limits<T>::digits + * 2 / 3 gave + full precision in all but one of the test cases (and that one was out by + just one bit). The maximum number of iterations remained 6, but in most + cases was reduced by one. +

+

+ Note also that the above code omits error handling, and does not handle + negative values of z correctly. That will be left as an exercise for the + reader! +

+

+ Now lets adapt the functor slightly to return the second derivative as + well: +

+
+template <class T>
+struct cbrt_functor
+{
+   cbrt_functor(T const& target) : a(target){}
+   std::tr1::tuple<T, T, T> operator()(T const& z)
+   {
+      T sqr = z * z;
+      return std::tr1::make_tuple(sqr * z - a, 3 * sqr, 6 * z);
+   }
+private:
+   T a;
+};
+
+

+ And then adapt the cbrt + function to use Halley iterations: +

+
+template <class T>
+T cbrt(T z)
+{
+   using namespace std;
+   int exp;
+   frexp(z, &exp);
+   T min = ldexp(0.5, exp/3);
+   T max = ldexp(2.0, exp/3);
+   T guess = ldexp(1.0, exp/3);
+   int digits = std::numeric_limits<T>::digits / 2;
+   return tools::halley_iterate(detail::cbrt_functor<T>(z), guess, min, max, digits);
+}
+
+

+ Note that the iterations are set to stop at just one-half of full precision, + and yet even so not one of the test cases had a single bit wrong. What's + more, the maximum number of iterations was now just 4. +

+

+ Just to complete the picture, we could have called schroeder_iterate + in the last example: and in fact it makes no difference to the accuracy + or number of iterations in this particular case. However, the relative + performance of these two methods may vary depending upon the nature of + f(x), and the accuracy to which the initial guess can be computed. There + appear to be no generalisations that can be made except "try them + and see". +

+

+ Finally, had we called cbrt with NTL::RR + set to 1000 bit precision, then full precision can be obtained with just + 7 iterations. To put that in perspective an increase in precision by a + factor of 20, has less than doubled the number of iterations. That just + goes to emphasise that most of the iterations are used up getting the first + few digits correct: after that these methods can churn out further digits + with remarkable efficiency. Or to put it another way: nothing + beats a really good initial guess! +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots2.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots2.html new file mode 100644 index 000000000..127fad3b7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/roots2.html @@ -0,0 +1,610 @@ + + + +Root Finding Without Derivatives + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/roots.hpp>
+
+

+

+
+namespace boost{ namespace math{ namespace tools{
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bracket_and_solve_root(
+      F f, 
+      const T& guess, 
+      const T& factor, 
+      bool rising, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   bracket_and_solve_root(
+      F f, 
+      const T& guess, 
+      const T& factor, 
+      bool rising, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      const T& fa, 
+      const T& fb, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      const T& fa, 
+      const T& fb, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+// Termination conditions:
+template <class T>
+struct eps_tolerance;
+
+struct equal_floor;
+struct equal_ceil;
+struct equal_nearest_integer;
+
+}}} // namespaces
+
+
+ + Description +
+

+ These functions solve the root of some function f(x) + without the need for the derivatives of f(x). The + functions here that use TOMS Algorithm 748 are asymptotically the most + efficient known, and have been shown to be optimal for a certain classes + of smooth functions. +

+

+ Alternatively, there is a simple bisection routine which can be useful + in its own right in some situations, or alternatively for narrowing down + the range containing the root, prior to calling a more advanced algorithm. +

+

+ All the algorithms in this section reduce the diameter of the enclosing + interval with the same asymptotic efficiency with which they locate the + root. This is in contrast to the derivative based methods which may never + significantly reduce the enclosing interval, even though they rapidly approach + the root. This is also in contrast to some other derivative-free methods + (for example the methods of Brent + or Dekker) which only reduce the enclosing interval on the final + step. Therefore these methods return a std::pair containing the enclosing + interval found, and accept a function object specifying the termination + condition. Three function objects are provided for ready-made termination + conditions: eps_tolerance causes termination when + the relative error in the enclosing interval is below a certain threshold, + while equal_floor and equal_ceil + are useful for certain statistical applications where the result is known + to be an integer. Other user-defined termination conditions are likely + to be used only rarely, but may be useful in some specific circumstances. +

+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   bisect(
+      F f, 
+      T min, 
+      T max, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+

+ These functions locate the root using bisection, function arguments are: +

+
+

+
+
f
+

+ A unary functor which is the function whose root is to be found. +

+
min
+

+ The left bracket of the interval known to contain the root. +

+
max
+

+ The right bracket of the interval known to contain the root. It is + a precondition that min < max and f(min)*f(max) + <= 0, the function signals evaluation error if these + preconditions are violated. The action taken is controlled by the evaluation + error policy. A best guess may be returned, perhaps significantly wrong. +

+
tol
+

+ A binary functor that specifies the termination condition: the function + will return the current brackets enclosing the root when tol(min,max) + becomes true. +

+
max_iter
+

+ The maximum number of invocations of f(x) to make + while searching for the root. +

+
+
+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Returns: a pair of values r that bracket the root + so that: +

+
+f(r.first) * f(r.second) <= 0
+
+

+ and either +

+
+tol(r.first, r.second) == true
+
+

+ or +

+
+max_iter >= m
+
+

+ where m is the initial value of max_iter + passed to the function. +

+

+ In other words, it's up to the caller to verify whether termination occurred + as a result of exceeding max_iter function invocations + (easily done by checking the value of max_iter when + the function returns), rather than because the termination condition tol + was satisfied. +

+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   bracket_and_solve_root(
+      F f, 
+      const T& guess, 
+      const T& factor, 
+      bool rising, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   bracket_and_solve_root(
+      F f, 
+      const T& guess, 
+      const T& factor, 
+      bool rising, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+

+ This is a convenience function that calls toms748_solve + internally to find the root of f(x). It's usable only + when f(x) is a monotonic function, and the location + of the root is known approximately, and in particular it is known whether + the root is occurs for positive or negative x. The + parameters are: +

+
+

+
+
f
+

+ A unary functor that is the function whose root is to be solved. f(x) + must be uniformly increasing or decreasing on x. +

+
guess
+

+ An initial approximation to the root +

+
factor
+

+ A scaling factor that is used to bracket the root: the value guess + is multiplied (or divided as appropriate) by factor + until two values are found that bracket the root. A value such as 2 + is a typical choice for factor. +

+
rising
+

+ Set to true if f(x) is rising + on x and false if f(x) + is falling on x. This value is used along with + the result of f(guess) to determine if guess + is above or below the root. +

+
tol
+

+ A binary functor that determines the termination condition for the + search for the root. tol is passed the current + brackets at each step, when it returns true then the current brackets + are returned as the result. +

+
max_iter
+

+ The maximum number of function invocations to perform in the search + for the root. +

+
+
+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Returns: a pair of values r that bracket the root + so that: +

+
+f(r.first) * f(r.second) <= 0
+
+

+ and either +

+
+tol(r.first, r.second) == true
+
+

+ or +

+
+max_iter >= m
+
+

+ where m is the initial value of max_iter + passed to the function. +

+

+ In other words, it's up to the caller to verify whether termination occurred + as a result of exceeding max_iter function invocations + (easily done by checking the value of max_iter when + the function returns), rather than because the termination condition tol + was satisfied. +

+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+template <class F, class T, class Tol>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      const T& fa, 
+      const T& fb, 
+      Tol tol, 
+      boost::uintmax_t& max_iter);
+      
+template <class F, class T, class Tol, class Policy>
+std::pair<T, T> 
+   toms748_solve(
+      F f, 
+      const T& a, 
+      const T& b, 
+      const T& fa, 
+      const T& fb, 
+      Tol tol, 
+      boost::uintmax_t& max_iter,
+      const Policy&);
+
+

+ These two functions implement TOMS Algorithm 748: it uses a mixture of + cubic, quadratic and linear (secant) interpolation to locate the root of + f(x). The two functions differ only by whether values + for f(a) and f(b) are already + available. The parameters are: +

+
+

+
+
f
+

+ A unary functor that is the function whose root is to be solved. f(x) + need not be uniformly increasing or decreasing on x + and may have multiple roots. +

+
a
+

+ The lower bound for the initial bracket of the root. +

+
b
+

+ The upper bound for the initial bracket of the root. It is a precondition + that a < b and that a + and b bracket the root to find so that f(a)*f(b) + < 0. +

+
fa
+

+ Optional: the value of f(a). +

+
fb
+

+ Optional: the value of f(b). +

+
tol
+

+ A binary functor that determines the termination condition for the + search for the root. tol is passed the current + brackets at each step, when it returns true then the current brackets + are returned as the result. +

+
max_iter
+

+ The maximum number of function invocations to perform in the search + for the root. On exit max_iter is set to actual + number of function invocations used. +

+
+
+

+

+

+ The final Policy argument + is optional and can be used to control the behaviour of the function: + how it handles errors, what level of precision to use etc. Refer to the + policy documentation for more details. +

+

+

+

+ Returns: a pair of values r that bracket the root + so that: +

+
+f(r.first) * f(r.second) <= 0
+
+

+ and either +

+
+tol(r.first, r.second) == true
+
+

+ or +

+
+max_iter >= m
+
+

+ where m is the initial value of max_iter + passed to the function. +

+

+ In other words, it's up to the caller to verify whether termination occurred + as a result of exceeding max_iter function invocations + (easily done by checking the value of max_iter), rather + than because the termination condition tol was satisfied. +

+
+template <class T>
+struct eps_tolerance
+{
+   eps_tolerance(int bits);
+   bool operator()(const T& a, const T& b)const;
+};
+
+

+ This is the usual termination condition used with these root finding functions. + Its operator() will return true when the relative distance between a + and b is less than twice the machine epsilon for T, + or 21-bits, whichever is the larger. In other words you set bits + to the number of bits of precision you want in the result. The minimal + tolerance of twice the machine epsilon of T is required to ensure that + we get back a bracketing interval: since this must clearly be at least + 1 epsilon in size. +

+
+struct equal_floor
+{
+   equal_floor();
+   template <class T> bool operator()(const T& a, const T& b)const;
+};
+
+

+ This termination condition is used when you want to find an integer result + that is the floor of the true root. It will terminate + as soon as both ends of the interval have the same floor. +

+
+struct equal_ceil
+{
+   equal_ceil();
+   template <class T> bool operator()(const T& a, const T& b)const;
+};
+
+

+ This termination condition is used when you want to find an integer result + that is the ceil of the true root. It will terminate + as soon as both ends of the interval have the same ceil. +

+
+struct equal_nearest_integer
+{
+   equal_nearest_integer();
+   template <class T> bool operator()(const T& a, const T& b)const;
+};
+
+

+ This termination condition is used when you want to find an integer result + that is the closest to the true root. It will terminate + as soon as both ends of the interval round to the same nearest integer. +

+
+ + Implementation +
+

+ The implementation of the bisection algorithm is extremely straightforward + and not detailed here. TOMS algorithm 748 is described in detail in: +

+

+ Algorithm 748: Enclosing Zeros of Continuous Functions, G. E. + Alefeld, F. A. Potra and Yixun Shi, ACM Transactions on Mathematica1 Software, + Vol. 21. No. 3. September 1995. Pages 327-344. +

+

+ The implementation here is a faithful translation of this paper into C++. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/series_evaluation.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/series_evaluation.html new file mode 100644 index 000000000..f68096d5e --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals1/series_evaluation.html @@ -0,0 +1,194 @@ + + + +Series Evaluation + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/series.hpp>
+
+

+

+
+namespace boost{ namespace math{ namespace tools{
+
+template <class Functor>
+typename Functor::result_type sum_series(Functor& func, int bits);
+
+template <class Functor>
+typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms);
+
+template <class Functor, class U>
+typename Functor::result_type sum_series(Functor& func, int bits, U init_value);
+
+template <class Functor, class U>
+typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms, U init_value);
+
+template <class Functor>
+typename Functor::result_type kahan_sum_series(Functor& func, int bits);
+
+template <class Functor>
+typename Functor::result_type kahan_sum_series(Functor& func, int bits, boost::uintmax_t& max_terms);
+
+}}} // namespaces
+
+
+ + Description +
+

+ These algorithms are intended for the summation + of infinite series. +

+

+ Each of the algorithms takes a nullary-function object as the first argument: + the function object will be repeatedly invoked to pull successive terms + from the series being summed. +

+

+ The second argument is the number of binary bits of precision required, + summation will stop when the next term is too small to have any effect + on the first bits bits of the result. +

+

+ The optional third argument max_terms sets an upper + limit on the number of terms of the series to evaluate. In addition, on + exit the function will set max_terms to the actual + number of terms of the series that were evaluated: this is particularly + useful for profiling the convergence properties of a new series. +

+

+ The final optional argument init_value is the initial + value of the sum to which the terms of the series should be added. This + is useful in two situations: +

+
    +
  • + Where the first value of the series has a different formula to successive + terms. In this case the first value in the series can be passed as the + last argument and the logic of the function object can then be simplified + to return subsequent terms. +
    • +
  • + Where the series is being added (or subtracted) from some other value: + termination of the series will likely occur much more rapidly if that + other value is passed as the last argument. For example, there are several + functions that can be expressed as 1 - S(z) where + S(z) is an infinite series. In this case, pass -1 as the last argument + and then negate the result of the summation to get the result of 1 + - S(z). +
    • +
+

+ The two kahan_sum_series variants of these algorithms + maintain a carry term that corrects for roundoff error during summation. + They are inspired by the Kahan + Summation Formula that appears in What + Every Computer Scientist Should Know About Floating-Point Arithmetic. + However, it should be pointed out that there are very few series that require + summation in this way. +

+
+ + Example +
+

+ Let's suppose we want to implement log(1+x) via its + infinite series, +

+

+ +

+

+ We begin by writing a small function object to return successive terms + of the series: +

+
+template <class T>
+struct log1p_series
+{
+   // we must define a result_type typedef:
+   typedef T result_type;
+
+   log1p_series(T x)
+      : k(0), m_mult(-x), m_prod(-1){}
+
+   T operator()()
+   {
+      // This is the function operator invoked by the summation
+      // algorithm, the first call to this operator should return
+      // the first term of the series, the second call the second 
+      // term and so on.
+      m_prod *= m_mult;
+      return m_prod / ++k; 
+   }
+
+private:
+   int k;
+   const T m_mult;
+   T m_prod;
+};
+
+

+ Implementing log(1+x) is now fairly trivial: +

+
+template <class T>
+T log1p(T x)
+{
+   // We really should add some error checking on x here!
+   assert(std::fabs(x) < 1);
+   
+   // construct the series functor:
+   log1p_series<T> s(x);
+   // and add it up, with enough digits for full machine precision
+   // plus a couple more for luck.... !
+   return tools::sum_series(s, tools::digits(x) + 2);
+}
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals2.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2.html new file mode 100644 index 000000000..2523f2900 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2.html @@ -0,0 +1,53 @@ + + + +Testing and Development + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+
Polynomials
+
Minimax Approximations + and the Remez Algorithm
+
Relative + Error and Testing
+
Graphing, + Profiling, and Generating Test Data for Special Functions
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/error_test.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/error_test.html new file mode 100644 index 000000000..b505d3cfe --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/error_test.html @@ -0,0 +1,241 @@ + + + +Relative Error and Testing + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/test.hpp>
+
+

+

+
+template <class T>
+T relative_error(T a, T b);
+
+template <class A, class F1, class F2>
+test_result<see-below> test(const A& a, F1 test_func, F2 expect_func);
+
+
+ + Description +
+
+template <class T>
+T relative_error(T a, T v);
+
+

+ Returns the relative error between a and v + using the usual formula: +

+

+ +

+

+ In addition the value returned is zero if: +

+
    +
  • + Both a and v are infinite. +
    • +
  • + Both a and v are denormalised + numbers or zero. +
    • +
+

+ Otherwise if only one of a and v + is zero then the value returned is 1. +

+
+template <class A, class F1, class F2>
+test_result<see-below> test(const A& a, F1 test_func, F2 expect_func);
+
+

+ This function is used for testing a function against tabulated test data. +

+

+ The return type contains statistical data on the relative errors (max, + mean, variance, and the number of test cases etc), as well as the row of + test data that caused the largest relative error. Public members of type + test_result are: +

+
+

+
+
unsigned + worst()const;
+

+ Returns the row at which the worst error occurred. +

+
T + min()const;
+

+ Returns the smallest relative error found. +

+
T + max()const;
+

+ Returns the largest relative error found. +

+
T + mean()const;
+

+ Returns the mean error found. +

+
boost::uintmax_t + count()const;
+

+ Returns the number of test cases. +

+
T + variance()const;
+

+ Returns the variance of the errors found. +

+
T + variance1()const;
+

+ Returns the unbiased variance of the errors found. +

+
T + rms()const
+

+ Returns the Root Mean Square, or quadratic mean of the errors. +

+
test_result& operator+=(const test_result& + t)
+

+ Combines two test_result's into a single result. +

+
+
+

+ The template parameter of test_result, is the same type as the values in + the two dimensional array passed to function test, + roughly that's A::value_type::value_type. +

+

+ Parameter a is a matrix of test data: and must be + a standard library Sequence type, that contains another Sequence type: + typically it will be a two dimensional instance of boost::array. + Each row of a should contain all the parameters that + are passed to the function under test as well as the expected result. +

+

+ Parameter test_func is the function under test, it + is invoked with each row of test data in a. Typically + type F1 is created with Boost.Lambda: see the example below. +

+

+ Parameter expect_func is a functor that extracts the + expected result from a row of test data in a. Typically + type F2 is created with Boost.Lambda: see the example below. +

+

+ If the function under test returns a non-finite value when a finite result + is expected, or if a gross error is found, then a message is sent to std::cerr, and a call to BOOST_ERROR() made + (which means that including this header requires you use Boost.Test). This + is mainly a debugging/development aid (and a good place for a breakpoint). +

+
+ + Example +
+

+ Suppose we want to test the tgamma and lgamma functions, we can create + a two dimensional matrix of test data, each row is one test case, and contains + three elements: the input value, and the expected results for the tgamma + and lgamma functions respectively. +

+
+static const boost::array<boost::array<TestType, 3>, NumberOfTests> 
+   factorials = {
+      /* big array of test data goes here */
+   };
+
+

+ Now we can invoke the test function to test tgamma: +

+
+using namespace boost::math::tools;
+using namespace boost::lambda;
+
+// get a pointer to the function under test:
+TestType (*funcp)(TestType) = boost::math::tgamma;
+
+// declare something to hold the result:
+test_result<TestType> result;
+//
+// and test tgamma against data:
+//
+result = test(
+   factorials, 
+   bind(funcp, ret<TestType>(_1[0])), // calls tgamma with factorials[row][0]
+   ret<TestType>(_1[1])               // extracts the expected result from factorials[row][1]
+);
+//
+// Print out some results:
+//
+std::cout << "The Mean was " << result.mean() << std::endl;
+std::cout << "The worst error was " << (result.max)() << std::endl;
+std::cout << "The worst error was at row " << result.worst_case() << std::endl;
+//
+// same again with lgamma this time:
+//
+funcp = boost::math::lgamma;
+result = test(
+   factorials, 
+   bind(funcp, ret<TestType>(_1[0])), // calls tgamma with factorials[row][0]
+   ret<TestType>(_1[2])               // extracts the expected result from factorials[row][2]
+);
+//
+// etc ...
+//
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/minimax.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/minimax.html new file mode 100644 index 000000000..6f0609275 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/minimax.html @@ -0,0 +1,273 @@ + + + +Minimax Approximations and the Remez Algorithm + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ The directory libs/math/minimax contains a command line driven program + for the generation of minimax approximations using the Remez algorithm. + Both polynomial and rational approximations are supported, although the + latter are tricky to converge: it is not uncommon for convergence of rational + forms to fail. No such limitations are present for polynomial approximations + which should always converge smoothly. +

+

+ It's worth stressing that developing rational approximations to functions + is often not an easy task, and one to which many books have been devoted. + To use this tool, you will need to have a reasonable grasp of what the + Remez algorithm is, and the general form of the approximation you want + to achieve. +

+

+ Unless you already familar with the Remez method you should first read + the brief background article + explaining the principals behind the Remez algorithm. +

+

+ The program consists of two parts: +

+
+

+
+
main.cpp
+

+ Contains the command line parser, and all the calls to the Remez code. +

+
f.cpp
+

+ Contains the function to approximate. +

+
+
+

+ Therefore to use this tool, you must modify f.cpp to return the function + to approximate. The tools supports multiple function approximations within + the same compiled program: each as a separate variant: +

+
+NTL::RR f(const NTL::RR& x, int variant);
+
+

+ Returns the value of the function variant at point + x. So if you wish you can just add the function to + approximate as a new variant after the existing examples. +

+

+ In addition to those two files, the program needs to be linked to a patched NTL library to compile. +

+

+ Note that the function f must return the rational + part of the approximation: for example if you are approximating a function + f(x) then it is quite common to use: +

+
+f(x) = g(x)(Y + R(x))
+
+

+ where g(x) is the dominant part of f(x), + Y is some constant, and R(x) + is the rational approximation part, usually optimised for a low absolute + error compared to |Y|. +

+

+ In this case you would define f to return f(x)/g(x) + and then set the y-offset of the approximation to Y + (see command line options below). +

+

+ Many other forms are possible, but in all cases the objective is to split + f(x) into a dominant part that you can evaluate easily + using standard math functions, and a smooth and slowly changing rational + approximation part. Refer to your favourite textbook for more examples. +

+

+ Command line options for the program are as follows: +

+
+

+
+
variant N
+

+ Sets the current function variant to N. This allows multiple functions + that are to be approximated to be compiled into the same executable. + Defaults to 0. +

+
range a b
+

+ Sets the domain for the approximation to the range [a,b], defaults + to [0,1]. +

+
relative
+

+ Sets the Remez code to optimise for relative error. This is the default + at program startup. Note that relative error can only be used if f(x) + has no roots over the range being optimised. +

+
absolute
+

+ Sets the Remez code to optimise for absolute error. +

+
pin [true|false]
+

+ "Pins" the code so that the rational approximation passes + through the origin. Obviously only set this to true + if R(0) must be zero. This is typically used when trying to preserve + a root at [0,0] while also optimising for relative error. +

+
order N D
+

+ Sets the order of the approximation to N in the + numerator and D in the denominator. If D + is zero then the result will be a polynomial approximation. There will + be N+D+2 coefficients in total, the first coefficient of the numerator + is zero if pin was set to true, and the first + coefficient of the denominator is always one. +

+
working-precision N
+

+ Sets the working precision of NTL::RR to N binary + digits. Defaults to 250. +

+
target-precision N
+

+ Sets the precision of printed output to N binary + digits: set to the same number of digits as the type that will be used + to evaluate the approximation. Defaults to 53 (for double precision). +

+
skew val
+

+ "Skews" the initial interpolated control points towards one + end or the other of the range. Positive values skew the initial control + points towards the left hand side of the range, and negative values + towards the right hand side. If an approximation won't converge (a + common situation) try adjusting the skew parameter until the first + step yields the smallest possible error. val should + be in the range [-100,+100], the default is zero. +

+
brake val
+

+ Sets a brake on each step so that the change in the control points + is braked by val%. Defaults to 50, try a higher + value if an approximation won't converge, or a lower value to get speedier + convergence. +

+
x-offset val
+

+ Sets the x-offset to val: the approximation will + be generated for f(x + X) + Y where X is + the x-offset and Y is the y-offset. Defaults to + zero. To avoid rounding errors, take care to specify a value that can + be exactly represented as a floating point number. +

+
y-offset val
+

+ Sets the y-offset to val: the approximation will + be generated for f(x + X) + Y where X is + the x-offset and Y is the y-offset. Defaults to + zero. To avoid rounding errors, take care to specify a value that can + be exactly represented as a floating point number. +

+
y-offset auto
+

+ Sets the y-offset to the average value of f(x) evaluated at the two + endpoints of the range plus the midpoint of the range. The calculated + value is deliberately truncated to float precision + (and should be stored as a float in your code). + The approximation will be generated for f(x + X) + Y where X is + the x-offset and Y is the y-offset. Defaults to + zero. +

+
graph N
+

+ Prints N evaluations of f(x) at evenly spaced points over the range + being optimised. If unspecified then N defaults + to 3. Use to check that f(x) is indeed smooth over the range of interest. +

+
step N
+

+ Performs N steps, or one step if N + is unspecified. After each step prints: the peek error at the extrema + of the error function of the approximation, the theoretical error term + solved for on the last step, and the maximum relative change in the + location of the Chebyshev control points. The approximation is converged + on the minimax solution when the two error terms are (approximately) + equal, and the change in the control points has decreased to a suitably + small value. +

+
test [float|double|long]
+

+ Tests the current approximation at float, double, or long double precision. + Useful to check for rounding errors in evaluating the approximation + at fixed precision. Tests are conducted at the extrema of the error + function of the approximation, and at the zeros of the error function. +

+
test [float|double|long] N
+

+ Tests the current approximation at float, double, or long double precision. + Useful to check for rounding errors in evaluating the approximation + at fixed precision. Tests are conducted at N evenly spaced points over + the range of the approximation. If none of [float|double|long] are + specified then tests using NTL::RR, this can be used to obtain the + error function of the approximation. +

+
rescale a b
+

+ Takes the current Chebeshev control points, and rescales them over + a new interval [a,b]. Sometimes this can be used to obtain starting + control points for an approximation that can not otherwise be converged. +

+
rotate
+

+ Moves one term from the numerator to the denominator, but keeps the + Chebyshev control points the same. Sometimes this can be used to obtain + starting control points for an approximation that can not otherwise + be converged. +

+
info
+

+ Prints out the current approximation: the location of the zeros of + the error function, the location of the Chebyshev control points, the + x and y offsets, and of course the coefficients of the polynomials. +

+
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/polynomials.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/polynomials.html new file mode 100644 index 000000000..8c063b194 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/polynomials.html @@ -0,0 +1,139 @@ + + + +Polynomials + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+
+ + Synopsis +
+

+ +

+
+#include <boost/math/tools/polynomial.hpp>
+
+

+

+
+namespace boost{ namespace math{ namespace tools{
+
+template <class T>
+class polynomial
+{
+public:
+   // typedefs:
+   typedef typename std::vector<T>::value_type value_type;
+   typedef typename std::vector<T>::size_type  size_type;
+
+   // construct:
+   polynomial(){}
+   template <class U>
+   polynomial(const U* data, unsigned order);
+   template <class U>
+   polynomial(const U& point);
+   
+   // access:
+   size_type size()const;
+   size_type degree()const;
+   value_type& operator[](size_type i);
+   const value_type& operator[](size_type i)const;
+
+   // operators:
+   template <class U>
+   polynomial& operator +=(const U& value);
+   template <class U>
+   polynomial& operator -=(const U& value);
+   template <class U>
+   polynomial& operator *=(const U& value);
+   template <class U>
+   polynomial& operator +=(const polynomial<U>& value);
+   template <class U>
+   polynomial& operator -=(const polynomial<U>& value);
+   template <class U>
+   polynomial& operator *=(const polynomial<U>& value);
+};
+
+template <class T>
+polynomial<T> operator + (const polynomial<T>& a, const polynomial<T>& b);
+template <class T>
+polynomial<T> operator - (const polynomial<T>& a, const polynomial<T>& b);
+template <class T>
+polynomial<T> operator * (const polynomial<T>& a, const polynomial<T>& b);
+
+template <class T, class U>
+polynomial<T> operator + (const polynomial<T>& a, const U& b);
+template <class T, class U>
+polynomial<T> operator - (const polynomial<T>& a, const U& b);
+template <class T, class U>
+polynomial<T> operator * (const polynomial<T>& a, const U& b);
+
+template <class U, class T>
+polynomial<T> operator + (const U& a, const polynomial<T>& b);
+template <class U, class T>
+polynomial<T> operator - (const U& a, const polynomial<T>& b);
+template <class U, class T>
+polynomial<T> operator * (const U& a, const polynomial<T>& b);
+
+template <class charT, class traits, class T>
+std::basic_ostream<charT, traits>& operator << 
+   (std::basic_ostream<charT, traits>& os, const polynomial<T>& poly);
+
+}}} // namespaces
+
+
+ + Description +
+

+ This is a fairly trivial class for polynomial manipulation. +

+

+ Implementation is currently of the "naive" variety, with O(N^2) + multiplication for example. This class should not be used in high-performance + computing environments: it is intended for the simple manipulation of small + polynomials, typically generated for special function approximation. +

+

+ Advanced manipulations: the FFT, division, GCD, factorisation etc are not + currently provided. Submissions for these are of course welcome :-) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/test_data.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/test_data.html new file mode 100644 index 000000000..0cef156ad --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals2/test_data.html @@ -0,0 +1,564 @@ + + + +Graphing, Profiling, and Generating Test Data for Special Functions + + + + + + + + + + + + + + + +
Boost C++ LibrariesHomeLibrariesPeopleFAQMore
+
+
+PrevUpHomeNext +
+
+
+

+ The class test_data and + associated helper functions are designed so that in just a few lines of + code you should be able to: +

+
    +
  • + Profile a continued fraction, or infinite series for convergence and + accuracy. +
    • +
  • + Generate csv data from a special function that can be imported into your + favorite graphing program (or spreadsheet) for further analysis. +
    • +
  • + Generate high precision test data. +
    • +
+
+ + Synopsis +
+
+namespace boost{ namespace math{ namespace tools{
+
+enum parameter_type
+{
+   random_in_range = 0,
+   periodic_in_range = 1,
+   power_series = 2,
+   dummy_param = 0x80,
+};
+
+template <class T>
+struct parameter_info;
+
+template <class T>
+parameter_info<T> make_random_param(T start_range, T end_range, int n_points);
+
+template <class T>
+parameter_info<T> make_periodic_param(T start_range, T end_range, int n_points);
+
+template <class T>
+parameter_info<T> make_power_param(T basis, int start_exponent, int end_exponent);
+
+template <class T>
+bool get_user_parameter_info(parameter_info<T>& info, const char* param_name);
+
+template <class T>
+class test_data
+{
+public:
+   typedef std::vector<T> row_type;
+   typedef row_type value_type;
+private:
+   typedef std::set<row_type> container_type;
+public:
+   typedef typename container_type::reference reference;
+   typedef typename container_type::const_reference const_reference;
+   typedef typename container_type::iterator iterator;
+   typedef typename container_type::const_iterator const_iterator;
+   typedef typename container_type::difference_type difference_type;
+   typedef typename container_type::size_type size_type;
+
+   // creation:
+   test_data(){}
+   template <class F>
+   test_data(F func, const parameter_info<T>& arg1);
+
+   // insertion:
+   template <class F>
+   test_data& insert(F func, const parameter_info<T>& arg1);
+
+   template <class F>
+   test_data& insert(F func, const parameter_info<T>& arg1, 
+                     const parameter_info<T>& arg2);
+
+   template <class F>
+   test_data& insert(F func, const parameter_info<T>& arg1, 
+                     const parameter_info<T>& arg2, 
+                     const parameter_info<T>& arg3);
+
+   void clear();
+
+   // access:
+   iterator begin();
+   iterator end();
+   const_iterator begin()const;
+   const_iterator end()const;
+   bool operator==(const test_data& d)const;
+   bool operator!=(const test_data& d)const;
+   void swap(test_data& other);
+   size_type size()const;
+   size_type max_size()const;
+   bool empty()const;
+
+   bool operator < (const test_data& dat)const;
+   bool operator <= (const test_data& dat)const;
+   bool operator > (const test_data& dat)const;
+   bool operator >= (const test_data& dat)const;
+};
+
+template <class charT, class traits, class T>
+std::basic_ostream<charT, traits>& write_csv(
+            std::basic_ostream<charT, traits>& os,
+            const test_data<T>& data);
+
+template <class charT, class traits, class T>
+std::basic_ostream<charT, traits>& write_csv(
+            std::basic_ostream<charT, traits>& os,
+            const test_data<T>& data,
+            const charT* separator);
+
+template <class T>
+std::ostream& write_code(std::ostream& os,
+                         const test_data<T>& data, 
+                         const char* name);
+                         
+}}} // namespaces
+
+
+ + Description +
+

+ This tool is best illustrated with the following series of examples. +

+

+ The functionality of test_data is split into the following parts: +

+
    +
  • + A functor that implements the function for which data is being generated: + this is the bit you have to write. +
    • +
  • + One of more parameters that are to be passed to the functor, these are + described in fairly abstract terms: give me N points distributed like + this etc. +
    • +
  • + The class test_data, that takes the functor and descriptions of the parameters + and computes how ever many output points have been requested, these are + stored in a sorted container. +
    • +
  • + Routines to iterate over the test_data container and output the data + in either csv format, or as C++ source code (as a table using Boost.Array). +
    • +
+
+ + Example + 1: Output Data for Graph Plotting +
+

+ For example, lets say we want to graph the lgamma function between -3 and + 100, one could do this like so: +

+
+#include <boost/math/tools/test_data.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+
+int main()
+{
+   using namespace boost::math::tools;
+   
+   // create an object to hold the data:
+   test_data<double> data;
+   
+   // insert 500 points at uniform intervals between just after -3 and 100:
+   double (*pf)(double) = boost::math::lgamma;
+   data.insert(pf, make_periodic_param(-3.0 + 0.00001, 100.0, 500));
+   
+   // print out in csv format:
+   write_csv(std::cout, data, ", ");
+   return 0;
+}
+
+

+ Which, when plotted, results in: +

+

+ lgamma +

+
+ + Example + 2: Creating Test Data +
+

+ As a second example, let's suppose we want to create highly accurate test + data for a special function. Since many special functions have two or more + independent parameters, it's very hard to effectively cover all of the + possible parameter space without generating gigabytes of data at great + computational expense. A second best approach is to provide the tools by + which a user (or the library maintainer) can quickly generate more data + on demand to probe the function over a particular domain of interest. +

+

+ In this example we'll generate test data for the beta function using NTL::RR at 1000 bit precision. + Rather than call our generic version of the beta function, we'll implement + a deliberately naive version of the beta function using lgamma, and rely + on the high precision of the data type used to get results accurate to + at least 128-bit precision. In this way our test data is independent of + whatever clever tricks we may wish to use inside the our beta function. +

+

+ To start with then, here's the function object that creates the test data: +

+
+#include <boost/math/tools/ntl.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+#include <boost/math/tools/test_data.hpp>
+#include <fstream>
+
+#include <boost/math/tools/test_data.hpp>
+
+using namespace boost::math::tools;
+
+struct beta_data_generator
+{
+   NTL::RR operator()(NTL::RR a, NTL::RR b)
+   {
+      //
+      // If we throw a domain error then test_data will
+      // ignore this input point. We'll use this to filter
+      // out all cases where a < b since the beta function
+      // is symmetrical in a and b:
+      //
+      if(a < b)
+         throw std::domain_error("");
+         
+      // very naively calculate spots with lgamma:
+      NTL::RR g1, g2, g3;
+      int s1, s2, s3;
+      g1 = boost::math::lgamma(a, &s1);
+      g2 = boost::math::lgamma(b, &s2);
+      g3 = boost::math::lgamma(a+b, &s3);
+      g1 += g2 - g3;
+      g1 = exp(g1);
+      g1 *= s1 * s2 * s3;
+      return g1;
+   }
+};
+
+

+ To create the data, we'll need to input the domains for a and b for which + the function will be tested: the function get_user_parameter_info + is designed for just that purpose. The start of main will look something + like: +

+
+// Set the precision on RR:
+NTL::RR::SetPrecision(1000); // bits.
+NTL::RR::SetOutputPrecision(40); // decimal digits.
+
+parameter_info<NTL::RR> arg1, arg2;
+test_data<NTL::RR> data;
+
+std::cout << "Welcome.\n"
+   "This program will generate spot tests for the beta function:\n"
+   "  beta(a, b)\n\n";
+
+bool cont;
+std::string line;
+
+do{
+   // prompt the user for the domain of a and b to test:
+   get_user_parameter_info(arg1, "a");
+   get_user_parameter_info(arg2, "b");
+   
+   // create the data:
+   data.insert(beta_data_generator(), arg1, arg2);
+   
+   // see if the user want's any more domains tested:
+   std::cout << "Any more data [y/n]?";
+   std::getline(std::cin, line);
+   boost::algorithm::trim(line);
+   cont = (line == "y");
+}while(cont);
+
+
+ + + + + +
[Caution]Caution

+ At this point one potential stumbling block should be mentioned: test_data<>::insert + will create a matrix of test data when there are two or more parameters, + so if we have two parameters and we're asked for a thousand points on + each, that's a million test points in total. Don't + say you weren't warned! +

+

+ There's just one final step now, and that's to write the test data to file: +

+
+std::cout << "Enter name of test data file [default=beta_data.ipp]";
+std::getline(std::cin, line);
+boost::algorithm::trim(line);
+if(line == "")
+   line = "beta_data.ipp";
+std::ofstream ofs(line.c_str());
+write_code(ofs, data, "beta_data");
+
+

+ The format of the test data looks something like: +

+
+#define SC_(x) static_cast<T>(BOOST_JOIN(x, L))
+   static const boost::array<boost::array<T, 3>, 1830>
+   beta_med_data = {
+      SC_(0.4883005917072296142578125),
+      SC_(0.4883005917072296142578125),
+      SC_(3.245912809500479157065104747353807392371), 
+      SC_(3.5808107852935791015625),
+      SC_(0.4883005917072296142578125),
+      SC_(1.007653173802923954909901438393379243537), 
+      /* ... lots of rows skipped */
+};
+
+

+ The first two values in each row are the input parameters that were passed + to our functor and the last value is the return value from the functor. + Had our functor returned a tuple rather than a value, then we would have + had one entry for each element in the tuple in addition to the input parameters. +

+

+ The first #define serves two purposes: +

+
    +
  • + It reduces the file sizes considerably: all those static_cast's + add up to a lot of bytes otherwise (they are needed to suppress compiler + warnings when T is narrower + than a long double). +
    • +
  • + It provides a useful customisation point: for example if we were testing + a user-defined type that has more precision than a long + double we could change it to: +
    • +
+

+ #define SC_(x) lexical_cast<T>(BOOST_STRINGIZE(x)) +

+

+ in order to ensure that no truncation of the values occurs prior to conversion + to T. Note that this isn't + used by default as it's rather hard on the compiler when the table is large. +

+
+ + Example + 3: Profiling a Continued Fraction for Convergence and Accuracy +
+

+ Alternatively, lets say we want to profile a continued fraction for convergence + and error. As an example, we'll use the continued fraction for the upper + incomplete gamma function, the following function object returns the next + aN and bN of the continued fraction each time it's invoked: +

+
+template <class T>
+struct upper_incomplete_gamma_fract
+{
+private:
+   T z, a;
+   int k;
+public:
+   typedef std::pair<T,T> result_type;
+
+   upper_incomplete_gamma_fract(T a1, T z1)
+      : z(z1-a1+1), a(a1), k(0)
+   {
+   }
+
+   result_type operator()()
+   {
+      ++k;
+      z += 2;
+      return result_type(k * (a - k), z);
+   }
+};
+
+

+ We want to measure both the relative error, and the rate of convergence + of this fraction, so we'll write a functor that returns both as a tuple: + class test_data will unpack the tuple for us, and create one column of + data for each element in the tuple (in addition to the input parameters): +

+
+#include <boost/math/tools/test_data.hpp>
+#include <boost/math/tools/test.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+#include <boost/math/tools/ntl.hpp>
+#include <boost/tr1/tuple.hpp>
+
+template <class T>
+struct profile_gamma_fraction
+{
+   typedef std::tr1::tuple<T, T> result_type;
+
+   result_type operator()(T val)
+   {
+      using namespace boost::math::tools;
+      // estimate the true value, using arbitary precision
+      // arithmetic and NTL::RR:
+      NTL::RR rval(val);
+      upper_incomplete_gamma_fract<NTL::RR> f1(rval, rval);
+      NTL::RR true_val = continued_fraction_a(f1, 1000);
+      //
+      // Now get the aproximation at double precision, along with the number of
+      // iterations required:
+      boost::uintmax_t iters = std::numeric_limits<boost::uintmax_t>::max();
+      upper_incomplete_gamma_fract<T> f2(val, val);
+      T found_val = continued_fraction_a(f2, std::numeric_limits<T>::digits, iters);
+      //
+      // Work out the relative error, as measured in units of epsilon:
+      T err = real_cast<T>(relative_error(true_val, NTL::RR(found_val)) / std::numeric_limits<T>::epsilon());
+      //
+      // now just return the results as a tuple:
+      return std::tr1::make_tuple(err, iters);
+   }
+};
+
+

+ Feeding that functor into test_data allows rapid output of csv data, for + whatever type T we may + be interested in: +

+
+int main()
+{
+   using namespace boost::math::tools;
+   // create an object to hold the data:
+   test_data<double> data;
+   // insert 500 points at uniform intervals between just after 0 and 100:
+   data.insert(profile_gamma_fraction<double>(), make_periodic_param(0.01, 100.0, 100));
+   // print out in csv format:
+   write_csv(std::cout, data, ", ");
+   return 0;
+}
+
+

+ This time there's no need to plot a graph, the first few rows are: +

+
+a and z,  Error/epsilon,  Iterations required
+
+0.01,     9723.14,        4726
+1.0099,   9.54818,        87
+2.0098,   3.84777,        40
+3.0097,   0.728358,       25
+4.0096,   2.39712,        21
+5.0095,   0.233263,       16
+
+

+ So it's pretty clear that this fraction shouldn't be used for small values + of a and z. +

+
+ + reference +
+

+

+

+ Most of this tool has been described already in the examples above, we'll + just add the following notes on the non-member functions: +

+
+template <class T>
+parameter_info<T> make_random_param(T start_range, T end_range, int n_points);
+
+

+ Tells class test_data to test n_points random values + in the range [start_range,end_range]. +

+
+template <class T>
+parameter_info<T> make_periodic_param(T start_range, T end_range, int n_points);
+
+

+ Tells class test_data to test n_points evenly spaced + values in the range [start_range,end_range]. +

+
+template <class T>
+parameter_info<T> make_power_param(T basis, int start_exponent, int end_exponent);
+
+

+ Tells class test_data to test points of the form basis + R * + 2expon for each expon in the range [start_exponent, + end_exponent], and R a random number in [0.5, 1]. +

+
+template <class T>
+bool get_user_parameter_info(parameter_info<T>& info, const char* param_name);
+
+

+ Prompts the user for the parameter range and form to use. +

+

+ Finally, if we don't want the parameter to be included in the output, we + can tell test_data by setting it a "dummy parameter": +

+
+parameter_info<double> p = make_random_param(2.0, 5.0, 10);
+p.type |= dummy_param;
+
+

+ This is useful when the functor used transforms the parameter in some way + before passing it to the function under test, usually the functor will + then return both the transformed input and the result in a tuple, so there's + no need for the original pseudo-parameter to be included in program output. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/toolkit/internals_overview.html b/doc/sf_and_dist/html/math_toolkit/toolkit/internals_overview.html new file mode 100644 index 000000000..dc941ea29 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/toolkit/internals_overview.html @@ -0,0 +1,58 @@ + + + +Overview + + + + + + + + + + + + + + + +
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+
+
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+
+
+

+ This section contains internal utilities used by the library's implementation + along with tools used in development and testing. These tools have only minimal + documentation, and crucially do not have stable interfaces. +

+

+ There is no doubt that these components can be improved, but they are also + largely incidental to the main purpose of this library. +

+

+ These tools are designed to "just get the job done", and receive + minimal documentation here, in the hopes that they will help stimulate further + submissions to this library. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/using_udt.html b/doc/sf_and_dist/html/math_toolkit/using_udt.html new file mode 100644 index 000000000..43cbf60f7 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/using_udt.html @@ -0,0 +1,55 @@ + + + +Use with User Defined Floating-Point Types + + + + + + + + + + + + + + + +
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+
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+
+
+
+
Using With NTL - a High-Precision + Floating-Point Library
+
Conceptual Requirements + for Real Number Types
+
Conceptual Requirements + for Distribution Types
+
Conceptual Archetypes + and Testing
+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/using_udt/archetypes.html b/doc/sf_and_dist/html/math_toolkit/using_udt/archetypes.html new file mode 100644 index 000000000..50986efd0 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/using_udt/archetypes.html @@ -0,0 +1,184 @@ + + + +Conceptual Archetypes and Testing + + + + + + + + + + + + + + + +
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+
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+
+
+

+ There are several concept archetypes available: +

+

+ +

+
+#include <boost/concepts/std_real_concept.hpp>
+

+

+
+namespace boost{
+namespace math{
+namespace concepts{
+
+class std_real_concept;
+
+}}} // namespaces
+
+

+ std_real_concept is an archetype + for the built-in Real types. +

+

+ The main purpose in providing this type is to verify that standard library + functions are found via a using declaration - bringing those functions into + the current scope - and not just because they happen to be in global scope. +

+

+ In order to ensure that a call to say pow + can be found either via argument dependent lookup, or failing that then in + the std namespace: all calls to standard library functions are unqualified, + with the std:: versions found via a using declaration to make them visible + in the current scope. Unfortunately it's all to easy to forget the using + declaration, and call the double version of the function that happens to + be in the global scope by mistake. +

+

+ For example if the code calls ::pow rather than std::pow, the code will cleanly + compile, but truncation of long doubles to double will cause a significant + loss of precision. In contrast a template instantiated with std_real_concept + will only compile if the all the standard + library functions used have been brought into the current scope with a using + declaration. +

+

+ There is a test program libs/math/test/std_real_concept_check.cpp + that instantiates every template in this library with type std_real_concept to verify it's usage of + standard library functions. +

+

+ +

+
+#include <boost/math/concepts/real_concept.hpp>
+

+

+
+namespace boost{ 
+namespace math{ 
+namespace concepts{
+
+class real_concept;
+
+}}} // namespaces
+
+

+ real_concept is an archetype + for user defined real types, + it declares it's standard library functions in it's own namespace: these + will only be found if they are called unqualified allowing argument dependent + lookup to locate them. In addition this type is useable at runtime: this + allows code that would not otherwise be exercised by the built-in floating + point types to be tested. There is no std::numeric_limits<> support + for this type, since this is not a conceptual requirement for RealType's. +

+

+ NTL RR is an example of a type meeting the requirements that this type models, + but note that use of a thin wrapper class is required: refer to "Using + With NTL - a High-Precision Floating-Point Library". +

+

+ There is no specific test case for type real_concept, + instead, since this type is usable at runtime, each individual test case + as well as testing float, double and long + double, also tests real_concept. +

+

+ +

+
+#include <boost/math/concepts/distribution.hpp>
+

+

+
+namespace boost{
+namespace math{
+namespace concepts{
+
+template <class RealType>
+class distribution_archetype;
+
+template <class Distribution>
+struct DistributionConcept;
+
+}}} // namespaces
+
+

+ The class template distribution_archetype + is a model of the Distribution + concept. +

+

+ The class template DistributionConcept + is a concept checking + class for distribution types. +

+

+ The test program distribution_concept_check.cpp + is responsible for using DistributionConcept + to verify that all the distributions in this library conform to the Distribution concept. +

+

+ The class template DistributionConcept + verifies the existence (but not proper function) of the non-member accessors + required by the Distribution + concept. These are checked by calls like +

+

+ v = pdf(dist, x); // (Result v is ignored). +

+

+ And in addition, those that accept two arguments do the right thing when + the arguments are of different types (the result type is always the same + as the distribution's value_type). (This is implemented by some additional + forwarding-functions in derived_accessors.hpp, so that there is no need for + any code changes. Likewise boilerplate versions of the hazard/chf/coefficient_of_variation + functions are implemented in there too.) +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/using_udt/concepts.html b/doc/sf_and_dist/html/math_toolkit/using_udt/concepts.html new file mode 100644 index 000000000..a59180ec4 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/using_udt/concepts.html @@ -0,0 +1,1296 @@ + + + +Conceptual Requirements for Real Number Types + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ The functions, and statistical distributions in this library can be used + with any type RealType that meets the conceptual requirements + given below. All the built in floating point types will meet these requirements. + User defined types that meet the requirements can also be used. For example, + with a thin wrapper class + one of the types provided with NTL (RR) + can be used. Submissions of binding to other extended precision types would + also be most welcome! +

+

+ The guiding principal behind these requirements, is that a RealType + behaves just like a built in floating point type. +

+
+ + Basic + Arithmetic Requirements +
+

+ These requirements are common to all of the functions in this library. +

+

+ In the following table r is an object of type RealType, cr and + cr2 are objects of type const + RealType, and ca + is an object of type const arithmetic-type (arithmetic types include all the + built in integers and floating point types). +

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Expression +

+ 		+

+ Result Type +

+ 		+

+ Notes +

+
+

+ RealType(cr) +

+ 		+

+ RealType +

+ 		+

+ RealType is copy constructible. +

+
+

+ RealType(ca) +

+ 		+

+ RealType +

+ 		+

+ RealType is copy constructible from the arithmetic types. +

+
+

+ r = + cr +

+ 		+

+ RealType& +

+ 		+

+ Assignment operator. +

+
+

+ r = + ca +

+ 		+

+ RealType& +

+ 		+

+ Assignment operator from the arithmetic types. +

+
+

+ r += + cr +

+ 		+

+ RealType& +

+ 		+

+ Adds cr to r. +

+
+

+ r += + ca +

+ 		+

+ RealType& +

+ 		+

+ Adds ar to r. +

+
+

+ r -= + cr +

+ 		+

+ RealType& +

+ 		+

+ Subtracts cr from r. +

+
+

+ r -= + ca +

+ 		+

+ RealType& +

+ 		+

+ Subtracts ca from r. +

+
+

+ r *= + cr +

+ 		+

+ RealType& +

+ 		+

+ Multiplies r by cr. +

+
+

+ r *= + ca +

+ 		+

+ RealType& +

+ 		+

+ Multiplies r by ca. +

+
+

+ r /= + cr +

+ 		+

+ RealType& +

+ 		+

+ Divides r by cr. +

+
+

+ r /= + ca +

+ 		+

+ RealType& +

+ 		+

+ Divides r by ca. +

+
+

+ -r +

+ 		+

+ RealType +

+ 		+

+ Unary Negation. +

+
+

+ +r +

+ 		+

+ RealType& +

+ 		+

+ Identity Operation. +

+
+

+ cr + + cr2 +

+ 		+

+ RealType +

+ 		+

+ Binary Addition +

+
+

+ cr + + ca +

+ 		+

+ RealType +

+ 		+

+ Binary Addition +

+
+

+ ca + + cr +

+ 		+

+ RealType +

+ 		+

+ Binary Addition +

+
+

+ cr - + cr2 +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ cr - + ca +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ ca - + cr +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ cr * + cr2 +

+ 		+

+ RealType +

+ 		+

+ Binary Multiplication +

+
+

+ cr * + ca +

+ 		+

+ RealType +

+ 		+

+ Binary Multiplication +

+
+

+ ca * + cr +

+ 		+

+ RealType +

+ 		+

+ Binary Multiplication +

+
+

+ cr / + cr2 +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ cr / + ca +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ ca / + cr +

+ 		+

+ RealType +

+ 		+

+ Binary Subtraction +

+
+

+ cr == + cr2 +

+ 		+

+ bool +

+ 		+

+ Equality Comparison +

+
+

+ cr == + ca +

+ 		+

+ bool +

+ 		+

+ Equality Comparison +

+
+

+ ca == + cr +

+ 		+

+ bool +

+ 		+

+ Equality Comparison +

+
+

+ cr != + cr2 +

+ 		+

+ bool +

+ 		+

+ Inequality Comparison +

+
+

+ cr != + ca +

+ 		+

+ bool +

+ 		+

+ Inequality Comparison +

+
+

+ ca != + cr +

+ 		+

+ bool +

+ 		+

+ Inequality Comparison +

+
+

+ cr <= + cr2 +

+ 		+

+ bool +

+ 		+

+ Less than equal to. +

+
+

+ cr <= + ca +

+ 		+

+ bool +

+ 		+

+ Less than equal to. +

+
+

+ ca <= + cr +

+ 		+

+ bool +

+ 		+

+ Less than equal to. +

+
+

+ cr >= + cr2 +

+ 		+

+ bool +

+ 		+

+ Greater than equal to. +

+
+

+ cr >= + ca +

+ 		+

+ bool +

+ 		+

+ Greater than equal to. +

+
+

+ ca >= + cr +

+ 		+

+ bool +

+ 		+

+ Greater than equal to. +

+
+

+ cr < + cr2 +

+ 		+

+ bool +

+ 		+

+ Less than comparison. +

+
+

+ cr < + ca +

+ 		+

+ bool +

+ 		+

+ Less than comparison. +

+
+

+ ca < + cr +

+ 		+

+ bool +

+ 		+

+ Less than comparison. +

+
+

+ cr > + cr2 +

+ 		+

+ bool +

+ 		+

+ Greater than comparison. +

+
+

+ cr > + ca +

+ 		+

+ bool +

+ 		+

+ Greater than comparison. +

+
+

+ ca > + cr +

+ 		+

+ bool +

+ 		+

+ Greater than comparison. +

+
+

+ boost::math::tools::digits<RealType>() +

+ 		+

+ int +

+ 		+

+ The number of digits in the significand of RealType. +

+
+

+ boost::math::tools::max_value<RealType>() +

+ 		+

+ RealType +

+ 		+

+ The largest representable number by type RealType. +

+
+

+ boost::math::tools::min_value<RealType>() +

+ 		+

+ RealType +

+ 		+

+ The smallest representable number by type RealType. +

+
+

+ boost::math::tools::log_max_value<RealType>() +

+ 		+

+ RealType +

+ 		+

+ The natural logarithm of the largest representable number by type + RealType. +

+
+

+ boost::math::tools::log_min_value<RealType>() +

+ 		+

+ RealType +

+ 		+

+ The natural logarithm of the smallest representable number by type + RealType. +

+
+

+ boost::math::tools::epsilon<RealType>() +

+ 		+

+ RealType +

+ 		+

+ The machine epsilon of RealType. +

+
+

+ Note that: +

+
    +
  1. + The functions log_max_value + and log_min_value can be + synthesised from the others, and so no explicit specialisation is required. +
    2. +
  2. + The function epsilon can + be synthesised from the others, so no explicit specialisation is required + provided the precision of RealType does not vary at runtime (see the header + boost/math/bindings/rr.hpp + for an example where the precision does vary at runtime). +
    3. +
  3. + The functions digits, + max_value and min_value, all get synthesised automatically + from std::numeric_limits. However, if numeric_limits is not specialised for + type RealType, then you will get a compiler error when code tries to use + these functions, unless you explicitly specialise + them. For example if the precision of RealType varies at runtime, then + numeric_limits support + may not be appropriate, see boost/math/tools/ntl.hpp + for examples. +
    4. +
+
+ + + + + +
[Warning]Warning
+

+ If std::numeric_limits<> + is not specialized for type RealType + then the default float precision of 6 decimal digits will be used by other + Boost programs including: +

+

+ Boost.Test: giving misleading error messages like +

+

+ "difference between {9.79796} and {9.79796} exceeds 5.42101e-19%". +

+

+ Boost.LexicalCast and Boost.Serialization when converting the number to + a string, causing potentially serious loss of accuracy on output. +

+

+ Although it might seem obvious that RealType should require std::numeric_limits to be specialized, this + is not sensible for NTL::RR + and similar classes where the number of digits is a runtime parameter (where + as for numeric_limits it + has to be fixed at compile time). +

+
+
+ + Standard + Library Support Requirements +
+

+ Many (though not all) of the functions in this library make calls to standard + library functions, the following table summarises the requirements. Note + that most of the functions in this library will only call a small subset + of the functions listed here, so if in doubt whether a user defined type + has enough standard library support to be useable the best advise is to try + it and see! +

+

+ In the following table r is an object of type RealType, cr1 and + cr2 are objects of type const + RealType, and i + is an object of type int. +

+
++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Expression +

+ 		+

+ Result Type +

+
+

+ fabs(cr1) +

+ 		+

+ RealType +

+
+

+ abs(cr1) +

+ 		+

+ RealType +

+
+

+ ceil(cr1) +

+ 		+

+ RealType +

+
+

+ floor(cr1) +

+ 		+

+ RealType +

+
+

+ exp(cr1) +

+ 		+

+ RealType +

+
+

+ pow(cr1, + cr2) +

+ 		+

+ RealType +

+
+

+ sqrt(cr1) +

+ 		+

+ RealType +

+
+

+ log(cr1) +

+ 		+

+ RealType +

+
+

+ frexp(cr1, + &i) +

+ 		+

+ RealType +

+
+

+ ldexp(cr1, + i) +

+ 		+

+ RealType +

+
+

+ cos(cr1) +

+ 		+

+ RealType +

+
+

+ sin(cr1) +

+ 		+

+ RealType +

+
+

+ asin(cr1) +

+ 		+

+ RealType +

+
+

+ tan(cr1) +

+ 		+

+ RealType +

+
+

+ atan(cr1) +

+ 		+

+ RealType +

+
+

+ Note that the table above lists only those standard library functions known + to be used (or likely to be used in the near future) by this library. The + following functions: acos, + atan2, fmod, + cosh, sinh, + tanh, modf + and log10 are not currently + used, but may be if further special functions are added. +

+

+ In addition, for efficient and accurate results, a Lanczos + approximation is highly desirable. You may be able to adapt an existing + approximation from boost/math/special_functions/lanczos.hpp + or libs/math/tools/ntl_rr_lanczos.hpp: + you will need change static_cast's to lexical_cast's, and the constants to + strings (in order to ensure the coefficients aren't + truncated to long double) and then specialise lanczos_traits + for type T. Otherwise you may have to hack libs/math/tools/lanczos_generator.cpp + to find a suitable approximation for your RealType. The code will still compile + if you don't do this, but both accuracy and efficiency will be greatly compromised + in any function that makes use of the gamma/beta/erf family of functions. +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
+
+PrevUpHomeNext +
+ + diff --git a/doc/sf_and_dist/html/math_toolkit/using_udt/dist_concept.html b/doc/sf_and_dist/html/math_toolkit/using_udt/dist_concept.html new file mode 100644 index 000000000..fec1930a1 --- /dev/null +++ b/doc/sf_and_dist/html/math_toolkit/using_udt/dist_concept.html @@ -0,0 +1,396 @@ + + + +Conceptual Requirements for Distribution Types + + + + + + + + + + + + + + + +
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+
+
+PrevUpHomeNext +
+
+
+

+ A DistributionType is a type that implements the following + conceptual requirements, and encapsulates a statistical distribution. +

+

+ Please note that this documentation should not be used as a substitute for + the reference documentation, + and tutorial of the statistical + distributions. +

+

+ In the following table, d is an object of type DistributionType, cd + is an object of type const DistributionType and cr + is an object of a type convertible to RealType. +

+
+++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+

+ Expression +

+ 		+

+ Result Type +

+ 		+

+ Notes +

+
+

+ DistributionType::value_type +

+ 		+

+ RealType +

+ 		+

+ The real-number type RealType upon which the + distribution operates. +

+
+

+ DistributionType::policy_type +

+ 		+

+ RealType +

+ 		+

+ The Policy to use when + evaluating functions that depend on this distribution. +

+
+

+ d = cd +

+ 		+

+ Distribution& +

+ 		+

+ Distribution types are assignable. +

+
+

+ Distribution(cd) +

+ 		+

+ Distribution +

+ 		+

+ Distribution types are copy constructible. +

+
+

+ pdf(cd, cr) +

+ 		+

+ RealType +

+ 		+

+ Returns the PDF of the distribution. +

+
+

+ cdf(cd, cr) +

+ 		+

+ RealType +

+ 		+

+ Returns the CDF of the distribution. +

+
+

+ cdf(complement(cd, cr)) +

+ 		+

+ RealType +

+ 		+

+ Returns the complement of the CDF of the distribution, the same as: + 1-cdf(cd, cr) +

+
+

+ quantile(cd, cr) +

+ 		+

+ RealType +

+ 		+

+ Returns the quantile of the distribution. +

+
+

+ quantile(complement(cd, cr)) +

+ 		+

+ RealType +

+ 		+

+ Returns the quantile of the distribution, starting from the complement + of the probability, the same as: quantile(cd, 1-cr) +

+
+

+ chf(cd, cr) +

+ 		+

+ RealType +

+ 		+

+ Returns the cumulative hazard function of the distribution. +

+
+

+ hazard(cd, cr) +

+ 		+

+ RealType +

+ 		+

+ Returns the hazard function of the distribution. +

+
+

+ kurtosis(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the kurtosis of the distribution. +

+
+

+ kurtosis_excess(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the kurtosis excess of the distribution. +

+
+

+ mean(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the mean of the distribution. +

+
+

+ mode(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the mode of the distribution. +

+
+

+ skewness(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the skewness of the distribution. +

+
+

+ standard_deviation(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the standard deviation of the distribution. +

+
+

+ variance(cd) +

+ 		+

+ RealType +

+ 		+

+ Returns the variance of the distribution. +

+
+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

+
+
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+ The special functions and tools in this library can be used with NTL::RR + (an arbitrary precision number type), via the bindings in boost/math/bindings/rr.hpp. + See also NTL: A Library for doing Number + Theory by Victor Shoup +

+

+ Unfortunately NTL::RR doesn't quite satisfy our conceptual + requirements, so there is a very thin wrapper class boost::math::ntl::RR defined in boost/math/bindings/rr.hpp + that you should use in place of NTL::RR. The + class is intended to be a drop-in replacement for the "real" NTL::RR + that adds some syntactic sugar to keep this library happy, plus some of the + standard library functions not implemented in NTL. +

+

+ Finally there is a high precision Lanczos + approximation suitable for use with boost::math::ntl::RR, used at 1000-bit precision in libs/math/tools/ntl_rr_lanczos.hpp. + The approximation has a theoretical precision of > 90 decimal digits, + and an experimental precision of > 100 decimal digits. To use that approximation, + just include that header before any of the special function headers (if you + don't use it, you'll get a slower, but fully generic implementation for all + of the gamma-like functions). +

+
+ + + +
Copyright © 2006 -2007 John Maddock, Paul A. Bristow, Hubert Holin + and Xiaogang Zhang

+ Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

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+ + diff --git a/doc/sf_and_dist/html4_symbols.qbk b/doc/sf_and_dist/html4_symbols.qbk new file mode 100644 index 000000000..43b472863 --- /dev/null +++ b/doc/sf_and_dist/html4_symbols.qbk @@ -0,0 +1,149 @@ +[/ Symbols and Greek letters (about 120) from HTML4 ] +[/ File HTML4_symbols.qbk] +[/ See http://www.htmlhelp.com/reference/html40/entities/symbols.html] +[/ http://www.alanwood.net/demos/ent4_frame.html] +[/ All (except 2 angle brackets) show OK on Firefox 2.0] + +[/ See also Latin-1 aka Western (ISO-8859-1) in latin1_symbols.qbk] +[/ http://www.htmlhelp.com/reference/html40/entities/latin1.html] + +[/ Also some miscellaneous math charaters added to this list - see the end.] + +[/ To use, enclose the template name in square brackets.] + +[template fnof[]'''ƒ'''] [/ ƒ Latin small f with hook = function = florin] +[template Alpha[]'''Α'''] [/ ? Greek capital letter alpha] +[template Beta[]'''Β'''] [/ ? Greek capital letter beta] +[template Gamma[]'''Γ'''] [/ G Greek capital letter gamma] +[template Delta[]'''Δ'''] [/ ? Greek capital letter delta] +[template Epsilon[]'''Ε'''] [/ ? Greek capital letter epsilon] +[template Zeta[]'''Ζ'''] [/ ? Greek capital letter zeta] +[template Eta[]'''Η'''] [/ ? Greek capital letter eta] +[template Theta[]'''Θ'''] [/ T Greek capital letter theta] +[template Iota[]'''Ι'''] [/ ? Greek capital letter iota] +[template Kappa[]'''Κ'''] [/ ? Greek capital letter kappa] +[template Lambda[]'''Λ'''] [/ ? Greek capital letter lambda] +[template Mu[]'''Μ'''] [/ ? Greek capital letter mu] +[template Nu[]'''Ν'''] [/ ? Greek capital letter nu] +[template Xi[]'''Ξ'''] [/ ? Greek capital letter xi] +[template Omicron[]'''Ο'''] [/ ? Greek capital letter omicron] +[template Pi[]'''Π'''] [/ ? Greek capital letter pi] +[template Rho[]'''Ρ'''] [/ ? Greek capital letter rho] +[template Sigma[]'''Σ'''] [/ S Greek capital letter sigma] +[template Tau[]'''Τ'''] [/ ? Greek capital letter tau] +[template Upsilon[]'''Υ'''] [/ ? Greek capital letter upsilon] +[template Phi[]'''Φ'''] [/ F Greek capital letter phi] +[template Chi[]'''Χ'''] [/ ? Greek capital letter chi] +[template Psi[]'''Ψ'''] [/ ? Greek capital letter psi] +[template Omega[]'''Ω'''] [/ O Greek capital letter omega] +[template alpha[]'''α'''] [/ a Greek small letter alpha] +[template beta[]'''β'''] [/ ß Greek small letter beta] +[template gamma[]'''γ'''] [/ ? Greek small letter gamma] +[template delta[]'''δ'''] [/ d Greek small letter delta] +[template epsilon[]'''ε'''] [/ e Greek small letter epsilon] +[template zeta[]'''ζ'''] [/ ? Greek small letter zeta] +[template eta[]'''η'''] [/ ? Greek small letter eta] +[template theta[]'''θ'''] [/ ? Greek small letter theta] +[template iota[]'''ι'''] [/ ? Greek small letter iota] +[template kappa[]'''κ'''] [/ ? Greek small letter kappa] +[template lambda[]'''λ'''] [/ ? Greek small letter lambda] +[template mu[]'''μ'''] [/ µ Greek small letter mu] +[template nu[]'''ν'''] [/ ? Greek small letter nu] +[template xi[]'''ξ'''] [/ ? Greek small letter xi] +[template omicron[]'''ο'''] [/ ? Greek small letter omicron] +[template pi[]'''π'''] [/ p Greek small letter pi] +[template rho[]'''ρ'''] [/ ? Greek small letter rho] +[template sigmaf[]'''ς'''] [/ ? Greek small letter final sigma] +[template sigma[]'''σ'''] [/ s Greek small letter sigma] +[template tau[]'''τ'''] [/ t Greek small letter tau] +[template upsilon[]'''υ'''] [/ ? Greek small letter upsilon] +[template phi[]'''φ'''] [/ f Greek small letter phi] +[template chi[]'''χ'''] [/ ? Greek small letter chi] +[template psi[]'''ψ'''] [/ ? Greek small letter psi] +[template omega[]'''ω'''] [/ ? Greek small letter omega] +[template thetasym[]'''ϑ'''] [/ ? Greek small letter theta symbol] +[template upsih[]'''ϒ'''] [/ ? Greek upsilon with hook symbol] +[template piv[]'''ϖ'''] [/ ? Greek pi symbol] +[template bull[]'''•'''] [/ • bullet = black small circle] +[template hellip[]'''…'''] [/ … horizontal ellipsis = three dot leader] +[template prime[]'''′'''] [/ ' prime = minutes = feet] +[template Prime[]'''″'''] [/ ? double prime = seconds = inches] +[template oline[]'''‾'''] [/ ? overline = spacing overscore] +[template frasl[]'''⁄'''] [/ / fraction slash] +[template weierp[]'''℘'''] [/ P script capital P = power set = Weierstrass p] +[template image[]'''ℑ'''] [/ I blackletter capital I = imaginary part] +[template real[]'''ℜ'''] [/ R blackletter capital R = real part symbol] +[template trade[]'''™'''] [/ ™ trade mark sign] +[template alefsym[]'''ℵ'''] [/ ? alef symbol = first transfinite cardinal] +[template larr[]'''←'''] [/ ? leftwards arrow] +[template uarr[]'''↑'''] [/ ? upwards arrow] +[template rarr[]'''→'''] [/ ? rightwards arrow] +[template darr[]'''↓'''] [/ ? downwards arrow] +[template harr[]'''↔'''] [/ ? left right arrow] +[template crarr[]'''↵'''] [/ ? downwards arrow with corner leftwards = CR] +[template lArr[]'''⇐'''] [/ ? leftwards double arrow] +[template uArr[]'''⇑'''] [/ ? upwards double arrow] +[template rArr[]'''⇒'''] [/ ? rightwards double arrow] +[template dArr[]'''⇓'''] [/ ? downwards double arrow] +[template hArr[]'''⇔'''] [/ ? left right double arrow] +[template forall[]'''∀'''] [/ ? for all] +[template part[]'''∂'''] [/ ? partial differential] +[template exist[]'''∃'''] [/ ? there exists] +[template empty[]'''∅'''] [/ Ø empty set = null set = diameter] +[template nabla[]'''∇'''] [/ ? nabla = backward difference] +[template isin[]'''∈'''] [/ ? element of] +[template notin[]'''∉'''] [/ ? not an element of] +[template ni[]'''∋'''] [/ ? contains as member] +[template prod[]'''∏'''] [/ ? n-ary product = product sign] +[template sum[]'''∑'''] [/ ? n-ary sumation] +[template minus[]'''−'''] [/ - minus sign] +[template lowast[]'''∗'''] [/ * asterisk operator] +[template radic[]'''√'''] [/ v square root = radical sign] +[template prop[]'''∝'''] [/ ? proportional to] +[template infin[]'''∞'''] [/ 8 infinity] +[template ang[]'''∠'''] [/ ? angle] +[template and[]'''∧'''] [/ ? logical and = wedge] +[template or[]'''∨'''] [/ ? logical or = vee] +[template cap[]'''∩'''] [/ n intersection = cap] +[template cup[]'''∪'''] [/ ? union = cup] +[template int[]'''∫'''] [/ ? integral] +[template there4[]'''∴'''] [/ ? therefore] +[template sim[]'''∼'''] [/ ~ tilde operator = varies with = similar to] +[template cong[]'''≅'''] [/ ? approximately equal to] +[template asymp[]'''≈'''] [/ ˜ almost equal to = asymptotic to] +[template ne[]'''≠'''] [/ ? not equal to] +[template equiv[]'''≡'''] [/ = identical to] +[template le[]'''≤'''] [/ = less-than or equal to] +[template ge[]'''≥'''] [/ = greater-than or equal to] +[template subset[]'''⊂'''] [/ ? subset of] +[template superset[]'''⊃'''] [/ ? superset of] +[template nsubset[]'''⊄'''] [/ ? not a subset of] +[template sube[]'''⊆'''] [/ ? subset of or equal to] +[template supe[]'''⊇'''] [/ ? superset of or equal to] +[template oplus[]'''⊕'''] [/ ? circled plus = direct sum] +[template otimes[]'''⊗'''] [/ ? circled times = vector product] +[template perp[]'''⊥'''] [/ ? up tack = orthogonal to = perpendicular] +[template sdot[]'''⋅'''] [/ · dot operator] +[template lceil[]'''⌈'''] [/ ? left ceiling = APL upstile] +[template rceil[]'''⌉'''] [/ ? right ceiling] +[template lfloor[]'''⌊'''] [/ ? left floor = APL downstile] +[template rfloor[]'''⌋'''] [/ ? right floor] +[template lang[]'''〈'''] [/ < left-pointing angle bracket = bra (Firefox shows ?)] +[template rang[]'''〉'''] [/ > right-pointing angle bracket = ket (Firefox shows ?)] +[template loz[]'''◊'''] [/ ? lozenge] +[template spades[]'''♠'''] [/ ? black spade suit] +[template clubs[]'''♣'''] [/ ? black club suit = shamrock] +[template hearts[]'''♥'''] [/ ? black heart suit = valentine] +[template diams[]'''♦'''] [/ ? black diamond suit] + +[/ Other symbols, not in the HTML4 list:] +[template space[]''' '''] +[template plusminus[]'''±'''] [/ ? plus or minus sign] + +[/ +Copyright 2007 Paul A. Bristow. +Distributed under the Boost Software License, Version 1.0. +(See accompanying file LICENSE_1_0.txt or copy at +http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/ibeta.qbk b/doc/sf_and_dist/ibeta.qbk new file mode 100644 index 000000000..73a1eb330 --- /dev/null +++ b/doc/sf_and_dist/ibeta.qbk @@ -0,0 +1,294 @@ +[section:ibeta_function Incomplete Beta Functions] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` ibeta(T1 a, T2 b, T3 x); + + template + ``__sf_result`` ibeta(T1 a, T2 b, T3 x, const ``__Policy``&); + + template + ``__sf_result`` ibetac(T1 a, T2 b, T3 x); + + template + ``__sf_result`` ibetac(T1 a, T2 b, T3 x, const ``__Policy``&); + + template + ``__sf_result`` beta(T1 a, T2 b, T3 x); + + template + ``__sf_result`` beta(T1 a, T2 b, T3 x, const ``__Policy``&); + + template + ``__sf_result`` betac(T1 a, T2 b, T3 x); + + template + ``__sf_result`` betac(T1 a, T2 b, T3 x, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +There are four [@http://en.wikipedia.org/wiki/Incomplete_beta_function incomplete beta functions] +: two are normalised versions (also known as ['regularized] beta functions) +that return values in the range [0, 1], and two are non-normalised and +return values in the range [0, __beta(a, b)]. Users interested in statistical +applications should use the normalised (or +[@http://mathworld.wolfram.com/RegularizedBetaFunction.html regularized] +) versions (ibeta and ibetac). + +All of these functions require /a > 0/, /b > 0/ and /0 <= x <= 1/. + +The return type of these functions is computed using the __arg_pomotion_rules +when T1, T2 and T3 are different types. + +[optional_policy] + + template + ``__sf_result`` ibeta(T1 a, T2 b, T3 x); + + template + ``__sf_result`` ibeta(T1 a, T2 b, T3 x, const ``__Policy``&); + +Returns the normalised incomplete beta function of a, b and x: + +[equation ibeta3] + +[$../graphs/ibeta.png] + + template + ``__sf_result`` ibetac(T1 a, T2 b, T3 x); + + template + ``__sf_result`` ibetac(T1 a, T2 b, T3 x, const ``__Policy``&); + +Returns the normalised complement of the incomplete beta function of a, b and x: + +[equation ibeta4] + + template + ``__sf_result`` beta(T1 a, T2 b, T3 x); + + template + ``__sf_result`` beta(T1 a, T2 b, T3 x, const ``__Policy``&); + +Returns the full (non-normalised) incomplete beta function of a, b and x: + +[equation ibeta1] + + template + ``__sf_result`` betac(T1 a, T2 b, T3 x); + + template + ``__sf_result`` betac(T1 a, T2 b, T3 x, const ``__Policy``&); + +Returns the full (non-normalised) complement of the incomplete beta function of a, b and x: + +[equation ibeta2] + +[h4 Accuracy] + +The following tables give peak and mean relative errors in over various domains of +a, b and x, along with comparisons to the __gsl and __cephes libraries. +Note that only results for the widest floating-point type on the system are given as +narrower types have __zero_error. + +Note that the results for 80 and 128-bit long doubles are noticeably higher than +for doubles: this is because the wider exponent range of these types allow +more extreme test cases to be tested. For example expected results that +are zero at double precision, may be finite but exceptionally small with +the wider exponent range of the long double types. + +[table Errors In the Function ibeta(a,b,x) +[[Significand Size] [Platform and Compiler] [0 < a,b < 10 + +and + +0 < x < 1] [0 < a,b < 100 + +and + +0 < x < 1][1x10[super -5] < a,b < 1x10[super 5] + +and + +0 < x < 1]] +[[53] [Win32, Visual C++ 8] + [Peak=42.3 Mean=2.9 + +(GSL Peak=682 Mean=32.5) + +(__cephes Peak=42.7 Mean=7.0)] + [Peak=108 Mean=16.6 + +(GSL Peak=690 Mean=151) + +(__cephes Peak=1545 Mean=218)] + [Peak=4x10[super 3][space] Mean=203 + +(GSL Peak~3x10[super 5][space] Mean~2x10[super 4][space]) + +(__cephes Peak~5x10[super 5][space] Mean~2x10[super 4][space])]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=21.9 Mean=3.1] + [Peak=270.7 Mean=26.8] [Peak~5x10[super 4][space] Mean=3x10[super 3][space] ]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=15.4 Mean=3.0] [Peak=112.9 Mean=14.3] + [Peak~5x10[super 4][space] Mean=3x10[super 3][space]]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=20.9 Mean=2.6] [Peak=88.1 Mean=14.3] + [Peak~2x10[super 4][space] Mean=1x10[super 3][space] ]] +] + +[table Errors In the Function ibetac(a,b,x) +[[Significand Size] [Platform and Compiler] [0 < a,b < 10 + +and + +0 < x < 1] [0 < a,b < 100 + +and + +0 < x < 1][1x10[super -5] < a,b < 1x10[super 5] + +and + +0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peak=13.9 Mean=2.0] + [Peak=56.2 Mean=14] [Peak=3x10[super 3][space] Mean=159]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=21.1 Mean=3.6] + [Peak=221.7 Mean=25.8] + [Peak~9x10[super 4][space] Mean=3x10[super 3][space] ]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=10.6 Mean=2.2] + [Peak=73.9 Mean=11.9] + [Peak~9x10[super 4][space] Mean=3x10[super 3][space] ]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=9.9 Mean=2.6] + [Peak=117.7 Mean=15.1] + [Peak~3x10[super 4][space] Mean=1x10[super 3][space] ]] +] + +[table Errors In the Function beta(a, b, x) +[[Significand Size] [Platform and Compiler] [0 < a,b < 10 + +and + +0 < x < 1] [0 < a,b < 100 + +and + +0 < x < 1][1x10[super -5] < a,b < 1x10[super 5] + +and + +0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peak=39 Mean=2.9] [Peak=91 Mean=12.7] [Peak=635 Mean=25]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=26 Mean=3.6] [Peak=180.7 Mean=30.1] [Peak~7x10[super 4][space] Mean=3x10[super 3][space] ]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=13 Mean=2.4] [Peak=67.1 Mean=13.4] [Peak~7x10[super 4][space] Mean=3x10[super 3][space] ]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=27.3 Mean=3.6] [Peak=49.8 Mean=9.1] [Peak~6x10[super 4][space] Mean=3x10[super 3][space] ]] +] + +[table Errors In the Function betac(a,b,x) +[[Significand Size] [Platform and Compiler] [0 < a,b < 10 + +and + +0 < x < 1] [0 < a,b < 100 + +and + +0 < x < 1][1x10[super -5] < a,b < 1x10[super 5] + +and + +0 < x < 1]] +[[53] [Win32, Visual C++ 8] [Peak=12.0 Mean=2.4] [Peak=91 Mean=15] [Peak=4x10[super 3][space] Mean=113]] +[[64] [Redhat Linux IA32, gcc-3.4.4] [Peak=19.8 Mean=3.8] [Peak=295.1 Mean=33.9] [Peak~1x10[super 5][space] Mean=5x10[super 3][space] ]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=11.2 Mean=2.4] [Peak=63.5 Mean=13.6] [Peak~1x10[super 5][space] Mean=5x10[super 3][space] ]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=15.6 Mean=3.5] [Peak=39.8 Mean=8.9] [Peak~9x10[super 4][space] Mean=5x10[super 3][space] ]] +] + +[h4 Testing] + +There are two sets of tests: spot tests compare values taken from +[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized Mathworld's online function evaluator] +with this implementation: they provide a basic "sanity check" +for the implementation, with one spot-test in each implementation-domain +(see implementation notes below). + +Accuracy tests use data generated at very high precision +(with [@http://shoup.net/ntl/doc/RR.txt NTL RR class] set at 1000-bit precision), +using the "textbook" continued fraction representation (refer to the first continued +fraction in the implementation discussion below). +Note that this continued fraction is /not/ used in the implementation, +and therefore we have test data that is fully independent of the code. + +[h4 Implementation] + +This implementation is closely based upon +[@http://portal.acm.org/citation.cfm?doid=131766.131776 "Algorithm 708; Significant digit computation of the incomplete beta function ratios", DiDonato and Morris, ACM, 1992.] + +All four of these functions share a common implementation: this is passed both +x and y, and can return either p or q where these are related by: + +[equation ibeta_inv5] + +so at any point we can swap a for b, x for y and p for q if this results in +a more favourable position. Generally such swaps are performed so that we always +compute a value less than 0.9: when required this can then be subtracted from 1 +without undue cancellation error. + +The following continued fraction representation is found in many textbooks +but is not used in this implementation - it's both slower and less accurate than +the alternatives - however it is used to generate test data: + +[equation ibeta5] + +The following continued fraction is due to [@http://portal.acm.org/citation.cfm?doid=131766.131776 Didonato and Morris], +and is used in this implementation when a and b are both greater than 1: + +[equation ibeta6] + +For smallish b and x then a series representation can be used: + +[equation ibeta7] + +When b << a then the transition from 0 to 1 occurs very close to x = 1 +and some care has to be taken over the method of computation, in that case +the following series representation is used: + +[equation ibeta8] +[/[equation ibeta9]] + +Where Q(a,x) is an [@http://functions.wolfram.com/GammaBetaErf/Gamma2/ incomplete gamma function]. +Note that this method relies +on keeping a table of all the p[sub n ] previously computed, which does limit +the precision of the method, depending upon the size of the table used. + +When /a/ and /b/ are both small integers, then we can relate the incomplete +beta to the binomial distribution and use the following finite sum: + +[equation ibeta12] + +Finally we can sidestep difficult areas, or move to an area with a more +efficient means of computation, by using the duplication formulae: + +[equation ibeta10] + +[equation ibeta11] + +The domains of a, b and x for which the various methods are used are identical +to those described in the +[@http://portal.acm.org/citation.cfm?doid=131766.131776 Didonato and Morris TOMS 708 paper]. + +[endsect][/section:ibeta_function The Incomplete Beta Function] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/ibeta_inv.qbk b/doc/sf_and_dist/ibeta_inv.qbk new file mode 100644 index 000000000..09d8b5fc1 --- /dev/null +++ b/doc/sf_and_dist/ibeta_inv.qbk @@ -0,0 +1,352 @@ +[section:ibeta_inv_function The Incomplete Beta Function Inverses] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, const ``__Policy``&); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py, const ``__Policy``&); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, const ``__Policy``&); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py, const ``__Policy``&); + + template + ``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p); + + template + ``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p, const ``__Policy``&); + + template + ``__sf_result`` ibetac_inva(T1 b, T2 x, T3 q); + + template + ``__sf_result`` ibetac_inva(T1 b, T2 x, T3 q, const ``__Policy``&); + + template + ``__sf_result`` ibeta_invb(T1 a, T2 x, T3 p); + + template + ``__sf_result`` ibeta_invb(T1 a, T2 x, T3 p, const ``__Policy``&); + + template + ``__sf_result`` ibetac_invb(T1 a, T2 x, T3 q); + + template + ``__sf_result`` ibetac_invb(T1 a, T2 x, T3 q, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + + +There are six [@http://functions.wolfram.com/GammaBetaErf/ incomplete beta function inverses] +which allow you solve for +any of the three parameters to the incomplete beta, starting from either +the result of the incomplete beta (p) or its complement (q). + +[optional_policy] + +[tip When people normally talk about the inverse of the incomplete +beta function, they are talking about inverting on parameter /x/. +These are implemented here as ibeta_inv and ibeta_inv, and are by +far the most efficient of the inverses presented here. + +The inverses on the /a/ and /b/ parameters find use in some statistical +applications, but have to be computed by rather brute force numerical +techniques and are consequently several times slower. +These are implemented here as ibeta_inva and ibeta_invb, +and complement versions ibetac_inva and ibetac_invb.] + +The return type of these functions is computed using the __arg_pomotion_rules +when called with arguments T1...TN of different types. + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, const ``__Policy``&); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py); + + template + ``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py, const ``__Policy``&); + +Returns a value /x/ such that: `p = ibeta(a, b, x);` +and sets `*py = 1 - x` when the `py` parameter is provided and is non-null. +Note that internally this function computes whichever is the smaller of +`x` and `1-x`, and therefore the value assigned to `*py` is free from +cancellation errors. That means that even if the function returns `1`, the +value stored in `*py` may be non-zero, albeit very small. + +Requires: /a,b > 0/ and /0 <= p <= 1/. + +[optional_policy] + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, const ``__Policy``&); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py); + + template + ``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py, const ``__Policy``&); + +Returns a value /x/ such that: `q = ibetac(a, b, x);` +and sets `*py = 1 - x` when the `py` parameter is provided and is non-null. +Note that internally this function computes whichever is the smaller of +`x` and `1-x`, and therefore the value assigned to `*py` is free from +cancellation errors. That means that even if the function returns `1`, the +value stored in `*py` may be non-zero, albeit very small. + +Requires: /a,b > 0/ and /0 <= q <= 1/. + +[optional_policy] + + template + ``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p); + + template + ``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p, const ``__Policy``&); + +Returns a value /a/ such that: `p = ibeta(a, b, x);` + +Requires: /b > 0/, /0 < x < 1/ and /0 <= p <= 1/. + +[optional_policy] + + template + ``__sf_result`` ibetac_inva(T1 b, T2 x, T3 p); + + template + ``__sf_result`` ibetac_inva(T1 b, T2 x, T3 p, const ``__Policy``&); + +Returns a value /a/ such that: `q = ibetac(a, b, x);` + +Requires: /b > 0/, /0 < x < 1/ and /0 <= q <= 1/. + +[optional_policy] + + template + ``__sf_result`` ibeta_invb(T1 b, T2 x, T3 p); + + template + ``__sf_result`` ibeta_invb(T1 b, T2 x, T3 p, const ``__Policy``&); + +Returns a value /b/ such that: `p = ibeta(a, b, x);` + +Requires: /a > 0/, /0 < x < 1/ and /0 <= p <= 1/. + +[optional_policy] + + template + ``__sf_result`` ibetac_invb(T1 b, T2 x, T3 p); + + template + ``__sf_result`` ibetac_invb(T1 b, T2 x, T3 p, const ``__Policy``&); + +Returns a value /b/ such that: `q = ibetac(a, b, x);` + +Requires: /a > 0/, /0 < x < 1/ and /0 <= q <= 1/. + +[optional_policy] + +[h4 Accuracy] + +The accuracy of these functions should closely follow that +of the regular forward incomplete beta functions. However, +note that in some parts of their domain, these functions can +be extremely sensitive to changes in input, particularly when +the argument /p/ (or it's complement /q/) is very close to `0` or `1`. + +[h4 Testing] + +There are two sets of tests: + +* Basic sanity checks attempt to "round-trip" from +/a, b/ and /x/ to /p/ or /q/ and back again. These tests have quite +generous tolerances: in general both the incomplete beta and its +inverses change so rapidly, that round tripping to more than a couple +of significant digits isn't possible. This is especially true when +/p/ or /q/ is very near one: in this case there isn't enough +"information content" in the input to the inverse function to get +back where you started. +* Accuracy checks using high precision test values. These measure +the accuracy of the result, given exact input values. + +[h4 Implementation of ibeta_inv and ibetac_inv] + +These two functions share a common implementation. + +First an initial approximation to x is computed then the +last few bits are cleaned up using +[@http://en.wikipedia.org/wiki/Simple_rational_approximation Halley iteration]. +The iteration limit is set to 1/2 of the number of bits in T, which by experiment is +sufficient to ensure that the inverses are at least as accurate as the normal +incomplete beta functions. Up to 5 iterations may be +required in extreme cases, although normally only one or two are required. +Further, the number of iterations required decreases with increasing /a/ and +/b/ (which generally form the more important use cases). + +The initial guesses used for iteration are obtained as follows: + +Firstly recall that: + +[equation ibeta_inv5] + +We may wish to start from either p or q, and to calculate either x or y. +In addition at +any stage we can exchange a for b, p for q, and x for y if it results in a +more manageable problem. + +For `a+b >= 5` the initial guess is computed using the methods described in: + +Asymptotic Inversion of the Incomplete Beta Function, +by N. M. [@http://homepages.cwi.nl/~nicot/ Temme]. +Journal of Computational and Applied Mathematics 41 (1992) 145-157. + +The nearly symmetrical case (section 2 of the paper) is used for + +[equation ibeta_inv2] + +and involves solving the inverse error function first. The method is accurate +to at least 2 decimal digits when [^a = 5] rising to at least 8 digits when +[^a = 10[super 5]]. + +The general error function case (section 3 of the paper) is used for + +[equation ibeta_inv3] + +and again expresses the inverse incomplete beta in terms of the +inverse of the error function. The method is accurate to at least +2 decimal digits when [^a+b = 5] rising to 11 digits when [^a+b = 10[super 5]]. +However, when the result is expected to be very small, and when a+b is +also small, then its accuracy tails off, in this case when p[super 1/a] < 0.0025 +then it is better to use the following as an initial estimate: + +[equation ibeta_inv4] + +Finally the for all other cases where `a+b > 5` the method of section +4 of the paper is used. This expresses the inverse incomplete beta in terms +of the inverse of the incomplete gamma function, and is therefore significantly +more expensive to compute than the other cases. However the method is accurate +to at least 3 decimal digits when [^a = 5] rising to at least 10 digits when +[^a = 10[super 5]]. This method is limited to a > b, and therefore we need to perform +an exchange a for b, p for q and x for y when this is not the case. In addition +when p is close to 1 the method is inaccurate should we actually want y rather +than x as output. Therefore when q is small ([^q[super 1/p] < 10[super -3]]) we use: + +[equation ibeta_inv6] + +which is both cheaper to compute than the full method, and a more accurate +estimate on q. + +When a and b are both small there is a distinct lack of information in the +literature on how to proceed. I am extremely grateful to Prof Nico Temme +who provided the following information with a great deal of patience and +explanation on his part. Any errors that follow are entirely my own, and not +Prof Temme's. + +When a and b are both less than 1, then there is a point of inflection in +the incomplete beta at point `xs = (1 - a) / (2 - a - b)`. Therefore if +[^p > I[sub x](a,b)] we swap a for b, p for q and x for y, so that now we always +look for a point x below the point of inflection `xs`, and on a convex curve. +An initial estimate for x is made with: + +[equation ibeta_inv7] + +which is provably below the true value for x: +[@http://en.wikipedia.org/wiki/Newton%27s_method Newton iteration] will +therefore smoothly converge on x without problems caused by overshooting etc. + +When a and b are both greater than 1, but a+b is too small to use the other +methods mentioned above, we proceed as follows. Observe that there is a point +of inflection in the incomplete beta at `xs = (1 - a) / (2 - a - b)`. Therefore +if [^p > I[sub x](a,b)] we swap a for b, p for q and x for y, so that now we always +look for a point x below the point of inflection `xs`, and on a concave curve. +An initial estimate for x is made with: + +[equation ibeta_inv4] + +which can be improved somewhat to: + +[equation ibeta_inv1] + +when b and x are both small (I've used b < a and x < 0.2). This +actually under-estimates x, which drops us on the wrong side of x for Newton +iteration to converge monotonically. However, use of higher derivatives +and Halley iteration keeps everything under control. + +The final case to be considered if when one of a and b is less than or equal +to 1, and the other greater that 1. Here, if b < a we swap a for b, p for q +and x for y. Now the curve of the incomplete beta is convex with no points +of inflection in [0,1]. For small p, x can be estimated using + +[equation ibeta_inv4] + +which under-estimates x, and drops us on the right side of the true value +for Newton iteration to converge monotonically. However, when p is large +this can quite badly underestimate x. This is especially an issue when we +really want to find y, in which case this method can be an arbitrary number +of order of magnitudes out, leading to very poor convergence during iteration. + +Things can be improved by considering the incomplete beta as a distorted +quarter circle, and estimating y from: + +[equation ibeta_inv8] + +This doesn't guarantee that we will drop in on the right side of x for +monotonic convergence, but it does get us close enough that Halley iteration +rapidly converges on the true value. + +[h4 Implementation of inverses on the a and b parameters] + +These four functions share a common implementation. + +First an initial approximation is computed for /a/ or /b/: +where possible this uses a Cornish-Fisher expansion for the +negative binomial distribution to get within around 1 of the +result. However, when /a/ or /b/ are very small the Cornish Fisher +expansion is not usable, in this case the initial approximation +is chosen so that +I[sub x](a, b) is near the middle of the range [0,1]. + +This initial guess is then +used as a starting value for a generic root finding algorithm. The +algorithm converges rapidly on the root once it has been +bracketed, but bracketing the root may take several iterations. +A better initial approximation for /a/ or /b/ would improve these +functions quite substantially: currently 10-20 incomplete beta +function invocations are required to find the root. + +[endsect][/section:ibeta_inv_function The Incomplete Beta Function Inverses] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/igamma.qbk b/doc/sf_and_dist/igamma.qbk new file mode 100644 index 000000000..9e0eac33b --- /dev/null +++ b/doc/sf_and_dist/igamma.qbk @@ -0,0 +1,386 @@ +[section:igamma Incomplete Gamma Functions] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` gamma_p(T1 a, T2 z); + + template + ``__sf_result`` gamma_p(T1 a, T2 z, const ``__Policy``&); + + template + ``__sf_result`` gamma_q(T1 a, T2 z); + + template + ``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&); + + template + ``__sf_result`` tgamma_lower(T1 a, T2 z); + + template + ``__sf_result`` tgamma_lower(T1 a, T2 z, const ``__Policy``&); + + template + ``__sf_result`` tgamma(T1 a, T2 z); + + template + ``__sf_result`` tgamma(T1 a, T2 z, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +There are four [@http://mathworld.wolfram.com/IncompleteGammaFunction.html +incomplete gamma functions]: +two are normalised versions (also known as /regularized/ incomplete gamma functions) +that return values in the range [0, 1], and two are non-normalised and +return values in the range [0, [Gamma](a)]. Users interested in statistical +applications should use the +[@http://mathworld.wolfram.com/RegularizedGammaFunction.html normalised versions (gamma_p and gamma_q)]. + +All of these functions require /a > 0/ and /z >= 0/, otherwise they return +the result of __domain_error. + +[optional_policy] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types, otherwise the return type is simply T1. + + template + ``__sf_result`` gamma_p(T1 a, T2 z); + + template + ``__sf_result`` gamma_p(T1 a, T2 z, const ``__Policy``&); + +Returns the normalised lower incomplete gamma function of a and z: + +[equation igamma4] + +This function changes rapidly from 0 to 1 around the point z == a: + +[$../graphs/gamma_p.png] + + template + ``__sf_result`` gamma_q(T1 a, T2 z); + + template + ``__sf_result`` gamma_q(T1 a, T2 z, const ``__Policy``&); + +Returns the normalised upper incomplete gamma function of a and z: + +[equation igamma3] + +This function changes rapidly from 1 to 0 around the point z == a: + +[$../graphs/gamma_q.png] + + template + ``__sf_result`` tgamma_lower(T1 a, T2 z); + + template + ``__sf_result`` tgamma_lower(T1 a, T2 z, const ``__Policy``&); + +Returns the full (non-normalised) lower incomplete gamma function of a and z: + +[equation igamma2] + + template + ``__sf_result`` tgamma(T1 a, T2 z); + + template + ``__sf_result`` tgamma(T1 a, T2 z, const ``__Policy``&); + +Returns the full (non-normalised) upper incomplete gamma function of a and z: + +[equation igamma1] + +[h4 Accuracy] + +The following tables give peak and mean relative errors in over various domains of +a and z, along with comparisons to the __gsl and __cephes libraries. +Note that only results for the widest floating point type on the system are given as +narrower types have __zero_error. + +Note that errors grow as /a/ grows larger. + +Note also that the higher error rates for the 80 and 128 bit +long double results are somewhat misleading: expected results that are +zero at 64-bit double precision may be non-zero - but exceptionally small - +with the larger exponent range of a long double. These results therefore +reflect the more extreme nature of the tests conducted for these types. + +All values are in units of epsilon. + +[table Errors In the Function gamma_p(a,z) +[[Significand Size] [Platform and Compiler] + [0.5 < a < 100 + + and + + 0.01*a < z < 100*a] + [1x10[super -12] < a < 5x10[super -2] + + and + + 0.01*a < z < 100*a] + [1e-6 < a < 1.7x10[super 6] + + and + + 1 < z < 100*a]] +[[53] [Win32, Visual C++ 8] + [Peak=36 Mean=9.1 + + (GSL Peak=342 Mean=46) + + (__cephes Peak=491 Mean=102)] + [Peak=4.5 Mean=1.4 + + (GSL Peak=4.8 Mean=0.76) + + (__cephes Peak=21 Mean=5.6)] + [Peak=244 Mean=21 + + (GSL Peak=1022 Mean=1054) + + (__cephes Peak~8x10[super 6] Mean~7x10[super 4])]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=241 Mean=36] [Peak=4.7 Mean=1.5] [Peak~30,220 Mean=1929]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peak=41 Mean=10] [Peak=4.7 Mean=1.4] [Peak~30,790 Mean=1864]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=40.2 Mean=10.2] [Peak=5 Mean=1.6] [Peak=5,476 Mean=440]] +] + +[table Errors In the Function gamma_q(a,z) +[[Significand Size] [Platform and Compiler] +[0.5 < a < 100 + +and + +0.01*a < z < 100*a] +[1x10[super -12] < a < 5x10[super -2] + +and + +0.01*a < z < 100*a] [1x10[super -6] < a < 1.7x10[super 6] + +and + +1 < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peak=28.3 Mean=7.2 + +(GSL Peak=201 Mean=13) + +(__cephes Peak=556 Mean=97)] [Peak=4.8 Mean=1.6 + +(GSL Peak~1.3x10[super 10] Mean=1x10[super +9]) + +(__cephes Peak~3x10[super 11] Mean=4x10[super 10])] [Peak=469 Mean=33 + +(GSL Peak=27,050 Mean=2159) + +(__cephes Peak~8x10[super 6] Mean~7x10[super 5])]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=280 Mean=33] [Peak=4.1 Mean=1.6] [Peak=11,490 Mean=732]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peak=32 Mean=9.4] [Peak=4.7 Mean=1.5] [Peak=6815 Mean=414]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=37 Mean=10] [Peak=11.2 Mean=2.0] [Peak=4,999 Mean=298]] +] + +[table Errors In the Function tgamma_lower(a,z) +[[Significand Size] [Platform and Compiler] [0.5 < a < 100 + +and + +0.01*a < z < 100*a] [1x10[super -12] < a < 5x10[super -2] + +and + +0.01*a < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peak=5.5 Mean=1.4] [Peak=3.6 Mean=0.78]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=402 Mean=79] [Peak=3.4 Mean=0.8]] +[[64] [Redhat Linux IA64, gcc-3.4] [Peak=6.8 Mean=1.4] [Peak=3.4 Mean=0.78]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=6.1 Mean=1.8] [Peak=3.7 Mean=0.89]] +] + +[table Errors In the Function tgamma(a,z) +[[Significand Size] [Platform and Compiler] [0.5 < a < 100 + +and + +0.01*a < z < 100*a] [1x10[super -12] < a < 5x10[super -2] + +and + +0.01*a < z < 100*a]] +[[53] [Win32, Visual C++ 8] [Peak=5.9 Mean=1.5] [Peak=1.8 Mean=0.6]] +[[64] [RedHat Linux IA32, gcc-3.3] [Peak=596 Mean=116] [Peak=3.2 Mean=0.84]] +[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=40.2 Mean=2.5] [Peak=3.2 Mean=0.8]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=364 Mean=17.6] [Peak=12.7 Mean=1.8]] +] + +[h4 Testing] + +There are two sets of tests: spot tests compare values taken from +[@http://functions.wolfram.com/GammaBetaErf/ Mathworld's online evaluator] +with this implementation to perform a basic "sanity check". +Accuracy tests use data generated at very high precision +(using NTL's RR class set at 1000-bit precision) using this implementation +with a very high precision 60-term __lanczos, and some but not all of the special +case handling disabled. +This is less than satisfactory: an independent method should really be used, +but apparently a complete lack of such methods are available. We can't even use a deliberately +naive implementation without special case handling since Legendre's continued fraction +(see below) is unstable for small a and z. + +[h4 Implementation] + +These four functions share a common implementation since +they are all related via: + +1) [equation igamma5] + +2) [equation igamma6] + +3) [equation igamma7] + +The lower incomplete gamma is computed from its series representation: + +4) [equation igamma8] + +Or by subtraction of the upper integral from either [Gamma](a) or 1 +when /x > a and x > 1.1/. + +The upper integral is computed from Legendre's continued fraction representation: + +5) [equation igamma9] + +When /x > 1.1/ or by subtraction of the lower integral from either [Gamma](a) or 1 +when /x < a/. + +For /x < 1.1/ computation of the upper integral is more complex as the continued +fraction representation is unstable in this area. However there is another +series representation for the lower integral: + +6) [equation igamma10] + +That lends itself to calculation of the upper integral via rearrangement +to: + +7) [equation igamma11] + +Refer to the documentation for __powm1 and __tgamma1pm1 for details +of their implementation. Note however that the precision of __tgamma1pm1 +is capped to either around 35 digits, or to that of the __lanczos associated with +type T - if there is one - whichever of the two is the greater. +That therefore imposes a similar limit on the precision of this +function in this region. + +For /x < 1.1/ the crossover point where the result is ~0.5 no longer +occurs for /x ~ y/. Using /x * 1.1 < a/ as the crossover criterion +for /0.5 < x <= 1.1/ keeps the maximum value computed (whether +it's the upper or lower interval) to around 0.6. Likewise for +/x <= 0.5/ then using /-0.4 / log(x) < a/ as the crossover criterion +keeps the maximum value computed to around 0.7 +(whether it's the upper or lower interval). + +There are two special cases used when a is an integer or half integer, +and the crossover conditions listed above indicate that we should compute +the upper integral Q. +If a is an integer in the range /1 <= a < 30/ then the following +finite sum is used: + +9) [equation igamma1f] + +While for half integers in the range /0.5 <= a < 30/ then the +following finite sum is used: + +10) [equation igamma2f] + +These are both more stable and more efficient than the continued fraction +alternative. + +When the argument /a/ is large, and /x ~ a/ then the series (4) and continued +fraction (5) above are very slow to converge. In this area an expansion due to +Temme is used: + +11) [equation igamma16] + +12) [equation igamma17] + +13) [equation igamma18] + +14) [equation igamma19] + +The double sum is truncated to a fixed number of terms - to give a specific +target precision - and evaluated as a polynomial-of-polynomials. There are +versions for up to 128-bit long double precision: types requiring +greater precision than that do not use these expansions. The +coefficients C[sub k][super n] are computed in advance using the recurrence +relations given by Temme. The zone where these expansions are used is + + (a > 20) && (a < 200) && fabs(x-a)/a < 0.4 + +And: + + (a > 200) && (fabs(x-a)/a < 4.5/sqrt(a)) + +The latter range is valid for all types up to 128-bit long doubles, and +is designed to ensure that the result is larger than 10[super -6], the +first range is used only for types up to 80-bit long doubles. These +domains are narrower than the ones recommended by either Temme or Didonato +and Morris. However, using a wider range results in large and inexact +(i.e. computed) values being passed to the `exp` and `erfc` functions +resulting in significantly larger error rates. In other words there is a +fine trade off here between efficiency and error. The current limits should +keep the number of terms required by (4) and (5) to no more than ~20 +at double precision. + +For the normalised incomplete gamma functions, calculation of the +leading power terms is central to the accuracy of the function. +For smallish a and x combining +the power terms with the __lanczos gives the greatest accuracy: + +15) [equation igamma12] + +In the event that this causes underflow/overflow then the exponent can +be reduced by a factor of /a/ and brought inside the power term. + +When a and x are large, we end up with a very large exponent with a base +near one: this will not be computed accurately via the pow function, +and taking logs simply leads to cancellation errors. The worst of the +errors can be avoided by using: + +16) [equation igamma13] + +when /a-x/ is small and a and x are large. There is still a subtraction +and therefore some cancellation errors - but the terms are small so the absolute +error will be small - and it is absolute rather than relative error that +counts in the argument to the /exp/ function. Note that for sufficiently +large a and x the errors will still get you eventually, although this does +delay the inevitable much longer than other methods. Use of /log(1+x)-x/ here +is inspired by Temme (see references below). + +[h4 References] + +* N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, +Probability in the Engineering and Informational Sciences, 8, 1994. +* N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions, +Siam J. Math Anal. Vol 10 No 4, July 1979, p757. +* A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma +Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986, p377. +* W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's Ideas +and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, +Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237. +[@http://citeseer.ist.psu.edu/gautschi98incomplete.html http://citeseer.ist.psu.edu/gautschi98incomplete.html] + +[endsect][/section:igamma The Incomplete Gamma Function] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/igamma_inv.qbk b/doc/sf_and_dist/igamma_inv.qbk new file mode 100644 index 000000000..24102c952 --- /dev/null +++ b/doc/sf_and_dist/igamma_inv.qbk @@ -0,0 +1,161 @@ +[section:igamma_inv Incomplete Gamma Function Inverses] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` gamma_q_inv(T1 a, T2 q); + + template + ``__sf_result`` gamma_q_inv(T1 a, T2 q, const ``__Policy``&); + + template + ``__sf_result`` gamma_p_inv(T1 a, T2 p); + + template + ``__sf_result`` gamma_p_inv(T1 a, T2 p, const ``__Policy``&); + + template + ``__sf_result`` gamma_q_inva(T1 x, T2 q); + + template + ``__sf_result`` gamma_q_inva(T1 x, T2 q, const ``__Policy``&); + + template + ``__sf_result`` gamma_p_inva(T1 x, T2 p); + + template + ``__sf_result`` gamma_p_inva(T1 x, T2 p, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +There are four [@http://mathworld.wolfram.com/IncompleteGammaFunction.html incomplete gamma function] +inverses which either compute +/x/ given /a/ and /p/ or /q/, +or else compute /a/ given /x/ and either /p/ or /q/. + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types, otherwise the return type is simply T1. + +[optional_policy] + +[tip When people normally talk about the inverse of the incomplete +gamma function, they are talking about inverting on parameter /x/. +These are implemented here as gamma_p_inv and gamma_q_inv, and are by +far the most efficient of the inverses presented here. + +The inverse on the /a/ parameter finds use in some statistical +applications but has to be computed by rather brute force numerical +techniques and is consequently several times slower. +These are implemented here as gamma_p_inva and gamma_q_inva.] + + + template + ``__sf_result`` gamma_q_inv(T1 a, T2 q); + + template + ``__sf_result`` gamma_q_inv(T1 a, T2 q, const ``__Policy``&); + +Returns a value x such that: `q = gamma_q(a, x);` + +Requires: /a > 0/ and /1 >= p,q >= 0/. + + template + ``__sf_result`` gamma_p_inv(T1 a, T2 p); + + template + ``__sf_result`` gamma_p_inv(T1 a, T2 p, const ``__Policy``&); + +Returns a value x such that: `p = gamma_p(a, x);` + +Requires: /a > 0/ and /1 >= p,q >= 0/. + + template + ``__sf_result`` gamma_q_inva(T1 x, T2 q); + + template + ``__sf_result`` gamma_q_inva(T1 x, T2 q, const ``__Policy``&); + +Returns a value a such that: `q = gamma_q(a, x);` + +Requires: /x > 0/ and /1 >= p,q >= 0/. + + template + ``__sf_result`` gamma_p_inva(T1 x, T2 p); + + template + ``__sf_result`` gamma_p_inva(T1 x, T2 p, const ``__Policy``&); + +Returns a value a such that: `p = gamma_p(a, x);` + +Requires: /x > 0/ and /1 >= p,q >= 0/. + +[h4 Accuracy] + +The accuracy of these functions doesn't vary much by platform or by +the type T. Given that these functions are computed by iterative methods, +they are deliberately "detuned" so as not to be too accurate: it is in +any case impossible for these function to be more accurate than the +regular forward incomplete gamma functions. In practice, the accuracy +of these functions is very similar to that of __gamma_p and __gamma_q +functions. + +[h4 Testing] + +There are two sets of tests: + +* Basic sanity checks attempt to "round-trip" from +/a/ and /x/ to /p/ or /q/ and back again. These tests have quite +generous tolerances: in general both the incomplete gamma, and its +inverses, change so rapidly that round tripping to more than a couple +of significant digits isn't possible. This is especially true when +/p/ or /q/ is very near one: in this case there isn't enough +"information content" in the input to the inverse function to get +back where you started. +* Accuracy checks using high precision test values. These measure +the accuracy of the result, given exact input values. + +[h4 Implementation] + +The functions gamma_p_inv and [@http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/ gamma_q_inv] +share a common implementation. + +First an initial approximation is computed using the methodology described +in: + +[@http://portal.acm.org/citation.cfm?id=23109&coll=portal&dl=ACM +A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma +Function Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.] + +Finally, the last few bits are cleaned up using Halley iteration, the iteration +limit is set to 2/3 of the number of bits in T, which by experiment is +sufficient to ensure that the inverses are at least as accurate as the normal +incomplete gamma functions. In testing, no more than 3 iterations are required +to produce a result as accurate as the forward incomplete gamma function, and +in many cases only one iteration is required. + +The functions gamma_p_inva and gamma_q_inva also share a common implementation +but are handled separately from gamma_p_inv and gamma_q_inv. + +An initial approximation for /a/ is computed very crudely so that +/gamma_p(a, x) ~ 0.5/, this value is then used as a starting point +for a generic derivative-free root finding algorithm. As a consequence, +these two functions are rather more expensive to compute than the +gamma_p_inv or gamma_q_inv functions. Even so, the root is usually found +in fewer than 10 iterations. + +[endsect][/section The Incomplete Gamma Function Inverses] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/implementation.qbk b/doc/sf_and_dist/implementation.qbk new file mode 100644 index 000000000..d3e8701c3 --- /dev/null +++ b/doc/sf_and_dist/implementation.qbk @@ -0,0 +1,559 @@ +[section:implementation Additional Implementation Notes] + +The majority of the implementation notes are included with the documentation +of each function or distribution. The notes here are of a more general nature, +and reflect more the general implementation philosophy used. + +[h4 Implemention philosophy] + +"First be right, then be fast." + +There will always be potential compromises +to be made between speed and accuracy. +It may be possible to find faster methods, +particularly for certain limited ranges of arguments, +but for most applications of math functions and distributions, +we judge that speed is rarely as important as accuracy. + +So our priority is accuracy. + +To permit evaluation of accuracy of the special functions, +production of extremely accurate tables of test values +has received considerable effort. + +(It also required much CPU effort - +there was some danger of molten plastic dripping from the bottom of JM's laptop, +so instead, PAB's Dual-core desktop was kept 50% busy for *days* +calculating some tables of test values!) + +For a specific RealType, say float or double, +it may be possible to find approximations for some functions +that are simpler and thus faster, but less accurate +(perhaps because there are no refining iterations, +for example, when calculating inverse functions). + +If these prove accurate enough to be "fit for his purpose", +then a user may substitute his custom specialization. + +For example, there are approximations dating back from times +when computation was a *lot* more expensive: + +H Goldberg and H Levine, Approximate formulas for +percentage points and normalisation of t and chi squared, +Ann. Math. Stat., 17(4), 216 - 225 (Dec 1946). + +A H Carter, Approximations to percentage points of the z-distribution, +Biometrika 34(2), 352 - 358 (Dec 1947). + +These could still provide sufficient accuracy for some speed-critical applications. + +[h4 Accuracy and Representation of Test Values] + +In order to be accurate enough for as many as possible real types, +constant values are given to 50 decimal digits if available +(though many sources proved only accurate near to 64-bit double precision). +Values are specified as long double types by appending L, +unless they are exactly representable, for example integers, or binary fractions like 0.125. +This avoids the risk of loss of accuracy converting from double, the default type. +Values are used after static_cast(1.2345L) +to provide the appropriate RealType for spot tests. + +Functions that return constants values, like kurtosis for example, are written as + +`static_cast(-3) / 5;` + +to provide the most accurate value +that the compiler can compute for the real type. +(The denominator is an integer and so will be promoted exactly). + +So tests for one third, *not* exactly representable with radix two floating-point, +(should) use, for example: + +`static_cast(1) / 3;` + +If a function is very sensitive to changes in input, +specifying an inexact value as input (such as 0.1) can throw +the result off by a noticeable amount: 0.1f is "wrong" +by ~1e-7 for example (because 0.1 has no exact binary representation). +That is why exact binary values - halves, quarters, and eighths etc - +are used in test code along with the occasional fraction `a/b` with `b` +a power of two (in order to ensure that the result is an exactly +representable binary value). + +[h4 Tolerance of Tests] + +The tolerances need to be set to the maximum of: + +* Some epsilon value. +* The accuracy of the data (often only near 64-bit double). + +Otherwise when long double has more digits than the test data, then no +amount of tweaking an epsilon based tolerance will work. + +A common problem is when tolerances that are suitable for implementations +like Microsoft VS.NET where double and long double are the same size: +tests fail on other systems where long double is more accurate than double. +Check first that the suffix L is present, and then that the tolerance is big enough. + +[h4 Handling Unsuitable Arguments] + +In +[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1665.pdf Errors in Mathematical Special Functions], J. Marraffino & M. Paterno +it is proposed that signalling a domain error is mandatory +when the argument would give an mathematically undefined result. + +*Guideline 1 + +[:A mathematical function is said to be defined at a point a = (a1, a2, . . .) +if the limits as x = (x1, x2, . . .) 'approaches a from all directions agree'. +The defined value may be any number, or +infinity, or -infinity.] + +Put crudely, if the function goes to + infinity +and then emerges 'round-the-back' with - infinity, +it is NOT defined. + +[:The library function which approximates a mathematical function shall signal a domain error +whenever evaluated with argument values for which the mathematical function is undefined.] + +*Guideline 2 + +[:The library function which approximates a mathematical function +shall signal a domain error whenever evaluated with argument values +for which the mathematical function obtains a non-real value.] + +This implementation is believed to follow these proposals and to assist compatibility with +['ISO/IEC 9899:1999 Programming languages - C] +and with the +[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Draft Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 5]. +[link math_toolkit.main_overview.error_handling See also domain_error]. + +See __policy_ref for details of the error handling policies that should allow +a user to comply with any of these recommendations, as well as other behaviour. + +See [link math_toolkit.main_overview.error_handling error handling] +for a detailed explanation of the mechanism, and +[link math_toolkit.dist.stat_tut.weg.error_eg error_handling example] +and +[@../../../example/error_handling_example.cpp error_handling_example.cpp] + +[caution If you enable throw but do NOT have try & catch block, +then the program will terminate with an uncaught exception and probably abort. +Therefore to get the benefit of helpful error messages, enabling *all* exceptions +*and* using try&catch is recommended for all applications. +However, for simplicity, this is not done for most examples.] + +[h4 Handling of Functions that are Not Mathematically defined] + +Functions that are not mathematically defined, +like the Cauchy mean, fail to compile by default. +[link math_toolkit.policy.pol_ref.assert_undefined A policy] +allows control of this. + +If the policy is to permit undefined functions, then calling them +throws a domain error, by default. But the error policy can be set +to not throw, and to return NaN instead. For example, + +`#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error` + +appears before the first Boost include, +then if the un-implemented function is called, +mean(cauchy<>()) will return std::numeric_limits::quiet_NaN(). + +[warning If `std::numeric_limits::has_quiet_NaN` is false +(for example T is a User-defined type), +then an exception will always be thrown when a domain error occurs. +Catching exceptions is therefore strongly recommended.] + +[h4 Median of distributions] + +There are many distributions for which we have been unable to find an analytic formula, +and this has deterred us from implementing +[@http://en.wikipedia.org/wiki/Median median functions], the mid-point in a list of values. + +However a useful median approximation for distribution `dist` may be available from + +`quantile(dist, 0.5)`. + +[@http://www.amstat.org/publications/jse/v13n2/vonhippel.html Mean, Median, and Skew, Paul T von Hippel] + +[@http://documents.wolfram.co.jp/teachersedition/MathematicaBook/24.5.html Descriptive Statistics,] + +[@http://documents.wolfram.co.jp/v5/Add-onsLinks/StandardPackages/Statistics/DescriptiveStatistics.html and ] + +[@http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/NumericalOperationsOnData/3.8.1.html +Mathematica Basic Statistics.] give more detail, in particular for discrete distributions. + + +[h4 Handling of Floating-Point Infinity] + +Some functions and distributions are well defined with + or - infinity as +argument(s), but after some experiments with handling infinite arguments +as special cases, we concluded that it was generally more useful to forbid this, +and instead to return the result of __domain_error. + +Handling infinity as special cases is additionally complicated +because, unlike built-in types on most - but not all - platforms, +not all User-Defined Types are +specialized to provide `std::numeric_limits::infinity()` +and would return zero rather than any representation of infinity. + +The rationale is that non-finiteness may happen because of error +or overflow in the users code, and it will be more helpful for this +to be diagnosed promptly rather than just continuing. +The code also became much more complicated, more error-prone, +much more work to test, and much less readable. + +However in a few cases, for example normal, where we felt it obvious, +we have permitted argument(s) to be infinity, +provided infinity is implemented for the realType on that implementation. + +Overflow, underflow, denorm can be handled using __error_policy. + +We have also tried to catch boundary cases where the mathematical specification +would result in divide by zero or overflow and signalling these similarly. +What happens at (and near), poles can be controlled through __error_policy. + +[h4 Scale, Shape and Location] + +We considered adding location and scale to the list of functions, for example: + + template + inline RealType scale(const triangular_distribution& dist) + { + RealType lower = dist.lower(); + RealType mode = dist.mode(); + RealType upper = dist.upper(); + RealType result; // of checks. + if(false == detail::check_triangular(BOOST_CURRENT_FUNCTION, lower, mode, upper, &result)) + { + return result; + } + return (upper - lower); + } + +but found that these concepts are not defined (or their definition too contentious) +for too many distributions to be generally applicable. +Because they are non-member functions, they can be added if required. + +[h4 Notes on Implementation of Specific Functions & Distributions] + +* Default parameters for the Triangular Distribution. +We are uncertain about the best default parameters. +Some sources suggest that the Standard Triangular Distribution has +lower = 0, mode = half and upper = 1. +However as a approximation for the normal distribution, +the most common usage, lower = -1, mode = 0 and upper = 1 would be more suitable. + +[h4 Rational Approximations Used] + +Some of the special functions in this library are implemented via +rational approximations. These are either taken from the literature, +or devised by John Maddock using +[link math_toolkit.toolkit.internals2.minimax our Remez code]. + +Rational rather than Polynomial approximations are used to ensure +accuracy: polynomial approximations are often wonderful up to +a certain level of accuracy, but then quite often fail to provide much greater +accuracy no matter how many more terms are added. + +Our own approximations were devised either for added accuracy +(to support 128-bit long doubles for example), or because +literature methods were unavailable or under non-BSL +compatible license. Our Remez code is known to produce good +agreement with literature results in fairly simple "toy" cases. +All approximations were checked +for convergence and to ensure that +they were not ill-conditioned (the coefficients can give a +theoretically good solution, but the resulting rational function +may be un-computable at fixed precision). + +Recomputing using different +Remez implementations may well produce differing coefficients: the +problem is well known to be ill conditioned in general, and our Remez implementation +often found a broad and ill-defined minima for many of these approximations +(of course for simple "toy" examples like approximating `exp` the minima +is well defined, and the coeffiecents should agree no matter whose Remez +implementation is used). This should not in general effect the validity +of the approximations: there's good literature supporting the idea that +coefficients can be "in error" without necessarily adversely effecting +the result. Note that "in error" has a special meaning in this context, +see [@http://front.math.ucdavis.edu/0101.5042 +"Approximate construction of rational approximations and the effect +of error autocorrection.", Grigori Litvinov, eprint arXiv:math/0101042]. +Therefore the coefficients still need to be accurately calculated, even if they can +be in error compared to the "true" minimax solution. + +[h4 Representation of Mathematical Constants] + +A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used +to provide high accuracy constants to mathematical functions and distributions, +since it is important to provide values uniformly for both built-in +float, double and long double types, +and for User Defined types like NTL::quad_float and NTL::RR. + +To permit calculations in this Math ToolKit and its tests, (and elsewhere) +at about 100 decimal digits with NTL::RR type, +it is obviously necessary to define constants to this accuracy. + +However, some compilers do not accept decimal digits strings as long as this. +So the constant is split into two parts, with the 1st containing at least +long double precision, and the 2nd zero if not needed or known. +The 3rd part permits an exponent to be provided if necessary (use zero if none) - +the other two parameters may only contain decimal digits (and sign and decimal point), +and may NOT include an exponent like 1.234E99 (nor a trailing F or L). +The second digit string is only used if T is a User-Defined Type, +when the constant is converted to a long string literal and lexical_casted to type T. +(This is necessary because you can't use a numeric constant +since even a long double might not have enough digits). + +For example, pi is defined: + + BOOST_DEFINE_MATH_CONSTANT(pi, + 3.141592653589793238462643383279502884197169399375105820974944, + 5923078164062862089986280348253421170679821480865132823066470938446095505, + 0) + +And used thus: + + using namespace boost::math::constants; + + double diameter = 1.; + double radius = diameter * pi(); + + or boost::math::constants::pi() + +Note that it is necessary (if inconvenient) to specify the type explicitly. + +So you cannot write + + double p = boost::math::constants::pi<>(); // could not deduce template argument for 'T' + +Neither can you write: + + double p = boost::math::constants::pi; // Context does not allow for disambiguation of overloaded function + double p = boost::math::constants::pi(); // Context does not allow for disambiguation of overloaded function + +[h4 Thread safety] + +Reporting of error by setting errno should be thread safe already +(otherwise none of the std lib math functions would be thread safe?). +If you turn on reporting of errors via exceptions, errno gets left unused anyway. + +Other than that, the code is intended to be thread safe *for built in +real-number types* : so float, double and long double are all thread safe. + +For non-built-in types - NTL::RR for example - initialisation of the various +constants used in the implementation is potentially *not* thread safe. +This most undesiable, but it would be a signficant challenge to fix it. +Some compilers may offer the option of having +static-constants initialised in a thread safe manner (Commeau, and maybe +others?), if that's the case then the problem is solved. This is a topic of +hot debate for the next C++ std revision, so hopefully all compilers +will be required to do the right thing here at some point. + +[h4 Sources of Test Data] + +We found a large number of sources of test data. +We have assumed that these are /"known good"/ +if they agree with the results from our test +and only consulted other sources for their /'vote'/ +in the case of serious disagreement. +The accuracy, actual and claimed, vary very widely. +Only [@http://functions.wolfram.com/ Wolfram Mathematica functions] +provided a higher accuracy than +C++ double (64-bit floating-point) and was regarded as +the most-trusted source by far. + +A useful index of sources is: +[@http://www.sal.hut.fi/Teaching/Resources/ProbStat/table.html +Web-oriented Teaching Resources in Probability and Statistics] + +[@http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm Statlet]: +Is a Javascript application that calculates and plots probability distributions, +and provides the most complete range of distributions: + +[:Bernoulli, Binomial, discrete uniform, geometric, hypergeometric, +negative binomial, Poisson, beta, Cauchy-Lorentz, chi-sequared, Erlang, +exponential, extreme value, Fisher, gamma, Laplace, logistic, +lognormal, normal, Parteo, Student's t, triangular, uniform, and Weibull.] + +It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, +& critical values for 5 tail values. + +It is also the only independent source found for the Weibull distribution; +unfortunately it appears to suffer from very poor accuracy in areas where +the underlying special function is known to be difficult to implement. + +[h4 Creating and Managing the Equations] + +The primary source for the equations is now +[@http://www.w3.org/Math/ MathML]: see the +*.mml files in libs\/math\/doc\/sf_and_dist\/equations\/. + +These are most easily edited by a GUI editor such as +[@http://mathcast.sourceforge.net/home.html Mathcast], +please note that the equation editor supplied with Open Office +currently mangles these files and should not currently be used. + +Convertion to SVG was achieved using +[@http://www.grigoriev.ru/svgmath/ SVGMath] and a command line +such as: + +[pre +$for file in *.mml; do +>/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \\ +>>$file > $(basename $file .mml).svg +>done +] + +Note that SVGMath requires that the mml files are *not* wrapped in an XHTML +XML wrapper - this is added by Mathcast by default - one workaround is to +copy an existing mml file and then edit it with Mathcast: the existing +format should then be preserved. This is a bug in the XML parser used by +SVGMath which the author is aware of. + +If neccessary the XHTML wrapper can be removed with: + +[pre cat filename | tr -d "\\r\\n" \| sed -e 's\/.*\\(\]\*>.\*<\/math>\\).\*\/\\1\/' > newfile] + +Setting up fonts for SVGMath is currently rather tricky, on a Windows XP system +JM's font setup is the same as the sample config file provided with SVGMath +but with: + +[pre + + +] + +changed to: + +[pre + + +] + +Note that unlike the sample config file supplied with SVGMath, this does not +make use of the Mathematica 7 font as this lacks sufficient Unicode information +for it to be used with either SVGMath or XEP "as is". + +Also note that the SVG files in the repository are almost certainly +Windows-specific since they reference various Windows Fonts. + +PNG files can be created from the SVG's using +[@http://xmlgraphics.apache.org/batik/tools/rasterizer.html Batik] +and a command such as: + +[pre java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg] + +The PDF is generated into \pdf\math.pdf +using a command from a shell or command window with current directory +\math_toolkit\libs\math\doc\sf_and_dist, typically: + +[pre bjam -a pdf] + +Note that XEP will have to be configured to *use and embed* +whatever fonts are used by the SVG equations +(if necessary editing the sample xep.xml provided by the XEP installation). + +(html is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html +using just bjam -a). + +JM's XEP config file has the following font configuration section added: + +[pre + + + <\/font> + <\/font> + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font> + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font> + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font> + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font> + <\/font> + <\/font> + <\/font\-family> + + + <\/font> + <\/font\-family> +] + +PAB had to alter his because the Lucida Sans Unicode font had a different name. +Changes are very likely to be required if you are not using Windows. + +XZ authored his equations using the venerable Latex, JM converted these to +MathML using [@http://gentoo-wiki.com/HOWTO_Convert_LaTeX_to_HTML_with_MathML mxlatex]. +This process is currently unreliable and required some manual intervention: +consequently Latex source is not considered a viable route for the automatic +production of SVG versions of equations. + +Equations are embedded in the quickbook source using the /equation/ +template defined in math.qbk. This outputs Docbook XML that looks like: + +[pre + + + + + + + + +] + +MathML is not currently present in the Docbook output, or in the +generated HTML: this needs further investigation. + +[h4 Producing Graphs] + +Graphs were mostly produced by a very laborious process entailing output +of columns of values from C++ programs to a .csv file, +use of [@http://www.rjsweb.fsnet.co.uk/graph/ RJS Graph] to arrange the display and axes, +and output to a .ps file, followed by conversion to .png using Adobe Photoshop, +or similar utility. This rigmarole is *not* recommended! + +We plan to carry out this process in a single step using the +[@http://code.google.com/soc/2007/boost/about.html Google Summer of Code 2007] +project of Jacob Voytko (whose work so far is at .\boost-sandbox\SOC\2007\visualization) +that should, when completed, allow output of annotated graphs as +Scalable Vector Graphic .svg files directly from C++ programs. + +[endsect] [/section:implementation Implementation Notes] + +[/ + Copyright 2006, 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/internals_overview.qbk b/doc/sf_and_dist/internals_overview.qbk new file mode 100644 index 000000000..41b7af6ac --- /dev/null +++ b/doc/sf_and_dist/internals_overview.qbk @@ -0,0 +1,22 @@ +[section:internals_overview Overview] + +This section contains internal utilities used by the library's implementation +along with tools used in development and testing. These tools have +only minimal documentation, and crucially +['do not have stable interfaces]. + +There is no doubt that these components can be improved, but they are also +largely incidental to the main purpose of this library. + +These tools are designed to "just get the job done", and receive minimal +documentation here, in the hopes that they will help stimulate further +submissions to this library. + +[endsect][/section:internals_overview Overview] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/inv_hyper.qbk b/doc/sf_and_dist/inv_hyper.qbk new file mode 100644 index 000000000..d353bb06a --- /dev/null +++ b/doc/sf_and_dist/inv_hyper.qbk @@ -0,0 +1,169 @@ +[/ math.qbk + Copyright 2006 Hubert Holin and John Maddock. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + +[def __form1 [^\[0;+'''∞'''\[]] +[def __form2 [^\]-'''∞''';+1\[]] +[def __form3 [^\]-'''∞''';-1\[]] +[def __form4 [^\]+1;+'''∞'''\[]] +[def __form5 [^\[-1;-1+'''ε'''\[]] +[def __form6 '''ε'''] +[def __form7 [^\]+1-'''ε''';+1\]]] + +[def __effects [*Effects: ]] +[def __formula [*Formula: ]] +[def __exm1 '''ex - 1'''[space]] +[def __ex '''ex'''] +[def __te '''2ε'''] + +[section:inv_hyper Inverse Hyperbolic Functions] + +[section:inv_hyper_over Inverse Hyperbolic Functions Overview] + +The exponential funtion is defined, for all objects for which this makes sense, +as the power series +[equation special_functions_blurb1], +with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]]. +In particular, the exponential function is well defined for real numbers, +complex number, quaternions, octonions, and matrices of complex numbers, +among others. + +[: ['[*Graph of exp on R]] ] + +[: [$../graphs/exp_on_r.png] ] + +[: ['[*Real and Imaginary parts of exp on C]]] +[: [$../graphs/im_exp_on_c.png]] + +The hyperbolic functions are defined as power series which +can be computed (for reals, complex, quaternions and octonions) as: + +Hyperbolic cosine: [equation special_functions_blurb5] + +Hyperbolic sine: [equation special_functions_blurb6] + +Hyperbolic tangent: [equation special_functions_blurb7] + +[: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]] +[: [$../graphs/trigonometric.png]] + +[: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]] +[: [$../graphs/hyperbolic.png]] + +The hyperbolic sine is one to one on the set of real numbers, +with range the full set of reals, while the hyperbolic tangent is +also one to one on the set of real numbers but with range __form1, and +therefore both have inverses. The hyperbolic cosine is one to one from __form2 +onto __form3 (and from __form4 onto __form3); the inverse function we use +here is defined on __form3 with range __form2. + +The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent, +and can be computed as [equation special_functions_blurb15]. + +The inverse of the hyperbolic sine is called the Argument hyperbolic sine, +and can be computed (for __form5) as [equation special_functions_blurb17]. + +The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine, +and can be computed as [equation special_functions_blurb18]. + +[endsect] + +[section:acosh acosh] + +`` +#include +`` + + template + ``__sf_result`` acosh(const T x); + + template + ``__sf_result`` acosh(const T x, const ``__Policy``&); + +Computes the reciprocal of (the restriction to the range of __form1) +[link math_toolkit.special.inv_hyper.inv_hyper_over +the hyperbolic cosine function], at x. Values returned are positive. Generalised +Taylor series are used near 1 and Laurent series are used near the +infinity to ensure accuracy. + +If x is in the range __form2 then returns the result of __domain_error. + +The return type of this function is computed using the __arg_pomotion_rules: +the return type is `double` when T is an integer type, and T otherwise. + +[optional_policy] + +[endsect] + +[section:asinh asinh] + +`` +#include +`` + + template + ``__sf_result`` asinh(const T x); + + template + ``__sf_result`` asinh(const T x, const ``__Policy``&); + +Computes the reciprocal of +[link math_toolkit.special.inv_hyper.inv_hyper_over +the hyperbolic sine function]. +Taylor series are used at the origin and Laurent series are used near the +infinity to ensure accuracy. + +The return type of this function is computed using the __arg_pomotion_rules: +the return type is `double` when T is an integer type, and T otherwise. + +[optional_policy] + +[endsect] + +[section:atanh atanh] + +`` +#include +`` + + template + ``__sf_result`` atanh(const T x); + + template + ``__sf_result`` atanh(const T x, const ``__Policy``&); + +Computes the reciprocal of +[link math_toolkit.special.inv_hyper.inv_hyper_over +the hyperbolic tangent function], at x. +Taylor series are used at the origin to ensure accuracy. + +[optional_policy] + +If x is in the range +__form3 +or in the range +__form4 +then returns the result of __domain_error. + +If x is in the range +__form5, +then the result of -__overflow_error is returned, with +__form6[space] +denoting numeric_limits::epsilon(). + +If x is in the range +__form7, +then the result of __overflow_error is returned, with +__form6[space] +denoting +numeric_limits::epsilon(). + +The return type of this function is computed using the __arg_pomotion_rules: +the return type is `double` when T is an integer type, and T otherwise. + +[endsect] + +[endsect] diff --git a/doc/sf_and_dist/issues.qbk b/doc/sf_and_dist/issues.qbk new file mode 100644 index 000000000..5ff84c47f --- /dev/null +++ b/doc/sf_and_dist/issues.qbk @@ -0,0 +1,90 @@ +[section:issues Known Issues, and Todo List] + +This section lists those issues that are known about. + +Predominantly this is a TODO list, or a list of possible +future enhancements. Items labled "High Priority" effect +the proper functioning of the component, and should be fixed +as soon as possible. Items labled "Medium Priority" are +desirable enhancements, often pertaining to the performance +of the component, but do not effect it's accuracy or functionality. +Items labled "Low Priority" should probably be investigated at +some point. Such classifications are obviously highly subjective. + +If you don't see a component listed here, then we don't have any known +issues with it. + +[h4 tgamma] + +* Can the __lanczos be optimized any further? (low priority) + +[h4 Incomplete Beta] + +* Investigate Didonato and Morris' asymptotic expansion for large a and b +(medium priority). + +[h4 Inverse Gamma] + +* Investigate whether we can skip iteration altogether if the first approximation +is good enough (Medium Priority). + +[h4 Polynomials] + +* The Legendre and Laguerre Polynomials have surprisingly different error +rates on different platforms, considering they are evaluated with only +basic arithmetic operations. Maybe this is telling us something, or maybe not +(Low Priority). + +[h4 Elliptic Integrals] + +* Carlson's algorithms are essentially unchanged from Xiaogang Zhang's +Google Summer of Code student project, and are based on Carlson's +original papers. However, Carlson has revised his algorithms since then +(refer to the references in the elliptic integral docs for a list), to +improve performance and accuracy, we may be able to take advantage +of these improvements too (Low Priority). +* [para Carlson's algorithms (mainly R[sub J]) are somewhat prone to +internal overflow/underflow when the arguments are very large or small. +The homogeneity relations:] +[para R[sub F](ka, kb, kc) = k[super -1/2] R[sub F](a, b, c)] +[para and] +[para R[sub J](ka, kb, kc, kr) = k[super -3/2] R[sub J](a, b, c, r)] +[para could be used to sidestep trouble here: provided the problem domains +can be accurately identified. (Medium Priority).] +* Carlson's R[sub C] can be reduced to elementary funtions (asin and log), +would it be more efficient evaluated this way, rather than by Carlson's +algorithms? (Low Priority). +* Should we add an implementation of Carlson's R[sub G]? It's not +required for the Legendre form integrals, but some people may find it +useful (Low Priority). +* There are a several other integrals: D([phi], k), Z([beta], k), +[Lambda][sub 0]([beta], k) and Bulirsch's ['el] functions that could +be implemented using Carlson's integrals (Low Priority). +* The integrals K(k) and E(k) could be implemented using rational +approximations (both for efficiency and accuracy), +assuming we can find them. (Medium Priority). +* There is a sub-domain of __ellint_3 that is unimplemented (see the docs +for details), currently +it's not clear how to solve this issue, or if it's ever likely +to be an real problem in practice - especially as most other implementations +don't support this domain either (Medium Priority). + +[h4 Inverse Hyperbolic Functions] + +* These functions are inherited from previous Boost versions, +before __log1p became widely available. Would they be better expressed +in terms of this function? This is probably only an issue +for very high precision types (Low Priority). + +[h4 Statistical distributions] + +* Student's t Perhaps switch to normal distribution as a better approximation for very large degrees of freedom? + +[endsect][/section:issues Known Issues, and Todo List] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/laguerre.qbk b/doc/sf_and_dist/laguerre.qbk new file mode 100644 index 000000000..6afc4af88 --- /dev/null +++ b/doc/sf_and_dist/laguerre.qbk @@ -0,0 +1,175 @@ +[section:laguerre Laguerre (and Associated) Polynomials] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` laguerre(unsigned n, T x); + + template + ``__sf_result`` laguerre(unsigned n, T x, const ``__Policy``&); + + template + ``__sf_result`` laguerre(unsigned n, unsigned m, T x); + + template + ``__sf_result`` laguerre(unsigned n, unsigned m, T x, const ``__Policy``&); + + template + ``__sf_result`` laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1); + + template + ``__sf_result`` laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1); + + + }} // namespaces + +[h4 Description] + +The return type of these functions is computed using the __arg_pomotion_rules: +note than when there is a single template argument the result is the same type +as that argument or `double` if the template argument is an integer type. + +[optional_policy] + + template + ``__sf_result`` laguerre(unsigned n, T x); + + template + ``__sf_result`` laguerre(unsigned n, T x, const ``__Policy``&); + +Returns the value of the Laguerre Polynomial of order /n/ at point /x/: + +[equation laguerre_0] + +The following graph illustrates the behaviour of the first few +Laguerre Polynomials: + +[$../graphs/laguerre.png] + + template + ``__sf_result`` laguerre(unsigned n, unsigned m, T x); + + template + ``__sf_result`` laguerre(unsigned n, unsigned m, T x, const ``__Policy``&); + +Returns the Associated Laguerre polynomial of degree /n/ +and order /m/ at point /x/: + +[equation laguerre_1] + + template + ``__sf_result`` laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1); + +Implements the three term recurrence relation for the Laguerre +polynomials, this function can be used to create a sequence of +values evaluated at the same /x/, and for rising /n/. + +[equation laguerre_2] + +For example we could produce a vector of the first 10 polynomial +values using: + + double x = 0.5; // Abscissa value + vector v; + v.push_back(laguerre(0, x)).push_back(laguerre(1, x)); + for(unsigned l = 1; l < 10; ++l) + v.push_back(laguerre_next(l, x, v[l], v[l-1])); + +Formally the arguments are: + +[variablelist +[[n][The degree /n/ of the last polynomial calculated.]] +[[x][The abscissa value]] +[[Ln][The value of the polynomial evaluated at degree /n/.]] +[[Lnm1][The value of the polynomial evaluated at degree /n-1/.]] +] + + template + ``__sf_result`` laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1); + +Implements the three term recurrence relation for the Associated Laguerre +polynomials, this function can be used to create a sequence of +values evaluated at the same /x/, and for rising degree /n/. + +[equation laguerre_3] + +For example we could produce a vector of the first 10 polynomial +values using: + + double x = 0.5; // Abscissa value + int m = 10; // order + vector v; + v.push_back(laguerre(0, m, x)).push_back(laguerre(1, m, x)); + for(unsigned l = 1; l < 10; ++l) + v.push_back(laguerre_next(l, m, x, v[l], v[l-1])); + +Formally the arguments are: + +[variablelist +[[n][The degree of the last polynomial calculated.]] +[[m][The order of the Associated Polynomial.]] +[[x][The abscissa value.]] +[[Ln][The value of the polynomial evaluated at degree /n/.]] +[[Lnm1][The value of the polynomial evaluated at degree /n-1/.]] +] + +[h4 Accuracy] + +The following table shows peak errors (in units of epsilon) +for various domains of input arguments. +Note that only results for the widest floating point type on the system are +given as narrower types have __zero_error. + +[table Peak Errors In the Laguerre Polynomial +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] ] +[[53] [Win32, Visual C++ 8] [Peak=3000 Mean=185] ] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=1x10[super 4] Mean=828]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=1x10[super 4] Mean=828] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=680 Mean=40]] +] + +[table Peak Errors In the Associated Laguerre Polynomial +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] ] +[[53] [Win32, Visual C++ 8] [Peak=433 Mean=11]] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=61.4 Mean=19.5]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=61.4 Mean=19.5] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=540 Mean=13.94] ] +] + +Note that the worst errors occur when the degree increases, values greater than +~120 are very unlikely to produce sensible results, especially in the associated +polynomial case when the order is also large. Further the relative errors +are likely to grow arbitrarily large when the function is very close to a root. + +[h4 Testing] + +A mixture of spot tests of values calculated using functions.wolfram.com, +and randomly generated test data are +used: the test data was computed using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision. + +[h4 Implementation] + +These functions are implemented using the stable three term +recurrence relations. These relations guarentee low absolute error +but cannot guarentee low relative error near one of the roots of the +polynomials. + +[endsect][/section:beta_function The Beta Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/lanczos.qbk b/doc/sf_and_dist/lanczos.qbk new file mode 100644 index 000000000..3fd253d84 --- /dev/null +++ b/doc/sf_and_dist/lanczos.qbk @@ -0,0 +1,246 @@ +[section:lanczos The Lanczos Approximation] + +[h4 Motivation] + +['Why base gamma and gamma-like functions on the Lanczos approximation?] + +First of all I should make clear that for the gamma function +over real numbers (as opposed to complex ones) +the Lanczos approximation (See [@http://en.wikipedia.org/wiki/Lanczos_approximation Wikipedia or ] +[@http://mathworld.wolfram.com/LanczosApproximation.html Mathworld]) +appears to offer no clear advantage over more traditional methods such as +[@http://en.wikipedia.org/wiki/Stirling_approximation Stirling's approximation]. +__pugh carried out an extensive comparison of the various methods available +and discovered that they were all very similar in terms of complexity +and relative error. However, the Lanczos approximation does have a couple of +properties that make it worthy of further consideration: + +* The approximation has an easy to compute truncation error that holds for +all /z > 0/. In practice that means we can use the same approximation for all +/z > 0/, and be certain that no matter how large or small /z/ is, the truncation +error will /at worst/ be bounded by some finite value. +* The approximation has a form that is particularly amenable to analytic +manipulation, in particular ratios of gamma or gamma-like functions +are particularly easy to compute without resorting to logarithms. + +It is the combination of these two properties that make the approximation +attractive: Stirling's approximation is highly accurate for large z, and +has some of the same analytic properties as the Lanczos approximation, but +can't easily be used across the whole range of z. + +As the simplest example, consider the ratio of two gamma functions: one could +compute the result via lgamma: + + exp(lgamma(a) - lgamma(b)); + +However, even if lgamma is uniformly accurate to 0.5ulp, the worst case +relative error in the above can easily be shown to be: + + Erel > a * log(a)/2 + b * log(b)/2 + +For small /a/ and /b/ that's not a problem, but to put the relationship another +way: ['each time a and b increase in magnitude by a factor of 10, at least one +decimal digit of precision will be lost.] + +In contrast, by analytically combining like power +terms in a ratio of Lanczos approximation's, these errors can be virtually eliminated +for small /a/ and /b/, and kept under control for very large (or very small +for that matter) /a/ and /b/. Of course, computing large powers is itself a +notoriously hard problem, but even so, analytic combinations of Lanczos +approximations can make the difference between obtaining a valid result, or +simply garbage. Refer to the implementation notes for the __beta function for +an example of this method in practice. The incomplete +[link math_toolkit.special.sf_gamma.igamma gamma_p gamma] and +[link math_toolkit.special.sf_beta.ibeta_function beta] functions +use similar analytic combinations of power terms, to combine gamma and beta +functions divided by large powers into single (simpler) expressions. + +[h4 The Approximation] + +The Lanczos Approximation to the Gamma Function is given by: + +[equation lanczos0] + +Where S[sub g](z) is an infinite sum, that is convergent for all z > 0, +and /g/ is an arbitrary parameter that controls the "shape" of the +terms in the sum which is given by: + +[equation lanczos0a] + +With individual coefficients defined in closed form by: + +[equation lanczos0b] + +However, evaluation of the sum in that form can lead to numerical instability +in the computation of the ratios of rising and falling factorials (effectively +we're multiplying by a series of numbers very close to 1, so roundoff errors +can accumulate quite rapidly). + +The Lanczos approximation is therefore often written in partial fraction form +with the leading constants absorbed by the coefficients in the sum: + +[equation lanczos1] + +where: + +[equation lanczos2] + +Again parameter /g/ is an arbitrarily chosen constant, and /N/ is an arbitrarily chosen +number of terms to evaluate in the "Lanczos sum" part. + +[note +Some authors +choose to define the sum from k=1 to N, and hence end up with N+1 coefficients. +This happens to confuse both the following discussion and the code (since C++ +deals with half open array ranges, rather than the closed range of the sum). +This convention is consistent with __godfrey, but not __pugh, so take care +when referring to the literature in this field.] + +[h4 Computing the Coefficients] + +The coefficients C0..CN-1 need to be computed from /N/ and /g/ +at high precision, and then stored as part of the program. +Calculation of the coefficients is performed via the method of __godfrey; +let the constants be contained in a column vector P, then: + +P = B D C F + +where B is an NxN matrix: + +[equation lanczos4] + +D is an NxN matrix: + +[equation lanczos3] + +C is an NxN matrix: + +[equation lanczos5] + +and F is an N element column vector: + +[equation lanczos6] + +Note than the matrices B, D and C contain all integer terms and depend +only on /N/, this product should be computed first, and then multiplied +by /F/ as the last step. + +[h4 Choosing the Right Parameters] + +The trick is to choose +/N/ and /g/ to give the desired level of accuracy: choosing a small value for +/g/ leads to a strictly convergent series, but one which converges only slowly. +Choosing a larger value of /g/ causes the terms in the series to be large +and\/or divergent for about the first /g-1/ terms, and to then suddenly converge +with a "crunch". + +__pugh has determined the optimal +value of /g/ for /N/ in the range /1 <= N <= 60/: unfortunately in practice choosing +these values leads to cancellation errors in the Lanczos sum as the largest +term in the (alternating) series is approximately 1000 times larger than the result. +These optimal values appear not to be useful in practice unless the evaluation +can be done with a number of guard digits /and/ the coefficients are stored +at higher precision than that desired in the result. These values are best +reserved for say, computing to float precision with double precision arithmetic. + +[table Optimal choices for N and g when computing with guard digits (source: Pugh) +[[Significand Size] [N] [g][Max Error]] +[[24] [6] [5.581][9.51e-12]] +[[53][13][13.144565][9.2213e-23]] +] + +The alternative described by __godfrey is to perform an exhaustive +search of the /N/ and /g/ parameter space to determine the optimal combination for +a given /p/ digit floating-point type. Repeating this work found a good +approximation for double precision arithmetic (close to the one __godfrey found), +but failed to find really +good approximations for 80 or 128-bit long doubles. Further it was observed +that the approximations obtained tended to optimised for the small values +of z (1 < z < 200) used to test the implementation against the factorials. +Computing ratios of gamma functions with large arguments were observed to +suffer from error resulting from the truncation of the Lancozos series. + +__pugh identified all the locations where the theoretical error of the +approximation were at a minimum, but unfortunately has published only the largest +of these minima. However, he makes the observation that the minima +coincide closely with the location where the first neglected term (a[sub N]) in the +Lanczos series S[sub g](z) changes sign. These locations are quite easy to +locate, albeit with considerable computer time. These "sweet spots" need +only be computed once, tabulated, and then searched when required for an +approximation that delivers the required precision for some fixed precision +type. + +Unfortunately, following this path failed to find a really good approximation +for 128-bit long doubles, and those found for 64 and 80-bit reals required an +excessive number of terms. There are two competing issues here: high precision +requires a large value of /g/, but avoiding cancellation errors in the evaluation +requires a small /g/. + +At this point note that the Lanczos sum can be converted into rational form +(a ratio of two polynomials, obtained from the partial-fraction form using +polynomial arithmetic), +and doing so changes the coefficients so that /they are all positive/. That +means that the sum in rational form can be evaluated without cancellation +error, albeit with double the number of coefficients for a given N. Repeating +the search of the "sweet spots", this time evaluating the Lanczos sum in +rational form, and testing only those "sweet spots" whose theoretical error +is less than the machine epsilon for the type being tested, yielded good +approximations for all the types tested. The optimal values found were quite +close to the best cases reported by __pugh (just slightly larger /N/ and slightly +smaller /g/ for a given precision than __pugh reports), and even though converting +to rational form doubles the number of stored coefficients, it should be +noted that half of them are integers (and therefore require less storage space) +and the approximations require a smaller /N/ than would otherwise be required, +so fewer floating point operations may be required overall. + +The following table shows the optimal values for /N/ and /g/ when computing +at fixed precision. These should be taken as work in progress: there are no +values for 106-bit significand machines (Darwin long doubles & NTL quad_float), +and further optimisation of the values of /g/ may be possible. +Errors given in the table +are estimates of the error due to truncation of the Lanczos infinite series +to /N/ terms. They are calculated from the sum of the first five neglected +terms - and are known to be rather pessimistic estimates - although it is noticeable +that the best combinations of /N/ and /g/ occurred when the estimated truncation error +almost exactly matches the machine epsilon for the type in question. + +[table Optimum value for N and g when computing at fixed precision +[[Significand Size][Platform/Compiler Used][N][g][Max Truncation Error]] +[[24][Win32, VC++ 7.1] [6] [1.428456135094165802001953125][9.41e-007]] +[[53][Win32, VC++ 7.1] [13] [6.024680040776729583740234375][3.23e-016]] +[[64][Suse Linux 9 IA64, gcc-3.3.3] [17] [12.2252227365970611572265625][2.34e-024]] +[[116][HP Tru64 Unix 5.1B \/ Alpha, Compaq C++ V7.1-006] [24] [20.3209821879863739013671875][4.75e-035]] +] + +Finally note that the Lanczos approximation can be written as follows +by removing a factor of exp(g) from the denominator, and then dividing +all the coefficients by exp(g): + +[equation lanczos7] + +This form is more convenient for calculating lgamma, but for the gamma +function the division by /e/ turns a possibly exact quality into an +inexact value: this reduces accuracy in the common case that +the input is exact, and so isn't used for the gamma function. + +[h4 References] + +# [#godfrey]Paul Godfrey, [@http://my.fit.edu/~gabdo/gamma.txt "A note on the computation of the convergent +Lanczos complex Gamma approximation"]. +# [#pugh]Glendon Ralph Pugh, +[@http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf +"An Analysis of the Lanczos Gamma Approximation"], +PhD Thesis November 2004. +# Viktor T. Toth, +[@http://www.rskey.org/gamma.htm "Calculators and the Gamma Function"]. +# Mathworld, [@http://mathworld.wolfram.com/LanczosApproximation.html +The Lanczos Approximation]. + +[endsect][/section:lanczos The Lanczos Approximation] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/latin1_symbols.qbk b/doc/sf_and_dist/latin1_symbols.qbk new file mode 100644 index 000000000..e5f0f427d --- /dev/null +++ b/doc/sf_and_dist/latin1_symbols.qbk @@ -0,0 +1,111 @@ +[/ Symbols and accented letters from Latin-1] +[/ File Latin1_symbols.qbk] +[/ http://www.htmlhelp.com/reference/html40/entities/latin1.html ] +[/ based on table Copyright 1998-2006 Liam Quinn.] +[/ Glyphs of the characters ] +[/ are available at the Unicode Consortium . ] + +[template nbsp[]''' '''] [/ no-break space = non-breaking space] +[template iexcl[]'''¡'''] [/ inverted exclamation mark ] +[template cent[]'''¢'''] [/ cent sign ] +[template pound[]'''£'''] [/ pound sign ] +[template curren[]'''¤'''] [/ currency sign ] +[template yen[]'''¥'''] [/ yen sign = yuan sign ] +[template brvbar[]'''¦'''] [/ broken vertical bar ] +[template sectsign[]'''§'''] [/ section sign ] +[template uml[]'''¨'''] [/ diaeresis ] +[template copy[]'''©'''] [/ copyright ] +[template ordf[]'''ª'''] [/ feminine ordinal indicator ] +[template laquo[]'''«'''] [/ left-pointing double angle quotation mark = left pointing guillemet ] +[template not[]'''¬'''] [/ not sign ] +[template shy[]'''­'''] [/ soft hyphen = discretionary hyphen ] +[template reg[]'''®'''] [/ registered sign = registered trade mark sign ] +[template macron[]'''¯'''] [/ macron = spacing macron = overline = APL overbar ] +[template deg[]'''°'''] [/ degree sign ] +[template plusmn[]'''±'''] [/ plus-minus sign = plus-or-minus sign ] +[template sup2[]'''²'''] [/ superscript two = superscript digit two = squared ] +[template cubed[]'''³'''] [/ superscript three = superscript digit three = cubed ] +[template acute[]'''´'''] [/ acute accent = spacing acute ] +[template micro[]'''µ'''] [/ micro sign ] +[template para[]'''¶'''] [/ pilcrow sign = paragraph sign ] +[template middot[]'''·'''] [/ middle dot = Georgian comma = Greek middle dot ] +[template cedil[]'''¸'''] [/ cedilla = spacing cedilla ] +[template sup1[]'''¹'''] [/ superscript one = superscript digit one ] +[template ordm[]'''º'''] [/ masculine ordinal indicator ] +[template raquo[]'''»'''] [/ right-pointing double angle quotation mark = right pointing guillemet ] +[template frac14[]'''¼'''] [/ vulgar fraction one quarter = fraction one quarter ] +[template frac12[]'''½'''] [/ vulgar fraction one half = fraction one half ] +[template frac34[]'''¾'''] [/vulgar fraction three quarters = fraction three quarters ] +[template iquest[]'''¿'''] [/ inverted question mark = turned question mark ] +[template Agrave[]'''À'''] [/ Latin capital letter A with grave = Latin capital letter A grave ] +[template Aacute[]'''Á'''] [/ Latin capital letter A with acute = Latin capital letter A acute ] +[template Acirc[]'''Â'''] [/ Latin capital letter A with circumflex ] +[template Atilde[]'''Ã'''] [/Latin capital letter A with tilde ] +[template Auml[]'''Ä'''] [/ Latin capital letter A with diaeresis ] +[template Aring[]'''Å'''] [/ Latin capital letter A with ring above = Latin capital letter A ring ] +[template AElig[]'''Æ'''] [/ Latin capital letter AE = Latin capital ligature AE ] +[template Ccedil[]'''Ç'''] [/ Latin capital letter C with cedilla ] +[template Egrave[]'''È'''] [/ Latin capital letter E with grave ] +[template Eacute[]'''É'''] [/ Latin capital letter E with acute ] +[template Ecirc[]'''Ê'''] [/ Latin capital letter E with circumflex ] +[template Euml[]'''Ë'''] [/ Latin capital letter E with diaeresis ] +[template Igrave[]'''Ì'''] [/ Latin capital letter I with grave ] +[template Iacute[]'''Í'''] [/ Latin capital letter I with acute ] +[template Icirc[]'''Î'''] [/ Latin capital letter I with circumflex ] +[template Iuml[]'''Ï'''] [/ Latin capital letter I with diaeresis ] +[template ETH[]'''Ð'''] [/ Latin capital letter ETH ] +[template Ntilde[]'''Ñ'''] [/ Latin capital letter N with tilde ] +[template Ograve[]'''Ò'''] [/ Latin capital letter O with grave] +[template Oacute[]'''Ó'''] [/ Latin capital letter O with acute ] +[template Ocirc[]'''Ô'''] [/ Latin capital letter O with circumflex ] +[template Otilde[]'''Õ'''] [/ Latin capital letter O with tilde ] +[template Ouml[]'''Ö'''] [/ Latin capital letter O with diaeresis ] +[template times[]'''×'''] [/ multiplication sign ] +[template Oslash[]'''Ø'''] [/ Latin capital letter O with stroke = Latin capital letter O slash ] +[template Ugrave[]'''Ù'''] [/ Latin capital letter U with grave ] +[template Uacute[]'''Ú'''] [/ Latin capital letter U with acute ] +[template Ucirc[]'''Û'''] [/ Latin capital letter U with circumflex ] +[template Uuml[]'''Ü'''] [/ Latin capital letter U with diaeresis ] +[template Yacute[]'''Ý'''] [/ Latin capital letter Y with acute ] +[template THORN[]'''Þ'''] [/ Latin capital letter THORN ] +[template szlig[]'''ß'''] [/ Latin small letter sharp s = ess-zed ] +[template agrave[]'''à'''] [/ Latin small letter a with grave = Latin small letter a grave ] +[template aacute[]'''á'''] [/ Latin small letter a with acute ] +[template acirc[]'''â'''] [/ Latin small letter a with circumflex ] +[template atilde[]'''ã'''] [/ Latin small letter a with tilde ] +[template auml[]'''ä'''] [/ Latin small letter a with diaeresis ] +[template aring[]'''å'''] [/ Latin small letter a with ring above = Latin small letter a ring ] +[template aelig[]'''æ'''] [/ Latin small letter ae = Latin small ligature ae ] +[template ccedil[]'''ç'''] [/ Latin small letter c with cedilla ] +[template egrave[]'''è'''] [/ Latin small letter e with grave ] +[template eacute[]'''é'''] [/ Latin small letter e with acute ] +[template ecirc[]'''ê'''] [/ Latin small letter e with circumflex ] +[template euml[]'''ë'''] [/ Latin small letter e with diaeresis ] +[template igrave[]'''ì'''] [/ Latin small letter i with grave ] +[template iacute[]'''í'''] [/ Latin small letter i with acute ] +[template icirc[]'''î'''] [/ Latin small letter i with circumflex ] +[template iuml[]'''ï'''] [/ Latin small letter i with diaeresis ] +[template eth[]'''ð'''] [/ Latin small letter eth ] +[template ntilde[]'''ñ'''] [/ Latin small letter n with tilde ] +[template ograve[]'''ò'''] [/Latin small letter o with grave ] +[template oacute[]'''ó'''] [/ Latin small letter o with acute ] +[template ocirc[]'''ô'''] [/ Latin small letter o with circumflex ] +[template otilde[]'''õ'''] [/ Latin small letter o with tilde ] +[template ouml[]'''ö'''] [/ Latin small letter o with diaeresis ] +[template divide[]'''÷'''] [/ division sign ] +[template oslash[]'''ø'''] [/ Latin small letter o with stroke = Latin small letter o slash ] +[template ugrave[]'''ù'''] [/ Latin small letter u with grave ] +[template uacute[]'''ú'''] [/ Latin small letter u with acute ] +[template ucirc[]'''û'''] [/ Latin small letter u with circumflex ] +[template uuml[]'''ü'''] [/ Latin small letter u with diaeresis ] +[template yacute[]'''ý'''] [/ Latin small letter y with acute ] +[template thorn[]'''þ'''] [/ Latin small letter thorn ] +[template yuml[]'''ÿ'''] [/ Latin small letter y with diaeresis ] + +[/ File Latin1_symbols.qbk +Copyright 2007 Paul A. Bristow. +Distributed under the Boost Software License, Version 1.0. +(See accompanying file LICENSE_1_0.txt or copy at +http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/legendre.qbk b/doc/sf_and_dist/legendre.qbk new file mode 100644 index 000000000..c7935689f --- /dev/null +++ b/doc/sf_and_dist/legendre.qbk @@ -0,0 +1,252 @@ +[section:legendre Legendre (and Associated) Polynomials] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` legendre_p(int n, T x); + + template + ``__sf_result`` legendre_p(int n, T x, const ``__Policy``&); + + template + ``__sf_result`` legendre_p(int n, int m, T x); + + template + ``__sf_result`` legendre_p(int n, int m, T x, const ``__Policy``&); + + template + ``__sf_result`` legendre_q(unsigned n, T x); + + template + ``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&); + + template + ``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1); + + template + ``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1); + + + }} // namespaces + +The return type of these functions is computed using the __arg_pomotion_rules: +note than when there is a single template argument the result is the same type +as that argument or `double` if the template argument is an integer type. + +[optional_policy] + +[h4 Description] + + template + ``__sf_result`` legendre_p(int l, T x); + + template + ``__sf_result`` legendre_p(int l, T x, const ``__Policy``&); + +Returns the Legendre Polynomial of the first kind: + +[equation legendre_0] + +Requires -1 <= x <= 1, otherwise returns the result of __domain_error. + +Negative orders are handled via the reflection formula: + +P[sub -l-1](x) = P[sub l](x) + +The following graph illustrates the behaviour of the first few +Legendre Polynomials: + +[$../graphs/legendre_p1.png] + + template + ``__sf_result`` legendre_p(int l, int m, T x); + + template + ``__sf_result`` legendre_p(int l, int m, T x, const ``__Policy``&); + +Returns the associated Legendre polynomial of the first kind: + +[equation legendre_1] + +Requires -1 <= x <= 1, otherwise returns the result of __domain_error. + +Negative values of /l/ and /m/ are handled via the identity relations: + +[equation legendre_3] + +[caution The definition of the associated Legendre polynomial used here +includes a leading Condon-Shortley phase term of (-1)[super m]. This +matches the definition given by Abramowitz and Stegun (8.6.6) and that +used by [@http://mathworld.wolfram.com/LegendrePolynomial.html Mathworld] +and [@http://documents.wolfram.com/mathematica/functions/LegendreP +Mathematica's LegendreP function]. However, uses in the literature +do not always include this phase term, and strangely the specification +for the associated Legendre function in the C++ TR1 (assoc_legendre) +also omits it, in spite of stating that it uses Abramowitz and Stegun +as the final arbiter on these matters. + +See: + +[@http://mathworld.wolfram.com/LegendrePolynomial.html +Weisstein, Eric W. "Legendre Polynomial." +From MathWorld--A Wolfram Web Resource]. + +Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" and +"Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook of +Mathematical Functions with Formulas, Graphs, and Mathematical Tables, +9th printing. New York: Dover, pp. 331-339 and 771-802, 1972. + ] + + template + ``__sf_result`` legendre_q(unsigned n, T x); + + template + ``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&); + +Returns the value of the Legendre polynomial that is the second solution +to the Legendre differential equation, for example: + +[equation legendre_2] + +Requires -1 <= x <= 1, otherwise __domain_error is called. + +The following graph illustrates the first few Legendre functions of the +second kind: + +[$../graphs/legendre_q.png] + + template + ``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1); + +Implements the three term recurrence relation for the Legendre +polynomials, this function can be used to create a sequence of +values evaluated at the same /x/, and for rising /l/. This recurrence +relation holds for Legendre Polynomials of both the first and second kinds. + +[equation legendre_4] + +For example we could produce a vector of the first 10 polynomial +values using: + + double x = 0.5; // Abscissa value + vector v; + v.push_back(legendre_p(0, x)).push_back(legendre_p(1, x)); + for(unsigned l = 1; l < 10; ++l) + v.push_back(legendre_next(l, x, v[l], v[l-1])); + +Formally the arguments are: + +[variablelist +[[l][The degree of the last polynomial calculated.]] +[[x][The abscissa value]] +[[Pl][The value of the polynomial evaluated at degree /l/.]] +[[Plm1][The value of the polynomial evaluated at degree /l-1/.]] +] + + template + ``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1); + +Implements the three term recurrence relation for the Associated Legendre +polynomials, this function can be used to create a sequence of +values evaluated at the same /x/, and for rising /l/. + +[equation legendre_5] + +For example we could produce a vector of the first m+10 polynomial +values using: + + double x = 0.5; // Abscissa value + int m = 10; // order + vector v; + v.push_back(legendre_p(m, m, x)).push_back(legendre_p(1 + m, m, x)); + for(unsigned l = 1 + m; l < m + 10; ++l) + v.push_back(legendre_next(l, m, x, v[l], v[l-1])); + +Formally the arguments are: + +[variablelist +[[l][The degree of the last polynomial calculated.]] +[[m][The order of the Associated Polynomial.]] +[[x][The abscissa value]] +[[Pl][The value of the polynomial evaluated at degree /l/.]] +[[Plm1][The value of the polynomial evaluated at degree /l-1/.]] +] + +[h4 Accuracy] + +The following table shows peak errors (in units of epsilon) +for various domains of input arguments. +Note that only results for the widest floating point type on the system are +given as narrower types have __zero_error. + +[table Peak Errors In the Legendre P Function +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] [Errors in range + +20 < l < 120]] +[[53] [Win32, Visual C++ 8] [Peak=211 Mean=20] [Peak=300 Mean=33]] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=70 Mean=10] [Peak=700 Mean=60]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=70 Mean=10] [Peak=700 Mean=60]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=35 Mean=6] [Peak=292 Mean=41]] +] + +[table Peak Errors In the Associated Legendre P Function +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] ] +[[53] [Win32, Visual C++ 8] [Peak=1200 Mean=7]] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=80 Mean=5]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=80 Mean=5] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=42 Mean=4] ] +] + +[table Peak Errors In the Legendre Q Function +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] [Errors in range + +20 < l < 120]] +[[53] [Win32, Visual C++ 8] [Peak=50 Mean=7] [Peak=4600 Mean=370]] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=51 Mean=8] [Peak=6000 Mean=480]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=51 Mean=8] [Peak=6000 Mean=480]] +[[113] [HPUX IA64, aCC A.06.06] [Peak=90 Mean=10] [Peak=1700 Mean=140]] +] + +Note that the worst errors occur when the order increases, values greater than +~120 are very unlikely to produce sensible results, especially in the associated +polynomial case when the degree is also large. Further the relative errors +are likely to grow arbitrarily large when the function is very close to a root. + +No comparisons to other libraries are shown here: there appears to be only one +viable implementation method for these functions, the comparisons to other +libraries that have been run show identical error rates to those given here. + +[h4 Testing] + +A mixture of spot tests of values calculated using functions.wolfram.com, +and randomly generated test data are +used: the test data was computed using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision. + +[h4 Implementation] + +These functions are implemented using the stable three term +recurrence relations. These relations guarentee low absolute error +but cannot guarentee low relative error near one of the roots of the +polynomials. + +[endsect][/section:beta_function The Beta Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/lgamma.qbk b/doc/sf_and_dist/lgamma.qbk new file mode 100644 index 000000000..b961d50b0 --- /dev/null +++ b/doc/sf_and_dist/lgamma.qbk @@ -0,0 +1,217 @@ +[section:lgamma Log Gamma] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` lgamma(T z); + + template + ``__sf_result`` lgamma(T z, const ``__Policy``&); + + template + ``__sf_result`` lgamma(T z, int* sign); + + template + ``__sf_result`` lgamma(T z, int* sign, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +The [@http://en.wikipedia.org/wiki/Gamma_function lgamma function] is defined by: + +[equation lgamm1] + +The second form of the function takes a pointer to an integer, +which if non-null is set on output to the sign of tgamma(z). + +[optional_policy] + +[$../graphs/lgamma.png] + +There are effectively two versions of this function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the significand +of T correspond to a certain __lanczos. In practice, any built-in +floating-point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp. + +The return type of these functions is computed using the __arg_pomotion_rules: +the result is of type `double` if T is an integer type, or type T otherwise. + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms +with various floating point types, along with comparisons to the +__gsl, __glibc, __hpc and +__cephes libraries. Unless otherwise specified any +floating point type that is narrower than the one shown will have +__zero_error. + +Note that while the relative errors near the positive roots of lgamma +are very low, the lgamma function has an infinite number of irrational +roots for negative arguments: very close to these negative roots only +a low absolute error can be guaranteed. + +[table +[[Significand Size] [Platform and Compiler] [Factorials and Half factorials] [Values Near Zero] [Values Near 1 or 2] [Values Near a Negative Pole]] +[[53] [Win32 Visual C++ 8] +[Peak=0.88 Mean=0.14 + +(GSL=33) (__cephes=1.5)] +[Peak=0.96 Mean=0.46 + +(GSL=5.2) (__cephes=1.1)] +[Peak=0.86 Mean=0.46 + +(GSL=1168) (__cephes~500000)] +[Peak=4.2 Mean=1.3 + +(GSL=25) (__cephes=1.6)] ] +[[64] [Linux IA32 / GCC] +[Peak=1.9 Mean=0.43 + +(__glibc Peak=1.7 Mean=0.49)] +[Peak=1.4 Mean=0.57 + +(__glibc Peak= 0.96 Mean=0.54)] +[Peak=0.86 Mean=0.35 + +(__glibc Peak=0.74 Mean=0.26)] + +[Peak=6.0 Mean=1.8 + +(__glibc Peak=3.0 Mean=0.86)] ] +[[64] [Linux IA64 / GCC] +[Peak=0.99 Mean=0.12 + +(__glibc Peak 0)] + +[Pek=1.2 Mean=0.6 + +(__glibc Peak 0)] +[Peak=0.86 Mean=0.16 + +(__glibc Peak 0)] +[Peak=2.3 Mean=0.69 + +(__glibc Peak 0)] ] +[[113] [HPUX IA64, aCC A.06.06] +[Peak=0.96 Mean=0.13 + +(__hpc Peak 0)] +[Peak=0.99 Mean=0.53 + +(__hpc Peak 0)] +[Peak=0.9 Mean=0.4 + +(__hpc Peak 0)] +[Peak=3.0 Mean=0.9 + +(__hpc Peak 0)] ] +] + +[h4 Testing] + +The main tests for this function involve comparisons against the logs of +the factorials which can be independently calculated to very high accuracy. + +Random tests in key problem areas are also used. + +[h4 Implementation] + +The generic version of this function is implemented by combining the series and +continued fraction representations for the incomplete gamma function: + +[equation lgamm2] + +where /l/ is an arbitrary integration limit: choosing [^l = max(10, a)] +seems to work fairly well. For negative /z/ the logarithm version of the +reflection formula is used: + +[equation lgamm3] + +For types of known precision, the __lanczos is used, a traits class +`boost::math::lanczos::lanczos_traits` maps type T to an appropriate +approximation. The logarithmic version of the __lanczos is: + +[equation lgamm4] + +Where L[sub e,g][space] is the Lanczos sum, scaled by e[super g]. + +As before the reflection formula is used for /z < 0/. + +When z is very near 1 or 2, then the logarithmic version of the __lanczos +suffers very badly from cancellation error: indeed for values sufficiently +close to 1 or 2, arbitrarily large relative errors can be obtained (even though +the absolute error is tiny). + +For types with up to 113 bits of precision +(up to and including 128-bit long doubles), root-preserving +rational approximations [jm_rationals] are used +over the intervals [1,2] and [2,3]. Over the interval [2,3] the approximation +form used is: + + lgamma(z) = (z-2)(z+1)(Y + R(z-2)); + +Where Y is a constant, and R(z-2) is the rational approximation: optimised +so that it's absolute error is tiny compared to Y. In addition +small values of z greater +than 3 can handled by argument reduction using the recurrence relation: + + lgamma(z+1) = log(z) + lgamma(z); + +Over the interval [1,2] two approximations have to be used, one for small z uses: + + lgamma(z) = (z-1)(z-2)(Y + R(z-1)); + +Once again Y is a constant, and R(z-1) is optimised for low absolute error +compared to Y. For z > 1.5 the above form wouldn't converge to a +minimax solution but this similar form does: + + lgamma(z) = (2-z)(1-z)(Y + R(2-z)); + +Finally for z < 1 the recurrence relation can be used to move to z > 1: + + lgamma(z) = lgamma(z+1) - log(z); + +Note that while this involves a subtraction, it appears not +to suffer from cancellation error: as z decreases from 1 +the `-log(z)` term grows positive much more +rapidly than the `lgamma(z+1)` term becomes negative. So in this +specific case, significant digits are preserved, rather than cancelled. + +For other types which do have a __lanczos defined for them +the current solution is as follows: imagine we +balance the two terms in the __lanczos by dividing the power term by its value +at /z = 1/, and then multiplying the Lanczos coefficients by the same value. +Now each term will take the value 1 at /z = 1/ and we can rearrange the power terms +in terms of log1p. Likewise if we subtract 1 from the Lanczos sum part +(algebraically, by subtracting the value of each term at /z = 1/), we obtain +a new summation that can be also be fed into log1p. Crucially, all of the +terms tend to zero, as /z -> 1/: + +[equation lgamm5] + +The C[sub k][space] terms in the above are the same as in the __lanczos. + +A similar rearrangement can be performed at /z = 2/: + +[equation lgamm6] + +[endsect][/section:lgamma The Log Gamma Function] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/license.qbk b/doc/sf_and_dist/license.qbk new file mode 100644 index 000000000..16860aa87 --- /dev/null +++ b/doc/sf_and_dist/license.qbk @@ -0,0 +1,6 @@ +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/math.qbk b/doc/sf_and_dist/math.qbk new file mode 100644 index 000000000..ad672056f --- /dev/null +++ b/doc/sf_and_dist/math.qbk @@ -0,0 +1,409 @@ +[article Math Toolkit + [quickbook 1.4] + [copyright 2006-2007 John Maddock, Paul A. Bristow, Hubert Holin and Xiaogang Zhang] + [purpose ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Maddock, John], [Bristow, Paul A.], [Holin, Hubert], [Zhang, Xiaogang]] + [category math] + [purpose mathematics] + [/last-revision $Date$] +] + +[template equation[name] ''' + + + + + + +'''] + +[include html4_symbols.qbk] [/ just for testing] +[/include latin1_symbols.qbk] [/ just for testing] +[include common_overviews.qbk][/ overviews that appear in more than one place!] +[include roadmap.qbk] [/ for history] + +[def __effects [*Effects: ]] +[def __formula [*Formula: ]] +[def __exm1 '''ex - 1'''] +[def __ex '''ex'''] +[def __te '''2ɛ'''] [/small Latin letter open e] + +[def __ceilR '''⌉'''] +[def __ceilL '''ऄ'''] +[def __floorR '''⌋'''] +[def __floorL '''⌊'''] +[def __infin '''∞'''] +[def __integral '''∫'''] +[def __aacute '''á'''] +[def __eacute '''é'''] +[def __quarter '''¼'''] +[def __nearequal '''≊'''] +[def __quarter '''¼'''] +[def __space '''​'''] + +[def __NTL_RR [@http://shoup.net/ntl/doc/RR.txt NTL::RR]] +[def __NTL_quad_float [@http://shoup.net/ntl/doc/quad_float.txt NTL::quad_float]] + +[def __godfrey [link godfrey Godfrey]] +[def __pugh [link pugh Pugh]] + +[def __caution This is now an official Boost library, but remains a library under + construction, the code is fully functional and robust, but + interfaces, library structure, and function and distribution names + may still be changed without notice.] + +[template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]] +[template jm_rationals[] [link math_toolkit.backgrounders.implementation.rational_approximations_used devised by JM]] +[def __domain_error [link domain_error domain_error]] +[def __pole_error [link pole_error pole_error]] +[def __overflow_error [link overflow_error overflow_error]] +[def __underflow_error [link underflow_error underflow_error]] +[def __denorm_error [link denorm_error denorm_error]] +[def __evaluation_error [link evaluation_error evaluation_error]] +[def __checked_narrowing_cast [link checked_narrowing_cast checked_narrowing_cast]] + +[def __arg_pomotion_rules [link math_toolkit.main_overview.result_type ['result type calculation rules]]] +[def __sf_result [link math_toolkit.main_overview.result_type ['calculated-result-type]]] + +[/ The following macros expand to links to the various special functions +and use the function's name as the link text] + +[/Misc] +[def __lanczos [link math_toolkit.backgrounders.lanczos Lanczos approximation]] +[def __zero_error [link zero_error effectively zero error]] +[def __relative_error [link math_toolkit.backgrounders.relative_error relative zero error]] + +[/gammas] +[def __lgamma [link math_toolkit.special.sf_gamma.lgamma lgamma]] +[def __digamma [link math_toolkit.special.sf_gamma.digamma digamma]] +[def __tgamma_ratio [link math_toolkit.special.sf_gamma.gamma_ratios tgamma_ratio]] +[def __tgamma_delta_ratio [link math_toolkit.special.sf_gamma.gamma_ratios tgamma_delta_ratio]] +[def __tgamma [link math_toolkit.special.sf_gamma.tgamma tgamma]] +[def __tgamma1pm1 [link math_toolkit.special.sf_gamma.tgamma tgamma1pm1]] +[def __tgamma_lower [link math_toolkit.special.sf_gamma.igamma tgamma_lower]] +[def __gamma_p [link math_toolkit.special.sf_gamma.igamma gamma_p]] +[def __gamma_q [link math_toolkit.special.sf_gamma.igamma gamma_q]] +[def __gamma_q_inv [link math_toolkit.special.sf_gamma.igamma_inv gamma_q_inv]] +[def __gamma_p_inv [link math_toolkit.special.sf_gamma.igamma_inv gamma_p_inv]] +[def __gamma_q_inva [link math_toolkit.special.sf_gamma.igamma_inv gamma_q_inva]] +[def __gamma_p_inva [link math_toolkit.special.sf_gamma.igamma_inv gamma_p_inva]] +[def __gamma_p_derivative [link math_toolkit.special.sf_gamma.gamma_derivatives gamma_p_derivative]] + +[/factorials] +[def __factorial [link math_toolkit.special.factorials.sf_factorial factorial]] +[def __unchecked_factorial [link math_toolkit.special.factorials.sf_factorial unchecked_factorial]] +[def __max_factorial [link math_toolkit.special.factorials.sf_factorial max_factorial]] +[def __double_factorial [link math_toolkit.special.factorials.sf_double_factorial double_factorial]] +[def __rising_factorial [link math_toolkit.special.factorials.sf_rising_factorial rising_factorial]] +[def __falling_factorial [link math_toolkit.special.factorials.sf_falling_factorial falling_factorial]] + +[/error functions] +[def __erf [link math_toolkit.special.sf_erf.error_function erf]] +[def __erfc [link math_toolkit.special.sf_erf.error_function erfc]] +[def __erf_inv [link math_toolkit.special.sf_erf.error_inv erf_inv]] +[def __erfc_inv [link math_toolkit.special.sf_erf.error_inv erfc_inv]] + +[/beta functions] +[def __beta [link math_toolkit.special.sf_beta.beta_function beta]] +[def __beta3 [link math_toolkit.special.sf_beta.ibeta_function beta]] +[def __betac [link math_toolkit.special.sf_beta.ibeta_function betac]] +[def __ibeta [link math_toolkit.special.sf_beta.ibeta_function ibeta]] +[def __ibetac [link math_toolkit.special.sf_beta.ibeta_function ibetac]] +[def __ibeta_inv [link math_toolkit.special.sf_beta.ibeta_inv_function ibeta_inv]] +[def __ibetac_inv [link math_toolkit.special.sf_beta.ibeta_inv_function ibetac_inv]] +[def __ibeta_inva [link math_toolkit.special.sf_beta.ibeta_inv_function ibeta_inva]] +[def __ibetac_inva [link math_toolkit.special.sf_beta.ibeta_inv_function ibetac_inva]] +[def __ibeta_invb [link math_toolkit.special.sf_beta.ibeta_inv_function ibeta_invb]] +[def __ibetac_invb [link math_toolkit.special.sf_beta.ibeta_inv_function ibetac_invb]] +[def __ibeta_derivative [link math_toolkit.special.sf_beta.beta_derivative ibeta_derivative]] + +[/elliptic integrals] +[def __ellint_rj [link math_toolkit.special.ellint.ellint_carlson ellint_rj]] +[def __ellint_rf [link math_toolkit.special.ellint.ellint_carlson ellint_rf]] +[def __ellint_rc [link math_toolkit.special.ellint.ellint_carlson ellint_rc]] +[def __ellint_rd [link math_toolkit.special.ellint.ellint_carlson ellint_rd]] +[def __ellint_1 [link math_toolkit.special.ellint.ellint_1 ellint_1]] +[def __ellint_2 [link math_toolkit.special.ellint.ellint_2 ellint_2]] +[def __ellint_3 [link math_toolkit.special.ellint.ellint_3 ellint_3]] + +[/Bessel functions] +[def __cyl_bessel_j [link math_toolkit.special.bessel.bessel cyl_bessel_j]] +[def __cyl_neumann [link math_toolkit.special.bessel.bessel cyl_neumann]] +[def __cyl_bessel_i [link math_toolkit.special.bessel.mbessel cyl_bessel_i]] +[def __cyl_bessel_k [link math_toolkit.special.bessel.mbessel cyl_bessel_k]] +[def __sph_bessel [link math_toolkit.special.bessel.sph_bessel sph_bessel]] +[def __sph_neumann [link math_toolkit.special.bessel.sph_bessel sph_neumann]] + +[/sinus cardinals] +[def __sinc_pi [link math_toolkit.special.sinc.sinc_pi sinc_pi]] +[def __sinhc_pi [link math_toolkit.special.sinc.sinhc_pi sinhc_pi]] + +[/Inverse hyperbolics] +[def __acosh [link math_toolkit.special.inv_hyper.acosh acosh]] +[def __asinh [link math_toolkit.special.inv_hyper.asinh asinh]] +[def __atanh [link math_toolkit.special.inv_hyper.atanh atanh]] + +[/classify] +[def __fpclassify [link math_toolkit.special.fpclass fpclassify]] +[def __isfinite [link math_toolkit.special.fpclass isfinite]] +[def __isnan [link math_toolkit.special.fpclass isnan]] +[def __isinf [link math_toolkit.special.fpclass isinf]] +[def __isnormal [link math_toolkit.special.fpclass isnormal]] + +[/powers etc] +[def __expm1 [link math_toolkit.special.powers.expm1 expm1]] +[def __log1p [link math_toolkit.special.powers.log1p log1p]] +[def __cbrt [link math_toolkit.special.powers.cbrt cbrt]] +[def __sqrt1pm1 [link math_toolkit.special.powers.sqrt1pm1 sqrt1pm1]] +[def __powm1 [link math_toolkit.special.powers.powm1 powm1]] +[def __hypot [link math_toolkit.special.powers.hypot hypot]] + +[/ distribution non-members] +[def __cdf [link math.dist.cdf Cumulative Distribution Function]] +[def __pdf [link math.dist.pdf Probability Density Function]] +[def __ccdf [link math.dist.ccdf Complement of the Cumulative Distribution Function]] +[def __quantile [link math.dist.quantile Quantile]] +[def __quantile_c [link math.dist.quantile_c Quantile from the complement of the probability]] +[def __mean [link math.dist.mean mean]] +[def __median [link math.dist.median median]] +[def __mode [link math.dist.mode mode]] +[def __skewness [link math.dist.skewness skewness]] +[def __kurtosis [link math.dist.kurtosis kurtosis]] +[def __kurtosis_excess [link math.dist.kurtosis_excess kurtosis_excess]] +[def __variance [link math.dist.variance variance]] +[def __sd [link math.dist.sd standard deviation]] +[def __hazard [link math.dist.hazard Hazard Function]] +[def __chf [link math.dist.chf Cumulative Hazard Function]] +[def __range [link math.dist.range range]] +[def __support [link math.dist.support support]] + +[/ distribution def names end in distrib to avoid clashes] +[def __beta_distrib [link math_toolkit.dist.dist_ref.dists.beta_dist Beta Distribution]] +[def __binomial_distrib [link math_toolkit.dist.dist_ref.dists.binomial_dist Binomial Distribution]] +[def __cauchy_distrib [link math_toolkit.dist.dist_ref.dists.cauchy_dist Cauchy Distribution]] +[def __chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.chi_squared_dist Chi Squared Distribution]] +[def __exp_distrib [link math_toolkit.dist.dist_ref.dists.exp_dist Exponential Distribution]] +[def __F_distrib [link math_toolkit.dist.dist_ref.dists.f_dist Fisher F Distribution]] +[def __gamma_distrib [link math_toolkit.dist.dist_ref.dists.gamma_dist Gamma Distribution]] +[def __lognormal_distrib [link math_toolkit.dist.dist_ref.dists.lognormal_dist Log-normal Distribution]] +[def __negative_binomial_distrib [link math_toolkit.dist.dist_ref.dists.negative_binomial_dist Negative Binomial Distribution]] +[def __normal_distrib [link math_toolkit.dist.dist_ref.dists.normal_dist Normal Distribution]] +[def __poisson_distrib [link math_toolkit.dist.dist_ref.dists.poisson_dist Poisson Distribution]] +[def __students_t_distrib [link math_toolkit.dist.dist_ref.dists.students_t_dist Students t Distribution]] +[def __weibull_distrib [link math_toolkit.dist.dist_ref.dists.weibull Weibull Distribution]] + +[/links to policy] +[def __Policy [link math_toolkit.policy Policy]] [/ Used in distribution template specifications] +[def __policy_section [link math_toolkit.policy Policies]] [/ Used in text to refer too.] +[def __policy_class [link math_toolkit.policy.pol_ref.pol_ref_ref policies::policy<>]] +[def __math_undefined [link math_toolkit.policy.pol_ref.assert_undefined mathematically undefined function]] +[def __policy_ref [link math_toolkit.policy.pol_ref policy reference]] +[def __math_discrete [link math_toolkit.policy.pol_ref.discrete_quant_ref discrete functions]] +[def __error_policy [link math_toolkit.policy.pol_ref.error_handling_policies error handling policies]] +[def __changing_policy_defaults [link math_toolkit.policy.pol_ref.policy_defaults changing policies defaults]] +[def __user_error_handling [link math_toolkit.policy.pol_tutorial.user_def_err_pol user error handling]] + +[def __usual_accessors __cdf, __pdf, __quantile, __hazard, + __chf, __mean, __median, __mode, __variance, __sd, __skewness, + __kurtosis, __kurtosis_excess, __range and __support] + +[def __gsl [@http://www.gnu.org/software/gsl/ GSL-1.9]] +[def __glibc [@http://www.gnu.org/software/libc/ GNU C Lib]] +[def __hpc [@http://docs.hp.com/en/B9106-90010/index.html HP-UX C Library]] +[def __cephes [@http://www.netlib.org/cephes/ Cephes]] + + +[def __why_complements [link why_complements why complements?]] +[def __complements [link complements complements]] + +[/ Some composite templates] +[template super[x]''''''[x]''''''] +[template sub[x]''''''[x]''''''] +[template floor[x]'''⌊'''[x]'''⌋'''] +[template floorlr[x][lfloor][x][rfloor]] +[template ceil[x] '''⌈'''[x]'''⌉'''] + +[template header_file[file] [@../../../../../[file] [file]]] + +[template optional_policy[] +The final __Policy argument is optional and can be used to +control the behaviour of the function: how it handles errors, +what level of precision to use etc. Refer to the +[link math_toolkit.policy policy documentation for more details].] + +[template discrete_quantile_warning[NAME] +[caution +The [NAME] distribution is a discrete distribution: internally +functions like the `cdf` and `pdf` are treated "as if" they are continuous +functions, but in reality the results returned from these functions +only have meaning if an integer value is provided for the random variate +argument. + +The quantile function will by default return an integer result that has been +/rounded outwards/. That is to say lower quantiles (where the probability is +less than 0.5) are rounded downward, and upper quantiles (where the probability +is greater than 0.5) are rounded upwards. This behaviour +ensures that if an X% quantile is requested, then /at least/ the requested +coverage will be present in the central region, and /no more than/ +the requested coverage will be present in the tails. + +This behaviour can be changed so that the quantile functions are rounded +differently, or even return a real-valued result using +[link math_toolkit.policy.pol_overview Policies]. It is strongly +recommended that you read the tutorial +[link math_toolkit.policy.pol_tutorial.understand_dis_quant +Understanding Quantiles of Discrete Distributions] before +using the quantile function on the [NAME] distribution. The +[link math_toolkit.policy.pol_ref.discrete_quant_ref reference docs] +describe how to change the rounding policy +for these distributions. +] [/ caution] +] [/ template discrete_quantile_warning] + +[section:main_overview Overview] + +[include overview.qbk] +[include structure.qbk] [/getting about] + +[include result_type_calc.qbk] +[include error_handling.qbk] + +[section:compilers_overview Compilers] +[compilers_overview] +[endsect] +[section:pol_overview Configuration and Policies] +[policy_overview] +[endsect] + +[include thread_safety.qbk] + +[section:perf_over Performance] +[performance_overview] +[endsect] +[section:history1 History and What's New] +[history] +[endsect] +[include contact_info.qbk] + +[endsect] [/section:main_overview Overview] + +[section:dist Statistical Distributions and Functions] +[include dist_tutorial.qbk] +[include dist_reference.qbk] +[endsect] [/section:dist Statistical Distributions and Functions] + +[section:special Special Functions] + +[section:sf_gamma Gamma Functions] +[include tgamma.qbk] +[include lgamma.qbk] +[include digamma.qbk] +[include gamma_ratios.qbk] +[include igamma.qbk] +[include igamma_inv.qbk] +[include gamma_derivatives.qbk] +[endsect] [/section:sf_gamma Gamma Functions] + +[include factorials.qbk] + +[section:sf_beta Beta Functions] +[include beta.qbk] +[include ibeta.qbk] +[include ibeta_inv.qbk] +[include beta_derivative.qbk] +[endsect] [/section:sf_beta Beta Functions] + +[section:sf_erf Error Functions] +[include erf.qbk] +[include erf_inv.qbk] +[endsect] [/section:sf_erf Error Functions] + +[section:sf_poly Polynomials] +[include legendre.qbk] +[include laguerre.qbk] +[include hermite.qbk] +[include spherical_harmonic.qbk] +[endsect] [/section:sf_poly Polynomials] + +[section:bessel Bessel Functions] +[include bessel_introduction.qbk] +[include bessel_jy.qbk] +[include bessel_ik.qbk] +[include bessel_spherical.qbk] +[endsect] [/section:bessel Bessel Functions] + +[section:ellint Elliptic Integrals] +[include ellint_introduction.qbk] +[include ellint_carlson.qbk] +[include ellint_legendre.qbk] +[endsect] [/section:ellint Elliptic Integrals] + +[include powers.qbk] +[include sinc.qbk] +[include inv_hyper.qbk] +[include fpclassify.qbk] +[endsect] [/section:special Special Functions] + +[section:toolkit Internal Details and Tools (Experimental)] + +[include internals_overview.qbk] + +[section:internals1 Reused Utilities] +[include series.qbk] +[include fraction.qbk] +[include rational.qbk] +[include roots.qbk] +[include roots_without_derivatives.qbk] +[include minima.qbk] +[endsect] [/section:internals1 Reused Utilities] + +[section:internals2 Testing and Development] +[include polynomial.qbk] +[include minimax.qbk] +[include relative_error.qbk] +[include test_data.qbk] +[endsect] [/section:internals2 Testing and Development] + +[endsect] [/section:toolkit Toolkit] + +[section:using_udt Use with User Defined Floating-Point Types] +[include concepts.qbk] +[endsect] [/section:using_udt Use with User Defined Floating-Point Types] + +[include policy.qbk] + +[include performance.qbk] + +[section:backgrounders Backgrounders] +[include implementation.qbk] +[include error.qbk] [/relative error NOT handling] +[include lanczos.qbk] +[include remez.qbk] +[include references.qbk] +[endsect] [/section:backgrounds Backgrounders] + +[section:status Library Status] +[section:history1 History and What's New] +[history] +[endsect] +[section:compilers Compilers] +[compilers_overview] +[endsect] +[include issues.qbk] +[include credits.qbk] +[/include test_HTML4_symbols.qbk] +[/include test_Latin1_symbols.qbk] + +[endsect] [/section:status Status and Roadmap] + +[/ math.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/minima.qbk b/doc/sf_and_dist/minima.qbk new file mode 100644 index 000000000..18e21940c --- /dev/null +++ b/doc/sf_and_dist/minima.qbk @@ -0,0 +1,64 @@ +[section:minima Locating Function Minima] + +[h4 synopsis] + +`` +#include +`` + + template + std::pair brent_find_minima(F f, T min, T max, int bits); + + template + std::pair brent_find_minima(F f, T min, T max, int bits, boost::uintmax_t& max_iter); + +[h4 Description] + +These two functions locate the minima of the continuous function /f/ using Brent's +algorithm. Parameters are: + +[variablelist +[[f] [The function to minimise. The function should be smooth over the + range \[min,max\], with no maxima occurring in that interval.]] +[[min] [The lower endpoint of the range in which to search + for the minima.]] +[[max] [The upper endpoint of the range in which to search + for the minima.]] +[[bits] [The number of bits precision to which the minima should be found. + Note that in principle, the minima can not be located to greater + accuracy than the square root of machine epsilon, therefore if /bits/ + is set to a value greater than one half of the bits in type T, then + the value will be ignored.]] +[[max_iter] [The maximum number of iterations to use + in the algorithm, if not provided the algorithm will just + keep on going until the minima is found.]] +] + +[*Returns:] a pair containing the value of the abscissa at the minima and the value +of f(x) at the minima. + +[h4 Implementation] + +This is a reasonably faithful implementation of Brent's algorithm, refer +to: + +Brent, R.P. 1973, Algorithms for Minimization without Derivatives +(Englewood Cliffs, NJ: Prentice-Hall), Chapter 5. + +Numerical Recipes in C, The Art of Scientific Computing, +Second Edition, William H. Press, Saul A. Teukolsky, +William T. Vetterling, and Brian P. Flannery. +Cambridge University Press. 1988, 1992. + +An algorithm with guaranteed convergence for finding a zero +of a function, R. P. Brent, The Computer Journal, Vol 44, 1971. + +[endsect][/section:minima Locating Function Minima] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/minimax.qbk b/doc/sf_and_dist/minimax.qbk new file mode 100644 index 000000000..59b9a078d --- /dev/null +++ b/doc/sf_and_dist/minimax.qbk @@ -0,0 +1,159 @@ +[section:minimax Minimax Approximations and the Remez Algorithm] + +The directory libs/math/minimax contains a command line driven +program for the generation of minimax approximations using the Remez +algorithm. Both polynomial and rational approximations are supported, +although the latter are tricky to converge: it is not uncommon for +convergence of rational forms to fail. No such limitations are present +for polynomial approximations which should always converge smoothly. + +It's worth stressing that developing rational approximations to functions +is often not an easy task, and one to which many books have been devoted. +To use this tool, you will need to have a reasonable grasp of what the Remez +algorithm is, and the general form of the approximation you want to achieve. + +Unless you already familar with the Remez method +you should first read the [link math_toolkit.backgrounders.remez +brief background article explaining the principals behind the +Remez algorithm]. + +The program consists of two parts: + +[variablelist +[[main.cpp][Contains the command line parser, and all the calls to the Remez code.]] +[[f.cpp][Contains the function to approximate.]] +] + +Therefore to use this tool, you must modify f.cpp to return the function to +approximate. The tools supports multiple function approximations within +the same compiled program: each as a separate variant: + + NTL::RR f(const NTL::RR& x, int variant); + +Returns the value of the function /variant/ at point /x/. So if you +wish you can just add the function to approximate as a new variant +after the existing examples. + +In addition to those two files, the program needs to be linked to +a [link math_toolkit.using_udt.use_ntl patched NTL library to compile]. + +Note that the function /f/ must return the rational part of the +approximation: for example if you are approximating a function +/f(x)/ then it is quite common to use: + + f(x) = g(x)(Y + R(x)) + +where /g(x)/ is the dominant part of /f(x)/, /Y/ is some constant, and +/R(x)/ is the rational approximation part, usually optimised for a low +absolute error compared to |Y|. + +In this case you would define /f/ to return ['f(x)/g(x)] and then set the +y-offset of the approximation to /Y/ (see command line options below). + +Many other forms are possible, but in all cases the objective is to +split /f(x)/ into a dominant part that you can evaluate easily using +standard math functions, and a smooth and slowly changing rational approximation +part. Refer to your favourite textbook for more examples. + +Command line options for the program are as follows: + +[variablelist +[[variant N][Sets the current function variant to N. This allows multiple functions + that are to be approximated to be compiled into the same executable. + Defaults to 0.]] +[[range a b][Sets the domain for the approximation to the range \[a,b\], defaults + to \[0,1\].]] +[[relative][Sets the Remez code to optimise for relative error. This is the default + at program startup. Note that relative error can only be used + if f(x) has no roots over the range being optimised.]] +[[absolute][Sets the Remez code to optimise for absolute error.]] +[[pin \[true|false\]]["Pins" the code so that the rational approximation + passes through the origin. Obviously only set this to + /true/ if R(0) must be zero. This is typically used when + trying to preserve a root at \[0,0\] while also optimising + for relative error.]] +[[order N D][Sets the order of the approximation to /N/ in the numerator and /D/ + in the denominator. If /D/ is zero then the result will be a polynomial + approximation. There will be N+D+2 coefficients in total, the first + coefficient of the numerator is zero if /pin/ was set to true, and the + first coefficient of the denominator is always one.]] +[[working-precision N][Sets the working precision of NTL::RR to /N/ binary digits. Defaults to 250.]] +[[target-precision N][Sets the precision of printed output to /N/ binary digits: + set to the same number of digits as the type that will be used to + evaluate the approximation. Defaults to 53 (for double precision).]] +[[skew val]["Skews" the initial interpolated control points towards one + end or the other of the range. Positive values skew the + initial control points towards the left hand side of the + range, and negative values towards the right hand side. + If an approximation won't converge (a common situation) + try adjusting the skew parameter until the first step yields + the smallest possible error. /val/ should be in the range + \[-100,+100\], the default is zero.]] +[[brake val][Sets a brake on each step so that the change in the + control points is braked by /val%/. Defaults to 50, + try a higher value if an approximation won't converge, + or a lower value to get speedier convergence.]] +[[x-offset val][Sets the x-offset to /val/: the approximation will + be generated for `f(x + X) + Y` where /X/ is the x-offset + and /Y/ is the y-offset. Defaults to zero. To avoid + rounding errors, take care to specify a value that can + be exactly represented as a floating point number.]] +[[y-offset val][Sets the y-offset to /val/: the approximation will + be generated for `f(x + X) + Y` where /X/ is the x-offset + and /Y/ is the y-offset. Defaults to zero. To avoid + rounding errors, take care to specify a value that can + be exactly represented as a floating point number.]] +[[y-offset auto][Sets the y-offset to the average value of f(x) + evaluated at the two endpoints of the range plus the midpoint + of the range. The calculated value is deliberately truncated + to /float/ precision (and should be stored as a /float/ + in your code). The approximation will + be generated for `f(x + X) + Y` where /X/ is the x-offset + and /Y/ is the y-offset. Defaults to zero.]] +[[graph N][Prints N evaluations of f(x) at evenly spaced points over the + range being optimised. If unspecified then /N/ defaults + to 3. Use to check that f(x) is indeed smooth over the range + of interest.]] +[[step N][Performs /N/ steps, or one step if /N/ is unspecified. + After each step prints: the peek error at the extrema of + the error function of the approximation, + the theoretical error term solved for on the last step, + and the maximum relative change in the location of the + Chebyshev control points. The approximation is converged on the + minimax solution when the two error terms are (approximately) + equal, and the change in the control points has decreased to + a suitably small value.]] +[[test \[float|double|long\]][Tests the current approximation at float, + double, or long double precision. Useful to check for rounding + errors in evaluating the approximation at fixed precision. + Tests are conducted at the extrema of the error function of the + approximation, and at the zeros of the error function.]] +[[test \[float|double|long\] N] [Tests the current approximation at float, + double, or long double precision. Useful to check for rounding + errors in evaluating the approximation at fixed precision. + Tests are conducted at N evenly spaced points over the range + of the approximation. If none of \[float|double|long\] are specified + then tests using NTL::RR, this can be used to obtain the error + function of the approximation.]] +[[rescale a b][Takes the current Chebeshev control points, and rescales them + over a new interval \[a,b\]. Sometimes this can be used to obtain + starting control points for an approximation that can not otherwise be + converged.]] +[[rotate][Moves one term from the numerator to the denominator, but keeps the + Chebyshev control points the same. Sometimes this can be used to obtain + starting control points for an approximation that can not otherwise be + converged.]] +[[info][Prints out the current approximation: the location of the zeros of the + error function, the location of the Chebyshev control points, the + x and y offsets, and of course the coefficients of the polynomials.]] +] + + +[endsect][/section:minimax Minimax Approximations and the Remez Algorithm] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/overview.qbk b/doc/sf_and_dist/overview.qbk new file mode 100644 index 000000000..69982aeb4 --- /dev/null +++ b/doc/sf_and_dist/overview.qbk @@ -0,0 +1,90 @@ +[section:intro About the Math Toolkit] + +This library is divided into three interconnected parts: + +[h4 Statistical Distributions] + +Provides a reasonably comprehensive set of +[link math_toolkit.dist statistical distributions], +upon which higher level statistical tests can be built. + +The initial focus is on the central +[@http://en.wikipedia.org/wiki/Univariate univariate ] +[@http://mathworld.wolfram.com/StatisticalDistribution.html distributions]. +Both [@http://mathworld.wolfram.com/ContinuousDistribution.html continuous] +(like [link math_toolkit.dist.dist_ref.dists.normal_dist normal] +& [link math_toolkit.dist.dist_ref.dists.f_dist Fisher]) +and [@http://mathworld.wolfram.com/DiscreteDistribution.html discrete] +(like [link math_toolkit.dist.dist_ref.dists.binomial_dist binomial] +& [link math_toolkit.dist.dist_ref.dists.poisson_dist Poisson]) +distributions are provided. + +A [link math_toolkit.dist.stat_tut comprehensive tutorial is provided], +along with a series of +[link math_toolkit.dist.stat_tut.weg worked examples] illustrating +how the library is used to conduct statistical tests. + +[h4 Mathematical Special Functions] + +Provides a small number of high quality +[link math_toolkit.special special functions], +initially these were concentrated on functions used in statistical applications +along with those in the [tr1]. + +The function families currently implemented are the gamma, beta & erf functions +along with the incomplete gamma and beta functions (four variants +of each) and all the possible inverses of these, plus digamma, +various factorial functions, +Bessel functions, elliptic integrals, sinus cardinals (along with their +hyperbolic variants), inverse hyperbolic functions, Legrendre/Laguerre/Hermite +polynomials and various +special power and logarithmic functions. + +All the implementations +are fully generic and support the use of arbitrary "real-number" types, +although they are optimised for use with types with known-about +[@http://en.wikipedia.org/wiki/Significand significand (or mantissa)] +sizes: typically `float`, `double` or `long double`. + +[h4 Implementation Toolkit] + +Provides [link math_toolkit.toolkit many of the tools] required to implement +mathematical special functions: hopefully the presence of +these will encourage other authors to contribute more special +function implementations in the future. These tools are currently +considered experimental: they are "exposed implementation details" +whose interfaces and\/or implementations may change. + +There are helpers for the +[link math_toolkit.toolkit.internals1.series_evaluation +evaluation of infinite series], +[link math_toolkit.toolkit.internals1.cf continued +fractions] and [link math_toolkit.toolkit.internals1.rational +rational approximations]. + +There is a fairly comprehensive set of root finding and +[link math_toolkit.toolkit.internals1.minima function minimisation +algorithms]: the root finding algorithms are both +[link math_toolkit.toolkit.internals1.roots with] and +[link math_toolkit.toolkit.internals1.roots2 without] derivative support. + +A [link math_toolkit.toolkit.internals2.minimax +Remez algorithm implementation] allows for the locating of minimax rational +approximations. + +There are also (experimental) classes for the +[link math_toolkit.toolkit.internals2.polynomials manipulation of polynomials], for +[link math_toolkit.toolkit.internals2.error_test +testing a special function against tabulated test data], and for +the [link math_toolkit.toolkit.internals2.test_data +rapid generation of test data] and/or data for output to an +external graphing application. + +[endsect] [/section:intro Introduction] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/performance.qbk b/doc/sf_and_dist/performance.qbk new file mode 100644 index 000000000..66b1000f0 --- /dev/null +++ b/doc/sf_and_dist/performance.qbk @@ -0,0 +1,369 @@ +[template perf[name value] [value]] +[template para[text] ''''''[text]''''''] + +[section:perf Performance] + +[section:perf_over Performance Overview] +[performance_overview] +[endsect] + +[section:interp Interpreting these Results] + +In all of the following tables, the best performing +result in each row, is assigned a relative value of "1" and shown +in bold, so a score of "2" means ['"twice as slow as the best +performing result".] Actual timings in seconds per function call +are also shown in parenthesis. + +Result were obtained on a system +with an Intel 2.8GHz Pentium 4 processor with 2Gb of RAM and running +either Windows XP or Mandriva Linux. + +[caution As usual +with performance results these should be taken with a large pinch +of salt: relative performance is known to shift quite a bit depending +upon the architecture of the particular test system used. Further +more, our performance results were obtained using our own test data: +these test values are designed to provide good coverage of our code and test +all the appropriate corner cases. They do not necessarily represent +"typical" usage: whatever that may be! +] + +[endsect] + +[section:getting_best Getting the Best Performance from this Library] + +By far the most important thing you can do when using this library +is turn on your compiler's optimisation options. As the following +table shows the penalty for using the library in debug mode can be +quite large. + +[table Performance Comparison of Release and Debug Settings +[[Function] + [Microsoft Visual C++ 8.0 + + Debug Settings: /Od /ZI + ] + [Microsoft Visual C++ 8.0 + + Release settings: /Ox /arch:SSE2 + ]] + +[[__erf][[perf msvc-debug-erf..[para 16.65][para (1.028e-006s)]]][[perf msvc-erf..[para *1.00*][para (6.173e-008s)]]]] +[[__erf_inv][[perf msvc-debug-erf_inv..[para 19.28][para (1.215e-006s)]]][[perf msvc-erf_inv..[para *1.00*][para (6.302e-008s)]]]] +[[__ibeta and __ibetac][[perf msvc-debug-ibeta..[para 8.32][para (1.540e-005s)]]][[perf msvc-ibeta..[para *1.00*][para (1.852e-006s)]]]] +[[__ibeta_inv and __ibetac_inv][[perf msvc-debug-ibeta_inv..[para 10.25][para (7.492e-005s)]]][[perf msvc-ibeta_inv..[para *1.00*][para (7.311e-006s)]]]] +[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf msvc-debug-ibeta_invab..[para 8.57][para (2.441e-004s)]]][[perf msvc-ibeta_invab..[para *1.00*][para (2.847e-005s)]]]] +[[__gamma_p and __gamma_q][[perf msvc-debug-igamma..[para 10.98][para (1.044e-005s)]]][[perf msvc-igamma..[para *1.00*][para (9.504e-007s)]]]] +[[__gamma_p_inv and __gamma_q_inv][[perf msvc-debug-igamma_inv..[para 10.25][para (3.721e-005s)]]][[perf msvc-igamma_inv..[para *1.00*][para (3.631e-006s)]]]] +[[__gamma_p_inva and __gamma_q_inva][[perf msvc-debug-igamma_inva..[para 11.26][para (1.124e-004s)]]][[perf msvc-igamma_inva..[para *1.00*][para (9.982e-006s)]]]] +] + +[endsect] + +[section:comp_compilers Comparing Compilers] + +After a good choice of build settings the next most important thing +you can do, is choose your compiler +- and the standard C library it sits on top of - very carefully. GCC-3.x +in particular has been found to be particularly bad at inlining code, +and performing the kinds of high level transformations that good C++ performance +demands (thankfully GCC-4.x is somewhat better in this respect). + +[table Performance Comparison of Various Windows Compilers +[[Function] + [Intel C++ 10.0 + + ( /Ox /Qipo /QxN ) + ] + [Microsoft Visual C++ 8.0 + + ( /Ox /arch:SSE2 ) + ] + [Cygwin G++ 3.4 + + ( /O3 ) + ]] +[[__erf][[perf intel-erf..[para *1.00*][para (4.118e-008s)]]][[perf msvc-erf..[para 1.50][para (6.173e-008s)]]][[perf gcc-erf..[para 3.24][para (1.336e-007s)]]]] +[[__erf_inv][[perf intel-erf_inv..[para *1.00*][para (4.439e-008s)]]][[perf msvc-erf_inv..[para 1.42][para (6.302e-008s)]]][[perf gcc-erf_inv..[para 7.88][para (3.500e-007s)]]]] +[[__ibeta and __ibetac][[perf intel-ibeta..[para *1.00*][para (1.631e-006s)]]][[perf msvc-ibeta..[para 1.14][para (1.852e-006s)]]][[perf gcc-ibeta..[para 3.05][para (4.975e-006s)]]]] +[[__ibeta_inv and __ibetac_inv][[perf intel-ibeta_inv..[para *1.00*][para (6.133e-006s)]]][[perf msvc-ibeta_inv..[para 1.19][para (7.311e-006s)]]][[perf gcc-ibeta_inv..[para 2.60][para (1.597e-005s)]]]] +[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf intel-ibeta_invab..[para *1.00*][para (2.453e-005s)]]][[perf msvc-ibeta_invab..[para 1.16][para (2.847e-005s)]]][[perf gcc-ibeta_invab..[para 2.83][para (6.947e-005s)]]]] +[[__gamma_p and __gamma_q][[perf intel-igamma..[para *1.00*][para (6.735e-007s)]]][[perf msvc-igamma..[para 1.41][para (9.504e-007s)]]][[perf gcc-igamma..[para 2.78][para (1.872e-006s)]]]] +[[__gamma_p_inv and __gamma_q_inv][[perf intel-igamma_inv..[para *1.00*][para (2.637e-006s)]]][[perf msvc-igamma_inv..[para 1.38][para (3.631e-006s)]]][[perf gcc-igamma_inv..[para 3.31][para (8.736e-006s)]]]] +[[__gamma_p_inva and __gamma_q_inva][[perf intel-igamma_inva..[para *1.00*][para (7.716e-006s)]]][[perf msvc-igamma_inva..[para 1.29][para (9.982e-006s)]]][[perf gcc-igamma_inva..[para 2.56][para (1.974e-005s)]]]] +] + +[endsect] + +[section:tuning Performance Tuning Macros] + +There are a small number of performance tuning options +that are determined by configuration macros. These should be set +in boost/math/tools/user.hpp; or else reported to the Boost-development +mailing list so that the appropriate option for a given compiler and +OS platform can be set automatically in our configuration setup. + +[table +[[Macro][Meaning]] +[[BOOST_MATH_POLY_METHOD] + [Determines how polynomials and most rational functions + are evaluated. Define to one + of the values 0, 1, 2 or 3: see below for the meaning of these values.]] +[[BOOST_MATH_RATIONAL_METHOD] + [Determines how symmetrical rational functions are evaluated: mostly + this only effects how the Lanczos approximation is evaluated, and how + the `evaluate_rational` function behaves. Define to one + of the values 0, 1, 2 or 3: see below for the meaning of these values. + ]] +[[BOOST_MATH_MAX_POLY_ORDER] + [The maximum order of polynomial or rational function that will + be evaluated by a method other than 0 (a simple "for" loop). + ]] +[[BOOST_MATH_INT_TABLE_TYPE(RT, IT)] + [Many of the coefficients to the polynomials and rational functions + used by this library are integers. Normally these are stored as tables + as integers, but if mixed integer / floating point arithmetic is much + slower than regular floating point arithmetic then they can be stored + as tables of floating point values instead. If mixed arithmetic is slow + then add: + + #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT + + to boost/math/tools/user.hpp, otherwise the default of: + + #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) IT + + Set in boost/math/config.hpp is fine, and may well result in smaller + code. + ]] +] + +The values to which `BOOST_MATH_POLY_METHOD` and `BOOST_MATH_RATIONAL_METHOD` +may be set are as follows: + +[table +[[Value][Effect]] +[[0][The polynomial or rational function is evaluated using Horner's + method, and a simple for-loop. + + Note that if the order of the polynomial + or rational function is a runtime parameter, or the order is + greater than the value of `BOOST_MATH_MAX_POLY_ORDER`, then + this method is always used, irrespective of the value + of `BOOST_MATH_POLY_METHOD` or `BOOST_MATH_RATIONAL_METHOD`.]] +[[1][The polynomial or rational function is evaluated without + the use of a loop, and using Horner's method. This only occurs + if the order of the polynomial is known at compile time and is less + than or equal to `BOOST_MATH_MAX_POLY_ORDER`. ]] +[[2][The polynomial or rational function is evaluated without + the use of a loop, and using a second order Horner's method. + In theory this permits two operations to occur in parallel + for polynomials, and four in parallel for rational functions. + This only occurs + if the order of the polynomial is known at compile time and is less + than or equal to `BOOST_MATH_MAX_POLY_ORDER`.]] +[[3][The polynomial or rational function is evaluated without + the use of a loop, and using a second order Horner's method. + In theory this permits two operations to occur in parallel + for polynomials, and four in parallel for rational functions. + This differs from method "2" in that the code is carefully ordered + to make the parallelisation more obvious to the compiler: rather than + relying on the compiler's optimiser to spot the parallelisation + opportunities. + This only occurs + if the order of the polynomial is known at compile time and is less + than or equal to `BOOST_MATH_MAX_POLY_ORDER`.]] +] + +To determine which +of these options is best for your particular compiler/platform build +the performance test application with your usual release settings, +and run the program with the --tune command line option. + +In practice the difference between methods is rather small at present, +as the following table shows. However, parallelisation /vectorisation +is likely to become more important in the future: quite likely the methods +currently supported will need to be supplemented or replaced by ones more +suited to highly vectorisable processors in the future. + +[table A Comparison of Polynomial Evaluation Methods +[[Compiler/platform][Method 0][Method 1][Method 2][Method 3]] +[[Microsoft C++ 8.0, Polynomial evaluation] [[perf msvc-Polynomial-method-0..[para 1.34][para (1.161e-007s)]]][[perf msvc-Polynomial-method-1..[para 1.13][para (9.777e-008s)]]][[perf msvc-Polynomial-method-2..[para 1.07][para (9.289e-008s)]]][[perf msvc-Polynomial-method-3..[para *1.00*][para (8.678e-008s)]]]] +[[Microsoft C++ 8.0, Rational evaluation] [[perf msvc-Rational-method-0..[para *1.00*][para (1.443e-007s)]]][[perf msvc-Rational-method-1..[para 1.03][para (1.492e-007s)]]][[perf msvc-Rational-method-2..[para 1.20][para (1.736e-007s)]]][[perf msvc-Rational-method-3..[para 1.07][para (1.540e-007s)]]]] +[[Intel C++ 10.0 (Windows), Polynomial evaluation] [[perf intel-Polynomial-method-0..[para 1.03][para (7.702e-008s)]]][[perf intel-Polynomial-method-1..[para 1.03][para (7.702e-008s)]]][[perf intel-Polynomial-method-2..[para *1.00*][para (7.446e-008s)]]][[perf intel-Polynomial-method-3..[para 1.03][para (7.690e-008s)]]]] +[[Intel C++ 10.0 (Windows), Rational evaluation] [[perf intel-Rational-method-0..[para *1.00*][para (1.245e-007s)]]][[perf intel-Rational-method-1..[para *1.00*][para (1.245e-007s)]]][[perf intel-Rational-method-2..[para 1.18][para (1.465e-007s)]]][[perf intel-Rational-method-3..[para 1.06][para (1.318e-007s)]]]] +[[GNU G++ 4.2 (Linux), Polynomial evaluation] [[perf gcc-4_2-ld-Polynomial-method-0..[para 1.61][para (1.220e-007s)]]][[perf gcc-4_2-ld-Polynomial-method-1..[para 1.68][para (1.269e-007s)]]][[perf gcc-4_2-ld-Polynomial-method-2..[para 1.23][para (9.275e-008s)]]][[perf gcc-4_2-ld-Polynomial-method-3..[para *1.00*][para (7.566e-008s)]]]] +[[GNU G++ 4.2 (Linux), Rational evaluation] [[perf gcc-4_2-ld-Rational-method-0..[para 1.26][para (1.660e-007s)]]][[perf gcc-4_2-ld-Rational-method-1..[para 1.33][para (1.758e-007s)]]][[perf gcc-4_2-ld-Rational-method-2..[para *1.00*][para (1.318e-007s)]]][[perf gcc-4_2-ld-Rational-method-3..[para 1.15][para (1.513e-007s)]]]] +[[Intel C++ 10.0 (Linux), Polynomial evaluation] [[perf intel-linux-Polynomial-method-0..[para 1.15][para (9.154e-008s)]]][[perf intel-linux-Polynomial-method-1..[para 1.15][para (9.154e-008s)]]][[perf intel-linux-Polynomial-method-2..[para *1.00*][para (7.934e-008s)]]][[perf intel-linux-Polynomial-method-3..[para *1.00*][para (7.934e-008s)]]]] +[[Intel C++ 10.0 (Linux), Rational evaluation] [[perf intel-linux-Rational-method-0..[para *1.00*][para (1.245e-007s)]]][[perf intel-linux-Rational-method-1..[para *1.00*][para (1.245e-007s)]]][[perf intel-linux-Rational-method-2..[para 1.35][para (1.684e-007s)]]][[perf intel-linux-Rational-method-3..[para 1.04][para (1.294e-007s)]]]] +] + +There is one final performance tuning option that is available as a compile time +[link math_toolkit.policy policy]. Normally when evaluating functions at `double` +precision, these are actually evaluated at `long double` precision internally: +this helps to ensure that as close to full `double` precision as possible is +achieved, but may slow down execution in some environments. The defaults for +this policy can be changed by +[link math_toolkit.policy.pol_ref.policy_defaults +defining the macro `BOOST_MATH_PROMOTE_DOUBLE_POLICY`] +to `false`, or +[link math_toolkit.policy.pol_ref.internal_promotion +by specifying a specific policy] when calling the special +functions or distributions. See also the +[link math_toolkit.policy.pol_tutorial policy tutorial]. + +[table Performance Comparison with and Without Internal Promotion to long double +[[Function] + [GCC 4.2 , Linux + + (with internal promotion of double to long double). + ] + [GCC 4.2, Linux + + (without promotion of double). + ] +] +[[__erf][[perf gcc-4_2-ld-erf..[para 1.48][para (1.387e-007s)]]][[perf gcc-4_2-erf..[para *1.00*][para (9.377e-008s)]]]] +[[__erf_inv][[perf gcc-4_2-ld-erf_inv..[para 1.11][para (4.009e-007s)]]][[perf gcc-4_2-erf_inv..[para *1.00*][para (3.598e-007s)]]]] +[[__ibeta and __ibetac][[perf gcc-4_2-ld-ibeta..[para 1.29][para (5.354e-006s)]]][[perf gcc-4_2-ibeta..[para *1.00*][para (4.137e-006s)]]]] +[[__ibeta_inv and __ibetac_inv][[perf gcc-4_2-ld-ibeta_inv..[para 1.44][para (2.220e-005s)]]][[perf gcc-4_2-ibeta_inv..[para *1.00*][para (1.538e-005s)]]]] +[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf gcc-4_2-ld-ibeta_invab..[para 1.25][para (7.009e-005s)]]][[perf gcc-4_2-ibeta_invab..[para *1.00*][para (5.607e-005s)]]]] +[[__gamma_p and __gamma_q][[perf gcc-4_2-ld-igamma..[para 1.26][para (3.116e-006s)]]][[perf gcc-4_2-igamma..[para *1.00*][para (2.464e-006s)]]]] +[[__gamma_p_inv and __gamma_q_inv][[perf gcc-4_2-ld-igamma_inv..[para 1.27][para (1.178e-005s)]]][[perf gcc-4_2-igamma_inv..[para *1.00*][para (9.291e-006s)]]]] +[[__gamma_p_inva and __gamma_q_inva][[perf gcc-4_2-ld-igamma_inva..[para 1.20][para (2.765e-005s)]]][[perf gcc-4_2-igamma_inva..[para *1.00*][para (2.311e-005s)]]]] +] + +[endsect] +[section:comparisons Comparisons to Other Open Source Libraries] + +We've run our performance tests both for our own code, and against other +open source implementations of the same functions. The results are +presented below to give you a rough idea of how they all compare. + +[caution +You should exercise extreme caution when interpreting +these results, relative performance may vary by platform, the tests use +data that gives good code coverage of /our/ code, but which may skew the +results towards the corner cases. Finally, remember that different +libraries make different choices with regard to performance verses +numerical stability. +] + +[heading Comparison to GSL-1.9 and Cephes] + +All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows XP +machine, with all the libraries compiled with Microsoft Visual C++ 2005 using +the `/Ox /arch:SSE2` options. + +[table +[[Function][Boost][GSL-1.9][Cephes]] +[[__tgamma][[perf msvc-gamma..[para 1.50][para (2.566e-007s)]]][[perf msvc-gamma-gsl..[para 1.54][para (2.627e-007s)]]][[perf msvc-gamma-cephes..[para *1.00*][para (1.709e-007s)]]]] +[[__lgamma][[perf msvc-lgamma..[para 1.73][para (2.688e-007s)]]][[perf msvc-lgamma-gsl..[para 3.61][para (5.621e-007s)]]][[perf msvc-lgamma-cephes..[para *1.00*][para (1.556e-007s)]]]] +[[__gamma_p and __gamma_q][[perf msvc-igamma..[para *1.00*][para (9.504e-007s)]]][[perf msvc-igamma-gsl..[para 2.15][para (2.042e-006s)]]][[perf msvc-igamma-cephes..[para 2.57][para (2.439e-006s)]]]] +[[__gamma_p_inv and __gamma_q_inv][[perf msvc-igamma_inv..[para *1.00*][para (3.631e-006s)]]][N\/A][+INF [footnote Cephes gets stuck in an infinite loop while trying to execute our test cases.]]] +[[__ibeta and __ibetac][[perf msvc-ibeta..[para *1.00*][para (1.852e-006s)]]][[perf msvc-ibeta-cephes..[para 1.07][para (1.974e-006s)]]][[perf msvc-ibeta-cephes..[para 1.07][para (1.974e-006s)]]]] +[[__ibeta_inv and __ibetac_inv][[perf msvc-ibeta_inv..[para *1.00*][para (7.311e-006s)]]][N\/A][[perf msvc-ibeta_inv-cephes..[para 2.24][para (1.637e-005s)]]]] +] + +[heading Comparison to the R Statistical Library on Windows] + +All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows XP +machine, with the test program compiled with Microsoft Visual C++ 2005, and +R-2.5.0 compiled in "standalone mode" with MinGW-3.4 +(R-2.5.0 appears not to be buildable with Visual C++). + +[table A Comparison to the R Statistical Library on Windows XP +[[Statistical Function][Boost][R]] +[[__beta_distrib CDF][[perf msvc-dist-beta-cdf..[para 1.20][para (1.916e-006s)]]][[perf msvc-dist-beta-R-cdf..[para *1.00*][para (1.597e-006s)]]]] +[[__beta_distrib Quantile][[perf msvc-dist-beta-quantile..[para *1.00*][para (6.570e-006s)]]][[perf msvc-dist-beta-R-quantile..[para 74.66[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]][para (4.905e-004s)]]]] +[[__binomial_distrib CDF][[perf msvc-dist-binomial-cdf..[para *1.00*][para (5.276e-007s)]]][[perf msvc-dist-binom-R-cdf..[para 2.45][para (1.293e-006s)]]]] +[[__binomial_distrib Quantile][[perf msvc-dist-binomial-quantile..[para *1.00*][para (4.013e-006s)]]][[perf msvc-dist-binom-R-quantile..[para 1.32][para (5.280e-006s)]]]] +[[__cauchy_distrib CDF][[perf msvc-dist-cauchy-cdf..[para *1.00*][para (1.231e-007s)]]][[perf msvc-dist-cauchy-R-cdf..[para 1.28][para (1.576e-007s)]]]] +[[__cauchy_distrib Quantile][[perf msvc-dist-cauchy-quantile..[para *1.00*][para (1.498e-007s)]]][[perf msvc-dist-cauchy-R-quantile..[para *1.00*][para (1.498e-007s)]]]] +[[__chi_squared_distrib CDF][[perf msvc-dist-chi_squared-cdf..[para *1.00*][para (7.889e-007s)]]][[perf msvc-dist-chisq-R-cdf..[para 2.48][para (1.955e-006s)]]]] +[[__chi_squared_distrib Quantile][[perf msvc-dist-chi_squared-quantile..[para *1.00*][para (4.303e-006s)]]][[perf msvc-dist-chisq-R-quantile..[para 1.61][para (6.925e-006s)]]]] +[[__exp_distrib CDF][[perf msvc-dist-exponential-cdf..[para *1.00*][para (1.955e-007s)]]][[perf msvc-dist-exp-R-cdf..[para 1.97][para (3.844e-007s)]]]] +[[__exp_distrib Quantile][[perf msvc-dist-exponential-quantile..[para 1.07][para (1.206e-007s)]]][[perf msvc-dist-exp-R-quantile..[para *1.00*][para (1.126e-007s)]]]] +[[__F_distrib CDF][[perf msvc-dist-fisher_f-cdf..[para *1.00*][para (1.309e-006s)]]][[perf msvc-dist-f-R-cdf..[para 2.12][para (2.780e-006s)]]]] +[[__F_distrib Quantile][[perf msvc-dist-fisher_f-quantile..[para *1.00*][para (7.204e-006s)]]][[perf msvc-dist-f-R-quantile..[para 1.78][para (1.280e-005s)]]]] +[[__gamma_distrib CDF][[perf msvc-dist-gamma-cdf..[para *1.00*][para (1.076e-006s)]]][[perf msvc-dist-gamma-R-cdf..[para 2.07][para (2.227e-006s)]]]] +[[__gamma_distrib Quantile][[perf msvc-dist-gamma-quantile..[para *1.00*][para (5.189e-006s)]]][[perf msvc-dist-gamma-R-quantile..[para 1.14][para (5.937e-006s)]]]] +[[__lognormal_distrib CDF][[perf msvc-dist-lognormal-cdf..[para *1.00*][para (2.078e-007s)]]][[perf msvc-dist-lnorm-R-cdf..[para 1.41][para (2.930e-007s)]]]] +[[__lognormal_distrib Quantile][[perf msvc-dist-lognormal-quantile..[para *1.00*][para (6.692e-007s)]]][[perf msvc-dist-lnorm-R-quantile..[para 1.63][para (1.090e-006s)]]]] +[[__negative_binomial_distrib CDF][[perf msvc-dist-negative_binomial-cdf..[para *1.00*][para (9.005e-007s)]]][[perf msvc-dist-nbinom-R-cdf..[para 2.42][para (2.178e-006s)]]]] +[[__negative_binomial_distrib Quantile][[perf msvc-dist-negative_binomial-quantile..[para *1.00*][para (9.601e-006s)]]][[perf msvc-dist-nbinom-R-quantile..[para 53.59[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in "typical" cases]][para (5.145e-004s)]]]] +[[__normal_distrib CDF][[perf msvc-dist-normal-cdf..[para *1.00*][para (5.926e-008s)]]][[perf msvc-dist-norm-R-cdf..[para 3.01][para (1.785e-007s)]]]] +[[__normal_distrib Quantile][[perf msvc-dist-normal-quantile..[para *1.00*][para (1.248e-007s)]]][[perf msvc-dist-norm-R-quantile..[para 1.05][para (1.311e-007s)]]]] +[[__poisson_distrib CDF][[perf msvc-dist-poisson-cdf..[para *1.00*][para (8.999e-007s)]]][[perf msvc-dist-pois-R-cdf..[para 2.42][para (2.175e-006s)]]]] +[[__poisson_distrib][[perf msvc-dist-poisson-quantile..[para *1.00*][para (1.853e-006s)]]][[perf msvc-dist-pois-R-quantile..[para 2.17][para (4.014e-006s)]]]] +[[__students_t_distrib CDF][[perf msvc-dist-students_t-cdf..[para *1.00*][para (1.223e-006s)]]][[perf msvc-dist-t-R-cdf..[para 1.13][para (1.376e-006s)]]]] +[[__students_t_distrib Quantile][[perf msvc-dist-students_t-quantile..[para *1.00*][para (2.570e-006s)]]][[perf msvc-dist-t-R-quantile..[para 1.04][para (2.668e-006s)]]]] +[[__weibull_distrib CDF][[perf msvc-dist-weibull-cdf..[para *1.00*][para (4.741e-007s)]]][[perf msvc-dist-weibull-R-cdf..[para 1.46][para (6.943e-007s)]]]] +[[__weibull_distrib Quantile][[perf msvc-dist-weibull-quantile..[para *1.00*][para (7.926e-007s)]]][[perf msvc-dist-weibull-R-quantile..[para 1.08][para (8.542e-007s)]]]] +] + +[heading Comparison to the R Statistical Library on Linux] + +All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Mandriva Linux +machine, with the test program and R-2.5.0 compiled with GNU G++ 4.2.0. + +[table A Comparison to the R Statistical Library on Linux +[[Statistical Function][Boost][R]] +[[__beta_distrib CDF][[perf gcc-4_2-dist-beta-cdf..[para 1.71][para (3.508e-006s)]]][[perf gcc-4_2-dist-beta-R-cdf..[para *1.00*][para (2.050e-006s)]]]] +[[__beta_distrib Quantile][[perf gcc-4_2-dist-beta-quantile..[para *1.00*][para (1.294e-005s)]]][[perf gcc-4_2-dist-beta-R-quantile..[para 44.06[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]][para (5.701e-004s)]]]] +[[__binomial_distrib CDF][[perf gcc-4_2-dist-binomial-cdf..[para 1.22][para (1.342e-006s)]]][[perf gcc-4_2-dist-binom-R-cdf..[para *1.00*][para (1.104e-006s)]]]] +[[__binomial_distrib Quantile][[perf gcc-4_2-dist-binomial-quantile..[para 1.36][para (7.083e-006s)]]][[perf gcc-4_2-dist-binom-R-quantile..[para *1.00*][para (5.194e-006s)]]]] +[[__cauchy_distrib CDF][[perf gcc-4_2-dist-cauchy-cdf..[para *1.00*][para (1.372e-007s)]]][[perf gcc-4_2-dist-cauchy-R-cdf..[para 1.47][para (2.017e-007s)]]]] +[[__cauchy_distrib Quantile][[perf gcc-4_2-dist-cauchy-quantile..[para *1.00*][para (1.542e-007s)]]][[perf gcc-4_2-dist-cauchy-R-quantile..[para 1.14][para (1.752e-007s)]]]] +[[__chi_squared_distrib CDF][[perf gcc-4_2-dist-chi_squared-cdf..[para 1.04][para (1.820e-006s)]]][[perf gcc-4_2-dist-chisq-R-cdf..[para *1.00*][para (1.753e-006s)]]]] +[[__chi_squared_distrib Quantile][[perf gcc-4_2-dist-chi_squared-quantile..[para 1.39][para (9.345e-006s)]]][[perf gcc-4_2-dist-chisq-R-quantile..[para *1.00*][para (6.728e-006s)]]]] +[[__exp_distrib CDF][[perf gcc-4_2-dist-exponential-cdf..[para *1.00*][para (2.195e-007s)]]][[perf gcc-4_2-dist-exp-R-cdf..[para 1.17][para (2.561e-007s)]]]] +[[__exp_distrib Quantile][[perf gcc-4_2-dist-exponential-quantile..[para *1.00*][para (1.123e-007s)]]][[perf gcc-4_2-dist-exp-R-quantile..[para 1.03][para (1.155e-007s)]]]] +[[__F_distrib CDF][[perf gcc-4_2-dist-fisher_f-cdf..[para *1.00*][para (2.744e-006s)]]][[perf gcc-4_2-dist-f-R-cdf..[para 1.08][para (2.970e-006s)]]]] +[[__F_distrib Quantile][[perf gcc-4_2-dist-fisher_f-quantile..[para 1.14][para (1.550e-005s)]]][[perf gcc-4_2-dist-f-R-quantile..[para *1.00*][para (1.359e-005s)]]]] +[[__gamma_distrib CDF][[perf gcc-4_2-dist-gamma-cdf..[para 1.29][para (2.578e-006s)]]][[perf gcc-4_2-dist-gamma-R-cdf..[para *1.00*][para (1.992e-006s)]]]] +[[__gamma_distrib Quantile][[perf gcc-4_2-dist-gamma-quantile..[para 1.77][para (1.020e-005s)]]][[perf gcc-4_2-dist-gamma-R-quantile..[para *1.00*][para (5.757e-006s)]]]] +[[__lognormal_distrib CDF][[perf gcc-4_2-dist-lognormal-cdf..[para *1.00*][para (1.782e-007s)]]][[perf gcc-4_2-dist-lnorm-R-cdf..[para 2.00][para (3.564e-007s)]]]] +[[__lognormal_distrib Quantile][[perf gcc-4_2-dist-lognormal-quantile..[para *1.00*][para (7.093e-007s)]]][[perf gcc-4_2-dist-lnorm-R-quantile..[para 1.07][para (7.607e-007s)]]]] +[[__negative_binomial_distrib CDF][[perf gcc-4_2-dist-negative_binomial-cdf..[para 1.03][para (2.209e-006s)]]][[perf gcc-4_2-dist-nbinom-R-cdf..[para *1.00*][para (2.141e-006s)]]]] +[[__negative_binomial_distrib Quantile][[perf gcc-4_2-dist-negative_binomial-quantile..[para *1.00*][para (1.826e-005s)]]][[perf gcc-4_2-dist-nbinom-R-quantile..[para 30.07[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in "typical" cases]][para (5.490e-004s)]]]] +[[__normal_distrib CDF][[perf gcc-4_2-dist-normal-cdf..[para *1.00*][para (8.542e-008s)]]][[perf gcc-4_2-dist-norm-R-cdf..[para 2.09][para (1.782e-007s)]]]] +[[__normal_distrib Quantile][[perf gcc-4_2-dist-normal-quantile..[para *1.00*][para (1.362e-007s)]]][[perf gcc-4_2-dist-norm-R-quantile..[para 1.26][para (1.722e-007s)]]]] +[[__poisson_distrib CDF][[perf gcc-4_2-dist-poisson-cdf..[para 1.10][para (1.953e-006s)]]][[perf gcc-4_2-dist-pois-R-cdf..[para *1.00*][para (1.775e-006s)]]]] +[[__poisson_distrib][[perf gcc-4_2-dist-poisson-quantile..[para 1.12][para (4.214e-006s)]]][[perf gcc-4_2-dist-pois-R-quantile..[para *1.00*][para (3.752e-006s)]]]] +[[__students_t_distrib CDF][[perf gcc-4_2-dist-students_t-cdf..[para 1.55][para (2.441e-006s)]]][[perf gcc-4_2-dist-t-R-cdf..[para *1.00*][para (1.576e-006s)]]]] +[[__students_t_distrib Quantile][[perf gcc-4_2-dist-students_t-quantile..[para 1.33][para (3.972e-006s)]]][[perf gcc-4_2-dist-t-R-quantile..[para *1.00*][para (2.990e-006s)]]]] +[[__weibull_distrib CDF][[perf gcc-4_2-dist-weibull-cdf..[para *1.00*][para (6.640e-007s)]]][[perf gcc-4_2-dist-weibull-R-cdf..[para 1.06][para (7.031e-007s)]]]] +[[__weibull_distrib Quantile][[perf gcc-4_2-dist-weibull-quantile..[para *1.00*][para (7.504e-007s)]]][[perf gcc-4_2-dist-weibull-R-quantile..[para 1.03][para (7.710e-007s)]]]] +] + +[endsect] + +[section:perf_test_app The Performance Test Application] + +Under ['boost-path]\/libs\/math\/performance you will find a +(fairly rudimentary) performance test application for this library. + +To run this application yourself, build the all the .cpp files in +['boost-path]\/libs\/math\/performance into an application using +your usual release-build settings. Run the application with --help +to see a full list of options, or with --all to test everything +(which takes quite a while), or with --tune to test the +[link math_toolkit.perf.tuning available performance tuning options]. + +If you want to use this application to test the effect of changing +any of the __policy_section, then you will need to build and run it twice: +once with the default __policy_section, and then a second time with the +__policy_section you want to test set as the default. + +[endsect] + +[endsect] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/poisson_optimisation.qbk b/doc/sf_and_dist/poisson_optimisation.qbk new file mode 100644 index 000000000..dc02cd5e3 --- /dev/null +++ b/doc/sf_and_dist/poisson_optimisation.qbk @@ -0,0 +1,121 @@ +[sect:optim Optimisation Examples} + +[h4 Poisson Distribution - Optimization and Accuracy is quite complicated. + +The general formula for calculating the CDF uses the incomplete gamma thus: + + return gamma_q(k+1, mean); + +But the case of small integral k is *very* common, so it is worth considering optimisation. + +The first obvious step is to use a finite sum of each pdf (Probability *density* function) +for each value of k to build up the cdf (*cumulative* distribution function). + +This could be done using the pdf function for the distribution, +for which there are two equivalent formulae: + + return exp(-mean + log(mean) * k - lgamma(k+1)); + + return gamma_p_derivative(k+1, mean); + +The pdf would probably be more accurate using gamma_p_derivative. + +The reason is that the expression: + + -mean + log(mean) * k - lgamma(k+1) + +Will produce a value much smaller than the largest of the terms, so you get +cancellation error: and then when you pass the result to exp() which +converts the absolute error in it's argument to a relative error in the +result (explanation available if required), you effectively amplify the +error further still. + +gamma_p_derivative is just a thin wrapper around some of the internals of +the incomplete gamma, it does its utmost to avoid issues like this, because +this function is responsible for virtually all of the error in the result. +Hopefully further advances in the future might improve things even further +(what is really needed is an 'accurate' pow(1+x) function, but that's a whole +other story!). + +But calling pdf function makes repeated, redundant, checks on the value of mean and k, + + result += pdf(dist, i); + +so it may be faster to substitute the formula for the pdf in a summation loop + + result += exp(-mean) * pow(mean, i) / unchecked_factorial(i); + +(simplified by removing casting from RealType). + +Of course, mean is unchanged during this summation, +so exp(mean) should only be calculated once, outside the loop. + +Optimising compilers 'might' do this, but one can easily ensure this. + +Obviously too, k must be small enough that unchecked_factorial is OK: +34 is an obvious choice as the limit for 32-bit float. +For larger k, the number of iterations is like to be uneconomic. +Only experiment can determine the optimum value of k +for any particular RealType (float, double...) + +But also note that + +The incomplete gamma already optimises this case +(argument "a" is a small int), +although only when the result q (1-p) would be < 0.5. + +And moreover, in the above series, each term can be calculated +from the previous one much more efficiently: + +cdf = sum from 0 to k of C[k] + +with: + +C[0] = exp(-mean) + +C[N+1] = C[N] * mean / (N+1) + +So hopefully that's: + + { + RealType result = exp(-mean); + RealType term = result; + for(int i = 1; i <= k; ++i) + { // cdf is sum of pdfs. + term *= mean / i; + result += term; + } + return result; + } + +This is exactly the same finite sum as used by gamma_p/gamma_q internally. + +As explained previously it's only used when the result + +p > 0.5 or 1-p = q < 0.5. + +The slight danger when using the sum directly like this, is that if +the mean is small and k is large then you're calculating a value ~1, so +conceivably you might overshoot slightly. For this and other reasons in the +case when p < 0.5 and q > 0.5 gamma_p/gamma_q use a different (infinite but +rapidly converging) sum, so that danger isn't present since you always +calculate the smaller of p and q. + +So... it's tempting to suggest that you just call gamma_p/gamma_q as +required. However, there is a slight benefit for the k = 0 case because you +avoid all the internal logic inside gamma_p/gamma_q trying to figure out +which method to use etc. + +For the incomplete beta function, there are no simple finite sums +available (that I know of yet anyway), so when there's a choice between a +finite sum of the PDF and an incomplete beta call, the finite sum may indeed +win out in that case. + +[endsect][/sect:optim Optimisation Examples} + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/policy.qbk b/doc/sf_and_dist/policy.qbk new file mode 100644 index 000000000..88b9f5d52 --- /dev/null +++ b/doc/sf_and_dist/policy.qbk @@ -0,0 +1,826 @@ +[section:policy Policies] + +[section:pol_overview Policy Overview] +[policy_overview] +[endsect][/section:pol_overview Policy Overview] + +[include policy_tutorial.qbk] + +[section:pol_ref Policy Reference] + +[section:error_handling_policies Error Handling Policies] + +There are two orthogonal aspects to error handling: + +* What to do (if anything) with the error. +* What kind of error is being raised. + +[h4 Available Actions When an Error is Raised] + +What to do with the error is encapsulated by an enumerated type: + + namespace boost { namespace math { namespace policies { + + enum error_policy_type + { + throw_on_error = 0, // throw an exception. + errno_on_error = 1, // set ::errno & return 0, NaN, infinity or best guess. + ignore_error = 2, // return 0, NaN, infinity or best guess. + user_error = 3 // call a user-defined error handler. + }; + + }}} // namespaces + +The various enumerated values have the following meanings: + +[h5 throw_on_error] + +Will throw one of the following exceptions, depending upon the + type of the error: + [table + [[Error Type][Exception]] + [[Domain Error][std::domain_error]] + [[Pole Error][std::domain_error]] + [[Overflow Error][std::overflow_error]] + [[Underflow Error][std::underflow_error]] + [[Denorm Error][std::underflow_error]] + [[Evaluation Error][boost::math::evaluation_error]] + ] + +[h5 errno_on_error] + +Will set global `::errno` to one of the following values depending +upon the error type, and then return the same value as if the error +had been ignored: + [table + [[Error Type][errno value]] + [[Domain Error][EDOM]] + [[Pole Error][EDOM]] + [[Overflow Error][ERANGE]] + [[Underflow Error][ERANGE]] + [[Denorm Error][ERANGE]] + [[Evaluation Error][EDOM]] + ] + +[h5 ignore_error] + +Will return a one of the values below depending on the error type (`::errno` is NOT changed):: + [table + [[Error Type][Returned Value]] + [[Domain Error][std::numeric_limits::quiet_NaN()]] + [[Pole Error][std::numeric_limits::quiet_NaN()]] + [[Overflow Error][std::numeric_limits::infinity()]] + [[Underflow Error][0]] + [[Denorm Error][The denormalised value.]] + [[Evaluation Error][The best guess as to the result: which + may be significantly in error.]] + ] + +[h5 user_error] + +Will call a user defined error handler: these are forward declared +in boost/math/policies/error_handling.hpp, but the actual definitions +must be provided by the user: + + namespace boost{ namespace math{ namespace policies{ + + template + T user_domain_error(const char* function, const char* message, const T& val); + + template + T user_pole_error(const char* function, const char* message, const T& val); + + template + T user_overflow_error(const char* function, const char* message, const T& val); + + template + T user_underflow_error(const char* function, const char* message, const T& val); + + template + T user_denorm_error(const char* function, const char* message, const T& val); + + template + T user_evaluation_error(const char* function, const char* message, const T& val); + + }}} // namespaces + +Note that the strings ['function] and ['message] may contain "%1%" format specifiers +designed to be used in conjunction with Boost.Format. +If these strings are to be presented to the program's end-user then +the "%1%" format specifier +should be replaced with the name of type T in the ['function] string, and +if there is a %1% specifier in the ['message] string then it +should be replaced with the value of ['val]. + +There is more information on user-defined error handlers in +the [link math_toolkit.policy.pol_tutorial.user_def_err_pol +tutorial here]. + +[h4 Kinds of Error Raised] + +There are five kinds of error reported by this library, +which are summarised in the following table: + +[table +[[Error Type] + [Policy Class] + [Description]] +[[Domain Error] + [boost::math::policies::domain_error<['action]>] + [Raised when more or more arguments are outside the + defined range of the function. + + Defaults to `boost::math::policies::domain_error` + + When the action is set to ['throw_on_error] + then throws `std::domain_error`]] +[[Pole Error] + [boost::math::policies::pole_error<['action]>] + [Raised when more or more arguments would cause the function + to be evaluated at a pole. + + Defaults to `boost::math::policies::pole_error` + + When the action is ['throw_on_error] then + throw a `std::domain_error`]] +[[Overflow Error] + [boost::math::policies::overflow_error<['action]>] + [Raised when the result of the function is outside + the representable range of the floating point type used. + + Defaults to `boost::math::policies::overflow_error`. + + When the action is ['throw_on_error] then throws a `std::overflow_error`.]] +[[Underflow Error] + [boost::math::policies::underflow_error<['action]>] + [Raised when the result of the function is too small + to be represented in the floating point type used. + + Defaults to `boost::math::policies::underflow_error` + + When the specified action is ['throw_on_error] then + throws a `std::underflow_error`]] +[[Denorm Error] + [boost::math::policies::denorm_error<['action]>] + [Raised when the result of the function is a + denormalised value. + + Defaults to `boost::math::policies::denorm_error` + + When the action is ['throw_on_error] then throws a `std::underflow_error`]] +[[Evaluation Error] + [boost::math::policies::evaluation_error<['action]>] + [Raised when the result of the function is well defined and + finite, but we were unable to compute it. Typically + this occurs when an iterative method fails to converge. + Of course ideally this error should never be raised: feel free + to report it as a bug if it is! + + Defaults to `boost::math::policies::evaluation_error` + + When the action is ['throw_on_error] then throws `boost::math::evaluation_error`]] +] + +[h4 Examples] + +Suppose we want a call to `tgamma` to behave in a C-compatible way and set global +`::errno` rather than throw an exception, we can achieve this at the call site +using: + +[import ../../example/policy_ref_snip1.cpp] + +[policy_ref_snip1] + +Suppose we want a statistical distribution to return infinities, +rather than throw exceptions, then we can use: + +[import ../../example/policy_ref_snip2.cpp] + +[policy_ref_snip2] + +[endsect] [/section:error_handling_policies Error Handling Policies] + +[section:internal_promotion Internal Promotion Policies] + +Normally when evaluating a function at say `float` precision, maximal +accuracy is assured by conducting the calculation at `double` precision +internally, and then rounding the result. There are two policies that +effect whether internal promotion takes place or not: + +[table +[[Policy][Meaning]] +[[`boost::math::policies::promote_float`] + [Indicates whether `float` arguments should be promoted to `double` + precision internally: defaults to `boost::math::policies::promote_float`]] +[[`boost::math::policies::promote_double`] + [Indicates whether `double` arguments should be promoted to `long double` + precision internally: defaults to `boost::math::policies::promote_double`]] +] + +[h4 Examples] + +Suppose we want `tgamma` to be evaluated without internal promotion to +`long double`, then we could use: + +[import ../../example/policy_ref_snip3.cpp] +[policy_ref_snip3] + +Alternatively, suppose we want a distribution to perform calculations +without promoting `float` to `double`, then we could use: + +[import ../../example/policy_ref_snip4.cpp] +[policy_ref_snip4] + +[endsect] [/section:internal_promotion Internal Promotion Policies] + +[section:assert_undefined Mathematically Undefined Function Policies] + +There are some functions that are generic +(they are present for all the statistical distributions supported) +but which may be mathematically undefined for certain distributions, but defined for others. + +For example, the Cauchy distribution does not have a mean, so what should + + mean(cauchy<>()); + +return, and should such an expression even compile at all? + +The default behaviour is for all such functions to not compile at all + - in fact they will raise a +[@http://www.boost.org/libs/static_assert/index.html static assertion] + - but by changing the policy +we can have them return the result of a domain error instead +(which may well throw an exception, depending on the error handling policy). + +This behaviour is controlled by the `assert_undefined<>` policy: + + namespace boost{ namespace math{ namespace policies { + + template + class assert_undefined; + + }}} //namespaces + +For example: + + #include + + using namespace boost::math::policies; + using namespace boost::math; + + // This will not compile, cauchy has no mean! + double m1 = mean(cauchy()); + + // This will compile, but raises a domain error! + double m2 = mean(cauchy_distribution > >()); + +[endsect][/section:assert_undefined Mathematically Undefined Function Policies] + +[section:discrete_quant_ref Discrete Quantile Policies] + +If a statistical distribution is ['discrete] then the random variable +can only have integer values - this leaves us with a problem when calculating +quantiles - we can either ignore the discreteness of the distribution and return +a real value, or we can round to an integer. As it happens, computing integer +values can be substantially faster than calculating a real value, so there are +definite advantages to returning an integer, but we do then need to decide +how best to round the result. The `discrete_quantile` policy defines how +discrete quantiles work, and how integer results are rounded: + + enum discrete_quantile_policy_type + { + real, + integer_round_outwards, // default + integer_round_inwards, + integer_round_down, + integer_round_up, + integer_round_nearest + }; + + template + struct discrete_quantile; + +The values that `discrete_quantile` can take have the following meanings: + +[h5 real] + +Ignores the discreteness of the distribution, and returns a real-valued +result. For example: + +[import ../../example/policy_ref_snip5.cpp] +[policy_ref_snip5] + +Results in `x = 27.3898` and `y = 68.1584`. + +[h5 integer_round_outwards] + +This is the default policy: an integer value is returned so that: + +* Lower quantiles (where the probability is less than 0.5) are rounded +down. +* Upper quantiles (where the probability is greater than 0.5) are rounded up. + +This is normally the safest rounding policy, since it ensures that both +one and two sided intervals are guaranteed to have ['at least] +the requested coverage. For example: + +[import ../../example/policy_ref_snip6.cpp] +[policy_ref_snip6] + +Results in `x = 27` (rounded down from 27.3898) and `y = 69` (rounded up from 68.1584). + +The variables x and y are now defined so that: + + cdf(negative_binomial(20), x) <= 0.05 + cdf(negative_binomial(20), y) >= 0.95 + +In other words we guarantee ['at least 90% coverage in the central region overall], +and also ['no more than 5% coverage in each tail]. + +[h5 integer_round_inwards] + +This is the opposite of ['integer_round_outwards]: an integer value is returned so that: + +* Lower quantiles (where the probability is less than 0.5) are rounded +['up]. +* Upper quantiles (where the probability is greater than 0.5) are rounded ['down]. + +For example: + +[import ../../example/policy_ref_snip7.cpp] + +[policy_ref_snip7] + +Results in `x = 28` (rounded up from 27.3898) and `y = 68` (rounded down from 68.1584). + +The variables x and y are now defined so that: + + cdf(negative_binomial(20), x) >= 0.05 + cdf(negative_binomial(20), y) <= 0.95 + +In other words we guarantee ['at no more than 90% coverage in the central region overall], +and also ['at least 5% coverage in each tail]. + +[h5 integer_round_down] + +Always rounds down to an integer value, no matter whether it's an upper +or a lower quantile. + +[h5 integer_round_up] + +Always rounds up to an integer value, no matter whether it's an upper +or a lower quantile. + +[h5 integer_round_nearest] + +Always rounds to the nearest integer value, no matter whether it's an upper +or a lower quantile. This will produce the requested coverage +['in the average case], but for any specific example may results in +either significantly more or less coverage than the requested amount. +For example: + +For example: + +[import ../../example/policy_ref_snip8.cpp] + +[policy_ref_snip8] + +Results in `x = 27` (rounded from 27.3898) and `y = 68` (rounded from 68.1584). + +[endsect][/section:discrete_quant_ref Discrete Quantile Policies] + +[section:precision_pol Precision Policies] + +There are two equivalent policies that effect the ['working precision] +used to calculate results, these policies both default to 0 - meaning +calculate to the maximum precision available in the type being used + - but can be set to other values to cause lower levels of precision +to be used. + + namespace boost{ namespace math{ namespace policies{ + + template + digits10; + + template + digits2; + + }}} // namespaces + +As you would expect, ['digits10] specifies the number of decimal digits +to use, and ['digits2] the number of binary digits. Internally, whichever +is used, the precision is always converted to ['binary digits]. + +These policies are specified at compile-time, because many of the special +functions use compile-time-dispatch to select which approximation to use +based on the precision requested and the numeric type being used. + +For example we could calculate `tgamma` to approximately 5 decimal digits using: + +[import ../../example/policy_ref_snip9.cpp] + +[policy_ref_snip9] + +Or again using ['make_policy]: + +[import ../../example/policy_ref_snip10.cpp] + +[policy_ref_snip10] + +And for a quantile of a distribution to approximately 25-bit precision: + +[import ../../example/policy_ref_snip11.cpp] + +[policy_ref_snip11] + +[endsect][/section:precision_pol Precision Policies] + +[section:iteration_pol Iteration Limits Policies] + +There are two policies that effect the iterative algorithms +used to implement the special functions in this library: + + template + class max_series_iterations; + + template + class max_root_iterations; + +The class `max_series_iterations` determines the maximum number of +iterations permitted in a series evaluation, before the special +function gives up and returns the result of __evaluation_error. + +The class `max_root_iterations` determines the maximum number +of iterations permitted in a root-finding algorithm before the special +function gives up and returns the result of __evaluation_error. + +[endsect][/section:iteration_pol Iteration Limits Policies] + +[section:policy_defaults Using macros to Change the Policy Defaults] + +You can use the various macros below to change any (or all) of the policies. + +You can make a local change by placing a macro definition *before* +a function or distribution #include. + +[caution There is a danger of One-Definition-Rule violations if you +add ad-hock macros to more than one source files: these must be set the same in *every +translation unit*.] + +[caution If you place it after the #include it will have no effect, +(and it will affect only any other following #includes). +This is probably not what you intend!] + +If you want to alter the defaults for any or all of +the policies for *all* functions and distributions, installation-wide, +then you can do so by defining various macros in +[@../../../../../boost/math/tools/user.hpp boost/math/tools/user.hpp]. + +[h5 BOOST_MATH_DOMAIN_ERROR_POLICY] + +Defines what happens when a domain error occurs, if not defined then +defaults to `throw_on_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_POLE_ERROR_POLICY] + +Defines what happens when a pole error occurs, if not defined then +defaults to `throw_on_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_OVERFLOW_ERROR_POLICY] + +Defines what happens when an overflow error occurs, if not defined then +defaults to `throw_on_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_EVALUATION_ERROR_POLICY] + +Defines what happens when an internal evaluation error occurs, if not defined then +defaults to `throw_on_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_UNDERFLOW_ERROR_POLICY] + +Defines what happens when an overflow error occurs, if not defined then +defaults to `ignore_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_DENORM_ERROR_POLICY] + +Defines what happens when a denormalisation error occurs, if not defined then +defaults to `ignore_error`, but can be set to any of the enumerated +actions for error handing: `throw_on_error`, `errno_on_error`, +`ignore_error` or `user_error`. + +[h5 BOOST_MATH_DIGITS10_POLICY] + +Defines how many decimal digits to use in internal computations: +defaults to `0` - meaning use all available digits - but can be set +to some other decimal value. Since setting this is likely to have +a substantial impact on accuracy, it's not generally recommended +that you change this from the default. + +[h5 BOOST_MATH_PROMOTE_FLOAT_POLICY] + +Determines whether `float` types get promoted to `double` +internally to ensure maximum precision in the result, defaults +to `true`, but can be set to `false` to turn promotion of +`float`'s off. + +[h5 BOOST_MATH_PROMOTE_DOUBLE_POLICY] + +Determines whether `double` types get promoted to `long double` +internally to ensure maximum precision in the result, defaults +to `true`, but can be set to `false` to turn promotion of +`double`'s off. + +[h5 BOOST_MATH_DISCRETE_QUANTILE_POLICY] + +Determines how discrete quantiles return their results: either +as an integer, or as a real value, can be set to one of the +enumerated values: `real`, `integer_round_outwards`, `integer_round_inwards`, +`integer_round_down`, `integer_round_up`, `integer_round_nearest`. Defaults to +`integer_round_outwards`. + +[h5 BOOST_MATH_ASSERT_UNDEFINED_POLICY] + +Determines whether functions that are mathematically undefined +for a specific distribution compile or raise a static (i.e. compile-time) +assertion. Defaults to `true`: meaning that any mathematically +undefined function will not compile. When set to `false` then the function +will compile but return the result of a domain error: this can be useful +for some generic code, that needs to work with all distributions and determine +at runtime whether or not a particular property is well defined. + +[h5 BOOST_MATH_MAX_SERIES_ITERATION_POLICY] + +Determines how many series iterations a special function is permitted +to perform before it gives up and returns an __evaluation_error: +Defaults to 1000000. + +[h5 BOOST_MATH_MAX_ROOT_ITERATION_POLICY] + +Determines how many root-finding iterations a special function is permitted +to perform before it gives up and returns an __evaluation_error: +Defaults to 200. + +[h5 Example] + +Suppose we want overflow errors to set `::errno` and return an infinity, +discrete quantiles to return a real-valued result (rather than round to +integer), and for mathematically undefined functions to compile, but return +a domain error. Then we could add the following to boost/math/tools/user.hpp: + + #define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error + #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real + #define BOOST_MATH_ASSERT_UNDEFINED_POLICY false + +or we could place these definitions *before* + + #include + using boost::math::normal_distribution; + +in a source .cpp file. + +[endsect][/section:policy_defaults Changing the Policy Defaults] + +[section:namespace_pol Setting Polices at Namespace Scope] + +Sometimes what you really want to do is bring all the special functions, +or all the distributions into a specific namespace-scope, along with +a specific policy to use with them. There are two macros defined to +assist with that: + + BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(Policy) + +and: + + BOOST_MATH_DECLARE_DISTRIBUTIONS(Type, Policy) + +You can use either of these macros after including any special function +or distribution header. For example: + +[import ../../example/policy_ref_snip12.cpp] + +[policy_ref_snip12] + +In this example, using BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS results in +a set of thin inline forwarding functions being defined: + + template + inline T tgamma(T a){ return ::boost::math::tgamma(a, mypolicy()); } + + template + inline T lgamma(T a) ( return ::boost::math::lgamma(a, mypolicy()); } + +and so on. Note that while a forwarding function is defined for all the special +functions, however, unless you include the specific header for the special +function you use (or boost/math/special_functions.hpp to include everything), +you will get linker errors from functions that are forward declared, but not +defined. + +We can do the same thing with the distributions, but this time we need to +specify the floating-point type to use: + +[import ../../example/policy_ref_snip13.cpp] + +[policy_ref_snip13] + +In this example the result of BOOST_MATH_DECLARE_DISTRIBUTIONS is to +declare a typedef for each distribution like this: + + typedef boost::math::cauchy_distribution cauchy; + tyepdef boost::math::gamma_distribution gamma; + +and so on. The name given to each typedef is the name of the distribution +with the "_distribution" suffix removed. + +[endsect][/section Changing the Policy Defaults] + +[section:pol_ref_ref Policy Class Reference] + +There's very little to say here, the `policy` class is just a rag-bag +compile-time container for a collection of policies: + +```#include ``` + + + namespace boost{ + namespace math{ + namespace policies + + template + struct policy + { + public: + typedef ``['computed-from-template-arguments]`` domain_error_type; + typedef ``['computed-from-template-arguments]`` pole_error_type; + typedef ``['computed-from-template-arguments]`` overflow_error_type; + typedef ``['computed-from-template-arguments]`` underflow_error_type; + typedef ``['computed-from-template-arguments]`` denorm_error_type; + typedef ``['computed-from-template-arguments]`` evaluation_error_type; + typedef ``['computed-from-template-arguments]`` precision_type; + typedef ``['computed-from-template-arguments]`` promote_float_type; + typedef ``['computed-from-template-arguments]`` promote_double_type; + typedef ``['computed-from-template-arguments]`` discrete_quantile_type; + typedef ``['computed-from-template-arguments]`` assert_undefined_type; + }; + + template <...argument list...> + typename normalise, A1>::type make_policy(...argument list..); + + template + struct normalise + { + typedef ``computed-from-template-arguments`` type; + }; + +The member typedefs of class `policy` are intended for internal use +but are documented briefly here for the sake of completeness. + + policy<...>::domain_error_type + +Specifies how domain errors are handled, will be an instance of +`boost::math::policies::domain_error<>` with the template argument to +`domain_error` one of the `error_policy_type` enumerated values. + + policy<...>::pole_error_type + +Specifies how pole-errors are handled, will be an instance of +`boost::math::policies::pole_error<>` with the template argument to +`pole_error` one of the `error_policy_type` enumerated values. + + policy<...>::overflow_error_type + +Specifies how overflow errors are handled, will be an instance of +`boost::math::policies::overflow_error<>` with the template argument to +`overflow_error` one of the `error_policy_type` enumerated values. + + policy<...>::underflow_error_type + +Specifies how underflow errors are handled, will be an instance of +`boost::math::policies::underflow_error<>` with the template argument to +`underflow_error` one of the `error_policy_type` enumerated values. + + policy<...>::denorm_error_type + +Specifies how denorm errors are handled, will be an instance of +`boost::math::policies::denorm_error<>` with the template argument to +`denorm_error` one of the `error_policy_type` enumerated values. + + policy<...>::evaluation_error_type + +Specifies how evaluation errors are handled, will be an instance of +`boost::math::policies::evaluation_error<>` with the template argument to +`evaluation_error` one of the `error_policy_type` enumerated values. + + policy<...>::precision_type + +Specifies the internal precision to use in binary digits (uses zero +to represent whatever the default precision is). Will be an instance +of `boost::math::policies::digits2` which in turn inherits from +`boost::mpl::int_`. + + policy<...>::promote_float_type + +Specifies whether or not to promote `float` arguments to `double` precision +internally. Will be an instance of `boost::math::policies::promote_float` +which in turn inherits from `boost::mpl::bool_`. + + policy<...>::promote_double_type + +Specifies whether or not to promote `double` arguments to `long double` precision +internally. Will be an instance of `boost::math::policies::promote_float` +which in turn inherits from `boost::mpl::bool_`. + + policy<...>::discrete_quantile_type + +Specifies how discrete quantiles are evaluated, will be an instance +of `boost::math::policies::discrete_quantile<>` instantiated with one of +the `discrete_quantile_policy_type` enumerated type. + + policy<...>::assert_undefined_type + +Specifies whether mathematically-undefined properties are +asserted as compile-time errors, or treated as runtime errors +instead. Will be an instance of `boost::math::policies::assert_undefined` +which in turn inherits from `boost::math::mpl::bool_`. + + + template <...argument list...> + typename normalise, A1>::type make_policy(...argument list..); + +`make_policy` is a helper function that converts a list of policies into +a normalised `policy` class. + + template + struct normalise + { + typedef ``computed-from-template-arguments`` type; + }; + +The `normalise` class template converts one instantiation of the +`policy` class into a normalised form. This is used internally +to reduce code bloat: so that instantiating a special function +on `policy` or `policy` actually both generate the same +code internally. + +Further more, `normalise` can be used to combine +a policy with one or more policies: for example many of the +special functions will use this to set policies which they don't +make use of to their default values, before forwarding to the actual +implementation. In this way code bloat is reduced, since the +actual implementation depends only on the policy types that they +actually use. + +[endsect][/section:pol_ref_ref Policy Class Reference] + +[endsect][/section:pol_ref Policy Reference] +[endsect][/section:policy Policies] + +[/ math.qbk + Copyright 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/policy_tutorial.qbk b/doc/sf_and_dist/policy_tutorial.qbk new file mode 100644 index 000000000..e0b4668a5 --- /dev/null +++ b/doc/sf_and_dist/policy_tutorial.qbk @@ -0,0 +1,504 @@ + +[section:pol_tutorial Policy Tutorial] + +[section:what_is_a_policy So Just What is a Policy Anyway?] + +A policy is a compile-time mechanism for customising the behaviour of a +special function, or a statistical distribution. With Policies you can +control: + +* What action to take when an error occurs. +* What happens when you call a function that is mathematically undefined +(for example if you ask for the mean of a Cauchy distribution). +* What happens when you ask for a quantile of a discrete distribution. +* Whether the library is allowed to internally promote `float` to `double` +and `double` to `long double` in order to improve precision. +* What precision to use when calculating the result. + +Some of these policies could arguably be runtime variables, but then we couldn't +use compile-time dispatch internally to select the best evaluation method +for the given policies. + +For this reason a Policy is a /type/: in fact it's an instance of the +class template `boost::math::policies::policy<>`. This class is just a +compile-time-container of user-selected policies (sometimes called a type-list): + + using namespace boost::math::policies; + // + // Define a policy that sets ::errno on overflow, and does + // not promote double to long double internally: + // + typedef policy, promote_double > mypolicy; + +[endsect][/section:what_is_a_policy So Just What is a Policy Anyway?] + +[section:policy_tut_defaults Policies Have Sensible Defaults] + +Most of the time you can just ignore the policy framework, the defaults for +the various policies are as follows, if these work OK for you then you can +stop reading now! + +[variablelist +[[Domain Error][Throws a `std::domain_error` exception.]] +[[Pole Error][Occurs when a function is evaluated at a pole: throws a `std::domain_error` exception.]] +[[Overflow Error][Throws a `std::overflow_error` exception.]] +[[Underflow][Ignores the underflow, and returns zero.]] +[[Denormalised Result][Ignores the fact that the result is denormalised, and returns it.]] +[[Internal Evaluation Error][Throws a `boost::math::evaluation_error` exception.]] +[[Promotion of float to double][Does occur by default - gives full float precision results.]] +[[Promotion of double to long double][Does occur by default if long double offers + more precision than double.]] +[[Precision of Approximation Used][By default uses an approximation that + will result in the lowest level of error for the type of the result.]] +[[Behaviour of Discrete Quantiles] + [ + The quantile function will by default return an integer result that has been + /rounded outwards/. That is to say lower quantiles (where the probability is + less than 0.5) are rounded downward, and upper quantiles (where the probability + is greater than 0.5) are rounded upwards. This behaviour + ensures that if an X% quantile is requested, then /at least/ the requested + coverage will be present in the central region, and /no more than/ + the requested coverage will be present in the tails. + +This behaviour can be changed so that the quantile functions are rounded + differently, or even return a real-valued result using + [link math_toolkit.policy.pol_overview Policies]. It is strongly + recommended that you read the tutorial + [link math_toolkit.policy.pol_tutorial.understand_dis_quant + Understanding Quantiles of Discrete Distributions] before + using the quantile function on a discrete distribution. The + [link math_toolkit.policy.pol_ref.discrete_quant_ref reference docs] + describe how to change the rounding policy + for these distributions. +]] +] + +What's more, if you define your own policy type, then it automatically +inherits the defaults for any policies not explicitly set, so given: + + using namespace boost::math::policies; + // + // Define a policy that sets ::errno on overflow, and does + // not promote double to long double internally: + // + typedef policy, promote_double > mypolicy; + +then `mypolicy` defines a policy where only the overflow error handling and +`double`-promotion policies differ from the defaults. + +[endsect][/section:policy_tut_defaults Policies Have Sensible Defaults] + +[section:policy_usage So How are Policies Used Anyway?] + +The details follow later, but basically policies can be set by either: + +* Defining some macros that change the default behaviour: [*this is the + recommended method for setting installation-wide policies]. +* By instantiating a distribution object with an explicit policy: + this is mainly reserved for ad hoc policy changes. +* By passing a policy to a special function as an optional final argument: + this is mainly reserved for ad hoc policy changes. +* By using some helper macros to define a set of functions or distributions +in the current namespace that use a specific policy: [*this is the +recommended method for setting policies on a project- or translation-unit-wide +basis]. + +The following sections introduce these methods in more detail. + +[endsect][/section:policy_usage So How are Policies Used Anyway?] + +[section:changing_policy_defaults Changing the Policy Defaults] + +The default policies used by the library are changed by the usual +configuration macro method. + +For example, passing `-DBOOST_MATH_DOMAIN_ERROR_POLICY=errno_on_error` to +your compiler will cause domain errors to set `::errno` and return a NaN +rather than the usual default behaviour of throwing a `std::domain_error` +exception. + +[tip For Microsoft Visual Studio,you can add to the Project Property Page, +C/C++, Preprocessor, Preprocessor definitions like: + +``BOOST_MATH_ASSERT_UNDEFINED_POLICY=0 +BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error`` + +This may be helpful to avoid complications with pre-compiled headers +that may mean that the equivalent definitions in source code: + +``#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false +#define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error`` + +may be ignored. + +The compiler command line shows: + +``/D "BOOST_MATH_ASSERT_UNDEFINED_POLICY=0" +/D "BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error"`` +] [/MSVC tip] + +There is however a very important caveat to this: + +[important +[*['Default policies changed by setting configuration macros must be changed +uniformly in every translation unit in the program.]] + +Failure to follow this rule may result in violations of the "One +Definition Rule (ODR)" and result in unpredictable program behaviour.] + +That means there are only two safe ways to use these macros: + +* Edit them in [@../../../../../boost/math/tools/user.hpp boost/math/tools/user.hpp], +so that the defaults are set on an installation-wide basis. +Unfortunately this may not be convenient if +you are using a pre-installed Boost distribution (on Linux for example). +* Set the defines in your project's Makefile or build environment, so that they +are set uniformly across all translation units. + +What you should *not* do is: + +* Set the defines in the source file using `#define` as doing so +almost certainly will break your program, unless you're absolutely +certain that the program is restricted to a single translation unit. + +And, yes, you will find examples in our test programs where we break this +rule: but only because we know there will always be a single +translation unit only: ['don't say that you weren't warned!] + +[error_handling_example] + +[endsect][/section:changing_policy_defaults Changing the Policy Defaults] + +[section:ad_hoc_dist_policies Setting Policies for Distributions on an Ad Hoc Basis] + +All of the statistical distributions in this library are class templates +that accept two template parameters, both with sensible defaults, for +example: + + namespace boost{ namespace math{ + + template > + class fisher_f_distribution; + + typedef fisher_f_distribution<> fisher_f; + + }} + +This policy gets used by all the accessor functions that accept +a distribution as an argument, and forwarded to all the functions called +by these. So if you use the shorthand-typedef for the distribution, then you get +`double` precision arithmetic and all the default policies. + +However, say for example we wanted to evaluate the quantile +of the binomial distribution at float precision, without internal +promotion to double, and with the result rounded to the /nearest/ +integer, then here's how it can be done: + +[import ../../example/policy_eg_3.cpp] + +[policy_eg_3] + +Which outputs: + +[pre quantile is: 40] + +[endsect][/section:ad_hoc_dist_policies Setting Policies for Distributions on an Ad Hoc Basis] + +[section:ad_hoc_sf_policies Changing the Policy on an Ad Hoc Basis for the Special Functions] + +All of the special functions in this library come in two overloaded forms, +one with a final "policy" parameter, and one without. For example: + + namespace boost{ namespace math{ + + template + RealType tgamma(RealType, const Policy&); + + template + RealType tgamma(RealType); + + }} // namespaces + +Normally, the second version is just a forwarding wrapper to the first +like this: + + template + inline RealType tgamma(RealType x) + { + return tgamma(x, policies::policy<>()); + } + +So calling a special function with a specific policy +is just a matter of defining the policy type to use +and passing it as the final parameter. For example, +suppose we want `tgamma` to behave in a C-compatible +fashion and set `::errno` when an error occurs, and never +throw an exception: + +[import ../../example/policy_eg_1.cpp] + +[policy_eg_1] + +which outputs: + +[pre +Result of tgamma(30000) is: 1.#INF +errno = 34 +Result of tgamma(-10) is: 1.#QNAN +errno = 33 +] + +Alternatively, for ad hoc use, we can use the `make_policy` +helper function to create a policy for us: this usage is more +verbose, so is probably only preferred when a policy is going +to be used once only: + +[import ../../example/policy_eg_2.cpp] + +[policy_eg_2] + +[endsect][/section:ad_hoc_sf_policies Changing the Policy on an Ad Hoc Basis for the Special Functions] + +[section:namespace_policies Setting Policies at Namespace or Translation Unit Scope] + +Sometimes what you want to do is just change a set of policies within +the current scope: the one thing you should not do in this situation +is use the configuration macros, as this can lead to "One Definition +Rule" violations. Instead this library provides a pair of macros +especially for this purpose. + +Let's consider the special functions first: we can declare a set of +forwarding functions that all use a specific policy using the +macro BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(['Policy]). This +macro should be used either inside a unique namespace set aside for the +purpose, or an unnamed namespace if you just want the functions +visible in global scope for the current file only. + +[import ../../example/policy_eg_4.cpp] + +[policy_eg_4] + +The same mechanism works well at file scope as well, by using an unnamed +namespace, we can ensure that these declarations don't conflict with any +alternate policies present in other translation units: + +[import ../../example/policy_eg_5.cpp] + +[policy_eg_5] + +Handling the statistical distributions is very similar except that now +the macro BOOST_MATH_DECLARE_DISTRIBUTIONS accepts two parameters: the +floating point type to use, and the policy type to apply. For example: + + BOOST_MATH_DECLARE_DISTRIBUTIONS(double, mypolicy) + +Results a set of typedefs being defined like this: + + typedef boost::math::normal_distribution normal; + +The name of each typedef is the same as the name of the distribution +class template, but without the "_distribution" suffix. + +[import ../../example/policy_eg_6.cpp] + +[policy_eg_6] + +[note +There is an important limitation to note: you can not use the macros +BOOST_MATH_DECLARE_DISTRIBUTIONS and BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS +['in the same namespace], as doing so creates ambiguities between functions +and distributions of the same name. +] + +As before, the same mechanism works well at file scope as well: by using an unnamed +namespace, we can ensure that these declarations don't conflict with any +alternate policies present in other translation units: + +[import ../../example/policy_eg_7.cpp] + +[policy_eg_7] + +[endsect][/section:namespace_policies Setting Policies at Namespace or Translation Unit Scope] + +[section:user_def_err_pol Calling User Defined Error Handlers] + +[import ../../example/policy_eg_8.cpp] + +[policy_eg_8] + +[import ../../example/policy_eg_9.cpp] + +[policy_eg_9] + +[endsect][/section:user_def_err_pol Calling User Defined Error Handlers] + +[section:understand_dis_quant Understanding Quantiles of Discrete Distributions] + +Discrete distributions present us with a problem when calculating the +quantile: we are starting from a continuous real-valued variable - the +probability - but the result (the value of the random variable) +should really be discrete. + +Consider for example a Binomial distribution, with a sample size of +50, and a success fraction of 0.5. There are a variety of ways +we can plot a discrete distribution, but if we plot the PDF +as a step-function then it looks something like this: + +[$../graphs/binomial_pdf.png] + +Now lets suppose that the user asks for a the quantile that corresponds +to a probability of 0.05, if we zoom in on the CDF for that region here's +what we see: + +[$../graphs/binomial_quantile_1.png] + +As can be seen there is no random variable that corresponds to +a probability of exactly 0.05, so we're left with two choices as +shown in the figure: + +* We could round the result down to 18. +* We could round the result up to 19. + +In fact there's actually a third choice as well: we could "pretend" that the +distribution was continuous and return a real valued result: in this case we +would calculate a result of approximately 18.701 (this accurately +reflects the fact that the result is nearer to 19 than 18). + +By using policies we can offer any of the above as options, but that +still leaves the question: ['What is actually the right thing to do?] + +And in particular: ['What policy should we use by default?] + +In coming to an answer we should realise that: + +* Calculating an integer result is often much faster than +calculating a real-valued result: in fact in our tests it +was up to 20 times faster. +* Normally people calculate quantiles so that they can perform +a test of some kind: ['"If the random variable is less than N +then we can reject our null-hypothesis with 90% confidence."] + +So there is a genuine benefit to calculating an integer result +as well as it being "the right thing to do" from a philosophical +point of view. What's more if someone asks for a quantile at 0.05, +then we can normally assume that they are asking for +['[*at least] 95% of the probability to the right of the value chosen, +and [*no more than] 5% of the probability to the left of the value chosen.] + +In the above binomial example we would therefore round the result down to 18. + +The converse applies to upper-quantiles: If the probability is greater than +0.5 we would want to round the quantile up, ['so that [*at least] the requested +probability is to the left of the value returned, and [*no more than] 1 - the +requested probability is to the right of the value returned.] + +Likewise for two-sided intervals, we would round lower quantiles down, +and upper quantiles up. This ensures that we have ['at least the requested +probability in the central region] and ['no more than 1 minus the requested +probability in the tail areas.] + +For example, taking our 50 sample binomial distribution with a success fraction +of 0.5, if we wanted a two sided 90% confidence interval, then we would ask +for the 0.05 and 0.95 quantiles with the results ['rounded outwards] so that +['at least 90% of the probability] is in the central area: + +[$../graphs/binomial_pdf_3.png] + +So far so good, but there is in fact a trap waiting for the unwary here: + + quantile(binomial(50, 0.5), 0.05); + +returns 18 as the result, which is what we would expect from the graph above, +and indeed there is no x greater than 18 for which: + + cdf(binomial(50, 0.5), x) <= 0.05; + +However: + + quantile(binomial(50, 0.5), 0.95); + +returns 31, and indeed while there is no x less than 31 for which: + + cdf(binomial(50, 0.5), x) >= 0.95; + +We might naively expect that for this symmetrical distribution the result +would be 32 (since 32 = 50 - 18), but we need to remember that the cdf of +the binomial is /inclusive/ of the random variable. So while the left tail +area /includes/ the quantile returned, the right tail area always excludes +an upper quantile value: since that "belongs" to the central area. + +Look at the graph above to see what's going on here: the lower quantile +of 18 belongs to the left tail, so any value <= 18 is in the left tail. +The upper quantile of 31 on the other hand belongs to the central area, +so the tail area actually starts at 32, so any value > 31 is in the +right tail. + +Therefore if U and L are the upper and lower quantiles respectively, then +a random variable X is in the tail area - where we would reject the null +hypothesis if: + + X <= L || X > U + +And the a variable X is inside the central region if: + + L < X <= U + +The moral here is to ['always be very careful with your comparisons +when dealing with a discrete distribution], and if in doubt, +['base your comparisons on CDF's instead]. + +[heading Other Rounding Policies are Available] + +As you would expect from a section on policies, you won't be surprised +to know that other rounding options are available: + +[variablelist + +[[integer_round_outwards] + [This is the default policy as described above: lower quantiles + are rounded down (probability < 0.5), and upper quantiles + (probability > 0.5) are rounded up. + + This gives /no more than/ the requested probability + in the tails, and /at least/ the requested probability + in the central area.]] +[[integer_round_inwards] + [This is the exact opposite of the default policy: + lower quantiles + are rounded up (probability < 0.5), + and upper quantiles (probability > 0.5) are rounded down. + + This gives /at least/ the requested probability + in the tails, and /no more than/ the requested probability + in the central area.]] +[[integer_round_down][This policy will always round the result down + no matter whether it is an upper or lower quantile]] +[[integer_round_up][This policy will always round the result up + no matter whether it is an upper or lower quantile]] +[[integer_round_nearest][This policy will always round the result + to the nearest integer + no matter whether it is an upper or lower quantile]] +[[real][This policy will return a real valued result + for the quantile of a discrete distribution: this is + generally much slower than finding an integer result + but does allow for more sophisticated rounding policies.]] + +] + +[import ../../example/policy_eg_10.cpp] + +[policy_eg_10] + +[endsect] + +[endsect][/section:pol_Tutorial Policy Tutorial] + + +[/ math.qbk + Copyright 2007 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/polynomial.qbk b/doc/sf_and_dist/polynomial.qbk new file mode 100644 index 000000000..fc451f370 --- /dev/null +++ b/doc/sf_and_dist/polynomial.qbk @@ -0,0 +1,94 @@ +[section:polynomials Polynomials] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + class polynomial + { + public: + // typedefs: + typedef typename std::vector::value_type value_type; + typedef typename std::vector::size_type size_type; + + // construct: + polynomial(){} + template + polynomial(const U* data, unsigned order); + template + polynomial(const U& point); + + // access: + size_type size()const; + size_type degree()const; + value_type& operator[](size_type i); + const value_type& operator[](size_type i)const; + + // operators: + template + polynomial& operator +=(const U& value); + template + polynomial& operator -=(const U& value); + template + polynomial& operator *=(const U& value); + template + polynomial& operator +=(const polynomial& value); + template + polynomial& operator -=(const polynomial& value); + template + polynomial& operator *=(const polynomial& value); + }; + + template + polynomial operator + (const polynomial& a, const polynomial& b); + template + polynomial operator - (const polynomial& a, const polynomial& b); + template + polynomial operator * (const polynomial& a, const polynomial& b); + + template + polynomial operator + (const polynomial& a, const U& b); + template + polynomial operator - (const polynomial& a, const U& b); + template + polynomial operator * (const polynomial& a, const U& b); + + template + polynomial operator + (const U& a, const polynomial& b); + template + polynomial operator - (const U& a, const polynomial& b); + template + polynomial operator * (const U& a, const polynomial& b); + + template + std::basic_ostream& operator << + (std::basic_ostream& os, const polynomial& poly); + + }}} // namespaces + +[h4 Description] + +This is a fairly trivial class for polynomial manipulation. + +Implementation is currently of the "naive" variety, with O(N^2) +multiplication for example. This class should not be used in +high-performance computing environments: it is intended for the +simple manipulation of small polynomials, typically generated +for special function approximation. + +Advanced manipulations: the FFT, division, GCD, factorisation etc are +not currently provided. Submissions for these are of course welcome :-) + +[endsect][/section:polynomials Polynomials] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/powers.qbk b/doc/sf_and_dist/powers.qbk new file mode 100644 index 000000000..ea6cf7854 --- /dev/null +++ b/doc/sf_and_dist/powers.qbk @@ -0,0 +1,260 @@ +[section:powers Logs, Powers, Roots and Exponentials] + +[section:log1p log1p] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` log1p(T x); + + template + ``__sf_result`` log1p(T x, const ``__Policy``&); + + }} // namespaces + +Returns the natural logarithm of `x+1`. + +The return type of this function is computed using the __arg_pomotion_rules: +the return is `double` when /x/ is an integer type and T otherwise. + +[optional_policy] + +There are many situations where it is desirable to compute `log(x+1)`. +However, for small `x` then `x+1` suffers from catastrophic cancellation errors +so that `x+1 == 1` and `log(x+1) == 0`, when in fact for very small x, the +best approximation to `log(x+1)` would be `x`. `log1p` calculates the best +approximation to `log(1+x)` using a Taylor series expansion for accuracy +(less than __te). +Alternatively note that there are faster methods available, +for example using the equivalence: + + log(1+x) == (log(1+x) * x) / ((1-x) - 1) + +However, experience has shown that these methods tend to fail quite spectacularly +once the compiler's optimizations are turned on, consequently they are +used only when known not to break with a particular compiler. +In contrast, the series expansion method seems to be reasonably +immune to optimizer-induced errors. + +Finally when BOOST_HAS_LOG1P is defined then the `float/double/long double` +specializations of this template simply forward to the platform's +native (POSIX) implementation of this function. + +[h4 Accuracy] + +For built in floating point types `log1p` +should have approximately 1 epsilon accuracy. + +[h4 Testing] + +A mixture of spot test sanity checks, and random high precision test values +calculated using NTL::RR at 1000-bit precision. + +[endsect] + +[section:expm1 expm1] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` expm1(T x); + + template + ``__sf_result`` expm1(T x, const ``__Policy``&); + + }} // namespaces + +Returns e[super x] - 1. + +The return type of this function is computed using the __arg_pomotion_rules: +the return is `double` when /x/ is an integer type and T otherwise. + +[optional_policy] + +For small x, then __ex is very close to 1, as a result calculating __exm1 results +in catastrophic cancellation errors when x is small. `expm1` calculates __exm1 using +rational approximations (for up to 128-bit long doubles), otherwise via +a series expansion when x is small (giving an accuracy of less than __te). + +Finally when BOOST_HAS_EXPM1 is defined then the `float/double/long double` +specializations of this template simply forward to the platform's +native (POSIX) implementation of this function. + +[h4 Accuracy] + +For built in floating point types `expm1` +should have approximately 1 epsilon accuracy. + +[h4 Testing] + +A mixture of spot test sanity checks, and random high precision test values +calculated using NTL::RR at 1000-bit precision. + +[endsect] + +[section:cbrt cbrt] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` cbrt(T x); + + template + ``__sf_result`` cbrt(T x, const ``__Policy``&); + + }} // namespaces + +Returns the cubed root of x: x[super 1/3]. + +The return type of this function is computed using the __arg_pomotion_rules: +the return is `double` when /x/ is an integer type and T otherwise. + +[optional_policy] + +Implemented using Halley iteration. + +[h4 Accuracy] + +For built in floating-point types `cbrt` +should have approximately 2 epsilon accuracy. + +[h4 Testing] + +A mixture of spot test sanity checks, and random high precision test values +calculated using NTL::RR at 1000-bit precision. + +[endsect] + +[section:sqrt1pm1 sqrt1pm1] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` sqrt1pm1(T x); + + template + ``__sf_result`` sqrt1pm1(T x, const ``__Policy``&); + + }} // namespaces + +Returns `sqrt(1+x) - 1`. + +The return type of this function is computed using the __arg_pomotion_rules: +the return is `double` when /x/ is an integer type and T otherwise. + +[optional_policy] + +This function is useful when you need the difference between sqrt(x) and 1, when +x is itself close to 1. + +Implemented in terms of `log1p` and `expm1`. + +[h4 Accuracy] + +For built in floating-point types `sqrt1pm1` +should have approximately 3 epsilon accuracy. + +[h4 Testing] + +A selection of random high precision test values +calculated using NTL::RR at 1000-bit precision. + +[endsect] + +[section:powm1 powm1] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` powm1(T1 x, T2 y); + + template + ``__sf_result`` powm1(T1 x, T2 y, const ``__Policy``&); + + }} // namespaces + +Returns x[super y ] - 1. + +The return type of this function is computed using the __arg_pomotion_rules +when T1 and T2 are dufferent types. + +[optional_policy] + +There are two domains where this is useful: when y is very small, or when +x is close to 1. + +Implemented in terms of `expm1`. + +[h4 Accuracy] + +Should have approximately 2-3 epsilon accuracy. + +[h4 Testing] + +A selection of random high precision test values +calculated using NTL::RR at 1000-bit precision. + +[endsect] + +[section:hypot hypot] + + template + ``__sf_result`` hypot(T1 x, T2 y); + + template + ``__sf_result`` hypot(T1 x, T2 y, const ``__Policy``&); + +__effects computes [equation hypot] +in such a way as to avoid undue underflow and overflow. + +The return type of this function is computed using the __arg_pomotion_rules +when T1 and T2 are of different types. + +[optional_policy] + +When calculating [equation hypot] it's quite easy for the intermediate terms to either +overflow or underflow, even though the result is in fact perfectly representable. + +[h4 Implementation] + +The function is even and symmetric in x and y, so first take assume +['x,y > 0] and ['x > y] (we can permute the arguments if this is not the case). + +Then if ['x * [epsilon][space] >= y] we can simply return /x/. + +Otherwise the result is given by: + +[equation hypot2] + +[endsect] + + +[endsect][/section:powers Logs, Powers, Roots and Exponentials] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/rational.qbk b/doc/sf_and_dist/rational.qbk new file mode 100644 index 000000000..61c82392e --- /dev/null +++ b/doc/sf_and_dist/rational.qbk @@ -0,0 +1,176 @@ +[section:rational Polynomial and Rational Function Evaluation] + +[h4 synopsis] + +`` +#include +`` + + // Polynomials: + template + V evaluate_polynomial(const T(&poly)[N], const V& val); + + template + V evaluate_polynomial(const boost::array& poly, const V& val); + + template + U evaluate_polynomial(const T* poly, U z, std::size_t count); + + // Even polynomials: + template + V evaluate_even_polynomial(const T(&poly)[N], const V& z); + + template + V evaluate_even_polynomial(const boost::array& poly, const V& z); + + template + U evaluate_even_polynomial(const T* poly, U z, std::size_t count); + + // Odd polynomials + template + V evaluate_odd_polynomial(const T(&a)[N], const V& z); + + template + V evaluate_odd_polynomial(const boost::array& a, const V& z); + + template + U evaluate_odd_polynomial(const T* poly, U z, std::size_t count); + + // Rational Functions: + template + V evaluate_rational(const T(&a)[N], const T(&b)[N], const V& z); + + template + V evaluate_rational(const boost::array& a, const boost::array& b, const V& z); + + template + V evaluate_rational(const T* num, const U* denom, V z, unsigned count); + +[h4 Description] + +Each of the functions come in three variants: a pair of overloaded functions +where the order of the polynomial or rational function is evaluated at +compile time, and an overload that accepts a runtime variable for the size +of the coefficient array. Generally speaking, compile time evaluation of the +array size results in better type safety, is less prone to programmer errors, +and may result in better optimised code. The polynomial evaluation functions +in particular, are specialised for various array sizes, allowing for +loop unrolling, and one hopes, optimal inline expansion. + + template + V evaluate_polynomial(const T(&poly)[N], const V& val); + + template + V evaluate_polynomial(const boost::array& poly, const V& val); + + template + U evaluate_polynomial(const T* poly, U z, std::size_t count); + +Evaluates the [@http://en.wikipedia.org/wiki/Polynomial polynomial] described by +the coefficients stored in /poly/. + +If the size of the array is specified at runtime, then the polynomial +most have order /count-1/ with /count/ coefficients. Otherwise it has +order /N-1/ with /N/ coefficients. + +Coefficients should be stored such that the coefficients for the x[super i ] terms +are in poly[i]. + +The types of the coefficients and of variable +/z/ may differ as long as /*poly/ is convertible to type /U/. +This allows, for example, for the coefficient table +to be a table of integers if this is appropriate. + + template + V evaluate_even_polynomial(const T(&poly)[N], const V& z); + + template + V evaluate_even_polynomial(const boost::array& poly, const V& z); + + template + U evaluate_even_polynomial(const T* poly, U z, std::size_t count); + +As above, but evaluates an even polynomial: one where all the powers +of /z/ are even numbers. Equivalent to calling +`evaluate_polynomial(poly, z*z, count)`. + + template + V evaluate_odd_polynomial(const T(&a)[N], const V& z); + + template + V evaluate_odd_polynomial(const boost::array& a, const V& z); + + template + U evaluate_odd_polynomial(const T* poly, U z, std::size_t count); + +As above but evaluates a polynomial where all the powers are odd numbers. +Equivalent to `evaluate_polynomial(poly+1, z*z, count-1) * z + poly[0]`. + + template + V evaluate_rational(const T(&num)[N], const U(&denom)[N], const V& z); + + template + V evaluate_rational(const boost::array& num, const boost::array& denom, const V& z); + + template + V evaluate_rational(const T* num, const U* denom, V z, unsigned count); + +Evaluates the rational function (the ratio of two polynomials) described by +the coefficients stored in /num/ and /demom/. + +If the size of the array is specified at runtime then both +polynomials most have order /count-1/ with /count/ coefficients. +Otherwise both polynomials have order /N-1/ with /N/ coefficients. + +Array /num/ describes the numerator, and /demon/ the denominator. + +Coefficients should be stored such that the coefficients for the x[super i ] terms +are in num[i] and denom[i]. + +The types of the coefficients and of variable +/v/ may differ as long as /*num/ and /*denom/ are convertible to type /V/. +This allows, for example, for one or both of the coefficient tables +to be a table of integers if this is appropriate. + +These functions are designed to safely evaluate the result, even when the value +/z/ is very large. As such they do not take advantage of compile time array +sizes to make any optimisations. These functions are best reserved for situations +where /z/ may be large: if you can be sure that numerical overflow will not occur +then polynomial evaluation with compile-time array sizes may offer slightly +better performance. + +[h4 Implementation] + +Polynomials are evaluated by +[@http://en.wikipedia.org/wiki/Horner_algorithm Horners method]. +If the array size is known at +compile time then the functions dispatch to size-specific implementations +that unroll the evaluation loop. + +Rational evaluation is by +[@http://en.wikipedia.org/wiki/Horner_algorithm Horners method]: +with the two polynomials being evaluated +in parallel to make the most of the processors floating-point pipeline. +If /v/ is greater than one, then the polynomials are evaluated in reverse +order as polynomials in ['1\/v]: this avoids unnecessary numerical overflow when the +coefficients are large. + +Both the polynomial and rational function evaluation algorithms can be +tuned using various configuration macros to provide optimal performance +for a particular combination of compiler and platform. This includes +support for second-order Horner's methods. The various options are +[link math_toolkit.perf.tuning documented here]. However, the performance +benefits to be gained from these are marginal on most current hardware, +consequently it's best to run the +[link math_toolkit.perf.perf_test_app performance test application] before +changing the default settings. + +[endsect][/section:rational Polynomial and Rational Function Evaluation] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/references.qbk b/doc/sf_and_dist/references.qbk new file mode 100644 index 000000000..af16d0d7e --- /dev/null +++ b/doc/sf_and_dist/references.qbk @@ -0,0 +1,81 @@ +[section:refs References] + +[h4 General references] + +(Specific detailed sources for individual functions and distributions +are given at the end of each individual section). + +[@http://dlmf.nist.gov/about/ DLMF (NIST Digital Library of Mathematical Functions)] +is intended to be a replacement for the legendary +Abramowitz and Stegun's Handbook of Mathematical Functions, +now scheduled to be completed in 2007. + +M. Abramowitz and I. A. Stegun (Eds.) (1964) +Handbook of Mathematical Functions with Formulas, Graphs, and +Mathematical Tables, +National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.. + +[@http://functions.wolfram.com/ The Wolfram Functions Site] +The Wolfram Functions Site - Providing +the mathematical and scientific community with the world's largest +(and most authorititive) collection of formulas and graphics about mathematical functions. + +[@http://www.itl.nist.gov/div898/handbook/index.htm NIST/SEMATECH e-Handbook of Statistical Methods] + +[@http://documents.wolfram.com/mathematica/Add-onsLinks/StandardPackages/Statistics/DiscreteDistributions.html Mathematica Documentation: DiscreteDistributions] +The Wolfram Research Documentation Center is a collection of online reference materials about Mathematica, CalculationCenter, and other Wolfram Research products. + +[@http://documents.wolfram.com/mathematica/Add-onsLinks/StandardPackages/Statistics/ContinuousDistributions.html Mathematica Documentation: ContinuousDistributions] +The Wolfram Research Documentation Center is a collection of online reference materials about Mathematica, CalculationCenter, and other Wolfram Research products. + +Statistical Distributions (Wiley Series in Probability & Statistics) (Paperback) +by N.A.J. Hastings, Brian Peacock, Merran Evans, ISBN: 0471371246, Wiley 2000. + +[@http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf pugh.pdf (application/pdf Object)] +Pugh Msc Thesis on the Lanczzos approximation to the gamma function. + +[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2003 N1514, 03-0097, A Proposal to Add Mathematical Special Functions to the C++ Standard Library (version 2), Walter E. Brown] + +[h4 Calculators* that we found (and used to cross-check - as far as their widely-varying accuracy allowed).] + +[@http://www.adsciengineering.com/bpdcalc/ http://www.adsciengineering.com/bpdcalc/] Binomial Probability Distribution Calculator. + +[h4 Other Libraries] + +[@http://www.moshier.net/#Cephes Cephes library] by Shephen Moshier and his book: + +Methods and programs for mathematical functions, Stephen L B Moshier, Ellis Horwood (1989) ISBN 0745802893 0470216093 provided inspiration. + +[@http://www.moshier.net/cephes28.zip 100-decimal digit calculator] provided some spot values. + +[@http://www.csit.fsu.edu/~burkardt/cpp_src/dcdflib/dcdflib.html C++ version]. + +[@lib.stat.cmu.edu/general/cdflib CDFLIB Library of Fortran Routines for Cumulative Distribution functions.] + +[@http://www.csit.fsu.edu/~burkardt/f_src/dcdflib/dcdflib.html DCDFLIB C++ version] +DCDFLIB is a library of C++ routines, using double precision arithmetic, for evaluating cumulative probability density functions. + +[@http://www.softintegration.com/docs/package/chnagstat/ http://www.softintegration.com/docs/package/chnagstat/] + +[@http://www.nag.com/numeric/numerical_libraries.asp NAG ]libraries. + +[@http://www.mathcad.com MathCAD] + +[@http://www.vni.com/products/imsl/jmsl/v30/api/com/imsl/stat/Cdf.html JMSL Numerical Library] (Java). + +John F Hart, Computer Approximations, (1978) ISBN 0 088275 642-7. + +William J Cody, Software Manual for the Elementary Functions, Prentice-Hall (1980) ISBN 0138220646. + +Nico Temme, Special Functions, An Introduction to the Classical Functions of Mathematical Physics, Wiley, ISBN: 0471-11313-1 (1996) who also gave valueable advice. + +[@http://www.cas.lancs.ac.uk/glossary_v1.1/prob.html#probdistn Statistics Glossary], Valerie Easton and John H. McColl. + +[endsect] [/section:references References] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/relative_error.qbk b/doc/sf_and_dist/relative_error.qbk new file mode 100644 index 000000000..4c669222a --- /dev/null +++ b/doc/sf_and_dist/relative_error.qbk @@ -0,0 +1,134 @@ +[section:error_test Relative Error and Testing] + +[h4 Synopsis] + +`` +#include +`` + + template + T relative_error(T a, T b); + + template + test_result test(const A& a, F1 test_func, F2 expect_func); + +[h4 Description] + + template + T relative_error(T a, T v); + +Returns the relative error between /a/ and /v/ using the usual formula: + +[equation error1] + +In addition the value returned is zero if: + +* Both /a/ and /v/ are infinite. +* Both /a/ and /v/ are denormalised numbers or zero. + +Otherwise if only one of /a/ and /v/ is zero then the value returned is 1. + + template + test_result test(const A& a, F1 test_func, F2 expect_func); + +This function is used for testing a function against tabulated test data. + +The return type contains statistical data on the relative errors (max, mean, +variance, and the number of test cases etc), as well as the row of test data that +caused the largest relative error. Public members of type test_result are: + +[variablelist +[[`unsigned worst()const;`][Returns the row at which the worst error occurred.]] +[[`T min()const;`][Returns the smallest relative error found.]] +[[`T max()const;`][Returns the largest relative error found.]] +[[`T mean()const;`][Returns the mean error found.]] +[[`boost::uintmax_t count()const;`][Returns the number of test cases.]] +[[`T variance()const;`][Returns the variance of the errors found.]] +[[`T variance1()const;`][Returns the unbiased variance of the errors found.]] +[[`T rms()const`][Returns the Root Mean Square, or quadratic mean of the errors.]] +[[`test_result& operator+=(const test_result& t)`][Combines two test_result's into +a single result.]] +] + +The template parameter of test_result, is the same type as the values in the two +dimensional array passed to function /test/, roughly that's +`A::value_type::value_type`. + +Parameter /a/ is a matrix of test data: and must be a standard library Sequence type, +that contains another Sequence type: +typically it will be a two dimensional instance of +[^boost::array]. Each row +of /a/ should contain all the parameters that are passed to the function +under test as well as the expected result. + +Parameter /test_func/ is the function under test, it is invoked with each row +of test data in /a/. Typically type F1 is created with Boost.Lambda: see +the example below. + +Parameter /expect_func/ is a functor that extracts the expected result +from a row of test data in /a/. Typically type F2 is created with Boost.Lambda: see +the example below. + +If the function under test returns a non-finite value when a finite result is +expected, or if a gross error is found, then a message is sent to `std::cerr`, +and a call to BOOST_ERROR() made (which means that including this header requires +you use Boost.Test). This is mainly a debugging/development aid +(and a good place for a breakpoint). + +[h4 Example] + +Suppose we want to test the tgamma and lgamma functions, we can create a +two dimensional matrix of test data, each row is one test case, and contains +three elements: the input value, and the expected results for the tgamma and +lgamma functions respectively. + + static const boost::array, NumberOfTests> + factorials = { + /* big array of test data goes here */ + }; + +Now we can invoke the test function to test tgamma: + + using namespace boost::math::tools; + using namespace boost::lambda; + + // get a pointer to the function under test: + TestType (*funcp)(TestType) = boost::math::tgamma; + + // declare something to hold the result: + test_result result; + // + // and test tgamma against data: + // + result = test( + factorials, + bind(funcp, ret(_1[0])), // calls tgamma with factorials[row][0] + ret(_1[1]) // extracts the expected result from factorials[row][1] + ); + // + // Print out some results: + // + std::cout << "The Mean was " << result.mean() << std::endl; + std::cout << "The worst error was " << (result.max)() << std::endl; + std::cout << "The worst error was at row " << result.worst_case() << std::endl; + // + // same again with lgamma this time: + // + funcp = boost::math::lgamma; + result = test( + factorials, + bind(funcp, ret(_1[0])), // calls tgamma with factorials[row][0] + ret(_1[2]) // extracts the expected result from factorials[row][2] + ); + // + // etc ... + // + +[endsect][/section:error_test Relative Error and Testing] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/remez.qbk b/doc/sf_and_dist/remez.qbk new file mode 100644 index 000000000..46d62e2f7 --- /dev/null +++ b/doc/sf_and_dist/remez.qbk @@ -0,0 +1,377 @@ +[section:remez The Remez Method] + +The [@http://en.wikipedia.org/wiki/Remez_algorithm Remez algorithm] +is a methodology for locating the minimax rational approximation +to a function. This short article gives a brief overview of the method, but +it should not be regarded as a thorough theoretical treatment, for that you +should consult your favorite textbook. + +Imagine that you want to approximate some function f(x) by way of a rational +function R(x), where R(x) may be either a polynomial P(x) or a ratio of two +polynomials P(x)/Q(x) (a rational function). Initially we'll concentrate on the +polynomial case, as it's by far the easier to deal with, later we'll extend +to the full rational function case. + +We want to find the "best" rational approximation, where +"best" is defined to be the approximation that has the least deviation +from f(x). We can measure the deviation by way of an error function: + +E[sub abs](x) = f(x) - R(x) + +which is expressed in terms of absolute error, but we can equally use +relative error: + +E[sub rel](x) = (f(x) - R(x)) / |f(x)| + +And indeed in general we can scale the error function in any way we want, it +makes no difference to the maths, although the two forms above cover almost +every practical case that you're likely to encounter. + +The minimax rational function R(x) is then defined to be the function that +yields the smallest maximal value of the error function. Chebyshev showed +that there is a unique minimax solution for R(x) that has the following +properties: + +* If R(x) is a polynomial of degree N, then there are N+2 unknowns: +the N+1 coefficients of the polynomial, and maximal value of the error +function. +* The error function has N+1 roots, and N+2 extrema (minima and maxima). +* The extrema alternate in sign, and all have the same magnitude. + +That means that if we know the location of the extrema of the error function +then we can write N+2 simultaneous equations: + +R(x[sub i]) + (-1)[super i]E = f(x[sub i]) + +where E is the maximal error term, and x[sub i] are the abscissa values of the +N+2 extrema of the error function. It is then trivial to solve the simultaneous +equations to obtain the polynomial coefficients and the error term. + +['Unfortunately we don't know where the extrema of the error function are located!] + +[h4 The Remez Method] + +The Remez method is an iterative technique which, given a broad range of +assumptions, will converge on the extrema of the error function, and therefore +the minimax solution. + +In the following discussion we'll use a concrete example to illustrate +the Remez method: an approximation to the function e[super x][space] over +the range \[-1, 1\]. + +Before we can begin the Remez method, we must obtain an initial value +for the location of the extrema of the error function. We could "guess" +these, but a much closer first approximation can be obtained by first +constructing an interpolated polynomial approximation to f(x). + +In order to obtain the N+1 coefficients of the interpolated polynomial +we need N+1 points (x[sub 0]...x[sub N]): with our interpolated form +passing through each of those points +that yields N+1 simultaneous equations: + +f(x[sub i]) = P(x[sub i]) = c[sub 0] + c[sub 1]x[sub i] ... + c[sub N]x[sub i][super N] + +Which can be solved for the coefficients c[sub 0]...c[sub N] in P(x). + +Obviously this is not a minimax solution, indeed our only guarantee is that f(x) and +P(x) touch at N+1 locations, away from those points the error may be arbitrarily +large. However, we would clearly like this initial approximation to be as close to +f(x) as possible, and it turns out that using the zeros of an orthogonal polynomial +as the initial interpolation points is a good choice. In our example we'll use the +zeros of a Chebyshev polynomial as these are particularly easy to calculate, +interpolating for a polynomial of degree 4, and measuring /relative error/ +we get the following error function: + +[$../graphs/remez-2.png] + +Which has a peak relative error of 1.2x10[super -3]. + +While this is a pretty good approximation already, judging by the +shape of the error function we can clearly do better. Before starting +on the Remez method propper, we have one more step to perform: locate +all the extrema of the error function, and store +these locations as our initial ['Chebyshev control points]. + +[note +In the simple case of a polynomial approximation, by interpolating through +the roots of a Chebyshev polynomial we have in fact created a ['Chebyshev +approximation] to the function: in terms of /absolute error/ +this is the best a priori choice for the interpolated form we can +achieve, and typically is very close to the minimax solution. + +However, if we want to optimise for /relative error/, or if the approximation +is a rational function, then the initial Chebyshev solution can be quite far +from the ideal minimax solution. + +A more technical discussion of the theory involved can be found in this +[@http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html online course].] + +[h4 Remez Step 1] + +The first step in the Remez method, given our current set of +N+2 Chebyshev control points x[sub i], is to solve the N+2 simultaneous +equations: + +P(x[sub i]) + (-1)[super i]E = f(x[sub i]) + +To obtain the error term E, and the coefficients of the polynomial P(x). + +This gives us a new approximation to f(x) that has the same error /E/ at +each of the control points, and whose error function ['alternates in sign] +at the control points. This is still not necessarily the minimax +solution though: since the control points may not be at the extrema of the error +function. After this first step here's what our approximation's error +function looks like: + +[$../graphs/remez-3.png] + +Clearly this is still not the minimax solution since the control points +are not located at the extrema, but the maximum relative error has now +dropped to 5.6x10[super -4]. + +[h4 Remez Step 2] + +The second step is to locate the extrema of the new approximation, which we do +in two stages: first, since the error function changes sign at each +control point, we must have N+1 roots of the error function located between +each pair of N+2 control points. Once these roots are found by standard root finding +techniques, we know that N extrema are bracketed between each pair of +roots, plus two more between the endpoints of the range and the first and last roots. +The N+2 extrema can then be found using standard function minimisation techniques. + +We now have a choice: multi-point exchange, or single point exchange. + +In single point exchange, we move the control point nearest to the largest extrema to +the absissa value of the extrema. + +In multi-point exchange we swap all the current control points, for the locations +of the extrema. + +In our example we perform multi-point exchange. + +[h4 Iteration] + +The Remez method then performs steps 1 and 2 above iteratively until the control +points are located at the extrema of the error function: this is then +the minimax solution. + +For our current example, two more iterations converges on a minimax +solution with a peak relative error of +5x10[super -4] and an error function that looks like: + +[$../graphs/remez-4.png] + +[h4 Rational Approximations] + +If we wish to extend the Remez method to a rational approximation of the form + +f(x) = R(x) = P(x) / Q(x) + +where P(x) and Q(x) are polynomials, then we proceed as before, except that now +we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M. This assumes +that Q(x) is normalised so that it's leading coefficient is 1, giving +N+M+1 polynomial coefficients in total, plus the error term E. + +The simultaneous equations to be solved are now: + +P(x[sub i]) / Q(x[sub i]) + (-1)[super i]E = f(x[sub i]) + +Evaluated at the N+M+2 control points x[sub i]. + +Unfortunately these equations are non-linear in the error term E: we can only +solve them if we know E, and yet E is one of the unknowns! + +The method usually adopted to solve these equations is an iterative one: we guess the +value of E, solve the equations to obtain a new value for E (as well as the polynomial +coefficients), then use the new value of E as the next guess. The method is +repeated until E converges on a stable value. + +These complications extend the running time required for the development +of rational approximations quite considerably. It is often desirable +to obtain a rational rather than polynomial approximation none the less: +rational approximations will often match more difficult to approximate +functions, to greater accuracy, and with greater efficiency, than their +polynomial alternatives. For example, if we takes our previous example +of an approximation to e[super x], we obtained 5x10[super -4] accuracy +with an order 4 polynomial. If we move two of the unknowns into the denominator +to give a pair of order 2 polynomials, and re-minimise, then the peak relative error drops +to 8.7x10[super -5]. That's a 5 fold increase in accuracy, for the same number +of terms overall. + +[h4 Practical Considerations] + +Most treatises on approximation theory stop at this point. However, from +a practical point of view, most of the work involves finding the right +approximating form, and then persuading the Remez method to converge +on a solution. + +So far we have used a direct approximation: + +f(x) = R(x) + +But this will converge to a useful approximation only if f(x) is smooth. In +addition round-off errors when evaluating the rational form mean that this +will never get closer than within a few epsilon of machine precision. +Therefore this form of direct approximation is often reserved for situations +where we want efficiency, rather than accuracy. + +The first step in improving the situation is generally to split f(x) into +a dominant part that we can compute accurately by another method, and a +slowly changing remainder which can be approximated by a rational approximation. +We might be tempted to write: + +f(x) = g(x) + R(x) + +where g(x) is the dominant part of f(x), but if f(x)\/g(x) is approximately +constant over the interval of interest then: + +f(x) = g(x)(c + R(x)) + +Will yield a much better solution: here /c/ is a constant that is the approximate +value of f(x)\/g(x) and R(x) is typically tiny compared to /c/. In this situation +if R(x) is optimised for absolute error, then as long as its error is small compared +to the constant /c/, that error will effectively get wiped out when R(x) is added to +/c/. + +The difficult part is obviously finding the right g(x) to extract from your +function: often the asymptotic behaviour of the function will give a clue, so +for example the function __erfc becomes proportional to +e[super -x[super 2]]\/x as x becomes large. Therefore using: + +erfc(z) = (C + R(x)) e[super -x[super 2]]/x + +as the approximating form seems like an obvious thing to try, and does indeed +yield a useful approximation. + +However, the difficulty then becomes one of converging the minimax solution. +Unfortunately, it is known that for some functions the Remez method can lead +to divergent behaviour, even when the initial starting approximation is quite good. +Furthermore, it is not uncommon for the solution obtained in the first Remez step +above to be a bad one: the equations to be solved are generally "stiff", often +very close to being singular, and assuming a solution is found at all, round-off +errors and a rapidly changing error function, can lead to a situation where the +error function does not in fact change sign at each control point as required. +If this occurs, it is fatal to the Remez method. It is also possible to +obtain solutions that are perfectly valid mathematically, but which are +quite useless computationally: either because there is an unavoidable amount +of roundoff error in the computation of the rational function, or because +the denominator has one or more roots over the interval of the approximation. +In the latter case while the approximation may have the correct limiting value at +the roots, the approximation is nonetheless useless. + +Assuming that the approximation does not have any fatal errors, and that the only +issue is converging adequately on the minimax solution, the aim is to +get as close as possible to the minimax solution before beginning the Remez method. +Using the zeros of a Chebyshev polynomial for the initial interpolation is a +good start, but may not be ideal when dealing with relative errors and\/or +rational (rather than polynomial) approximations. One approach is to skew +the initial interpolation points to one end: for example if we raise the +roots of the Chebyshev polynomial to a positive power greater than 1 +then the roots will be skewed towards the middle of the \[-1,1\] interval, +while a positive power less than one +will skew them towards either end. More usefully, if we initially rescale the +points over \[0,1\] and then raise to a positive power, we can skew them to the left +or right. Returning to our example of e[super x][space] over \[-1,1\], the initial +interpolated form was some way from the minimax solution: + +[$../graphs/remez-2.png] + +However, if we first skew the interpolation points to the left (rescale them +to \[0, 1\], raise to the power 1.3, and then rescale back to \[-1,1\]) we +reduce the error from 1.3x10[super -3][space]to 6x10[super -4]: + +[$../graphs/remez-5.png] + +It's clearly still not ideal, but it is only a few percent away from +our desired minimax solution (5x10[super -4]). + +[h4 Remez Method Checklist] + +The following lists some of the things to check if the Remez method goes wrong, +it is by no means an exhaustive list, but is provided in the hopes that it will +prove useful. + +* Is the function smooth enough? Can it be better separated into +a rapidly changing part, and an asymptotic part? +* Does the function being approximated have any "blips" in it? Check +for problems as the function changes computation method, or +if a root, or an infinity has been divided out. The telltale +sign is if there is a narrow region where the Remez method will +not converge. +* Check you have enough accuracy in your calculations: remember that +the Remez method works on the difference between the approximation +and the function being approximated: so you must have more digits of +precision available than the precision of the approximation +being constructed. So for example at double precision, you +shouldn't expect to be able to get better than a float precision +approximation. +* Try skewing the initial interpolated approximation to minimise the +error before you begin the Remez steps. +* If the approximation won't converge or is ill-conditioned from one starting +location, try starting from a different location. +* If a rational function won't converge, one can minimise a polynomial +(which presents no problems), then rotate one term from the numerator to +the denominator and minimise again. In theory one can continue moving +terms one at a time from numerator to denominator, and then re-minimising, +retaining the last set of control points at each stage. +* Try using a smaller interval. It may also be possible to optimise over +one (small) interval, rescale the control points over a larger interval, +and then re-minimise. +* Keep absissa values small: use a change of variable to keep the abscissa +over, say \[0, b\], for some smallish value /b/. + +[h4 References] + +The original references for the Remez Method and it's extension +to rational functions are unfortunately in Russian: + +Remez, E.Ya., ['Fundamentals of numerical methods for Chebyshev approximations], +"Naukova Dumka", Kiev, 1969. + +Remez, E.Ya., Gavrilyuk, V.T., ['Computer development of certain approaches +to the approximate construction of solutions of Chebyshev problems +nonlinearly depending on parameters], Ukr. Mat. Zh. 12 (1960), 324-338. + +Gavrilyuk, V.T., ['Generalization of the first polynomial algorithm of +E.Ya.Remez for the problem of constructing rational-fractional +Chebyshev approximations], Ukr. Mat. Zh. 16 (1961), 575-585. + +Some English language sources include: + +Fraser, W., Hart, J.F., ['On the computation of rational approximations +to continuous functions], Comm. of the ACM 5 (1962), 401-403, 414. + +Ralston, A., ['Rational Chebyshev approximation by Remes' algorithms], +Numer.Math. 7 (1965), no. 4, 322-330. + +A. Ralston, ['Rational Chebyshev approximation, Mathematical +Methods for Digital Computers v. 2] (Ralston A., Wilf H., eds.), +Wiley, New York, 1967, pp. 264-284. + +Hart, J.F. e.a., ['Computer approximations], Wiley, New York a.o., 1968. + +Cody, W.J., Fraser, W., Hart, J.F., ['Rational Chebyshev approximation +using linear equations], Numer.Math. 12 (1968), 242-251. + +Cody, W.J., ['A survey of practical rational and polynomial +approximation of functions], SIAM Review 12 (1970), no. 3, 400-423. + +Barrar, R.B., Loeb, H.J., ['On the Remez algorithm for non-linear +families], Numer.Math. 15 (1970), 382-391. + +Dunham, Ch.B., ['Convergence of the Fraser-Hart algorithm for rational +Chebyshev approximation], Math. Comp. 29 (1975), no. 132, 1078-1082. + +G. L. Litvinov, ['Approximate construction of rational +approximations and the effect of error autocorrection], +Russian Journal of Mathematical Physics, vol.1, No. 3, 1994. + +[endsect][/section:remez The Remez Method] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/result_type_calc.qbk b/doc/sf_and_dist/result_type_calc.qbk new file mode 100644 index 000000000..dd128106e --- /dev/null +++ b/doc/sf_and_dist/result_type_calc.qbk @@ -0,0 +1,85 @@ + +[section:result_type Calculation of the Type of the Result] + +The functions in this library are all overloaded to accept +mixed floating point (or mixed integer and floating point type) +arguments. So for example: + + foo(1.0, 2.0); + foo(1.0f, 2); + foo(1.0, 2L); + +etc, are all valid calls, as long as "foo" is a function taking two +floating-point arguments. But that leaves the question: + +[blurb ['"Given a special function with N arguments of +types T1, T2, T3 ... TN, then what type is the result?"]] + +[*If all the arguments are of the same (floating point) type then the +result is the same type as the arguments.] + +Otherwise, the type of the result +is computed using the following logic: + +# Any arguments that are not template arguments are disregarded from +further analysis. +# For each type in the argument list, if that type is an integer type +then it is treated as if it were of type double for the purposes of +further analysis. +# If any of the arguments is a user-defined class type, then the result type +is the first such class type that is constructible from all of the other +argument types. +# If any of the arguments is of type `long double`, then the result is of type +`long double`. +# If any of the arguments is of type `double`, then the result is of type +`double`. +# Otherwise the result is of type `float`. + +For example: + + cyl_bessel(2, 3.0); + +Returns a `double` result, as does: + + cyl_bessel(2, 3.0f); + +as in this case the integer first argument is treated as a `double` and takes +precedence over the `float` second argument. To get a `float` result we would need +all the arguments to be of type float: + + cyl_bessel_j(2.0f, 3.0f); + +When one or more of the arguments is not a template argument then it +doesn't effect the return type at all, for example: + + sph_bessel(2, 3.0f); + +returns a `float`, since the first argument is not a template argument and +so doesn't effect the result: without this rule functions that take +explicitly integer arguments could never return `float`. + +And for user defined types, all of the following return an NTL::RR result: + + cyl_bessel_j(0, NTL::RR(2)); + + cyl_bessel_j(NTL::RR(2), 3); + + cyl_bessel_j(NTL::quad_float(2), NTL::RR(3)); + +In the last case, quad_float is convertible to RR, but not vice-versa, so +the result will be an NTL::RR. Note that this assumes that you are using +a [link math_toolkit.using_udt.use_ntl patched NTL library]. + +These rules are chosen to be compatible with the behaviour of +['ISO/IEC 9899:1999 Programming languages - C] +and with the +[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Draft Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 5]. + +[endsect] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/roadmap.qbk b/doc/sf_and_dist/roadmap.qbk new file mode 100644 index 000000000..0344f2c4c --- /dev/null +++ b/doc/sf_and_dist/roadmap.qbk @@ -0,0 +1,71 @@ +[template history[] +[h4 Milestone 5: Post Review First Official Release] + +* Added Policy based framework that allows fine grained control +over function behaviour. +* [*Breaking change:] Changed default behaviour for domain, pole and overflow errors +to throw an exception (based on review feedback), this +behaviour can be customised using __Policy's. +* [*Breaking change:] Changed exception thrown when an internal evaluation error +occurs to boost::math::evaluation_error. +* [*Breaking change:] Changed discrete quantiles to return an integer result: +this is anything up to 20 times faster than finding the true root, this +behaviour can be customised using __Policy's. +* Polynomial/rational function evaluation is now customisable and hopefully +faster than before. +* Added performance test program. + +[h4 Milestone 4: Second Review Candidate (1st March 2007)] + +* Moved Xiaogang Zhang's Bessel Functions code into the library, +and brought them into line with the rest of the code. +* Added C# "Distribution Explorer" demo application. + +[h4 Milestone 3: First Review Candidate (31st Dec 2006)] + +* Implemented the main probability distribution and density functions. +* Implemented digamma. +* Added more factorial functions. +* Implemented the Hermite, Legendre and Laguerre polynomials plus the +spherical harmonic functions from TR1. +* Moved Xiaogang Zhang's elliptic integral code into the library, +and brought them into line with the rest of the code. +* Moved Hubert Holin's existing Boost.Math special functions +into this library and brought them into line with the rest of the code. + +[h4 Milestone 2: Released September 10th 2006] + +* Implement preview release of the statistical distributions. +* Added statistical distributions tutorial. +* Implemented root finding algorithms. +* Implemented the inverses of the incomplete gamma and beta functions. +* Rewrite erf/erfc as rational approximations (valid to 128-bit precision). +* Integrated the statistical results generated from +the test data with Boost.Test: uses a database of expected +results, indexed by test, floating point type, platform, and compiler. +* Improved lgamma near 1 and 2 (rational approximations). +* Improved erf/erfc inverses (rational approximations). +* Implemented Rational function generation (the Remez method). + +[h4 Milestone 1: Released March 31st 2006] + +* Implement gamma/beta/erf functions along with their incomplete counterparts. +* Generate high quality test data, against which future improvements can be judged. +* Provide tools for the evaluation of infinite series, continued fractions, and +rational functions. +* Provide tools for testing against tabulated test data, and collecting statistics +on error rates. +* Provide sufficient docs for people to be able to find their way around the library. + +SVN Revisions: + +Sandbox and trunk last synchonised at revision: 41065. + +] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/roots.qbk b/doc/sf_and_dist/roots.qbk new file mode 100644 index 000000000..3ec65ed1f --- /dev/null +++ b/doc/sf_and_dist/roots.qbk @@ -0,0 +1,245 @@ +[section:roots Root Finding With Derivatives] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + T newton_raphson_iterate(F f, T guess, T min, T max, int digits); + + template + T newton_raphson_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + template + T halley_iterate(F f, T guess, T min, T max, int digits); + + template + T halley_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + template + T schroeder_iterate(F f, T guess, T min, T max, int digits); + + template + T schroeder_iterate(F f, T guess, T min, T max, int digits, boost::uintmax_t& max_iter); + + }}} // namespaces + +[h4 Description] + +These functions all perform iterative root finding: `newton_raphson_iterate` +performs second order [link newton Newton Raphson iteration], while `halley_iterate` and +`schroeder_iterate` perform third order [link halley Halley] and +[link schroeder Schroeder] iteration respectively. + +The functions all take the same parameters: + +[variablelist Parameters of the root finding functions +[[F f] [Type F must be a callable function object that accepts one parameter and + returns a tuple: + +For the second order iterative methods (Newton Raphson) + the tuple should have two elements containing the evaluation + of the function and it's first derivative. + +For the third order methods (Halley and Schroeder) the tuple + should have three elements containing the evaluation of + the function and its first and second derivatives.]] +[[T guess] [The initial starting value.]] +[[T min] [The minimum possible value for the result, this is used as an initial lower bracket.]] +[[T max] [The maximum possible value for the result, this is used as an initial upper bracket.]] +[[int digits] [The desired number of binary digits.]] +[[uintmax_t max_iter] [An optional maximum number of iterations to perform.]] +] + +When using these functions you should note that: + +* They may be very sensitive to the initial guess, typically they converge very rapidly +if the initial guess has two or three decimal digits correct. However convergence +can be no better than bisection, or in some rare cases even worse than bisection if the +initial guess is a long way from the correct value and the derivatives are close to zero. +* These functions include special cases to handle zero first (and second where appropriate) +derivatives, and fall back to bisection in this case. However, it is helpful +if F is defined to return an arbitrarily small value ['of the correct sign] rather +than zero. +* If the derivative at the current best guess for the result is infinite (or +very close to being infinite) then these functions may terminate prematurely. +A large first derivative leads to a very small next step, triggering the termination +condition. Derivative based iteration may not be appropriate in such cases. +* These functions fall back to bisection if the next computed step would take the +next value out of bounds. The bounds are updated after each step to ensure this leads +to convergence. However, a good initial guess backed up by asymptotically-tight +bounds will improve performance no end rather than relying on bisection. +* The value of /digits/ is crucial to good performance of these functions, +if it is set too high then at best you will get one extra (unnecessary) +iteration, and at worst the last few steps will proceed by bisection. +Remember that the returned value can never be more accurate than f(x) can be +evaluated, and that if f(x) suffers from cancellation errors as it +tends to zero then the computed steps will be effectively random. The +value of /digits/ should be set so that iteration terminates before this point: +remember that for second and third order methods the number of correct +digits in the result is increasing quite +substantially with each iteration, /digits/ should be set by experiment so that the final +iteration just takes the next value into the zone where f(x) becomes inaccurate. +* Finally: you may well be able to do better than these functions by hand-coding +the heuristics used so that they are tailored to a specific function. You may also +be able to compute the ratio of derivatives used by these methods more efficiently +than computing the derivatives themselves. As ever, algebraic simplification can +be a big win. + +[#newton] +[h4 Newton Raphson Method] +Given an initial guess x0 the subsequent values are computed using: + +[equation roots1] + +Out of bounds steps revert to bisection of the current bounds. + +Under ideal conditions, the number of correct digits doubles with each iteration. + +[#halley] +[h4 Halley's Method] + +Given an initial guess x0 the subsequent values are computed using: + +[equation roots2] + +Over-compensation by the second derivative (one which would proceed +in the wrong direction) causes the method to +revert to a Newton-Raphson step. + +Out of bounds steps revert to bisection of the current bounds. + +Under ideal conditions, the number of correct digits trebles with each iteration. + +[#schroeder] +[h4 Schroeder's Method] + +Given an initial guess x0 the subsequent values are computed using: + +[equation roots3] + +Over-compensation by the second derivative (one which would proceed +in the wrong direction) causes the method to +revert to a Newton-Raphson step. Likewise a Newton step is used +whenever that Newton step would change the next value by more than 10%. + +Out of bounds steps revert to bisection of the current bounds. + +Under ideal conditions, the number of correct digits trebles with each iteration. + +[h4 Example] + +Lets suppose we want to find the cube root of a number, the equation we want to +solve along with its derivatives are: + +[equation roots4] + +To begin with lets solve the problem using Newton Raphson iterations, we'll +begin be defining a function object that returns the evaluation of the function +to solve, along with its first derivative: + + template + struct cbrt_functor + { + cbrt_functor(T const& target) : a(target){} + std::tr1::tuple operator()(T const& z) + { + T sqr = z * z; + return std::tr1::make_tuple(sqr * z - a, 3 * sqr); + } + private: + T a; + }; + +Implementing the cube root is fairly trivial now, the hardest part is finding +a good approximation to begin with: in this case we'll just divide the exponent +by three: + + template + T cbrt(T z) + { + using namespace std; + int exp; + frexp(z, &exp); + T min = ldexp(0.5, exp/3); + T max = ldexp(2.0, exp/3); + T guess = ldexp(1.0, exp/3); + int digits = std::numeric_limits::digits; + return tools::newton_raphson_iterate(detail::cbrt_functor(z), guess, min, max, digits); + } + +Using the test data in libs/math/test/cbrt_test.cpp this found the cube root +exact to the last digit in every case, and in no more than 6 iterations at double +precision. However, you will note that a high precision was used in this +example, exactly what was warned against earlier on in these docs! In this +particular case its possible to compute f(x) exactly and without undue +cancellation error, so a high limit is not too much of an issue. However, +reducing the limit to `std::numeric_limits::digits * 2 / 3` gave full +precision in all but one of the test cases (and that one was out by just one bit). +The maximum number of iterations remained 6, but in most cases was reduced by one. + +Note also that the above code omits error handling, and does not handle +negative values of z correctly. That will be left as an exercise for the reader! + +Now lets adapt the functor slightly to return the second derivative as well: + + template + struct cbrt_functor + { + cbrt_functor(T const& target) : a(target){} + std::tr1::tuple operator()(T const& z) + { + T sqr = z * z; + return std::tr1::make_tuple(sqr * z - a, 3 * sqr, 6 * z); + } + private: + T a; + }; + +And then adapt the `cbrt` function to use Halley iterations: + + template + T cbrt(T z) + { + using namespace std; + int exp; + frexp(z, &exp); + T min = ldexp(0.5, exp/3); + T max = ldexp(2.0, exp/3); + T guess = ldexp(1.0, exp/3); + int digits = std::numeric_limits::digits / 2; + return tools::halley_iterate(detail::cbrt_functor(z), guess, min, max, digits); + } + +Note that the iterations are set to stop at just one-half of full precision, +and yet even so not one of the test cases had a single bit wrong. +What's more, the maximum number of iterations was now just 4. + +Just to complete the picture, we could have called `schroeder_iterate` in the last +example: and in fact it makes no difference to the accuracy or number of iterations +in this particular case. However, the relative performance of these two methods +may vary depending upon the nature of f(x), and the accuracy to which the initial +guess can be computed. There appear to be no generalisations that can be made +except "try them and see". + +Finally, had we called cbrt with [@http://shoup.net/ntl/doc/RR.txt NTL::RR] +set to 1000 bit precision, then full precision can be obtained with just 7 iterations. +To put that in perspective +an increase in precision by a factor of 20, has less than doubled the number of +iterations. That just goes to emphasise that most of the iterations are used +up getting the first few digits correct: after that these methods can churn out +further digits with remarkable efficiency. Or to put it another way: ['nothing beats +a really good initial guess!] + +[endsect][/section:roots Root Finding With Derivatives] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/roots_without_derivatives.qbk b/doc/sf_and_dist/roots_without_derivatives.qbk new file mode 100644 index 000000000..6f32ebdb0 --- /dev/null +++ b/doc/sf_and_dist/roots_without_derivatives.qbk @@ -0,0 +1,419 @@ +[section:roots2 Root Finding Without Derivatives] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol); + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + + template + std::pair + bracket_and_solve_root( + F f, + const T& guess, + const T& factor, + bool rising, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + bracket_and_solve_root( + F f, + const T& guess, + const T& factor, + bool rising, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + const T& fa, + const T& fb, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + const T& fa, + const T& fb, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + + // Termination conditions: + template + struct eps_tolerance; + + struct equal_floor; + struct equal_ceil; + struct equal_nearest_integer; + + }}} // namespaces + +[h4 Description] + +These functions solve the root of some function /f(x)/ without the +need for the derivatives of /f(x)/. The functions here that use TOMS +Algorithm 748 are asymptotically the most efficient known, and have +been shown to be optimal for a certain classes of smooth functions. + +Alternatively, there is a simple bisection routine which can be useful +in its own right in some situations, or alternatively for narrowing +down the range containing the root, prior to calling a more advanced +algorithm. + +All the algorithms in this section reduce the diameter of the enclosing +interval with the same asymptotic efficiency with which they locate the +root. This is in contrast to the derivative based methods which may /never/ +significantly reduce the enclosing interval, even though they rapidly approach +the root. This is also in contrast to some other derivative-free methods +(for example the methods of [@http://en.wikipedia.org/wiki/Brent%27s_method Brent or Dekker)] +which only reduce the enclosing interval on the final step. +Therefore these methods return a std::pair containing the enclosing interval found, +and accept a function object specifying the termination condition. +Three function objects are provided for ready-made termination conditions: +/eps_tolerance/ causes termination when the relative error in the enclosing +interval is below a certain threshold, while /equal_floor/ and /equal_ceil/ are +useful for certain statistical applications where the result is known to be +an integer. Other user-defined termination conditions are likely to be used +only rarely, but may be useful in some specific circumstances. + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol); + + template + std::pair + bisect( + F f, + T min, + T max, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + +These functions locate the root using bisection, function arguments are: + +[variablelist +[[f] [A unary functor which is the function whose root is to be found.]] +[[min] [The left bracket of the interval known to contain the root.]] +[[max] [The right bracket of the interval known to contain the root. + It is a precondition that /min < max/ and /f(min)*f(max) <= 0/, + the function signals evaluation error if these preconditions are violated. + The action taken is controlled by the evaluation error policy. + A best guess may be returned, perhaps significantly wrong.]] +[[tol] [A binary functor that specifies the termination condition: the function + will return the current brackets enclosing the root when /tol(min,max)/ becomes true.]] +[[max_iter][The maximum number of invocations of /f(x)/ to make while searching for the root.]] +] + +[optional_policy] + +Returns: a pair of values /r/ that bracket the root so that: + + f(r.first) * f(r.second) <= 0 + +and either + + tol(r.first, r.second) == true + +or + + max_iter >= m + +where /m/ is the initial value of /max_iter/ passed to the function. + +In other words, it's up to the caller to verify whether termination occurred +as a result of exceeding /max_iter/ function invocations (easily done by +checking the value of /max_iter/ when the function returns), rather than +because the termination condition /tol/ was satisfied. + + template + std::pair + bracket_and_solve_root( + F f, + const T& guess, + const T& factor, + bool rising, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + bracket_and_solve_root( + F f, + const T& guess, + const T& factor, + bool rising, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + +This is a convenience function that calls /toms748_solve/ internally +to find the root of /f(x)/. It's usable only when /f(x)/ is a monotonic +function, and the location of the root is known approximately, and in +particular it is known whether the root is occurs for positive or negative +/x/. The parameters are: + +[variablelist +[[f][A unary functor that is the function whose root is to be solved. + f(x) must be uniformly increasing or decreasing on /x/.]] +[[guess][An initial approximation to the root]] +[[factor][A scaling factor that is used to bracket the root: the value + /guess/ is multiplied (or divided as appropriate) by /factor/ + until two values are found that bracket the root. A value + such as 2 is a typical choice for /factor/.]] +[[rising][Set to /true/ if /f(x)/ is rising on /x/ and /false/ if /f(x)/ + is falling on /x/. This value is used along with the result + of /f(guess)/ to determine if /guess/ is + above or below the root.]] +[[tol] [A binary functor that determines the termination condition for the search + for the root. /tol/ is passed the current brackets at each step, + when it returns true then the current brackets are returned as the result.]] +[[max_iter] [The maximum number of function invocations to perform in the search + for the root.]] +] + +[optional_policy] + +Returns: a pair of values /r/ that bracket the root so that: + + f(r.first) * f(r.second) <= 0 + +and either + + tol(r.first, r.second) == true + +or + + max_iter >= m + +where /m/ is the initial value of /max_iter/ passed to the function. + +In other words, it's up to the caller to verify whether termination occurred +as a result of exceeding /max_iter/ function invocations (easily done by +checking the value of /max_iter/ when the function returns), rather than +because the termination condition /tol/ was satisfied. + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + const T& fa, + const T& fb, + Tol tol, + boost::uintmax_t& max_iter); + + template + std::pair + toms748_solve( + F f, + const T& a, + const T& b, + const T& fa, + const T& fb, + Tol tol, + boost::uintmax_t& max_iter, + const ``__Policy``&); + +These two functions implement TOMS Algorithm 748: it uses a mixture of +cubic, quadratic and linear (secant) interpolation to locate the root of +/f(x)/. The two functions differ only by whether values for /f(a)/ and +/f(b)/ are already available. The parameters are: + +[variablelist +[[f] [A unary functor that is the function whose root is to be solved. + f(x) need not be uniformly increasing or decreasing on /x/ and + may have multiple roots.]] +[[a] [ The lower bound for the initial bracket of the root.]] +[[b] [The upper bound for the initial bracket of the root. + It is a precondition that /a < b/ and that /a/ and /b/ + bracket the root to find so that /f(a)*f(b) < 0/.]] +[[fa] [Optional: the value of /f(a)/.]] +[[fb] [Optional: the value of /f(b)/.]] +[[tol] [A binary functor that determines the termination condition for the search + for the root. /tol/ is passed the current brackets at each step, + when it returns true then the current brackets are returned as the result.]] +[[max_iter] [The maximum number of function invocations to perform in the search + for the root. On exit /max_iter/ is set to actual number of function + invocations used.]] +] + +[optional_policy] + +Returns: a pair of values /r/ that bracket the root so that: + + f(r.first) * f(r.second) <= 0 + +and either + + tol(r.first, r.second) == true + +or + + max_iter >= m + +where /m/ is the initial value of /max_iter/ passed to the function. + +In other words, it's up to the caller to verify whether termination occurred +as a result of exceeding /max_iter/ function invocations (easily done by +checking the value of /max_iter/), rather than because the termination +condition /tol/ was satisfied. + + template + struct eps_tolerance + { + eps_tolerance(int bits); + bool operator()(const T& a, const T& b)const; + }; + +This is the usual termination condition used with these root finding functions. +Its operator() will return true when the relative distance between /a/ and /b/ +is less than twice the machine epsilon for T, or 2[super 1-bits], whichever is +the larger. In other words you set /bits/ to the number of bits of precision you +want in the result. The minimal tolerance of twice the machine epsilon of T is +required to ensure that we get back a bracketing interval: since this must clearly +be at least 1 epsilon in size. + + struct equal_floor + { + equal_floor(); + template bool operator()(const T& a, const T& b)const; + }; + +This termination condition is used when you want to find an integer result +that is the /floor/ of the true root. It will terminate as soon as both ends +of the interval have the same /floor/. + + struct equal_ceil + { + equal_ceil(); + template bool operator()(const T& a, const T& b)const; + }; + +This termination condition is used when you want to find an integer result +that is the /ceil/ of the true root. It will terminate as soon as both ends +of the interval have the same /ceil/. + + struct equal_nearest_integer + { + equal_nearest_integer(); + template bool operator()(const T& a, const T& b)const; + }; + +This termination condition is used when you want to find an integer result +that is the /closest/ to the true root. It will terminate as soon as both ends +of the interval round to the same nearest integer. + +[h4 Implementation] + +The implementation of the bisection algorithm is extremely straightforward +and not detailed here. TOMS algorithm 748 is described in detail in: + +['Algorithm 748: Enclosing Zeros of Continuous Functions, +G. E. Alefeld, F. A. Potra and Yixun Shi, +ACM Transactions on Mathematica1 Software, Vol. 21. No. 3. September 1995. +Pages 327-344.] + +The implementation here is a faithful translation of this paper into C++. + +[endsect][/section:roots2 Root Finding Without Derivatives] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/series.qbk b/doc/sf_and_dist/series.qbk new file mode 100644 index 000000000..04b02a7a6 --- /dev/null +++ b/doc/sf_and_dist/series.qbk @@ -0,0 +1,129 @@ +[section:series_evaluation Series Evaluation] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ namespace tools{ + + template + typename Functor::result_type sum_series(Functor& func, int bits); + + template + typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms); + + template + typename Functor::result_type sum_series(Functor& func, int bits, U init_value); + + template + typename Functor::result_type sum_series(Functor& func, int bits, boost::uintmax_t& max_terms, U init_value); + + template + typename Functor::result_type kahan_sum_series(Functor& func, int bits); + + template + typename Functor::result_type kahan_sum_series(Functor& func, int bits, boost::uintmax_t& max_terms); + + }}} // namespaces + +[h4 Description] + +These algorithms are intended for the +[@http://en.wikipedia.org/wiki/Series_%28mathematics%29 summation of infinite series]. + +Each of the algorithms takes a nullary-function object as the first argument: +the function object will be repeatedly invoked to pull successive terms from +the series being summed. + +The second argument is the number of binary bits of +precision required, summation will stop when the next term is too small to +have any effect on the first /bits/ bits of the result. + +The optional third argument /max_terms/ sets an upper limit on the number +of terms of the series to evaluate. In addition, on exit the function will +set /max_terms/ to the actual number of terms of the series that were +evaluated: this is particularly useful for profiling the convergence +properties of a new series. + +The final optional argument /init_value/ is the initial value of the sum +to which the terms of the series should be added. This is useful in two situations: + +* Where the first value of the series has a different formula to successive +terms. In this case the first value in the series can be passed as the +last argument and the logic of the function object can then be simplified +to return subsequent terms. +* Where the series is being added (or subtracted) from some other value: +termination of the series will likely occur much more rapidly if that other +value is passed as the last argument. For example, there are several functions +that can be expressed as /1 - S(z)/ where S(z) is an infinite series. In this +case, pass -1 as the last argument and then negate the result of the summation +to get the result of /1 - S(z)/. + +The two /kahan_sum_series/ variants of these algorithms maintain a carry term +that corrects for roundoff error during summation. +They are inspired by the +[@http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm /Kahan Summation Formula/] +that appears in +[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic]. +However, it should be pointed out that there are very few series that require +summation in this way. + +[h4 Example] + +Let's suppose we want to implement /log(1+x)/ via its infinite series, + +[equation log1pseries] + +We begin by writing a small function object to return successive terms +of the series: + + template + struct log1p_series + { + // we must define a result_type typedef: + typedef T result_type; + + log1p_series(T x) + : k(0), m_mult(-x), m_prod(-1){} + + T operator()() + { + // This is the function operator invoked by the summation + // algorithm, the first call to this operator should return + // the first term of the series, the second call the second + // term and so on. + m_prod *= m_mult; + return m_prod / ++k; + } + + private: + int k; + const T m_mult; + T m_prod; + }; + +Implementing log(1+x) is now fairly trivial: + + template + T log1p(T x) + { + // We really should add some error checking on x here! + assert(std::fabs(x) < 1); + + // construct the series functor: + log1p_series s(x); + // and add it up, with enough digits for full machine precision + // plus a couple more for luck.... ! + return tools::sum_series(s, tools::digits(x) + 2); + } + +[endsect][/section Series Evaluation] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/sinc.qbk b/doc/sf_and_dist/sinc.qbk new file mode 100644 index 000000000..3d7cfbf16 --- /dev/null +++ b/doc/sf_and_dist/sinc.qbk @@ -0,0 +1,103 @@ +[/ math.qbk + Copyright 2006 Hubert Holin and John Maddock. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + +[section:sinc Sinus Cardinal and Hyperbolic Sinus Cardinal Functions] + +[section:sinc_overview Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview] + +The [@http://mathworld.wolfram.com/SincFunction.html Sinus Cardinal family of functions] +(indexed by the family of indices [^a > 0]) +is defined by +[equation special_functions_blurb20]; +it sees heavy use in signal processing tasks. + +By analogy, the +[@http://mathworld.wolfram.com/SinhcFunction.htm Hyperbolic Sinus Cardinal] +family of functions +(also indexed by the family of indices [^a > 0]) is defined by +[equation special_functions_blurb22]. + +These two families of functions are composed of entire functions. + +These functions (__sinc_pi and __sinhc_pi) are needed by +[@http://www.boost.org/libs/math/quaternion/quaternion.html our implementation] +of [@http://mathworld.wolfram.com/Quaternion.html quaternions] +and [@http://mathworld.wolfram.com/Octonion.html octonions]. + +[: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]] +[: [$../graphs/sinc_pi_and_sinhc_pi_on_r.png]] + +[endsect] + +[section sinc_pi] + +`` +#include +`` + + template + ``__sf_result`` sinc_pi(const T x); + + template + ``__sf_result`` sinc_pi(const T x, const ``__Policy``&); + + template class U> + U sinc_pi(const U x); + + template class U, class ``__Policy``> + U sinc_pi(const U x, const ``__Policy``&); + +Computes +[link math_toolkit.special.sinc.sinc_overview +the Sinus Cardinal] of x: + + sinc_pi(x) = sin(x) / x + +The second form is for complex numbers, +quaternions, octonions etc. Taylor series are used at the origin +to ensure accuracy. + +[optional_policy] + +[endsect] + +[section sinhc_pi] + +`` +#include +`` + + template + ``__sf_result`` sinhc_pi(const T x); + + template + ``__sf_result`` sinhc_pi(const T x, const ``__Policy``&); + + template class U> + U sinhc_pi(const U x); + + template class U, class ``__Policy``> + U sinhc_pi(const U x, const ``__Policy``&); + +Computes http://mathworld.wolfram.com/SinhcFunction.html +[link math_toolkit.special.sinc.sinc_overview +the Hyperbolic Sinus Cardinal] of x: + + sinhc_pi(x) = sinh(x) / x + +The second form is for +complex numbers, quaternions, octonions etc. Taylor series are used at the origin +to ensure accuracy. + +The return type of the first form is computed using the __arg_pomotion_rules +when T is an integer type. + +[optional_policy] + +[endsect] + +[endsect] diff --git a/doc/sf_and_dist/spherical_harmonic.qbk b/doc/sf_and_dist/spherical_harmonic.qbk new file mode 100644 index 000000000..3cfc6946c --- /dev/null +++ b/doc/sf_and_dist/spherical_harmonic.qbk @@ -0,0 +1,150 @@ +[section:sph_harm Spherical Harmonics] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi); + + template + std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + + template + ``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi); + + template + ``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + + template + ``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi); + + template + ``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +The return type of these functions is computed using the __arg_pomotion_rules +when T1 and T2 are different types. + +[optional_policy] + + template + std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi); + + template + std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + +Returns the value of the Spherical Harmonic Y[sub n][super m](theta, phi): + +[equation spherical_0] + +The spherical harmonics Y[sub n][super m](theta, phi) are the angular +portion of the solution to Laplace's equation in spherical coordinates +where azimuthal symmetry is not present. + +[caution Care must be taken in correctly identifying the arguments to this +function: [theta][space] is taken as the polar (colatitudinal) coordinate +with [theta][space] in \[0, [pi]\], and [phi][space] as the azimuthal (longitudinal) +coordinate with [phi][space] in \[0,2[pi]). This is the convention used in Physics, +and matches the definition used by +[@http://documents.wolfram.com/mathematica/functions/SphericalHarmonicY +Mathematica in the function SpericalHarmonicY], +but is opposite to the usual mathematical conventions. + +Some other sources include an additional Condon-Shortley phase term of +(-1)[super m] in the definition of this function: note however that our +definition of the associated Legendre polynomial already includes this term. + +This implementation returns zero for m > n + +For [theta][space] outside \[0, [pi]\] and [phi][space] outside \[0, 2[pi]\] this +implementation follows the convention used by Mathematica: +the function is periodic with period [pi][space] in [theta][space] and 2[pi][space] in +[phi]. Please note that this is not the behaviour one would get +from a casual application of the function's definition. Cautious users +should keep [theta][space] and [phi][space] to the range \[0, [pi]\] and +\[0, 2[pi]\] respectively. + +See: [@http://mathworld.wolfram.com/SphericalHarmonic.html +Weisstein, Eric W. "Spherical Harmonic." +From MathWorld--A Wolfram Web Resource]. ] + + template + ``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi); + + template + ``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + +Returns the real part of Y[sub n][super m](theta, phi): + +[equation spherical_1] + + template + ``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi); + + template + ``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&); + +Returns the imaginary part of Y[sub n][super m](theta, phi): + +[equation spherical_2] + +[h4 Accuracy] + +The following table shows peak errors for various domains of input arguments. +Note that only results for the widest floating point type on the system are +given as narrower types have __zero_error. Peak errors are the same +for both the real and imaginary parts, as the error is dominated by +calculation of the associated Legendre polynomials: especially near the +roots of the associated Legendre function. + +All values are in units of epsilon. + +[table Peak Errors In the Sperical Harmonic Functions +[[Significand Size] [Platform and Compiler] [Errors in range + +0 < l < 20] ] +[[53] [Win32, Visual C++ 8] [Peak=2x10[super 4] Mean=700] ] +[[64] [SUSE Linux IA32, g++ 4.1] [Peak=2900 Mean=100]] +[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=2900 Mean=100] ] +[[113] [HPUX IA64, aCC A.06.06] [Peak=6700 Mean=230]] +] + +Note that the worst errors occur when the degree increases, values greater than +~120 are very unlikely to produce sensible results, especially +when the order is also large. Further the relative errors +are likely to grow arbitrarily large when the function is very close to a root. + +[h4 Testing] + +A mixture of spot tests of values calculated using functions.wolfram.com, +and randomly generated test data are +used: the test data was computed using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision. + +[h4 Implementation] + +These functions are implemented fairly naively using the formulae +given above. Some extra care is taken to prevent roundoff error +when converting from polar coordinates (so for example the +['1-x[super 2]] term used by the associated Legendre functions is calculated +without roundoff error using ['x = cos(theta)], and +['1-x[super 2] = sin[super 2](theta)]). The limiting factor in the error +rates for these functions is the need to calculate values near the roots +of the associated Legendre functions. + +[endsect][/section:beta_function The Beta Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/structure.qbk b/doc/sf_and_dist/structure.qbk new file mode 100644 index 000000000..785b340b5 --- /dev/null +++ b/doc/sf_and_dist/structure.qbk @@ -0,0 +1,127 @@ +[section:navigation Navigation] + +Used in combination with the configured browser key, the +following keys act as handy shortcuts for common navigation tasks. + +[h5 Shortcuts] +[: +[^[*p]] - Previous page + +[^[*n]] - Next page + +[^[*h]] - home + +[^[*u]] - Up +] + +The following table shows how to access these from common browsers: + +[table +[[Browser][Access Method]] +[[Internet Explorer] + [Alt+Key highlights the link only, so for example to move to the next topic + you would need "Alt+n" followed by "Enter".]] +[[Firefox 2.0 and later][Alt+Shift+Key follows the link, so for example + "Alt+Shift+n" will take you to the next topic.]] +[[Opera][Press Shift+Esc followed by the access key.]] +[[Konqueror][Press and release the Ctrl key, followed by the access key]] +] + +Some browsers also make these links available in their site-navigation +toolbars: in Opera for example you can use Ctrl plus the left and right +arrow keys to move between "next" and "previous" topics. + +[endsect] + +[section:directories Directory and File Structure] + +[h4 boost\/math] + +[variablelist +[[\/concepts\/] + [Prototype defining the *essential* features of a RealType + class (see real_concept.hpp). Most applications will use `double` + as the RealType (and short `typedef` names of distributions are + reserved for this type where possible), a few will use `float` or + `long double`, but it is also possible to use higher precision types + like __NTL_RR that conform to the requirements specified by real_concept.]] + +[[\/constants\/] + [Templated definition of some highly accurate math + constants (in constants.hpp).]] + +[[\/distributions\/] + [Distributions used in mathematics and, especially, statistics: + Gaussian, Students-t, Fisher, Binomial etc]] + +[[\/policies\/] + [Policy framework, for handling user requested behaviour modifications.]] + +[[\/special_functions\/] + [Math functions generally regarded as 'special', like beta, + cbrt, erf, gamma, lgamma, tgamma ... (Some of these are specified in + C++, and C99\/TR1, and perhaps TR2).]] + +[[\/tools\/] + [Tools used by functions, like evaluating polynomials, continued fractions, + root finding, precision and limits, and by tests. Some will + find application outside this package.]] +] + +[h4 boost\/libs] + +[variablelist +[[\/doc\/] + [Documentation source files in Quickbook format processed into + html and pdf formats.]] + +[[\/examples\/] + [Examples and demos of using math functions and distributions.]] + +[[\/performance\/] + [Performance testing and tuning program.]] + +[[\/test\/] + [Test files, in various .cpp files, most using Boost.Test + (some with test data as .ipp files, usually generated using NTL RR + type with ample precision for the type, often for precisions + suitable for up to 256-bit significand real types).]] + +[[\/tools\/] + [Programs used to generate test data. Also changes to the + [@http://shoup.net/ntl/ NTL] released package to provide a few additional + (and vital) extra features.]] +] + +[endsect] + +[section:namespaces Namespaces] + +All math functions and distributions are in `namespace boost::math` + +So, for example, the Students-t distribution template in `namespace boost::math` is + + template class students_t_distribution + +and can be instantiated with the help of the reserved name `students_t`(for `RealType double`) + + typedef students_t_distribution students_t; + + student_t mydist(10); + +Functions not intended for use by applications are in `boost::math::detail`. + +Functions that may have more general use, like `digits` +(significand), `max_value`, `min_value` and `epsilon` are in +`boost::math::tools`. + +__Policy and configuration information is in namespace `boost::math::policies`. + +[endsect] + +[/ structure.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/test_data.qbk b/doc/sf_and_dist/test_data.qbk new file mode 100644 index 000000000..dcc0480fd --- /dev/null +++ b/doc/sf_and_dist/test_data.qbk @@ -0,0 +1,438 @@ +[section:test_data Graphing, Profiling, and Generating Test Data for Special Functions] + +The class `test_data` and associated helper functions are designed so that in just +a few lines of code you should be able to: + +* Profile a continued fraction, or infinite series for convergence and accuracy. +* Generate csv data from a special function that can be imported into your favorite +graphing program (or spreadsheet) for further analysis. +* Generate high precision test data. + +[h4 Synopsis] + + namespace boost{ namespace math{ namespace tools{ + + enum parameter_type + { + random_in_range = 0, + periodic_in_range = 1, + power_series = 2, + dummy_param = 0x80, + }; + + template + struct parameter_info; + + template + parameter_info make_random_param(T start_range, T end_range, int n_points); + + template + parameter_info make_periodic_param(T start_range, T end_range, int n_points); + + template + parameter_info make_power_param(T basis, int start_exponent, int end_exponent); + + template + bool get_user_parameter_info(parameter_info& info, const char* param_name); + + template + class test_data + { + public: + typedef std::vector row_type; + typedef row_type value_type; + private: + typedef std::set container_type; + public: + typedef typename container_type::reference reference; + typedef typename container_type::const_reference const_reference; + typedef typename container_type::iterator iterator; + typedef typename container_type::const_iterator const_iterator; + typedef typename container_type::difference_type difference_type; + typedef typename container_type::size_type size_type; + + // creation: + test_data(){} + template + test_data(F func, const parameter_info& arg1); + + // insertion: + template + test_data& insert(F func, const parameter_info& arg1); + + template + test_data& insert(F func, const parameter_info& arg1, + const parameter_info& arg2); + + template + test_data& insert(F func, const parameter_info& arg1, + const parameter_info& arg2, + const parameter_info& arg3); + + void clear(); + + // access: + iterator begin(); + iterator end(); + const_iterator begin()const; + const_iterator end()const; + bool operator==(const test_data& d)const; + bool operator!=(const test_data& d)const; + void swap(test_data& other); + size_type size()const; + size_type max_size()const; + bool empty()const; + + bool operator < (const test_data& dat)const; + bool operator <= (const test_data& dat)const; + bool operator > (const test_data& dat)const; + bool operator >= (const test_data& dat)const; + }; + + template + std::basic_ostream& write_csv( + std::basic_ostream& os, + const test_data& data); + + template + std::basic_ostream& write_csv( + std::basic_ostream& os, + const test_data& data, + const charT* separator); + + template + std::ostream& write_code(std::ostream& os, + const test_data& data, + const char* name); + + }}} // namespaces + +[h4 Description] + +This tool is best illustrated with the following series of examples. + +The functionality of test_data is split into the following parts: + +* A functor that implements the function for which data is being generated: +this is the bit you have to write. +* One of more parameters that are to be passed to the functor, these are +described in fairly abstract terms: give me N points distributed like /this/ etc. +* The class test_data, that takes the functor and descriptions of the parameters +and computes how ever many output points have been requested, these are stored +in a sorted container. +* Routines to iterate over the test_data container and output the data in either +csv format, or as C++ source code (as a table using Boost.Array). + +[h5 Example 1: Output Data for Graph Plotting] + +For example, lets say we want to graph the lgamma function between -3 and 100, +one could do this like so: + + #include + #include + + int main() + { + using namespace boost::math::tools; + + // create an object to hold the data: + test_data data; + + // insert 500 points at uniform intervals between just after -3 and 100: + double (*pf)(double) = boost::math::lgamma; + data.insert(pf, make_periodic_param(-3.0 + 0.00001, 100.0, 500)); + + // print out in csv format: + write_csv(std::cout, data, ", "); + return 0; + } + +Which, when plotted, results in: + +[$../graphs/lgamma.png] + +[h5 Example 2: Creating Test Data] + +As a second example, let's suppose we want to create highly accurate test +data for a special function. Since many special functions have two or +more independent parameters, it's very hard to effectively cover all of +the possible parameter space without generating gigabytes of data at +great computational expense. A second best approach is to provide the tools +by which a user (or the library maintainer) can quickly generate more data +on demand to probe the function over a particular domain of interest. + +In this example we'll generate test data for the beta function using +[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000 bit precision. +Rather than call our generic +version of the beta function, we'll implement a deliberately naive version +of the beta function using lgamma, and rely on the high precision of the +data type used to get results accurate to at least 128-bit precision. In this +way our test data is independent of whatever clever tricks we may wish to +use inside the our beta function. + +To start with then, here's the function object that creates the test data: + + #include + #include + #include + #include + + #include + + using namespace boost::math::tools; + + struct beta_data_generator + { + NTL::RR operator()(NTL::RR a, NTL::RR b) + { + // + // If we throw a domain error then test_data will + // ignore this input point. We'll use this to filter + // out all cases where a < b since the beta function + // is symmetrical in a and b: + // + if(a < b) + throw std::domain_error(""); + + // very naively calculate spots with lgamma: + NTL::RR g1, g2, g3; + int s1, s2, s3; + g1 = boost::math::lgamma(a, &s1); + g2 = boost::math::lgamma(b, &s2); + g3 = boost::math::lgamma(a+b, &s3); + g1 += g2 - g3; + g1 = exp(g1); + g1 *= s1 * s2 * s3; + return g1; + } + }; + +To create the data, we'll need to input the domains for a and b +for which the function will be tested: the function `get_user_parameter_info` +is designed for just that purpose. The start of main will look something like: + + // Set the precision on RR: + NTL::RR::SetPrecision(1000); // bits. + NTL::RR::SetOutputPrecision(40); // decimal digits. + + parameter_info arg1, arg2; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the beta function:\n" + " beta(a, b)\n\n"; + + bool cont; + std::string line; + + do{ + // prompt the user for the domain of a and b to test: + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "b"); + + // create the data: + data.insert(beta_data_generator(), arg1, arg2); + + // see if the user want's any more domains tested: + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + +[caution At this point one potential stumbling block should be mentioned: +test_data<>::insert will create a matrix of test data when there are two +or more parameters, so if we have two parameters and we're asked for +a thousand points on each, that's a ['million test points in total]. +Don't say you weren't warned!] + +There's just one final step now, and that's to write the test data to file: + + std::cout << "Enter name of test data file [default=beta_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "beta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "beta_data"); + +The format of the test data looks something like: + + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1830> + beta_med_data = { + SC_(0.4883005917072296142578125), + SC_(0.4883005917072296142578125), + SC_(3.245912809500479157065104747353807392371), + SC_(3.5808107852935791015625), + SC_(0.4883005917072296142578125), + SC_(1.007653173802923954909901438393379243537), + /* ... lots of rows skipped */ + }; + +The first two values in each row are the input parameters that were passed +to our functor and the last value is the return value from the functor. +Had our functor returned a tuple rather than a value, then we would have had +one entry for each element in the tuple in addition to the input parameters. + +The first #define serves two purposes: + +* It reduces the file sizes considerably: all those `static_cast`'s add up to a lot +of bytes otherwise (they are needed to suppress compiler warnings when `T` is +narrower than a `long double`). +* It provides a useful customisation point: for example if we were testing +a user-defined type that has more precision than a `long double` we could change +it to: + +[^#define SC_(x) lexical_cast(BOOST_STRINGIZE(x))] + +in order to ensure that no truncation of the values occurs prior to conversion +to `T`. Note that this isn't used by default as it's rather hard on the compiler +when the table is large. + +[h5 Example 3: Profiling a Continued Fraction for Convergence and Accuracy] + +Alternatively, lets say we want to profile a continued fraction for +convergence and error. As an example, we'll use the continued fraction +for the upper incomplete gamma function, the following function object +returns the next a[sub N ] and b[sub N ] of the continued fraction +each time it's invoked: + + template + struct upper_incomplete_gamma_fract + { + private: + T z, a; + int k; + public: + typedef std::pair result_type; + + upper_incomplete_gamma_fract(T a1, T z1) + : z(z1-a1+1), a(a1), k(0) + { + } + + result_type operator()() + { + ++k; + z += 2; + return result_type(k * (a - k), z); + } + }; + +We want to measure both the relative error, and the rate of convergence +of this fraction, so we'll write a functor that returns both as a tuple: +class test_data will unpack the tuple for us, and create one column of data +for each element in the tuple (in addition to the input parameters): + + #include + #include + #include + #include + #include + + template + struct profile_gamma_fraction + { + typedef std::tr1::tuple result_type; + + result_type operator()(T val) + { + using namespace boost::math::tools; + // estimate the true value, using arbitary precision + // arithmetic and NTL::RR: + NTL::RR rval(val); + upper_incomplete_gamma_fract f1(rval, rval); + NTL::RR true_val = continued_fraction_a(f1, 1000); + // + // Now get the aproximation at double precision, along with the number of + // iterations required: + boost::uintmax_t iters = std::numeric_limits::max(); + upper_incomplete_gamma_fract f2(val, val); + T found_val = continued_fraction_a(f2, std::numeric_limits::digits, iters); + // + // Work out the relative error, as measured in units of epsilon: + T err = real_cast(relative_error(true_val, NTL::RR(found_val)) / std::numeric_limits::epsilon()); + // + // now just return the results as a tuple: + return std::tr1::make_tuple(err, iters); + } + }; + +Feeding that functor into test_data allows rapid output of csv data, +for whatever type `T` we may be interested in: + + int main() + { + using namespace boost::math::tools; + // create an object to hold the data: + test_data data; + // insert 500 points at uniform intervals between just after 0 and 100: + data.insert(profile_gamma_fraction(), make_periodic_param(0.01, 100.0, 100)); + // print out in csv format: + write_csv(std::cout, data, ", "); + return 0; + } + +This time there's no need to plot a graph, the first few rows are: + + a and z, Error/epsilon, Iterations required + + 0.01, 9723.14, 4726 + 1.0099, 9.54818, 87 + 2.0098, 3.84777, 40 + 3.0097, 0.728358, 25 + 4.0096, 2.39712, 21 + 5.0095, 0.233263, 16 + +So it's pretty clear that this fraction shouldn't be used for small values +of a and z. + +[h4 reference] +[#test_data_reference] + +Most of this tool has been described already in the examples above, we'll +just add the following notes on the non-member functions: + + template + parameter_info make_random_param(T start_range, T end_range, int n_points); + +Tells class test_data to test /n_points/ random values in the range +[start_range,end_range]. + + template + parameter_info make_periodic_param(T start_range, T end_range, int n_points); + +Tells class test_data to test /n_points/ evenly spaced values in the range +[start_range,end_range]. + + template + parameter_info make_power_param(T basis, int start_exponent, int end_exponent); + +Tells class test_data to test points of the form ['basis + R * 2[super expon]] for each +/expon/ in the range [start_exponent, end_exponent], and /R/ a random number in \[0.5, 1\]. + + template + bool get_user_parameter_info(parameter_info& info, const char* param_name); + +Prompts the user for the parameter range and form to use. + +Finally, if we don't want the parameter to be included in the output, we can tell +test_data by setting it a "dummy parameter": + + parameter_info p = make_random_param(2.0, 5.0, 10); + p.type |= dummy_param; + +This is useful when the functor used transforms the parameter in some way +before passing it to the function under test, usually the functor will then +return both the transformed input and the result in a tuple, so there's no +need for the original pseudo-parameter to be included in program output. + +[endsect][/section:test_data Graphing, Profiling, and Generating Test Data for Special Functions] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/test_html4_symbols.qbk b/doc/sf_and_dist/test_html4_symbols.qbk new file mode 100644 index 000000000..f19684d10 --- /dev/null +++ b/doc/sf_and_dist/test_html4_symbols.qbk @@ -0,0 +1,144 @@ +[section:test_HTML_4_Symbols test HTML4 symbols] +[/ Examples of using all the Greek and Math symbols defined in HTML4_symbols.qbk] +[/ See http://www.htmlhelp.com/reference/html40/entities/symbols.html] + +[/ Also some miscellaneous math charaters added to this list - see the end.] + +[/ To use, enclose the template name in square brackets.] + + +[fnof], +[Alpha], +[Beta], +[Gamma], +[Delta], +[Epsilon], +[Zeta], +[Eta], +[Theta], +[Iota], +[Kappa], +[Lambda], +[Mu], +[Nu], +[Xi], +[Omicron], +[Pi], +[Rho], +[Sigma], +[Tau], +[Upsilon], +[Phi], +[Chi], +[Psi], +[Omega], +[alpha], +[beta], +[gamma], +[delta], +[epsilon], +[zeta], +[eta], +[theta], +[iota], +[kappa], +[lambda], +[mu], +[nu], +[xi], +[omicron], +[pi], +[rho], +[sigmaf], +[sigma], +[tau], +[upsilon], +[phi], +[chi], +[psi], +[omega], +[thetasym], +[upsih], +[piv], +[bull], +[hellip], +[prime], +[Prime], +[oline], +[frasl], +[weierp], +[image], +[real], +[trade], +[alefsym], +[larr], +[uarr], +[rarr], +[darr], +[harr], +[crarr], +[lArr], +[uArr], +[rArr], +[dArr], +[hArr], +[forall], +[part], +[exist], +[empty], +[nabla], +[isin], +[notin], +[ni], +[prod], +[sum], +[minus], +[lowast], +[radic], +[prop], +[infin], +[ang], +[and], +[or], +[cap], +[cup], +[int], +[there4], +[sim], +[cong], +[asymp], +[ne], +[equiv], +[le], +[ge], +[subset], +[superset], +[nsubset], +[sube], +[supe], +[oplus], +[otimes], +[perp], +[sdot], +[lceil], +[rceil], +[lfloor], +[rfloor], +[lang], +[rang], +[loz], +[spades], +[clubs], +[hearts], +[diams], + +[endsect] + +[/ test_MTML4_symbols.qbk + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + diff --git a/doc/sf_and_dist/tgamma.qbk b/doc/sf_and_dist/tgamma.qbk new file mode 100644 index 000000000..a660eaf0a --- /dev/null +++ b/doc/sf_and_dist/tgamma.qbk @@ -0,0 +1,202 @@ +[section:tgamma Gamma] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` tgamma(T z); + + template + ``__sf_result`` tgamma(T z, const ``__Policy``&); + + template + ``__sf_result`` tgamma1pm1(T dz); + + template + ``__sf_result`` tgamma1pm1(T dz, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + + template + ``__sf_result`` tgamma(T z); + + template + ``__sf_result`` tgamma(T z, const ``__Policy``&); + +Returns the "true gamma" (hence name tgamma) of value z: + +[equation gamm1] + +[$../graphs/gamma.png] + +[optional_policy] + +There are effectively two versions of the [@http://en.wikipedia.org/wiki/Gamma_function tgamma] +function internally: a fully +generic version that is slow, but reasonably accurate, and a much more +efficient approximation that is used where the number of digits in the significand +of T correspond to a certain __lanczos. In practice any built in +floating point type you will encounter has an appropriate __lanczos +defined for it. It is also possible, given enough machine time, to generate +further __lanczos's using the program libs/math/tools/lanczos_generator.cpp. + +The return type of this function is computed using the __arg_pomotion_rules: +the result is `double` when T is an integer type, and T otherwise. + + template + ``__sf_result`` tgamma1pm1(T dz); + + template + ``__sf_result`` tgamma1pm1(T dz, const ``__Policy``&); + +Returns `tgamma(dz + 1) - 1`. Internally the implementation does not make +use of the addition and subtraction implied by the definition, leading to +accurate results even for very small `dz`. However, the implementation is +capped to either 35 digit accuracy, or to the precision of the __lanczos +associated with type T, whichever is more accurate. + +The return type of this function is computed using the __arg_pomotion_rules: +the result is `double` when T is an integer type, and T otherwise. + +[optional_policy] + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types, +along with comparisons to the __gsl, __glibc, __hpc and __cephes libraries. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table +[[Significand Size] [Platform and Compiler] [Factorials and Half factorials] [Values Near Zero] [Values Near 1 or 2] [Values Near a Negative Pole]] +[[53] [Win32 Visual C++ 8] +[Peak=1.9 Mean=0.7 + +(GSL=3.9) + +(__cephes=3.0)] +[Peak=2.0 Mean=1.1 + +(GSL=4.5) + +(__cephes=1)] +[Peak=2.0 Mean=1.1 + +(GSL=7.9) + +(__cephes=1.0)] +[Peak=2.6 Mean=1.3 + +(GSL=2.5) + +(__cephes=2.7)] ] +[[64] [Linux IA32 / GCC] +[Peak=300 Mean=49.5 + +(__glibc Peak=395 Mean=89)] +[Peak=3.0 Mean=1.4 + +(__glibc Peak=11 Mean=3.3)] +[Peak=5.0 Mean=1.8 + +(__glibc Peak=0.92 Mean=0.2)] +[Peak=157 Mean=65 + +(__glibc Peak=205 Mean=108)] ] +[[64] [Linux IA64 / GCC] +[__glibc Peak 2.8 Mean=0.9 + +(__glibc Peak 0.7)] +[Peak=4.8 Mean=1.5 + +(__glibc Peak 0)] +[Peak=4.8 Mean=1.5 + +(__glibc Peak 0)] + +[Peak=5.0 Mean=1.7 +(__glibc Peak 0)] ] +[[113] [HPUX IA64, aCC A.06.06] +[Peak=2.5 Mean=1.1 + +(__hpc Peak 0)] +[Peak=3.5 Mean=1.7 + +(__hpc Peak 0)] +[Peak=3.5 Mean=1.6 + +(__hpc Peak 0)] +[Peak=5.2 Mean=1.92 + +(__hpc Peak 0)] ] +] + +[h4 Testing] + +The gamma is relatively easy to test: factorials and half-integer factorials +can be calculated exactly by other means and compared with the gamma function. +In addition, some accuracy tests in known tricky areas were computed at high precision +using the generic version of this function. + +The function `tgamma1pm1` is tested against values calculated very naively +using the formula `tgamma(1+dz)-1` with a lanczos approximation accurate +to around 100 decimal digits. + +[h4 Implementation] + +The generic version of the `tgamma` function is implemented by combining the series and +continued fraction representations for the incomplete gamma function: + +[equation gamm2] + +where /l/ is an arbitrary integration limit: choosing [^l = max(10, a)] +seems to work fairly well. + +For types of known precision the __lanczos is used, a traits class +`boost::math::lanczos::lanczos_traits` maps type T to an appropriate +approximation. + +For z in the range -20 < z < 1 then recursion is used to shift to z > 1 via: + +[equation gamm3] + +For very small z, this helps to preserve the identity: + +[equation gamm4] + +For z < -20 the reflection formula: + +[equation gamm5] + +is used. Particular care has to be taken to evaluate the `z * sin([pi][space] * z)` part: +a special routine is used to reduce z prior to multiplying by [pi][space] to ensure that the +result in is the range [0, [pi]/2]. Without this an excessive amount of error occurs +in this region (which is hard enough already, as the rate of change near a negative pole +is /exceptionally/ high). + +Finally if the argument is a small integer then table lookup of the factorial +is used. + +The function `tgamma1pm1` is implemented using rational approximations [jm_rationals] in the +region `-0.5 < dz < 2`. These are the same approximations (and internal routines) +that are used for __lgamma, and so aren't detailed further here. The result of +the approximation is `log(tgamma(dz+1))` which can fed into __expm1 to give +the desired result. Outside the range `-0.5 < dz < 2` then the naive formula +`tgamma1pm1(dz) = tgamma(dz+1)-1` can be used directly. + +[endsect][/section:tgamma The Gamma Function] +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf_and_dist/thread_safety.qbk b/doc/sf_and_dist/thread_safety.qbk new file mode 100644 index 000000000..ef825854d --- /dev/null +++ b/doc/sf_and_dist/thread_safety.qbk @@ -0,0 +1,30 @@ +[section:threads Thread Safety] + +The library is fully thread safe and re-entrant provided the function +and class templates in the library are instantiated with +built-in floating point types: i.e. the types `float`, `double` +and `long double`. + +However, the library [*is not thread safe] when +used with user-defined (i.e. class type) numeric types. + +The reason for the latter limitation is the need to +initialise symbolic constants using constructs such as: + + static const T coefficient_array = { ... list of values ... }; + +Which is always thread safe when T is a built-in floating point type, +but not when T is a user defined type: as in this case there +is a need for T's constructors to be run, leading to potential +race conditions. + +This limitation may be addressed in a future release. + +[endsect] [/section:threads Thread Safety] + +[/ + Copyright 2006 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf_and_dist/win32_nmake.mak b/doc/sf_and_dist/win32_nmake.mak new file mode 100644 index 000000000..be932f32d --- /dev/null +++ b/doc/sf_and_dist/win32_nmake.mak @@ -0,0 +1,73 @@ +# Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. +# Distributed under the Boost Software License, Version 1.0. +# (See accompanying file LICENSE_1_0.txt or copy at +# http://www.boost.org/LICENSE_1_0.txt). +# +# Example makefile that builds the docs. +# Note that all the following paths will have to be changed to match +# your actual installation paths. +# + +# Path to quickbook executable: +QB="C:/download/open/xml/bin/quickbook.exe" + +# Path to xsltproc: +XSLTPROC="C:/download/open/xml/bin/xsltproc-win32/xsltproc.exe" + +# Path to Boost Trunc: +BOOST=c:/data/boost/boost/trunk + +# Path to FO processor (XEP): +FO=C:/Progra~1/xep/xep.bat + +# Configuration options: +COMMON_XSL_PARAM=--stringparam admon.graphics "1" --stringparam body.start.indent "0pt" --stringparam chunk.first.sections "1" --stringparam chunk.section.depth "10" --stringparam fop.extensions "0" --stringparam generate.section.toc.level "10" --stringparam html.stylesheet "../../../../../../trunk/doc/html/boostbook.css" --stringparam navig.graphics "1" --stringparam page.margin.inner "0.5in" --stringparam page.margin.outer "0.5in" --stringparam paper.type "A4" --stringparam toc.max.depth "4" --stringparam toc.section.depth "10" --stringparam xep.extensions "1" +PDF_XSL_PARAM=--stringparam admon.graphics.extension ".svg" --stringparam use.role.for.mediaobject 1 --stringparam preferred.mediaobject.role print --stringparam admon.graphics.path "../html/images/" +HTML_XSL_PARAM= +PROJECT_NAME=math + +all : pdf html + +pdf : pdf/$(PROJECT_NAME).pdf +html : html/index.html + +xml/$(PROJECT_NAME).xml : + -mkdir xml + $(QB) --output-file=xml\$(PROJECT_NAME).xml $(PROJECT_NAME).qbk + +xml/$(PROJECT_NAME).docbook : xml\$(PROJECT_NAME).xml xml/catalog.xml + set XML_CATALOG_FILES=xml/catalog.xml + $(XSLTPROC) $(COMMON_XSL_PARAM) --xinclude -o "xml\$(PROJECT_NAME).docbook" "$(BOOST)\tools\boostbook\xsl\docbook.xsl" "xml\$(PROJECT_NAME).xml" + +xml/$(PROJECT_NAME).fo : xml\$(PROJECT_NAME).docbook xml/catalog.xml + set XML_CATALOG_FILES=xml/catalog.xml + $(XSLTPROC) $(COMMON_XSL_PARAM) $(PDF_XSL_PARAM) --xinclude -o "xml\$(PROJECT_NAME).fo" "$(BOOST)\tools\boostbook\xsl\fo.xsl" "xml\$(PROJECT_NAME).docbook" + +pdf/$(PROJECT_NAME).pdf : xml\$(PROJECT_NAME).fo + -mkdir pdf + set JAVA_HOME=C:/PROGRA~1/Java/j2re1.4.2_12 + call $(FO) xml\$(PROJECT_NAME).fo pdf\$(PROJECT_NAME).pdf + +html/index.html : xml\$(PROJECT_NAME).fo + -mkdir html + set XML_CATALOG_FILES=xml/catalog.xml + $(XSLTPROC) $(COMMON_XSL_PARAM) $(HTML_XSL_PARAM) --xinclude -o "html/" "$(BOOST)\tools\boostbook\xsl\html.xsl" "xml\$(PROJECT_NAME).docbook" + +xml/catalog.xml : + @echo < + + + + + + +<< + + + + + + diff --git a/doc/svg-admon/caution.svg b/doc/svg-admon/caution.svg new file mode 100644 index 000000000..4bd586a08 --- /dev/null +++ b/doc/svg-admon/caution.svg @@ -0,0 +1,68 @@ + + + + + + Attenzione + + + + pulsante + + + + + Open Clip Art Library + + + + + Architetto Francesco Rollandin + + + + + Architetto Francesco Rollandin + + + + image/svg+xml + + + en + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/doc/svg-admon/home.svg b/doc/svg-admon/home.svg new file mode 100644 index 000000000..e803a3178 --- /dev/null +++ b/doc/svg-admon/home.svg @@ -0,0 +1,26 @@ + + + + + + + + +]> + + + + + + + + + + + + + + diff --git a/doc/svg-admon/important.svg b/doc/svg-admon/important.svg new file mode 100644 index 000000000..dd84f3fe3 --- /dev/null +++ b/doc/svg-admon/important.svg @@ -0,0 +1,25 @@ + + + + + + + + +]> + + + + + + + + + + + + + + + diff --git a/doc/svg-admon/next.svg b/doc/svg-admon/next.svg new file mode 100644 index 000000000..75fa83ed8 --- /dev/null +++ b/doc/svg-admon/next.svg @@ -0,0 +1,19 @@ + + + + + + +]> + + + + + + + + + + + diff --git a/doc/svg-admon/note.svg b/doc/svg-admon/note.svg new file mode 100644 index 000000000..648299d26 --- /dev/null +++ b/doc/svg-admon/note.svg @@ -0,0 +1,33 @@ + + + + + + + + + + + + +]> + + + + + + + + + + + + + + + + + + + diff --git a/doc/svg-admon/prev.svg b/doc/svg-admon/prev.svg new file mode 100644 index 000000000..6d88ffdd0 --- /dev/null +++ b/doc/svg-admon/prev.svg @@ -0,0 +1,19 @@ + + + + + + +]> + + + + + + + + + + + diff --git a/doc/svg-admon/tip.svg b/doc/svg-admon/tip.svg new file mode 100644 index 000000000..cd437a5e8 --- /dev/null +++ b/doc/svg-admon/tip.svg @@ -0,0 +1,84 @@ + + + + + + lamp + + + + office + + lamp + + + + + Open Clip Art Library + + + + + Sergio Luiz Araujo Silva + + + + + Public Domain + + + set 2005 + image/svg+xml + + + en + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/doc/svg-admon/up.svg b/doc/svg-admon/up.svg new file mode 100644 index 000000000..d31aa9c80 --- /dev/null +++ b/doc/svg-admon/up.svg @@ -0,0 +1,19 @@ + + + + + + +]> + + + + + + + + + + + diff --git a/doc/svg-admon/warning.svg b/doc/svg-admon/warning.svg new file mode 100644 index 000000000..fc8d7484c --- /dev/null +++ b/doc/svg-admon/warning.svg @@ -0,0 +1,23 @@ + + + + + + + + +]> + + + + + + + + + + + + + diff --git a/example/Jamfile.v2 b/example/Jamfile.v2 new file mode 100644 index 000000000..2d4917225 --- /dev/null +++ b/example/Jamfile.v2 @@ -0,0 +1,79 @@ +# \math_toolkit\libs\math\example\jamfile.v2 +# Runs statistics examples +# Copyright 2007 John Maddock and Paul A. Bristow. +# Distributed under the Boost Software License, Version 1.0. +# (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +# bring in the rules for testing +import testing ; + +project + : requirements + gcc:-Wno-missing-braces + darwin:-Wno-missing-braces + acc:+W2068,2461,2236,4070 + intel:-Qwd264,239 + msvc:all + msvc:on + msvc:/wd4996 + msvc:/wd4512 + msvc:/wd4610 + msvc:/wd4510 + msvc:/wd4127 + msvc:/wd4701 + msvc:/wd4127 + msvc:/wd4305 + ../../.. + ; + +run binomial_confidence_limits.cpp ; +run binomial_example3.cpp ; +run binomial_sample_sizes.cpp ; +run binomial_example_nag.cpp ; +run c_error_policy_example.cpp ; +run chi_square_std_dev_test.cpp ; +run distribution_construction.cpp ; +run error_handling_example.cpp ; +run error_policies_example.cpp ; +run error_policy_example.cpp ; +run f_test.cpp ; +run neg_binom_confidence_limits.cpp ; +run neg_binomial_sample_sizes.cpp ; +# run negative_binomial_construction_examples.cpp ; delete for now +run negative_binomial_example1.cpp ; +run negative_binomial_example2.cpp ; +# run negative_binomial_example3.cpp ; +run policy_eg_1.cpp ; +run policy_eg_2.cpp ; +run policy_eg_3.cpp ; +run policy_eg_4.cpp ; +run policy_eg_5.cpp ; +run policy_eg_6.cpp ; +run policy_eg_7.cpp ; +run policy_eg_8.cpp ; +run policy_eg_9.cpp ; +run policy_eg_10.cpp ; +run policy_ref_snip1.cpp ; +run policy_ref_snip10.cpp ; +run policy_ref_snip11.cpp ; +run policy_ref_snip12.cpp ; +run policy_ref_snip13.cpp ; +run policy_ref_snip2.cpp ; +run policy_ref_snip3.cpp ; +run policy_ref_snip4.cpp ; +run policy_ref_snip5.cpp ; +run policy_ref_snip6.cpp ; +run policy_ref_snip7.cpp ; +run policy_ref_snip8.cpp ; +run policy_ref_snip9.cpp ; +#run statistics_functions_example1.cpp ; +run students_t_example1.cpp ; +run students_t_example2.cpp ; +run students_t_example3.cpp ; +run students_t_single_sample.cpp ; +run students_t_two_samples.cpp ; + + + + + diff --git a/example/binomial_coinflip_example.cpp b/example/binomial_coinflip_example.cpp new file mode 100644 index 000000000..939b44624 --- /dev/null +++ b/example/binomial_coinflip_example.cpp @@ -0,0 +1,237 @@ +// Copyright Paul A. 2007 +// Copyright John Maddock 2006 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Simple example of computing probabilities and quantiles for +// a Bernoulli random variable representing the flipping of a coin. + +// http://mathworld.wolfram.com/CoinTossing.html +// http://en.wikipedia.org/wiki/Bernoulli_trial +// Weisstein, Eric W. "Dice." From MathWorld--A Wolfram Web Resource. +// http://mathworld.wolfram.com/Dice.html +// http://en.wikipedia.org/wiki/Bernoulli_distribution +// http://mathworld.wolfram.com/BernoulliDistribution.html +// +// An idealized coin consists of a circular disk of zero thickness which, +// when thrown in the air and allowed to fall, will rest with either side face up +// ("heads" H or "tails" T) with equal probability. A coin is therefore a two-sided die. +// Despite slight differences between the sides and nonzero thickness of actual coins, +// the distribution of their tosses makes a good approximation to a p==1/2 Bernoulli distribution. + +//[binomial_coinflip_example1 + +/*`An example of a [@http://en.wikipedia.org/wiki/Bernoulli_process Bernoulli process] +is coin flipping. +A variable in such a sequence may be called a Bernoulli variable. + +This example shows using the Binomial distribution to predict the probability +of heads and tails when throwing a coin. + +The number of correct answers (say heads), +X, is distributed as a binomial random variable +with binomial distribution parameters number of trials (flips) n = 10 and probability (success_fraction) of getting a head p = 0.5 (a 'fair' coin). + +(Our coin is assumed fair, but we could easily change the success_fraction parameter p +from 0.5 to some other value to simulate an unfair coin, +say 0.6 for one with chewing gum on the tail, +so it is more likely to fall tails down and heads up). + +First we need some includes and using statements to be able to use the binomial distribution, some std input and output, and get started: +*/ + +#include + using boost::math::binomial; + +#include + using std::cout; using std::endl; using std::left; +#include + using std::setw; + +int main() +{ + cout << "Using Binomial distribution to predict how many heads and tails." << endl; + try + { +/*` +See note [link coinflip_eg_catch with the catch block] +about why a try and catch block is always a good idea. + +First, construct a binomial distribution with parameters success_fraction +1/2, and how many flips. +*/ + const double success_fraction = 0.5; // = 50% = 1/2 for a 'fair' coin. + int flips = 10; + binomial flip(flips, success_fraction); + + cout.precision(4); +/*` + Then some examples of using Binomial moments (and echoing the parameters). +*/ + cout << "From " << flips << " one can expect to get on average " + << mean(flip) << " heads (or tails)." << endl; + cout << "Mode is " << mode(flip) << endl; + cout << "Standard deviation is " << standard_deviation(flip) << endl; + cout << "So about 2/3 will lie within 1 standard deviation and get between " + << ceil(mean(flip) - standard_deviation(flip)) << " and " + << floor(mean(flip) + standard_deviation(flip)) << " correct." << endl; + cout << "Skewness is " << skewness(flip) << endl; + // Skewness of binomial distributions is only zero (symmetrical) + // if success_fraction is exactly one half, + // for example, when flipping 'fair' coins. + cout << "Skewness if success_fraction is " << flip.success_fraction() + << " is " << skewness(flip) << endl << endl; // Expect zero for a 'fair' coin. +/*` +Now we show a variety of predictions on the probability of heads: +*/ + cout << "For " << flip.trials() << " coin flips: " << endl; + cout << "Probability of getting no heads is " << pdf(flip, 0) << endl; + cout << "Probability of getting at least one head is " << 1. - pdf(flip, 0) << endl; +/*` +When we want to calculate the probability for a range or values we can sum the PDF's: +*/ + cout << "Probability of getting 0 or 1 heads is " + << pdf(flip, 0) + pdf(flip, 1) << endl; // sum of exactly == probabilities +/*` +Or we can use the cdf. +*/ + cout << "Probability of getting 0 or 1 (<= 1) heads is " << cdf(flip, 1) << endl; + cout << "Probability of getting 9 or 10 heads is " << pdf(flip, 9) + pdf(flip, 10) << endl; +/*` +Note that using +*/ + cout << "Probability of getting 9 or 10 heads is " << 1. - cdf(flip, 8) << endl; +/*` +is less accurate than using the complement +*/ + cout << "Probability of getting 9 or 10 heads is " << cdf(complement(flip, 8)) << endl; +/*` +Since the subtraction may involve +[@http://docs.sun.com/source/806-3568/ncg_goldberg.html cancellation error], +where as `cdf(complement(flip, 8))` +does not use such a subtraction internally, and so does not exhibit the problem. + +To get the probability for a range of heads, we can either add the pdfs for each number of heads +*/ + cout << "Probability of between 4 and 6 heads (4 or 5 or 6) is " + // P(X == 4) + P(X == 5) + P(X == 6) + << pdf(flip, 4) + pdf(flip, 5) + pdf(flip, 6) << endl; +/*` +But this is probably less efficient than using the cdf +*/ + cout << "Probability of between 4 and 6 heads (4 or 5 or 6) is " + // P(X <= 6) - P(X <= 3) == P(X < 4) + << cdf(flip, 6) - cdf(flip, 3) << endl; +/*` +Certainly for a bigger range like, 3 to 7 +*/ + cout << "Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is " + // P(X <= 7) - P(X <= 2) == P(X < 3) + << cdf(flip, 7) - cdf(flip, 2) << endl; + cout << endl; + +/*` +Finally, print two tables of probability for the /exactly/ and /at least/ a number of heads. +*/ + // Print a table of probability for the exactly a number of heads. + cout << "Probability of getting exactly (==) heads" << endl; + for (int successes = 0; successes <= flips; successes++) + { // Say success means getting a head (or equally success means getting a tail). + double probability = pdf(flip, successes); + cout << left << setw(2) << successes << " " << setw(10) + << probability << " or 1 in " << 1. / probability + << ", or " << probability * 100. << "%" << endl; + } // for i + cout << endl; + + // Tabulate the probability of getting between zero heads and 0 upto 10 heads. + cout << "Probability of getting upto (<=) heads" << endl; + for (int successes = 0; successes <= flips; successes++) + { // Say success means getting a head + // (equally success could mean getting a tail). + double probability = cdf(flip, successes); // P(X <= heads) + cout << setw(2) << successes << " " << setw(10) << left + << probability << " or 1 in " << 1. / probability << ", or " + << probability * 100. << "%"<< endl; + } // for i +/*` +The last (0 to 10 heads) must, of course, be 100% probability. +*/ + } + catch(const std::exception& e) + { + // + /*` + [#coinflip_eg_catch] + It is always essential to include try & catch blocks because + default policies are to throw exceptions on arguments that + are out of domain or cause errors like numeric-overflow. + + Lacking try & catch blocks, the program will abort, whereas the + message below from the thrown exception will give some helpful + clues as to the cause of the problem. + */ + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } +//] [binomial_coinflip_example1] + return 0; +} // int main() + +// Output: + +//[binomial_coinflip_example_output +/*` + +[pre +Using Binomial distribution to predict how many heads and tails. +From 10 one can expect to get on average 5 heads (or tails). +Mode is 5 +Standard deviation is 1.581 +So about 2/3 will lie within 1 standard deviation and get between 4 and 6 correct. +Skewness is 0 +Skewness if success_fraction is 0.5 is 0 + +For 10 coin flips: +Probability of getting no heads is 0.0009766 +Probability of getting at least one head is 0.999 +Probability of getting 0 or 1 heads is 0.01074 +Probability of getting 0 or 1 (<= 1) heads is 0.01074 +Probability of getting 9 or 10 heads is 0.01074 +Probability of getting 9 or 10 heads is 0.01074 +Probability of getting 9 or 10 heads is 0.01074 +Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6562 +Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6563 +Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is 0.8906 + +Probability of getting exactly (==) heads +0 0.0009766 or 1 in 1024, or 0.09766% +1 0.009766 or 1 in 102.4, or 0.9766% +2 0.04395 or 1 in 22.76, or 4.395% +3 0.1172 or 1 in 8.533, or 11.72% +4 0.2051 or 1 in 4.876, or 20.51% +5 0.2461 or 1 in 4.063, or 24.61% +6 0.2051 or 1 in 4.876, or 20.51% +7 0.1172 or 1 in 8.533, or 11.72% +8 0.04395 or 1 in 22.76, or 4.395% +9 0.009766 or 1 in 102.4, or 0.9766% +10 0.0009766 or 1 in 1024, or 0.09766% + +Probability of getting upto (<=) heads +0 0.0009766 or 1 in 1024, or 0.09766% +1 0.01074 or 1 in 93.09, or 1.074% +2 0.05469 or 1 in 18.29, or 5.469% +3 0.1719 or 1 in 5.818, or 17.19% +4 0.377 or 1 in 2.653, or 37.7% +5 0.623 or 1 in 1.605, or 62.3% +6 0.8281 or 1 in 1.208, or 82.81% +7 0.9453 or 1 in 1.058, or 94.53% +8 0.9893 or 1 in 1.011, or 98.93% +9 0.999 or 1 in 1.001, or 99.9% +10 1 or 1 in 1, or 100% +] +*/ +//][/binomial_coinflip_example_output] diff --git a/example/binomial_confidence_limits.cpp b/example/binomial_confidence_limits.cpp new file mode 100644 index 000000000..3e6f46fb8 --- /dev/null +++ b/example/binomial_confidence_limits.cpp @@ -0,0 +1,159 @@ +// Copyright John Maddock 2006 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +#include +#include + +void confidence_limits_on_frequency(unsigned trials, unsigned successes) +{ + // + // trials = Total number of trials. + // successes = Total number of observed successes. + // + // Calculate confidence limits for an observed + // frequency of occurrence that follows a binomial + // distribution. + // + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "___________________________________________\n" + "2-Sided Confidence Limits For Success Ratio\n" + "___________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << trials << "\n"; + cout << setw(40) << left << "Number of successes" << "= " << successes << "\n"; + cout << setw(40) << left << "Sample frequency of occurrence" << "= " << double(successes) / trials << "\n"; + // + // Define a table of significance levels: + // + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Print table header: + // + cout << "\n\n" + "_______________________________________________________________________\n" + "Confidence Lower CP Upper CP Lower JP Upper JP\n" + " Value (%) Limit Limit Limit Limit\n" + "_______________________________________________________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // Calculate Clopper Pearson bounds: + double l = binomial_distribution<>::find_lower_bound_on_p(trials, successes, alpha[i]/2); + double u = binomial_distribution<>::find_upper_bound_on_p(trials, successes, alpha[i]/2); + // Print Clopper Pearson Limits: + cout << fixed << setprecision(5) << setw(15) << right << l; + cout << fixed << setprecision(5) << setw(15) << right << u; + // Calculate Jeffreys Prior Bounds: + l = binomial_distribution<>::find_lower_bound_on_p(trials, successes, alpha[i]/2, binomial_distribution<>::jeffreys_prior_interval); + u = binomial_distribution<>::find_upper_bound_on_p(trials, successes, alpha[i]/2, binomial_distribution<>::jeffreys_prior_interval); + // Print Jeffreys Prior Limits: + cout << fixed << setprecision(5) << setw(15) << right << l; + cout << fixed << setprecision(5) << setw(15) << right << u << std::endl; + } + cout << endl; +} // void confidence_limits_on_frequency() + +int main() +{ + confidence_limits_on_frequency(20, 4); + confidence_limits_on_frequency(200, 40); + confidence_limits_on_frequency(2000, 400); + + return 0; +} // int main() + +/* + +------ Build started: Project: binomial_confidence_limits, Configuration: Debug Win32 ------ +Compiling... +binomial_confidence_limits.cpp +Linking... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\binomial_confidence_limits.exe" +___________________________________________ +2-Sided Confidence Limits For Success Ratio +___________________________________________ + +Number of Observations = 20 +Number of successes = 4 +Sample frequency of occurrence = 0.2 + + +_______________________________________________________________________ +Confidence Lower CP Upper CP Lower JP Upper JP + Value (%) Limit Limit Limit Limit +_______________________________________________________________________ + 50.000 0.12840 0.29588 0.14974 0.26916 + 75.000 0.09775 0.34633 0.11653 0.31861 + 90.000 0.07135 0.40103 0.08734 0.37274 + 95.000 0.05733 0.43661 0.07152 0.40823 + 99.000 0.03576 0.50661 0.04655 0.47859 + 99.900 0.01905 0.58632 0.02634 0.55960 + 99.990 0.01042 0.64997 0.01530 0.62495 + 99.999 0.00577 0.70216 0.00901 0.67897 + +___________________________________________ +2-Sided Confidence Limits For Success Ratio +___________________________________________ + +Number of Observations = 200 +Number of successes = 40 +Sample frequency of occurrence = 0.2000000 + + +_______________________________________________________________________ +Confidence Lower CP Upper CP Lower JP Upper JP + Value (%) Limit Limit Limit Limit +_______________________________________________________________________ + 50.000 0.17949 0.22259 0.18190 0.22001 + 75.000 0.16701 0.23693 0.16934 0.23429 + 90.000 0.15455 0.25225 0.15681 0.24956 + 95.000 0.14689 0.26223 0.14910 0.25951 + 99.000 0.13257 0.28218 0.13468 0.27940 + 99.900 0.11703 0.30601 0.11902 0.30318 + 99.990 0.10489 0.32652 0.10677 0.32366 + 99.999 0.09492 0.34485 0.09670 0.34197 + +___________________________________________ +2-Sided Confidence Limits For Success Ratio +___________________________________________ + +Number of Observations = 2000 +Number of successes = 400 +Sample frequency of occurrence = 0.2000000 + + +_______________________________________________________________________ +Confidence Lower CP Upper CP Lower JP Upper JP + Value (%) Limit Limit Limit Limit +_______________________________________________________________________ + 50.000 0.19382 0.20638 0.19406 0.20613 + 75.000 0.18965 0.21072 0.18990 0.21047 + 90.000 0.18537 0.21528 0.18561 0.21503 + 95.000 0.18267 0.21821 0.18291 0.21796 + 99.000 0.17745 0.22400 0.17769 0.22374 + 99.900 0.17150 0.23079 0.17173 0.23053 + 99.990 0.16658 0.23657 0.16681 0.23631 + 99.999 0.16233 0.24169 0.16256 0.24143 + +*/ + + + diff --git a/example/binomial_example3.cpp b/example/binomial_example3.cpp new file mode 100644 index 000000000..2b0509070 --- /dev/null +++ b/example/binomial_example3.cpp @@ -0,0 +1,91 @@ +// Copyright Paul A. 2006 +// Copyright John Maddock 2006 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Simple example of computing probabilities for a binomial random variable. +// Replication of source nag_binomial_dist (g01bjc). + +// Shows how to replace NAG C library calls by Boost Math Toolkit C++ calls. + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +using std::cout; +using std::endl; +using std::ios; +using std::showpoint; +#include +using std::fixed; +using std::setw; +#include + +int main() +{ + cout << "Example 3 of using the binomial distribution to replicate a NAG library call." << endl; + using boost::math::binomial_distribution; + + // This replicates the computation of the examples of using nag-binomial_dist + // using g01bjc in section g01 Sample Calculations on Statistical Data. + // http://www.nag.co.uk/numeric/cl/manual/pdf/G01/g01bjc.pdf + // Program results section 8.3 page 3.g01bjc.3 + //8.2. Program Data + //g01bjc Example Program Data + //4 0.50 2 : n, p, k + //19 0.44 13 + //100 0.75 67 + //2000 0.33 700 + //8.3. Program Results + //g01bjc Example Program Results + //n p k plek pgtk peqk + //4 0.500 2 0.68750 0.31250 0.37500 + //19 0.440 13 0.99138 0.00862 0.01939 + //100 0.750 67 0.04460 0.95540 0.01700 + //2000 0.330 700 0.97251 0.02749 0.00312 + + cout.setf(ios::showpoint); // Trailing zeros to show significant decimal digits. + cout.precision(5); // Should be able to calculate this? + cout << fixed; + // Binomial distribution. + + // Note that cdf(dist, k) is equivalent to NAG library plek probability of <= k + // cdf(complement(dist, k)) is equivalent to NAG library pgtk probability of > k + // pdf(dist, k) is equivalent to NAG library peqk probability of == k + + cout << " n p k plek pgtk peqk " << endl; + binomial_distribution<>my_dist(4, 0.5); + cout << setw(4) << (int)my_dist.trials() << " " << my_dist.success_fraction() << " "<< 2 << " " + << cdf(my_dist, 2) << " " << cdf(complement(my_dist, 2)) << " " << pdf(my_dist, 2) << endl; + binomial_distribution<>two(19, 0.440); + cout << setw(4) << (int)two.trials() << " " << two.success_fraction() << " " << 13 << " " + << cdf(two, 13) << " " << cdf(complement(two, 13)) << " " << pdf(two, 13) << endl; + binomial_distribution<>three(100, 0.750); + cout << setw(4) << (int)three.trials() << " " << three.success_fraction() << " " << 67 << " " + << cdf(three, 67) << " " << cdf(complement(three, 67)) << " " << pdf(three, 67) << endl; + binomial_distribution<>four(2000, 0.330); + cout << setw(4) << (int)four.trials() << " " << four.success_fraction() << " " << 700 << " " + << cdf(four, 700) << " " << cdf(complement(four, 700)) << " " << pdf(four, 700) << endl; + + return 0; +} // int main() + +/* + +Autorun "i:\boost-sandbox\math_toolkit\libs\math\test\msvc80\debug\binomial_example3.exe" +Example 3 of using the binomial distribution. + n p k plek pgtk peqk + 4 0.50000 2 0.68750 0.31250 0.37500 + 19 0.44000 13 0.99138 0.00862 0.01939 + 100 0.75000 67 0.04460 0.95540 0.01700 +2000 0.33000 700 0.97251 0.02749 0.00312 + + + + */ + diff --git a/example/binomial_example_nag.cpp b/example/binomial_example_nag.cpp new file mode 100644 index 000000000..1020a8cf5 --- /dev/null +++ b/example/binomial_example_nag.cpp @@ -0,0 +1,91 @@ +// Copyright Paul A. 2007 +// Copyright John Maddock 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Simple example of computing probabilities for a binomial random variable. +// Replication of source nag_binomial_dist (g01bjc). + +// Shows how to replace NAG C library calls by Boost Math Toolkit C++ calls. +// Note that the default policy does not replicate the way that NAG +// library calls handle 'bad' arguments, but you can define policies that do, +// as well as other policies that may suit your application even better. +// See the examples of changing default policies for details. + +#include + +#include + using std::cout; using std::endl; using std::ios; using std::showpoint; +#include + using std::fixed; using std::setw; + +int main() +{ + cout << "Using the binomial distribution to replicate a NAG library call." << endl; + using boost::math::binomial_distribution; + + // This replicates the computation of the examples of using nag-binomial_dist + // using g01bjc in section g01 Somple Calculations on Statistical Data. + // http://www.nag.co.uk/numeric/cl/manual/pdf/G01/g01bjc.pdf + // Program results section 8.3 page 3.g01bjc.3 + //8.2. Program Data + //g01bjc Example Program Data + //4 0.50 2 : n, p, k + //19 0.44 13 + //100 0.75 67 + //2000 0.33 700 + //8.3. Program Results + //g01bjc Example Program Results + //n p k plek pgtk peqk + //4 0.500 2 0.68750 0.31250 0.37500 + //19 0.440 13 0.99138 0.00862 0.01939 + //100 0.750 67 0.04460 0.95540 0.01700 + //2000 0.330 700 0.97251 0.02749 0.00312 + + cout.setf(ios::showpoint); // Trailing zeros to show significant decimal digits. + cout.precision(5); // Might calculate this from trials in distribution? + cout << fixed; + // Binomial distribution. + + // Note that cdf(dist, k) is equivalent to NAG library plek probability of <= k + // cdf(complement(dist, k)) is equivalent to NAG library pgtk probability of > k + // pdf(dist, k) is equivalent to NAG library peqk probability of == k + + cout << " n p k plek pgtk peqk " << endl; + binomial_distribution<>my_dist(4, 0.5); + cout << setw(4) << (int)my_dist.trials() << " " << my_dist.success_fraction() + << " " << 2 << " " << cdf(my_dist, 2) << " " + << cdf(complement(my_dist, 2)) << " " << pdf(my_dist, 2) << endl; + + binomial_distribution<>two(19, 0.440); + cout << setw(4) << (int)two.trials() << " " << two.success_fraction() + << " " << 13 << " " << cdf(two, 13) << " " + << cdf(complement(two, 13)) << " " << pdf(two, 13) << endl; + + binomial_distribution<>three(100, 0.750); + cout << setw(4) << (int)three.trials() << " " << three.success_fraction() + << " " << 67 << " " << cdf(three, 67) << " " << cdf(complement(three, 67)) + << " " << pdf(three, 67) << endl; + binomial_distribution<>four(2000, 0.330); + cout << setw(4) << (int)four.trials() << " " << four.success_fraction() + << " " << 700 << " " + << cdf(four, 700) << " " << cdf(complement(four, 700)) + << " " << pdf(four, 700) << endl; + + return 0; +} // int main() + +/* + +Example of using the binomial distribution to replicate a NAG library call. + n p k plek pgtk peqk + 4 0.50000 2 0.68750 0.31250 0.37500 + 19 0.44000 13 0.99138 0.00862 0.01939 + 100 0.75000 67 0.04460 0.95540 0.01700 +2000 0.33000 700 0.97251 0.02749 0.00312 + + + */ + diff --git a/example/binomial_quiz_example.cpp b/example/binomial_quiz_example.cpp new file mode 100644 index 000000000..a4be2dcdb --- /dev/null +++ b/example/binomial_quiz_example.cpp @@ -0,0 +1,520 @@ +// Copyright Paul A. Bristow 2007 +// Copyright John Maddock 2006 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// binomial_examples_quiz.cpp + +// Simple example of computing probabilities and quantiles for a binomial random variable +// representing the correct guesses on a multiple-choice test. + +// source http://www.stat.wvu.edu/SRS/Modules/Binomial/test.html + +//[binomial_quiz_example1 +/*` +A multiple choice test has four possible answers to each of 16 questions. +A student guesses the answer to each question, +so the probability of getting a correct answer on any given question is +one in four, a quarter, 1/4, 25% or fraction 0.25. +The conditions of the binomial experiment are assumed to be met: +n = 16 questions constitute the trials; +each question results in one of two possible outcomes (correct or incorrect); +the probability of being correct is 0.25 and is constant if no knowledge about the subject is assumed; +the questions are answered independently if the student's answer to a question +in no way influences his/her answer to another question. + +First, we need to be able to use the binomial distribution constructor +(and some std input/output, of course). +*/ + +#include + using boost::math::binomial; + +#include + using std::cout; using std::endl; + using std::ios; using std::flush; using std::left; using std::right; using std::fixed; +#include + using std::setw; using std::setprecision; +//][/binomial_quiz_example1] + +int main() +{ + try + { + cout << "Binomial distribution example - guessing in a quiz." << endl; +//[binomial_quiz_example2 +/*` +The number of correct answers, X, is distributed as a binomial random variable +with binomial distribution parameters: questions n = 16 and success fraction probability p = 0.25. +So we construct a binomial distribution: +*/ + int questions = 16; // All the questions in the quiz. + int answers = 4; // Possible answers to each question. + double success_fraction = (double)answers / (double)questions; // If a random guess. + // Caution: = answers / questions would be zero (because they are integers)! + binomial quiz(questions, success_fraction); +/*` +and display the distribution parameters we used thus: +*/ + cout << "In a quiz with " << quiz.trials() + << " questions and with a probability of guessing right of " + << quiz.success_fraction() * 100 << " %" + << " or 1 in " << static_cast(1. / quiz.success_fraction()) << endl; +/*` +Show a few probabilities of just guessing: +*/ + cout << "Probability of getting none right is " << pdf(quiz, 0) << endl; // 0.010023 + cout << "Probability of getting exactly one right is " << pdf(quiz, 1) << endl; + cout << "Probability of getting exactly two right is " << pdf(quiz, 2) << endl; + int pass_score = 11; + cout << "Probability of getting exactly " << pass_score << " answers right by chance is " + << pdf(quiz, questions) << endl; +/*` +[pre +Probability of getting none right is 0.0100226 +Probability of getting exactly one right is 0.0534538 +Probability of getting exactly two right is 0.133635 +Probability of getting exactly 11 answers right by chance is 2.32831e-010 +] +These don't give any encouragement to guessers! + +We can tabulate the 'getting exactly right' ( == ) probabilities thus: +*/ + cout << "\n" "Guessed Probability" << right << endl; + for (int successes = 0; successes <= questions; successes++) + { + double probability = pdf(quiz, successes); + cout << setw(2) << successes << " " << probability << endl; + } + cout << endl; +/*` +[pre +Guessed Probability + 0 0.0100226 + 1 0.0534538 + 2 0.133635 + 3 0.207876 + 4 0.225199 + 5 0.180159 + 6 0.110097 + 7 0.0524273 + 8 0.0196602 + 9 0.00582526 +10 0.00135923 +11 0.000247132 +12 3.43239e-005 +13 3.5204e-006 +14 2.51457e-007 +15 1.11759e-008 +16 2.32831e-010 +] +Then we can add the probabilities of some 'exactly right' like this: +*/ + cout << "Probability of getting none or one right is " << pdf(quiz, 0) + pdf(quiz, 1) << endl; + +/*` +[pre +Probability of getting none or one right is 0.0634764 +] +But if more than a couple of scores are involved, it is more convenient (and may be more accurate) +to use the Cumulative Distribution Function (cdf) instead: +*/ + cout << "Probability of getting none or one right is " << cdf(quiz, 1) << endl; +/*` +[pre +Probability of getting none or one right is 0.0634764 +] +Since the cdf is inclusive, we can get the probability of getting up to 10 right ( <= ) +*/ + cout << "Probability of getting <= 10 right (to fail) is " << cdf(quiz, 10) << endl; +/*` +[pre +Probability of getting <= 10 right (to fail) is 0.999715 +] +To get the probability of getting 11 or more right (to pass), +it is tempting to use ``1 - cdf(quiz, 10)`` to get the probability of > 10 +*/ + cout << "Probability of getting > 10 right (to pass) is " << 1 - cdf(quiz, 10) << endl; +/*` +[pre +Probability of getting > 10 right (to pass) is 0.000285239 +] +But this should be resisted in favor of using the complement function. [link why_complements Why complements?] +*/ + cout << "Probability of getting > 10 right (to pass) is " << cdf(complement(quiz, 10)) << endl; +/*` +[pre +Probability of getting > 10 right (to pass) is 0.000285239 +] +And we can check that these two, <= 10 and > 10, add up to unity. +*/ +BOOST_ASSERT((cdf(quiz, 10) + cdf(complement(quiz, 10))) == 1.); +/*` +If we want a < rather than a <= test, because the CDF is inclusive, we must subtract one from the score. +*/ + cout << "Probability of getting less than " << pass_score + << " (< " << pass_score << ") answers right by guessing is " + << cdf(quiz, pass_score -1) << endl; +/*` +[pre +Probability of getting less than 11 (< 11) answers right by guessing is 0.999715 +] +and similarly to get a >= rather than a > test +we also need to subtract one from the score (and can again check the sum is unity). +This is because if the cdf is /inclusive/, +then its complement must be /exclusive/ otherwise there would be one possible +outcome counted twice! +*/ + cout << "Probability of getting at least " << pass_score + << "(>= " << pass_score << ") answers right by guessing is " + << cdf(complement(quiz, pass_score-1)) + << ", only 1 in " << 1/cdf(complement(quiz, pass_score-1)) << endl; + + BOOST_ASSERT((cdf(quiz, pass_score -1) + cdf(complement(quiz, pass_score-1))) == 1); + +/*` +[pre +Probability of getting at least 11 (>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83 +] +Finally we can tabulate some probabilities: +*/ + cout << "\n" "At most (<=)""\n""Guessed OK Probability" << right << endl; + for (int score = 0; score <= questions; score++) + { + cout << setw(2) << score << " " << setprecision(10) + << cdf(quiz, score) << endl; + } + cout << endl; +/*` +[pre +At most (<=) +Guessed OK Probability + 0 0.01002259576 + 1 0.0634764398 + 2 0.1971110499 + 3 0.4049871101 + 4 0.6301861752 + 5 0.8103454274 + 6 0.9204427481 + 7 0.9728700437 + 8 0.9925302796 + 9 0.9983555346 +10 0.9997147608 +11 0.9999618928 +12 0.9999962167 +13 0.9999997371 +14 0.9999999886 +15 0.9999999998 +16 1 +] +*/ + cout << "\n" "At least (>)""\n""Guessed OK Probability" << right << endl; + for (int score = 0; score <= questions; score++) + { + cout << setw(2) << score << " " << setprecision(10) + << cdf(complement(quiz, score)) << endl; + } +/*` +[pre +At least (>) +Guessed OK Probability + 0 0.9899774042 + 1 0.9365235602 + 2 0.8028889501 + 3 0.5950128899 + 4 0.3698138248 + 5 0.1896545726 + 6 0.07955725188 + 7 0.02712995629 + 8 0.00746972044 + 9 0.001644465374 +10 0.0002852391917 +11 3.810715862e-005 +12 3.783265129e-006 +13 2.628657967e-007 +14 1.140870154e-008 +15 2.328306437e-010 +16 0 +] +We now consider the probabilities of *ranges* of correct guesses. + +First, calculate the probability of getting a range of guesses right, +by adding the exact probabilities of each from low ... high. +*/ + int low = 3; // Getting at least 3 right. + int high = 5; // Getting as most 5 right. + double sum = 0.; + for (int i = low; i <= high; i++) + { + sum += pdf(quiz, i); + } + cout.precision(4); + cout << "Probability of getting between " + << low << " and " << high << " answers right by guessing is " + << sum << endl; // 0.61323 +/*` +[pre +Probability of getting between 3 and 5 answers right by guessing is 0.6132 +] +Or, usually better, we can use the difference of cdfs instead: +*/ + cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is " + << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 0.61323 +/*` +[pre +Probability of getting between 3 and 5 answers right by guessing is 0.6132 +] +And we can also try a few more combinations of high and low choices: +*/ + low = 1; high = 6; + cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is " + << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 and 6 P= 0.91042 + low = 1; high = 8; + cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is " + << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 <= x 8 P = 0.9825 + low = 4; high = 4; + cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is " + << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 4 <= x 4 P = 0.22520 + +/*` +[pre +Probability of getting between 1 and 6 answers right by guessing is 0.9104 +Probability of getting between 1 and 8 answers right by guessing is 0.9825 +Probability of getting between 4 and 4 answers right by guessing is 0.2252 +] +[h4 Using Binomial distribution moments] +Using moments of the distribution, we can say more about the spread of results from guessing. +*/ + cout << "By guessing, on average, one can expect to get " << mean(quiz) << " correct answers." << endl; + cout << "Standard deviation is " << standard_deviation(quiz) << endl; + cout << "So about 2/3 will lie within 1 standard deviation and get between " + << ceil(mean(quiz) - standard_deviation(quiz)) << " and " + << floor(mean(quiz) + standard_deviation(quiz)) << " correct." << endl; + cout << "Mode (the most frequent) is " << mode(quiz) << endl; + cout << "Skewness is " << skewness(quiz) << endl; + +/*` +[pre +By guessing, on average, one can expect to get 4 correct answers. +Standard deviation is 1.732 +So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct. +Mode (the most frequent) is 4 +Skewness is 0.2887 +] +[h4 Quantiles] +The quantiles (percentiles or percentage points) for a few probability levels: +*/ + cout << "Quartiles " << quantile(quiz, 0.25) << " to " + << quantile(complement(quiz, 0.25)) << endl; // Quartiles + cout << "1 standard deviation " << quantile(quiz, 0.33) << " to " + << quantile(quiz, 0.67) << endl; // 1 sd + cout << "Deciles " << quantile(quiz, 0.1) << " to " + << quantile(complement(quiz, 0.1))<< endl; // Deciles + cout << "5 to 95% " << quantile(quiz, 0.05) << " to " + << quantile(complement(quiz, 0.05))<< endl; // 5 to 95% + cout << "2.5 to 97.5% " << quantile(quiz, 0.025) << " to " + << quantile(complement(quiz, 0.025)) << endl; // 2.5 to 97.5% + cout << "2 to 98% " << quantile(quiz, 0.02) << " to " + << quantile(complement(quiz, 0.02)) << endl; // 2 to 98% + + cout << "If guessing then percentiles 1 to 99% will get " << quantile(quiz, 0.01) + << " to " << quantile(complement(quiz, 0.01)) << " right." << endl; +/*` +Notice that these output integral values because the default policy is `integer_round_outwards`. +[pre +Quartiles 2 to 5 +1 standard deviation 2 to 5 +Deciles 1 to 6 +5 to 95% 0 to 7 +2.5 to 97.5% 0 to 8 +2 to 98% 0 to 8 +] +*/ + +//] [/binomial_quiz_example2] + +//[discrete_quantile_real +/*` +Quantiles values are controlled by the +[link math_toolkit.policy.pol_ref.discrete_quant_ref discrete quantile policy] +chosen. +The default is `integer_round_outwards`, +so the lower quantile is rounded down, and the upper quantile is rounded up. + +But we might believe that the real values tell us a little more - see +[link math_toolkit.policy.pol_tutorial.understand_dis_quant Understanding Discrete Quantile Policy]. + +We could control the policy for *all* distributions by + + #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real + + at the head of the program would make this policy apply +to this *one, and only*, translation unit. + +Or we can now create a (typedef for) policy that has discrete quantiles real +(here avoiding any 'using namespaces ...' statements): +*/ + using boost::math::policies::policy; + using boost::math::policies::discrete_quantile; + using boost::math::policies::real; + using boost::math::policies::integer_round_outwards; // Default. + typedef boost::math::policies::policy > real_quantile_policy; +/*` +Add a custom binomial distribution called ``real_quantile_binomial`` that uses ``real_quantile_policy`` +*/ + using boost::math::binomial_distribution; + typedef binomial_distribution real_quantile_binomial; +/*` +Construct an object of this custom distribution: +*/ + real_quantile_binomial quiz_real(questions, success_fraction); +/*` +And use this to show some quantiles - that now have real rather than integer values. +*/ + cout << "Quartiles " << quantile(quiz, 0.25) << " to " + << quantile(complement(quiz_real, 0.25)) << endl; // Quartiles 2 to 4.6212 + cout << "1 standard deviation " << quantile(quiz_real, 0.33) << " to " + << quantile(quiz_real, 0.67) << endl; // 1 sd 2.6654 4.194 + cout << "Deciles " << quantile(quiz_real, 0.1) << " to " + << quantile(complement(quiz_real, 0.1))<< endl; // Deciles 1.3487 5.7583 + cout << "5 to 95% " << quantile(quiz_real, 0.05) << " to " + << quantile(complement(quiz_real, 0.05))<< endl; // 5 to 95% 0.83739 6.4559 + cout << "2.5 to 97.5% " << quantile(quiz_real, 0.025) << " to " + << quantile(complement(quiz_real, 0.025)) << endl; // 2.5 to 97.5% 0.42806 7.0688 + cout << "2 to 98% " << quantile(quiz_real, 0.02) << " to " + << quantile(complement(quiz_real, 0.02)) << endl; // 2 to 98% 0.31311 7.7880 + + cout << "If guessing, then percentiles 1 to 99% will get " << quantile(quiz_real, 0.01) + << " to " << quantile(complement(quiz_real, 0.01)) << " right." << endl; +/*` +[pre +Real Quantiles +Quartiles 2 to 4.621 +1 standard deviation 2.665 to 4.194 +Deciles 1.349 to 5.758 +5 to 95% 0.8374 to 6.456 +2.5 to 97.5% 0.4281 to 7.069 +2 to 98% 0.3131 to 7.252 +If guessing then percentiles 1 to 99% will get 0 to 7.788 right. +] +*/ + +//] [/discrete_quantile_real] + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because + // default policies are to throw exceptions on arguments that cause + // errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + + + +/* + +Output is: + +Binomial distribution example - guessing in a quiz. +In a quiz with 16 questions and with a probability of guessing right of 25 % or 1 in 4 +Probability of getting none right is 0.0100226 +Probability of getting exactly one right is 0.0534538 +Probability of getting exactly two right is 0.133635 +Probability of getting exactly 11 answers right by chance is 2.32831e-010 +Guessed Probability + 0 0.0100226 + 1 0.0534538 + 2 0.133635 + 3 0.207876 + 4 0.225199 + 5 0.180159 + 6 0.110097 + 7 0.0524273 + 8 0.0196602 + 9 0.00582526 +10 0.00135923 +11 0.000247132 +12 3.43239e-005 +13 3.5204e-006 +14 2.51457e-007 +15 1.11759e-008 +16 2.32831e-010 +Probability of getting none or one right is 0.0634764 +Probability of getting none or one right is 0.0634764 +Probability of getting <= 10 right (to fail) is 0.999715 +Probability of getting > 10 right (to pass) is 0.000285239 +Probability of getting > 10 right (to pass) is 0.000285239 +Probability of getting less than 11 (< 11) answers right by guessing is 0.999715 +Probability of getting at least 11(>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83 +At most (<=) +Guessed OK Probability + 0 0.01002259576 + 1 0.0634764398 + 2 0.1971110499 + 3 0.4049871101 + 4 0.6301861752 + 5 0.8103454274 + 6 0.9204427481 + 7 0.9728700437 + 8 0.9925302796 + 9 0.9983555346 +10 0.9997147608 +11 0.9999618928 +12 0.9999962167 +13 0.9999997371 +14 0.9999999886 +15 0.9999999998 +16 1 +At least (>=) +Guessed OK Probability + 0 0.9899774042 + 1 0.9365235602 + 2 0.8028889501 + 3 0.5950128899 + 4 0.3698138248 + 5 0.1896545726 + 6 0.07955725188 + 7 0.02712995629 + 8 0.00746972044 + 9 0.001644465374 +10 0.0002852391917 +11 3.810715862e-005 +12 3.783265129e-006 +13 2.628657967e-007 +14 1.140870154e-008 +15 2.328306437e-010 +16 0 +Probability of getting between 3 and 5 answers right by guessing is 0.6132 +Probability of getting between 3 and 5 answers right by guessing is 0.6132 +Probability of getting between 1 and 6 answers right by guessing is 0.9104 +Probability of getting between 1 and 8 answers right by guessing is 0.9825 +Probability of getting between 4 and 4 answers right by guessing is 0.2252 +By guessing, on average, one can expect to get 4 correct answers. +Standard deviation is 1.732 +So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct. +Mode (the most frequent) is 4 +Skewness is 0.2887 +Quartiles 2 to 5 +1 standard deviation 2 to 5 +Deciles 1 to 6 +5 to 95% 0 to 7 +2.5 to 97.5% 0 to 8 +2 to 98% 0 to 8 +If guessing then percentiles 1 to 99% will get 0 to 8 right. +Quartiles 2 to 4.621 +1 standard deviation 2.665 to 4.194 +Deciles 1.349 to 5.758 +5 to 95% 0.8374 to 6.456 +2.5 to 97.5% 0.4281 to 7.069 +2 to 98% 0.3131 to 7.252 +If guessing, then percentiles 1 to 99% will get 0 to 7.788 right. + +*/ + diff --git a/example/binomial_sample_sizes.cpp b/example/binomial_sample_sizes.cpp new file mode 100644 index 000000000..8d3a16069 --- /dev/null +++ b/example/binomial_sample_sizes.cpp @@ -0,0 +1,76 @@ +// (C) Copyright John Maddock 2006 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +#include +#include + +void find_max_sample_size(double p, unsigned successes) +{ + // + // p = success ratio. + // successes = Total number of observed successes. + // + // Calculate how many trials we can have to ensure the + // maximum number of successes does not exceed "successes". + // A typical use would be failure analysis, where you want + // zero or fewer "successes" with some probability. + // + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "________________________\n" + "Maximum Number of Trials\n" + "________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Success ratio" << "= " << p << "\n"; + cout << setw(40) << left << "Maximum Number of \"successes\" permitted" << "= " << successes << "\n"; + // + // Define a table of confidence intervals: + // + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Print table header: + // + cout << "\n\n" + "____________________________\n" + "Confidence Max Number\n" + " Value (%) Of Trials \n" + "____________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate trials: + double t = binomial_distribution<>::find_maximum_number_of_trials(successes, p, alpha[i]); + t = floor(t); + // Print Trials: + cout << fixed << setprecision(0) << setw(15) << right << t << endl; + } + cout << endl; +} + +int main() +{ + find_max_sample_size(1.0/1000, 0); + find_max_sample_size(1.0/10000, 0); + find_max_sample_size(1.0/100000, 0); + find_max_sample_size(1.0/1000000, 0); + + return 0; +} + diff --git a/example/c_error_policy_example.cpp b/example/c_error_policy_example.cpp new file mode 100644 index 000000000..f05c9ddef --- /dev/null +++ b/example/c_error_policy_example.cpp @@ -0,0 +1,74 @@ +// C_error_policy_example.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Suppose we want a call to tgamma to behave in a C-compatible way +// and set global ::errno rather than throw an exception. + +#include +using boost::math::tgamma; + +using boost::math::policies::policy; +// Possible errors +using boost::math::policies::overflow_error; +using boost::math::policies::underflow_error; +using boost::math::policies::domain_error; +using boost::math::policies::pole_error; +using boost::math::policies::denorm_error; +using boost::math::policies::evaluation_error; + +using boost::math::policies::errno_on_error; +//using boost::math::policies::ignore_error; + +//using namespace boost::math::policies; +//using namespace boost::math; + +// Define a policy: +typedef policy< + domain_error, // 'bad' arguments. + pole_error, // argument is pole value. + overflow_error, // argument value causes overflow. + evaluation_error // evaluation does not converge and may be inaccurate, or worse. + > C_error_policy; + +// std +#include + using std::cout; + using std::endl; + +int main() +{ + // We can achieve this at the function call site + // with the previously defined policy C_error_policy. + double t = tgamma(4., C_error_policy()); + cout << "tgamma(4., C_error_policy() = " << t << endl; // 6 + + // Alternatively we could use the function make_policy, + // provided for convenience, + // and define everything at the call site: + t = tgamma(4., make_policy( + domain_error(), + pole_error(), + overflow_error(), + evaluation_error() + )); + cout << "tgamma(4., make_policy( ...) = " << t << endl; // 6 + + return 0; +} // int main() + +/* + +Output + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\c_error_policy_example.exe" +tgamma(4., C_error_policy() = 6 +tgamma(4., make_policy( ...) = 6 + +*/ diff --git a/example/chi_square_std_dev_test.cpp b/example/chi_square_std_dev_test.cpp new file mode 100644 index 000000000..005c861a0 --- /dev/null +++ b/example/chi_square_std_dev_test.cpp @@ -0,0 +1,552 @@ +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +using std::cout; using std::endl; +using std::left; using std::fixed; using std::right; using std::scientific; +#include +using std::setw; +using std::setprecision; +#include + + +void confidence_limits_on_std_deviation( + double Sd, // Sample Standard Deviation + unsigned N) // Sample size +{ + // Calculate confidence intervals for the standard deviation. + // For example if we set the confidence limit to + // 0.95, we know that if we repeat the sampling + // 100 times, then we expect that the true standard deviation + // will be between out limits on 95 occations. + // Note: this is not the same as saying a 95% + // confidence interval means that there is a 95% + // probability that the interval contains the true standard deviation. + // The interval computed from a given sample either + // contains the true standard deviation or it does not. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm + + // using namespace boost::math; + using boost::math::chi_squared; + using boost::math::quantile; + using boost::math::complement; + + // Print out general info: + cout << + "________________________________________________\n" + "2-Sided Confidence Limits For Standard Deviation\n" + "________________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << N << "\n"; + cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n"; + // + // Define a table of significance/risk levels: + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Start by declaring the distribution we'll need: + chi_squared dist(N - 1); + // + // Print table header: + // + cout << "\n\n" + "_____________________________________________\n" + "Confidence Lower Upper\n" + " Value (%) Limit Limit\n" + "_____________________________________________\n"; + // + // Now print out the data for the table rows. + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // Calculate limits: + double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha[i] / 2))); + double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha[i] / 2)); + // Print Limits: + cout << fixed << setprecision(5) << setw(15) << right << lower_limit; + cout << fixed << setprecision(5) << setw(15) << right << upper_limit << endl; + } + cout << endl; +} // void confidence_limits_on_std_deviation + +void confidence_limits_on_std_deviation_alpha( + double Sd, // Sample Standard Deviation + double alpha // confidence + ) +{ // Calculate confidence intervals for the standard deviation. + // for the alpha parameter, for a range number of observations, + // from a mere 2 up to a million. + // O. L. Davies, Statistical Methods in Research and Production, ISBN 0 05 002437 X, + // 4.33 Page 68, Table H, pp 452 459. + + // using namespace std; + // using namespace boost::math; + using boost::math::chi_squared; + using boost::math::quantile; + using boost::math::complement; + + // Define a table of numbers of observations: + unsigned int obs[] = {2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 30, 40 , 50, 60, 100, 120, 1000, 10000, 50000, 100000, 1000000}; + + cout << // Print out heading: + "________________________________________________\n" + "2-Sided Confidence Limits For Standard Deviation\n" + "________________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Confidence level (two-sided) " << "= " << alpha << "\n"; + cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n"; + + cout << "\n\n" // Print table header: + "_____________________________________________\n" + "Observations Lower Upper\n" + " Limit Limit\n" + "_____________________________________________\n"; + for(unsigned i = 0; i < sizeof(obs)/sizeof(obs[0]); ++i) + { + unsigned int N = obs[i]; // Observations + // Start by declaring the distribution with the appropriate : + chi_squared dist(N - 1); + + // Now print out the data for the table row. + cout << fixed << setprecision(3) << setw(10) << right << N; + // Calculate limits: (alpha /2 because it is a two-sided (upper and lower limit) test. + double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha / 2))); + double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha / 2)); + // Print Limits: + cout << fixed << setprecision(4) << setw(15) << right << lower_limit; + cout << fixed << setprecision(4) << setw(15) << right << upper_limit << endl; + } + cout << endl; +}// void confidence_limits_on_std_deviation_alpha + +void chi_squared_test( + double Sd, // Sample std deviation + double D, // True std deviation + unsigned N, // Sample size + double alpha) // Significance level +{ + // + // A Chi Squared test applied to a single set of data. + // We are testing the null hypothesis that the true + // standard deviation of the sample is D, and that any variation is down + // to chance. We can also test the alternative hypothesis + // that any difference is not down to chance. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm + // + // using namespace boost::math; + using boost::math::chi_squared; + using boost::math::quantile; + using boost::math::complement; + using boost::math::cdf; + + // Print header: + cout << + "______________________________________________\n" + "Chi Squared test for sample standard deviation\n" + "______________________________________________\n\n"; + cout << setprecision(5); + cout << setw(55) << left << "Number of Observations" << "= " << N << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n"; + cout << setw(55) << left << "Expected True Standard Deviation" << "= " << D << "\n\n"; + // + // Now we can calculate and output some stats: + // + // test-statistic: + double t_stat = (N - 1) * (Sd / D) * (Sd / D); + cout << setw(55) << left << "Test Statistic" << "= " << t_stat << "\n"; + // + // Finally define our distribution, and get the probability: + // + chi_squared dist(N - 1); + double p = cdf(dist, t_stat); + cout << setw(55) << left << "CDF of test statistic: " << "= " + << setprecision(3) << scientific << p << "\n"; + double ucv = quantile(complement(dist, alpha)); + double ucv2 = quantile(complement(dist, alpha / 2)); + double lcv = quantile(dist, alpha); + double lcv2 = quantile(dist, alpha / 2); + cout << setw(55) << left << "Upper Critical Value at alpha: " << "= " + << setprecision(3) << scientific << ucv << "\n"; + cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << ucv2 << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha: " << "= " + << setprecision(3) << scientific << lcv << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << lcv2 << "\n\n"; + // + // Finally print out results of alternative hypothesis: + // + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + cout << "Standard Deviation != " << setprecision(3) << fixed << D << " "; + if((ucv2 < t_stat) || (lcv2 > t_stat)) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Standard Deviation < " << setprecision(3) << fixed << D << " "; + if(lcv > t_stat) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Standard Deviation > " << setprecision(3) << fixed << D << " "; + if(ucv < t_stat) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; +} // void chi_squared_test + +void chi_squared_sample_sized( + double diff, // difference from variance to detect + double variance) // true variance +{ + using namespace std; + // using boost::math; + using boost::math::chi_squared; + using boost::math::quantile; + using boost::math::complement; + using boost::math::cdf; + + try + { + cout << // Print out general info: + "_____________________________________________________________\n" + "Estimated sample sizes required for various confidence levels\n" + "_____________________________________________________________\n\n"; + cout << setprecision(5); + cout << setw(40) << left << "True Variance" << "= " << variance << "\n"; + cout << setw(40) << left << "Difference to detect" << "= " << diff << "\n"; + // + // Define a table of significance levels: + // + double alpha[] = { 0.5, 0.33333333333333333333333, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Print table header: + // + cout << "\n\n" + "_______________________________________________________________\n" + "Confidence Estimated Estimated\n" + " Value (%) Sample Size Sample Size\n" + " (lower one- (upper one-\n" + " sided test) sided test)\n" + "_______________________________________________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate df for a lower single-sided test: + double df = chi_squared::find_degrees_of_freedom( + -diff, alpha[i], alpha[i], variance); + // convert to sample size: + double size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size; + // calculate df for an upper single-sided test: + df = chi_squared::find_degrees_of_freedom( + diff, alpha[i], alpha[i], variance); + // convert to sample size: + size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size << endl; + } + cout << endl; + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } +} // chi_squared_sample_sized + +int main() +{ + // Run tests for Gear data + // see http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm + // Tests measurements of gear diameter. + // + confidence_limits_on_std_deviation(0.6278908E-02, 100); + chi_squared_test(0.6278908E-02, 0.1, 100, 0.05); + chi_squared_sample_sized(0.1 - 0.6278908E-02, 0.1); + // + // Run tests for silicon wafer fabrication data. + // see http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm + // A supplier of 100 ohm.cm silicon wafers claims that his fabrication + // process can produce wafers with sufficient consistency so that the + // standard deviation of resistivity for the lot does not exceed + // 10 ohm.cm. A sample of N = 10 wafers taken from the lot has a + // standard deviation of 13.97 ohm.cm + // + confidence_limits_on_std_deviation(13.97, 10); + chi_squared_test(13.97, 10.0, 10, 0.05); + chi_squared_sample_sized(13.97 * 13.97 - 100, 100); + chi_squared_sample_sized(55, 100); + chi_squared_sample_sized(1, 100); + + // List confidence interval multipliers for standard deviation + // for a range of numbers of observations from 2 to a million, + // and for a few alpha values, 0.1, 0.05, 0.01 for condfidences 90, 95, 99 % + confidence_limits_on_std_deviation_alpha(1., 0.1); + confidence_limits_on_std_deviation_alpha(1., 0.05); + confidence_limits_on_std_deviation_alpha(1., 0.01); + + return 0; +} + +/* + +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ +Number of Observations = 100 +Standard Deviation = 0.006278908 +_____________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +_____________________________________________ + 50.000 0.00601 0.00662 + 75.000 0.00582 0.00685 + 90.000 0.00563 0.00712 + 95.000 0.00551 0.00729 + 99.000 0.00530 0.00766 + 99.900 0.00507 0.00812 + 99.990 0.00489 0.00855 + 99.999 0.00474 0.00895 +______________________________________________ +Chi Squared test for sample standard deviation +______________________________________________ +Number of Observations = 100 +Sample Standard Deviation = 0.00628 +Expected True Standard Deviation = 0.10000 +Test Statistic = 0.39030 +CDF of test statistic: = 1.438e-099 +Upper Critical Value at alpha: = 1.232e+002 +Upper Critical Value at alpha/2: = 1.284e+002 +Lower Critical Value at alpha: = 7.705e+001 +Lower Critical Value at alpha/2: = 7.336e+001 +Results for Alternative Hypothesis and alpha = 0.0500 +Alternative Hypothesis Conclusion +Standard Deviation != 0.100 NOT REJECTED +Standard Deviation < 0.100 NOT REJECTED +Standard Deviation > 0.100 REJECTED +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ +True Variance = 0.10000 +Difference to detect = 0.09372 +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (lower one- (upper one- + sided test) sided test) +_______________________________________________________________ + 50.000 2 2 + 66.667 2 5 + 75.000 2 10 + 90.000 4 32 + 95.000 5 52 + 99.000 8 102 + 99.900 13 178 + 99.990 18 257 + 99.999 23 337 +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ +Number of Observations = 10 +Standard Deviation = 13.9700000 +_____________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +_____________________________________________ + 50.000 12.41880 17.25579 + 75.000 11.23084 19.74131 + 90.000 10.18898 22.98341 + 95.000 9.60906 25.50377 + 99.000 8.62898 31.81825 + 99.900 7.69466 42.51593 + 99.990 7.04085 55.93352 + 99.999 6.54517 73.00132 +______________________________________________ +Chi Squared test for sample standard deviation +______________________________________________ +Number of Observations = 10 +Sample Standard Deviation = 13.97000 +Expected True Standard Deviation = 10.00000 +Test Statistic = 17.56448 +CDF of test statistic: = 9.594e-001 +Upper Critical Value at alpha: = 1.692e+001 +Upper Critical Value at alpha/2: = 1.902e+001 +Lower Critical Value at alpha: = 3.325e+000 +Lower Critical Value at alpha/2: = 2.700e+000 +Results for Alternative Hypothesis and alpha = 0.0500 +Alternative Hypothesis Conclusion +Standard Deviation != 10.000 REJECTED +Standard Deviation < 10.000 REJECTED +Standard Deviation > 10.000 NOT REJECTED +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ +True Variance = 100.00000 +Difference to detect = 95.16090 +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (lower one- (upper one- + sided test) sided test) +_______________________________________________________________ + 50.000 2 2 + 66.667 2 5 + 75.000 2 10 + 90.000 4 32 + 95.000 5 51 + 99.000 7 99 + 99.900 11 174 + 99.990 15 251 + 99.999 20 330 +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ +True Variance = 100.00000 +Difference to detect = 55.00000 +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (lower one- (upper one- + sided test) sided test) +_______________________________________________________________ + 50.000 2 2 + 66.667 4 10 + 75.000 8 21 + 90.000 23 71 + 95.000 36 115 + 99.000 71 228 + 99.900 123 401 + 99.990 177 580 + 99.999 232 762 +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ +True Variance = 100.00000 +Difference to detect = 1.00000 +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (lower one- (upper one- + sided test) sided test) +_______________________________________________________________ + 50.000 2 2 + 66.667 14696 14993 + 75.000 36033 36761 + 90.000 130079 132707 + 95.000 214283 218612 + 99.000 428628 437287 + 99.900 756333 771612 + 99.990 1095435 1117564 + 99.999 1440608 1469711 +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ +Confidence level (two-sided) = 0.1000000 +Standard Deviation = 1.0000000 +_____________________________________________ +Observations Lower Upper + Limit Limit +_____________________________________________ + 2 0.5102 15.9472 + 3 0.5778 4.4154 + 4 0.6196 2.9200 + 5 0.6493 2.3724 + 6 0.6720 2.0893 + 7 0.6903 1.9154 + 8 0.7054 1.7972 + 9 0.7183 1.7110 + 10 0.7293 1.6452 + 15 0.7688 1.4597 + 20 0.7939 1.3704 + 30 0.8255 1.2797 + 40 0.8454 1.2320 + 50 0.8594 1.2017 + 60 0.8701 1.1805 + 100 0.8963 1.1336 + 120 0.9045 1.1203 + 1000 0.9646 1.0383 + 10000 0.9885 1.0118 + 50000 0.9948 1.0052 + 100000 0.9963 1.0037 + 1000000 0.9988 1.0012 +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ +Confidence level (two-sided) = 0.0500000 +Standard Deviation = 1.0000000 +_____________________________________________ +Observations Lower Upper + Limit Limit +_____________________________________________ + 2 0.4461 31.9102 + 3 0.5207 6.2847 + 4 0.5665 3.7285 + 5 0.5991 2.8736 + 6 0.6242 2.4526 + 7 0.6444 2.2021 + 8 0.6612 2.0353 + 9 0.6755 1.9158 + 10 0.6878 1.8256 + 15 0.7321 1.5771 + 20 0.7605 1.4606 + 30 0.7964 1.3443 + 40 0.8192 1.2840 + 50 0.8353 1.2461 + 60 0.8476 1.2197 + 100 0.8780 1.1617 + 120 0.8875 1.1454 + 1000 0.9580 1.0459 + 10000 0.9863 1.0141 + 50000 0.9938 1.0062 + 100000 0.9956 1.0044 + 1000000 0.9986 1.0014 +________________________________________________ +2-Sided Confidence Limits For Standard Deviation +________________________________________________ +Confidence level (two-sided) = 0.0100000 +Standard Deviation = 1.0000000 +_____________________________________________ +Observations Lower Upper + Limit Limit +_____________________________________________ + 2 0.3562 159.5759 + 3 0.4344 14.1244 + 4 0.4834 6.4675 + 5 0.5188 4.3960 + 6 0.5464 3.4848 + 7 0.5688 2.9798 + 8 0.5875 2.6601 + 9 0.6036 2.4394 + 10 0.6177 2.2776 + 15 0.6686 1.8536 + 20 0.7018 1.6662 + 30 0.7444 1.4867 + 40 0.7718 1.3966 + 50 0.7914 1.3410 + 60 0.8065 1.3026 + 100 0.8440 1.2200 + 120 0.8558 1.1973 + 1000 0.9453 1.0609 + 10000 0.9821 1.0185 + 50000 0.9919 1.0082 + 100000 0.9943 1.0058 + 1000000 0.9982 1.0018 +*/ + diff --git a/example/distribution_construction.cpp b/example/distribution_construction.cpp new file mode 100644 index 000000000..fb7b54aac --- /dev/null +++ b/example/distribution_construction.cpp @@ -0,0 +1,201 @@ +// distribution_construction.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Caution: this file contains Quickbook markup as well as code +// and comments, don't change any of the special comment markups! + +//[distribution_construction1 + +/*` + +The structure of distributions is rather different from some other statistical libraries, +for example in less object-oriented language like FORTRAN and C, +that provide a few arguments to each free function. +This library provides each distribution as a template C++ class. +A distribution is constructed with a few arguments, and then +member and non-member functions are used to find values of the +distribution, often a function of a random variate. + +First we need some includes to access the negative binomial distribution +(and the binomial, beta and gamma too). + +*/ + +#include // for negative_binomial_distribution + using boost::math::negative_binomial_distribution; // default type is double. + using boost::math::negative_binomial; // typedef provides default type is double. +#include // for binomial_distribution. +#include // for beta_distribution. +#include // for gamma_distribution. +#include // for normal_distribution. +/*` +Several examples of constructing distributions follow: +*/ +//] [/distribution_construction1 end of Quickbook in C++ markup] + + + + +int main() +{ +//[distribution_construction2 +/*` +First, a negative binomial distribution with 8 successes +and a success fraction 0.25, 25% or 1 in 4, is constructed like this: +*/ + boost::math::negative_binomial_distribution mydist0(8., 0.25); + /*` + But this is inconveniently long, so by writing + */ + using namespace boost::math; + /*` + or + */ + using boost::math::negative_binomial_distribution; + /*` + we can reduce typing. + + Since the vast majority of applications use will be using double precision, + the template argument to the distribution (RealType) defaults + to type double, so we can also write: + */ + + negative_binomial_distribution<> mydist9(8., 0.25); // Uses default RealType = double. + + /*` + But the name "negative_binomial_distribution" is still inconveniently long, + so for most distributions, a convenience typedef is provided, for example: + + typedef negative_binomial_distribution negative_binomial; // Reserved name of type double. + + [caution + This convenience typedef is /not/ provided if a clash would occur + with the name of a function: currently only "beta" and "gamma" + fall into this category. + ] + + So, after a using statement, + */ + + using boost::math::negative_binomial; + + /*` + we have a convenient typedef to `negative_binomial_distribution`: + */ + negative_binomial mydist(8., 0.25); + + /*` + Some more examples using the convenience typedef: + */ + negative_binomial mydist10(5., 0.4); // Both arguments double. + /*` + And automatic conversion takes place, so you can use integers and floats: + */ + negative_binomial mydist11(5, 0.4); // Using provided typedef double, int and double arguments. + /*` + This is probably the most common usage. + */ + negative_binomial mydist12(5., 0.4F); // Double and float arguments. + negative_binomial mydist13(5, 1); // Both arguments integer. + + /*` + Similarly for most other distributions like the binomial. + */ + binomial mybinomial(1, 0.5); // is more concise than + binomial_distribution<> mybinomd1(1, 0.5); + + /*` + For cases when the typdef distribution name would clash with a math special function + (currently only beta and gamma) + the typedef is deliberately not provided, and the longer version of the name + must be used. For example do not use: + + using boost::math::beta; + beta mybetad0(1, 0.5); // Error beta is a math FUNCTION! + + Which produces the error messages: + + [pre + error C2146: syntax error : missing ';' before identifier 'mybetad0' + warning C4551: function call missing argument list + error C3861: 'mybetad0': identifier not found + ] + + Instead you should use: + */ + using boost::math::beta_distribution; + beta_distribution<> mybetad1(1, 0.5); + /*` + or for the gamma distribution: + */ + gamma_distribution<> mygammad1(1, 0.5); + + /*` + We can, of course, still provide the type explicitly thus: + */ + + // Explicit double precision: + negative_binomial_distribution mydist1(8., 0.25); + + // Explicit float precision, double arguments are truncated to float: + negative_binomial_distribution mydist2(8., 0.25); + + // Explicit float precision, integer & double arguments converted to float. + negative_binomial_distribution mydist3(8, 0.25); + + // Explicit float precision, float arguments, so no conversion: + negative_binomial_distribution mydist4(8.F, 0.25F); + + // Explicit float precision, integer arguments promoted to float. + negative_binomial_distribution mydist5(8, 1); + + // Explicit double precision: + negative_binomial_distribution mydist6(8., 0.25); + + // Explicit long double precision: + negative_binomial_distribution mydist7(8., 0.25); + + /*` + And if you have your own RealType called MyFPType, + for example NTL RR (an arbitrary precision type), then we can write: + + negative_binomial_distribution mydist6(8, 1); // Integer arguments -> MyFPType. + + [heading Default arguments to distribution constructors.] + + Note that default constructor arguments are only provided for some distributions. + So if you wrongly assume a default argument you will get an error message, for example: + + negative_binomial_distribution<> mydist8; + + [pre error C2512 no appropriate default constructor available.] + + No default constructors are provided for the negative binomial, + because it is difficult to chose any sensible default values for this distribution. + For other distributions, like the normal distribution, + it is obviously very useful to provide 'standard' + defaults for the mean and standard deviation thus: + + normal_distribution(RealType mean = 0, RealType sd = 1); + + So in this case we can write: + */ + using boost::math::normal; + + normal norm1; // Standard normal distribution. + normal norm2(2); // Mean = 2, std deviation = 1. + normal norm3(2, 3); // Mean = 2, std deviation = 3. + + return 0; +} // int main() + +/*`There is no useful output from this program, of course. */ + +//] [/end of distribution_construction2] + diff --git a/example/error_handling_example.cpp b/example/error_handling_example.cpp new file mode 100644 index 000000000..5b06a407e --- /dev/null +++ b/example/error_handling_example.cpp @@ -0,0 +1,143 @@ +// example_error_handling.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook markup as well as code +// and comments, don't change any of the special comment markups! + +//[error_handling_example +/*` +The following example demonstrates the effect of +setting the macro BOOST_MATH_DOMAIN_ERROR_POLICY +when an invalid argument is encountered. For the +purposes of this example, we'll pass a negative +degrees of freedom parameter to the student's t +distribution. + +Since we know that this is a single file program we could +just add: + + #define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error + +to the top of the source file to change the default policy +to one that simply returns a NaN when a domain error occurs. +Alternatively we could use: + + #define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error + +To ensure the `::errno` is set when a domain error occurs +as well as returning a NaN. + +This is safe provided the program consists of a single +translation unit /and/ we place the define /before/ any +#includes. Note that should we add the define after the includes +then it will have no effect! A warning such as: + +[pre warning C4005: 'BOOST_MATH_OVERFLOW_ERROR_POLICY' : macro redefinition] + +is a certain sign that it will /not/ have the desired effect. + +We'll begin our sample program with the needed includes: +*/ + +// Boost +#include + using boost::math::students_t; // Probability of students_t(df, t). + +// std +#include + using std::cout; + using std::endl; + +#include + using std::exception; + +/*` + +Next we'll define the program's main() to call the student's t +distribution with an invalid degrees of freedom parameter, the program +is set up to handle either an exception or a NaN: + +*/ + +int main() +{ + cout << "Example error handling using Student's t function. " << endl; + cout << "BOOST_MATH_DOMAIN_ERROR_POLICY is set to: " + << BOOST_STRINGIZE(BOOST_MATH_DOMAIN_ERROR_POLICY) << endl; + + double degrees_of_freedom = -1; // A bad argument! + double t = 10; + + try + { + errno = 0; + students_t dist(degrees_of_freedom); // exception is thrown here if enabled + double p = cdf(dist, t); + // test for error reported by other means: + if((boost::math::isnan)(p)) + { + cout << "cdf returned a NaN!" << endl; + cout << "errno is set to: " << errno << endl; + } + else + cout << "Probability of Student's t is " << p << endl; + } + catch(const std::exception& e) + { + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + + return 0; +} // int main() + +/*` + +Here's what the program output looks like with a default build +(one that does throw exceptions): + +[pre +Example error handling using Student's t function. +BOOST_MATH_DOMAIN_ERROR_POLICY is set to: throw_on_error + +Message from thrown exception was: + Error in function boost::math::students_t_distribution::students_t_distribution: + Degrees of freedom argument is -1, but must be > 0 ! +] + +Alternatively let's build with: + + #define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error + +Now the program output is: + +[pre +Example error handling using Student's t function. +BOOST_MATH_DOMAIN_ERROR_POLICY is set to: ignore_error +cdf returned a NaN! +errno is set to: 0 +] + +And finally let's build with: + + #define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error + +Which gives the output: + +[pre +Example error handling using Student's t function. +BOOST_MATH_DOMAIN_ERROR_POLICY is set to: errno_on_error +cdf returned a NaN! +errno is set to: 33 +] + +*/ + +//] [error_handling_eg end quickbook markup] diff --git a/example/error_policies_example.cpp b/example/error_policies_example.cpp new file mode 100644 index 000000000..c726b5417 --- /dev/null +++ b/example/error_policies_example.cpp @@ -0,0 +1,97 @@ +// error_policies_example.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include + using boost::math::normal_distribution; + +#include + using boost::math::students_t; // Probability of students_t(df, t). + using boost::math::students_t_distribution; + +// using namespace boost::math; +//.\error_policy_normal.cpp(30) : error C2872: 'policy' : ambiguous symbol +// could be 'I:\Boost-sandbox\math_toolkit\boost/math/policies/policy.hpp(392) : boost::math::policies::policy' +// or 'boost::math::policies' + + // So can't use this using namespace command. +// Suppose we want a statistical distribution to return infinities, +// rather than throw exceptions (the default policy), then we can use: + +// std +#include + using std::cout; + using std::endl; + +using boost::math::policies::policy; +// Possible errors +using boost::math::policies::overflow_error; +using boost::math::policies::underflow_error; +using boost::math::policies::domain_error; +using boost::math::policies::pole_error; +using boost::math::policies::denorm_error; +using boost::math::policies::evaluation_error; + +using boost::math::policies::ignore_error; + +// Define a custom policy to ignore just overflow: +typedef policy< +overflow_error + > my_policy; + +// Define another custom policy (perhaps ill-advised?) +// to ignore all errors: domain, pole, overflow, underflow, denorm & evaluation: +typedef policy< +domain_error, +pole_error, +overflow_error, +underflow_error, +denorm_error, +evaluation_error + > my_ignoreall_policy; + +// Define a new distribution with a custom policy to ignore_error +// (& thus perhaps return infinity for some arguments): +typedef boost::math::normal_distribution my_normal; +// Note: uses default parameters zero mean and unit standard deviation. + +// We could also do the same for another distribution, for example: +using boost::math::students_t_distribution; +typedef students_t_distribution my_students_t; + +int main() +{ + cout << "quantile(my_normal(), 0.05); = " << quantile(my_normal(), 0.05) << endl; // 0.05 is argument within normal range. + cout << "quantile(my_normal(), 0.); = " << quantile(my_normal(), 0.) << endl; // argument zero, so expect infinity. + cout << "quantile(my_normal(), 0.); = " << quantile(my_normal(), 0.F) << endl; // argument zero, so expect infinity. + + cout << "quantile(my_students_t(), 0.); = " << quantile(my_students_t(-1), 0.F) << endl; // 'bad' argument negative, so expect NaN. + + // Construct a (0, 1) normal distribution that ignores all errors, + // returning NaN, infinity, zero, or best guess, + // and NOT setting errno. + normal_distribution my_normal2(0.L, 1.L); // explicit parameters for distribution. + cout << "quantile(my_normal2(), 0.); = " << quantile(my_normal2, 0.01) << endl; // argument 0.01, so result finite. + cout << "quantile(my_normal2(), 0.); = " << quantile(my_normal2, 0.) << endl; // argument zero, so expect infinity. + + return 0; +} + +/* + +Output: + +quantile(my_normal(), 0.05); = -1.64485 +quantile(my_normal(), 0.); = -1.#INF +quantile(my_normal(), 0.); = -1.#INF +quantile(my_students_t(), 0.); = 1.#QNAN +quantile(my_normal2(), 0.); = -2.32635 +quantile(my_normal2(), 0.); = -1.#INF + +*/ diff --git a/example/error_policy_example.cpp b/example/error_policy_example.cpp new file mode 100644 index 000000000..cbe875481 --- /dev/null +++ b/example/error_policy_example.cpp @@ -0,0 +1,88 @@ +// example_policy_handling.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// See error_handling_example.cpp for use of +// macro definition to change policy for +// domain_error - negative degrees of freedom argument +// for student's t distribution CDF, +// and catching the exception. + +// See error_handling_policies for more examples. + +// Boost +#include +using boost::math::students_t_distribution; // Probability of students_t(df, t). +using boost::math::students_t; // Probability of students_t(df, t) convenience typedef for double. + +// std +#include + using std::cout; + using std::endl; + +#include + using std::exception; + +using boost::math::policies::policy; +using boost::math::policies::domain_error; +using boost::math::policies::ignore_error; + +// Define a (bad?) policy to ignore domain errors ('bad' arguments): +typedef policy< + domain_error + > my_policy; + +// Define my distribution with this different domain error policy: +typedef students_t_distribution my_students_t; + +int main() +{ // Example of error handling of bad argument(s) to a distribution. + cout << "Example error handling using Student's t function. " << endl; + + double degrees_of_freedom = -1; double t = -1.; // Two 'bad' arguments! + + try + { + cout << "Probability of ignore_error Student's t is " << cdf(my_students_t(degrees_of_freedom), t) << endl; + cout << "Probability of default error policy Student's t is " << endl; + // BY contrast the students_t distribution default domain error policy is to throw, + cout << cdf(students_t(-1), -1) << endl; // so this will throw. + /*` + Message from thrown exception was: + Error in function boost::math::students_t_distribution::students_t_distribution: + Degrees of freedom argument is -1, but must be > 0 ! + */ + + // We could also define a 'custom' distribution + // with an "ignore overflow error policy" in a single statement: + using boost::math::policies::overflow_error; + students_t_distribution > > students_t_no_throw(-1); + + } + catch(const std::exception& e) + { + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + + return 0; +} // int main() + +/* + +Output: + +Example error handling using Student's t function. +Probability of ignore_error Student's t is 1.#QNAN +Probability of default error policy Student's t is +Message from thrown exception was: + Error in function boost::math::students_t_distribution::students_t_distribution: + Degrees of freedom argument is -1, but must be > 0 ! + +*/ diff --git a/example/f_test.cpp b/example/f_test.cpp new file mode 100644 index 000000000..186741c3e --- /dev/null +++ b/example/f_test.cpp @@ -0,0 +1,184 @@ +// Copyright John Maddock 2006 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +#include +#include + +void f_test( + double sd1, // Sample 1 std deviation + double sd2, // Sample 2 std deviation + double N1, // Sample 1 size + double N2, // Sample 2 size + double alpha) // Significance level +{ + // + // An F test applied to two sets of data. + // We are testing the null hypothesis that the + // standard deviation of the samples is equal, and + // that any variation is down to chance. We can + // also test the alternative hypothesis that any + // difference is not down to chance. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm + // + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "____________________________________\n" + "F test for equal standard deviations\n" + "____________________________________\n\n"; + cout << setprecision(5); + cout << "Sample 1:\n"; + cout << setw(55) << left << "Number of Observations" << "= " << N1 << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd1 << "\n\n"; + cout << "Sample 2:\n"; + cout << setw(55) << left << "Number of Observations" << "= " << N2 << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd2 << "\n\n"; + // + // Now we can calculate and output some stats: + // + // F-statistic: + double F = (sd1 / sd2); + F *= F; + cout << setw(55) << left << "Test Statistic" << "= " << F << "\n\n"; + // + // Finally define our distribution, and get the probability: + // + fisher_f dist(N1 - 1, N2 - 1); + double p = cdf(dist, F); + cout << setw(55) << left << "CDF of test statistic: " << "= " + << setprecision(3) << scientific << p << "\n"; + double ucv = quantile(complement(dist, alpha)); + double ucv2 = quantile(complement(dist, alpha / 2)); + double lcv = quantile(dist, alpha); + double lcv2 = quantile(dist, alpha / 2); + cout << setw(55) << left << "Upper Critical Value at alpha: " << "= " + << setprecision(3) << scientific << ucv << "\n"; + cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << ucv2 << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha: " << "= " + << setprecision(3) << scientific << lcv << "\n"; + cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= " + << setprecision(3) << scientific << lcv2 << "\n\n"; + // + // Finally print out results of null and alternative hypothesis: + // + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + cout << "Standard deviations are unequal (two sided test) "; + if((ucv2 < F) || (lcv2 > F)) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Standard deviation 1 is less than standard deviation 2 "; + if(lcv > F) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Standard deviation 1 is greater than standard deviation 2 "; + if(ucv < F) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; +} + +int main() +{ + // + // Run tests for ceramic strength data: + // see http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm + // The data for this case study were collected by Said Jahanmir of the + // NIST Ceramics Division in 1996 in connection with a NIST/industry + // ceramics consortium for strength optimization of ceramic strength. + // + f_test(65.54909, 61.85425, 240, 240, 0.05); + // + // And again for the process change comparison: + // see http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm + // A new procedure to assemble a device is introduced and tested for + // possible improvement in time of assembly. The question being addressed + // is whether the standard deviation of the new assembly process (sample 2) is + // better (i.e., smaller) than the standard deviation for the old assembly + // process (sample 1). + // + f_test(4.9082, 2.5874, 11, 9, 0.05); + return 0; +} + +/* + +Output: + +____________________________________ +F test for equal standard deviations +____________________________________ + +Sample 1: +Number of Observations = 240 +Sample Standard Deviation = 65.549 + +Sample 2: +Number of Observations = 240 +Sample Standard Deviation = 61.854 + +Test Statistic = 1.123 + +CDF of test statistic: = 8.148e-001 +Upper Critical Value at alpha: = 1.238e+000 +Upper Critical Value at alpha/2: = 1.289e+000 +Lower Critical Value at alpha: = 8.080e-001 +Lower Critical Value at alpha/2: = 7.756e-001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Standard deviations are unequal (two sided test) REJECTED +Standard deviation 1 is less than standard deviation 2 REJECTED +Standard deviation 1 is greater than standard deviation 2 REJECTED + + +____________________________________ +F test for equal standard deviations +____________________________________ + +Sample 1: +Number of Observations = 11.00000 +Sample Standard Deviation = 4.90820 + +Sample 2: +Number of Observations = 9.00000 +Sample Standard Deviation = 2.58740 + +Test Statistic = 3.59847 + +CDF of test statistic: = 9.589e-001 +Upper Critical Value at alpha: = 3.347e+000 +Upper Critical Value at alpha/2: = 4.295e+000 +Lower Critical Value at alpha: = 3.256e-001 +Lower Critical Value at alpha/2: = 2.594e-001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Standard deviations are unequal (two sided test) REJECTED +Standard deviation 1 is less than standard deviation 2 REJECTED +Standard deviation 1 is greater than standard deviation 2 NOT REJECTED + +*/ + diff --git a/example/find_location_example.cpp b/example/find_location_example.cpp new file mode 100644 index 000000000..ab47dc6ea --- /dev/null +++ b/example/find_location_example.cpp @@ -0,0 +1,170 @@ +// find_location.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example of finding location (mean) +// for normal (Gaussian) & Cauchy distribution. + +// Note that this file contains Quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[find_location1 +/*` +First we need some includes to access the normal distribution, +the algorithms to find location (and some std output of course). +*/ + +#include // for normal_distribution + using boost::math::normal; // typedef provides default type is double. +#include // for cauchy_distribution + using boost::math::cauchy; // typedef provides default type is double. +#include + using boost::math::find_location; + using boost::math::complement; // Needed if you want to use the complement version. + using boost::math::policies::policy; + +#include + using std::cout; using std::endl; +#include + using std::setw; using std::setprecision; +#include + using std::numeric_limits; +//] [/find_location1] + +int main() +{ + cout << "Example: Find location (mean)." << endl; + try + { +//[find_location2 +/*` +For this example, we will use the standard normal distribution, +with mean (location) zero and standard deviation (scale) unity. +This is also the default for this implementation. +*/ + normal N01; // Default 'standard' normal distribution with zero mean and + double sd = 1.; // normal default standard deviation is 1. +/*`Suppose we want to find a different normal distribution whose mean is shifted +so that only fraction p (here 0.001 or 0.1%) are below a certain chosen limit +(here -2, two standard deviations). +*/ + double z = -2.; // z to give prob p + double p = 0.001; // only 0.1% below z + + cout << "Normal distribution with mean = " << N01.location() + << ", standard deviation " << N01.scale() + << ", has " << "fraction <= " << z + << ", p = " << cdf(N01, z) << endl; + cout << "Normal distribution with mean = " << N01.location() + << ", standard deviation " << N01.scale() + << ", has " << "fraction > " << z + << ", p = " << cdf(complement(N01, z)) << endl; // Note: uses complement. +/*` +[pre +Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501 +Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725 +] +We can now use ''find_location'' to give a new offset mean. +*/ + double l = find_location(z, p, sd); + cout << "offset location (mean) = " << l << endl; +/*` +that outputs: +[pre +offset location (mean) = 1.09023 +] +showing that we need to shift the mean just over one standard deviation from its previous value of zero. + +Then we can check that we have achieved our objective +by constructing a new distribution +with the offset mean (but same standard deviation): +*/ + normal np001pc(l, sd); // Same standard_deviation (scale) but with mean (location) shifted. +/*` +And re-calculating the fraction below our chosen limit. +*/ +cout << "Normal distribution with mean = " << l + << " has " << "fraction <= " << z + << ", p = " << cdf(np001pc, z) << endl; + cout << "Normal distribution with mean = " << l + << " has " << "fraction > " << z + << ", p = " << cdf(complement(np001pc, z)) << endl; +/*` +[pre +Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001 +Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999 +] + +[h4 Controlling Error Handling from find_location] +We can also control the policy for handling various errors. +For example, we can define a new (possibly unwise) +policy to ignore domain errors ('bad' arguments). + +Unless we are using the boost::math namespace, we will need: +*/ + using boost::math::policies::policy; + using boost::math::policies::domain_error; + using boost::math::policies::ignore_error; + +/*` +Using a typedef is often convenient, especially if it is re-used, +although it is not required, as the various examples below show. +*/ + typedef policy > ignore_domain_policy; + // find_location with new policy, using typedef. + l = find_location(z, p, sd, ignore_domain_policy()); + // Default policy policy<>, needs "using boost::math::policies::policy;" + l = find_location(z, p, sd, policy<>()); + // Default policy, fully specified. + l = find_location(z, p, sd, boost::math::policies::policy<>()); + // A new policy, ignoring domain errors, without using a typedef. + l = find_location(z, p, sd, policy >()); +/*` +If we want to use a probability that is the +[link complements complement of our probability], +we should not even think of writing `find_location(z, 1 - p, sd)`, +but, [link why_complements to avoid loss of accuracy], use the complement version. +*/ + z = 2.; + double q = 0.95; // = 1 - p; // complement. + l = find_location(complement(z, q, sd)); + + normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(location) shifted + cout << "Normal distribution with mean = " << l << " has " + << "fraction <= " << z << " = " << cdf(np95pc, z) << endl; + cout << "Normal distribution with mean = " << l << " has " + << "fraction > " << z << " = " << cdf(complement(np95pc, z)) << endl; + //] [/find_location2] + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + + + +//[find_location_example_output +/*` +[pre +Example: Find location (mean). +Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501 +Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725 +offset location (mean) = 1.09023 +Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001 +Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999 +Normal distribution with mean = 0.355146 has fraction <= 2 = 0.95 +Normal distribution with mean = 0.355146 has fraction > 2 = 0.05 +] +*/ +//] [/find_location_example_output] diff --git a/example/find_mean_and_sd_normal.cpp b/example/find_mean_and_sd_normal.cpp new file mode 100644 index 000000000..adef44697 --- /dev/null +++ b/example/find_mean_and_sd_normal.cpp @@ -0,0 +1,408 @@ +// find_mean_and_sd_normal.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example of finding mean or sd for normal distribution. + +// Note that this file contains Quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[normal_std +/*` +First we need some includes to access the normal distribution, +the algorithms to find location and scale +(and some std output of course). +*/ + +#include // for normal_distribution + using boost::math::normal; // typedef provides default type is double. +#include // for cauchy_distribution + using boost::math::cauchy; // typedef provides default type is double. +#include + using boost::math::find_location; +#include + using boost::math::find_scale; + using boost::math::complement; + using boost::math::policies::policy; + +#include + using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint; +#include + using std::setw; using std::setprecision; +#include + using std::numeric_limits; +//] [/normal_std Quickbook] + +int main() +{ + cout << "Find_location (mean) and find_scale (standard deviation) examples." << endl; + try + { + +//[normal_find_location_and_scale_eg + +/*` +[h4 Using find_location and find_scale to meet dispensing and measurement specifications] + +Consider an example from K Krishnamoorthy, +Handbook of Statistical Distributions with Applications, +ISBN 1-58488-635-8, (2006) p 126, example 10.3.7. + +"A machine is set to pack 3 kg of ground beef per pack. +Over a long period of time it is found that the average packed was 3 kg +with a standard deviation of 0.1 kg. +Assume the packing is normally distributed." + +We start by constructing a normal distribution with the given parameters: +*/ + +double mean = 3.; // kg +double standard_deviation = 0.1; // kg +normal packs(mean, standard_deviation); +/*`We can then find the fraction (or %) of packages that weigh more than 3.1 kg. +*/ + +double max_weight = 3.1; // kg +cout << "Percentage of packs > " << max_weight << " is " +<< cdf(complement(packs, max_weight)) * 100. << endl; // P(X > 3.1) + +/*`We might want to ensure that 95% of packs are over a minimum weight specification, +then we want the value of the mean such that P(X < 2.9) = 0.05. + +Using the mean of 3 kg, we can estimate +the fraction of packs that fail to meet the specification of 2.9 kg. +*/ + +double minimum_weight = 2.9; +cout <<"Fraction of packs <= " << minimum_weight << " with a mean of " << mean + << " is " << cdf(complement(packs, minimum_weight)) << endl; +// fraction of packs <= 2.9 with a mean of 3 is 0.841345 + +/*`This is 0.84 - more than the target fraction of 0.95. +If we want 95% to be over the minimum weight, +what should we set the mean weight to be? + +Using the KK StatCalc program supplied with the book and the method given on page 126 gives 3.06449. + +We can confirm this by constructing a new distribution which we call 'xpacks' +with a safety margin mean of 3.06449 thus: +*/ +double over_mean = 3.06449; +normal xpacks(over_mean, standard_deviation); +cout << "Fraction of packs >= " << minimum_weight +<< " with a mean of " << xpacks.mean() + << " is " << cdf(complement(xpacks, minimum_weight)) << endl; +// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005 + +/*`Using this Math Toolkit, we can calculate the required mean directly thus: +*/ +double under_fraction = 0.05; // so 95% are above the minimum weight mean - sd = 2.9 +double low_limit = standard_deviation; +double offset = mean - low_limit - quantile(packs, under_fraction); +double nominal_mean = mean + offset; +// mean + (mean - low_limit - quantile(packs, under_fraction)); + +normal nominal_packs(nominal_mean, standard_deviation); +cout << "Setting the packer to " << nominal_mean << " will mean that " + << "fraction of packs >= " << minimum_weight + << " is " << cdf(complement(nominal_packs, minimum_weight)) << endl; +// Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95 + +/*` +This calculation is generalized as the free function called +[link math_toolkit.dist.dist_ref.dist_algorithms find_location]. + +To use this we will need to +*/ + +#include + using boost::math::find_location; +/*`and then use find_location function to find safe_mean, +& construct a new normal distribution called 'goodpacks'. +*/ +double safe_mean = find_location(minimum_weight, under_fraction, standard_deviation); +normal good_packs(safe_mean, standard_deviation); +/*`with the same confirmation as before: +*/ +cout << "Setting the packer to " << nominal_mean << " will mean that " + << "fraction of packs >= " << minimum_weight + << " is " << cdf(complement(good_packs, minimum_weight)) << endl; +// Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95 + +/*` +[h4 Using Cauchy-Lorentz instead of normal distribution] + +After examining the weight distribution of a large number of packs, we might decide that, +after all, the assumption of a normal distribution is not really justified. +We might find that the fit is better to a __cauchy_distrib. +This distribution has wider 'wings', so that whereas most of the values +are closer to the mean than the normal, there are also more values than 'normal' +that lie further from the mean than the normal. + +This might happen because a larger than normal lump of meat is either included or excluded. + +We first create a __cauchy_distrib with the original mean and standard deviation, +and estimate the fraction that lie below our minimum weight specification. +*/ + +cauchy cpacks(mean, standard_deviation); +cout << "Cauchy Setting the packer to " << mean << " will mean that " + << "fraction of packs >= " << minimum_weight + << " is " << cdf(complement(cpacks, minimum_weight)) << endl; +// Cauchy Setting the packer to 3 will mean that fraction of packs >= 2.9 is 0.75 + +/*`Note that far fewer of the packs meet the specification, only 75% instead of 95%. +Now we can repeat the find_location, using the cauchy distribution as template parameter, +in place of the normal used above. +*/ + +double lc = find_location(minimum_weight, under_fraction, standard_deviation); +cout << "find_location(minimum_weight, over fraction, standard_deviation); " << lc << endl; +// find_location(minimum_weight, over fraction, packs.standard_deviation()); 3.53138 +/*`Note that the safe_mean setting needs to be much higher, 3.53138 instead of 3.06449, +so we will make rather less profit. + +And again confirm that the fraction meeting specification is as expected. +*/ +cauchy goodcpacks(lc, standard_deviation); +cout << "Cauchy Setting the packer to " << lc << " will mean that " + << "fraction of packs >= " << minimum_weight + << " is " << cdf(complement(goodcpacks, minimum_weight)) << endl; +// Cauchy Setting the packer to 3.53138 will mean that fraction of packs >= 2.9 is 0.95 + +/*`Finally we could estimate the effect of a much tighter specification, +that 99% of packs met the specification. +*/ + +cout << "Cauchy Setting the packer to " + << find_location(minimum_weight, 0.99, standard_deviation) + << " will mean that " + << "fraction of packs >= " << minimum_weight + << " is " << cdf(complement(goodcpacks, minimum_weight)) << endl; + +/*`Setting the packer to 3.13263 will mean that fraction of packs >= 2.9 is 0.99, +but will more than double the mean loss from 0.0644 to 0.133 kg per pack. + +Of course, this calculation is not limited to packs of meat, it applies to dispensing anything, +and it also applies to a 'virtual' material like any measurement. + +The only caveat is that the calculation assumes that the standard deviation (scale) is known with +a reasonably low uncertainty, something that is not so easy to ensure in practice. +And that the distribution is well defined, __normal_distrib or __cauchy_distrib, or some other. + +If one is simply dispensing a very large number of packs, +then it may be feasible to measure the weight of hundreds or thousands of packs. +With a healthy 'degrees of freedom', the confidence intervals for the standard deviation +are not too wide, typically about + and - 10% for hundreds of observations. + +For other applications, where it is more difficult or expensive to make many observations, +the confidence intervals are depressingly wide. + +See [link math_toolkit.dist.stat_tut.weg.cs_eg.chi_sq_intervals Confidence Intervals on the standard deviation] +for a worked example +[@../../example/chi_square_std_dev_test.cpp chi_square_std_dev_test.cpp] +of estimating these intervals. + + +[h4 Changing the scale or standard deviation] + +Alternatively, we could invest in a better (more precise) packer +(or measuring device) with a lower standard deviation, or scale. + +This might cost more, but would reduce the amount we have to 'give away' +in order to meet the specification. + +To estimate how much better (how much smaller standard deviation) it would have to be, +we need to get the 5% quantile to be located at the under_weight limit, 2.9 +*/ +double p = 0.05; // wanted p th quantile. +cout << "Quantile of " << p << " = " << quantile(packs, p) + << ", mean = " << packs.mean() << ", sd = " << packs.standard_deviation() << endl; +/*` +Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 + +With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551 kg, +a little below our target of 2.9 kg. +So we know that the standard deviation is going to have to be smaller. + +Let's start by guessing that it (now 0.1) needs to be halved, to a standard deviation of 0.05 kg. +*/ +normal pack05(mean, 0.05); +cout << "Quantile of " << p << " = " << quantile(pack05, p) + << ", mean = " << pack05.mean() << ", sd = " << pack05.standard_deviation() << endl; +// Quantile of 0.05 = 2.91776, mean = 3, sd = 0.05 + +cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean + << " and standard deviation of " << pack05.standard_deviation() + << " is " << cdf(complement(pack05, minimum_weight)) << endl; +// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.97725 +/*` +So 0.05 was quite a good guess, but we are a little over the 2.9 target, +so the standard deviation could be a tiny bit more. So we could do some +more guessing to get closer, say by increasing standard deviation to 0.06 kg, +constructing another new distribution called pack06. +*/ +normal pack06(mean, 0.06); +cout << "Quantile of " << p << " = " << quantile(pack06, p) + << ", mean = " << pack06.mean() << ", sd = " << pack06.standard_deviation() << endl; +// Quantile of 0.05 = 2.90131, mean = 3, sd = 0.06 + +cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean + << " and standard deviation of " << pack06.standard_deviation() + << " is " << cdf(complement(pack06, minimum_weight)) << endl; +// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95221 +/*` +Now we are getting really close, but to do the job properly, +we might need to use root finding method, for example the tools provided, +and used elsewhere, in the Math Toolkit, see +[link math_toolkit.toolkit.internals1.roots2 Root Finding Without Derivatives]. + +But in this (normal) distribution case, we can and should be even smarter +and make a direct calculation. +*/ + +/*`Our required limit is minimum_weight = 2.9 kg, often called the random variate z. +For a standard normal distribution, then probability p = N((minimum_weight - mean) / sd). + +We want to find the standard deviation that would be required to meet this limit, +so that the p th quantile is located at z (minimum_weight). +In this case, the 0.05 (5%) quantile is at 2.9 kg pack weight, when the mean is 3 kg, +ensuring that 0.95 (95%) of packs are above the minimum weight. + +Rearranging, we can directly calculate the required standard deviation: +*/ +normal N01; // standard normal distribution with meamn zero and unit standard deviation. +p = 0.05; +double qp = quantile(N01, p); +double sd95 = (minimum_weight - mean) / qp; + +cout << "For the "<< p << "th quantile to be located at " + << minimum_weight << ", would need a standard deviation of " << sd95 << endl; +// For the 0.05th quantile to be located at 2.9, would need a standard deviation of 0.0607957 + +/*`We can now construct a new (normal) distribution pack95 for the 'better' packer, +and check that our distribution will meet the specification. +*/ + +normal pack95(mean, sd95); +cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean + << " and standard deviation of " << pack95.standard_deviation() + << " is " << cdf(complement(pack95, minimum_weight)) << endl; +// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95 + +/*`This calculation is generalized in the free function find_scale, +as shown below, giving the same standard deviation. +*/ +double ss = find_scale(minimum_weight, under_fraction, packs.mean()); +cout << "find_scale(minimum_weight, under_fraction, packs.mean()); " << ss << endl; +// find_scale(minimum_weight, under_fraction, packs.mean()); 0.0607957 + +/*`If we had defined an over_fraction, or percentage that must pass specification +*/ +double over_fraction = 0.95; +/*`And (wrongly) written + + double sso = find_scale(minimum_weight, over_fraction, packs.mean()); + +With the default policy, we would get a message like + +[pre +Message from thrown exception was: + Error in function boost::math::find_scale(double, double, double, Policy): + Computed scale (-0.060795683191176959) is <= 0! Was the complement intended? +] + +But this would return a *negative* standard deviation - obviously impossible. +The probability should be 1 - over_fraction, not over_fraction, thus: +*/ + +double ss1o = find_scale(minimum_weight, 1 - over_fraction, packs.mean()); +cout << "find_scale(minimum_weight, under_fraction, packs.mean()); " << ss1o << endl; +// find_scale(minimum_weight, under_fraction, packs.mean()); 0.0607957 + +/*`But notice that using '1 - over_fraction' - will lead to a +[link why_complements loss of accuracy, especially if over_fraction was close to unity.] +In this (very common) case, we should instead use the __complements, +giving the most accurate result. +*/ + +double ssc = find_scale(complement(minimum_weight, over_fraction, packs.mean())); +cout << "find_scale(complement(minimum_weight, over_fraction, packs.mean())); " << ssc << endl; +// find_scale(complement(minimum_weight, over_fraction, packs.mean())); 0.0607957 + +/*`Note that our guess of 0.06 was close to the accurate value of 0.060795683191176959. + +We can again confirm our prediction thus: +*/ + +normal pack95c(mean, ssc); +cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean + << " and standard deviation of " << pack95c.standard_deviation() + << " is " << cdf(complement(pack95c, minimum_weight)) << endl; +// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95 + +/*`Notice that these two deceptively simple questions: + +* Do we over-fill to make sure we meet a minimum specification (or under-fill to avoid an overdose)? + +and/or + +* Do we measure better? + +are actually extremely common. + +The weight of beef might be replaced by a measurement of more or less anything, +from drug tablet content, Apollo landing rocket firing, X-ray treatment doses... + +The scale can be variation in dispensing or uncertainty in measurement. +*/ +//] [/normal_find_location_and_scale_eg Quickbook end] + + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + + +/* + +Output is: + +Find_location and find_scale examples. +Percentage of packs > 3.1 is 15.8655 +Fraction of packs <= 2.9 with a mean of 3 is 0.841345 +Fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005 +Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95 +Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95 +Cauchy Setting the packer to 3 will mean that fraction of packs >= 2.9 is 0.75 +find_location(minimum_weight, over fraction, standard_deviation); 3.53138 +Cauchy Setting the packer to 3.53138 will mean that fraction of packs >= 2.9 is 0.95 +Cauchy Setting the packer to -0.282052 will mean that fraction of packs >= 2.9 is 0.95 +Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 +Quantile of 0.05 = 2.91776, mean = 3, sd = 0.05 +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.97725 +Quantile of 0.05 = 2.90131, mean = 3, sd = 0.06 +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95221 +For the 0.05th quantile to be located at 2.9, would need a standard deviation of 0.0607957 +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95 +find_scale(minimum_weight, under_fraction, packs.mean()); 0.0607957 +find_scale(minimum_weight, under_fraction, packs.mean()); 0.0607957 +find_scale(complement(minimum_weight, over_fraction, packs.mean())); 0.0607957 +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95 + +*/ + + + diff --git a/example/find_root_example.cpp b/example/find_root_example.cpp new file mode 100644 index 000000000..abcb936cb --- /dev/null +++ b/example/find_root_example.cpp @@ -0,0 +1,239 @@ +// find_root_example.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example of using root finding. + +// Note that this file contains Quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[root_find1 +/*` +First we need some includes to access the normal distribution +(and some std output of course). +*/ + +#include // root finding. + +#include // for normal_distribution + using boost::math::normal; // typedef provides default type is double. + +#include + using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint; +#include + using std::setw; using std::setprecision; +#include + using std::numeric_limits; +//] //[/root_find1] + +namespace boost{ namespace math { namespace tools +{ + +template +inline std::pair bracket_and_solve_root(F f, // functor + const T& guess, + const T& factor, + bool rising, + Tol tol, // binary functor specifying termination when tol(min, max) becomes true. + // eps_tolerance most suitable for this continuous function + boost::uintmax_t& max_iter); // explicit (rather than default) max iterations. +// return interval as a pair containing result. + +namespace detail +{ + + // Functor for finding standard deviation: + template + struct standard_deviation_functor + { + standard_deviation_functor(RealType m, RealType s, RealType d) + : mean(m), standard_deviation(s) + { + } + RealType operator()(const RealType& sd) + { // Unary functor - the function whose root is to be found. + if(sd <= tools::min_value()) + { // + return 1; + } + normal_distribution t(mean, sd); + RealType qa = quantile(complement(t, alpha)); + RealType qb = quantile(complement(t, beta)); + qa += qb; + qa *= qa; + qa *= ratio; + qa -= (df + 1); + return qa; + } // operator() + RealType mean; + RealType standard_deviation; + }; // struct standard_deviation_functor +} // namespace detail + +template +RealType normal_distribution::find_standard_deviation( + RealType difference_from_mean, + RealType mean, + RealType sd, + RealType hint) // Best guess available - current sd if none better? +{ + static const char* function = "boost::math::normal_distribution<%1%>::find_standard_deviation"; + + // Check for domain errors: + RealType error_result; + if(false == detail::check_probability( + function, sd, &error_result, Policy()) + ) + return error_result; + + if(hint <= 0) + { // standard deviation can never be negative. + hint = 1; + } + + detail::standard_deviation_functor f(mean, sd, difference_from_mean); + tools::eps_tolerance tol(policies::digits()); + boost::uintmax_t max_iter = 100; + std::pair r = tools::bracket_and_solve_root(f, hint, RealType(2), false, tol, max_iter, Policy()); + RealType result = r.first + (r.second - r.first) / 2; + if(max_iter == 100) + { + policies::raise_evaluation_error(function, "Unable to locate solution in a reasonable time:" + " either there is no answer to how many degrees of freedom are required" + " or the answer is infinite. Current best guess is %1%", result, Policy()); + } + return result; +} // find_standard_deviation +} // namespace tools +} // namespace math +} // namespace boost + + +int main() +{ + cout << "Example: Normal distribution, root finding."; + try + { + +//[root_find2 + +/*`A machine is set to pack 3 kg of ground beef per pack. +Over a long period of time it is found that the average packed was 3 kg +with a standard deviation of 0.1 kg. +Assuming the packing is normally distributed, +we can find the fraction (or %) of packages that weigh more than 3.1 kg. +*/ + +double mean = 3.; // kg +double standard_deviation = 0.1; // kg +normal packs(mean, standard_deviation); + +double max_weight = 3.1; // kg +cout << "Percentage of packs > " << max_weight << " is " +<< cdf(complement(packs, max_weight)) << endl; // P(X > 3.1) + +double under_weight = 2.9; +cout <<"fraction of packs <= " << under_weight << " with a mean of " << mean + << " is " << cdf(complement(packs, under_weight)) << endl; +// fraction of packs <= 2.9 with a mean of 3 is 0.841345 +// This is 0.84 - more than the target 0.95 +// Want 95% to be over this weight, so what should we set the mean weight to be? +// KK StatCalc says: +double over_mean = 3.0664; +normal xpacks(over_mean, standard_deviation); +cout << "fraction of packs >= " << under_weight +<< " with a mean of " << xpacks.mean() + << " is " << cdf(complement(xpacks, under_weight)) << endl; +// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005 +double under_fraction = 0.05; // so 95% are above the minimum weight mean - sd = 2.9 +double low_limit = standard_deviation; +double offset = mean - low_limit - quantile(packs, under_fraction); +double nominal_mean = mean + offset; + +normal nominal_packs(nominal_mean, standard_deviation); +cout << "Setting the packer to " << nominal_mean << " will mean that " + << "fraction of packs >= " << under_weight + << " is " << cdf(complement(nominal_packs, under_weight)) << endl; + +/*` +Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95. + +Setting the packer to 3.13263 will mean that fraction of packs >= 2.9 is 0.99, +but will more than double the mean loss from 0.0644 to 0.133. + +Alternatively, we could invest in a better (more precise) packer with a lower standard deviation. + +To estimate how much better (how much smaller standard deviation) it would have to be, +we need to get the 5% quantile to be located at the under_weight limit, 2.9 +*/ +double p = 0.05; // wanted p th quantile. +cout << "Quantile of " << p << " = " << quantile(packs, p) + << ", mean = " << packs.mean() << ", sd = " << packs.standard_deviation() << endl; // +/*` +Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 + +With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551 kg, +a little below our target of 2.9 kg. +So we know that the standard deviation is going to have to be smaller. + +Let's start by guessing that it (now 0.1) needs to be halved, to a standard deviation of 0.05 +*/ +normal pack05(mean, 0.05); +cout << "Quantile of " << p << " = " << quantile(pack05, p) + << ", mean = " << pack05.mean() << ", sd = " << pack05.standard_deviation() << endl; + +cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean + << " and standard deviation of " << pack05.standard_deviation() + << " is " << cdf(complement(pack05, under_weight)) << endl; +// +/*` +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.9772 + +So 0.05 was quite a good guess, but we are a little over the 2.9 target, +so the standard deviation could be a tiny bit more. So we could do some +more guessing to get closer, say by increasing to 0.06 +*/ + +normal pack06(mean, 0.06); +cout << "Quantile of " << p << " = " << quantile(pack06, p) + << ", mean = " << pack06.mean() << ", sd = " << pack06.standard_deviation() << endl; + +cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean + << " and standard deviation of " << pack06.standard_deviation() + << " is " << cdf(complement(pack06, under_weight)) << endl; +/*` +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.9522 + +Now we are getting really close, but to do the job properly, +we could use root finding method, for example the tools provided, and used elsewhere, +in the Math Toolkit, see +[link math_toolkit.toolkit.internals1.roots2 Root Finding Without Derivatives]. + +But in this normal distribution case, we could be even smarter and make a direct calculation. +*/ +//] [/root_find2] + + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + +/* + +Output is: + + + +*/ diff --git a/example/find_scale_example.cpp b/example/find_scale_example.cpp new file mode 100644 index 000000000..83d12fd08 --- /dev/null +++ b/example/find_scale_example.cpp @@ -0,0 +1,180 @@ +// find_scale.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example of finding scale (standard deviation) for normal (Gaussian). + +// Note that this file contains Quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[find_scale1 +/*` +First we need some includes to access the __normal_distrib, +the algorithms to find scale (and some std output of course). +*/ + +#include // for normal_distribution + using boost::math::normal; // typedef provides default type is double. +#include + using boost::math::find_scale; + using boost::math::complement; // Needed if you want to use the complement version. + using boost::math::policies::policy; // Needed to specify the error handling policy. + +#include + using std::cout; using std::endl; +#include + using std::setw; using std::setprecision; +#include + using std::numeric_limits; +//] [/find_scale1] + +int main() +{ + cout << "Example: Find scale (standard deviation)." << endl; + try + { +//[find_scale2 +/*` +For this example, we will use the standard __normal_distrib, +with location (mean) zero and standard deviation (scale) unity. +Conveniently, this is also the default for this implementation's constructor. +*/ + normal N01; // Default 'standard' normal distribution with zero mean + double sd = 1.; // and standard deviation is 1. +/*`Suppose we want to find a different normal distribution with standard deviation +so that only fraction p (here 0.001 or 0.1%) are below a certain chosen limit +(here -2. standard deviations). +*/ + double z = -2.; // z to give prob p + double p = 0.001; // only 0.1% below z = -2 + + cout << "Normal distribution with mean = " << N01.location() // aka N01.mean() + << ", standard deviation " << N01.scale() // aka N01.standard_deviation() + << ", has " << "fraction <= " << z + << ", p = " << cdf(N01, z) << endl; + cout << "Normal distribution with mean = " << N01.location() + << ", standard deviation " << N01.scale() + << ", has " << "fraction > " << z + << ", p = " << cdf(complement(N01, z)) << endl; // Note: uses complement. +/*` +[pre +Normal distribution with mean = 0 has fraction <= -2, p = 0.0227501 +Normal distribution with mean = 0 has fraction > -2, p = 0.97725 +] +Noting that p = 0.02 instead of our target of 0.001, +we can now use `find_scale` to give a new standard deviation. +*/ + double l = N01.location(); + double s = find_scale(z, p, l); + cout << "scale (standard deviation) = " << s << endl; +/*` +that outputs: +[pre +scale (standard deviation) = 0.647201 +] +showing that we need to reduce the standard deviation from 1. to 0.65. + +Then we can check that we have achieved our objective +by constructing a new distribution +with the new standard deviation (but same zero mean): +*/ + normal np001pc(N01.location(), s); +/*` +And re-calculating the fraction below (and above) our chosen limit. +*/ + cout << "Normal distribution with mean = " << l + << " has " << "fraction <= " << z + << ", p = " << cdf(np001pc, z) << endl; + cout << "Normal distribution with mean = " << l + << " has " << "fraction > " << z + << ", p = " << cdf(complement(np001pc, z)) << endl; +/*` +[pre +Normal distribution with mean = 0 has fraction <= -2, p = 0.001 +Normal distribution with mean = 0 has fraction > -2, p = 0.999 +] + +[h4 Controlling how Errors from find_scale are handled] +We can also control the policy for handling various errors. +For example, we can define a new (possibly unwise) +policy to ignore domain errors ('bad' arguments). + +Unless we are using the boost::math namespace, we will need: +*/ + using boost::math::policies::policy; + using boost::math::policies::domain_error; + using boost::math::policies::ignore_error; + +/*` +Using a typedef is convenient, especially if it is re-used, +although it is not required, as the various examples below show. +*/ + typedef policy > ignore_domain_policy; + // find_scale with new policy, using typedef. + l = find_scale(z, p, l, ignore_domain_policy()); + // Default policy policy<>, needs using boost::math::policies::policy; + + l = find_scale(z, p, l, policy<>()); + // Default policy, fully specified. + l = find_scale(z, p, l, boost::math::policies::policy<>()); + // New policy, without typedef. + l = find_scale(z, p, l, policy >()); +/*` +If we want to express a probability, say 0.999, that is a complement, `1 - p` +we should not even think of writing `find_scale(z, 1 - p, l)`, +but [link why_complements instead], use the __complements version. +*/ + z = -2.; + double q = 0.999; // = 1 - p; // complement of 0.001. + sd = find_scale(complement(z, q, l)); + + normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(scale) shifted + cout << "Normal distribution with mean = " << l << " has " + << "fraction <= " << z << " = " << cdf(np95pc, z) << endl; + cout << "Normal distribution with mean = " << l << " has " + << "fraction > " << z << " = " << cdf(complement(np95pc, z)) << endl; + +/*` +Sadly, it is all too easy to get probabilities the wrong way round, +when you may get a warning like this: +[pre +Message from thrown exception was: + Error in function boost::math::find_scale(complement(double, double, double, Policy)): + Computed scale (-0.48043523852179076) is <= 0! Was the complement intended? +] +The default error handling policy is to throw an exception with this message, +but if you chose a policy to ignore the error, +the (impossible) negative scale is quietly returned. +*/ +//] [/find_scale2] + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + +//[find_scale_example_output +/*` +[pre +Example: Find scale (standard deviation). +Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501 +Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725 +scale (standard deviation) = 0.647201 +Normal distribution with mean = 0 has fraction <= -2, p = 0.001 +Normal distribution with mean = 0 has fraction > -2, p = 0.999 +Normal distribution with mean = 0.946339 has fraction <= -2 = 0.001 +Normal distribution with mean = 0.946339 has fraction > -2 = 0.999 +] +*/ +//] [/find_scale_example_output] diff --git a/example/neg_binom_confidence_limits.cpp b/example/neg_binom_confidence_limits.cpp new file mode 100644 index 000000000..3908a323c --- /dev/null +++ b/example/neg_binom_confidence_limits.cpp @@ -0,0 +1,179 @@ +// neg_binomial_confidence_limits.cpp + +// Copyright John Maddock 2006 +// Copyright Paul A. Bristow 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Caution: this file contains quickbook markup as well as code +// and comments, don't change any of the special comment markups! + +//[neg_binomial_confidence_limits + +/*` + +First we need some includes to access the negative binomial distribution +(and some basic std output of course). + +*/ + +#include +using boost::math::negative_binomial; + +#include +using std::cout; using std::endl; +#include +using std::setprecision; +using std::setw; using std::left; using std::fixed; using std::right; + +/*` +First define a table of significance levels: these are the +probabilities that the true occurrence frequency lies outside the calculated +interval: +*/ + + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + +/*` + +Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence +that the true occurence frequency lies *inside* the calculated interval. + +We need a function to calculate and print confidence limits +for an observed frequency of occurrence +that follows a negative binomial distribution. + +*/ + +void confidence_limits_on_frequency(unsigned trials, unsigned successes) +{ + // trials = Total number of trials. + // successes = Total number of observed successes. + // failures = trials - successes. + // success_fraction = successes /trials. + // Print out general info: + cout << + "______________________________________________\n" + "2-Sided Confidence Limits For Success Fraction\n" + "______________________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of trials" << " = " << trials << "\n"; + cout << setw(40) << left << "Number of successes" << " = " << successes << "\n"; + cout << setw(40) << left << "Number of failures" << " = " << trials - successes << "\n"; + cout << setw(40) << left << "Observed frequency of occurrence" << " = " << double(successes) / trials << "\n"; + + // Print table header: + cout << "\n\n" + "___________________________________________\n" + "Confidence Lower Upper\n" + " Value (%) Limit Limit\n" + "___________________________________________\n"; + + +/*` +And now for the important part - the bounds themselves. +For each value of /alpha/, we call `find_lower_bound_on_p` and +`find_upper_bound_on_p` to obtain lower and upper bounds respectively. +Note that since we are calculating a two-sided interval, +we must divide the value of alpha in two. Had we been calculating a +single-sided interval, for example: ['"Calculate a lower bound so that we are P% +sure that the true occurrence frequency is greater than some value"] +then we would *not* have divided by two. + +*/ + + // Now print out the upper and lower limits for the alpha table values. + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // Calculate bounds: + double lower = negative_binomial::find_lower_bound_on_p(trials, successes, alpha[i]/2); + double upper = negative_binomial::find_upper_bound_on_p(trials, successes, alpha[i]/2); + // Print limits: + cout << fixed << setprecision(5) << setw(15) << right << lower; + cout << fixed << setprecision(5) << setw(15) << right << upper << endl; + } + cout << endl; +} // void confidence_limits_on_frequency(unsigned trials, unsigned successes) + +/*` + +And then call confidence_limits_on_frequency with increasing numbers of trials, +but always the same success fraction 0.1, or 1 in 10. + +*/ + +int main() +{ + confidence_limits_on_frequency(20, 2); // 20 trials, 2 successes, 2 in 20, = 1 in 10 = 0.1 success fraction. + confidence_limits_on_frequency(200, 20); // More trials, but same 0.1 success fraction. + confidence_limits_on_frequency(2000, 200); // Many more trials, but same 0.1 success fraction. + + return 0; +} // int main() + +//] [/negative_binomial_confidence_limits_eg end of Quickbook in C++ markup] + +/* + +______________________________________________ +2-Sided Confidence Limits For Success Fraction +______________________________________________ +Number of trials = 20 +Number of successes = 2 +Number of failures = 18 +Observed frequency of occurrence = 0.1 +___________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +___________________________________________ + 50.000 0.04812 0.13554 + 75.000 0.03078 0.17727 + 90.000 0.01807 0.22637 + 95.000 0.01235 0.26028 + 99.000 0.00530 0.33111 + 99.900 0.00164 0.41802 + 99.990 0.00051 0.49202 + 99.999 0.00016 0.55574 +______________________________________________ +2-Sided Confidence Limits For Success Fraction +______________________________________________ +Number of trials = 200 +Number of successes = 20 +Number of failures = 180 +Observed frequency of occurrence = 0.1000000 +___________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +___________________________________________ + 50.000 0.08462 0.11350 + 75.000 0.07580 0.12469 + 90.000 0.06726 0.13695 + 95.000 0.06216 0.14508 + 99.000 0.05293 0.16170 + 99.900 0.04343 0.18212 + 99.990 0.03641 0.20017 + 99.999 0.03095 0.21664 +______________________________________________ +2-Sided Confidence Limits For Success Fraction +______________________________________________ +Number of trials = 2000 +Number of successes = 200 +Number of failures = 1800 +Observed frequency of occurrence = 0.1000000 +___________________________________________ +Confidence Lower Upper + Value (%) Limit Limit +___________________________________________ + 50.000 0.09536 0.10445 + 75.000 0.09228 0.10776 + 90.000 0.08916 0.11125 + 95.000 0.08720 0.11352 + 99.000 0.08344 0.11802 + 99.900 0.07921 0.12336 + 99.990 0.07577 0.12795 + 99.999 0.07282 0.13206 +*/ \ No newline at end of file diff --git a/example/neg_binomial_sample_sizes.cpp b/example/neg_binomial_sample_sizes.cpp new file mode 100644 index 000000000..def7b1553 --- /dev/null +++ b/example/neg_binomial_sample_sizes.cpp @@ -0,0 +1,202 @@ +// neg_binomial_sample_sizes.cpp + +// Copyright Paul A. Bristow 2007 +// Copyright John Maddock 2006 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +using boost::math::negative_binomial; + +// Default RealType is double so this permits use of: +double find_minimum_number_of_trials( +double k, // number of failures (events), k >= 0. +double p, // fraction of trails for which event occurs, 0 <= p <= 1. +double probability); // probability threshold, 0 <= probability <= 1. + +#include +using std::cout; +using std::endl; +using std::fixed; +using std::right; +#include +using std::setprecision; +using std::setw; + +//[neg_binomial_sample_sizes + +/*` +It centres around a routine that prints out +a table of minimum sample sizes for various probability thresholds: +*/ + void find_number_of_trials(double failures, double p); +/*` +First define a table of significance levels: these are the maximum +acceptable probability that /failure/ or fewer events will be observed. +*/ + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; +/*` +Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence +that the desired number of failures will be observed. + +Much of the rest of the program is pretty-printing, the important part +is in the calculation of minimum number of trials required for each +value of alpha using: + + (int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]); + +*/ + +/*` + +find_minimum_number_of_trials returns a double, +so ceil rounds this up to ensure we have an integral minimum number of trials. + +*/ + +void find_number_of_trials(double failures, double p) +{ + // trials = number of trials + // failures = number of failures before achieving required success(es). + // p = success fraction (0 <= p <= 1.). + // + // Calculate how many trials we need to ensure the + // required number of failures DOES exceed "failures". + + cout << "\n""Target number of failures = " << failures; + cout << ", Success fraction = " << 100 * p << "%" << endl; + // Print table header: + cout << "\n\n" + "____________________________\n" + "Confidence Min Number\n" + " Value (%) Of Trials \n" + "____________________________\n"; + // Now print out the data for the alpha table values. + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { // Confidence values %: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]) << " " + // find_minimum_number_of_trials + << setw(6) << right + << (int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i])) + << endl; + } + cout << endl; +} // void find_number_of_trials(double failures, double p) + +/*` finally we can produce some tables of minimum trials for the chosen confidence levels: +*/ + +int main() +{ + find_number_of_trials(5, 0.5); + find_number_of_trials(50, 0.5); + find_number_of_trials(500, 0.5); + find_number_of_trials(50, 0.1); + find_number_of_trials(500, 0.1); + find_number_of_trials(5, 0.9); + + return 0; +} // int main() + +//] [/neg_binomial_sample_sizes.cpp end of Quickbook in C++ markup] + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\neg_binomial_sample_sizes.exe" +Target number of failures = 5, Success fraction = 50% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 11 + 75.000 14 + 90.000 17 + 95.000 18 + 99.000 22 + 99.900 27 + 99.990 31 + 99.999 36 +Target number of failures = 50.000, Success fraction = 50.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 101 + 75.000 109 + 90.000 115 + 95.000 119 + 99.000 128 + 99.900 137 + 99.990 146 + 99.999 154 +Target number of failures = 500.000, Success fraction = 50.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 1001 + 75.000 1023 + 90.000 1043 + 95.000 1055 + 99.000 1078 + 99.900 1104 + 99.990 1126 + 99.999 1146 +Target number of failures = 50.000, Success fraction = 10.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 56 + 75.000 58 + 90.000 60 + 95.000 61 + 99.000 63 + 99.900 66 + 99.990 68 + 99.999 71 +Target number of failures = 500.000, Success fraction = 10.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 556 + 75.000 562 + 90.000 567 + 95.000 570 + 99.000 576 + 99.900 583 + 99.990 588 + 99.999 594 +Target number of failures = 5.000, Success fraction = 90.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 57 + 75.000 73 + 90.000 91 + 95.000 103 + 99.000 127 + 99.900 159 + 99.990 189 + 99.999 217 +Target number of failures = 5.000, Success fraction = 40.000% +____________________________ +Confidence Min Number + Value (%) Of Trials +____________________________ + 50.000 10 + 75.000 11 + 90.000 13 + 95.000 15 + 99.000 18 + 99.900 21 + 99.990 25 + 99.999 28 + +*/ diff --git a/example/negative_binomial_example1.cpp b/example/negative_binomial_example1.cpp new file mode 100644 index 000000000..ce5c50999 --- /dev/null +++ b/example/negative_binomial_example1.cpp @@ -0,0 +1,521 @@ +// negative_binomial_example1.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example 1 of using negative_binomial distribution. + +//[negative_binomial_eg1_1 + +/*` +Based on [@http://en.wikipedia.org/wiki/Negative_binomial_distribution +a problem by Dr. Diane Evans, +Professor of Mathematics at Rose-Hulman Institute of Technology]. + +Pat is required to sell candy bars to raise money for the 6th grade field trip. +There are thirty houses in the neighborhood, +and Pat is not supposed to return home until five candy bars have been sold. +So the child goes door to door, selling candy bars. +At each house, there is a 0.4 probability (40%) of selling one candy bar +and a 0.6 probability (60%) of selling nothing. + +What is the probability mass (density) function (pdf) for selling the last (fifth) +candy bar at the nth house? + +The Negative Binomial(r, p) distribution describes the probability of k failures +and r successes in k+r Bernoulli(p) trials with success on the last trial. +(A [@http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli trial] +is one with only two possible outcomes, success of failure, +and p is the probability of success). +See also [@ http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution] +and [@http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml Bernoulli applications]. + +In this example, we will deliberately produce a variety of calculations +and outputs to demonstrate the ways that the negative binomial distribution +can be implemented with this library: it is also deliberately over-commented. + +First we need to #define macros to control the error and discrete handling policies. +For this simple example, we want to avoid throwing +an exception (the default policy) and just return infinity. +We want to treat the distribution as if it was continuous, +so we choose a discrete_quantile policy of real, +rather than the default policy integer_round_outwards. +*/ +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error +#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real +/*` +After that we need some includes to provide easy access to the negative binomial distribution, +[caution It is vital to #include distributions etc *after* the above #defines] +and we need some std library iostream, of course. +*/ +#include + // for negative_binomial_distribution + using boost::math::negative_binomial; // typedef provides default type is double. + using ::boost::math::pdf; // Probability mass function. + using ::boost::math::cdf; // Cumulative density function. + using ::boost::math::quantile; + +#include + using std::cout; using std::endl; + using std::noshowpoint; using std::fixed; using std::right; using std::left; +#include + using std::setprecision; using std::setw; + +#include + using std::numeric_limits; +//] [negative_binomial_eg1_1] + +int main() +{ + cout <<"Selling candy bars - using the negative binomial distribution." + << "\nby Dr. Diane Evans," + "\nProfessor of Mathematics at Rose-Hulman Institute of Technology," + << "\nsee http://en.wikipedia.org/wiki/Negative_binomial_distribution\n" + << endl; + cout << endl; + cout.precision(5); + // None of the values calculated have a useful accuracy as great this, but + // INF shows wrongly with < 5 ! + // https://connect.microsoft.com/VisualStudio/feedback/ViewFeedback.aspx?FeedbackID=240227 +//[negative_binomial_eg1_2 +/*` +It is always sensible to use try and catch blocks because defaults policies are to +throw an exception if anything goes wrong. + +A simple catch block (see below) will ensure that you get a +helpful error message instead of an abrupt program abort. +*/ + try + { +/*` +Selling five candy bars means getting five successes, so successes r = 5. +The total number of trials (n, in this case, houses visited) this takes is therefore + = sucesses + failures or k + r = k + 5. +*/ + double sales_quota = 5; // Pat's sales quota - successes (r). +/*` +At each house, there is a 0.4 probability (40%) of selling one candy bar +and a 0.6 probability (60%) of selling nothing. +*/ + double success_fraction = 0.4; // success_fraction (p) - so failure_fraction is 0.6. +/*` +The Negative Binomial(r, p) distribution describes the probability of k failures +and r successes in k+r Bernoulli(p) trials with success on the last trial. +(A [@http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli trial] +is one with only two possible outcomes, success of failure, +and p is the probability of success). + +We therefore start by constructing a negative binomial distribution +with parameters sales_quota (required successes) and probability of success. +*/ + negative_binomial nb(sales_quota, success_fraction); // type double by default. +/*` +To confirm, display the success_fraction & successes parameters of the distribution. +*/ + cout << "Pat has a sales per house success rate of " << success_fraction + << ".\nTherefore he would, on average, sell " << nb.success_fraction() * 100 + << " bars after trying 100 houses." << endl; + + int all_houses = 30; // The number of houses on the estate. + + cout << "With a success rate of " << nb.success_fraction() + << ", he might expect, on average,\n" + "to need to visit about " << success_fraction * all_houses + << " houses in order to sell all " << nb.successes() << " bars. " << endl; +/*` +[pre +Pat has a sales per house success rate of 0.4. +Therefore he would, on average, sell 40 bars after trying 100 houses. +With a success rate of 0.4, he might expect, on average, +to need to visit about 12 houses in order to sell all 5 bars. +] + +The random variable of interest is the number of houses +that must be visited to sell five candy bars, +so we substitute k = n - 5 into a negative_binomial(5, 0.4) +and obtain the [link math.dist.pdf probability mass (density) function (pdf or pmf)] +of the distribution of houses visited. +Obviously, the best possible case is that Pat makes sales on all the first five houses. + +We calculate this using the pdf function: +*/ + cout << "Probability that Pat finishes on the " << sales_quota << "th house is " + << pdf(nb, 5 - sales_quota) << endl; // == pdf(nb, 0) +/*` +Of course, he could not finish on fewer than 5 houses because he must sell 5 candy bars. +So the 5th house is the first that he could possibly finish on. + +To finish on or before the 8th house, Pat must finish at the 5th, 6th, 7th or 8th house. +The probability that he will finish on *exactly* ( == ) on any house +is the Probability Density Function (pdf). +*/ + cout << "Probability that Pat finishes on the 6th house is " + << pdf(nb, 6 - sales_quota) << endl; + cout << "Probability that Pat finishes on the 7th house is " + << pdf(nb, 7 - sales_quota) << endl; + cout << "Probability that Pat finishes on the 8th house is " + << pdf(nb, 8 - sales_quota) << endl; +/*` +[pre +Probability that Pat finishes on the 6th house is 0.03072 +Probability that Pat finishes on the 7th house is 0.055296 +Probability that Pat finishes on the 8th house is 0.077414 +] + +The sum of the probabilities for these houses is the Cumulative Distribution Function (cdf). +We can calculate it by adding the individual probabilities. +*/ + cout << "Probability that Pat finishes on or before the 8th house is sum " + "\n" << "pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = " + // Sum each of the mass/density probabilities for houses sales_quota = 5, 6, 7, & 8. + << pdf(nb, 5 - sales_quota) // 0 failures. + + pdf(nb, 6 - sales_quota) // 1 failure. + + pdf(nb, 7 - sales_quota) // 2 failures. + + pdf(nb, 8 - sales_quota) // 3 failures. + << endl; +/*`[pre +pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367 +] + +Or, usually better, by using the negative binomial *cumulative* distribution function. +*/ + cout << "\nProbability of selling his quota of " << sales_quota + << " bars\non or before the " << 8 << "th house is " + << cdf(nb, 8 - sales_quota) << endl; +/*`[pre +Probability of selling his quota of 5 bars on or before the 8th house is 0.17367 +]*/ + cout << "\nProbability that Pat finishes exactly on the 10th house is " + << pdf(nb, 10 - sales_quota) << endl; + cout << "\nProbability of selling his quota of " << sales_quota + << " bars\non or before the " << 10 << "th house is " + << cdf(nb, 10 - sales_quota) << endl; +/*` +[pre +Probability that Pat finishes exactly on the 10th house is 0.10033 +Probability of selling his quota of 5 bars on or before the 10th house is 0.3669 +]*/ + cout << "Probability that Pat finishes exactly on the 11th house is " + << pdf(nb, 11 - sales_quota) << endl; + cout << "\nProbability of selling his quota of " << sales_quota + << " bars\non or before the " << 11 << "th house is " + << cdf(nb, 11 - sales_quota) << endl; +/*`[pre +Probability that Pat finishes on the 11th house is 0.10033 +Probability of selling his quota of 5 candy bars +on or before the 11th house is 0.46723 +]*/ + cout << "Probability that Pat finishes exactly on the 12th house is " + << pdf(nb, 12 - sales_quota) << endl; + + cout << "\nProbability of selling his quota of " << sales_quota + << " bars\non or before the " << 12 << "th house is " + << cdf(nb, 12 - sales_quota) << endl; +/*`[pre +Probability that Pat finishes on the 12th house is 0.094596 +Probability of selling his quota of 5 candy bars +on or before the 12th house is 0.56182 +] +Finally consider the risk of Pat not selling his quota of 5 bars +even after visiting all the houses. +Calculate the probability that he /will/ sell on +or before the last house: +Calculate the probability that he would sell all his quota on the very last house. +*/ + cout << "Probability that Pat finishes on the " << all_houses + << " house is " << pdf(nb, all_houses - sales_quota) << endl; +/*` +Probability of selling his quota of 5 bars on the 30th house is +[pre +Probability that Pat finishes on the 30 house is 0.00069145 +] +when he'd be very unlucky indeed! + +What is the probability that Pat exhausts all 30 houses in the neighborhood, +and *still* doesn't sell the required 5 candy bars? +*/ + cout << "\nProbability of selling his quota of " << sales_quota + << " bars\non or before the " << all_houses << "th house is " + << cdf(nb, all_houses - sales_quota) << endl; +/*` +[pre +Probability of selling his quota of 5 bars +on or before the 30th house is 0.99849 +] + +/*`So the risk of failing even after visiting all the houses is 1 - this probability, + ``1 - cdf(nb, all_houses - sales_quota`` +But using this expression may cause serious inaccuracy, +so it would be much better to use the complement of the cdf: +So the risk of failing even at, or after, the 31th (non-existent) houses is 1 - this probability, + ``1 - cdf(nb, all_houses - sales_quota)`` +But using this expression may cause serious inaccuracy. +So it would be much better to use the complement of the cdf. +[link why_complements Why complements?] +*/ + cout << "\nProbability of failing to sell his quota of " << sales_quota + << " bars\neven after visiting all " << all_houses << " houses is " + << cdf(complement(nb, all_houses - sales_quota)) << endl; +/*` +[pre +Probability of failing to sell his quota of 5 bars +even after visiting all 30 houses is 0.0015101 +] +We can also use the quantile (percentile), the inverse of the cdf, to +predict which house Pat will finish on. So for the 8th house: +*/ + double p = cdf(nb, (8 - sales_quota)); + cout << "Probability of meeting sales quota on or before 8th house is "<< p << endl; +/*` +[pre +Probability of meeting sales quota on or before 8th house is 0.174 +] +*/ + cout << "If the confidence of meeting sales quota is " << p + << ", then the finishing house is " << quantile(nb, p) + sales_quota << endl; + + cout<< " quantile(nb, p) = " << quantile(nb, p) << endl; +/*` +[pre +If the confidence of meeting sales quota is 0.17367, then the finishing house is 8 +] +Demanding absolute certainty that all 5 will be sold, +implies an infinite number of trials. +(Of course, there are only 30 houses on the estate, +so he can't ever be *certain* of selling his quota). +*/ + cout << "If the confidence of meeting sales quota is " << 1. + << ", then the finishing house is " << quantile(nb, 1) + sales_quota << endl; + // 1.#INF == infinity. +/*`[pre +If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF +] +And similarly for a few other probabilities: +*/ + cout << "If the confidence of meeting sales quota is " << 0. + << ", then the finishing house is " << quantile(nb, 0.) + sales_quota << endl; + + cout << "If the confidence of meeting sales quota is " << 0.5 + << ", then the finishing house is " << quantile(nb, 0.5) + sales_quota << endl; + + cout << "If the confidence of meeting sales quota is " << 1 - 0.00151 // 30 th + << ", then the finishing house is " << quantile(nb, 1 - 0.00151) + sales_quota << endl; +/*` +[pre +If the confidence of meeting sales quota is 0, then the finishing house is 5 +If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337 +If the confidence of meeting sales quota is 0.99849, then the finishing house is 30 +] + +Notice that because we chose a discrete quantile policy of real, +the result can be an 'unreal' fractional house. + +If the opposite is true, we don't want to assume any confidence, then this is tantamount +to assuming that all the first sales_quota trials will be successful sales. +*/ + cout << "If confidence of meeting quota is zero\n(we assume all houses are successful sales)" + ", then finishing house is " << sales_quota << endl; +/*` +[pre +If confidence of meeting quota is zero (we assume all houses are successful sales), then finishing house is 5 +If confidence of meeting quota is 0, then finishing house is 5 +] +We can list quantiles for a few probabilities: +*/ + + double ps[] = {0., 0.001, 0.01, 0.05, 0.1, 0.5, 0.9, 0.95, 0.99, 0.999, 1.}; + // Confidence as fraction = 1-alpha, as percent = 100 * (1-alpha[i]) % + cout.precision(3); + for (int i = 0; i < sizeof(ps)/sizeof(ps[0]); i++) + { + cout << "If confidence of meeting quota is " << ps[i] + << ", then finishing house is " << quantile(nb, ps[i]) + sales_quota + << endl; + } + +/*` +[pre +If confidence of meeting quota is 0, then finishing house is 5 +If confidence of meeting quota is 0.001, then finishing house is 5 +If confidence of meeting quota is 0.01, then finishing house is 5 +If confidence of meeting quota is 0.05, then finishing house is 6.2 +If confidence of meeting quota is 0.1, then finishing house is 7.06 +If confidence of meeting quota is 0.5, then finishing house is 11.3 +If confidence of meeting quota is 0.9, then finishing house is 17.8 +If confidence of meeting quota is 0.95, then finishing house is 20.1 +If confidence of meeting quota is 0.99, then finishing house is 24.8 +If confidence of meeting quota is 0.999, then finishing house is 31.1 +If confidence of meeting quota is 1, then finishing house is 1.#INF +] + +We could have applied a ceil function to obtain a 'worst case' integer value for house. +``ceil(quantile(nb, ps[i]))`` + +Or, if we had used the default discrete quantile policy, integer_outside, by omitting +``#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real`` +we would have achieved the same effect. + +The real result gives some suggestion which house is most likely. +For example, compare the real and integer_outside for 95% confidence. + +[pre +If confidence of meeting quota is 0.95, then finishing house is 20.1 +If confidence of meeting quota is 0.95, then finishing house is 21 +] +The real value 20.1 is much closer to 20 than 21, so integer_outside is pessimistic. +We could also use integer_round_nearest policy to suggest that 20 is more likely. + +Finally, we can tabulate the probability for the last sale being exactly on each house. +*/ + cout << "\nHouse for " << sales_quota << "th (last) sale. Probability (%)" << endl; + cout.precision(5); + for (int i = (int)sales_quota; i < all_houses+1; i++) + { + cout << left << setw(3) << i << " " << setw(8) << cdf(nb, i - sales_quota) << endl; + } + cout << endl; +/*` +[pre +House for 5 th (last) sale. Probability (%) +5 0.01024 +6 0.04096 +7 0.096256 +8 0.17367 +9 0.26657 +10 0.3669 +11 0.46723 +12 0.56182 +13 0.64696 +14 0.72074 +15 0.78272 +16 0.83343 +17 0.874 +18 0.90583 +19 0.93039 +20 0.94905 +21 0.96304 +22 0.97342 +23 0.98103 +24 0.98655 +25 0.99053 +26 0.99337 +27 0.99539 +28 0.99681 +29 0.9978 +30 0.99849 +] + +As noted above, using a catch block is always a good idea, even if you do not expect to use it. +*/ + } + catch(const std::exception& e) + { // Since we have set an overflow policy of ignore_error, + // an overflow exception should never be thrown. + std::cout << "\nMessage from thrown exception was:\n " << e.what() << std::endl; +/*` +For example, without a ignore domain error policy, if we asked for ``pdf(nb, -1)`` for example, we would get: +[pre +Message from thrown exception was: + Error in function boost::math::pdf(const negative_binomial_distribution&, double): + Number of failures argument is -1, but must be >= 0 ! +] +*/ +//] [/ negative_binomial_eg1_2] + } + return 0; +} // int main() + + +/* + +Output is: + +Selling candy bars - using the negative binomial distribution. +by Dr. Diane Evans, +Professor of Mathematics at Rose-Hulman Institute of Technology, +see http://en.wikipedia.org/wiki/Negative_binomial_distribution +Pat has a sales per house success rate of 0.4. +Therefore he would, on average, sell 40 bars after trying 100 houses. +With a success rate of 0.4, he might expect, on average, +to need to visit about 12 houses in order to sell all 5 bars. +Probability that Pat finishes on the 5th house is 0.01024 +Probability that Pat finishes on the 6th house is 0.03072 +Probability that Pat finishes on the 7th house is 0.055296 +Probability that Pat finishes on the 8th house is 0.077414 +Probability that Pat finishes on or before the 8th house is sum +pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367 +Probability of selling his quota of 5 bars +on or before the 8th house is 0.17367 +Probability that Pat finishes exactly on the 10th house is 0.10033 +Probability of selling his quota of 5 bars +on or before the 10th house is 0.3669 +Probability that Pat finishes exactly on the 11th house is 0.10033 +Probability of selling his quota of 5 bars +on or before the 11th house is 0.46723 +Probability that Pat finishes exactly on the 12th house is 0.094596 +Probability of selling his quota of 5 bars +on or before the 12th house is 0.56182 +Probability that Pat finishes on the 30 house is 0.00069145 +Probability of selling his quota of 5 bars +on or before the 30th house is 0.99849 +Probability of failing to sell his quota of 5 bars +even after visiting all 30 houses is 0.0015101 +Probability of meeting sales quota on or before 8th house is 0.17367 +If the confidence of meeting sales quota is 0.17367, then the finishing house is 8 + quantile(nb, p) = 3 +If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF +If the confidence of meeting sales quota is 0, then the finishing house is 5 +If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337 +If the confidence of meeting sales quota is 0.99849, then the finishing house is 30 +If confidence of meeting quota is zero +(we assume all houses are successful sales), then finishing house is 5 +If confidence of meeting quota is 0, then finishing house is 5 +If confidence of meeting quota is 0.001, then finishing house is 5 +If confidence of meeting quota is 0.01, then finishing house is 5 +If confidence of meeting quota is 0.05, then finishing house is 6.2 +If confidence of meeting quota is 0.1, then finishing house is 7.06 +If confidence of meeting quota is 0.5, then finishing house is 11.3 +If confidence of meeting quota is 0.9, then finishing house is 17.8 +If confidence of meeting quota is 0.95, then finishing house is 20.1 +If confidence of meeting quota is 0.99, then finishing house is 24.8 +If confidence of meeting quota is 0.999, then finishing house is 31.1 +If confidence of meeting quota is 1, then finishing house is 1.#J +House for 5th (last) sale. Probability (%) +5 0.01024 +6 0.04096 +7 0.096256 +8 0.17367 +9 0.26657 +10 0.3669 +11 0.46723 +12 0.56182 +13 0.64696 +14 0.72074 +15 0.78272 +16 0.83343 +17 0.874 +18 0.90583 +19 0.93039 +20 0.94905 +21 0.96304 +22 0.97342 +23 0.98103 +24 0.98655 +25 0.99053 +26 0.99337 +27 0.99539 +28 0.99681 +29 0.9978 +30 0.99849 + +*/ + + + + + + diff --git a/example/negative_binomial_example2.cpp b/example/negative_binomial_example2.cpp new file mode 100644 index 000000000..aba8350c3 --- /dev/null +++ b/example/negative_binomial_example2.cpp @@ -0,0 +1,182 @@ +// negative_binomial_example2.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Simple example demonstrating use of the Negative Binomial Distribution. + +#include + using boost::math::negative_binomial_distribution; + using boost::math::negative_binomial; // typedef + +// In a sequence of trials or events +// (Bernoulli, independent, yes or no, succeed or fail) +// with success_fraction probability p, +// negative_binomial is the probability that k or fewer failures +// preceed the r th trial's success. + +#include +using std::cout; +using std::endl; +using std::setprecision; +using std::showpoint; +using std::setw; +using std::left; +using std::right; +#include +using std::numeric_limits; + +int main() +{ + cout << "Negative_binomial distribution - simple example 2" << endl; + // Construct a negative binomial distribution with: + // 8 successes (r), success fraction (p) 0.25 = 25% or 1 in 4 successes. + negative_binomial mynbdist(8, 0.25); // Shorter method using typedef. + + // Display (to check) properties of the distribution just constructed. + cout << "mean(mynbdist) = " << mean(mynbdist) << endl; // 24 + cout << "mynbdist.successes() = " << mynbdist.successes() << endl; // 8 + // r th successful trial, after k failures, is r + k th trial. + cout << "mynbdist.success_fraction() = " << mynbdist.success_fraction() << endl; + // success_fraction = failures/successes or k/r = 0.25 or 25%. + cout << "mynbdist.percent success = " << mynbdist.success_fraction() * 100 << "%" << endl; + // Show as % too. + // Show some cumulative distribution function values for failures k = 2 and 8 + cout << "cdf(mynbdist, 2.) = " << cdf(mynbdist, 2.) << endl; // 0.000415802001953125 + cout << "cdf(mynbdist, 8.) = " << cdf(mynbdist, 8.) << endl; // 0.027129956288263202 + cout << "cdf(complement(mynbdist, 8.)) = " << cdf(complement(mynbdist, 8.)) << endl; // 0.9728700437117368 + // Check that cdf plus its complement is unity. + cout << "cdf + complement = " << cdf(mynbdist, 8.) + cdf(complement(mynbdist, 8.)) << endl; // 1 + // Note: No complement for pdf! + + // Compare cdf with sum of pdfs. + double sum = 0.; // Calculate the sum of all the pdfs, + int k = 20; // for 20 failures + for(signed i = 0; i <= k; ++i) + { + sum += pdf(mynbdist, double(i)); + } + // Compare with the cdf + double cdf8 = cdf(mynbdist, static_cast(k)); + double diff = sum - cdf8; // Expect the diference to be very small. + cout << setprecision(17) << "Sum pdfs = " << sum << ' ' // sum = 0.40025683281803698 + << ", cdf = " << cdf(mynbdist, static_cast(k)) // cdf = 0.40025683281803687 + << ", difference = " // difference = 0.50000000000000000 + << setprecision(1) << diff/ (std::numeric_limits::epsilon() * sum) + << " in epsilon units." << endl; + + // Note: Use boost::math::tools::epsilon rather than std::numeric_limits + // to cover RealTypes that do not specialize numeric_limits. + +//[neg_binomial_example2 + + // Print a table of values that can be used to plot + // using Excel, or some other superior graphical display tool. + + cout.precision(17); // Use max_digits10 precision, the maximum available for a reference table. + cout << showpoint << endl; // include trailing zeros. + // This is a maximum possible precision for the type (here double) to suit a reference table. + int maxk = static_cast(2. * mynbdist.successes() / mynbdist.success_fraction()); + // This maxk shows most of the range of interest, probability about 0.0001 to 0.999. + cout << "\n"" k pdf cdf""\n" << endl; + for (int k = 0; k < maxk; k++) + { + cout << right << setprecision(17) << showpoint + << right << setw(3) << k << ", " + << left << setw(25) << pdf(mynbdist, static_cast(k)) + << left << setw(25) << cdf(mynbdist, static_cast(k)) + << endl; + } + cout << endl; +//] [/ neg_binomial_example2] + return 0; +} // int main() + +/* + +Output is: + +negative_binomial distribution - simple example 2 +mean(mynbdist) = 24 +mynbdist.successes() = 8 +mynbdist.success_fraction() = 0.25 +mynbdist.percent success = 25% +cdf(mynbdist, 2.) = 0.000415802001953125 +cdf(mynbdist, 8.) = 0.027129956288263202 +cdf(complement(mynbdist, 8.)) = 0.9728700437117368 +cdf + complement = 1 +Sum pdfs = 0.40025683281803692 , cdf = 0.40025683281803687, difference = 0.25 in epsilon units. + +//[neg_binomial_example2_1 + k pdf cdf + 0, 1.5258789062500000e-005 1.5258789062500003e-005 + 1, 9.1552734375000000e-005 0.00010681152343750000 + 2, 0.00030899047851562522 0.00041580200195312500 + 3, 0.00077247619628906272 0.0011882781982421875 + 4, 0.0015932321548461918 0.0027815103530883789 + 5, 0.0028678178787231476 0.0056493282318115234 + 6, 0.0046602040529251142 0.010309532284736633 + 7, 0.0069903060793876605 0.017299838364124298 + 8, 0.0098301179241389001 0.027129956288263202 + 9, 0.013106823898851871 0.040236780187115073 + 10, 0.016711200471036140 0.056947980658151209 + 11, 0.020509200578089786 0.077457181236241013 + 12, 0.024354675686481652 0.10181185692272265 + 13, 0.028101548869017230 0.12991340579173993 + 14, 0.031614242477644432 0.16152764826938440 + 15, 0.034775666725408917 0.19630331499479325 + 16, 0.037492515688331451 0.23379583068312471 + 17, 0.039697957787645101 0.27349378847076977 + 18, 0.041352039362130305 0.31484582783290005 + 19, 0.042440250924291580 0.35728607875719176 + 20, 0.042970754060845245 0.40025683281803687 + 21, 0.042970754060845225 0.44322758687888220 + 22, 0.042482450037426581 0.48571003691630876 + 23, 0.041558918514873783 0.52726895543118257 + 24, 0.040260202311284021 0.56752915774246648 + 25, 0.038649794218832620 0.60617895196129912 + 26, 0.036791631035234917 0.64297058299653398 + 27, 0.034747651533277427 0.67771823452981139 + 28, 0.032575923312447595 0.71029415784225891 + 29, 0.030329307911589130 0.74062346575384819 + 30, 0.028054609818219924 0.76867807557206813 + 31, 0.025792141284492545 0.79447021685656061 + 32, 0.023575629142856460 0.81804584599941710 + 33, 0.021432390129869489 0.83947823612928651 + 34, 0.019383705779220189 0.85886194190850684 + 35, 0.017445335201298231 0.87630727710980494 + 36, 0.015628112784496322 0.89193538989430121 + 37, 0.013938587078064250 0.90587397697236549 + 38, 0.012379666154859701 0.91825364312722524 + 39, 0.010951243136991251 0.92920488626421649 + 40, 0.0096507830144735539 0.93885566927869002 + 41, 0.0084738582566109364 0.94732952753530097 + 42, 0.0074146259745345548 0.95474415350983555 + 43, 0.0064662435824429246 0.96121039709227851 + 44, 0.0056212231142827853 0.96683162020656122 + 45, 0.0048717266990450708 0.97170334690560634 + 46, 0.0042098073105878630 0.97591315421619418 + 47, 0.0036275999165703964 0.97954075413276465 + 48, 0.0031174686783026818 0.98265822281106729 + 49, 0.0026721160099737302 0.98533033882104104 + 50, 0.0022846591885275322 0.98761499800956853 + 51, 0.0019486798960970148 0.98956367790566557 + 52, 0.0016582516423517923 0.99122192954801736 + 53, 0.0014079495076571762 0.99262987905567457 + 54, 0.0011928461106539983 0.99382272516632852 + 55, 0.0010084971662802015 0.99483122233260868 + 56, 0.00085091948404891532 0.99568214181665760 + 57, 0.00071656377604119542 0.99639870559269883 + 58, 0.00060228420831048650 0.99700098980100937 + 59, 0.00050530624256557675 0.99750629604357488 + 60, 0.00042319397814867202 0.99792949002172360 + 61, 0.00035381791615708398 0.99828330793788067 + 62, 0.00029532382517950324 0.99857863176306016 + 63, 0.00024610318764958566 0.99882473495070978 +//] [neg_binomial_example2_1 end of Quickbook] + +*/ diff --git a/example/normal_misc_examples.cpp b/example/normal_misc_examples.cpp new file mode 100644 index 000000000..04be5b08c --- /dev/null +++ b/example/normal_misc_examples.cpp @@ -0,0 +1,509 @@ +// negative_binomial_example3.cpp + +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example of using normal distribution. + +// Note that this file contains Quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[normal_basic1 +/*` +First we need some includes to access the normal distribution +(and some std output of course). +*/ + +#include // for normal_distribution + using boost::math::normal; // typedef provides default type is double. + +#include + using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint; +#include + using std::setw; using std::setprecision; +#include + using std::numeric_limits; + +int main() +{ + cout << "Example: Normal distribution, Miscellaneous Applications."; + + try + { + { // Traditional tables and values. +/*`Let's start by printing some traditional tables. +*/ + double step = 1.; // in z + double range = 4; // min and max z = -range to +range. + int precision = 17; // traditional tables are only computed to much lower precision. + + // Construct a standard normal distribution s + normal s; // (default mean = zero, and standard deviation = unity) + cout << "Standard normal distribution, mean = "<< s.mean() + << ", standard deviation = " << s.standard_deviation() << endl; + +/*` First the probability distribution function (pdf). +*/ + cout << "Probability distribution function values" << endl; + cout << " z " " pdf " << endl; + cout.precision(5); + for (double z = -range; z < range + step; z += step) + { + cout << left << setprecision(3) << setw(6) << z << " " + << setprecision(precision) << setw(12) << pdf(s, z) << endl; + } + cout.precision(6); // default + /*`And the area under the normal curve from -[infin] up to z, + the cumulative distribution function (cdf). +*/ + // For a standard normal distribution + cout << "Standard normal mean = "<< s.mean() + << ", standard deviation = " << s.standard_deviation() << endl; + cout << "Integral (area under the curve) from - infinity up to z " << endl; + cout << " z " " cdf " << endl; + for (double z = -range; z < range + step; z += step) + { + cout << left << setprecision(3) << setw(6) << z << " " + << setprecision(precision) << setw(12) << cdf(s, z) << endl; + } + cout.precision(6); // default + +/*`And all this you can do with a nanoscopic amount of work compared to +the team of *human computers* toiling with Milton Abramovitz and Irene Stegen +at the US National Bureau of Standards (now [@http://www.nist.gov NIST]). +Starting in 1938, their "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables", +was eventually published in 1964, and has been reprinted numerous times since. +(A major replacement is planned at [@http://dlmf.nist.gov Digital Library of Mathematical Functions]). + +Pretty-printing a traditional 2-dimensional table is left as an exercise for the student, +but why bother now that the Math Toolkit lets you write +*/ + double z = 2.; + cout << "Area for z = " << z << " is " << cdf(s, z) << endl; // to get the area for z. +/*` +Correspondingly, we can obtain the traditional 'critical' values for significance levels. +For the 95% confidence level, the significance level usually called alpha, +is 0.05 = 1 - 0.95 (for a one-sided test), so we can write +*/ + cout << "95% of area has a z below " << quantile(s, 0.95) << endl; + // 95% of area has a z below 1.64485 +/*`and a two-sided test (a comparison between two levels, rather than a one-sided test) + +*/ + cout << "95% of area has a z between " << quantile(s, 0.975) + << " and " << -quantile(s, 0.975) << endl; + // 95% of area has a z between 1.95996 and -1.95996 +/*` + +First, define a table of significance levels: these are the probabilities +that the true occurrence frequency lies outside the calculated interval. + +It is convenient to have an alpha level for the probability that z lies outside just one standard deviation. +This will not be some nice neat number like 0.05, but we can easily calculate it, +*/ + double alpha1 = cdf(s, -1) * 2; // 0.3173105078629142 + cout << setprecision(17) << "Significance level for z == 1 is " << alpha1 << endl; +/*` + and place in our array of favorite alpha values. +*/ + double alpha[] = {0.3173105078629142, // z for 1 standard deviation. + 0.20, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; +/*` + +Confidence value as % is (1 - alpha) * 100 (so alpha 0.05 == 95% confidence) +that the true occurrence frequency lies *inside* the calculated interval. + +*/ + cout << "level of significance (alpha)" << setprecision(4) << endl; + cout << "2-sided 1 -sided z(alpha) " << endl; + for (int i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + cout << setw(15) << alpha[i] << setw(15) << alpha[i] /2 << setw(10) << quantile(complement(s, alpha[i]/2)) << endl; + // Use quantile(complement(s, alpha[i]/2)) to avoid potential loss of accuracy from quantile(s, 1 - alpha[i]/2) + } + cout << endl; + +/*`Notice the distinction between one-sided (also called one-tailed) +where we are using a > *or* < test (and not both) +and considering the area of the tail (integral) from z up to +[infin], +and a two-sided test where we are using two > *and* < tests, and thus considering two tails, +from -[infin] up to z low and z high up to +[infin]. + +So the 2-sided values alpha[i] are calculated using alpha[i]/2. + +If we consider a simple example of alpha = 0.05, then for a two-sided test, +the lower tail area from -[infin] up to -1.96 is 0.025 (alpha/2) +and the upper tail area from +z up to +1.96 is also 0.025 (alpha/2), +and the area between -1.96 up to 12.96 is alpha = 0.95. +and the sum of the two tails is 0.025 + 0.025 = 0.05, + +*/ +//] [/[normal_basic1] + +//[normal_basic2 + +/*`Armed with the cumulative distribution function, we can easily calculate the +easy to remember proportion of values that lie within 1, 2 and 3 standard deviations from the mean. + +*/ + cout.precision(3); + cout << showpoint << "cdf(s, s.standard_deviation()) = " + << cdf(s, s.standard_deviation()) << endl; // from -infinity to 1 sd + cout << "cdf(complement(s, s.standard_deviation())) = " + << cdf(complement(s, s.standard_deviation())) << endl; + cout << "Fraction 1 standard deviation within either side of mean is " + << 1 - cdf(complement(s, s.standard_deviation())) * 2 << endl; + cout << "Fraction 2 standard deviations within either side of mean is " + << 1 - cdf(complement(s, 2 * s.standard_deviation())) * 2 << endl; + cout << "Fraction 3 standard deviations within either side of mean is " + << 1 - cdf(complement(s, 3 * s.standard_deviation())) * 2 << endl; + +/*` +To a useful precision, the 1, 2 & 3 percentages are 68, 95 and 99.7, +and these are worth memorising as useful 'rules of thumb', as, for example, in +[@http://en.wikipedia.org/wiki/Standard_deviation standard deviation]: + +[pre +Fraction 1 standard deviation within either side of mean is 0.683 +Fraction 2 standard deviations within either side of mean is 0.954 +Fraction 3 standard deviations within either side of mean is 0.997 +] + +We could of course get some really accurate values for these +[@http://en.wikipedia.org/wiki/Confidence_interval confidence intervals] +by using cout.precision(15); + +[pre +Fraction 1 standard deviation within either side of mean is 0.682689492137086 +Fraction 2 standard deviations within either side of mean is 0.954499736103642 +Fraction 3 standard deviations within either side of mean is 0.997300203936740 +] + +But before you get too excited about this impressive precision, +don't forget that the *confidence intervals of the standard deviation* are surprisingly wide, +especially if you have estimated the standard deviation from only a few measurements. +*/ +//] [/[normal_basic2] + + +//[normal_bulbs_example1 +/*` +Examples from K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, +ISBN 1 58488 635 8, page 125... implemented using the Math Toolkit library. + +A few very simple examples are shown here: +*/ +// K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, + // ISBN 1 58488 635 8, page 125, example 10.3.5 +/*`Mean lifespan of 100 W bulbs is 1100 h with standard deviation of 100 h. +Assuming, perhaps with little evidence and much faith, that the distribution is normal, +we construct a normal distribution called /bulbs/ with these values: +*/ + double mean_life = 1100.; + double life_standard_deviation = 100.; + normal bulbs(mean_life, life_standard_deviation); + double expected_life = 1000.; + +/*`The we can use the Cumulative distribution function to predict fractions +(or percentages, if * 100) that will last various lifetimes. +*/ + cout << "Fraction of bulbs that will last at best (<=) " // P(X <= 1000) + << expected_life << " is "<< cdf(bulbs, expected_life) << endl; + cout << "Fraction of bulbs that will last at least (>) " // P(X > 1000) + << expected_life << " is "<< cdf(complement(bulbs, expected_life)) << endl; + double min_life = 900; + double max_life = 1200; + cout << "Fraction of bulbs that will last between " + << min_life << " and " << max_life << " is " + << cdf(bulbs, max_life) // P(X <= 1200) + - cdf(bulbs, min_life) << endl; // P(X <= 900) +/*` +[note Real-life failures are often very ab-normal, +with a significant number that 'dead-on-arrival' or suffer failure very early in their life: +the lifetime of the survivors of 'early mortality' may be well described by the normal distribution.] +*/ +//] [/normal_bulbs_example1 Quickbook end] + } + { + // K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, + // ISBN 1 58488 635 8, page 125, Example 10.3.6 + +//[normal_bulbs_example3 +/*`Weekly demand for 5 lb sacks of onions at a store is normally distributed with mean 140 sacks and standard deviation 10. +*/ + double mean = 140.; // sacks per week. + double standard_deviation = 10; + normal sacks(mean, standard_deviation); + + double stock = 160.; // per week. + cout << "Percentage of weeks overstocked " + << cdf(sacks, stock) * 100. << endl; // P(X <=160) + // Percentage of weeks overstocked 97.7 + +/*`So there will be lots of mouldy onions! +So we should be able to say what stock level will meet demand 95% of the weeks. +*/ + double stock_95 = quantile(sacks, 0.95); + cout << "Store should stock " << int(stock_95) << " sacks to meet 95% of demands." << endl; +/*`And it is easy to estimate how to meet 80% of demand, and waste even less. +*/ + double stock_80 = quantile(sacks, 0.80); + cout << "Store should stock " << int(stock_80) << " sacks to meet 8 out of 10 demands." << endl; +//] [/normal_bulbs_example3 Quickbook end] + } + { // K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, + // ISBN 1 58488 635 8, page 125, Example 10.3.7 + +//[normal_bulbs_example4 + +/*`A machine is set to pack 3 kg of ground beef per pack. +Over a long period of time it is found that the average packed was 3 kg +with a standard deviation of 0.1 kg. +Assuming the packing is normally distributed, +we can find the fraction (or %) of packages that weigh more than 3.1 kg. +*/ + +double mean = 3.; // kg +double standard_deviation = 0.1; // kg +normal packs(mean, standard_deviation); + +double max_weight = 3.1; // kg +cout << "Percentage of packs > " << max_weight << " is " +<< cdf(complement(packs, max_weight)) << endl; // P(X > 3.1) + +double under_weight = 2.9; +cout <<"fraction of packs <= " << under_weight << " with a mean of " << mean + << " is " << cdf(complement(packs, under_weight)) << endl; +// fraction of packs <= 2.9 with a mean of 3 is 0.841345 +// This is 0.84 - more than the target 0.95 +// Want 95% to be over this weight, so what should we set the mean weight to be? +// KK StatCalc says: +double over_mean = 3.0664; +normal xpacks(over_mean, standard_deviation); +cout << "fraction of packs >= " << under_weight +<< " with a mean of " << xpacks.mean() + << " is " << cdf(complement(xpacks, under_weight)) << endl; +// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005 +double under_fraction = 0.05; // so 95% are above the minimum weight mean - sd = 2.9 +double low_limit = standard_deviation; +double offset = mean - low_limit - quantile(packs, under_fraction); +double nominal_mean = mean + offset; + +normal nominal_packs(nominal_mean, standard_deviation); +cout << "Setting the packer to " << nominal_mean << " will mean that " + << "fraction of packs >= " << under_weight + << " is " << cdf(complement(nominal_packs, under_weight)) << endl; + +/*` +Setting the packer to 3.06449 will mean that fraction of packs >= 2.9 is 0.95. + +Setting the packer to 3.13263 will mean that fraction of packs >= 2.9 is 0.99, +but will more than double the mean loss from 0.0644 to 0.133. + +Alternatively, we could invest in a better (more precise) packer with a lower standard deviation. + +To estimate how much better (how much smaller standard deviation) it would have to be, +we need to get the 5% quantile to be located at the under_weight limit, 2.9 +*/ +double p = 0.05; // wanted p th quantile. +cout << "Quantile of " << p << " = " << quantile(packs, p) + << ", mean = " << packs.mean() << ", sd = " << packs.standard_deviation() << endl; // +/*` +Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 + +With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551 kg, +a little below our target of 2.9 kg. +So we know that the standard deviation is going to have to be smaller. + +Let's start by guessing that it (now 0.1) needs to be halved, to a standard deviation of 0.05 +*/ +normal pack05(mean, 0.05); +cout << "Quantile of " << p << " = " << quantile(pack05, p) + << ", mean = " << pack05.mean() << ", sd = " << pack05.standard_deviation() << endl; + +cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean + << " and standard deviation of " << pack05.standard_deviation() + << " is " << cdf(complement(pack05, under_weight)) << endl; +// +/*` +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.9772 + +So 0.05 was quite a good guess, but we are a little over the 2.9 target, +so the standard deviation could be a tiny bit more. So we could do some +more guessing to get closer, say by increasing to 0.06 +*/ + +normal pack06(mean, 0.06); +cout << "Quantile of " << p << " = " << quantile(pack06, p) + << ", mean = " << pack06.mean() << ", sd = " << pack06.standard_deviation() << endl; + +cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean + << " and standard deviation of " << pack06.standard_deviation() + << " is " << cdf(complement(pack06, under_weight)) << endl; +/*` +Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.9522 + +Now we are getting really close, but to do the job properly, +we could use root finding method, for example the tools provided, and used elsewhere, +in the Math Toolkit, see +[link math_toolkit.toolkit.internals1.roots2 Root Finding Without Derivatives]. + +But in this normal distribution case, we could be even smarter and make a direct calculation. +*/ + +normal s; // For standard normal distribution, +double sd = 0.1; +double x = 2.9; // Our required limit. +// then probability p = N((x - mean) / sd) +// So if we want to find the standard deviation that would be required to meet this limit, +// so that the p th quantile is located at x, +// in this case the 0.95 (95%) quantile at 2.9 kg pack weight, when the mean is 3 kg. + +double prob = pdf(s, (x - mean) / sd); +double qp = quantile(s, 0.95); +cout << "prob = " << prob << ", quantile(p) " << qp << endl; // p = 0.241971, quantile(p) 1.64485 +// Rearranging, we can directly calculate the required standard deviation: +double sd95 = abs((x - mean)) / qp; + +cout << "If we want the "<< p << " th quantile to be located at " + << x << ", would need a standard deviation of " << sd95 << endl; + +normal pack95(mean, sd95); // Distribution of the 'ideal better' packer. +cout <<"Fraction of packs >= " << under_weight << " with a mean of " << mean + << " and standard deviation of " << pack95.standard_deviation() + << " is " << cdf(complement(pack95, under_weight)) << endl; + +// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0608 is 0.95 + +/*`Notice that these two deceptively simple questions +(do we over-fill or measure better) are actually very common. +The weight of beef might be replaced by a measurement of more or less anything. +But the calculations rely on the accuracy of the standard deviation - something +that is almost always less good than we might wish, +especially if based on a few measurements. +*/ + +//] [/normal_bulbs_example4 Quickbook end] + } + + { // K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, + // ISBN 1 58488 635 8, page 125, example 10.3.8 +//[normal_bulbs_example5 +/*`A bolt is usable if between 3.9 and 4.1 long. +From a large batch of bolts, a sample of 50 show a +mean length of 3.95 with standard deviation 0.1. +Assuming a normal distribution, what proportion is usable? +The true sample mean is unknown, +but we can use the sample mean and standard deviation to find approximate solutions. +*/ + + normal bolts(3.95, 0.1); + double top = 4.1; + double bottom = 3.9; + +cout << "Fraction long enough [ P(X <= " << top << ") ] is " << cdf(bolts, top) << endl; +cout << "Fraction too short [ P(X <= " << bottom << ") ] is " << cdf(bolts, bottom) << endl; +cout << "Fraction OK -between " << bottom << " and " << top + << "[ P(X <= " << top << ") - P(X<= " << bottom << " ) ] is " + << cdf(bolts, top) - cdf(bolts, bottom) << endl; + +cout << "Fraction too long [ P(X > " << top << ") ] is " + << cdf(complement(bolts, top)) << endl; + +cout << "95% of bolts are shorter than " << quantile(bolts, 0.95) << endl; + +//] [/normal_bulbs_example5 Quickbook end] + } + } + catch(const std::exception& e) + { // Always useful to include try & catch blocks because default policies + // are to throw exceptions on arguments that cause errors like underflow, overflow. + // Lacking try & catch blocks, the program will abort without a message below, + // which may give some helpful clues as to the cause of the exception. + std::cout << + "\n""Message from thrown exception was:\n " << e.what() << std::endl; + } + return 0; +} // int main() + + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\normal_misc_examples.exe" +Example: Normal distribution, Miscellaneous Applications.Standard normal distribution, mean = 0, standard deviation = 1 +Probability distribution function values + z pdf +-4 0.00013383022576488537 +-3 0.0044318484119380075 +-2 0.053990966513188063 +-1 0.24197072451914337 +0 0.3989422804014327 +1 0.24197072451914337 +2 0.053990966513188063 +3 0.0044318484119380075 +4 0.00013383022576488537 +Standard normal mean = 0, standard deviation = 1 +Integral (area under the curve) from - infinity up to z + z cdf +-4 3.1671241833119979e-005 +-3 0.0013498980316300959 +-2 0.022750131948179219 +-1 0.1586552539314571 +0 0.5 +1 0.84134474606854293 +2 0.97724986805182079 +3 0.9986501019683699 +4 0.99996832875816688 +Area for z = 2 is 0.97725 +95% of area has a z below 1.64485 +95% of area has a z between 1.95996 and -1.95996 +Significance level for z == 1 is 0.3173105078629142 +level of significance (alpha) +2-sided 1 -sided z(alpha) +0.3173 0.1587 1 +0.2 0.1 1.282 +0.1 0.05 1.645 +0.05 0.025 1.96 +0.01 0.005 2.576 +0.001 0.0005 3.291 +0.0001 5e-005 3.891 +1e-005 5e-006 4.417 +cdf(s, s.standard_deviation()) = 0.841 +cdf(complement(s, s.standard_deviation())) = 0.159 +Fraction 1 standard deviation within either side of mean is 0.683 +Fraction 2 standard deviations within either side of mean is 0.954 +Fraction 3 standard deviations within either side of mean is 0.997 +Fraction of bulbs that will last at best (<=) 1.00e+003 is 0.159 +Fraction of bulbs that will last at least (>) 1.00e+003 is 0.841 +Fraction of bulbs that will last between 900. and 1.20e+003 is 0.819 +Percentage of weeks overstocked 97.7 +Store should stock 156 sacks to meet 95% of demands. +Store should stock 148 sacks to meet 8 out of 10 demands. +Percentage of packs > 3.10 is 0.159 +fraction of packs <= 2.90 with a mean of 3.00 is 0.841 +fraction of packs >= 2.90 with a mean of 3.07 is 0.952 +Setting the packer to 3.06 will mean that fraction of packs >= 2.90 is 0.950 +Quantile of 0.0500 = 2.84, mean = 3.00, sd = 0.100 +Quantile of 0.0500 = 2.92, mean = 3.00, sd = 0.0500 +Fraction of packs >= 2.90 with a mean of 3.00 and standard deviation of 0.0500 is 0.977 +Quantile of 0.0500 = 2.90, mean = 3.00, sd = 0.0600 +Fraction of packs >= 2.90 with a mean of 3.00 and standard deviation of 0.0600 is 0.952 +prob = 0.242, quantile(p) 1.64 +If we want the 0.0500 th quantile to be located at 2.90, would need a standard deviation of 0.0608 +Fraction of packs >= 2.90 with a mean of 3.00 and standard deviation of 0.0608 is 0.950 +Fraction long enough [ P(X <= 4.10) ] is 0.933 +Fraction too short [ P(X <= 3.90) ] is 0.309 +Fraction OK -between 3.90 and 4.10[ P(X <= 4.10) - P(X<= 3.90 ) ] is 0.625 +Fraction too long [ P(X > 4.10) ] is 0.0668 +95% of bolts are shorter than 4.11 + +*/ + + + diff --git a/example/policy_eg_1.cpp b/example/policy_eg_1.cpp new file mode 100644 index 000000000..4950517c9 --- /dev/null +++ b/example/policy_eg_1.cpp @@ -0,0 +1,35 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include + +//[policy_eg_1 + +#include +// +// Define the policy to use: +// +using namespace boost::math::policies; +typedef policy< + domain_error, + pole_error, + overflow_error, + evaluation_error +> c_policy; +// +// Now use the policy when calling tgamma: +// +int main() +{ + errno = 0; + std::cout << "Result of tgamma(30000) is: " + << boost::math::tgamma(30000, c_policy()) << std::endl; + std::cout << "errno = " << errno << std::endl; + std::cout << "Result of tgamma(-10) is: " + << boost::math::tgamma(-10, c_policy()) << std::endl; + std::cout << "errno = " << errno << std::endl; +} + +//] \ No newline at end of file diff --git a/example/policy_eg_10.cpp b/example/policy_eg_10.cpp new file mode 100644 index 000000000..0a352020c --- /dev/null +++ b/example/policy_eg_10.cpp @@ -0,0 +1,171 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_eg_10 + +/*` + +To understand how the rounding policies for +the discrete distributions can be used, we'll +use the 50-sample binomial distribution with a +success fraction of 0.5 once again, and calculate +all the possible quantiles at 0.05 and 0.95. + +Begin by including the needed headers: + +*/ + +#include +#include + +/*` + +Next we'll bring the needed declarations into scope, and +define distribution types for all the available rounding policies: + +*/ + +using namespace boost::math::policies; +using namespace boost::math; + +typedef binomial_distribution< + double, + policy > > + binom_round_outwards; + +typedef binomial_distribution< + double, + policy > > + binom_round_inwards; + +typedef binomial_distribution< + double, + policy > > + binom_round_down; + +typedef binomial_distribution< + double, + policy > > + binom_round_up; + +typedef binomial_distribution< + double, + policy > > + binom_round_nearest; + +typedef binomial_distribution< + double, + policy > > + binom_real_quantile; + +/*` + +Now let's set to work calling those quantiles: + +*/ + +int main() +{ + std::cout << + "Testing rounding policies for a 50 sample binomial distribution,\n" + "with a success fraction of 0.5.\n\n" + "Lower quantiles are calculated at p = 0.05\n\n" + "Upper quantiles at p = 0.95.\n\n"; + + std::cout << std::setw(25) << std::right + << "Policy"<< std::setw(18) << std::right + << "Lower Quantile" << std::setw(18) << std::right + << "Upper Quantile" << std::endl; + + // Test integer_round_outwards: + std::cout << std::setw(25) << std::right + << "integer_round_outwards" + << std::setw(18) << std::right + << quantile(binom_round_outwards(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_round_outwards(50, 0.5), 0.95) + << std::endl; + + // Test integer_round_inwards: + std::cout << std::setw(25) << std::right + << "integer_round_inwards" + << std::setw(18) << std::right + << quantile(binom_round_inwards(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_round_inwards(50, 0.5), 0.95) + << std::endl; + + // Test integer_round_down: + std::cout << std::setw(25) << std::right + << "integer_round_down" + << std::setw(18) << std::right + << quantile(binom_round_down(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_round_down(50, 0.5), 0.95) + << std::endl; + + // Test integer_round_up: + std::cout << std::setw(25) << std::right + << "integer_round_up" + << std::setw(18) << std::right + << quantile(binom_round_up(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_round_up(50, 0.5), 0.95) + << std::endl; + + // Test integer_round_nearest: + std::cout << std::setw(25) << std::right + << "integer_round_nearest" + << std::setw(18) << std::right + << quantile(binom_round_nearest(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_round_nearest(50, 0.5), 0.95) + << std::endl; + + // Test real: + std::cout << std::setw(25) << std::right + << "real" + << std::setw(18) << std::right + << quantile(binom_real_quantile(50, 0.5), 0.05) + << std::setw(18) << std::right + << quantile(binom_real_quantile(50, 0.5), 0.95) + << std::endl; +} + +/*` + +Which produces the program output: + +[pre +Testing rounding policies for a 50 sample binomial distribution, +with a success fraction of 0.5. + +Lower quantiles are calculated at p = 0.05 + +Upper quantiles at p = 0.95. + +Testing rounding policies for a 50 sample binomial distribution, +with a success fraction of 0.5. + +Lower quantiles are calculated at p = 0.05 + +Upper quantiles at p = 0.95. + + Policy Lower Quantile Upper Quantile + integer_round_outwards 18 31 + integer_round_inwards 19 30 + integer_round_down 18 30 + integer_round_up 19 31 + integer_round_nearest 19 30 + real 18.701 30.299 +] + +*/ + +//] ends quickbook import + diff --git a/example/policy_eg_2.cpp b/example/policy_eg_2.cpp new file mode 100644 index 000000000..1f9988656 --- /dev/null +++ b/example/policy_eg_2.cpp @@ -0,0 +1,44 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include + +//[policy_eg_2 + +#include + +int main() +{ + using namespace boost::math::policies; + errno = 0; + std::cout << "Result of tgamma(30000) is: " + << boost::math::tgamma( + 30000, + make_policy( + domain_error(), + pole_error(), + overflow_error(), + evaluation_error() + ) + ) << std::endl; + // Check errno was set: + std::cout << "errno = " << errno << std::endl; + // and again with evaluation at a pole: + std::cout << "Result of tgamma(-10) is: " + << boost::math::tgamma( + -10, + make_policy( + domain_error(), + pole_error(), + overflow_error(), + evaluation_error() + ) + ) << std::endl; + // Check errno was set: + std::cout << "errno = " << errno << std::endl; +} + +//] + diff --git a/example/policy_eg_3.cpp b/example/policy_eg_3.cpp new file mode 100644 index 000000000..946f6a848 --- /dev/null +++ b/example/policy_eg_3.cpp @@ -0,0 +1,42 @@ +// Copyright John Maddock 2007. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning (disable : 4305) // 'initializing' : truncation from 'long double' to 'const eval_type' +# pragma warning (disable : 4244) // conversion from 'long double' to 'const eval_type' +#endif + +#include + +//[policy_eg_3 + +#include + +// +// Begin by defining a policy type, that gives the +// behaviour we want: +// +using namespace boost::math::policies; +typedef policy< + promote_float, + discrete_quantile +> mypolicy; +// +// Then define a distribution that uses it: +// +typedef boost::math::binomial_distribution mybinom; +// +// And now use it to get the quantile: +// +int main() +{ + std::cout << "quantile is: " << + quantile(mybinom(200, 0.25), 0.05) << std::endl; +} + +//] + diff --git a/example/policy_eg_4.cpp b/example/policy_eg_4.cpp new file mode 100644 index 000000000..fbb90fb4a --- /dev/null +++ b/example/policy_eg_4.cpp @@ -0,0 +1,102 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#include + +//[policy_eg_4 + +/*` +Suppose we want `C::foo()` to behave in a C-compatible way and set +`::errno` on error rather than throwing any exceptions. + +We'll begin by including the needed header: +*/ + +#include + +/*` + +Open up the "C" namespace that we'll use for our functions, and +define the policy type we want: in this case one that sets +::errno rather than throwing exceptions. Any policies we don't +specify here will inherit the defaults: + +*/ + +namespace C{ + +using namespace boost::math::policies; + +typedef policy< + domain_error, + pole_error, + overflow_error, + evaluation_error +> c_policy; + +/*` + +All we need do now is invoke the BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS +macro passing our policy type as the single argument: + +*/ + +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(c_policy) + +} // close namespace C + +/*` + +We now have a set of forwarding functions defined in namespace C +that all look something like this: + +`` +template +inline typename boost::math::tools::promote_args::type + tgamma(RT z) +{ + return boost::math::tgamma(z, c_policy()); +} +`` + +So that when we call `C::tgamma(z)` we really end up calling +`boost::math::tgamma(z, C::c_policy())`: + +*/ + + +int main() +{ + errno = 0; + std::cout << "Result of tgamma(30000) is: " + << C::tgamma(30000) << std::endl; + std::cout << "errno = " << errno << std::endl; + std::cout << "Result of tgamma(-10) is: " + << C::tgamma(-10) << std::endl; + std::cout << "errno = " << errno << std::endl; +} + +/*` + +Which outputs: + +[pre +Result of C::tgamma(30000) is: 1.#INF +errno = 34 +Result of C::tgamma(-10) is: 1.#QNAN +errno = 33 +] + +This mechanism is particularly useful when we want to define a +project-wide policy, and don't want to modify the Boost source +or set - possibly fragile and easy to forget - project wide +build macros. + +*/ +//] + diff --git a/example/policy_eg_5.cpp b/example/policy_eg_5.cpp new file mode 100644 index 000000000..5cdbf2203 --- /dev/null +++ b/example/policy_eg_5.cpp @@ -0,0 +1,41 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#include + +//[policy_eg_5 + +#include + +namespace { + +using namespace boost::math::policies; + +typedef policy< + domain_error, + pole_error, + overflow_error, + evaluation_error +> c_policy; + +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(c_policy) + +} // close unnamed namespace + +int main() +{ + errno = 0; + std::cout << "Result of tgamma(30000) is: " + << tgamma(30000) << std::endl; + std::cout << "errno = " << errno << std::endl; + std::cout << "Result of tgamma(-10) is: " + << tgamma(-10) << std::endl; + std::cout << "errno = " << errno << std::endl; +} + +//] diff --git a/example/policy_eg_6.cpp b/example/policy_eg_6.cpp new file mode 100644 index 000000000..d5acfae64 --- /dev/null +++ b/example/policy_eg_6.cpp @@ -0,0 +1,119 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#include + +//[policy_eg_6 + +/*` +Suppose we want a set of distributions to behave as follows: + +* Return infinity on overflow, rather than throwing an exception. +* Don't perform any promotion from double to long double internally. +* Return the closest integer result from the quantiles of discrete +distributions. + +We'll begin by including the needed header: +*/ + +#include + +/*` + +Open up an appropriate namespace for our distributions, and +define the policy type we want. Any policies we don't +specify here will inherit the defaults: + +*/ + +namespace my_distributions{ + +using namespace boost::math::policies; + +typedef policy< + // return infinity and set errno rather than throw: + overflow_error, + // Don't promote double -> long double internally: + promote_double, + // Return the closest integer result for discrete quantiles: + discrete_quantile +> my_policy; + +/*` + +All we need do now is invoke the BOOST_MATH_DECLARE_DISTRIBUTIONS +macro passing the floating point and policy types as arguments: + +*/ + +BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy) + +} // close namespace my_namespace + +/*` + +We now have a set of typedefs defined in namespace my_namespace +that all look something like this: + +`` +typedef boost::math::normal_distribution normal; +typedef boost::math::cauchy_distribution cauchy; +typedef boost::math::gamma_distribution gamma; +// etc +`` + +So that when we use my_namespace::normal we really end up using +`boost::math::normal_distribution`: + +*/ + + +int main() +{ + // + // Start with something we know will overflow: + // + my_distributions::normal norm(10, 2); + errno = 0; + std::cout << "Result of quantile(norm, 0) is: " + << quantile(norm, 0) << std::endl; + std::cout << "errno = " << errno << std::endl; + errno = 0; + std::cout << "Result of quantile(norm, 1) is: " + << quantile(norm, 1) << std::endl; + std::cout << "errno = " << errno << std::endl; + // + // Now try a discrete distribution: + // + my_distributions::binomial binom(20, 0.25); + std::cout << "Result of quantile(binom, 0.05) is: " + << quantile(binom, 0.05) << std::endl; + std::cout << "Result of quantile(complement(binom, 0.05)) is: " + << quantile(complement(binom, 0.05)) << std::endl; +} + +/*` + +Which outputs: + +[pre +Result of quantile(norm, 0) is: -1.#INF +errno = 34 +Result of quantile(norm, 1) is: 1.#INF +errno = 34 +Result of quantile(binom, 0.05) is: 1 +Result of quantile(complement(binom, 0.05)) is: 8 +] + +This mechanism is particularly useful when we want to define a +project-wide policy, and don't want to modify the Boost source +or set - possibly fragile and easy to forget - project wide +build macros. + +*/ //] ends quickbook imported section + diff --git a/example/policy_eg_7.cpp b/example/policy_eg_7.cpp new file mode 100644 index 000000000..9f0cbd3e9 --- /dev/null +++ b/example/policy_eg_7.cpp @@ -0,0 +1,57 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#include + +//[policy_eg_7 + +#include + +namespace { + +using namespace boost::math::policies; + +typedef policy< + // return infinity and set errno rather than throw: + overflow_error, + // Don't promote double -> long double internally: + promote_double, + // Return the closest integer result for discrete quantiles: + discrete_quantile +> my_policy; + +BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy) + +} // close namespace my_namespace + +int main() +{ + // + // Start with something we know will overflow: + // + normal norm(10, 2); + errno = 0; + std::cout << "Result of quantile(norm, 0) is: " + << quantile(norm, 0) << std::endl; + std::cout << "errno = " << errno << std::endl; + errno = 0; + std::cout << "Result of quantile(norm, 1) is: " + << quantile(norm, 1) << std::endl; + std::cout << "errno = " << errno << std::endl; + // + // Now try a discrete distribution: + // + binomial binom(20, 0.25); + std::cout << "Result of quantile(binom, 0.05) is: " + << quantile(binom, 0.05) << std::endl; + std::cout << "Result of quantile(complement(binom, 0.05)) is: " + << quantile(complement(binom, 0.05)) << std::endl; +} + +//] ends quickbook imported section + diff --git a/example/policy_eg_8.cpp b/example/policy_eg_8.cpp new file mode 100644 index 000000000..6b9bc5d6e --- /dev/null +++ b/example/policy_eg_8.cpp @@ -0,0 +1,135 @@ +// Copyright John Maddock 2007. +// Copyright Paul a. Bristow 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#ifdef _MSC_VER +# pragma warning (disable : 4100) // 'unreferenced formal parameter +#endif + + +#include + +//[policy_eg_8 + +/*` + +Suppose we want our own user-defined error handlers rather than the +any of the default ones supplied by the library to be used. If +we set the policy for a specific type of error to `user_error` +then the library will call a user-supplied error handler. +These are forward declared, but not defined in +boost/math/policies/error_handling.hpp like this: + + namespace boost{ namespace math{ namespace policy{ + + template + T user_domain_error(const char* function, const char* message, const T& val); + template + T user_pole_error(const char* function, const char* message, const T& val); + template + T user_overflow_error(const char* function, const char* message, const T& val); + template + T user_underflow_error(const char* function, const char* message, const T& val); + template + T user_denorm_error(const char* function, const char* message, const T& val); + template + T user_evaluation_error(const char* function, const char* message, const T& val); + + }}} // namespaces + +So out first job is to include the header we want to use, and then +provide definitions for the user-defined error handlers we want to use: + +*/ + +#include +#include + +namespace boost{ namespace math{ namespace policies{ + +template +T user_domain_error(const char* function, const char* message, const T& val) +{ + std::cerr << "Domain Error." << std::endl; + return std::numeric_limits::quiet_NaN(); +} + +template +T user_pole_error(const char* function, const char* message, const T& val) +{ + std::cerr << "Pole Error." << std::endl; + return std::numeric_limits::quiet_NaN(); +} + + +}}} // namespaces + + +/*` + +Now we'll need to define a suitable policy that will call these handlers, +and define some forwarding functions that make use of the policy: + +*/ + +namespace{ + +using namespace boost::math::policies; + +typedef policy< + domain_error, + pole_error +> user_error_policy; + +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(user_error_policy) + +} // close unnamed namespace + +/*` + +We now have a set of forwarding functions defined in an unnamed namespace +that all look something like this: + +`` +template +inline typename boost::math::tools::promote_args::type + tgamma(RT z) +{ + return boost::math::tgamma(z, user_error_policy()); +} +`` + +So that when we call `tgamma(z)` we really end up calling +`boost::math::tgamma(z, user_error_policy())`, and any +errors will get directed to our own error handlers: + +*/ + + +int main() +{ + std::cout << "Result of erf_inv(-10) is: " + << erf_inv(-10) << std::endl; + std::cout << "Result of tgamma(-10) is: " + << tgamma(-10) << std::endl; +} + +/*` + +Which outputs: + +[pre +Domain Error. +Result of erf_inv(-10) is: 1.#QNAN +Pole Error. +Result of tgamma(-10) is: 1.#QNAN +] +*/ + +//] + diff --git a/example/policy_eg_9.cpp b/example/policy_eg_9.cpp new file mode 100644 index 000000000..2ac39c497 --- /dev/null +++ b/example/policy_eg_9.cpp @@ -0,0 +1,316 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#include + +//[policy_eg_9 + +/*` + +The previous example was all well and good, but the custom error handlers +didn't really do much of any use. In this example we'll implement all +the custom handlers and show how the information provided to them can be +used to generate nice formatted error messages. + +Each error handler has the general form: + + template + T user_``['error_type]``( + const char* function, + const char* message, + const T& val); + +and accepts three arguments: + +[variablelist +[[const char* function] + [The name of the function that raised the error, this string + contains one or more %1% format specifiers that should be + replaced by the name of type T.]] +[[const char* message] + [A message associated with the error, normally this + contains a %1% format specifier that should be replaced with + the value of ['value]: however note that overflow and underflow messages + do not contain this %1% specifier (since the value of ['value] is + immaterial in these cases).]] +[[const T& value] + [The value that caused the error: either an argument to the function + if this is a domain or pole error, the tentative result + if this is a denorm or evaluation error, or zero or infinity for + underflow or overflow errors.]] +] + +As before we'll include the headers we need first: + +*/ + +#include +#include + +/*` + +Next we'll implement the error handlers for each type of error, +starting with domain errors: + +*/ + +namespace boost{ namespace math{ namespace policies{ + +template +T user_domain_error(const char* function, const char* message, const T& val) +{ + /*` + We'll begin with a bit of defensive programming: + */ + if(function == 0) + function = "Unknown function with arguments of type %1%"; + if(message == 0) + message = "Cause unknown with bad argument %1%"; + /*` + Next we'll format the name of the function with the name of type T: + */ + std::string msg("Error in function "); + msg += (boost::format(function) % typeid(T).name()).str(); + /*` + Then likewise format the error message with the value of parameter /val/, + making sure we output all the digits of /val/: + */ + msg += ": \n"; + int prec = 2 + (std::numeric_limits::digits * 30103UL) / 100000UL; + msg += (boost::format(message) % boost::io::group(std::setprecision(prec), val)).str(); + /*` + Now we just have to do something with the message, we could throw an + exception, but for the purposes of this example we'll just dump the message + to std::cerr: + */ + std::cerr << msg << std::endl; + /*` + Finally the only sensible value we can return from a domain error is a NaN: + */ + return std::numeric_limits::quiet_NaN(); +} + +/*` +Pole errors are essentially a special case of domain errors, +so in this example we'll just return the result of a domain error: +*/ + +template +T user_pole_error(const char* function, const char* message, const T& val) +{ + return user_domain_error(function, message, val); +} + +/*` +Overflow errors are very similar to domain errors, except that there's +no %1% format specifier in the /message/ parameter: +*/ +template +T user_overflow_error(const char* function, const char* message, const T& val) +{ + if(function == 0) + function = "Unknown function with arguments of type %1%"; + if(message == 0) + message = "Result of function is too large to represent"; + + std::string msg("Error in function "); + msg += (boost::format(function) % typeid(T).name()).str(); + + msg += ": \n"; + msg += message; + + std::cerr << msg << std::endl; + + // Value passed to the function is an infinity, just return it: + return val; +} + +/*` +Underflow errors are much the same as overflow: +*/ + +template +T user_underflow_error(const char* function, const char* message, const T& val) +{ + if(function == 0) + function = "Unknown function with arguments of type %1%"; + if(message == 0) + message = "Result of function is too small to represent"; + + std::string msg("Error in function "); + msg += (boost::format(function) % typeid(T).name()).str(); + + msg += ": \n"; + msg += message; + + std::cerr << msg << std::endl; + + // Value passed to the function is zero, just return it: + return val; +} + +/*` +Denormalised results are much the same as underflow: +*/ + +template +T user_denorm_error(const char* function, const char* message, const T& val) +{ + if(function == 0) + function = "Unknown function with arguments of type %1%"; + if(message == 0) + message = "Result of function is denormalised"; + + std::string msg("Error in function "); + msg += (boost::format(function) % typeid(T).name()).str(); + + msg += ": \n"; + msg += message; + + std::cerr << msg << std::endl; + + // Value passed to the function is denormalised, just return it: + return val; +} + +/*` +Which leaves us with evaluation errors, these occur when an internal +error occurs that prevents the function being fully evaluated. +The parameter /val/ contains the closest approximation to the result +found so far: +*/ + +template +T user_evaluation_error(const char* function, const char* message, const T& val) +{ + if(function == 0) + function = "Unknown function with arguments of type %1%"; + if(message == 0) + message = "An internal evaluation error occured with " + "the best value calculated so far of %1%"; + + std::string msg("Error in function "); + msg += (boost::format(function) % typeid(T).name()).str(); + + msg += ": \n"; + int prec = 2 + (std::numeric_limits::digits * 30103UL) / 100000UL; + msg += (boost::format(message) % boost::io::group(std::setprecision(prec), val)).str(); + + std::cerr << msg << std::endl; + + // What do we return here? This is generally a fatal error, + // that should never occur, just return a NaN for the purposes + // of the example: + return std::numeric_limits::quiet_NaN(); +} + +}}} // namespaces + + +/*` + +Now we'll need to define a suitable policy that will call these handlers, +and define some forwarding functions that make use of the policy: + +*/ + +namespace{ + +using namespace boost::math::policies; + +typedef policy< + domain_error, + pole_error, + overflow_error, + underflow_error, + denorm_error, + evaluation_error +> user_error_policy; + +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(user_error_policy) + +} // close unnamed namespace + +/*` + +We now have a set of forwarding functions defined in an unnamed namespace +that all look something like this: + +`` +template +inline typename boost::math::tools::promote_args::type + tgamma(RT z) +{ + return boost::math::tgamma(z, user_error_policy()); +} +`` + +So that when we call `tgamma(z)` we really end up calling +`boost::math::tgamma(z, user_error_policy())`, and any +errors will get directed to our own error handlers: + +*/ + + +int main() +{ + // Raise a domain error: + std::cout << "Result of erf_inv(-10) is: " + << erf_inv(-10) << std::endl << std::endl; + // Raise a pole error: + std::cout << "Result of tgamma(-10) is: " + << tgamma(-10) << std::endl << std::endl; + // Raise an overflow error: + std::cout << "Result of tgamma(3000) is: " + << tgamma(3000) << std::endl << std::endl; + // Raise an underflow error: + std::cout << "Result of tgamma(-190.5) is: " + << tgamma(-190.5) << std::endl << std::endl; + // Unfortunately we can't predicably raise a denormalised + // result, nor can we raise an evaluation error in this example + // since these should never really occur! +} + +/*` + +Which outputs: + +[pre +Error in function boost::math::erf_inv(double, double): +Argument outside range \[-1, 1\] in inverse erf function (got p=-10). +Result of erf_inv(-10) is: 1.#QNAN + +Error in function boost::math::tgamma(long double): +Evaluation of tgamma at a negative integer -10. +Result of tgamma(-10) is: 1.#QNAN + +Error in function boost::math::tgamma(long double): +Result of tgamma is too large to represent. +Error in function boost::math::tgamma(double): +Result of function is too large to represent +Result of tgamma(3000) is: 1.#INF + +Error in function boost::math::tgamma(long double): +Result of tgamma is too large to represent. +Error in function boost::math::tgamma(long double): +Result of tgamma is too small to represent. +Result of tgamma(-190.5) is: 0 +] + +Notice how some of the calls result in an error handler being called more +than once, or for more than one handler to be called: this is an artefact +of the fact that many functions are implemented in terms of one or more +sub-routines each of which may have it's own error handling. For example +`tgamma(-190.5)` is implemented in terms of `tgamma(190.5)` - which overflows - +the reflection formula for `tgamma` then notices that it's dividing by +infinity and underflows. + +*/ + +//] + diff --git a/example/policy_ref_snip1.cpp b/example/policy_ref_snip1.cpp new file mode 100644 index 000000000..89542dd09 --- /dev/null +++ b/example/policy_ref_snip1.cpp @@ -0,0 +1,44 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +double some_value = 0; + +//[policy_ref_snip1 + +#include + +using namespace boost::math::policies; +using namespace boost::math; + +// Define a policy: +typedef policy< + domain_error, + pole_error, + overflow_error, + policies::evaluation_error + > my_policy; + +// call the function: +double t1 = tgamma(some_value, my_policy()); + +// Alternatively we could use make_policy and define everything at the call site: +double t2 = tgamma(some_value, make_policy( + domain_error(), + pole_error(), + overflow_error(), + policies::evaluation_error() + )); + +//] + +#include + +int main() +{ + std::cout << t1 << " " << t2 << std::endl; +} diff --git a/example/policy_ref_snip10.cpp b/example/policy_ref_snip10.cpp new file mode 100644 index 000000000..18c5a6152 --- /dev/null +++ b/example/policy_ref_snip10.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip10 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +double t = tgamma(12, policy >()); + +//] + +#include + +int main() +{ + std::cout << t << std::endl; +} diff --git a/example/policy_ref_snip11.cpp b/example/policy_ref_snip11.cpp new file mode 100644 index 000000000..5a4b4e665 --- /dev/null +++ b/example/policy_ref_snip11.cpp @@ -0,0 +1,27 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip11 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +double q = quantile( + normal_distribution > >(), + 0.05); + +//] + +#include + +int main() +{ + std::cout << q << std::endl; +} diff --git a/example/policy_ref_snip12.cpp b/example/policy_ref_snip12.cpp new file mode 100644 index 000000000..7b764db75 --- /dev/null +++ b/example/policy_ref_snip12.cpp @@ -0,0 +1,38 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip12 + +#include + +namespace myspace{ + +using namespace boost::math::policies; + +// Define a policy that does not throw on overflow: +typedef policy > my_policy; + +// Define the special functions in this scope to use the policy: +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(my_policy) + +} + +// +// Now we can use myspace::tgamma etc. +// They will automatically use "my_policy": +// +double t = myspace::tgamma(30.0); // will not throw on overflow + +//] + +#include + +int main() +{ + std::cout << t << std::endl; +} diff --git a/example/policy_ref_snip13.cpp b/example/policy_ref_snip13.cpp new file mode 100644 index 000000000..de294a799 --- /dev/null +++ b/example/policy_ref_snip13.cpp @@ -0,0 +1,54 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#ifdef _MSC_VER +#pragma warning (disable : 4189) // 'd' : local variable is initialized but not referenced +#endif + +//[policy_ref_snip13 + +#include + +namespace myspace{ + +using namespace boost::math::policies; + +// Define a policy to use, in this case we want all the distribution +// accessor functions to compile, even if they are mathematically +// undefined: +typedef policy > my_policy; + +BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy) + +} + +// Now we can use myspace::cauchy etc, which will use policy +// myspace::mypolicy: +// +// This compiles but raises a domain error at runtime: +// +void test_cauchy() +{ + try + { + double d = mean(myspace::cauchy()); + } + catch(const std::domain_error& e) + { + std::cout << e.what() << std::endl; + } +} + +//] + +#include + +int main() +{ + test_cauchy(); +} diff --git a/example/policy_ref_snip2.cpp b/example/policy_ref_snip2.cpp new file mode 100644 index 000000000..39892475f --- /dev/null +++ b/example/policy_ref_snip2.cpp @@ -0,0 +1,34 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip2 + +#include + +using namespace boost::math::policies; +using namespace boost::math; + +// Define a policy: +typedef policy< + overflow_error + > my_policy; + +// Define the distribution: +typedef normal_distribution my_norm; + +// Get a quantile: +double q = quantile(my_norm(), 0.05); + +//] + +#include + +int main() +{ + std::cout << q << std::endl; +} diff --git a/example/policy_ref_snip3.cpp b/example/policy_ref_snip3.cpp new file mode 100644 index 000000000..a9edf8608 --- /dev/null +++ b/example/policy_ref_snip3.cpp @@ -0,0 +1,36 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +double some_value = 2; + +//[policy_ref_snip3 + +#include + +using namespace boost::math::policies; +using namespace boost::math; + +// Define a policy: +typedef policy< + promote_double + > my_policy; + +// Call the function: +double t1 = tgamma(some_value, my_policy()); + +// Alternatively we could use make_policy and define everything at the call site: +double t2 = tgamma(some_value, make_policy(promote_double())); + +//] + +#include + +int main() +{ + std::cout << t1 << " " << t2 << std::endl; +} diff --git a/example/policy_ref_snip4.cpp b/example/policy_ref_snip4.cpp new file mode 100644 index 000000000..8ce529e05 --- /dev/null +++ b/example/policy_ref_snip4.cpp @@ -0,0 +1,40 @@ +// Copyright John Maddock 2007. +// Copyright Paul A. Bristow 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +#ifdef _MSC_VER +# pragma warning (disable : 4305) // 'initializing' : truncation from 'long double' to 'const eval_type' +# pragma warning (disable : 4244) // 'conversion' : truncation from 'long double' to 'const eval_type' +#endif + +//[policy_ref_snip4 + +#include + +using namespace boost::math::policies; +using namespace boost::math; + +// Define a policy: +typedef policy< + promote_float + > my_policy; + +// Define the distribution: +typedef normal_distribution my_norm; + +// Get a quantile: +float q = quantile(my_norm(), 0.05f); + +//] [policy_ref_snip4] + +#include + +int main() +{ + std::cout << q << std::endl; +} diff --git a/example/policy_ref_snip5.cpp b/example/policy_ref_snip5.cpp new file mode 100644 index 000000000..5cf9c0a87 --- /dev/null +++ b/example/policy_ref_snip5.cpp @@ -0,0 +1,33 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip5 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +typedef negative_binomial_distribution< + double, + policy > + > dist_type; + +// Lower quantile: +double x = quantile(dist_type(20, 0.3), 0.05); +// Upper quantile: +double y = quantile(complement(dist_type(20, 0.3), 0.05)); + +//] + +#include + +int main() +{ + std::cout << x << " " << y << std::endl; +} diff --git a/example/policy_ref_snip6.cpp b/example/policy_ref_snip6.cpp new file mode 100644 index 000000000..4ca04e79f --- /dev/null +++ b/example/policy_ref_snip6.cpp @@ -0,0 +1,27 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip6 + +#include + +using namespace boost::math; + +// Lower quantile rounded down: +double x = quantile(negative_binomial(20, 0.3), 0.05); +// Upper quantile rounded up: +double y = quantile(complement(negative_binomial(20, 0.3), 0.05)); + +//] + +#include + +int main() +{ + std::cout << x << " " << y << std::endl; +} diff --git a/example/policy_ref_snip7.cpp b/example/policy_ref_snip7.cpp new file mode 100644 index 000000000..e7ae56dd0 --- /dev/null +++ b/example/policy_ref_snip7.cpp @@ -0,0 +1,33 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip7 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +typedef negative_binomial_distribution< + double, + policy > + > dist_type; + +// Lower quantile rounded up: +double x = quantile(dist_type(20, 0.3), 0.05); +// Upper quantile rounded down: +double y = quantile(complement(dist_type(20, 0.3), 0.05)); + +//] + +#include + +int main() +{ + std::cout << x << " " << y << std::endl; +} diff --git a/example/policy_ref_snip8.cpp b/example/policy_ref_snip8.cpp new file mode 100644 index 000000000..00be08afa --- /dev/null +++ b/example/policy_ref_snip8.cpp @@ -0,0 +1,33 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip8 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +typedef negative_binomial_distribution< + double, + policy > + > dist_type; + +// Lower quantile rounded up: +double x = quantile(dist_type(20, 0.3), 0.05); +// Upper quantile rounded down: +double y = quantile(complement(dist_type(20, 0.3), 0.05)); + +//] + +#include + +int main() +{ + std::cout << x << " " << y << std::endl; +} diff --git a/example/policy_ref_snip9.cpp b/example/policy_ref_snip9.cpp new file mode 100644 index 000000000..a27b176a3 --- /dev/null +++ b/example/policy_ref_snip9.cpp @@ -0,0 +1,27 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Note that this file contains quickbook mark-up as well as code +// and comments, don't change any of the special comment mark-ups! + +//[policy_ref_snip9 + +#include + +using namespace boost::math; +using namespace boost::math::policies; + +typedef policy > pol; + +double t = tgamma(12, pol()); + +//] + +#include + +int main() +{ + std::cout << t << std::endl; +} diff --git a/example/students_t_example1.cpp b/example/students_t_example1.cpp new file mode 100644 index 000000000..c86b89d5d --- /dev/null +++ b/example/students_t_example1.cpp @@ -0,0 +1,101 @@ +// students_t_example1.cpp + +// Copyright Paul A. Bristow 2006, 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example 1 of using Student's t + +// http://en.wikipedia.org/wiki/Student's_t-test says: +// The t statistic was invented by William Sealy Gosset +// for cheaply monitoring the quality of beer brews. +// "Student" was his pen name. +// WS Gosset was statistician for Guinness brewery in Dublin, Ireland, +// hired due to Claude Guinness's innovative policy of recruiting the +// best graduates from Oxford and Cambridge for applying biochemistry +// and statistics to Guinness's industrial processes. +// Gosset published the t test in Biometrika in 1908, +// but was forced to use a pen name by his employer who regarded the fact +// that they were using statistics as a trade secret. +// In fact, Gosset's identity was unknown not only to fellow statisticians +// but to his employer - the company insisted on the pseudonym +// so that it could turn a blind eye to the breach of its rules. + +// Data for this example from: +// P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. +// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 +// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907 + +// Determination of mercury by cold-vapour atomic absorption, +// the following values were obtained fusing a trusted +// Standard Reference Material containing 38.9% mercury, +// which we assume is correct or 'true'. +double standard = 38.9; + +const int values = 3; +double value[values] = {38.9, 37.4, 37.1}; + +// Is there any evidence for systematic error? + +// The Students't distribution function is described at +// http://en.wikipedia.org/wiki/Student%27s_t_distribution +#include + using boost::math::students_t; // Probability of students_t(df, t). + +#include + using std::cout; using std::endl; +#include + using std::setprecision; +#include + using std::sqrt; + +int main() +{ + cout << "Example 1 using Student's t function. " << endl; + + // Example/test using tabulated value + // (deliberately coded as naively as possible). + + // Null hypothesis is that there is no difference (greater or less) + // between measured and standard. + + double degrees_of_freedom = values-1; // 3-1 = 2 + cout << "Measurement 1 = " << value[0] << ", measurement 2 = " << value[1] << ", measurement 3 = " << value[2] << endl; + double mean = (value[0] + value[1] + value[2]) / static_cast(values); + cout << "Standard = " << standard << ", mean = " << mean << ", (mean - standard) = " << mean - standard << endl; + double sd = sqrt(((value[0] - mean) * (value[0] - mean) + (value[1] - mean) * (value[1] - mean) + (value[2] - mean) * (value[2] - mean))/ static_cast(values-1)); + cout << "Standard deviation = " << sd << endl; + if (sd == 0.) + { + cout << "Measured mean is identical to SRM value," << endl; + cout << "so probability of no difference between measured and standard (the 'null hypothesis') is unity." << endl; + return 0; + } + + double t = (mean - standard) * std::sqrt(static_cast(values)) / sd; + cout << "Student's t = " << t << endl; + cout.precision(2); // Useful accuracy is only a few decimal digits. + cout << "Probability of Student's t is " << cdf(students_t(degrees_of_freedom), std::abs(t)) << endl; + // 0.91, is 1 tailed. + // So there is insufficient evidence of a difference to meet a 95% (1 in 20) criterion. + + return 0; +} // int main() + +/* + +Output is: + +Example 1 using Student's t function. +Measurement 1 = 38.9, measurement 2 = 37.4, measurement 3 = 37.1 +Standard = 38.9, mean = 37.8, (mean - standard) = -1.1 +Standard deviation = 0.964365 +Student's t = -1.97566 +Probability of Student's t is 0.91 + +*/ + + diff --git a/example/students_t_example2.cpp b/example/students_t_example2.cpp new file mode 100644 index 000000000..4ae727a4e --- /dev/null +++ b/example/students_t_example2.cpp @@ -0,0 +1,125 @@ +// students_t_example2.cpp + +// Copyright Paul A. Bristow 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example 2 of using Student's t + +// A general guide to Student's t is at +// http://en.wikipedia.org/wiki/Student's_t-test +// (and many other elementary and advanced statistics texts). +// It says: +// The t statistic was invented by William Sealy Gosset +// for cheaply monitoring the quality of beer brews. +// "Student" was his pen name. +// Gosset was statistician for Guinness brewery in Dublin, Ireland, +// hired due to Claude Guinness's innovative policy of recruiting the +// best graduates from Oxford and Cambridge for applying biochemistry +// and statistics to Guinness's industrial processes. +// Gosset published the t test in Biometrika in 1908, +// but was forced to use a pen name by his employer who regarded the fact +// that they were using statistics as a trade secret. +// In fact, Gosset's identity was unknown not only to fellow statisticians +// but to his employer - the company insisted on the pseudonym +// so that it could turn a blind eye to the breach of its rules. + +// The Students't distribution function is described at +// http://en.wikipedia.org/wiki/Student%27s_t_distribution + +#include + using boost::math::students_t; // Probability of students_t(df, t). + +#include + using std::cout; + using std::endl; +#include + using std::setprecision; + using std::setw; +#include + using std::sqrt; + +// This example of a one-sided test is from: +// +// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 59-60 +// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. + +// An acid-base titrimetric method has a significant indicator error and +// thus tends to give results with a positive systematic error (+bias). +// To test this an exactly 0.1 M solution of acid is used to titrate +// 25.00 ml of exactly 0.1 M solution of alkali, +// with the following results (ml): + +double reference = 25.00; // 'True' result. +const int values = 6; // titrations. +double data [values] = {25.06, 25.18, 24.87, 25.51, 25.34, 25.41}; + +int main() +{ + cout << "Example2 using Student's t function. "; +#if defined(__FILE__) && defined(__TIMESTAMP__) + cout << " " << __FILE__ << ' ' << __TIMESTAMP__ << ' '<< _MSC_FULL_VER; +#endif + cout << endl; + + double sum = 0.; + for (int value = 0; value < values; value++) + { // Echo data and calculate mean. + sum += data[value]; + cout << setw(4) << value << ' ' << setw(14) << data[value] << endl; + } + double mean = sum /static_cast(values); + cout << "Mean = " << mean << endl; // 25.2283 + + double sd = 0.; + for (int value = 0; value < values; value++) + { // Calculate standard deviation. + sd +=(data[value] - mean) * (data[value] - mean); + } + int degrees_of_freedom = values - 1; // Use the n-1 formula. + sd /= degrees_of_freedom; // == variance. + sd= sqrt(sd); + cout << "Standard deviation = " << sd<< endl; // = 0.238279 + + double t = (mean - reference) * sqrt(static_cast(values))/ sd; // + cout << "Student's t = " << t << ", with " << degrees_of_freedom << " degrees of freedom." << endl; // = 2.34725 + + cout << "Probability of positive bias is " << cdf(students_t(degrees_of_freedom), t) << "."<< endl; // = 0.967108. + // A 1-sided test because only testing for a positive bias. + // If > 0.95 then greater than 1 in 20 conventional (arbitrary) requirement. + + return 0; +} // int main() + +/* + +Output is: + +------ Build started: Project: students_t_example2, Configuration: Debug Win32 ------ +Compiling... +students_t_example2.cpp +Linking... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\students_t_example2.exe" +Example2 using Student's t function. ..\..\..\..\..\..\boost-sandbox\libs\math_functions\example\students_t_example2.cpp Sat Aug 12 16:55:59 2006 140050727 + 0 25.06 + 1 25.18 + 2 24.87 + 3 25.51 + 4 25.34 + 5 25.41 +Mean = 25.2283 +Standard deviation = 0.238279 +Student's t = 2.34725, with 5 degrees of freedom. +Probability of positive bias is 0.967108. +Build Time 0:03 +Build log was saved at "file://i:\boost-06-05-03-1300\libs\math\test\Math_test\students_t_example2\Debug\BuildLog.htm" +students_t_example2 - 0 error(s), 0 warning(s) +========== Build: 1 succeeded, 0 failed, 0 up-to-date, 0 skipped ========== + +*/ + + + + diff --git a/example/students_t_example3.cpp b/example/students_t_example3.cpp new file mode 100644 index 000000000..289daec09 --- /dev/null +++ b/example/students_t_example3.cpp @@ -0,0 +1,120 @@ +// students_t_example3.cpp +// Copyright Paul A. Bristow 2006, 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Example 3 of using Student's t. + +// A general guide to Student's t is at +// http://en.wikipedia.org/wiki/Student's_t-test +// (and many other elementary and advanced statistics texts). +// It says: +// The t statistic was invented by William Sealy Gosset +// for cheaply monitoring the quality of beer brews. +// "Student" was his pen name. +// Gosset was statistician for Guinness brewery in Dublin, Ireland, +// hired due to Claude Guinness's innovative policy of recruiting the +// best graduates from Oxford and Cambridge for applying biochemistry +// and statistics to Guinness's industrial processes. +// Gosset published the t test in Biometrika in 1908, +// but was forced to use a pen name by his employer who regarded the fact +// that they were using statistics as a trade secret. +// In fact, Gosset's identity was unknown not only to fellow statisticians +// but to his employer - the company insisted on the pseudonym +// so that it could turn a blind eye to the breach of its rules. + +// The Students't distribution function is described at +// http://en.wikipedia.org/wiki/Student%27s_t_distribution + +#include + using boost::math::students_t; // Probability of students_t(df, t). + +#include + using std::cout; using std::endl; +#include + using std::setprecision; using std::setw; +#include + using std::sqrt; + +// This example of a two-sided test is from: +// B. M. Smith & M. B. Griffiths, Analyst, 1982, 107, 253, +// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 58-59 +// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907 + +// Concentrations of lead (ug/l) determined by two different methods +// for each of four test portions, +// the concentration of each portion is significantly different, +// the values may NOT be pooled. +// (Called a 'paired test' by Miller and Miller +// because each portion analysed has a different concentration.) + +// Portion Wet oxidation Direct Extraction +// 1 71 76 +// 2 61 68 +// 3 50 48 +// 4 60 57 + +const int portions = 4; +const int methods = 2; +float data [portions][methods] = {{71, 76}, {61,68}, {50, 48}, {60, 57}}; +float diffs[portions]; + +int main() +{ + cout << "Example3 using Student's t function. " << endl; + float mean_diff = 0.f; + cout << "\n""Portion wet_oxidation Direct_extraction difference" << endl; + for (int portion = 0; portion < portions; portion++) + { // Echo data and differences. + diffs[portion] = data[portion][0] - data[portion][1]; + mean_diff += diffs[portion]; + cout << setw(4) << portion << ' ' << setw(14) << data[portion][0] << ' ' << setw(18)<< data[portion][1] << ' ' << setw(9) << diffs[portion] << endl; + } + mean_diff /= portions; + cout << "Mean difference = " << mean_diff << endl; // -1.75 + + float sd_diffs = 0.f; + for (int portion = 0; portion < portions; portion++) + { // Calculate standard deviation of differences. + sd_diffs +=(diffs[portion] - mean_diff) * (diffs[portion] - mean_diff); + } + int degrees_of_freedom = portions-1; // Use the n-1 formula. + sd_diffs /= degrees_of_freedom; + sd_diffs = sqrt(sd_diffs); + cout << "Standard deviation of differences = " << sd_diffs << endl; // 4.99166 + // Standard deviation of differences = 4.99166 + double t = mean_diff * sqrt(static_cast(portions))/ sd_diffs; // -0.70117 + cout << "Student's t = " << t << ", if " << degrees_of_freedom << " degrees of freedom." << endl; + // Student's t = -0.70117, if 3 degrees of freedom. + cout << "Probability of the means being different is " + << 2.F * cdf(students_t(degrees_of_freedom), t) << "."<< endl; // 0.266846 * 2 = 0.533692 + // Double the probability because using a 'two-sided test' because + // mean for 'Wet oxidation' could be either + // greater OR LESS THAN for 'Direct extraction'. + + return 0; +} // int main() + +/* + +Output is: + +Example3 using Student's t function. +Portion wet_oxidation Direct_extraction difference + 0 71 76 -5 + 1 61 68 -7 + 2 50 48 2 + 3 60 57 3 +Mean difference = -1.75 +Standard deviation of differences = 4.99166 +Student's t = -0.70117, if 3 degrees of freedom. +Probability of the means being different is 0.533692. + +*/ + + + + diff --git a/example/students_t_single_sample.cpp b/example/students_t_single_sample.cpp new file mode 100644 index 000000000..101835ae4 --- /dev/null +++ b/example/students_t_single_sample.cpp @@ -0,0 +1,427 @@ +// Copyright John Maddock 2006 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +#include +#include + +void confidence_limits_on_mean(double Sm, double Sd, unsigned Sn) +{ + // + // Sm = Sample Mean. + // Sd = Sample Standard Deviation. + // Sn = Sample Size. + // + // Calculate confidence intervals for the mean. + // For example if we set the confidence limit to + // 0.95, we know that if we repeat the sampling + // 100 times, then we expect that the true mean + // will be between out limits on 95 occations. + // Note: this is not the same as saying a 95% + // confidence interval means that there is a 95% + // probability that the interval contains the true mean. + // The interval computed from a given sample either + // contains the true mean or it does not. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm + + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "__________________________________\n" + "2-Sided Confidence Limits For Mean\n" + "__________________________________\n\n"; + cout << setprecision(7); + cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n"; + cout << setw(40) << left << "Mean" << "= " << Sm << "\n"; + cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n"; + // + // Define a table of significance/risk levels: + // + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Start by declaring the distribution we'll need: + // + students_t dist(Sn - 1); + // + // Print table header: + // + cout << "\n\n" + "_______________________________________________________________\n" + "Confidence T Interval Lower Upper\n" + " Value (%) Value Width Limit Limit\n" + "_______________________________________________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate T: + double T = quantile(complement(dist, alpha[i] / 2)); + // Print T: + cout << fixed << setprecision(3) << setw(10) << right << T; + // Calculate width of interval (one sided): + double w = T * Sd / sqrt(double(Sn)); + // Print width: + if(w < 0.01) + cout << scientific << setprecision(3) << setw(17) << right << w; + else + cout << fixed << setprecision(3) << setw(17) << right << w; + // Print Limits: + cout << fixed << setprecision(5) << setw(15) << right << Sm - w; + cout << fixed << setprecision(5) << setw(15) << right << Sm + w << endl; + } + cout << endl; +} + +void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha) +{ + // + // M = true mean. + // Sm = Sample Mean. + // Sd = Sample Standard Deviation. + // Sn = Sample Size. + // alpha = Significance Level. + // + // A Students t test applied to a single set of data. + // We are testing the null hypothesis that the true + // mean of the sample is M, and that any variation is down + // to chance. We can also test the alternative hypothesis + // that any difference is not down to chance. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm + // + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "__________________________________\n" + "Student t test for a single sample\n" + "__________________________________\n\n"; + cout << setprecision(5); + cout << setw(55) << left << "Number of Observations" << "= " << Sn << "\n"; + cout << setw(55) << left << "Sample Mean" << "= " << Sm << "\n"; + cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n"; + cout << setw(55) << left << "Expected True Mean" << "= " << M << "\n\n"; + // + // Now we can calculate and output some stats: + // + // Difference in means: + double diff = Sm - M; + cout << setw(55) << left << "Sample Mean - Expected Test Mean" << "= " << diff << "\n"; + // Degrees of freedom: + unsigned v = Sn - 1; + cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n"; + // t-statistic: + double t_stat = diff * sqrt(double(Sn)) / Sd; + cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n"; + // + // Finally define our distribution, and get the probability: + // + students_t dist(v); + double q = cdf(complement(dist, fabs(t_stat))); + cout << setw(55) << left << "Probability that difference is due to chance" << "= " + << setprecision(3) << scientific << 2 * q << "\n\n"; + // + // Finally print out results of alternative hypothesis: + // + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + cout << "Mean != " << setprecision(3) << fixed << M << " "; + if(q < alpha / 2) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Mean < " << setprecision(3) << fixed << M << " "; + if(cdf(dist, t_stat) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Mean > " << setprecision(3) << fixed << M << " "; + if(cdf(complement(dist, t_stat)) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; +} + +void single_sample_find_df(double M, double Sm, double Sd) +{ + // + // M = true mean. + // Sm = Sample Mean. + // Sd = Sample Standard Deviation. + // + using namespace std; + using namespace boost::math; + + // Print out general info: + cout << + "_____________________________________________________________\n" + "Estimated sample sizes required for various confidence levels\n" + "_____________________________________________________________\n\n"; + cout << setprecision(5); + cout << setw(40) << left << "True Mean" << "= " << M << "\n"; + cout << setw(40) << left << "Sample Mean" << "= " << Sm << "\n"; + cout << setw(40) << left << "Sample Standard Deviation" << "= " << Sd << "\n"; + // + // Define a table of significance intervals: + // + double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 }; + // + // Print table header: + // + cout << "\n\n" + "_______________________________________________________________\n" + "Confidence Estimated Estimated\n" + " Value (%) Sample Size Sample Size\n" + " (one sided test) (two sided test)\n" + "_______________________________________________________________\n"; + // + // Now print out the data for the table rows. + // + for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) + { + // Confidence value: + cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); + // calculate df for single sided test: + double df = students_t::find_degrees_of_freedom( + fabs(M - Sm), alpha[i], alpha[i], Sd); + // convert to sample size: + double size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size; + // calculate df for two sided test: + df = students_t::find_degrees_of_freedom( + fabs(M - Sm), alpha[i]/2, alpha[i], Sd); + // convert to sample size: + size = ceil(df) + 1; + // Print size: + cout << fixed << setprecision(0) << setw(16) << right << size << endl; + } + cout << endl; +} + +int main() +{ + // + // Run tests for Heat Flow Meter data + // see http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm + // The data was collected while calibrating a heat flow meter + // against a known value. + // + confidence_limits_on_mean(9.261460, 0.2278881e-01, 195); + single_sample_t_test(5, 9.261460, 0.2278881e-01, 195, 0.05); + single_sample_find_df(5, 9.261460, 0.2278881e-01); + + // + // Data for this example from: + // P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. + // from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 + // J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907 + // + // Determination of mercury by cold-vapour atomic absorption, + // the following values were obtained fusing a trusted + // Standard Reference Material containing 38.9% mercury, + // which we assume is correct or 'true'. + // + confidence_limits_on_mean(37.8, 0.964365, 3); + // 95% test: + single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.05); + // 90% test: + single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.1); + // parameter estimate: + single_sample_find_df(38.9, 37.8, 0.964365); + + return 0; +} + +/* + +Output: + +------ Build started: Project: students_t_single_sample, Configuration: Debug Win32 ------ +Compiling... +students_t_single_sample.cpp +Linking... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\students_t_single_sample.exe" +__________________________________ +2-Sided Confidence Limits For Mean +__________________________________ + +Number of Observations = 195 +Mean = 9.26146 +Standard Deviation = 0.02278881 + + +_______________________________________________________________ +Confidence T Interval Lower Upper + Value (%) Value Width Limit Limit +_______________________________________________________________ + 50.000 0.676 1.103e-003 9.26036 9.26256 + 75.000 1.154 1.883e-003 9.25958 9.26334 + 90.000 1.653 2.697e-003 9.25876 9.26416 + 95.000 1.972 3.219e-003 9.25824 9.26468 + 99.000 2.601 4.245e-003 9.25721 9.26571 + 99.900 3.341 5.453e-003 9.25601 9.26691 + 99.990 3.973 6.484e-003 9.25498 9.26794 + 99.999 4.537 7.404e-003 9.25406 9.26886 + +__________________________________ +Student t test for a single sample +__________________________________ + +Number of Observations = 195 +Sample Mean = 9.26146 +Sample Standard Deviation = 0.02279 +Expected True Mean = 5.00000 + +Sample Mean - Expected Test Mean = 4.26146 +Degrees of Freedom = 194 +T Statistic = 2611.28380 +Probability that difference is due to chance = 0.000e+000 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Mean != 5.000 NOT REJECTED +Mean < 5.000 REJECTED +Mean > 5.000 NOT REJECTED + + +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ + +True Mean = 5.00000 +Sample Mean = 9.26146 +Sample Standard Deviation = 0.02279 + + +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (one sided test) (two sided test) +_______________________________________________________________ + 50.000 2 2 + 75.000 2 2 + 90.000 2 2 + 95.000 2 2 + 99.000 2 2 + 99.900 3 3 + 99.990 3 3 + 99.999 4 4 + +__________________________________ +2-Sided Confidence Limits For Mean +__________________________________ + +Number of Observations = 3 +Mean = 37.8000000 +Standard Deviation = 0.9643650 + + +_______________________________________________________________ +Confidence T Interval Lower Upper + Value (%) Value Width Limit Limit +_______________________________________________________________ + 50.000 0.816 0.455 37.34539 38.25461 + 75.000 1.604 0.893 36.90717 38.69283 + 90.000 2.920 1.626 36.17422 39.42578 + 95.000 4.303 2.396 35.40438 40.19562 + 99.000 9.925 5.526 32.27408 43.32592 + 99.900 31.599 17.594 20.20639 55.39361 + 99.990 99.992 55.673 -17.87346 93.47346 + 99.999 316.225 176.067 -138.26683 213.86683 + +__________________________________ +Student t test for a single sample +__________________________________ + +Number of Observations = 3 +Sample Mean = 37.80000 +Sample Standard Deviation = 0.96437 +Expected True Mean = 38.90000 + +Sample Mean - Expected Test Mean = -1.10000 +Degrees of Freedom = 2 +T Statistic = -1.97566 +Probability that difference is due to chance = 1.869e-001 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Mean != 38.900 REJECTED +Mean < 38.900 REJECTED +Mean > 38.900 REJECTED + + +__________________________________ +Student t test for a single sample +__________________________________ + +Number of Observations = 3 +Sample Mean = 37.80000 +Sample Standard Deviation = 0.96437 +Expected True Mean = 38.90000 + +Sample Mean - Expected Test Mean = -1.10000 +Degrees of Freedom = 2 +T Statistic = -1.97566 +Probability that difference is due to chance = 1.869e-001 + +Results for Alternative Hypothesis and alpha = 0.1000 + +Alternative Hypothesis Conclusion +Mean != 38.900 REJECTED +Mean < 38.900 NOT REJECTED +Mean > 38.900 REJECTED + + +_____________________________________________________________ +Estimated sample sizes required for various confidence levels +_____________________________________________________________ + +True Mean = 38.90000 +Sample Mean = 37.80000 +Sample Standard Deviation = 0.96437 + + +_______________________________________________________________ +Confidence Estimated Estimated + Value (%) Sample Size Sample Size + (one sided test) (two sided test) +_______________________________________________________________ + 50.000 2 2 + 75.000 3 4 + 90.000 7 9 + 95.000 11 13 + 99.000 20 22 + 99.900 35 37 + 99.990 50 53 + 99.999 66 68 + +Build Time 0:03 +Build log was saved at "file://i:\boost-06-05-03-1300\libs\math\test\Math_test\students_t_single_sample\Debug\BuildLog.htm" +students_t_single_sample - 0 error(s), 0 warning(s) +========== Build: 1 succeeded, 0 failed, 0 up-to-date, 0 skipped ========== + + +*/ + diff --git a/example/students_t_two_samples.cpp b/example/students_t_two_samples.cpp new file mode 100644 index 000000000..e3a4fa742 --- /dev/null +++ b/example/students_t_two_samples.cpp @@ -0,0 +1,261 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4512) // assignment operator could not be generated. +# pragma warning(disable: 4510) // default constructor could not be generated. +# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required. +#endif + +#include +#include +#include + +void two_samples_t_test_equal_sd( + double Sm1, + double Sd1, + unsigned Sn1, + double Sm2, + double Sd2, + unsigned Sn2, + double alpha) +{ + // + // Sm1 = Sample Mean 1. + // Sd1 = Sample Standard Deviation 1. + // Sn1 = Sample Size 1. + // Sm2 = Sample Mean 2. + // Sd2 = Sample Standard Deviation 2. + // Sn2 = Sample Size 2. + // alpha = Significance Level. + // + // A Students t test applied to two sets of data. + // We are testing the null hypothesis that the two + // samples have the same mean and that any difference + // if due to chance. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm + // + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "_______________________________________________\n" + "Student t test for two samples (equal variances)\n" + "_______________________________________________\n\n"; + cout << setprecision(5); + cout << setw(55) << left << "Number of Observations (Sample 1)" << "= " << Sn1 << "\n"; + cout << setw(55) << left << "Sample 1 Mean" << "= " << Sm1 << "\n"; + cout << setw(55) << left << "Sample 1 Standard Deviation" << "= " << Sd1 << "\n"; + cout << setw(55) << left << "Number of Observations (Sample 2)" << "= " << Sn2 << "\n"; + cout << setw(55) << left << "Sample 2 Mean" << "= " << Sm2 << "\n"; + cout << setw(55) << left << "Sample 2 Standard Deviation" << "= " << Sd2 << "\n"; + // + // Now we can calculate and output some stats: + // + // Degrees of freedom: + double v = Sn1 + Sn2 - 2; + cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n"; + // Pooled variance: + double sp = sqrt(((Sn1-1) * Sd1 * Sd1 + (Sn2-1) * Sd2 * Sd2) / v); + cout << setw(55) << left << "Pooled Standard Deviation" << "= " << v << "\n"; + // t-statistic: + double t_stat = (Sm1 - Sm2) / (sp * sqrt(1.0 / Sn1 + 1.0 / Sn2)); + cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n"; + // + // Define our distribution, and get the probability: + // + students_t dist(v); + double q = cdf(complement(dist, fabs(t_stat))); + cout << setw(55) << left << "Probability that difference is due to chance" << "= " + << setprecision(3) << scientific << 2 * q << "\n\n"; + // + // Finally print out results of alternative hypothesis: + // + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + cout << "Sample 1 Mean != Sample 2 Mean " ; + if(q < alpha / 2) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Sample 1 Mean < Sample 2 Mean "; + if(cdf(dist, t_stat) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Sample 1 Mean > Sample 2 Mean "; + if(cdf(complement(dist, t_stat)) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; +} + +void two_samples_t_test_unequal_sd( + double Sm1, + double Sd1, + unsigned Sn1, + double Sm2, + double Sd2, + unsigned Sn2, + double alpha) +{ + // + // Sm1 = Sample Mean 1. + // Sd1 = Sample Standard Deviation 1. + // Sn1 = Sample Size 1. + // Sm2 = Sample Mean 2. + // Sd2 = Sample Standard Deviation 2. + // Sn2 = Sample Size 2. + // alpha = Significance Level. + // + // A Students t test applied to two sets of data. + // We are testing the null hypothesis that the two + // samples have the same mean and that any difference + // if due to chance. + // See http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm + // + using namespace std; + using namespace boost::math; + + // Print header: + cout << + "_________________________________________________\n" + "Student t test for two samples (unequal variances)\n" + "_________________________________________________\n\n"; + cout << setprecision(5); + cout << setw(55) << left << "Number of Observations (Sample 1)" << "= " << Sn1 << "\n"; + cout << setw(55) << left << "Sample 1 Mean" << "= " << Sm1 << "\n"; + cout << setw(55) << left << "Sample 1 Standard Deviation" << "= " << Sd1 << "\n"; + cout << setw(55) << left << "Number of Observations (Sample 2)" << "= " << Sn2 << "\n"; + cout << setw(55) << left << "Sample 2 Mean" << "= " << Sm2 << "\n"; + cout << setw(55) << left << "Sample 2 Standard Deviation" << "= " << Sd2 << "\n"; + // + // Now we can calculate and output some stats: + // + // Degrees of freedom: + double v = Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2; + v *= v; + double t1 = Sd1 * Sd1 / Sn1; + t1 *= t1; + t1 /= (Sn1 - 1); + double t2 = Sd2 * Sd2 / Sn2; + t2 *= t2; + t2 /= (Sn2 - 1); + v /= (t1 + t2); + cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n"; + // t-statistic: + double t_stat = (Sm1 - Sm2) / sqrt(Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2); + cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n"; + // + // Define our distribution, and get the probability: + // + students_t dist(v); + double q = cdf(complement(dist, fabs(t_stat))); + cout << setw(55) << left << "Probability that difference is due to chance" << "= " + << setprecision(3) << scientific << 2 * q << "\n\n"; + // + // Finally print out results of alternative hypothesis: + // + cout << setw(55) << left << + "Results for Alternative Hypothesis and alpha" << "= " + << setprecision(4) << fixed << alpha << "\n\n"; + cout << "Alternative Hypothesis Conclusion\n"; + cout << "Sample 1 Mean != Sample 2 Mean " ; + if(q < alpha / 2) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Sample 1 Mean < Sample 2 Mean "; + if(cdf(dist, t_stat) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << "Sample 1 Mean > Sample 2 Mean "; + if(cdf(complement(dist, t_stat)) < alpha) + cout << "NOT REJECTED\n"; + else + cout << "REJECTED\n"; + cout << endl << endl; +} + +int main() +{ + // + // Run tests for Car Mileage sample data + // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm + // from the NIST website http://www.itl.nist.gov. The data compares + // miles per gallon of US cars with miles per gallon of Japanese cars. + // + two_samples_t_test_equal_sd(20.14458, 6.414700, 249, 30.48101, 6.107710, 79, 0.05); + two_samples_t_test_unequal_sd(20.14458, 6.414700, 249, 30.48101, 6.107710, 79, 0.05); + + return 0; +} // int main() + +/* +Output is + +------ Build started: Project: students_t_two_samples, Configuration: Debug Win32 ------ +Compiling... +students_t_two_samples.cpp +Linking... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\students_t_two_samples.exe" +_______________________________________________ +Student t test for two samples (equal variances) +_______________________________________________ + +Number of Observations (Sample 1) = 249 +Sample 1 Mean = 20.145 +Sample 1 Standard Deviation = 6.4147 +Number of Observations (Sample 2) = 79 +Sample 2 Mean = 30.481 +Sample 2 Standard Deviation = 6.1077 +Degrees of Freedom = 326 +Pooled Standard Deviation = 326 +T Statistic = -12.621 +Probability that difference is due to chance = 5.273e-030 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Sample 1 Mean != Sample 2 Mean NOT REJECTED +Sample 1 Mean < Sample 2 Mean NOT REJECTED +Sample 1 Mean > Sample 2 Mean REJECTED + + +_________________________________________________ +Student t test for two samples (unequal variances) +_________________________________________________ + +Number of Observations (Sample 1) = 249 +Sample 1 Mean = 20.14458 +Sample 1 Standard Deviation = 6.41470 +Number of Observations (Sample 2) = 79 +Sample 2 Mean = 30.48101 +Sample 2 Standard Deviation = 6.10771 +Degrees of Freedom = 136.87499 +T Statistic = -12.94627 +Probability that difference is due to chance = 1.571e-025 + +Results for Alternative Hypothesis and alpha = 0.0500 + +Alternative Hypothesis Conclusion +Sample 1 Mean != Sample 2 Mean NOT REJECTED +Sample 1 Mean < Sample 2 Mean NOT REJECTED +Sample 1 Mean > Sample 2 Mean REJECTED + +Build Time 0:03 +Build log was saved at "file://i:\boost-06-05-03-1300\libs\math\test\Math_test\students_t_two_samples\Debug\BuildLog.htm" +students_t_two_samples - 0 error(s), 0 warning(s) +========== Build: 1 succeeded, 0 failed, 0 up-to-date, 0 skipped ========== + +*/ + diff --git a/index.html b/index.html index 002cf0762..bc4416553 100644 --- a/index.html +++ b/index.html @@ -1,13 +1,14 @@ - + Automatic redirection failed, please go to -../../doc/html/boost_math.html. +doc/html/index.html.

Copyright Daryle Walker, Hubert Holin and John Maddock 2006

Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at www.boost.org/LICENSE_1_0.txt).

+ diff --git a/minimax/f.cpp b/minimax/f.cpp new file mode 100644 index 000000000..86f55b87e --- /dev/null +++ b/minimax/f.cpp @@ -0,0 +1,246 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define L22 +#include "../tools/ntl_rr_lanczos.hpp" +#include "../tools/ntl_rr_digamma.hpp" +#include +#include +#include + +#include + + +boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant) +{ + static const boost::math::ntl::RR tiny = boost::math::tools::min_value() * 64; + switch(variant) + { + case 0: + return boost::math::erfc(x) * x / exp(-x * x); + case 1: + return boost::math::erf(x); + case 2: + { + boost::math::ntl::RR x_ = x == 0 ? 1e-80 : x; + return boost::math::erf(x_) / x_; + } + case 3: + { + boost::math::ntl::RR y(x); + if(y == 0) + y += tiny; + return boost::math::lgamma(y+2) / y - 0.5; + } + case 4: + // + // lgamma in the range [2,3], use: + // + // lgamma(x) = (x-2) * (x + 1) * (c + R(x - 2)) + // + // Works well at 80-bit long double precision, but doesn't + // stretch to 128-bit precision. + // + if(x == 0) + { + return boost::lexical_cast("0.42278433509846713939348790991759756895784066406008") / 3; + } + return boost::math::lgamma(x+2) / (x * (x+3)); + case 5: + { + // + // lgamma in the range [1,2], use: + // + // lgamma(x) = (x - 1) * (x - 2) * (c + R(x - 1)) + // + // works well over [1, 1.5] but not near 2 :-( + // + boost::math::ntl::RR r1 = boost::lexical_cast("0.57721566490153286060651209008240243104215933593992"); + boost::math::ntl::RR r2 = boost::lexical_cast("0.42278433509846713939348790991759756895784066406008"); + if(x == 0) + { + return r1; + } + if(x == 1) + { + return r2; + } + return boost::math::lgamma(x+1) / (x * (x - 1)); + } + case 6: + { + // + // lgamma in the range [1.5,2], use: + // + // lgamma(x) = (2 - x) * (1 - x) * (c + R(2 - x)) + // + // works well over [1.5, 2] but not near 1 :-( + // + boost::math::ntl::RR r1 = boost::lexical_cast("0.57721566490153286060651209008240243104215933593992"); + boost::math::ntl::RR r2 = boost::lexical_cast("0.42278433509846713939348790991759756895784066406008"); + if(x == 0) + { + return r2; + } + if(x == 1) + { + return r1; + } + return boost::math::lgamma(2-x) / (x * (x - 1)); + } + case 7: + { + // + // erf_inv in range [0, 0.5] + // + boost::math::ntl::RR y = x; + if(y == 0) + y = boost::math::tools::epsilon() / 64; + return boost::math::erf_inv(y) / (y * (y+10)); + } + case 8: + { + // + // erfc_inv in range [0.25, 0.5] + // Use an y-offset of 0.25, and range [0, 0.25] + // abs error, auto y-offset. + // + boost::math::ntl::RR y = x; + if(y == 0) + y = boost::lexical_cast("1e-5000"); + return sqrt(-2 * log(y)) / boost::math::erfc_inv(y); + } + case 9: + { + boost::math::ntl::RR x2 = x; + if(x2 == 0) + x2 = boost::lexical_cast("1e-5000"); + boost::math::ntl::RR y = exp(-x2*x2); // sqrt(-log(x2)) - 5; + return boost::math::erfc_inv(y) / x2; + } + case 10: + { + // + // Digamma over the interval [1,2], set x-offset to 1 + // and optimise for absolute error over [0,1]. + // + int current_precision = boost::math::ntl::RR::precision(); + if(current_precision < 1000) + boost::math::ntl::RR::SetPrecision(1000); + // + // This value for the root of digamma is calculated using our + // differentiated lanczos approximation. It agrees with Cody + // to ~ 25 digits and to Morris to 35 digits. See: + // TOMS ALGORITHM 708 (Didonato and Morris). + // and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher. + // + //boost::math::ntl::RR root = boost::lexical_cast("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871"); + // + // Actually better to calculate the root on the fly, it appears to be more + // accurate: convergence is easier with the 1000-bit value, the approximation + // produced agrees with functions.mathworld.com values to 35 digits even quite + // near the root. + // + static boost::math::tools::eps_tolerance tol(1000); + static boost::uintmax_t max_iter = 1000; + static const boost::math::ntl::RR root = boost::math::tools::bracket_and_solve_root(&boost::math::digamma, boost::math::ntl::RR(1.4), boost::math::ntl::RR(1.5), true, tol, max_iter).first; + + boost::math::ntl::RR x2 = x; + double lim = 1e-65; + if(fabs(x2 - root) < lim) + { + // + // This is a problem area: + // x2-root suffers cancellation error, so does digamma. + // That gets compounded again when Remez calculates the error + // function. This cludge seems to stop the worst of the problems: + // + static const boost::math::ntl::RR a = boost::math::digamma(root - lim) / -lim; + static const boost::math::ntl::RR b = boost::math::digamma(root + lim) / lim; + boost::math::ntl::RR fract = (x2 - root + lim) / (2*lim); + boost::math::ntl::RR r = (1-fract) * a + fract * b; + std::cout << "In root area: " << r; + return r; + } + boost::math::ntl::RR result = boost::math::digamma(x2) / (x2 - root); + if(current_precision < 1000) + boost::math::ntl::RR::SetPrecision(current_precision); + return result; + } + case 11: + // expm1: + if(x == 0) + { + static boost::math::ntl::RR lim = 1e-80; + static boost::math::ntl::RR a = boost::math::expm1(-lim); + static boost::math::ntl::RR b = boost::math::expm1(lim); + static boost::math::ntl::RR l = (b-a) / (2 * lim); + return l; + } + return boost::math::expm1(x) / x; + case 12: + // demo, and test case: + return exp(x); + case 13: + // K(k): + { + // x = k^2 + boost::math::ntl::RR k2 = x; + if(k2 > boost::math::ntl::RR(1) - 1e-40) + k2 = boost::math::ntl::RR(1) - 1e-40; + /*if(k < 1e-40) + k = 1e-40;*/ + boost::math::ntl::RR p2 = boost::math::constants::pi() / 2; + return (boost::math::ellint_1(sqrt(k2))) / (p2 - boost::math::log1p(-k2)); + } + case 14: + // K(k) + { + // x = 1 - k^2 + boost::math::ntl::RR mp = x; + if(mp < 1e-20) + mp = 1e-20; + boost::math::ntl::RR k = sqrt(1 - mp); + static const boost::math::ntl::RR l4 = log(boost::math::ntl::RR(4)); + boost::math::ntl::RR p2 = boost::math::constants::pi() / 2; + return (boost::math::ellint_1(k) + 1) / (1 + l4 - log(mp)); + } + case 15: + // E(k) + { + // x = 1-k^2 + boost::math::ntl::RR z = 1 - x * log(x); + return boost::math::ellint_2(sqrt(1-x)) / z; + } + case 16: + // Bessel I0(x) over [0,16]: + { + return boost::math::cyl_bessel_i(0, sqrt(x)); + } + case 17: + // Bessel I0(x) over [16,INF] + { + boost::math::ntl::RR z = 1 / (boost::math::ntl::RR(1)/16 - x); + return boost::math::cyl_bessel_i(0, z) * sqrt(z) / exp(z); + } + } + return 0; +} + +void show_extra( + const boost::math::tools::polynomial& n, + const boost::math::tools::polynomial& d, + const boost::math::ntl::RR& x_offset, + const boost::math::ntl::RR& y_offset, + int variant) +{ + switch(variant) + { + default: + // do nothing here... + ; + } +} + diff --git a/minimax/main.cpp b/minimax/main.cpp new file mode 100644 index 000000000..16551df86 --- /dev/null +++ b/minimax/main.cpp @@ -0,0 +1,520 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_UBLAS_TYPE_CHECK_EPSILON (type_traits::type_sqrt (boost::math::tools::epsilon ())) +#define BOOST_UBLAS_TYPE_CHECK_MIN (type_traits::type_sqrt ( boost::math::tools::min_value())) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include // for test_main + +extern boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant); +extern void show_extra( + const boost::math::tools::polynomial& n, + const boost::math::tools::polynomial& d, + const boost::math::ntl::RR& x_offset, + const boost::math::ntl::RR& y_offset, + int variant); + +using namespace boost::spirit; + +boost::math::ntl::RR a(0), b(1); // range to optimise over +bool rel_error(true); +bool pin(false); +int orderN(3); +int orderD(1); +int target_precision(53); +int working_precision(250); +bool started(false); +int variant(0); +int skew(0); +int brake(50); +boost::math::ntl::RR x_offset(0), y_offset(0); +bool auto_offset_y; + +boost::shared_ptr > p_remez; + +boost::math::ntl::RR the_function(const boost::math::ntl::RR& val) +{ + return f(val + x_offset, variant) + y_offset; +} + +void step_some(unsigned count) +{ + try{ + NTL::RR::SetPrecision(working_precision); + if(!started) + { + // + // If we have an automatic y-offset calculate it now: + // + if(auto_offset_y) + { + boost::math::ntl::RR fa, fb, fm; + fa = f(a + x_offset, variant); + fb = f(b + x_offset, variant); + fm = f((a+b)/2 + x_offset, variant); + y_offset = -(fa + fb + fm) / 3; + NTL::RR::SetOutputPrecision(5); + std::cout << "Setting auto-y-offset to " << y_offset << std::endl; + } + // + // Truncate offsets to float precision: + // + x_offset = NTL::RoundToPrecision(x_offset.value(), 20); + y_offset = NTL::RoundToPrecision(y_offset.value(), 20); + // + // Construct new Remez state machine: + // + p_remez.reset(new boost::math::tools::remez_minimax( + &the_function, + orderN, orderD, + a, b, + pin, + rel_error, + skew, + working_precision)); + std::cout << "Max error in interpolated form: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast(p_remez->max_error()) << std::endl; + // + // Signal that we've started: + // + started = true; + } + unsigned i; + for(i = 0; i < count; ++i) + { + std::cout << "Stepping..." << std::endl; + p_remez->set_brake(brake); + boost::math::ntl::RR r = p_remez->iterate(); + NTL::RR::SetOutputPrecision(3); + std::cout + << "Maximum Deviation Found: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast(p_remez->max_error()) << std::endl + << "Expected Error Term: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast(p_remez->error_term()) << std::endl + << "Maximum Relative Change in Control Points: " << std::setprecision(3) << std::scientific << boost::math::tools::real_cast(r) << std::endl; + } + } + catch(const std::exception& e) + { + std::cout << "Step failed with exception: " << e.what() << std::endl; + } +} + +void step(const char*, const char*) +{ + step_some(1); +} + +void show(const char*, const char*) +{ + NTL::RR::SetPrecision(working_precision); + if(started) + { + boost::math::tools::polynomial n = p_remez->numerator(); + boost::math::tools::polynomial d = p_remez->denominator(); + int prec = 2 + (target_precision * 3010LL)/10000; + std::cout << std::scientific << std::setprecision(prec); + NTL::RR::SetOutputPrecision(prec); + boost::numeric::ublas::vector v = p_remez->zero_points(); + + std::cout << " Zeros = {\n"; + unsigned i; + for(i = 0; i < v.size(); ++i) + { + std::cout << " " << v[i] << std::endl; + } + std::cout << " }\n"; + + v = p_remez->chebyshev_points(); + std::cout << " Chebeshev Control Points = {\n"; + for(i = 0; i < v.size(); ++i) + { + std::cout << " " << v[i] << std::endl; + } + std::cout << " }\n"; + + std::cout << "X offset: " << x_offset << std::endl; + std::cout << "Y offset: " << y_offset << std::endl; + + std::cout << "P = {"; + for(i = 0; i < n.size(); ++i) + { + std::cout << " " << n[i] << std::endl; + } + std::cout << " }\n"; + + std::cout << "Q = {"; + for(i = 0; i < d.size(); ++i) + { + std::cout << " " << d[i] << std::endl; + } + std::cout << " }\n"; + + show_extra(n, d, x_offset, y_offset, variant); + } + else + { + std::cerr << "Nothing to display" << std::endl; + } +} + +void do_graph(unsigned points) +{ + NTL::RR::SetPrecision(working_precision); + boost::math::ntl::RR step = (b - a) / (points - 1); + boost::math::ntl::RR x = a; + while(points > 1) + { + NTL::RR::SetOutputPrecision(10); + std::cout << std::setprecision(10) << std::setw(30) << std::left + << boost::lexical_cast(x) << the_function(x) << std::endl; + --points; + x += step; + } + std::cout << std::setprecision(10) << std::setw(30) << std::left + << boost::lexical_cast(b) << the_function(b) << std::endl; +} + +void graph(const char*, const char*) +{ + do_graph(3); +} + +template +void do_test(T, const char* name) +{ + boost::math::ntl::RR::SetPrecision(working_precision); + if(started) + { + // + // We want to test the approximation at fixed precision: + // either float, double or long double. Begin by getting the + // polynomials: + // + boost::math::tools::polynomial n, d; + n = p_remez->numerator(); + d = p_remez->denominator(); + // + // We'll test at the Chebeshev control points which is where + // (in theory) the largest deviation should occur. For good + // measure we'll test at the zeros as well: + // + boost::numeric::ublas::vector + zeros(p_remez->zero_points()), + cheb(p_remez->chebyshev_points()); + + boost::math::ntl::RR max_error(0); + + // + // Do the tests at the zeros: + // + std::cout << "Starting tests at " << name << " precision...\n"; + std::cout << "Absissa Error\n"; + unsigned i; + for(i = 0; i < zeros.size(); ++i) + { + boost::math::ntl::RR true_result = the_function(zeros[i]); + T absissa = boost::math::tools::real_cast(zeros[i]); + boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); + boost::math::ntl::RR err; + if(rel_error) + { + err = boost::math::tools::relative_error(test_result, true_result); + } + else + { + err = fabs(test_result - true_result); + } + if(err > max_error) + max_error = err; + std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa + << boost::math::tools::real_cast(err) << std::endl; + } + // + // Do the tests at the Chebeshev control points: + // + for(i = 0; i < cheb.size(); ++i) + { + boost::math::ntl::RR true_result = the_function(cheb[i]); + T absissa = boost::math::tools::real_cast(cheb[i]); + boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); + boost::math::ntl::RR err; + if(rel_error) + { + err = boost::math::tools::relative_error(test_result, true_result); + } + else + { + err = fabs(test_result - true_result); + } + if(err > max_error) + max_error = err; + std::cout << std::setprecision(6) << std::setw(15) << std::left << absissa + << boost::math::tools::real_cast(err) << std::endl; + } + std::cout << "Max error found: " << std::setprecision(6) << boost::math::tools::real_cast(max_error) << std::endl; + } + else + { + std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; + } +} + +void test_float(const char*, const char*) +{ + do_test(float(0), "float"); +} + +void test_double(const char*, const char*) +{ + do_test(double(0), "double"); +} + +void test_long(const char*, const char*) +{ + do_test((long double)(0), "long double"); +} + +void test_all(const char*, const char*) +{ + do_test(float(0), "float"); + do_test(double(0), "double"); + do_test((long double)(0), "long double"); +} + +template +void do_test_n(T, const char* name, unsigned count) +{ + boost::math::ntl::RR::SetPrecision(working_precision); + if(started) + { + // + // We want to test the approximation at fixed precision: + // either float, double or long double. Begin by getting the + // polynomials: + // + boost::math::tools::polynomial n, d; + n = p_remez->numerator(); + d = p_remez->denominator(); + + boost::math::ntl::RR max_error(0); + boost::math::ntl::RR step = (b - a) / count; + + // + // Do the tests at the zeros: + // + std::cout << "Starting tests at " << name << " precision...\n"; + std::cout << "Absissa Error\n"; + for(boost::math::ntl::RR x = a; x <= b; x += step) + { + boost::math::ntl::RR true_result = the_function(x); + T absissa = boost::math::tools::real_cast(x); + boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); + boost::math::ntl::RR err; + if(rel_error) + { + err = boost::math::tools::relative_error(test_result, true_result); + } + else + { + err = fabs(test_result - true_result); + } + if(err > max_error) + max_error = err; + std::cout << std::setprecision(6) << std::setw(15) << std::left << boost::math::tools::real_cast(absissa) + << (test_result < true_result ? "-" : "") << boost::math::tools::real_cast(err) << std::endl; + } + std::cout << "Max error found: " << std::setprecision(6) << boost::math::tools::real_cast(max_error) << std::endl; + } + else + { + std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; + } +} + +void test_n(unsigned n) +{ + do_test_n(boost::math::ntl::RR(), "boost::math::ntl::RR", n); +} + +void test_float_n(unsigned n) +{ + do_test_n(float(0), "float", n); +} + +void test_double_n(unsigned n) +{ + do_test_n(double(0), "double", n); +} + +void test_long_n(unsigned n) +{ + do_test_n((long double)(0), "long double", n); +} + +void rotate(const char*, const char*) +{ + if(p_remez) + { + p_remez->rotate(); + } + else + { + std::cerr << "Nothing to rotate" << std::endl; + } +} + +void rescale(const char*, const char*) +{ + if(p_remez) + { + p_remez->rescale(a, b); + } + else + { + std::cerr << "Nothing to rescale" << std::endl; + } +} + +void graph_poly(const char*, const char*) +{ + int i = 50; + boost::math::ntl::RR::SetPrecision(working_precision); + if(started) + { + // + // We want to test the approximation at fixed precision: + // either float, double or long double. Begin by getting the + // polynomials: + // + boost::math::tools::polynomial n, d; + n = p_remez->numerator(); + d = p_remez->denominator(); + + boost::math::ntl::RR max_error(0); + boost::math::ntl::RR step = (b - a) / i; + + std::cout << "Evaluating Numerator...\n"; + boost::math::ntl::RR val; + for(val = a; val <= b; val += step) + std::cout << n.evaluate(val) << std::endl; + std::cout << "Evaluating Denominator...\n"; + for(val = a; val <= b; val += step) + std::cout << d.evaluate(val) << std::endl; + } + else + { + std::cout << "Nothing to test: try converging an approximation first!!!" << std::endl; + } +} + +int test_main(int, char* []) +{ + std::string line; + while(std::getline(std::cin, line)) + { + if(parse(line.c_str(), str_p("quit"), space_p).full) + return 0; + if(false == parse(line.c_str(), + ( + + str_p("range")[assign_a(started, false)] && real_p[assign_a(a)] && real_p[assign_a(b)] + || + str_p("relative")[assign_a(started, false)][assign_a(rel_error, true)] + || + str_p("absolute")[assign_a(started, false)][assign_a(rel_error, false)] + || + str_p("pin")[assign_a(started, false)] && str_p("true")[assign_a(pin, true)] + || + str_p("pin")[assign_a(started, false)] && str_p("false")[assign_a(pin, false)] + || + str_p("pin")[assign_a(started, false)] && str_p("1")[assign_a(pin, true)] + || + str_p("pin")[assign_a(started, false)] && str_p("0")[assign_a(pin, false)] + || + str_p("pin")[assign_a(started, false)][assign_a(pin, true)] + || + str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)] && uint_p[assign_a(orderD)] + || + str_p("order")[assign_a(started, false)] && uint_p[assign_a(orderN)] + || + str_p("target-precision") && uint_p[assign_a(target_precision)] + || + str_p("working-precision")[assign_a(started, false)] && uint_p[assign_a(working_precision)] + || + str_p("variant")[assign_a(started, false)] && int_p[assign_a(variant)] + || + str_p("skew")[assign_a(started, false)] && int_p[assign_a(skew)] + || + str_p("brake") && int_p[assign_a(brake)] + || + str_p("step") && int_p[&step_some] + || + str_p("step")[&step] + || + str_p("poly")[&graph_poly] + || + str_p("info")[&show] + || + str_p("graph") && uint_p[&do_graph] + || + str_p("graph")[&graph] + || + str_p("x-offset") && real_p[assign_a(x_offset)] + || + str_p("y-offset") && str_p("auto")[assign_a(auto_offset_y, true)] + || + str_p("y-offset") && real_p[assign_a(y_offset)][assign_a(auto_offset_y, false)] + || + str_p("test") && str_p("float") && uint_p[&test_float_n] + || + str_p("test") && str_p("float")[&test_float] + || + str_p("test") && str_p("double") && uint_p[&test_double_n] + || + str_p("test") && str_p("double")[&test_double] + || + str_p("test") && str_p("long") && uint_p[&test_long_n] + || + str_p("test") && str_p("long")[&test_long] + || + str_p("test") && str_p("all")[&test_all] + || + str_p("test") && uint_p[&test_n] + || + str_p("rotate")[&rotate] + || + str_p("rescale") && real_p[assign_a(a)] && real_p[assign_a(b)] && epsilon_p[&rescale] + + ), space_p).full) + { + std::cout << "Unable to parse directive: \"" << line << "\"" << std::endl; + } + else + { + std::cout << "a = " << a << std::endl; + std::cout << "b = " << b << std::endl; + std::cout << "Relative Error = " << rel_error << std::endl; + std::cout << "Pin to Origin = " << pin << std::endl; + std::cout << "Order (Numerator) = " << orderN << std::endl; + std::cout << "Order (Denominator) = " << orderD << std::endl; + std::cout << "Target Precision = " << target_precision << std::endl; + std::cout << "Working Precision = " << working_precision << std::endl; + std::cout << "Variant = " << variant << std::endl; + std::cout << "Skew = " << skew << std::endl; + std::cout << "Brake = " << brake << std::endl; + std::cout << "X Offset = " << x_offset << std::endl; + std::cout << "Y Offset = " << y_offset << std::endl; + std::cout << "Automatic Y Offset = " << auto_offset_y << std::endl; + } + } + return 0; +} diff --git a/octonion/octonion_test.cpp b/octonion/octonion_test.cpp index d49d8a476..848e4de38 100644 --- a/octonion/octonion_test.cpp +++ b/octonion/octonion_test.cpp @@ -33,15 +33,20 @@ template<> struct string_type_name \ DEFINE_TYPE_NAME(float); DEFINE_TYPE_NAME(double); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS DEFINE_TYPE_NAME(long double); +#endif - +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS typedef boost::mpl::list test_types; +#else +typedef boost::mpl::list test_types; +#endif // Apple GCC 4.0 uses the "double double" format for its long double, // which means that epsilon is VERY small but useless for // comparisons. So, don't do those comparisons. -#if defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4 +#if (defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4) || defined(BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS) typedef boost::mpl::list near_eps_test_types; #else typedef boost::mpl::list near_eps_test_types; diff --git a/performance/Jamfile.v2 b/performance/Jamfile.v2 new file mode 100644 index 000000000..b20c97be2 --- /dev/null +++ b/performance/Jamfile.v2 @@ -0,0 +1,27 @@ +# Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang. +# Distributed under the Boost Software License, Version 1.0. +# (See accompanying file LICENSE_1_0.txt or copy at +# http://www.boost.org/LICENSE_1_0.txt). + + +exe math_performance : +distributions.cpp +main.cpp +test_erf.cpp +test_gamma.cpp +test_ibeta.cpp +test_igamma.cpp +test_polynomial.cpp +test_reference.cpp ; + +install dist-bin + : + math_performance + : + EXE + ./bin + : + release + ; + + diff --git a/performance/distributions.cpp b/performance/distributions.cpp new file mode 100644 index 000000000..1c483749f --- /dev/null +++ b/performance/distributions.cpp @@ -0,0 +1,452 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +double probabilities[] = { + 1e-5, + 1e-4, + 1e-3, + 1e-2, + 0.05, + 0.1, + 0.2, + 0.3, + 0.4, + 0.5, + 0.6, + 0.7, + 0.8, + 0.9, + 0.95, + 1-1e-5, + 1-1e-4, + 1-1e-3, + 1-1e-2 +}; + +int int_values[] = { + 1, + 2, + 3, + 5, + 10, + 20, + 50, + 100, + 1000, + 10000, + 100000 +}; + +double real_values[] = { + 1e-5, + 1e-4, + 1e-2, + 1e-1, + 1, + 10, + 100, + 1000, + 10000, + 100000 +}; + +#define BOOST_MATH_DISTRIBUTION2_TEST(name, param1_table, param2_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" BOOST_STRINGIZE(name) "-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += cdf(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i], param2_table[j]), random_variable_table[k]);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_pdf_, name), "dist-" BOOST_STRINGIZE(name) "-pdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += pdf(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i], param2_table[j]), random_variable_table[k]);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" BOOST_STRINGIZE(name) "-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += quantile(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i], param2_table[j]), probability_table[k]);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + } + +#define BOOST_MATH_DISTRIBUTION1_TEST(name, param1_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" BOOST_STRINGIZE(name) "-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += cdf(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i]), random_variable_table[k]);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_pdf_, name), "dist-" BOOST_STRINGIZE(name) "-pdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += pdf(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i]), random_variable_table[k]);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" BOOST_STRINGIZE(name) "-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += quantile(boost::math:: BOOST_JOIN(name, _distribution) <>(param1_table[i]), probability_table[k]);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + } + +BOOST_MATH_DISTRIBUTION2_TEST(beta, probabilities, probabilities, probabilities, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(binomial, int_values, probabilities, int_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(cauchy, int_values, real_values, int_values, probabilities) +BOOST_MATH_DISTRIBUTION1_TEST(chi_squared, int_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION1_TEST(exponential, real_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(fisher_f, int_values, int_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(gamma, real_values, real_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(lognormal, real_values, real_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(negative_binomial, int_values, probabilities, int_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(normal, real_values, real_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION1_TEST(poisson, real_values, int_values, probabilities) +BOOST_MATH_DISTRIBUTION1_TEST(students_t, int_values, real_values, probabilities) +BOOST_MATH_DISTRIBUTION2_TEST(weibull, real_values, real_values, real_values, probabilities) + +#ifdef TEST_R + +#define MATHLIB_STANDALONE 1 + +extern "C" { +#include "Rmath.h" +} + +#define BOOST_MATH_R_DISTRIBUTION2_TEST(name, param1_table, param2_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" #name "-R-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += p##name (random_variable_table[k], param1_table[i], param2_table[j], 1, 0);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_pdf_, name), "dist-" #name "-R-pdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += d##name (random_variable_table[k], param1_table[i], param2_table[j], 0);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" #name "-R-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += q##name (probability_table[k], param1_table[i], param2_table[j], 1, 0);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + } + +#define BOOST_MATH_R_DISTRIBUTION1_TEST(name, param1_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" #name "-R-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += p##name (random_variable_table[k], param1_table[i], 1, 0);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_pdf_, name), "dist-" #name "-R-pdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += d##name (random_variable_table[k], param1_table[i], 0);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" #name "-R-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += q##name (probability_table[k], param1_table[i], 1, 0);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + } + +BOOST_MATH_R_DISTRIBUTION2_TEST(beta, probabilities, probabilities, probabilities, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(binom, int_values, probabilities, int_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(cauchy, int_values, real_values, int_values, probabilities) +BOOST_MATH_R_DISTRIBUTION1_TEST(chisq, int_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION1_TEST(exp, real_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(f, int_values, int_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(gamma, real_values, real_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(lnorm, real_values, real_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(nbinom, int_values, probabilities, int_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(norm, real_values, real_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION1_TEST(pois, real_values, int_values, probabilities) +BOOST_MATH_R_DISTRIBUTION1_TEST(t, int_values, real_values, probabilities) +BOOST_MATH_R_DISTRIBUTION2_TEST(weibull, real_values, real_values, real_values, probabilities) + +#endif + +#ifdef TEST_CEPHES + +extern "C"{ + +double bdtr(int k, int n, double p); +double bdtri(int k, int n, double p); + +double chdtr(double df, double x); +double chdtri(double df, double p); + +double fdtr(int k, int n, double p); +double fdtri(int k, int n, double p); + +double nbdtr(int k, int n, double p); +double nbdtri(int k, int n, double p); + +} + +#define BOOST_MATH_CEPHES_DISTRIBUTION2_TEST(name, param1_table, param2_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" #name "-cephes-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += name##dtr (param1_table[i], param2_table[j], random_variable_table[k]);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" #name "-cephes-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned b_size = sizeof(param2_table)/sizeof(param2_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned j = 0; j < b_size; ++j)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += name##dtri (param1_table[i], param2_table[j], probability_table[k]);\ + }\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * b_size * c_size);\ + } + +#define BOOST_MATH_CEPHES_DISTRIBUTION1_TEST(name, param1_table, random_variable_table, probability_table) \ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist, name), "dist-" #name "-cephes-cdf")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(random_variable_table)/sizeof(random_variable_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += name##dtr (param1_table[i], random_variable_table[k]);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + }\ + BOOST_MATH_PERFORMANCE_TEST(BOOST_JOIN(dist_quant, name), "dist-" #name "-cephes-quantile")\ + {\ + double result = 0;\ + unsigned a_size = sizeof(param1_table)/sizeof(param1_table[0]);\ + unsigned c_size = sizeof(probability_table)/sizeof(probability_table[0]);\ + \ + for(unsigned i = 0; i < a_size; ++i)\ + {\ + for(unsigned k = 0; k < c_size; ++k)\ + {\ + result += name##dtri (param1_table[i], probability_table[k]);\ + }\ + }\ + \ + consume_result(result);\ + set_call_count(a_size * c_size);\ + } +// Cephes inverse doesn't actually calculate the quantile!!! +// BOOST_MATH_CEPHES_DISTRIBUTION2_TEST(b, int_values, int_values, probabilities, probabilities) +BOOST_MATH_CEPHES_DISTRIBUTION1_TEST(ch, int_values, real_values, probabilities) +BOOST_MATH_CEPHES_DISTRIBUTION2_TEST(f, int_values, int_values, real_values, probabilities) +// Cephes inverse doesn't calculate the quantile!!! +// BOOST_MATH_CEPHES_DISTRIBUTION2_TEST(nb, int_values, int_values, probabilities, probabilities) + +#endif + diff --git a/performance/main.cpp b/performance/main.cpp new file mode 100644 index 000000000..c70f1182a --- /dev/null +++ b/performance/main.cpp @@ -0,0 +1,212 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include +#include +#include +#include +#include +#include +#include +#include +#include "performance_measure.hpp" +#include + +#ifdef TEST_GSL +#include +#include +#endif + + +extern void reference_evaluate(); + +std::map times; +std::set tests; +double total = 0; +int call_count = 0; + +std::set& all_tests() +{ + static std::set i; + return i; +} + +void add_new_test(test_info i) +{ + all_tests().insert(i); +} + +void set_call_count(int i) +{ + call_count = i; +} + +void show_help() +{ + std::cout << "Specify on the command line which functions to test.\n" + "Available options are:\n"; + std::set::const_iterator i(all_tests().begin()), j(all_tests().end()); + while(i != j) + { + std::cout << " --" << (*i).name << std::endl; + ++i; + } + std::cout << "Or use --all to test everything." << std::endl; +} + +void run_tests() +{ + // Get time for empty proceedure: + double reference_time = performance_measure(reference_evaluate); + + std::set::const_iterator i, j; + for(i = tests.begin(), j = tests.end(); i != j; ++i) + { + set_call_count(1); + std::cout << "Testing " << std::left << std::setw(40) << i->name << std::flush; + double time = performance_measure(i->proc) - reference_time; + time /= call_count; + std::cout << std::setprecision(3) << std::scientific << time << std::endl; + } +} + +bool add_named_test(const char* name) +{ + bool found = false; + std::set::const_iterator a(all_tests().begin()), b(all_tests().end()); + while(a != b) + { + if(std::strcmp(name, (*a).name) == 0) + { + found = true; + tests.insert(*a); + break; + } + ++a; + } + return found; +} + +void print_current_config() +{ + std::cout << "Currently, polynomial evaluation uses method " << BOOST_MATH_POLY_METHOD << std::endl; + std::cout << "Currently, rational function evaluation uses method " << BOOST_MATH_RATIONAL_METHOD << std::endl; + if(BOOST_MATH_POLY_METHOD + BOOST_MATH_RATIONAL_METHOD > 0) + std::cout << "Currently, the largest order of polynomial or rational function" + " that uses a method other than 0, is " << BOOST_MATH_MAX_POLY_ORDER << std::endl; + bool uses_mixed_tables = boost::is_same::value; + if(uses_mixed_tables) + std::cout << "Currently, rational functions with integer coefficients are evaluated using mixed integer/real arithmetic" << std::endl; + else + std::cout << "Currently, rational functions with integer coefficients are evaluated using all real arithmetic (integer coefficients are actually stored as reals)" << std::endl << std::endl; + std::cout << "Policies are currently set as follows:\n\n"; + std::cout << "Policy Value\n"; + std::cout << "BOOST_MATH_DOMAIN_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_DOMAIN_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_POLE_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_POLE_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_OVERFLOW_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_OVERFLOW_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_UNDERFLOW_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_UNDERFLOW_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_DENORM_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_DENORM_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_EVALUATION_ERROR_POLICY " << BOOST_STRINGIZE(BOOST_MATH_EVALUATION_ERROR_POLICY) << std::endl; + std::cout << "BOOST_MATH_DIGITS10_POLICY " << BOOST_STRINGIZE(BOOST_MATH_DIGITS10_POLICY) << std::endl; + std::cout << "BOOST_MATH_PROMOTE_FLOAT_POLICY " << BOOST_STRINGIZE(BOOST_MATH_PROMOTE_FLOAT_POLICY) << std::endl; + std::cout << "BOOST_MATH_PROMOTE_DOUBLE_POLICY " << BOOST_STRINGIZE(BOOST_MATH_PROMOTE_DOUBLE_POLICY) << std::endl; + std::cout << "BOOST_MATH_DISCRETE_QUANTILE_POLICY " << BOOST_STRINGIZE(BOOST_MATH_DISCRETE_QUANTILE_POLICY) << std::endl; + std::cout << "BOOST_MATH_ASSERT_UNDEFINED_POLICY " << BOOST_STRINGIZE(BOOST_MATH_ASSERT_UNDEFINED_POLICY) << std::endl; + std::cout << "BOOST_MATH_MAX_SERIES_ITERATION_POLICY " << BOOST_STRINGIZE(BOOST_MATH_MAX_SERIES_ITERATION_POLICY) << std::endl; + std::cout << "BOOST_MATH_MAX_ROOT_ITERATION_POLICY " << BOOST_STRINGIZE(BOOST_MATH_MAX_ROOT_ITERATION_POLICY) << std::endl; +} + +int main(int argc, const char** argv) +{ + try{ + +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + + if(argc >= 2) + { + for(int i = 1; i < argc; ++i) + { + if(std::strcmp(argv[i], "--help") == 0) + { + show_help(); + } + else if(std::strcmp(argv[i], "--tune") == 0) + { + print_current_config(); + add_named_test("Polynomial-method-0"); + add_named_test("Polynomial-method-1"); + add_named_test("Polynomial-method-2"); + add_named_test("Polynomial-method-3"); + add_named_test("Polynomial-mixed-method-0"); + add_named_test("Polynomial-mixed-method-1"); + add_named_test("Polynomial-mixed-method-2"); + add_named_test("Polynomial-mixed-method-3"); + add_named_test("Rational-method-0"); + add_named_test("Rational-method-1"); + add_named_test("Rational-method-2"); + add_named_test("Rational-method-3"); + add_named_test("Rational-mixed-method-0"); + add_named_test("Rational-mixed-method-1"); + add_named_test("Rational-mixed-method-2"); + add_named_test("Rational-mixed-method-3"); + } + else if(std::strcmp(argv[i], "--all") == 0) + { + print_current_config(); + std::set::const_iterator a(all_tests().begin()), b(all_tests().end()); + while(a != b) + { + tests.insert(*a); + ++a; + } + } + else + { + bool found = false; + if((argv[i][0] == '-') && (argv[i][1] == '-')) + { + found = add_named_test(argv[i] + 2); + } + if(!found) + { + std::cerr << "Unknown option: " << argv[i] << std::endl; + show_help(); + return 1; + } + } + } + run_tests(); + } + else + { + show_help(); + } + // + // This is just to confuse the optimisers: + // + if(argc == 100000) + { + std::cerr << total << std::endl; + } + + } + catch(const std::exception& e) + { + std::cerr << e.what() << std::endl; + } + + return 0; +} + +void consume_result(double x) +{ + // Do nothing proceedure, don't let whole program optimisation + // optimise this away - doing so may cause false readings.... + total += x; +} diff --git a/performance/performance_measure.hpp b/performance/performance_measure.hpp new file mode 100644 index 000000000..894426798 --- /dev/null +++ b/performance/performance_measure.hpp @@ -0,0 +1,86 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_PERFORMANCE_MEASURE_HPP +#define BOOST_MATH_PERFORMANCE_MEASURE_HPP + +#include +#include +#include + +template +double performance_measure(F f) +{ + unsigned count = 1; + double time, result; + // + // Begin by figuring out how many times to repeat + // the function call in order to get a measureable time: + // + do + { + boost::timer t; + for(unsigned i = 0; i < count; ++i) + f(); + time = t.elapsed(); + count *= 2; + t.restart(); + }while(time < 0.5); + + count /= 2; + result = time; + // + // Now repeat the measurement over and over + // and take the shortest measured time as the + // result, generally speaking this gives us + // consistent results: + // + for(unsigned i = 0; i < 10u;++i) + { + boost::timer t; + for(unsigned i = 0; i < count; ++i) + f(); + time = t.elapsed(); + if(time < result) + result = time; + t.restart(); + } + return result / count; +} + +struct test_info +{ + test_info(const char* n, void (*p)()); + const char* name; + void (*proc)(); +}; + +inline bool operator < (const test_info& a, const test_info& b) +{ + return std::strcmp(a.name, b.name) < 0; +} + +extern void consume_result(double); +extern void set_call_count(int i); +extern void add_new_test(test_info i); + +inline test_info::test_info(const char* n, void (*p)()) +{ + name = n; + proc = p; + add_new_test(*this); +} + +#define BOOST_MATH_PERFORMANCE_TEST(name, string) \ + void BOOST_JOIN(name, __LINE__) ();\ + namespace{\ + test_info BOOST_JOIN(BOOST_JOIN(name, _init), __LINE__)(string, BOOST_JOIN(name, __LINE__));\ + }\ + void BOOST_JOIN(name, __LINE__) () + + + +#endif // BOOST_MATH_PERFORMANCE_MEASURE_HPP + diff --git a/performance/required_defines.hpp b/performance/required_defines.hpp new file mode 100644 index 000000000..ba0e72a56 --- /dev/null +++ b/performance/required_defines.hpp @@ -0,0 +1,9 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error +#define BOOST_MATH_POLE_ERROR_POLICY errno_on_error +#define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error + diff --git a/performance/test_erf.cpp b/performance/test_erf.cpp new file mode 100644 index 000000000..1052c1a05 --- /dev/null +++ b/performance/test_erf.cpp @@ -0,0 +1,109 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" +#include "performance_measure.hpp" + +#include +#include + +#define T double +#include "../test/erf_data.ipp" +#include "../test/erf_large_data.ipp" +#include "../test/erf_small_data.ipp" + +template +double erf_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::erf(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(erf_test, "erf") +{ + double result = erf_evaluate2(erf_data); + result += erf_evaluate2(erf_large_data); + result += erf_evaluate2(erf_small_data); + + consume_result(result); + set_call_count((sizeof(erf_data) + sizeof(erf_large_data) + sizeof(erf_small_data)) / sizeof(erf_data[0])); +} + +template +double erf_inv_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::erf_inv(data[i][1]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(erf_inv_test, "erf_inv") +{ + double result = erf_inv_evaluate2(erf_data); + result += erf_inv_evaluate2(erf_large_data); + result += erf_inv_evaluate2(erf_small_data); + + consume_result(result); + set_call_count((sizeof(erf_data) + sizeof(erf_large_data) + sizeof(erf_small_data)) / sizeof(erf_data[0])); +} + +#ifdef TEST_CEPHES + +extern "C" { + +double erf(double); + +} + +template +double erf_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += erf(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(erf_test, "erf-cephes") +{ + double result = erf_evaluate_cephes(erf_data); + result += erf_evaluate_cephes(erf_large_data); + result += erf_evaluate_cephes(erf_small_data); + + consume_result(result); + set_call_count((sizeof(erf_data) + sizeof(erf_large_data) + sizeof(erf_small_data)) / sizeof(erf_data[0])); +} + +#endif + +#ifdef TEST_GSL + +#include + +template +double erf_evaluate_gsl(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gsl_sf_erf (data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(erf_test, "erf-gsl") +{ + double result = erf_evaluate_gsl(erf_data); + result += erf_evaluate_gsl(erf_large_data); + result += erf_evaluate_gsl(erf_small_data); + + consume_result(result); + set_call_count((sizeof(erf_data) + sizeof(erf_large_data) + sizeof(erf_small_data)) / sizeof(erf_data[0])); +} + +#endif + + diff --git a/performance/test_gamma.cpp b/performance/test_gamma.cpp new file mode 100644 index 000000000..2d17fd514 --- /dev/null +++ b/performance/test_gamma.cpp @@ -0,0 +1,200 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +#include +#include + +#define T double +#include "../test/test_gamma_data.ipp" + +template +double gamma_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::tgamma(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(gamma_test, "gamma") +{ + double result = gamma_evaluate2(factorials); + result += gamma_evaluate2(near_1); + result += gamma_evaluate2(near_2); + result += gamma_evaluate2(near_0); + result += gamma_evaluate2(near_m10); + result += gamma_evaluate2(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +template +double lgamma_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::lgamma(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(lgamma_test, "lgamma") +{ + double result = lgamma_evaluate2(factorials); + result += lgamma_evaluate2(near_1); + result += lgamma_evaluate2(near_2); + result += lgamma_evaluate2(near_0); + result += lgamma_evaluate2(near_m10); + result += lgamma_evaluate2(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +#ifdef TEST_CEPHES + +extern "C" { + +double gamma(double); +double lgam(double); + +} + +template +double gamma_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gamma(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(gamma_test, "gamma-cephes") +{ + double result = gamma_evaluate_cephes(factorials); + result += gamma_evaluate_cephes(near_1); + result += gamma_evaluate_cephes(near_2); + result += gamma_evaluate_cephes(near_0); + result += gamma_evaluate_cephes(near_m10); + result += gamma_evaluate_cephes(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +template +double lgamma_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += lgam(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(lgamma_test, "lgamma-cephes") +{ + double result = lgamma_evaluate_cephes(factorials); + result += lgamma_evaluate_cephes(near_1); + result += lgamma_evaluate_cephes(near_2); + result += lgamma_evaluate_cephes(near_0); + result += lgamma_evaluate_cephes(near_m10); + result += lgamma_evaluate_cephes(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +#endif + +#ifdef TEST_GSL + +#include + +template +double gamma_evaluate_gsl(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gsl_sf_gamma(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(gamma_test, "gamma-gsl") +{ + double result = gamma_evaluate_gsl(factorials); + result += gamma_evaluate_gsl(near_1); + result += gamma_evaluate_gsl(near_2); + result += gamma_evaluate_gsl(near_0); + result += gamma_evaluate_gsl(near_m10); + result += gamma_evaluate_gsl(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +template +double lgamma_evaluate_gsl(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gsl_sf_lngamma(data[i][0]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(lgamma_test, "lgamma-gsl") +{ + double result = lgamma_evaluate_gsl(factorials); + result += lgamma_evaluate_gsl(near_1); + result += lgamma_evaluate_gsl(near_2); + result += lgamma_evaluate_gsl(near_0); + result += lgamma_evaluate_gsl(near_m10); + result += lgamma_evaluate_gsl(near_m55); + + consume_result(result); + set_call_count( + (sizeof(factorials) + + sizeof(near_1) + + sizeof(near_2) + + sizeof(near_0) + + sizeof(near_m10) + + sizeof(near_m55)) / sizeof(factorials[0])); +} + +#endif + diff --git a/performance/test_ibeta.cpp b/performance/test_ibeta.cpp new file mode 100644 index 000000000..a9a33e1de --- /dev/null +++ b/performance/test_ibeta.cpp @@ -0,0 +1,191 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +#include +#include + +#define T double +#include "../test/ibeta_data.ipp" +#include "../test/ibeta_int_data.ipp" +#include "../test/ibeta_large_data.ipp" +#include "../test/ibeta_small_data.ipp" + +template +double ibeta_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::ibeta(data[i][0], data[i][1], data[i][2]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_test, "ibeta") +{ + double result = ibeta_evaluate2(ibeta_data); + result += ibeta_evaluate2(ibeta_int_data); + result += ibeta_evaluate2(ibeta_large_data); + result += ibeta_evaluate2(ibeta_small_data); + + consume_result(result); + set_call_count( + (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +template +double ibeta_inv_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_inv_test, "ibeta_inv") +{ + double result = ibeta_inv_evaluate2(ibeta_data); + result += ibeta_inv_evaluate2(ibeta_int_data); + result += ibeta_inv_evaluate2(ibeta_large_data); + result += ibeta_inv_evaluate2(ibeta_small_data); + + consume_result(result); + set_call_count( + (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +template +double ibeta_invab_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + { + //std::cout << "ibeta_inva(" << data[i][1] << "," << data[i][2] << "," << data[i][5] << ");" << std::endl; + result += boost::math::ibeta_inva(data[i][1], data[i][2], data[i][5]); + //std::cout << "ibeta_invb(" << data[i][0] << "," << data[i][2] << "," << data[i][5] << ");" << std::endl; + result += boost::math::ibeta_invb(data[i][0], data[i][2], data[i][5]); + //std::cout << "ibetac_inva(" << data[i][1] << "," << data[i][2] << "," << data[i][6] << ");" << std::endl; + result += boost::math::ibetac_inva(data[i][1], data[i][2], data[i][6]); + //std::cout << "ibetac_invb(" << data[i][0] << "," << data[i][2] << "," << data[i][6] << ");" << std::endl; + result += boost::math::ibetac_invb(data[i][0], data[i][2], data[i][6]); + } + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_test, "ibeta_invab") +{ + double result = ibeta_invab_evaluate2(ibeta_data); + result += ibeta_invab_evaluate2(ibeta_int_data); + result += ibeta_invab_evaluate2(ibeta_large_data); + result += ibeta_invab_evaluate2(ibeta_small_data); + + consume_result(result); + set_call_count( + 4 * (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +#ifdef TEST_CEPHES + +extern "C" { + +double incbet(double a, double b, double x); +double incbi(double a, double b, double y); + +} + +template +double ibeta_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += incbet(data[i][0], data[i][1], data[i][2]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_test, "ibeta-cephes") +{ + double result = ibeta_evaluate_cephes(ibeta_data); + result += ibeta_evaluate_cephes(ibeta_int_data); + result += ibeta_evaluate_cephes(ibeta_large_data); + result += ibeta_evaluate_cephes(ibeta_small_data); + + consume_result(result); + set_call_count( + (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +template +double ibeta_inv_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += incbi(data[i][0], data[i][1], data[i][5]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_inv_test, "ibeta_inv-cephes") +{ + double result = ibeta_inv_evaluate_cephes(ibeta_data); + result += ibeta_inv_evaluate_cephes(ibeta_int_data); + result += ibeta_inv_evaluate_cephes(ibeta_large_data); + result += ibeta_inv_evaluate_cephes(ibeta_small_data); + + consume_result(result); + set_call_count( + (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +#endif + +#ifdef TEST_GSL +// +// This test segfaults inside GSL.... +// + +#include + +template +double ibeta_evaluate_gsl(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gsl_sf_beta_inc(data[i][0], data[i][1], data[i][2]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(ibeta_test, "ibeta-gsl") +{ + double result = ibeta_evaluate_gsl(ibeta_data); + result += ibeta_evaluate_gsl(ibeta_int_data); + result += ibeta_evaluate_gsl(ibeta_large_data); + result += ibeta_evaluate_gsl(ibeta_small_data); + + consume_result(result); + set_call_count( + (sizeof(ibeta_data) + + sizeof(ibeta_int_data) + + sizeof(ibeta_large_data) + + sizeof(ibeta_small_data)) / sizeof(ibeta_data[0])); +} + +#endif + diff --git a/performance/test_igamma.cpp b/performance/test_igamma.cpp new file mode 100644 index 000000000..b81de2bfa --- /dev/null +++ b/performance/test_igamma.cpp @@ -0,0 +1,190 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +#include +#include + +#define T double +#include "../test/igamma_big_data.ipp" +#include "../test/igamma_int_data.ipp" +#include "../test/igamma_med_data.ipp" +#include "../test/igamma_small_data.ipp" + +template +double igamma_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + { + result += boost::math::gamma_p(data[i][0], data[i][1]); + result += boost::math::gamma_q(data[i][0], data[i][1]); + } + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(igamma_test, "igamma") +{ + double result = igamma_evaluate2(igamma_big_data); + result += igamma_evaluate2(igamma_int_data); + result += igamma_evaluate2(igamma_med_data); + result += igamma_evaluate2(igamma_small_data); + + consume_result(result); + set_call_count( + 2 * (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} + +template +double igamma_inv_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + { + result += boost::math::gamma_p_inv(data[i][0], data[i][5]); + result += boost::math::gamma_q_inv(data[i][0], data[i][3]); + } + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(igamma_inv_test, "igamma_inv") +{ + double result = igamma_inv_evaluate2(igamma_big_data); + result += igamma_inv_evaluate2(igamma_int_data); + result += igamma_inv_evaluate2(igamma_med_data); + result += igamma_inv_evaluate2(igamma_small_data); + + consume_result(result); + set_call_count( + 2 * (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} + +template +double igamma_inva_evaluate2(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + { + result += boost::math::gamma_p_inva(data[i][1], data[i][5]); + result += boost::math::gamma_q_inva(data[i][1], data[i][3]); + } + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(igamma_inva_test, "igamma_inva") +{ + double result = igamma_inva_evaluate2(igamma_big_data); + result += igamma_inva_evaluate2(igamma_int_data); + result += igamma_inva_evaluate2(igamma_med_data); + result += igamma_inva_evaluate2(igamma_small_data); + + consume_result(result); + set_call_count( + 2 * (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} + +#ifdef TEST_CEPHES + +extern "C" { + +double igam(double, double); +double igami(double, double); + +} + +template +double igamma_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += igam(data[i][0], data[i][1]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(igamma_test, "igamma-cephes") +{ + double result = igamma_evaluate_cephes(igamma_big_data); + result += igamma_evaluate_cephes(igamma_int_data); + result += igamma_evaluate_cephes(igamma_med_data); + result += igamma_evaluate_cephes(igamma_small_data); + + consume_result(result); + set_call_count( + (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} + +template +double igamma_inv_evaluate_cephes(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += igami(data[i][0], data[i][3]); // note needs complement of probability!! + return result; +} + +// +// This test does not run to completion, gets stuck +// in infinite loop inside cephes.... +// +BOOST_MATH_PERFORMANCE_TEST(igamma_inv_test, "igamma_inv-cephes") +{ + double result = igamma_inv_evaluate_cephes(igamma_big_data); + result += igamma_inv_evaluate_cephes(igamma_int_data); + result += igamma_inv_evaluate_cephes(igamma_med_data); + result += igamma_inv_evaluate_cephes(igamma_small_data); + + consume_result(result); + set_call_count( + (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} +#endif + +#ifdef TEST_GSL + +#include + +template +double igamma_evaluate_gsl(const boost::array, N>& data) +{ + double result = 0; + for(unsigned i = 0; i < N; ++i) + result += gsl_sf_gamma_inc_P(data[i][0], data[i][1]); + return result; +} + +BOOST_MATH_PERFORMANCE_TEST(igamma_test, "igamma-gsl") +{ + double result = igamma_evaluate_gsl(igamma_big_data); + result += igamma_evaluate_gsl(igamma_int_data); + result += igamma_evaluate_gsl(igamma_med_data); + result += igamma_evaluate_gsl(igamma_small_data); + + consume_result(result); + set_call_count( + (sizeof(igamma_big_data) + + sizeof(igamma_int_data) + + sizeof(igamma_med_data) + + sizeof(igamma_small_data)) / sizeof(igamma_big_data[0])); +} + +#endif diff --git a/performance/test_polynomial.cpp b/performance/test_polynomial.cpp new file mode 100644 index 000000000..3104b7a8c --- /dev/null +++ b/performance/test_polynomial.cpp @@ -0,0 +1,422 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +#include +#include + +static const double num[13] = { + static_cast(56906521.91347156388090791033559122686859L), + static_cast(103794043.1163445451906271053616070238554L), + static_cast(86363131.28813859145546927288977868422342L), + static_cast(43338889.32467613834773723740590533316085L), + static_cast(14605578.08768506808414169982791359218571L), + static_cast(3481712.15498064590882071018964774556468L), + static_cast(601859.6171681098786670226533699352302507L), + static_cast(75999.29304014542649875303443598909137092L), + static_cast(6955.999602515376140356310115515198987526L), + static_cast(449.9445569063168119446858607650988409623L), + static_cast(19.51992788247617482847860966235652136208L), + static_cast(0.5098416655656676188125178644804694509993L), + static_cast(0.006061842346248906525783753964555936883222L) +}; +static const double denom[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u) +}; +static const boost::uint32_t denom_int[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u) +}; + +template +U evaluate_polynomial_0(const T* poly, U const& z, std::size_t count) +{ + U sum = static_cast(poly[count - 1]); + for(int i = static_cast(count) - 2; i >= 0; --i) + { + sum *= z; + sum += static_cast(poly[i]); + } + return sum; +} + +template +inline V evaluate_polynomial_1(const T* a, const V& x) +{ + return static_cast((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]); +} + +template +inline V evaluate_polynomial_2(const T* a, const V& x) +{ + V x2 = x * x; + return static_cast((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x); +} + +template +inline V evaluate_polynomial_3(const T* a, const V& x) +{ + V x2 = x * x; + V t[2]; + t[0] = static_cast(a[12] * x2 + a[10]); + t[1] = static_cast(a[11] * x2 + a[9]); + t[0] *= x2; + t[1] *= x2; + t[0] += a[8]; + t[1] += a[7]; + t[0] *= x2; + t[1] *= x2; + t[0] += a[6]; + t[1] += a[5]; + t[0] *= x2; + t[1] *= x2; + t[0] += a[4]; + t[1] += a[3]; + t[0] *= x2; + t[1] *= x2; + t[0] += a[2]; + t[1] += a[1]; + t[0] *= x2; + t[0] += a[0]; + t[1] *= x; + return t[0] + t[1]; +} + +template +V evaluate_rational_0(const T* num, const U* denom, const V& z_, std::size_t count) +{ + V z(z_); + V s1, s2; + if(z <= 1) + { + s1 = static_cast(num[count-1]); + s2 = static_cast(denom[count-1]); + for(int i = (int)count - 2; i >= 0; --i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += denom[i]; + } + } + else + { + z = 1 / z; + s1 = static_cast(num[0]); + s2 = static_cast(denom[0]); + for(unsigned i = 1; i < count; ++i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += denom[i]; + } + } + return s1 / s2; +} + +template +inline V evaluate_rational_1(const T* a, const U* b, const V& x) +{ + if(x <= 1) + return static_cast(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0])); + else + { + V z = 1 / x; + return static_cast(((((((((((((a[0] * z + a[1]) * z + a[2]) * z + a[3]) * z + a[4]) * z + a[5]) * z + a[6]) * z + a[7]) * z + a[8]) * z + a[9]) * z + a[10]) * z + a[11]) * z + a[12]) / ((((((((((((b[0] * z + b[1]) * z + b[2]) * z + b[3]) * z + b[4]) * z + b[5]) * z + b[6]) * z + b[7]) * z + b[8]) * z + b[9]) * z + b[10]) * z + b[11]) * z + b[12])); + } +} + +template +inline V evaluate_rational_2(const T* a, const U* b, const V& x) +{ + if(x <= 1) + { + V x2 = x * x; + return static_cast(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x)); + } + else + { + V z = 1 / x; + V z2 = 1 / (x * x); + return static_cast(((((((a[0] * z2 + a[2]) * z2 + a[4]) * z2 + a[6]) * z2 + a[8]) * z2 + a[10]) * z2 + a[12] + (((((a[1] * z2 + a[3]) * z2 + a[5]) * z2 + a[7]) * z2 + a[9]) * z2 + a[11]) * z) / ((((((b[0] * z2 + b[2]) * z2 + b[4]) * z2 + b[6]) * z2 + b[8]) * z2 + b[10]) * z2 + b[12] + (((((b[1] * z2 + b[3]) * z2 + b[5]) * z2 + b[7]) * z2 + b[9]) * z2 + b[11]) * z)); + } +} + +template +inline V evaluate_rational_3(const T* a, const U* b, const V& x) +{ + if(x <= 1) + { + V x2 = x * x; + V t[4]; + t[0] = a[12] * x2 + a[10]; + t[1] = a[11] * x2 + a[9]; + t[2] = b[12] * x2 + b[10]; + t[3] = b[11] * x2 + b[9]; + t[0] *= x2; + t[1] *= x2; + t[2] *= x2; + t[3] *= x2; + t[0] += a[8]; + t[1] += a[7]; + t[2] += b[8]; + t[3] += b[7]; + t[0] *= x2; + t[1] *= x2; + t[2] *= x2; + t[3] *= x2; + t[0] += a[6]; + t[1] += a[5]; + t[2] += b[6]; + t[3] += b[5]; + t[0] *= x2; + t[1] *= x2; + t[2] *= x2; + t[3] *= x2; + t[0] += a[4]; + t[1] += a[3]; + t[2] += b[4]; + t[3] += b[3]; + t[0] *= x2; + t[1] *= x2; + t[2] *= x2; + t[3] *= x2; + t[0] += a[2]; + t[1] += a[1]; + t[2] += b[2]; + t[3] += b[1]; + t[0] *= x2; + t[2] *= x2; + t[0] += a[0]; + t[2] += b[0]; + t[1] *= x; + t[3] *= x; + return (t[0] + t[1]) / (t[2] + t[3]); + } + else + { + V z = 1 / x; + V z2 = 1 / (x * x); + V t[4]; + t[0] = a[0] * z2 + a[2]; + t[1] = a[1] * z2 + a[3]; + t[2] = b[0] * z2 + b[2]; + t[3] = b[1] * z2 + b[3]; + t[0] *= z2; + t[1] *= z2; + t[2] *= z2; + t[3] *= z2; + t[0] += a[4]; + t[1] += a[5]; + t[2] += b[4]; + t[3] += b[5]; + t[0] *= z2; + t[1] *= z2; + t[2] *= z2; + t[3] *= z2; + t[0] += a[6]; + t[1] += a[7]; + t[2] += b[6]; + t[3] += b[7]; + t[0] *= z2; + t[1] *= z2; + t[2] *= z2; + t[3] *= z2; + t[0] += a[8]; + t[1] += a[9]; + t[2] += b[8]; + t[3] += b[9]; + t[0] *= z2; + t[1] *= z2; + t[2] *= z2; + t[3] *= z2; + t[0] += a[10]; + t[1] += a[11]; + t[2] += b[10]; + t[3] += b[11]; + t[0] *= z2; + t[2] *= z2; + t[0] += a[12]; + t[2] += b[12]; + t[1] *= z; + t[3] *= z; + return (t[0] + t[1]) / (t[2] + t[3]); + } +} + + + +BOOST_MATH_PERFORMANCE_TEST(polynomial_evaluate_0, "Polynomial-method-0") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_0(num, i, 13) / evaluate_polynomial_0(denom, i, 13); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_evaluate_1, "Polynomial-method-1") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_1(num, i) / evaluate_polynomial_1(denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_evaluate_2, "Polynomial-method-2") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_2(num, i) / evaluate_polynomial_2(denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_evaluate_3, "Polynomial-method-3") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_3(num, i) / evaluate_polynomial_3(denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_m_evaluate_0, "Polynomial-mixed-method-0") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_0(num, i, 13) / evaluate_polynomial_0(denom_int, i, 13); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_m_evaluate_1, "Polynomial-mixed-method-1") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_1(num, i) / evaluate_polynomial_1(denom_int, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_m_evaluate_2, "Polynomial-mixed-method-2") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_2(num, i) / evaluate_polynomial_2(denom_int, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(polynomial_m_evaluate_3, "Polynomial-mixed-method-3") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_polynomial_3(num, i) / evaluate_polynomial_3(denom_int, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_evaluate_0, "Rational-method-0") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_0(num, denom, i, 13); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_evaluate_1, "Rational-method-1") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_1(num, denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_evaluate_2, "Rational-method-2") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_2(num, denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_evaluate_3, "Rational-method-3") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_3(num, denom, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_m_evaluate_0, "Rational-mixed-method-0") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_0(num, denom_int, i, 13); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_m_evaluate_1, "Rational-mixed-method-1") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_1(num, denom_int, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_m_evaluate_2, "Rational-mixed-method-2") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_2(num, denom_int, i); + consume_result(result); + set_call_count(1000); +} + +BOOST_MATH_PERFORMANCE_TEST(rational_m_evaluate_3, "Rational-mixed-method-3") +{ + double result = 0; + for(double i = 1; i < 1000; i += 0.5) + result += evaluate_rational_3(num, denom_int, i); + consume_result(result); + set_call_count(1000); +} + + diff --git a/performance/test_reference.cpp b/performance/test_reference.cpp new file mode 100644 index 000000000..6e6a784e1 --- /dev/null +++ b/performance/test_reference.cpp @@ -0,0 +1,16 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "required_defines.hpp" + +#include "performance_measure.hpp" + +void reference_evaluate() +{ + consume_result(2.0); + set_call_count(1); +} + + diff --git a/quaternion/quaternion_test.cpp b/quaternion/quaternion_test.cpp index 362b05196..6da75db40 100644 --- a/quaternion/quaternion_test.cpp +++ b/quaternion/quaternion_test.cpp @@ -35,12 +35,16 @@ DEFINE_TYPE_NAME(double); DEFINE_TYPE_NAME(long double); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS typedef boost::mpl::list test_types; +#else +typedef boost::mpl::list test_types; +#endif // Apple GCC 4.0 uses the "double double" format for its long double, // which means that epsilon is VERY small but useless for // comparisons. So, don't do those comparisons. -#if defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4 +#if (defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4) || defined(BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS) typedef boost::mpl::list near_eps_test_types; #else typedef boost::mpl::list near_eps_test_types; diff --git a/test/Jamfile.v2 b/test/Jamfile.v2 index 541410dab..95fb0a168 100644 --- a/test/Jamfile.v2 +++ b/test/Jamfile.v2 @@ -1,12 +1,384 @@ -# Copyright Daryle Walker, Hubert Holin and John Maddock 2006 +# Copyright Daryle Walker, Hubert Holin, John Maddock and Paul A. Bristow 2006 - 2007 # Distributed under the Boost Software License, Version 1.0. # (See accompanying file LICENSE_1_0.txt or copy at # http://www.boost.org/LICENSE_1_0.txt. - +# \math_toolkit\libs\math\test\jamfile.v2 +# Runs all math toolkit tests, functions & distributions. # bring in the rules for testing import testing ; +project + : requirements + gcc:-Wno-missing-braces + darwin:-Wno-missing-braces + acc:+W2068,2461,2236,4070,4069 + intel:-Qwd264,239 + intel:/nologo + intel-linux:-wd239 + intel:/nologo + msvc:all + msvc:on + msvc:/wd4996 + msvc:/wd4512 + msvc:/wd4610 + msvc:/wd4510 + msvc:/wd4127 + msvc:/wd4701 # needed for lexical cast - temporary. + msvc-7.1:../vc71_fix//vc_fix + borland:static + borland:static + # msvc:/wd4506 has no effect? + # suppress xstring(237) : warning C4506: no definition for inline function + ../../.. + ../../regex/build//boost_regex + shared:BOOST_REGEX_DYN_LINK=1 + # Sunpro has problems building regex as a shared lib: + sun:static + BOOST_ALL_NO_LIB=1 + ; + +run hypot_test.cpp ; +run log1p_expm1_test.cpp ; +run powm1_sqrtp1m1_test.cpp ; +run special_functions_test.cpp ../../test/build//boost_unit_test_framework/static ; +run test_bernoulli.cpp ; +run test_constants.cpp ; +run test_beta.cpp ; +run test_beta_dist.cpp ; +run test_binomial.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_binomial_float ; +run test_binomial.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_binomial_double ; +run test_binomial.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_binomial_long_double ; +run test_binomial.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_binomial_real_concept ; +run test_binomial_coeff.cpp ; +run test_carlson.cpp ; +run test_cauchy.cpp ; +run test_cbrt.cpp ; +run test_chi_squared.cpp ; +run test_classify.cpp ; +run test_digamma.cpp ; +run test_dist_overloads.cpp ; +run test_ellint_1.cpp ; +run test_ellint_2.cpp ; +run test_ellint_3.cpp ; +run test_erf.cpp ; +run test_error_handling.cpp ; +run test_exponential_dist.cpp ; +run test_extreme_value.cpp ; +run test_factorials.cpp ; +run test_find_location.cpp ; +run test_find_scale.cpp ; +run test_fisher_f.cpp ; +run test_gamma.cpp ; +run test_gamma_dist.cpp ; +run test_hermite.cpp ; +run test_ibeta.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_ibeta_float ; +run test_ibeta.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_ibeta_double ; +run test_ibeta.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_ibeta_long_double ; +run test_ibeta.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_ibeta_real_concept ; +run test_ibeta_inv.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_ibeta_inv_float ; +run test_ibeta_inv.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_ibeta_inv_double ; +run test_ibeta_inv.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_ibeta_inv_long_double ; +run test_ibeta_inv.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_ibeta_inv_real_concept ; +run test_ibeta_inv_ab.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_ibeta_inv_ab_float ; +run test_ibeta_inv_ab.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_ibeta_inv_ab_double ; +run test_ibeta_inv_ab.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_ibeta_inv_ab_long_double ; +run test_ibeta_inv_ab.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_ibeta_inv_ab_real_concept ; +run test_igamma.cpp ; +run test_igamma_inv.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_igamma_inv_float ; +run test_igamma_inv.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_igamma_inv_double ; +run test_igamma_inv.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_igamma_inv_long_double ; +run test_igamma_inv.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_igamma_inv_real_concept ; +run test_igamma_inva.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_igamma_inva_float ; +run test_igamma_inva.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_igamma_inva_double ; +run test_igamma_inva.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_igamma_inva_long_double ; +run test_igamma_inva.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_igamma_inva_real_concept ; +run test_instantiate1.cpp test_instantiate2.cpp ; +run test_laguerre.cpp ; +run test_legendre.cpp ; +run test_lognormal.cpp ; +run test_minima.cpp ; +run test_negative_binomial.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_negative_binomial_float ; +run test_negative_binomial.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_negative_binomial_double ; +run test_negative_binomial.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_negative_binomial_long_double ; +run test_negative_binomial.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_negative_binomial_real_concept ; +run test_normal.cpp ; +run test_pareto.cpp ; +run test_poisson.cpp + : # command line + : # input files + : # requirements + TEST_FLOAT + : test_poisson_float ; +run test_poisson.cpp + : # command line + : # input files + : # requirements + TEST_DOUBLE + : test_poisson_double ; +run test_poisson.cpp + : # command line + : # input files + : # requirements + TEST_LDOUBLE + : test_poisson_long_double ; +run test_poisson.cpp + : # command line + : # input files + : # requirements + TEST_REAL_CONCEPT + : test_poisson_real_concept ; +run test_rayleigh.cpp ; + +run test_rationals.cpp +test_rational_instances/test_rational_double1.cpp +test_rational_instances/test_rational_double2.cpp +test_rational_instances/test_rational_double3.cpp +test_rational_instances/test_rational_double4.cpp +test_rational_instances/test_rational_double5.cpp +test_rational_instances/test_rational_float1.cpp +test_rational_instances/test_rational_float2.cpp +test_rational_instances/test_rational_float3.cpp +test_rational_instances/test_rational_float4.cpp +test_rational_instances/test_rational_ldouble1.cpp +test_rational_instances/test_rational_ldouble2.cpp +test_rational_instances/test_rational_ldouble3.cpp +test_rational_instances/test_rational_ldouble4.cpp +test_rational_instances/test_rational_ldouble5.cpp +test_rational_instances/test_rational_real_concept1.cpp +test_rational_instances/test_rational_real_concept2.cpp +test_rational_instances/test_rational_real_concept3.cpp +test_rational_instances/test_rational_real_concept4.cpp +test_rational_instances/test_rational_real_concept5.cpp +; + +run test_remez.cpp ; +run test_roots.cpp ; +run test_spherical_harmonic.cpp ; +run test_students_t.cpp ; +run test_tgamma_ratio.cpp ; +run test_toms748_solve.cpp ; +run test_triangular.cpp ; +run test_uniform.cpp ; +run test_weibull.cpp ; +run test_policy.cpp ; +run test_policy_2.cpp ; +run test_policy_sf.cpp ; + +run test_bessel_j.cpp ; +run test_bessel_y.cpp ; +run test_bessel_i.cpp ; +run test_bessel_k.cpp ; + +compile compile_test/dist_bernoulli_incl_test.cpp ; +compile compile_test/dist_beta_incl_test.cpp ; +compile compile_test/dist_binomial_incl_test.cpp ; +compile compile_test/dist_cauchy_incl_test.cpp ; +compile compile_test/dist_chi_squared_incl_test.cpp ; +compile compile_test/dist_complement_incl_test.cpp ; +compile compile_test/dist_exponential_incl_test.cpp ; +compile compile_test/dist_extreme_val_incl_test.cpp ; +compile compile_test/dist_fisher_f_incl_test.cpp ; +compile compile_test/dist_gamma_incl_test.cpp ; +compile compile_test/dist_lognormal_incl_test.cpp ; +compile compile_test/dist_neg_binom_incl_test.cpp ; +compile compile_test/dist_normal_incl_test.cpp ; +compile compile_test/dist_poisson_incl_test.cpp ; +compile compile_test/dist_students_t_incl_test.cpp ; +compile compile_test/dist_triangular_incl_test.cpp ; +compile compile_test/dist_uniform_incl_test.cpp ; +compile compile_test/dist_weibull_incl_test.cpp ; +compile compile_test/distribution_concept_check.cpp ; +compile compile_test/sf_beta_incl_test.cpp ; +compile compile_test/sf_bessel_incl_test.cpp ; +compile compile_test/sf_binomial_incl_test.cpp ; +compile compile_test/sf_cbrt_incl_test.cpp ; +compile compile_test/sf_cos_pi_incl_test.cpp ; +compile compile_test/sf_digamma_incl_test.cpp ; +compile compile_test/sf_ellint_1_incl_test.cpp ; +compile compile_test/sf_ellint_2_incl_test.cpp ; +compile compile_test/sf_ellint_3_incl_test.cpp ; +compile compile_test/sf_ellint_rc_incl_test.cpp ; +compile compile_test/sf_ellint_rd_incl_test.cpp ; +compile compile_test/sf_ellint_rf_incl_test.cpp ; +compile compile_test/sf_ellint_rj_incl_test.cpp ; +compile compile_test/sf_erf_incl_test.cpp ; +compile compile_test/sf_expm1_incl_test.cpp ; +compile compile_test/sf_factorials_incl_test.cpp ; +compile compile_test/sf_fpclassify_incl_test.cpp ; +compile compile_test/sf_gamma_incl_test.cpp ; +compile compile_test/sf_hermite_incl_test.cpp ; +compile compile_test/sf_hypot_incl_test.cpp ; +compile compile_test/sf_laguerre_incl_test.cpp ; +compile compile_test/sf_lanczos_incl_test.cpp ; +compile compile_test/sf_legendre_incl_test.cpp ; +compile compile_test/sf_log1p_incl_test.cpp ; +compile compile_test/sf_math_fwd_incl_test.cpp ; +compile compile_test/sf_powm1_incl_test.cpp ; +compile compile_test/sf_sign_incl_test.cpp ; +compile compile_test/sf_sin_pi_incl_test.cpp ; +compile compile_test/sf_sinc_incl_test.cpp ; +compile compile_test/sf_sinhc_incl_test.cpp ; +compile compile_test/sf_sph_harm_incl_test.cpp ; +compile compile_test/sf_sqrt1pm1_incl_test.cpp ; +compile compile_test/std_real_concept_check.cpp ; +compile compile_test/test_traits.cpp ; +compile compile_test/tools_config_inc_test.cpp ; +compile compile_test/tools_fraction_inc_test.cpp ; +compile compile_test/tools_minima_inc_test.cpp ; +compile compile_test/tools_polynomial_inc_test.cpp ; +compile compile_test/tools_precision_inc_test.cpp ; +compile compile_test/tools_rational_inc_test.cpp ; +compile compile_test/tools_real_cast_inc_test.cpp ; +compile compile_test/tools_remez_inc_test.cpp ; +compile compile_test/tools_roots_inc_test.cpp ; +compile compile_test/tools_series_inc_test.cpp ; +compile compile_test/tools_solve_inc_test.cpp ; +compile compile_test/tools_stats_inc_test.cpp ; +compile compile_test/tools_test_data_inc_test.cpp ; +compile compile_test/tools_test_inc_test.cpp ; +compile compile_test/tools_toms748_inc_test.cpp ; + run ../test/common_factor_test.cpp ../../test/build//boost_unit_test_framework/static ; @@ -16,17 +388,16 @@ run ../octonion/octonion_test.cpp run ../quaternion/quaternion_test.cpp ../../test/build//boost_unit_test_framework/static ; -run ../special_functions/special_functions_test.cpp - ../../test/build//boost_unit_test_framework/static ; - run ../quaternion/quaternion_mult_incl_test.cpp ../quaternion/quaternion_mi1.cpp ../quaternion/quaternion_mi2.cpp ../../test/build//boost_unit_test_framework/static ; run complex_test.cpp ; -run hypot_test.cpp ; -run log1p_expm1_test.cpp ; + + + + diff --git a/test/acosh_test.hpp b/test/acosh_test.hpp new file mode 100644 index 000000000..fa3f8c6ae --- /dev/null +++ b/test/acosh_test.hpp @@ -0,0 +1,90 @@ +// unit test file acosh.hpp for the special functions test suite + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include +#include +#include + + +#include + + +#include + + +template +T acosh_error_evaluator(T x) +{ + using ::std::abs; + using ::std::sinh; + using ::std::cosh; + + using ::std::numeric_limits; + + using ::boost::math::acosh; + + + static T const epsilon = numeric_limits::epsilon(); + + T y = cosh(x); + T z = acosh(y); + + T absolute_error = abs(z-abs(x)); + T relative_error = absolute_error*abs(sinh(x)); + T scaled_error = relative_error/epsilon; + + return(scaled_error); +} + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(acosh_test, T) +{ + BOOST_MESSAGE("Testing acosh in the real domain for " + << string_type_name::_() << "."); + + for (int i = 0; i <= 100; i++) + { + T x = static_cast(i-50)/static_cast(5); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (acosh_error_evaluator(x)) + (static_cast(4))); + } +} + + +void acosh_manual_check() +{ + BOOST_MESSAGE(" "); + BOOST_MESSAGE("acosh"); + + for (int i = 0; i <= 100; i++) + { + float xf = static_cast(i-50)/static_cast(5); + double xd = static_cast(i-50)/static_cast(5); + long double xl = + static_cast(i-50)/static_cast(5); + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << acosh_error_evaluator(xf) + << ::std::setw(15) + << acosh_error_evaluator(xd) + << ::std::setw(15) + << acosh_error_evaluator(xl)); +#else + BOOST_MESSAGE( ::std::setw(15) + << acosh_error_evaluator(xf) + << ::std::setw(15) + << acosh_error_evaluator(xd)); +#endif + } + + BOOST_MESSAGE(" "); +} + diff --git a/test/asinh_test.hpp b/test/asinh_test.hpp new file mode 100644 index 000000000..0cb4b2315 --- /dev/null +++ b/test/asinh_test.hpp @@ -0,0 +1,90 @@ +// unit test file asinh.hpp for the special functions test suite + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include +#include +#include + + +#include + + +#include + + +template +T asinh_error_evaluator(T x) +{ + using ::std::abs; + using ::std::sinh; + using ::std::cosh; + + using ::std::numeric_limits; + + using ::boost::math::asinh; + + + static T const epsilon = numeric_limits::epsilon(); + + T y = sinh(x); + T z = asinh(y); + + T absolute_error = abs(z-x); + T relative_error = absolute_error*cosh(x); + T scaled_error = relative_error/epsilon; + + return(scaled_error); +} + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(asinh_test, T) +{ + BOOST_MESSAGE("Testing asinh in the real domain for " + << string_type_name::_() << "."); + + for (int i = 0; i <= 80; i++) + { + T x = static_cast(i-40)/static_cast(4); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (asinh_error_evaluator(x)) + (static_cast(4))); + } +} + + +void asinh_manual_check() +{ + BOOST_MESSAGE(" "); + BOOST_MESSAGE("asinh"); + + for (int i = 0; i <= 80; i++) + { + float xf = static_cast(i-40)/static_cast(4); + double xd = static_cast(i-40)/static_cast(4); + long double xl = + static_cast(i-40)/static_cast(4); + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << asinh_error_evaluator(xf) + << ::std::setw(15) + << asinh_error_evaluator(xd) + << ::std::setw(15) + << asinh_error_evaluator(xl)); +#else + BOOST_MESSAGE( ::std::setw(15) + << asinh_error_evaluator(xf) + << ::std::setw(15) + << asinh_error_evaluator(xd)); +#endif + } + + BOOST_MESSAGE(" "); +} + diff --git a/test/assoc_legendre_p.ipp b/test/assoc_legendre_p.ipp new file mode 100644 index 000000000..456ba6d49 --- /dev/null +++ b/test/assoc_legendre_p.ipp @@ -0,0 +1,410 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 400> assoc_legendre_p = { + SC_(3.755727291107177734375), SC_(-3), SC_(0.264718532562255859375), SC_(0.018682285998021253444483874168352748715136623066073), + SC_(3.755727291107177734375), SC_(-3), SC_(0.67001712322235107421875), SC_(0.0085227010576716344744906184676725516467064196759033), + SC_(3.755727291107177734375), SC_(-3), SC_(0.91501367092132568359375), SC_(0.0013678553914178941694232605088191864494205442141689), + SC_(3.755727291107177734375), SC_(-3), SC_(0.93538987636566162109375), SC_(0.00092121779437596438014780141399846622556782069609554), + SC_(3.755727291107177734375), SC_(-2), SC_(-0.804919183254241943359375), SC_(-0.035427019537072229468317544697039851719910785732282), + SC_(3.755727291107177734375), SC_(-2), SC_(-0.62323606014251708984375), SC_(-0.047644590452413896385001513687947050090798484234256), + SC_(3.755727291107177734375), SC_(-2), SC_(0.62944734096527099609375), SC_(0.047507226872528274797147731134568104938153965122183), + SC_(3.755727291107177734375), SC_(-2), SC_(0.92977702617645263671875), SC_(0.015749804707019642230489140711548190498803023729124), + SC_(3.755727291107177734375), SC_(-2), SC_(0.98576259613037109375), SC_(0.0034836978383022116191949923980075709550874307751656), + SC_(3.755727291107177734375), SC_(-1), SC_(0.09376299381256103515625), SC_(-0.11897883801786023759113449248494493928619190467444), + SC_(3.755727291107177734375), SC_(0), SC_(-0.72904598712921142578125), SC_(0.12483445078630286253952851342405305778981983166886), + SC_(3.755727291107177734375), SC_(0), SC_(0.0944411754608154296875), SC_(-0.13955592906053357908142302859499928047171124489978), + SC_(3.755727291107177734375), SC_(0), SC_(0.826751708984375), SC_(0.1726224256643575927228084765374660491943359375), + SC_(3.755727291107177734375), SC_(0), SC_(0.99292266368865966796875), SC_(0.95791076106518834501582130387317692843396343960194), + SC_(3.755727291107177734375), SC_(1), SC_(-0.5579319000244140625), SC_(-0.69267329476836900305232758748436796165065650726706), + SC_(3.755727291107177734375), SC_(1), SC_(-0.38366591930389404296875), SC_(0.36569809882896106974883774476288881583212189972924), + SC_(3.755727291107177734375), SC_(2), SC_(-0.74602639675140380859375), SC_(-4.9623208282269009166469230084559050020232007227605), + SC_(3.755727291107177734375), SC_(2), SC_(-0.44300353527069091796875), SC_(-5.340947203111010720410977109344485835862315070699), + SC_(3.755727291107177734375), SC_(2), SC_(0.81158387660980224609375), SC_(4.1552884837362426955817673026802316904593226354336), + SC_(3.755727291107177734375), SC_(2), SC_(0.93773555755615234375), SC_(1.6970953962729668417983019956807311245938763022423), + SC_(4.285762786865234375), SC_(-4), SC_(0.0944411754608154296875), SC_(0.0025579199993109637443137263005669146428545187165902), + SC_(4.285762786865234375), 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SC_(19.9363040924072265625), SC_(-19), SC_(0.264718532562255859375), SC_(0.78629720841879126459468443827603688509992408472136e-23), + SC_(19.9363040924072265625), SC_(-17), SC_(-0.44300353527069091796875), SC_(0.58251104165678394898744857711146365095314417691743e-21), + SC_(19.9363040924072265625), SC_(-16), SC_(-0.62323606014251708984375), SC_(-0.29751304424052425636113962115386812591456175870573e-20), + SC_(19.9363040924072265625), SC_(-14), SC_(0.91501367092132568359375), SC_(0.12709748787166211089363084338252308414817659872819e-20), + SC_(19.9363040924072265625), SC_(-14), SC_(0.92977702617645263671875), SC_(0.3871138758064428596131493508699859591733838577277e-21), + SC_(19.9363040924072265625), SC_(-12), SC_(0.93773555755615234375), SC_(0.89408958662499196674261505271428146255253342324884e-18), + SC_(19.9363040924072265625), SC_(-11), SC_(-0.38366591930389404296875), SC_(0.26026063256932547239295328669255586707226242612148e-14), + SC_(19.9363040924072265625), SC_(-10), SC_(-0.74602639675140380859375), SC_(-0.34765506905243672613499520824866410956137457719745e-13), + SC_(19.9363040924072265625), SC_(-10), SC_(0.81158387660980224609375), SC_(0.63172556307492864748930303721096827403984649074523e-13), + SC_(19.9363040924072265625), SC_(-4), SC_(0.0944411754608154296875), SC_(-0.12660004211205826580535251463738122438007427377005e-5), + SC_(19.9363040924072265625), SC_(-4), SC_(0.62944734096527099609375), SC_(-0.54962149013781842129875877015678395599272910286725e-6), + SC_(19.9363040924072265625), SC_(2), SC_(0.09376299381256103515625), SC_(66.642893417106666543810431512449143935500286088633), + SC_(19.9363040924072265625), SC_(3), SC_(0.99292266368865966796875), SC_(-1337.0443049910513329145515282976202263100750256813), + SC_(19.9363040924072265625), SC_(8), SC_(0.826751708984375), SC_(3010989900.8893375803525358367041329372033717978545), + SC_(19.9363040924072265625), SC_(10), SC_(-0.72904598712921142578125), SC_(-451944141173.12209414901560544187498153708108846852), + SC_(19.9363040924072265625), SC_(11), SC_(0.67001712322235107421875), SC_(-3818836562168.0063397346459855637481748033411304683), + SC_(19.9363040924072265625), SC_(13), SC_(0.98576259613037109375), SC_(-555684705.08358775884287423663622045616768375133224), + SC_(19.9363040924072265625), SC_(15), SC_(-0.804919183254241943359375), SC_(-43129486142739936.248977475359797313525493031517393), + SC_(19.9363040924072265625), SC_(19), SC_(0.93538987636566162109375), SC_(-21677629202869.188161110548407561929925083445523615) + }; +#undef SC_ + diff --git a/test/atanh_test.hpp b/test/atanh_test.hpp new file mode 100644 index 000000000..32a628e2f --- /dev/null +++ b/test/atanh_test.hpp @@ -0,0 +1,173 @@ +// unit test file atanh.hpp for the special functions test suite + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include +#include +//#include + + +#include + + +#include +#include + +template +T atanh_error_evaluator(T x) +{ + using ::std::abs; + using ::std::tanh; + using ::std::cosh; + + using ::std::numeric_limits; + + using ::boost::math::atanh; + + + static T const epsilon = numeric_limits::epsilon(); + + T y = tanh(x); + T z = atanh(y); + + T absolute_error = abs(z-x); + T relative_error = absolute_error/(cosh(x)*cosh(x)); + T scaled_error = relative_error/epsilon; + + return(scaled_error); +} + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(atanh_test, T) +{ + using ::std::abs; + using ::std::tanh; + using ::std::log; + + using ::std::numeric_limits; + + using ::boost::math::atanh; + + + BOOST_MESSAGE("Testing atanh in the real domain for " + << string_type_name::_() << "."); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(atanh(static_cast(0)))) + (numeric_limits::epsilon())); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(atanh(static_cast(3)/5) - log(static_cast(2)))) + (numeric_limits::epsilon())); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(atanh(static_cast(-3)/5) + log(static_cast(2)))) + (numeric_limits::epsilon())); + + for (int i = 0; i <= 100; i++) + { + T x = static_cast(i-50)/static_cast(5); + T y = tanh(x); + + if ( + (abs(y-static_cast(1)) >= numeric_limits::epsilon())&& + (abs(y+static_cast(1)) >= numeric_limits::epsilon()) + ) + { + BOOST_CHECK_PREDICATE(::std::less_equal(), + (atanh_error_evaluator(x)) + (static_cast(4))); + } + } +} + + +void atanh_manual_check() +{ + using ::std::abs; + using ::std::tanh; + + using ::std::numeric_limits; + + + BOOST_MESSAGE(" "); + BOOST_MESSAGE("atanh"); + + for (int i = 0; i <= 100; i++) + { + float xf = static_cast(i-50)/static_cast(5); + double xd = static_cast(i-50)/static_cast(5); + long double xl = + static_cast(i-50)/static_cast(5); + + float yf = tanh(xf); + double yd = tanh(xd); + (void) &yd; // avoid "unused variable" warning + long double yl = tanh(xl); + (void) &yl; // avoid "unused variable" warning + + if ( + std::numeric_limits::has_infinity && + std::numeric_limits::has_infinity && + std::numeric_limits::has_infinity + ) + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << atanh_error_evaluator(xf) + << ::std::setw(15) + << atanh_error_evaluator(xd) + << ::std::setw(15) + << atanh_error_evaluator(xl)); +#else + BOOST_MESSAGE( ::std::setw(15) + << atanh_error_evaluator(xf) + << ::std::setw(15) + << atanh_error_evaluator(xd)); +#endif + } + else + { + if ( + (abs(yf-static_cast(1)) < + numeric_limits::epsilon())|| + (abs(yf+static_cast(1)) < + numeric_limits::epsilon())|| + (abs(yf-static_cast(1)) < + numeric_limits::epsilon())|| + (abs(yf+static_cast(1)) < + numeric_limits::epsilon())|| + (abs(yf-static_cast(1)) < + numeric_limits::epsilon())|| + (abs(yf+static_cast(1)) < + numeric_limits::epsilon()) + ) + { + BOOST_MESSAGE("Platform's numerics may lack precision."); + } + else + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << atanh_error_evaluator(xf) + << ::std::setw(15) + << atanh_error_evaluator(xd) + << ::std::setw(15) + << atanh_error_evaluator(xl)); +#else + BOOST_MESSAGE( ::std::setw(15) + << atanh_error_evaluator(xf) + << ::std::setw(15) + << atanh_error_evaluator(xd)); +#endif + } + } + } + + BOOST_MESSAGE(" "); +} + diff --git a/test/bessel_i_data.ipp b/test/bessel_i_data.ipp new file mode 100644 index 000000000..0d9ea06b2 --- /dev/null +++ b/test/bessel_i_data.ipp @@ -0,0 +1,236 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 225> bessel_i_data = { + SC_(-0.8049192047119140625e2), SC_(0.24750102996826171875e2), SC_(0.4186983670927603951456761016615805321694e29), + SC_(-0.8049192047119140625e2), SC_(0.637722015380859375e2), SC_(0.2032477564839492547710163141114983376107e7), + SC_(-0.7460263824462890625e2), SC_(0.24750102996826171875e2), SC_(0.7209773983625557491285409746913864781653e24), + SC_(-0.7460263824462890625e2), SC_(0.637722015380859375e2), SC_(0.8754902581281478276585360070503782597115e9), + SC_(-0.7290460205078125e2), SC_(0.24750102996826171875e2), SC_(0.1045354686776007657866756333678839303562e23), + SC_(-0.7290460205078125e2), SC_(0.637722015380859375e2), SC_(0.4716204450814114467834563893626508706616e10), + SC_(-0.62323604583740234375e2), SC_(0.24750102996826171875e2), SC_(0.3651466017937929635806605426551713742921e15), + SC_(-0.62323604583740234375e2), SC_(0.637722015380859375e2), SC_(0.8566833579732135640262696128535710188301e14), + SC_(-0.5579319000244140625e2), SC_(0.24750102996826171875e2), SC_(-0.7703641421675666973721209829630518563931e10), + SC_(-0.5579319000244140625e2), SC_(0.637722015380859375e2), SC_(0.1959694422054590004110877333169097220352e17), + SC_(-0.4430035400390625e2), SC_(0.95070552825927734375e1), SC_(0.2934779151682606205400047286649505699432e23), + SC_(-0.4430035400390625e2), SC_(0.24750102996826171875e2), SC_(0.6405675645065933411553551945741721563518e3), + SC_(-0.4430035400390625e2), SC_(0.637722015380859375e2), SC_(0.805557170418531409978537810934369410633e20), + SC_(-0.383665924072265625e2), SC_(0.51139926910400390625e1), SC_(0.289105444853831500904637353663660192819e28), + SC_(-0.383665924072265625e2), SC_(0.95070552825927734375e1), SC_(0.8806324212322684518238175986513600832194e17), + SC_(-0.383665924072265625e2), SC_(0.24750102996826171875e2), SC_(0.3890036496141556574166698709019531196928e0), + SC_(-0.383665924072265625e2), SC_(0.637722015380859375e2), SC_(0.3063025493893607419003502570490592505508e22), + SC_(0.93762989044189453125e1), SC_(0.177219114266335964202880859375e-2), SC_(0.2803618162597370401397229001288041823179e-34), + SC_(0.93762989044189453125e1), SC_(0.22177286446094512939453125e-2), SC_(0.229588528336937186404069781250920318956e-33), + SC_(0.93762989044189453125e1), SC_(0.7444499991834163665771484375e-2), SC_(0.1959749964125288689721223617361145955449e-28), + SC_(0.93762989044189453125e1), SC_(0.1433600485324859619140625e-1), SC_(0.9133296812900414557505943321740418454784e-26), + SC_(0.93762989044189453125e1), SC_(0.1760916970670223236083984375e-1), SC_(0.6281058749227662133654486730221725796013e-25), + SC_(0.93762989044189453125e1), SC_(0.6152711808681488037109375e-1), SC_(0.7806782632836478900912324974768292604178e-20), + SC_(0.93762989044189453125e1), SC_(0.11958599090576171875e0), SC_(0.3969270921873940596846801055093851364249e-17), + SC_(0.93762989044189453125e1), SC_(0.15262925624847412109375e0), SC_(0.3911054564312661834266087273264220427648e-16), + SC_(0.93762989044189453125e1), SC_(0.408089816570281982421875e0), SC_(0.3968059264413538749627636752562708954912e-12), + SC_(0.93762989044189453125e1), SC_(0.6540834903717041015625e0), SC_(0.3328913174482554006724623549002084389481e-10), + SC_(0.93762989044189453125e1), SC_(0.1097540378570556640625e1), SC_(0.4345996628731497806117301412025701409681e-8), + SC_(0.93762989044189453125e1), SC_(0.30944411754608154296875e1), SC_(0.8821763783526972687094781579187606888861e-4), + SC_(0.93762989044189453125e1), SC_(0.51139926910400390625e1), SC_(0.1440790625360570911757430559538249450382e-1), + SC_(0.93762989044189453125e1), SC_(0.95070552825927734375e1), SC_(0.1943226931700462962154856194109129706652e2), + SC_(0.93762989044189453125e1), SC_(0.24750102996826171875e2), SC_(0.7543097629536504483214121556669228286985e9), + SC_(0.93762989044189453125e1), SC_(0.637722015380859375e2), SC_(0.1242187462089386651190442543425398784833e27), + SC_(0.944411754608154296875e1), SC_(0.177219114266335964202880859375e-2), SC_(0.148982667418264205724082321914073417083e-34), + SC_(0.944411754608154296875e1), SC_(0.22177286446094512939453125e-2), SC_(0.1238718017352724275257612770297545398575e-33), + SC_(0.944411754608154296875e1), SC_(0.7444499991834163665771484375e-2), SC_(0.1147864422785975785379486658863341965539e-28), + SC_(0.944411754608154296875e1), SC_(0.1433600485324859619140625e-1), SC_(0.559265688248480010364759849989645260096e-26), + SC_(0.944411754608154296875e1), SC_(0.1760916970670223236083984375e-1), SC_(0.3900141694706919558831468837845481277321e-25), + SC_(0.944411754608154296875e1), SC_(0.6152711808681488037109375e-1), SC_(0.5276757393757080096863772555264582929817e-20), + SC_(0.944411754608154296875e1), SC_(0.11958599090576171875e0), SC_(0.2806586679017851973719099727210813946663e-17), + SC_(0.944411754608154296875e1), SC_(0.15262925624847412109375e0), SC_(0.2811556875187931972313758314199441466104e-16), + SC_(0.944411754608154296875e1), SC_(0.408089816570281982421875e0), SC_(0.3049214538640230343488562864541510399665e-12), + SC_(0.944411754608154296875e1), SC_(0.6540834903717041015625e0), SC_(0.2641125801178791553193637198564322320001e-10), + SC_(0.944411754608154296875e1), SC_(0.1097540378570556640625e1), SC_(0.3570822482377035574485486720053965916982e-8), + SC_(0.944411754608154296875e1), SC_(0.30944411754608154296875e1), SC_(0.7766278241102208401170874203781603111746e-4), + SC_(0.944411754608154296875e1), SC_(0.51139926910400390625e1), SC_(0.1309298511678054464902027194534409538646e-1), + SC_(0.944411754608154296875e1), SC_(0.95070552825927734375e1), SC_(0.1827999074103902886609783889480769570524e2), + SC_(0.944411754608154296875e1), SC_(0.24750102996826171875e2), SC_(0.7351908192175552948778255155781258947671e9), + SC_(0.944411754608154296875e1), SC_(0.637722015380859375e2), SC_(0.1229766721665510769829680551623499922036e27), + SC_(0.264718532562255859375e2), SC_(0.177219114266335964202880859375e-2), SC_(0.8225184464794312214427441884602794315635e-108), + SC_(0.264718532562255859375e2), SC_(0.22177286446094512939453125e-2), SC_(0.3114921421876925970246827842503222114267e-105), + SC_(0.264718532562255859375e2), SC_(0.7444499991834163665771484375e-2), SC_(0.2604432142391172275446430144882881083067e-91), + SC_(0.264718532562255859375e2), SC_(0.1433600485324859619140625e-1), SC_(0.8900355057385521217383601748827962581012e-84), + SC_(0.264718532562255859375e2), SC_(0.1760916970670223236083984375e-1), SC_(0.2058868164797190179746749314402013402828e-81), + SC_(0.264718532562255859375e2), SC_(0.6152711808681488037109375e-1), SC_(0.4972024613715337297288025064797981454153e-67), + SC_(0.264718532562255859375e2), SC_(0.11958599090576171875e0), SC_(0.2171276940537634328598470815758727598254e-59), + SC_(0.264718532562255859375e2), 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SC_(0.98576263427734375e2), SC_(0.1760916970670223236083984375e-1), SC_(0.187601891562751902089087302157938087049e-357), + SC_(0.98576263427734375e2), SC_(0.6152711808681488037109375e-1), SC_(0.6799154838478458841616493729745617835238e-304), + SC_(0.98576263427734375e2), SC_(0.11958599090576171875e0), SC_(0.1918358571470383749967177500137981440808e-275), + SC_(0.98576263427734375e2), SC_(0.15262925624847412109375e0), SC_(0.5343578537582739461840941918655264976087e-265), + SC_(0.98576263427734375e2), SC_(0.408089816570281982421875e0), SC_(0.6787062833732678279692755114282109142003e-223), + SC_(0.98576263427734375e2), SC_(0.6540834903717041015625e0), SC_(0.1066640939494771247816497353023791644911e-202), + SC_(0.98576263427734375e2), SC_(0.1097540378570556640625e1), SC_(0.1540136528087774369513263265420126735826e-180), + SC_(0.98576263427734375e2), SC_(0.30944411754608154296875e1), SC_(0.3732038034481165129234044502070414699443e-136), + SC_(0.98576263427734375e2), SC_(0.51139926910400390625e1), SC_(0.1250755987062674994392330438153044634978e-114), + SC_(0.98576263427734375e2), SC_(0.95070552825927734375e1), SC_(0.5155910228741865630530748597345526561019e-88), + SC_(0.98576263427734375e2), SC_(0.24750102996826171875e2), SC_(0.173088587965140287251789752425440165731e-46), + SC_(0.98576263427734375e2), SC_(0.637722015380859375e2), SC_(0.2139803586023867005821680732168695726563e-2), + SC_(0.99292266845703125e2), SC_(0.177219114266335964202880859375e-2), SC_(0.2261194572892689144464257783997217591003e-459), + SC_(0.99292266845703125e2), SC_(0.22177286446094512939453125e-2), SC_(0.1059718907506904441739516643053360474766e-449), + SC_(0.99292266845703125e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1760708480112020640215899721963204640613e-397), + SC_(0.99292266845703125e2), SC_(0.1433600485324859619140625e-1), SC_(0.3188104531258877486126505505288677161449e-369), + SC_(0.99292266845703125e2), SC_(0.1760916970670223236083984375e-1), SC_(0.2351789542752206704101133488009018496105e-360), + SC_(0.99292266845703125e2), SC_(0.6152711808681488037109375e-1), SC_(0.2087568008199554926652533137918586792421e-306), + SC_(0.99292266845703125e2), SC_(0.11958599090576171875e0), SC_(0.9479028425669903092514832066160512980969e-278), + SC_(0.99292266845703125e2), SC_(0.15262925624847412109375e0), SC_(0.3144358707762065981613586172661430942717e-267), + SC_(0.99292266845703125e2), SC_(0.408089816570281982421875e0), SC_(0.8076041994016289659837261163786154124696e-225), + SC_(0.99292266845703125e2), SC_(0.6540834903717041015625e0), SC_(0.1779204438764558957337121273226739006915e-204), + SC_(0.99292266845703125e2), SC_(0.1097540378570556640625e1), SC_(0.3721424984805355820747068117876586135294e-182), + SC_(0.99292266845703125e2), SC_(0.30944411754608154296875e1), SC_(0.1893861897195351427223854481637771374334e-137), + SC_(0.99292266845703125e2), SC_(0.51139926910400390625e1), SC_(0.9092055250182067057531632044074461256826e-116), + SC_(0.99292266845703125e2), SC_(0.95070552825927734375e1), SC_(0.5835883537392851212536191805017949717878e-89), + SC_(0.99292266845703125e2), SC_(0.24750102996826171875e2), SC_(0.3851491419574571630736597349656750764496e-47), + SC_(0.99292266845703125e2), SC_(0.637722015380859375e2), SC_(0.8891727416380076726409331943293544170762e-3) + }; +#undef SC_ + + diff --git a/test/bessel_i_int_data.ipp b/test/bessel_i_int_data.ipp new file mode 100644 index 000000000..31bd6d477 --- /dev/null +++ b/test/bessel_i_int_data.ipp @@ -0,0 +1,506 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 495> bessel_i_int_data = { + SC_(0), SC_(0.177219114266335964202880859375e-2), SC_(0.1000000785165515654790765567976560128368e1), + SC_(0), SC_(0.22177286446094512939453125e-2), SC_(0.1000001229580463247166647265657459686513e1), + SC_(0), SC_(0.7444499991834163665771484375e-2), SC_(0.1000013855193023439561602999309818071557e1), + SC_(0), SC_(0.1433600485324859619140625e-1), SC_(0.100005138091877460790808253311381371617e1), + SC_(0), SC_(0.1760916970670223236083984375e-1), SC_(0.1000077522216818092267597684468025144708e1), + SC_(0), SC_(0.6152711808681488037109375e-1), SC_(0.1000946620505179168127565717529250425225e1), + SC_(0), SC_(0.11958599090576171875e0), SC_(0.1003578399092795290959506558979668000986e1), + SC_(0), SC_(0.15262925624847412109375e0), SC_(0.1005832407473077490942708958509537944332e1), + SC_(0), SC_(0.408089816570281982421875e0), SC_(0.1042069688780878266908906813963061201236e1), + SC_(0), SC_(0.6540834903717041015625e0), SC_(0.1109850431257999241542412962070021625239e1), + SC_(0), SC_(0.1097540378570556640625e1), SC_(0.1324594459649893673687924138468203300537e1), + SC_(0), SC_(0.30944411754608154296875e1), SC_(0.5270503008520823311035815313268274322688e1), + SC_(0), SC_(0.51139926910400390625e1), SC_(0.3016460113324413124385977969424656719006e2), + SC_(0), SC_(0.95070552825927734375e1), SC_(0.1765221230864037183559483071058505259281e4), + SC_(0), SC_(0.24750102996826171875e2), SC_(0.4520582716299093122418957895745838047526e10), + SC_(0), SC_(0.637722015380859375e2), SC_(0.248523599457696979135532431768725854655e27), + SC_(0.1e1), SC_(0.177219114266335964202880859375e-2), SC_(0.8860959191975001520307655146348128981208e-3), + SC_(0.1e1), SC_(0.22177286446094512939453125e-2), SC_(0.1108865004023609343492425973877677708278e-2), + SC_(0.1e1), SC_(0.7444499991834163665771484375e-2), SC_(0.3722275782133396649790935284938128705517e-2), + SC_(0.1e1), SC_(0.1433600485324859619140625e-1), SC_(0.7168186575111060974651037851905127311172e-2), + SC_(0.1e1), SC_(0.1760916970670223236083984375e-1), SC_(0.8804926126614467969087386987026758424778e-2), + SC_(0.1e1), SC_(0.6152711808681488037109375e-1), SC_(0.307781186030490619563031243121873491391e-1), + SC_(0.1e1), SC_(0.11958599090576171875e0), SC_(0.5989994518931145664339323232552863277613e-1), + SC_(0.1e1), SC_(0.15262925624847412109375e0), SC_(0.7653706917131299515526663760316122678536e-1), + SC_(0.1e1), SC_(0.408089816570281982421875e0), SC_(0.2083221213089251435227012018452735830071e0), + SC_(0.1e1), SC_(0.6540834903717041015625e0), SC_(0.3448458972066057120326871217765728528051e0), + SC_(0.1e1), SC_(0.1097540378570556640625e1), SC_(0.6356539293968716593197570138590683019623e0), + SC_(0.1e1), SC_(0.30944411754608154296875e1), SC_(0.4304586767845429639244166020309162877221e1), + SC_(0.1e1), SC_(0.51139926910400390625e1), SC_(0.2702631060982870150322284389310299253322e2), + SC_(0.1e1), SC_(0.95070552825927734375e1), SC_(0.1669630395093490324472744646621461529409e4), + SC_(0.1e1), SC_(0.24750102996826171875e2), SC_(0.4428295881367777978681563662886643036499e10), + SC_(0.1e1), SC_(0.637722015380859375e2), SC_(0.2465673119157042745018873572402000151201e27), + SC_(0.4e1), SC_(0.177219114266335964202880859375e-2), SC_(0.2568686424002909205919386454789887238475e-13), + SC_(0.4e1), SC_(0.22177286446094512939453125e-2), SC_(0.629944815795979203946014830217428242984e-13), + SC_(0.4e1), SC_(0.7444499991834163665771484375e-2), SC_(0.7998565658415621562337103146472082554028e-11), + SC_(0.4e1), SC_(0.1433600485324859619140625e-1), SC_(0.1099982550547901452676110099180043770797e-9), + SC_(0.4e1), SC_(0.1760916970670223236083984375e-1), SC_(0.250398097475533981445277556832601328223e-9), + SC_(0.4e1), SC_(0.6152711808681488037109375e-1), SC_(0.3732650011579987791541719677211396374467e-7), + SC_(0.4e1), SC_(0.11958599090576171875e0), SC_(0.5329672477129410019119113447394893295596e-6), + SC_(0.4e1), SC_(0.15262925624847412109375e0), SC_(0.141489997131418627857066995095369938696e-5), + SC_(0.4e1), SC_(0.408089816570281982421875e0), SC_(0.7282921216911263304294810852632628068137e-4), + SC_(0.4e1), SC_(0.6540834903717041015625e0), SC_(0.4869396484162391070562181449261259775595e-3), + SC_(0.4e1), SC_(0.1097540378570556640625e1), SC_(0.4012161862208555247000124967949382739361e-2), + SC_(0.4e1), SC_(0.30944411754608154296875e1), SC_(0.3787138783037547310647400896688635049979e0), + SC_(0.4e1), SC_(0.51139926910400390625e1), SC_(0.5868359078308841151534796428362771527491e1), + SC_(0.4e1), SC_(0.95070552825927734375e1), SC_(0.7357193068805439491961808818463893925223e3), + SC_(0.4e1), SC_(0.24750102996826171875e2), SC_(0.3252313835173492963089917060960028522653e10), + SC_(0.4e1), SC_(0.637722015380859375e2), SC_(0.2190135725660260903026664293135379072029e27), + SC_(0.7e1), SC_(0.177219114266335964202880859375e-2), SC_(0.8510076576264637138875125872170131898096e-25), + SC_(0.7e1), SC_(0.22177286446094512939453125e-2), SC_(0.4089953800055587124509461216692582736447e-24), + SC_(0.7e1), SC_(0.7444499991834163665771484375e-2), SC_(0.1964305273103342680910343954984308619768e-20), + SC_(0.7e1), SC_(0.1433600485324859619140625e-1), SC_(0.1929120066647206188436854652387114727807e-18), + SC_(0.7e1), SC_(0.1760916970670223236083984375e-1), SC_(0.8138340472907783210251698976282644477818e-18), + SC_(0.7e1), SC_(0.6152711808681488037109375e-1), SC_(0.5174601186268260520938521328073083909595e-14), + SC_(0.7e1), SC_(0.11958599090576171875e0), SC_(0.542395054389276152812978441225258183457e-12), + SC_(0.7e1), SC_(0.15262925624847412109375e0), SC_(0.299323047925904744170833613652767758086e-11), + SC_(0.7e1), SC_(0.408089816570281982421875e0), SC_(0.2937036464827401981806966262136490086826e-8), + SC_(0.7e1), SC_(0.6540834903717041015625e0), SC_(0.8046255455349868199127611214199975918109e-7), + SC_(0.7e1), SC_(0.1097540378570556640625e1), SC_(0.3087574366778968153326883525133715811508e-5), + SC_(0.7e1), SC_(0.30944411754608154296875e1), SC_(0.5653447566166178294962968673241498039082e-2), + SC_(0.7e1), SC_(0.51139926910400390625e1), SC_(0.3104877913933256744262177732064662628008e0), + SC_(0.7e1), SC_(0.95070552825927734375e1), SC_(0.1318882152673454326811262619237084813807e3), + SC_(0.7e1), SC_(0.24750102996826171875e2), SC_(0.1657171203518438763471041373689390646813e10), + SC_(0.7e1), SC_(0.637722015380859375e2), SC_(0.1687969369952926235916289436609130146855e27), + SC_(0.1e2), SC_(0.177219114266335964202880859375e-2), SC_(0.8223234174016769910080935836038629086129e-37), + SC_(0.1e2), SC_(0.22177286446094512939453125e-2), SC_(0.7744994585234402998026715482431864809177e-36), + SC_(0.1e2), SC_(0.7444499991834163665771484375e-2), SC_(0.140699610530434722230776594136166384761e-30), + SC_(0.1e2), SC_(0.1433600485324859619140625e-1), SC_(0.9867802131594229578497672216212381206342e-28), + SC_(0.1e2), SC_(0.1760916970670223236083984375e-1), SC_(0.7714874409767011750174102200723123371757e-27), + SC_(0.1e2), SC_(0.6152711808681488037109375e-1), SC_(0.2092377824573518684672948738047686808098e-21), + SC_(0.1e2), SC_(0.11958599090576171875e0), SC_(0.161020520917686222303614853893004305517e-18), + SC_(0.1e2), SC_(0.15262925624847412109375e0), SC_(0.1847331437121988828659943855683029847574e-17), + SC_(0.1e2), SC_(0.408089816570281982421875e0), SC_(0.3460494035292646148422797589423437304445e-13), + SC_(0.1e2), SC_(0.6540834903717041015625e0), SC_(0.3894842946330373264656841132859435813454e-11), + SC_(0.1e2), SC_(0.1097540378570556640625e1), SC_(0.7014848610606988360553920593019620983893e-9), + SC_(0.1e2), SC_(0.30944411754608154296875e1), SC_(0.2688034653083270014812653556923722759657e-4), + SC_(0.1e2), SC_(0.51139926910400390625e1), SC_(0.5883608669420951460953858890820260901642e-2), + SC_(0.1e2), SC_(0.95070552825927734375e1), SC_(0.1093507510277669423960436015414354945337e2), + SC_(0.1e2), SC_(0.24750102996826171875e2), SC_(0.5917457915288690964012606008596472223686e9), + SC_(0.1e2), SC_(0.637722015380859375e2), SC_(0.1129466336865796349694848290694042226462e27), + SC_(0.13e2), SC_(0.177219114266335964202880859375e-2), SC_(0.3334011286277694028669539285669885563166e-49), + SC_(0.13e2), SC_(0.22177286446094512939453125e-2), SC_(0.6153738758043090277122480232744400184223e-48), + SC_(0.13e2), SC_(0.7444499991834163665771484375e-2), SC_(0.4228556218314688954181791384498131827781e-41), + SC_(0.13e2), SC_(0.1433600485324859619140625e-1), SC_(0.2117859499568276103587689359158579219165e-37), + SC_(0.13e2), SC_(0.1760916970670223236083984375e-1), SC_(0.3068580854652108160425423230103324526003e-36), + SC_(0.13e2), SC_(0.6152711808681488037109375e-1), SC_(0.354996892487294926273047608799400251176e-29), + SC_(0.13e2), 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SC_(0.67e2), SC_(0.11958599090576171875e0), SC_(0.2976080454780070131652687845254578786778e-176), + SC_(0.67e2), SC_(0.15262925624847412109375e0), SC_(0.3739472000953063909501613448588687457119e-169), + SC_(0.67e2), SC_(0.408089816570281982421875e0), SC_(0.1548613210047140776105471826780937390242e-140), + SC_(0.67e2), SC_(0.6540834903717041015625e0), SC_(0.8263193751493687283195945930118436171448e-127), + SC_(0.67e2), SC_(0.1097540378570556640625e1), SC_(0.9531001594119617180115060699004781472022e-112), + SC_(0.67e2), SC_(0.30944411754608154296875e1), SC_(0.1423429375392353585102712760378566415253e-81), + SC_(0.67e2), SC_(0.51139926910400390625e1), SC_(0.6276199872704888189953779431744609272855e-67), + SC_(0.67e2), SC_(0.95070552825927734375e1), SC_(0.8751710351217371349774248758780557808399e-49), + SC_(0.67e2), SC_(0.24750102996826171875e2), SC_(0.399443014585480721312551749526965287716e-20), + SC_(0.67e2), SC_(0.637722015380859375e2), SC_(0.1303190285326414512188004605064925401042e13), + SC_(0.7e2), SC_(0.177219114266335964202880859375e-2), SC_(0.175887342640394106189151976112543057962e-313), + SC_(0.7e2), SC_(0.7444499991834163665771484375e-2), SC_(0.7550621788530834428425404682332589500832e-270), + SC_(0.7e2), SC_(0.1433600485324859619140625e-1), SC_(0.6301810450616594782432087551850756373668e-250), + SC_(0.7e2), SC_(0.1760916970670223236083984375e-1), SC_(0.1125167731015917733561916583817457784138e-243), + SC_(0.7e2), SC_(0.6152711808681488037109375e-1), SC_(0.1213839287680701613177842094136722210551e-205), + SC_(0.7e2), SC_(0.11958599090576171875e0), SC_(0.1937040012939367633269395257833394549994e-185), + SC_(0.7e2), SC_(0.15262925624847412109375e0), SC_(0.5060297469126818749198761907151271684388e-178), + SC_(0.7e2), SC_(0.408089816570281982421875e0), SC_(0.4005465233840990833504194622177985321517e-148), + SC_(0.7e2), SC_(0.6540834903717041015625e0), SC_(0.8799796018332889235436642767544430577921e-134), + SC_(0.7e2), SC_(0.1097540378570556640625e1), SC_(0.4794825121102189651189070515356607000206e-118), + SC_(0.7e2), SC_(0.30944411754608154296875e1), SC_(0.1602843216269687242254187608050533636733e-86), + SC_(0.7e2), SC_(0.51139926910400390625e1), SC_(0.3181779993739836562974728016162666336895e-71), + SC_(0.7e2), SC_(0.95070552825927734375e1), SC_(0.2822466596594199323749265377605991268801e-52), + SC_(0.7e2), SC_(0.24750102996826171875e2), SC_(0.2104365489296975788813919994670728216215e-22), + SC_(0.7e2), SC_(0.637722015380859375e2), SC_(0.7844318561221098998292957969567564923615e11), + SC_(0.73e2), SC_(0.177219114266335964202880859375e-2), SC_(0.3279159790515129976232600411030994939645e-328), + SC_(0.73e2), SC_(0.22177286446094512939453125e-2), SC_(0.4224790877225924823386404075215067890494e-321), + SC_(0.73e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1043483587404467322105724165693680432988e-282), + SC_(0.73e2), SC_(0.1433600485324859619140625e-1), SC_(0.6219365054628790184510838867649950741467e-262), + SC_(0.73e2), SC_(0.1760916970670223236083984375e-1), SC_(0.2057928083398578974258661409116904762983e-255), + SC_(0.73e2), SC_(0.6152711808681488037109375e-1), SC_(0.9470152726761824765032042182394011675392e-216), + SC_(0.73e2), SC_(0.11958599090576171875e0), SC_(0.1109621677740774224342517916855017900658e-194), + SC_(0.73e2), SC_(0.15262925624847412109375e0), SC_(0.6026765241850857913163429093708050618843e-187), + SC_(0.73e2), SC_(0.408089816570281982421875e0), SC_(0.9118139298541229295297403648253683737698e-156), + SC_(0.73e2), SC_(0.6540834903717041015625e0), SC_(0.8247872764458903831476787586065099347135e-141), + SC_(0.73e2), SC_(0.1097540378570556640625e1), SC_(0.2123029007982407474621311100276015417167e-124), + SC_(0.73e2), SC_(0.30944411754608154296875e1), SC_(0.1588696020280731697227952432556978774771e-91), + SC_(0.73e2), SC_(0.51139926910400390625e1), SC_(0.1420130918879929727408563849006120293294e-75), + SC_(0.73e2), SC_(0.95070552825927734375e1), SC_(0.8020381326448500037051674847470025971321e-56), + SC_(0.73e2), SC_(0.24750102996826171875e2), SC_(0.9826710538792853844402410449812254638335e-25), + SC_(0.73e2), SC_(0.637722015380859375e2), SC_(0.4293839369539487481259198718773671804064e10), + SC_(0.76e2), SC_(0.177219114266335964202880859375e-2), SC_(0.5408759889893148190520203529735474973187e-343), + SC_(0.76e2), SC_(0.22177286446094512939453125e-2), SC_(0.1365632588234360977374366339815548997377e-335), + SC_(0.76e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1275838367710964746430025849813881543957e-295), + SC_(0.76e2), SC_(0.1433600485324859619140625e-1), SC_(0.5430425912637421039353201453109713889038e-274), + SC_(0.76e2), SC_(0.1760916970670223236083984375e-1), SC_(0.3330046136342914006909942281157109406199e-267), + SC_(0.76e2), SC_(0.6152711808681488037109375e-1), SC_(0.6536720629329230577091818914690091163553e-226), + SC_(0.76e2), SC_(0.11958599090576171875e0), SC_(0.5623652442375975181466641168529807893221e-204), + SC_(0.76e2), SC_(0.15262925624847412109375e0), SC_(0.6350380451255890749856575267278817991626e-196), + SC_(0.76e2), SC_(0.408089816570281982421875e0), SC_(0.183640054565323298205707300952041465441e-163), + SC_(0.76e2), SC_(0.6540834903717041015625e0), SC_(0.6839439254682699613485852891479608181366e-148), + SC_(0.76e2), SC_(0.1097540378570556640625e1), SC_(0.8316719082770017371804750065668802553579e-131), + SC_(0.76e2), SC_(0.30944411754608154296875e1), SC_(0.1393297633857098951754512416010366098401e-96), + SC_(0.76e2), SC_(0.51139926910400390625e1), SC_(0.5609441790373577387000154824729897241791e-80), + SC_(0.76e2), SC_(0.95070552825927734375e1), SC_(0.2018365197172710168614503238313291896062e-59), + SC_(0.76e2), SC_(0.24750102996826171875e2), SC_(0.4085433998985207824201599332796010854695e-27), + SC_(0.76e2), SC_(0.637722015380859375e2), SC_(0.2142103160400899021062036922016113667452e9), + SC_(0.79e2), SC_(0.177219114266335964202880859375e-2), SC_(0.7930982305179989580158425245351456663586e-358), + SC_(0.79e2), SC_(0.22177286446094512939453125e-2), SC_(0.3924250147523408219992167518859050391414e-350), + SC_(0.79e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1386755351840733324704199171711908798229e-308), + SC_(0.79e2), SC_(0.1433600485324859619140625e-1), SC_(0.4215177758474358774763535279898355939459e-286), + SC_(0.79e2), SC_(0.1760916970670223236083984375e-1), SC_(0.4790319326701827605381983744427272324794e-279), + SC_(0.79e2), SC_(0.6152711808681488037109375e-1), SC_(0.4011040414527164043809376106988478947351e-236), + SC_(0.79e2), SC_(0.11958599090576171875e0), SC_(0.253370605560492432574321731615139964047e-213), + SC_(0.79e2), SC_(0.15262925624847412109375e0), SC_(0.5948527457011269418607542342494814436562e-205), + SC_(0.79e2), SC_(0.408089816570281982421875e0), SC_(0.3287936896920888923522197582153467750847e-171), + SC_(0.79e2), SC_(0.6540834903717041015625e0), SC_(0.5041909325640412505781517484471102307831e-155), + SC_(0.79e2), SC_(0.1097540378570556640625e1), SC_(0.2896326090972206182393839086729167150143e-137), + SC_(0.79e2), SC_(0.30944411754608154296875e1), SC_(0.1086380977296709087208790251658448356189e-101), + SC_(0.79e2), SC_(0.51139926910400390625e1), SC_(0.1970229546359244151730974118220967814952e-84), + SC_(0.79e2), SC_(0.95070552825927734375e1), SC_(0.4519411101705219203865562848704682354613e-63), + SC_(0.79e2), SC_(0.24750102996826171875e2), SC_(0.1518463586382666158992394326311743396678e-29), + SC_(0.79e2), SC_(0.637722015380859375e2), SC_(0.9760490072697396207873117894195488769342e7), + SC_(0.82e2), SC_(0.177219114266335964202880859375e-2), SC_(0.1038436202803785426388133600179459730169e-372), + SC_(0.82e2), SC_(0.22177286446094512939453125e-2), SC_(0.1006938753460328462474122574448053771171e-364), + SC_(0.82e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1345945900323246361787623818357538331624e-321), + SC_(0.82e2), SC_(0.1433600485324859619140625e-1), SC_(0.292160471475819054556258294299816966923e-298), + SC_(0.82e2), SC_(0.1760916970670223236083984375e-1), SC_(0.6153217141248195324459719214887728749464e-291), + SC_(0.82e2), SC_(0.6152711808681488037109375e-1), SC_(0.2197747157001515487720136860444751064868e-246), + SC_(0.82e2), SC_(0.11958599090576171875e0), SC_(0.1019336562451156029299163377789061540366e-222), + SC_(0.82e2), SC_(0.15262925624847412109375e0), SC_(0.4975570065935101711688291197874212937039e-214), + SC_(0.82e2), SC_(0.408089816570281982421875e0), SC_(0.5256585491004637978784987888275707136183e-179), + SC_(0.82e2), SC_(0.6540834903717041015625e0), SC_(0.3318904542226912390049207214862611260169e-162), + SC_(0.82e2), SC_(0.1097540378570556640625e1), SC_(0.9006810914088316982393635835954853735918e-144), + SC_(0.82e2), SC_(0.30944411754608154296875e1), SC_(0.7564505234377340524197089618341840082876e-107), + SC_(0.82e2), SC_(0.51139926910400390625e1), SC_(0.6180693935663269590148206442946860331756e-89), + SC_(0.82e2), SC_(0.95070552825927734375e1), SC_(0.9043352144198374855605316967339611243444e-67), + SC_(0.82e2), SC_(0.24750102996826171875e2), SC_(0.5065035140645080960031365557479815802671e-32), + SC_(0.82e2), SC_(0.637722015380859375e2), SC_(0.4070462729680846577801828424629699736782e6), + SC_(0.85e2), SC_(0.177219114266335964202880859375e-2), SC_(0.1219116513647314073579445140073826698962e-387), + SC_(0.85e2), SC_(0.22177286446094512939453125e-2), SC_(0.2316658303570511364023325655883461576972e-379), + SC_(0.85e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1171299375215890527495299558278169129332e-334), + SC_(0.85e2), SC_(0.1433600485324859619140625e-1), SC_(0.1815681105495951303009870770747843010446e-310), + SC_(0.85e2), SC_(0.1760916970670223236083984375e-1), SC_(0.708683910208375847232135348016178171053e-303), + SC_(0.85e2), SC_(0.6152711808681488037109375e-1), SC_(0.1079719594775910837503589945246873551394e-256), + SC_(0.85e2), SC_(0.11958599090576171875e0), SC_(0.3676982627128526557346333692497334159336e-232), + SC_(0.85e2), SC_(0.15262925624847412109375e0), SC_(0.3731546483902763830031124428268564725393e-223), + SC_(0.85e2), SC_(0.408089816570281982421875e0), SC_(0.7535240432188811873875509550682990477762e-187), + SC_(0.85e2), SC_(0.6540834903717041015625e0), SC_(0.1958883058289393975902194279609190904406e-169), + SC_(0.85e2), SC_(0.1097540378570556640625e1), SC_(0.2511373518429964095251580585112467258299e-150), + SC_(0.85e2), SC_(0.30944411754608154296875e1), SC_(0.4723067963574572650298919057601100049465e-112), + SC_(0.85e2), SC_(0.51139926910400390625e1), SC_(0.1738839346363387627371195241710367004536e-93), + SC_(0.85e2), SC_(0.95070552825927734375e1), SC_(0.1623663257276872998800059443147901760439e-70), + SC_(0.85e2), SC_(0.24750102996826171875e2), SC_(0.1521743437932867300705177236196212703582e-34), + SC_(0.85e2), SC_(0.637722015380859375e2), SC_(0.1556799959129105832955966318088587426661e5), + SC_(0.88e2), SC_(0.177219114266335964202880859375e-2), SC_(0.1288209613593357571142720689583005763762e-402), + SC_(0.88e2), SC_(0.22177286446094512939453125e-2), SC_(0.4797299543775689080036441106208914106219e-394), + SC_(0.88e2), SC_(0.7444499991834163665771484375e-2), SC_(0.9174536812416633326341915508185869175459e-348), + SC_(0.88e2), SC_(0.1433600485324859619140625e-1), SC_(0.1015625556161420986205394396723116662882e-322), + SC_(0.88e2), SC_(0.1760916970670223236083984375e-1), SC_(0.7346472096657986490100705296779319049912e-315), + SC_(0.88e2), SC_(0.6152711808681488037109375e-1), SC_(0.4774415143129565154141226398131338281536e-267), + SC_(0.88e2), SC_(0.11958599090576171875e0), SC_(0.119382735471855848143552750474098416176e-241), + SC_(0.88e2), SC_(0.15262925624847412109375e0), SC_(0.2518899646481384301970914279883481939291e-232), + SC_(0.88e2), SC_(0.408089816570281982421875e0), SC_(0.9722253412784373652536121925492657625805e-195), + SC_(0.88e2), SC_(0.6540834903717041015625e0), SC_(0.1040637574745703228070355356117916659304e-176), + SC_(0.88e2), SC_(0.1097540378570556640625e1), SC_(0.6302766367711271660992485559308335862541e-157), + SC_(0.88e2), SC_(0.30944411754608154296875e1), SC_(0.2654441873858763831864401113742713122992e-117), + SC_(0.88e2), SC_(0.51139926910400390625e1), SC_(0.4403904099603861384280491749834790369519e-98), + SC_(0.88e2), SC_(0.95070552825927734375e1), SC_(0.262551184129831686961964273106436690334e-74), + SC_(0.88e2), SC_(0.24750102996826171875e2), SC_(0.4131930234604820436302932418803598894853e-37), + SC_(0.88e2), SC_(0.637722015380859375e2), SC_(0.5471250742874498784335752261273680101442e3) + }; +#undef SC_ + + diff --git a/test/bessel_j_data.ipp b/test/bessel_j_data.ipp new file mode 100644 index 000000000..4200c23b5 --- /dev/null +++ b/test/bessel_j_data.ipp @@ -0,0 +1,371 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 360> bessel_j_data = { + SC_(0.4430477856658399105072021484375e-3), SC_(0.553809732082299888134002685546875e-4), SC_(0.9956156809860747445801192500664050062602e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.69304020144045352935791015625e-4), SC_(0.9957146107589140226790508756099801328105e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.23264062474481761455535888671875e-3), SC_(0.9962489699005580621378590557841736186861e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.4480001516640186309814453125e-3), SC_(0.9965382149919219697926696508657333563376e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.5502865533344447612762451171875e-3), SC_(0.9966289891095531217170976766456612311928e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.19227224402129650115966796875e-2), SC_(0.9971807065952682162692966068389032708599e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.37370622158050537109375e-2), SC_(0.9974717909688313208758556750808530574188e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.47696642577648162841796875e-2), SC_(0.997577426859888822291081757361344556863e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.1275280676782131195068359375e-1), SC_(0.9979773067040365116413896458097958587041e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.20440109074115753173828125e-1), SC_(0.9981222651757299264986454464589469235334e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.3429813683032989501953125e-1), SC_(0.9981619333713874456307490264513905997672e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.96701286733150482177734375e-1), SC_(0.9965812006107194383507063599312965181976e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.159812271595001220703125e0), SC_(0.9927699251072058869431015490774723070186e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.297095477581024169921875e0), SC_(0.9774885373571948238761430963137133013962e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.77344071865081787109375e0), SC_(0.8558678749290511288516105356098172615967e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.1992881298065185546875e1), SC_(0.2283524941992771533363988837045423832888e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.3915013790130615234375e1), SC_(-0.401360440349915932533731778088781472183e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.79858455657958984375e1), SC_(0.1751115157942275375093377805999900633628e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.1571910858154296875e2), SC_(-0.1426234585704966197380052262204215409713e0), + SC_(0.4430477856658399105072021484375e-3), SC_(0.31483119964599609375e2), SC_(0.1066310031567526427476684436417433539043e0), + SC_(0.554432161152362823486328125e-3), SC_(0.553809732082299888134002685546875e-4), SC_(0.9945164185569587230213539870667972875692e0), + SC_(0.554432161152362823486328125e-3), SC_(0.69304020144045352935791015625e-4), SC_(0.9946400847213057054537534776624837878946e0), + SC_(0.554432161152362823486328125e-3), SC_(0.23264062474481761455535888671875e-3), SC_(0.9953081108718330939070656828660938331342e0), + SC_(0.554432161152362823486328125e-3), SC_(0.4480001516640186309814453125e-3), SC_(0.9956697541222540255647650289685550550287e0), + SC_(0.554432161152362823486328125e-3), SC_(0.5502865533344447612762451171875e-3), SC_(0.9957832579906576456563607224027456916251e0), + SC_(0.554432161152362823486328125e-3), SC_(0.19227224402129650115966796875e-2), SC_(0.9964733546229106923992847311503904080801e0), + SC_(0.554432161152362823486328125e-3), SC_(0.37370622158050537109375e-2), SC_(0.9968380173570457415468434756113694340878e0), + SC_(0.554432161152362823486328125e-3), SC_(0.47696642577648162841796875e-2), SC_(0.9969706788099270079524162776507030382138e0), + SC_(0.554432161152362823486328125e-3), SC_(0.1275280676782131195068359375e-1), SC_(0.9974795810399399980819372250486803819028e0), + SC_(0.554432161152362823486328125e-3), SC_(0.20440109074115753173828125e-1), SC_(0.99767689617703742799168249521337668749e0), + SC_(0.554432161152362823486328125e-3), SC_(0.3429813683032989501953125e-1), SC_(0.997774089399990647214931657780519793888e0), + SC_(0.554432161152362823486328125e-3), SC_(0.96701286733150482177734375e-1), SC_(0.9963092188861358436843828984354699213759e0), + SC_(0.554432161152362823486328125e-3), SC_(0.159812271595001220703125e0), SC_(0.9925549713148663125876374807339340488473e0), + SC_(0.554432161152362823486328125e-3), SC_(0.297095477581024169921875e0), SC_(0.9773461257455259400626571167158074279339e0), + SC_(0.554432161152362823486328125e-3), SC_(0.77344071865081787109375e0), SC_(0.8558479783471517562239377557029974866496e0), + SC_(0.554432161152362823486328125e-3), SC_(0.1992881298065185546875e1), SC_(0.2284416118095148413691988171034857034998e0), + SC_(0.554432161152362823486328125e-3), SC_(0.3915013790130615234375e1), SC_(-0.4013573659865744373419228392808844392131e0), + SC_(0.554432161152362823486328125e-3), SC_(0.79858455657958984375e1), SC_(0.1751502024658724728242948804818811752031e0), + SC_(0.554432161152362823486328125e-3), SC_(0.1571910858154296875e2), SC_(-0.1425986283388769556266104448492207320811e0), + SC_(0.554432161152362823486328125e-3), SC_(0.31483119964599609375e2), SC_(0.1066145438340245056954574795725125118984e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.553809732082299888134002685546875e-4), SC_(0.9817093353081942469816526285068995358612e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.69304020144045352935791015625e-4), SC_(0.9821191746862947666429796292865148899257e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.23264062474481761455535888671875e-3), SC_(0.9843351663379732777023812441892374028968e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.4480001516640186309814453125e-3), SC_(0.9855363506977195696759812664468424040665e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.5502865533344447612762451171875e-3), SC_(0.9859135941901516415721117055805978018756e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.19227224402129650115966796875e-2), SC_(0.9882110091337713422555556649984351858334e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.37370622158050537109375e-2), SC_(0.9894314739398718125445096933051219189052e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.47696642577648162841796875e-2), SC_(0.9898786773968001423681957632107335237307e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.1275280676782131195068359375e-1), SC_(0.9916575656584680781858750626325680648059e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.20440109074115753173828125e-1), SC_(0.9924654079000974647945374252001453718364e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.3429813683032989501953125e-1), SC_(0.9932338692418333674513222303931472051914e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.96701286733150482177734375e-1), SC_(0.9931225190063597669143919008613986546829e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.159812271595001220703125e0), SC_(0.9900352203814238880722659452483734427322e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.297095477581024169921875e0), SC_(0.9756754635104943903830166510741270074568e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.77344071865081787109375e0), SC_(0.8556130162510752278666332268905263436046e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.1992881298065185546875e1), SC_(0.2294863665034258649591045326467545257489e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.3915013790130615234375e1), SC_(-0.4013203973956146567802727002608565844354e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.79858455657958984375e1), SC_(0.1756036258093201067742480797203249679677e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.1571910858154296875e2), SC_(-0.1423070182265885360371078443389686783143e0), + SC_(0.186112499795854091644287109375e-2), SC_(0.31483119964599609375e2), SC_(0.1064212124432705753393290287137815754319e0), + SC_(0.35840012133121490478515625e-2), SC_(0.553809732082299888134002685546875e-4), SC_(0.9650707876272019770134952457282249740739e0), + SC_(0.35840012133121490478515625e-2), SC_(0.69304020144045352935791015625e-4), SC_(0.9658467964630894046501757076218342927801e0), + SC_(0.35840012133121490478515625e-2), SC_(0.23264062474481761455535888671875e-3), SC_(0.9700478597469915082824318140123835180869e0), + SC_(0.35840012133121490478515625e-2), SC_(0.4480001516640186309814453125e-3), SC_(0.9723287476406665086749967542371269571234e0), + SC_(0.35840012133121490478515625e-2), SC_(0.5502865533344447612762451171875e-3), SC_(0.9730456263943909567987631022793046602603e0), + SC_(0.35840012133121490478515625e-2), SC_(0.19227224402129650115966796875e-2), SC_(0.9774175323141709829066403138557060923786e0), + SC_(0.35840012133121490478515625e-2), SC_(0.37370622158050537109375e-2), SC_(0.9797457898312132593210429697126849835518e0), + SC_(0.35840012133121490478515625e-2), SC_(0.47696642577648162841796875e-2), SC_(0.9806007189648424616267897116538953898626e0), + SC_(0.35840012133121490478515625e-2), SC_(0.1275280676782131195068359375e-1), SC_(0.9840289261106419257463048917206926912931e0), + SC_(0.35840012133121490478515625e-2), SC_(0.20440109074115753173828125e-1), SC_(0.9856314206424828761679016945348314437259e0), + SC_(0.35840012133121490478515625e-2), SC_(0.3429813683032989501953125e-1), SC_(0.9872749179206597028494401845390763149145e0), + SC_(0.35840012133121490478515625e-2), SC_(0.96701286733150482177734375e-1), SC_(0.988932179910175424589294675055202246355e0), + SC_(0.35840012133121490478515625e-2), SC_(0.159812271595001220703125e0), SC_(0.9867184436488505284336133268536754401756e0), + SC_(0.35840012133121490478515625e-2), SC_(0.297095477581024169921875e0), SC_(0.9734727764095169166354550030712991853188e0), + SC_(0.35840012133121490478515625e-2), SC_(0.77344071865081787109375e0), SC_(0.8552988701003771721146721734619235915706e0), + SC_(0.35840012133121490478515625e-2), SC_(0.1992881298065185546875e1), SC_(0.2308618505494366249905008758109493850602e0), + SC_(0.35840012133121490478515625e-2), SC_(0.3915013790130615234375e1), SC_(-0.4012691147847807908816435949894823168188e0), + SC_(0.35840012133121490478515625e-2), SC_(0.79858455657958984375e1), SC_(0.1762002663351808150861701687808126150305e0), + SC_(0.35840012133121490478515625e-2), SC_(0.1571910858154296875e2), SC_(-0.1419216381983571259124926406653600173921e0), + SC_(0.35840012133121490478515625e-2), SC_(0.31483119964599609375e2), SC_(0.1061656276058522622372436370913678330655e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.553809732082299888134002685546875e-4), SC_(0.9572656431203538165112750398451169576976e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.69304020144045352935791015625e-4), SC_(0.9582112064201335261630997112342307287719e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.23264062474481761455535888671875e-3), SC_(0.9633331943281124926230864636692210997818e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.4480001516640186309814453125e-3), SC_(0.9661162109066795212283435150515623357621e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.5502865533344447612762451171875e-3), SC_(0.9669912189279992303843912808888387671473e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.19227224402129650115966796875e-2), SC_(0.9723308178532373593917839240834483621504e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.37370622158050537109375e-2), SC_(0.9751771194142703096718649204073811795873e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.47696642577648162841796875e-2), SC_(0.9762229407760513009389469194046638514027e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.1275280676782131195068359375e-1), SC_(0.9804245691270382739486838438029335140198e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.20440109074115753173828125e-1), SC_(0.9824004047135082246753499791797478879785e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.3429813683032989501953125e-1), SC_(0.984455534869400551529392629224463906169e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.96701286733150482177734375e-1), SC_(0.9869464641414984822321264268443877210374e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.159812271595001220703125e0), SC_(0.9851453236176178426246334004315599658192e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.297095477581024169921875e0), SC_(0.9724266380293597530681971818509394699107e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.77344071865081787109375e0), SC_(0.8551479374909868558451585757714077368885e0), + SC_(0.44022924266755580902099609375e-2), SC_(0.1992881298065185546875e1), SC_(0.2315143366636852187122636336456783890548e0), + 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SC_(0.1992881298065185546875e1), SC_(0.9905301122130665466055237290421121475096e-89), + SC_(0.638867645263671875e2), SC_(0.3915013790130615234375e1), SC_(0.5149539828266276213224902637532044041335e-70), + SC_(0.638867645263671875e2), SC_(0.79858455657958984375e1), SC_(0.2564876610067021281696593047029893911603e-50), + SC_(0.638867645263671875e2), SC_(0.1571910858154296875e2), SC_(0.7745090555307013050588237027600011961354e-32), + SC_(0.638867645263671875e2), SC_(0.31483119964599609375e2), SC_(0.7349468722273911955644415269211426919788e-14) + }; +#undef SC_ + + diff --git a/test/bessel_j_int_data.ipp b/test/bessel_j_int_data.ipp new file mode 100644 index 000000000..2646fc4e7 --- /dev/null +++ b/test/bessel_j_int_data.ipp @@ -0,0 +1,235 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 224> bessel_j_int_data = { + SC_(0), SC_(0.1730655412757187150418758392333984375e-5), SC_(0.999999999999251207960573702716351702855e0), + SC_(0), SC_(0.216575062950141727924346923828125e-5), SC_(0.9999999999988273810527038974545295287162e0), + SC_(0), SC_(0.72700195232755504548549652099609375e-5), SC_(0.9999999999867867040328417323221266089012e0), + SC_(0), SC_(0.14000004739500582218170166015625e-4), SC_(0.9999999999509999668240905595691881208684e0), + SC_(0), SC_(0.17196454791701398789882659912109375e-4), SC_(0.9999999999260704856506088935623108659333e0), + SC_(0), SC_(0.60085076256655156612396240234375e-4), SC_(0.9999999990974459030116346626772683432438e0), + SC_(0), SC_(0.116783194243907928466796875e-3), SC_(0.9999999965904213884537241019212664253672e0), + SC_(0), SC_(0.149052008055150508880615234375e-3), SC_(0.9999999944458747313939138453006009257493e0), + SC_(0), SC_(0.3985252114944159984588623046875e-3), SC_(0.9999999602944143449661387732367591439684e0), + SC_(0), SC_(0.63875340856611728668212890625e-3), SC_(0.999999897998523362367123063793585129913e0), + SC_(0), SC_(0.10718167759478092193603515625e-2), SC_(0.999999712802220319854399500739679836503e0), + SC_(0), SC_(0.302191521041095256805419921875e-2), SC_(0.9999977170084182855294667634808613841995e0), + SC_(0), SC_(0.499413348734378814697265625e-2), SC_(0.9999937646673975145784086810883184951737e0), + SC_(0), SC_(0.928423367440700531005859375e-2), SC_(0.9999784508673620039917059777810212577274e0), + SC_(0), SC_(0.241700224578380584716796875e-1), SC_(0.9998539578359781705487991537766139314417e0), + SC_(0), SC_(0.6227754056453704833984375e-1), SC_(0.9990306120021848197073447709696842002313e0), + SC_(0), SC_(0.12234418094158172607421875e0), SC_(0.9962614745793813396644750956695984663693e0), + SC_(0), SC_(0.249557673931121826171875e0), SC_(0.9844907414458467656383892255940207494775e0), + SC_(0), SC_(0.4912221431732177734375e0), SC_(0.9405788968099850208754506401326367253837e0), + SC_(0), SC_(0.98384749889373779296875e0), SC_(0.7722629604360946893704391598535972411534e0), + SC_(0), SC_(0.11576130390167236328125e1), SC_(0.69201921296869073470763460902488953195e0), + SC_(0), SC_(0.3451677799224853515625e1), SC_(-0.3729999217923199419780408144861949608561e0), + SC_(0), SC_(0.788237094879150390625e1), SC_(0.1981996630505566917360863478152806154613e0), + SC_(0), SC_(0.15848876953125e2), SC_(-0.1592255309532503835966920565319209366427e0), + SC_(0.1e1), SC_(0.553809732082299888134002685546875e-4), SC_(0.276904865934989734483098712606972348472e-4), + SC_(0.1e1), SC_(0.69304020144045352935791015625e-4), SC_(0.3465201005121827144301015458593023209872e-4), + SC_(0.1e1), SC_(0.23264062474481761455535888671875e-3), SC_(0.1163203115854777552927670265943683111663e-3), + SC_(0.1e1), SC_(0.4480001516640186309814453125e-3), SC_(0.224000070212291655064298940047799298227e-3), + SC_(0.1e1), SC_(0.5502865533344447612762451171875e-3), SC_(0.2751432662525235957510064043924201271744e-3), + SC_(0.1e1), SC_(0.19227224402129650115966796875e-2), SC_(0.9613607758541307952398436842542503763082e-3), + SC_(0.1e1), SC_(0.37370622158050537109375e-2), SC_(0.1868527846001727517468459203726183793639e-2), + SC_(0.1e1), SC_(0.47696642577648162841796875e-2), SC_(0.2384825347112756329700145763683641710414e-2), + SC_(0.1e1), SC_(0.1275280676782131195068359375e-1), SC_(0.6376273757226442847790070686813990559145e-2), + SC_(0.1e1), SC_(0.20440109074115753173828125e-1), SC_(0.102195208064807658405217565669384689625e-1), + SC_(0.1e1), SC_(0.3429813683032989501953125e-1), SC_(0.1714654684930302330423492169172745899688e-1), + SC_(0.1e1), SC_(0.96701286733150482177734375e-1), SC_(0.4829414868544472860291893155827576361067e-1), + SC_(0.1e1), SC_(0.159812271595001220703125e0), SC_(0.7965130716107429048780909632824084802407e-1), + SC_(0.1e1), SC_(0.297095477581024169921875e0), SC_(0.1469147961900711346572957228234698754028e0), + SC_(0.1e1), SC_(0.77344071865081787109375e0), SC_(0.3585147006385292414727602076757818503575e0), + SC_(0.1e1), SC_(0.1992881298065185546875e1), SC_(0.5771736137851551766101983376499833110574e0), + SC_(0.1e1), SC_(0.3915013790130615234375e1), SC_(-0.3315763536880066424018546732060494035118e-1), + SC_(0.1e1), SC_(0.79858455657958984375e1), SC_(0.2325970018905132078980792992006627584192e0), + SC_(0.1e1), SC_(0.1571910858154296875e2), SC_(0.1373449658609949106151643744232618350175e0), + SC_(0.1e1), SC_(0.31483119964599609375e2), SC_(-0.9230877498012981860222086238303332873573e-1), + SC_(0.4e1), SC_(0.553809732082299888134002685546875e-4), SC_(0.2449689884414968964700618717685524683171e-19), + SC_(0.4e1), SC_(0.69304020144045352935791015625e-4), SC_(0.6007620436965929637530930219443973179641e-19), + SC_(0.4e1), SC_(0.23264062474481761455535888671875e-3), SC_(0.7628005478429104492672012434718507350272e-17), + SC_(0.4e1), SC_(0.4480001516640186309814453125e-3), SC_(0.1049014316653821463646665194851778246401e-15), + SC_(0.4e1), SC_(0.5502865533344447612762451171875e-3), SC_(0.2387945284954097519631184175866877863264e-15), + SC_(0.4e1), SC_(0.19227224402129650115966796875e-2), SC_(0.3559058080985751663501872629042355217152e-13), + SC_(0.4e1), SC_(0.37370622158050537109375e-2), SC_(0.5079135337601192358742525894357081341335e-12), + SC_(0.4e1), SC_(0.47696642577648162841796875e-2), SC_(0.1347781590846459623440987284254345472313e-11), + SC_(0.4e1), SC_(0.1275280676782131195068359375e-1), SC_(0.6887924230376394398803439333202892325557e-10), + SC_(0.4e1), SC_(0.20440109074115753173828125e-1), SC_(0.4545613845374467200233204286239376175147e-9), + SC_(0.4e1), SC_(0.3429813683032989501953125e-1), SC_(0.3603506796835261381553751482886494436605e-8), + SC_(0.4e1), SC_(0.96701286733150482177734375e-1), SC_(0.2276117732048184291309579825404846223101e-6), + SC_(0.4e1), SC_(0.159812271595001220703125e0), SC_(0.1696502961325871806891524286212542749233e-5), + SC_(0.4e1), SC_(0.297095477581024169921875e0), SC_(0.2019926506375704595965844529208340579256e-4), + SC_(0.4e1), SC_(0.77344071865081787109375e0), SC_(0.9043871099795724296870802463629379435165e-3), + SC_(0.4e1), SC_(0.1992881298065185546875e1), SC_(0.3356363239882888737614531598548044804195e-1), + SC_(0.4e1), SC_(0.3915013790130615234375e1), SC_(0.2683403769106168710689421055914199402732e0), + SC_(0.4e1), SC_(0.79858455657958984375e1), SC_(-0.1019714387996719292888771314960256244029e0), + SC_(0.4e1), SC_(0.1571910858154296875e2), SC_(-0.1970617873757679766451608652880933303855e0), + SC_(0.4e1), SC_(0.31483119964599609375e2), SC_(0.1274270375173584709656744808050405443416e0), + SC_(0.7e1), SC_(0.553809732082299888134002685546875e-4), SC_(0.2476758015173796525820991581584477984471e-35), + SC_(0.7e1), SC_(0.69304020144045352935791015625e-4), SC_(0.1190333036597015037788087962950721904204e-34), + SC_(0.7e1), SC_(0.23264062474481761455535888671875e-3), SC_(0.5716870852680826502230510205353692076762e-31), + SC_(0.7e1), SC_(0.4480001516640186309814453125e-3), SC_(0.561444224078444008798450658023844580616e-29), + SC_(0.7e1), SC_(0.5502865533344447612762451171875e-3), SC_(0.2368545839293556062239215049355950751334e-28), + SC_(0.7e1), SC_(0.19227224402129650115966796875e-2), SC_(0.1505828833808392807660517316033033280684e-24), + SC_(0.7e1), SC_(0.37370622158050537109375e-2), SC_(0.1577871390165051118291058054008029867733e-22), + SC_(0.7e1), SC_(0.47696642577648162841796875e-2), SC_(0.8705101764884624738964439520067718158933e-22), + SC_(0.7e1), SC_(0.1275280676782131195068359375e-1), SC_(0.8503500497157549233110373576875716028951e-19), + SC_(0.7e1), SC_(0.20440109074115753173828125e-1), SC_(0.2310661278306557800428208671429361931635e-17), + SC_(0.7e1), SC_(0.3429813683032989501953125e-1), SC_(0.8654405276396266715021848331684138021568e-16), + SC_(0.7e1), SC_(0.96701286733150482177734375e-1), SC_(0.1225344911694889641798963820236998213373e-12), + SC_(0.7e1), SC_(0.159812271595001220703125e0), SC_(0.4123668359663111441088023707096475138751e-11), + SC_(0.7e1), SC_(0.297095477581024169921875e0), SC_(0.3158156744651217917953769670307028606449e-9), + SC_(0.7e1), SC_(0.77344071865081787109375e0), SC_(0.251896150858704683595843840399835306166e-6), + SC_(0.7e1), SC_(0.1992881298065185546875e1), SC_(0.1707853166377511336641576065559156632457e-3), + SC_(0.7e1), SC_(0.3915013790130615234375e1), SC_(0.1335220837033977061080155342538527713965e-1), + SC_(0.7e1), SC_(0.79858455657958984375e1), SC_(0.3197732082032941630469007231253521918143e0), + SC_(0.7e1), SC_(0.1571910858154296875e2), SC_(0.1523464020656448663148336960516768275145e0), + SC_(0.7e1), SC_(0.31483119964599609375e2), SC_(-0.8465752063992111585817831175823765390968e-2), + SC_(0.1e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.7303698611664865207490004658067936181905e-52), + SC_(0.1e2), SC_(0.69304020144045352935791015625e-4), SC_(0.6878936281621022752851314176441951633029e-51), + SC_(0.1e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.1249662001772833555544044498679159797243e-45), + SC_(0.1e2), SC_(0.4480001516640186309814453125e-3), SC_(0.87643279255017773967807642045361302034e-43), + SC_(0.1e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.6852136624817352900940351580328586337538e-42), + SC_(0.1e2), SC_(0.19227224402129650115966796875e-2), SC_(0.185824479115402751843849340758426235854e-36), + SC_(0.1e2), SC_(0.37370622158050537109375e-2), SC_(0.1429684321223564675360504319868530544464e-33), + SC_(0.1e2), SC_(0.47696642577648162841796875e-2), SC_(0.1639890621444124727733839773586431172039e-32), + SC_(0.1e2), SC_(0.1275280676782131195068359375e-1), SC_(0.306191547516836368220168038780704053939e-28), + SC_(0.1e2), SC_(0.20440109074115753173828125e-1), SC_(0.3425823439848955129914050150766420797628e-26), + SC_(0.1e2), SC_(0.3429813683032989501953125e-1), SC_(0.6062205935674056071761047413439550035471e-24), + SC_(0.1e2), SC_(0.96701286733150482177734375e-1), SC_(0.1923832425983654541213096146081906580397e-19), + SC_(0.1e2), SC_(0.159812271595001220703125e0), SC_(0.2922712946223807891584380230043171121063e-17), + SC_(0.1e2), SC_(0.297095477581024169921875e0), SC_(0.143888516723819817784418267406968074987e-14), + SC_(0.1e2), SC_(0.77344071865081787109375e0), SC_(0.2033758816108020473655970022244540004408e-10), + SC_(0.1e2), SC_(0.1992881298065185546875e1), SC_(0.242885590788180736747780657352089100395e-6), + SC_(0.1e2), SC_(0.3915013790130615234375e1), SC_(0.1598475267744058553357687546625030272909e-3), + SC_(0.1e2), SC_(0.79858455657958984375e1), SC_(0.60056972136838046146179075502942356573e-1), + SC_(0.1e2), SC_(0.1571910858154296875e2), SC_(-0.1842232590397531945311647766302288337231e0), + SC_(0.1e2), SC_(0.31483119964599609375e2), SC_(-0.9317261959891319658027121347121001915601e-1), + SC_(0.13e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.90368551918725968389007265700993262299e-69), + SC_(0.13e2), SC_(0.69304020144045352935791015625e-4), SC_(0.1667974085958167624378345556115399941477e-67), + SC_(0.13e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.1146151324192891686284768890985563371943e-60), + SC_(0.13e2), SC_(0.4480001516640186309814453125e-3), SC_(0.5740448587965102107018898696167719687858e-57), + SC_(0.13e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.8317358972729155842535186608238358987262e-56), + SC_(0.13e2), SC_(0.19227224402129650115966796875e-2), SC_(0.9621558980749171469364024498494388216795e-49), + SC_(0.13e2), SC_(0.37370622158050537109375e-2), SC_(0.5435304733189136019244042387714022185558e-45), + SC_(0.13e2), SC_(0.47696642577648162841796875e-2), SC_(0.1296197228784403471129989731886036646855e-43), + SC_(0.13e2), SC_(0.1275280676782131195068359375e-1), SC_(0.4625978489147388510480567448018715189249e-38), + SC_(0.13e2), SC_(0.20440109074115753173828125e-1), SC_(0.2131121530101465666113363379817102327031e-35), + SC_(0.13e2), SC_(0.3429813683032989501953125e-1), SC_(0.1781711933682478816925141204995206372152e-32), + SC_(0.13e2), SC_(0.96701286733150482177734375e-1), SC_(0.1267291457925455878167599468504869327715e-26), + SC_(0.13e2), SC_(0.159812271595001220703125e0), SC_(0.8690870700737698910678361841372229152816e-24), + SC_(0.13e2), SC_(0.297095477581024169921875e0), SC_(0.2749753181518427631420704990495067851464e-20), + SC_(0.13e2), SC_(0.77344071865081787109375e0), SC_(0.6874485945761929731188960123719668004339e-15), + SC_(0.13e2), SC_(0.1992881298065185546875e1), SC_(0.1427954947280088715244535777966915467592e-9), + SC_(0.13e2), SC_(0.3915013790130615234375e1), SC_(0.7548925903672885635112989058254310604162e-6), + SC_(0.13e2), SC_(0.79858455657958984375e1), SC_(0.3214379121675310487741347594390187230881e-2), + SC_(0.13e2), SC_(0.1571910858154296875e2), SC_(0.2562871752817527993867029711304921017067e0), + SC_(0.13e2), SC_(0.31483119964599609375e2), SC_(0.135452434319841135048770973964056570279e0), + SC_(0.16e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.5710443102859793594686693457798971291695e-86), + SC_(0.16e2), SC_(0.69304020144045352935791015625e-4), SC_(0.2065548122783126765855808525012370612477e-84), + SC_(0.16e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.5368702794457035352466622003877382747158e-76), + SC_(0.16e2), SC_(0.4480001516640186309814453125e-3), SC_(0.1920220273734407205690092457873836811378e-71), + SC_(0.16e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.5156118426330840464377189920132922032504e-70), + SC_(0.16e2), SC_(0.19227224402129650115966796875e-2), SC_(0.2544286257456308178600141145465753228241e-61), + SC_(0.16e2), SC_(0.37370622158050537109375e-2), SC_(0.1055323568504563665150514946643052879562e-56), + SC_(0.16e2), SC_(0.47696642577648162841796875e-2), SC_(0.5232452376007528048513570920264122109991e-55), + SC_(0.16e2), SC_(0.1275280676782131195068359375e-1), SC_(0.3569372759580808303489639154371020886878e-48), + SC_(0.16e2), SC_(0.20440109074115753173828125e-1), SC_(0.6770631088264124985080490485409279874034e-45), + SC_(0.16e2), SC_(0.3429813683032989501953125e-1), SC_(0.2674369558860517619389933396711866115561e-41), + SC_(0.16e2), SC_(0.96701286733150482177734375e-1), SC_(0.4263407059281994448315250268426546365419e-34), + SC_(0.16e2), SC_(0.159812271595001220703125e0), SC_(0.1319773345595717550011326018119444329018e-30), + SC_(0.16e2), SC_(0.297095477581024169921875e0), SC_(0.2683325671969811781305622600203263139886e-26), + SC_(0.16e2), SC_(0.77344071865081787109375e0), SC_(0.1185527072917230718908112159994283095826e-19), + SC_(0.16e2), SC_(0.1992881298065185546875e1), SC_(0.4257913764415272936226850400127709560274e-13), + SC_(0.16e2), SC_(0.3915013790130615234375e1), SC_(0.1770596374018406424364588196143722326332e-8), + SC_(0.16e2), SC_(0.79858455657958984375e1), SC_(0.7609462131466514170142031981046135155552e-4), + SC_(0.16e2), SC_(0.1571910858154296875e2), SC_(0.1598242323049512757290685523586043599273e0), + SC_(0.16e2), SC_(0.31483119964599609375e2), SC_(-0.146616929844240669724265072627562678578e0), + SC_(0.19e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.2085386435150050836259354457009000018077e-103), + SC_(0.19e2), SC_(0.69304020144045352935791015625e-4), SC_(0.1478242165594325382694493377958682108845e-101), + SC_(0.19e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.1453319211512903213205953436626128066329e-91), + SC_(0.19e2), SC_(0.4480001516640186309814453125e-3), SC_(0.3712107285598227397108662690263089748287e-86), + SC_(0.19e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.1847245305223723518947855245367191928343e-84), + SC_(0.19e2), SC_(0.19227224402129650115966796875e-2), SC_(0.3888219252235519964245528947552171853277e-74), + SC_(0.19e2), SC_(0.37370622158050537109375e-2), SC_(0.1184163364577840228862020640686836979404e-68), + SC_(0.19e2), SC_(0.47696642577648162841796875e-2), SC_(0.1220685316325757321764623390231984663671e-66), + SC_(0.19e2), SC_(0.1275280676782131195068359375e-1), SC_(0.1591638310631743076506768378115511870553e-58), + SC_(0.19e2), SC_(0.20440109074115753173828125e-1), SC_(0.1243123177590827411934225161624922624915e-54), + SC_(0.19e2), SC_(0.3429813683032989501953125e-1), SC_(0.2319899218435133188992825576466373416419e-50), + SC_(0.19e2), SC_(0.96701286733150482177734375e-1), SC_(0.8288911503438383816925041924523248045712e-42), + SC_(0.19e2), SC_(0.159812271595001220703125e0), SC_(0.115821498784548972023307498969902449394e-37), + SC_(0.19e2), SC_(0.297095477581024169921875e0), SC_(0.1513146789392266259407994247748973151373e-32), + SC_(0.19e2), SC_(0.77344071865081787109375e0), SC_(0.1180867041363348335303085045154147913528e-24), + SC_(0.19e2), SC_(0.1992881298065185546875e1), SC_(0.7309651660473666120687072805302589654808e-17), + SC_(0.19e2), SC_(0.3915013790130615234375e1), SC_(0.2364159870071755910484486348587559990813e-11), + SC_(0.19e2), SC_(0.79858455657958984375e1), SC_(0.9689789089192189055495551850283928885267e-6), + SC_(0.19e2), SC_(0.1571910858154296875e2), SC_(0.28988183636629975718412063879767316272e-1), + SC_(0.19e2), SC_(0.31483119964599609375e2), SC_(0.1404351655526923900119491455044344208598e0), + SC_(0.22e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.4791884438276096956171867526066745487233e-121), + SC_(0.22e2), SC_(0.69304020144045352935791015625e-4), SC_(0.6656698862399030252104289006319409158079e-119), + SC_(0.22e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.2475458914860796896424360165211198402136e-107), + SC_(0.22e2), SC_(0.4480001516640186309814453125e-3), SC_(0.4515366894275771826117281426200239790956e-101), + SC_(0.22e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.4164178287432561100733955741346219769583e-99), + SC_(0.22e2), SC_(0.19227224402129650115966796875e-2), SC_(0.3738854595795469765521076338466195179729e-87), + SC_(0.22e2), SC_(0.37370622158050537109375e-2), SC_(0.8360661820524098792008512761880716972576e-81), + SC_(0.22e2), SC_(0.47696642577648162841796875e-2), SC_(0.1791864672483119204014438872980421431477e-78), + SC_(0.22e2), SC_(0.1275280676782131195068359375e-1), SC_(0.4465806266019354303051149660509529893234e-69), + SC_(0.22e2), SC_(0.20440109074115753173828125e-1), SC_(0.1436157691171859834871864549854120137171e-64), + SC_(0.22e2), SC_(0.3429813683032989501953125e-1), SC_(0.1266250352189026976463935184908537726894e-59), + SC_(0.22e2), SC_(0.96701286733150482177734375e-1), SC_(0.1014002264944669915985752126502534659735e-49), + SC_(0.22e2), SC_(0.159812271595001220703125e0), SC_(0.6395517348407679453466968468982951315152e-45), + SC_(0.22e2), SC_(0.297095477581024169921875e0), SC_(0.5368707332343228506070497601994580522396e-39), + SC_(0.22e2), SC_(0.77344071865081787109375e0), SC_(0.7398504152815100288293238816236231544857e-30), + SC_(0.22e2), SC_(0.1992881298065185546875e1), SC_(0.7877708523407809449378036883914301743673e-21), + SC_(0.22e2), SC_(0.3915013790130615234375e1), SC_(0.1968332845994952491334209201789498977283e-14), + SC_(0.22e2), SC_(0.79858455657958984375e1), SC_(0.7448464245775065475039475038584227288619e-8), + SC_(0.22e2), SC_(0.1571910858154296875e2), SC_(0.253851170309502550500960913045849958251e-2), + SC_(0.22e2), SC_(0.31483119964599609375e2), SC_(-0.8447893849187497539115391467530479019211e-1), + SC_(0.25e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.7372571830959398154087818458405571713543e-139), + SC_(0.25e2), SC_(0.69304020144045352935791015625e-4), SC_(0.2007082018704239928877827931765826257223e-136), + SC_(0.25e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.282321085875648461583736420544653624557e-123), + SC_(0.25e2), SC_(0.4480001516640186309814453125e-3), SC_(0.3677548883718449906250285120192357358526e-116), + SC_(0.25e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.6285313494759937735919237336354498066612e-114), + SC_(0.25e2), SC_(0.19227224402129650115966796875e-2), SC_(0.240723943542042702532722602099509196343e-100), + SC_(0.25e2), SC_(0.37370622158050537109375e-2), SC_(0.3952415336587423864493642493686435620558e-93), + SC_(0.25e2), SC_(0.47696642577648162841796875e-2), SC_(0.1761163090097583722535523692248334183587e-90), + SC_(0.25e2), SC_(0.1275280676782131195068359375e-1), SC_(0.838973477506519849649546640592049467827e-80), + SC_(0.25e2), SC_(0.20440109074115753173828125e-1), SC_(0.1110920317250862811627432685131862921107e-74), + SC_(0.25e2), SC_(0.3429813683032989501953125e-1), SC_(0.4627673735504195531564801614260252853823e-69), + SC_(0.25e2), SC_(0.96701286733150482177734375e-1), SC_(0.830561324288197807079744142399604541678e-58), + SC_(0.25e2), SC_(0.159812271595001220703125e0), SC_(0.236456302063194497551168460901108346276e-52), + SC_(0.25e2), SC_(0.297095477581024169921875e0), SC_(0.1275371942824651350612111302983155850293e-45), + SC_(0.25e2), SC_(0.77344071865081787109375e0), SC_(0.3103000869941400473392593242837244578633e-35), + SC_(0.25e2), SC_(0.1992881298065185546875e1), SC_(0.5676014969386446596753686198525691279888e-25), + SC_(0.25e2), SC_(0.3915013790130615234375e1), SC_(0.1090825721430814617531511126945649824391e-17), + SC_(0.25e2), SC_(0.79858455657958984375e1), SC_(0.3734200809691738263209576777640843925841e-10), + SC_(0.25e2), SC_(0.1571910858154296875e2), SC_(0.1291120784936711314245621877597952164143e-3), + SC_(0.25e2), SC_(0.31483119964599609375e2), SC_(-0.7461962297933840624621930608252220218503e-1), + SC_(0.28e2), SC_(0.553809732082299888134002685546875e-4), SC_(0.7963713582106584412097140308116466491608e-157), + SC_(0.28e2), SC_(0.69304020144045352935791015625e-4), SC_(0.4248692230753312125913578638497606609336e-154), + SC_(0.28e2), SC_(0.23264062474481761455535888671875e-3), SC_(0.2260553821923174448109780057299133388939e-139), + SC_(0.28e2), SC_(0.4480001516640186309814453125e-3), SC_(0.2102847637959861796417097898716140319015e-131), + SC_(0.28e2), SC_(0.5502865533344447612762451171875e-3), SC_(0.6660525810149796134788024971104638208402e-129), + SC_(0.28e2), SC_(0.19227224402129650115966796875e-2), SC_(0.108813792122284445419802118573486294617e-113), + SC_(0.28e2), SC_(0.37370622158050537109375e-2), SC_(0.1311802444742152354234077144037248874522e-105), + SC_(0.28e2), SC_(0.47696642577648162841796875e-2), SC_(0.1215284288581256977996490020569407519253e-102), + SC_(0.28e2), SC_(0.1275280676782131195068359375e-1), SC_(0.1106574120737639697400622354620027455407e-90), + SC_(0.28e2), SC_(0.20440109074115753173828125e-1), SC_(0.6033198112217245150379653252026122716124e-85), + SC_(0.28e2), SC_(0.3429813683032989501953125e-1), SC_(0.1187379908091547229246861752037218903857e-78), + SC_(0.28e2), SC_(0.96701286733150482177734375e-1), SC_(0.4776253533970672888906190995310047058597e-66), + SC_(0.28e2), SC_(0.159812271595001220703125e0), SC_(0.6137721383009946630525985606230109474371e-60), + SC_(0.28e2), SC_(0.297095477581024169921875e0), SC_(0.2127051389870186274825552068510907743209e-52), + SC_(0.28e2), SC_(0.77344071865081787109375e0), SC_(0.9135584911756594671676404048881870523782e-41), + SC_(0.28e2), SC_(0.1992881298065185546875e1), SC_(0.2868280579406050057547310818758384973257e-29), + SC_(0.28e2), SC_(0.3915013790130615234375e1), SC_(0.4227077578718476642466348871890940509389e-21), + SC_(0.28e2), SC_(0.79858455657958984375e1), SC_(0.1291218836633488658985438527652338235293e-12), + SC_(0.28e2), SC_(0.1571910858154296875e2), SC_(0.4222984862431763788622725393431592171924e-5), + SC_(0.28e2), SC_(0.31483119964599609375e2), SC_(0.1996895117755311022877682593279374977116e0) + }; +#undef SC_ + + diff --git a/test/bessel_j_large_data.ipp b/test/bessel_j_large_data.ipp new file mode 100644 index 000000000..28867761e --- /dev/null +++ b/test/bessel_j_large_data.ipp @@ -0,0 +1,81 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 70> bessel_j_large_data = { + SC_(0.725889492034912109375e1), SC_(0.725889492034912109375e1), SC_(0.2307850131230964328215158373258970201933e0), + SC_(0.725889492034912109375e1), SC_(0.90838165283203125e1), SC_(0.331270090805281544869225081474258975783e0), + SC_(0.725889492034912109375e1), SC_(0.30492671966552734375e2), SC_(0.1392044369140870127137144436194280008198e0), + SC_(0.725889492034912109375e1), SC_(0.5872027587890625e2), SC_(-0.1034372792561996393994604565745500159306e0), + SC_(0.725889492034912109375e1), SC_(0.7212715911865234375e2), SC_(-0.7703453431004507456085858477671512851731e-1), + SC_(0.725889492034912109375e1), SC_(0.25201507568359375e3), SC_(0.1959284506006571693892495807977692143684e-1), + SC_(0.725889492034912109375e1), SC_(0.48982421875e3), SC_(0.3554545809249573170321495025893136154241e-1), + SC_(0.725889492034912109375e1), SC_(0.62516943359375e3), SC_(-0.2922687275543058917680294015955588486854e-1), + SC_(0.725889492034912109375e1), SC_(0.1671535888671875e4), SC_(0.1607375857460556311523938125070882107754e-1), + SC_(0.725889492034912109375e1), SC_(0.26791259765625e4), SC_(-0.1488078416329900906439423002725322373542e-1), + SC_(0.725889492034912109375e1), SC_(0.4495525390625e4), SC_(-0.1140024419570086978904733358200459324628e-1), + SC_(0.725889492034912109375e1), SC_(0.126748310546875e5), SC_(-0.3122722904155054704789731376588374939888e-2), + SC_(0.725889492034912109375e1), SC_(0.209469140625e5), SC_(0.3655895534999378610724150587348023983602e-2), + SC_(0.725889492034912109375e1), SC_(0.389408984375e5), SC_(-0.1325113645945403697015197310014126427368e-2), + SC_(0.90838165283203125e1), SC_(0.725889492034912109375e1), SC_(0.6868013625729672215673766261089739717914e-1), + SC_(0.90838165283203125e1), SC_(0.90838165283203125e1), SC_(0.2142196934269988501665220845763503388042e0), + SC_(0.90838165283203125e1), SC_(0.30492671966552734375e2), SC_(-0.6854165161240333873733646364329358362341e-1), + SC_(0.90838165283203125e1), SC_(0.5872027587890625e2), SC_(0.9706365272921634392020123576878096972323e-1), + SC_(0.90838165283203125e1), SC_(0.7212715911865234375e2), SC_(0.4318373430471325672009672522183904089517e-1), + SC_(0.90838165283203125e1), SC_(0.25201507568359375e3), SC_(-0.3323948817558893756314993595448795803537e-2), + SC_(0.90838165283203125e1), SC_(0.48982421875e3), SC_(-0.3208784009186279824559250801925804018465e-1), + SC_(0.90838165283203125e1), SC_(0.62516943359375e3), SC_(0.2415910850536018880522735141575176512777e-1), + SC_(0.90838165283203125e1), SC_(0.1671535888671875e4), SC_(-0.1232973481208096922064985021712946282434e-1), + SC_(0.90838165283203125e1), SC_(0.26791259765625e4), SC_(0.1541293552432680038181584370193664107473e-1), + SC_(0.90838165283203125e1), SC_(0.4495525390625e4), SC_(0.1002388002178436742775036807597412212988e-1), + SC_(0.90838165283203125e1), SC_(0.126748310546875e5), SC_(0.4739251557631330719337662497765662043458e-2), + SC_(0.90838165283203125e1), SC_(0.209469140625e5), SC_(-0.4641170982395964367850932352617365960394e-2), + SC_(0.90838165283203125e1), SC_(0.389408984375e5), SC_(0.2364161829230146293999578445914286938024e-3), + SC_(0.30492671966552734375e2), SC_(0.725889492034912109375e1), SC_(0.5389197635915806233390841938474448086338e-16), + SC_(0.30492671966552734375e2), SC_(0.90838165283203125e1), SC_(0.3952459558161787055937696735475346378253e-13), + SC_(0.30492671966552734375e2), SC_(0.30492671966552734375e2), SC_(0.1431568992863962918440953734528390055733e0), + SC_(0.30492671966552734375e2), SC_(0.5872027587890625e2), SC_(0.8578089143574435265609743500241838450074e-1), + SC_(0.30492671966552734375e2), SC_(0.7212715911865234375e2), SC_(0.1410641386118672963443531500494113618346e-1), + SC_(0.30492671966552734375e2), SC_(0.25201507568359375e3), SC_(-0.2831444492336412069372220459848882424058e-1), + SC_(0.30492671966552734375e2), SC_(0.48982421875e3), SC_(-0.2314909531023927534092551249336846514091e-1), + SC_(0.30492671966552734375e2), SC_(0.62516943359375e3), SC_(0.2170848675622913630936132875332654675284e-1), + SC_(0.30492671966552734375e2), SC_(0.1671535888671875e4), SC_(-0.9325685551001457451901386831696429723505e-2), + SC_(0.30492671966552734375e2), SC_(0.26791259765625e4), SC_(-0.6947960924639771467867037818061032165711e-2), + SC_(0.30492671966552734375e2), SC_(0.4495525390625e4), SC_(0.2522782171073615304290664333918949926448e-3), + SC_(0.30492671966552734375e2), SC_(0.126748310546875e5), SC_(-0.7032986459719800448169085872312178500116e-2), + SC_(0.30492671966552734375e2), SC_(0.209469140625e5), SC_(0.5122201654532113914038286303179605549787e-2), + SC_(0.30492671966552734375e2), SC_(0.389408984375e5), SC_(0.3118763746002366768916876665037341197225e-2), + SC_(0.5872027587890625e2), SC_(0.725889492034912109375e1), SC_(0.1355459268067898658151384249990581267605e-46), + SC_(0.5872027587890625e2), SC_(0.90838165283203125e1), SC_(0.6263805301401313877353888242546513061021e-41), + SC_(0.5872027587890625e2), SC_(0.30492671966552734375e2), SC_(0.1202543691751492070742433226952612421088e-11), + SC_(0.5872027587890625e2), SC_(0.5872027587890625e2), SC_(0.1150759176943336800292116211528794948354e0), + SC_(0.5872027587890625e2), SC_(0.7212715911865234375e2), SC_(-0.1047430440042659670890981946453377232975e-2), + SC_(0.5872027587890625e2), SC_(0.25201507568359375e3), SC_(-0.4086747435836167042364362710950743904056e-1), + SC_(0.5872027587890625e2), SC_(0.48982421875e3), SC_(-0.8183965550065315841115618961272005737216e-2), + SC_(0.5872027587890625e2), SC_(0.62516943359375e3), SC_(0.2146006391006581305906226294562488601133e-1), + SC_(0.5872027587890625e2), SC_(0.1671535888671875e4), SC_(-0.1521907719127834149642973426124244709606e-1), + SC_(0.5872027587890625e2), SC_(0.26791259765625e4), SC_(-0.5359901862639028443437495673595828567149e-2), + SC_(0.5872027587890625e2), SC_(0.4495525390625e4), SC_(-0.6687847432133282099926443966258547872974e-3), + SC_(0.5872027587890625e2), SC_(0.126748310546875e5), SC_(-0.7023133406228636526878960014610930229604e-2), + SC_(0.5872027587890625e2), SC_(0.209469140625e5), SC_(0.5494684780269989275230320303506992388372e-2), + SC_(0.5872027587890625e2), SC_(0.389408984375e5), SC_(0.2133199464416083487500787075862183481877e-2), + SC_(0.7212715911865234375e2), SC_(0.725889492034912109375e1), SC_(0.18957209567263693085259721077483394587e-63), + SC_(0.7212715911865234375e2), SC_(0.90838165283203125e1), SC_(0.18129450849783963899751662717930347867e-56), + SC_(0.7212715911865234375e2), SC_(0.30492671966552734375e2), SC_(0.799552979179394942566748705329500029705e-20), + SC_(0.7212715911865234375e2), SC_(0.5872027587890625e2), SC_(0.1552366431895120369074625091623203198931e-3), + SC_(0.7212715911865234375e2), SC_(0.7212715911865234375e2), SC_(0.1074534531973475595081469483471348487375e0), + SC_(0.7212715911865234375e2), SC_(0.25201507568359375e3), SC_(-0.4021841759432511100331858349923571437074e-1), + SC_(0.7212715911865234375e2), SC_(0.48982421875e3), SC_(-0.2170518020212747257681494905588835722404e-1), + SC_(0.7212715911865234375e2), SC_(0.62516943359375e3), SC_(0.3200282452240180595697263016953209545847e-1), + SC_(0.7212715911865234375e2), SC_(0.1671535888671875e4), SC_(0.1388345457492732160509267345077835744377e-1), + SC_(0.7212715911865234375e2), SC_(0.26791259765625e4), SC_(-0.1211552634023650615661641900044154973817e-1), + SC_(0.7212715911865234375e2), SC_(0.4495525390625e4), SC_(-0.1044260623905249384981833094590727496255e-1), + SC_(0.7212715911865234375e2), SC_(0.126748310546875e5), SC_(0.4589655631278176598474380175755836047107e-2), + SC_(0.7212715911865234375e2), SC_(0.209469140625e5), SC_(-0.2719723241223086004132685612975137005354e-2), + SC_(0.7212715911865234375e2), SC_(0.389408984375e5), SC_(-0.4035830877477336799210808243472842844322e-2) + }; +#undef SC_ + + diff --git a/test/bessel_k_data.ipp b/test/bessel_k_data.ipp new file mode 100644 index 000000000..9c465cfed --- /dev/null +++ b/test/bessel_k_data.ipp @@ -0,0 +1,274 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 263> bessel_k_data = { + SC_(-0.8049192047119140625e2), SC_(0.24750102996826171875e2), SC_(0.6579017807810652710369517871806355927214e29), + SC_(-0.8049192047119140625e2), SC_(0.637722015380859375e2), SC_(0.2395518238062557960566710371847643552469e-8), + SC_(-0.8049192047119140625e2), SC_(0.1252804412841796875e3), SC_(0.3069043255911758700865294859650240330974e-44), + SC_(-0.8049192047119140625e2), SC_(0.25554705810546875e3), SC_(0.2303430936664631154413247069375132759954e-106), + SC_(-0.8049192047119140625e2), SC_(0.503011474609375e3), SC_(0.1203148508747254149682895744594491240807e-216), + SC_(-0.8049192047119140625e2), SC_(0.10074598388671875e4), SC_(0.2865368119939400701179862849573503322931e-437), + SC_(-0.8049192047119140625e2), SC_(0.1185395751953125e4), SC_(0.8632633219300624004437758135158135952472e-515), + SC_(-0.8049192047119140625e2), SC_(0.353451806640625e4), SC_(0.5013665804582944405266048580134316878986e-1536), + SC_(-0.8049192047119140625e2), SC_(0.80715478515625e4), SC_(0.7765547631230743133384730763696548377855e-3507), + SC_(-0.8049192047119140625e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.639546878366615050472401588575857541732e-7050)), + SC_(-0.8049192047119140625e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5074028894875745794984647078151040612894e-13928)), + SC_(-0.8049192047119140625e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2862328185162412476566225413964872968853e-15796)), + SC_(-0.7460263824462890625e2), SC_(0.24750102996826171875e2), SC_(0.1194046640827563151857444163209777353211e25), + SC_(-0.7460263824462890625e2), SC_(0.637722015380859375e2), SC_(0.5818966684329205041972653154218173748165e-11), + SC_(-0.7460263824462890625e2), SC_(0.1252804412841796875e3), SC_(0.9892143938422535628101195141323126645363e-46), + SC_(-0.7460263824462890625e2), SC_(0.25554705810546875e3), SC_(0.3972603961730133195379956336197334288476e-107), + SC_(-0.7460263824462890625e2), SC_(0.503011474609375e3), SC_(0.4874624060193139320406839502988832481089e-217), + SC_(-0.7460263824462890625e2), SC_(0.10074598388671875e4), SC_(0.1822212069789909176095875838528811873338e-437), + SC_(-0.7460263824462890625e2), SC_(0.1185395751953125e4), SC_(0.5875055967970574458131259176159286617499e-515), + SC_(-0.7460263824462890625e2), SC_(0.353451806640625e4), SC_(0.4406079158432466047722722836894011978239e-1536), + SC_(-0.7460263824462890625e2), SC_(0.80715478515625e4), SC_(0.7338395057162425548486505792810413989371e-3507), + SC_(-0.7460263824462890625e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6218012346099611746045494400088987852165e-7050)), + SC_(-0.7460263824462890625e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5002276251033884106325883264873018499132e-13928)), + SC_(-0.7460263824462890625e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2826609186886966995318007844162294906315e-15796)), + SC_(-0.7290460205078125e2), SC_(0.24750102996826171875e2), SC_(0.5561803915497248563365929946842781443078e23), + SC_(-0.7290460205078125e2), SC_(0.637722015380859375e2), SC_(0.1094524924593545154904194423989731358977e-11), + SC_(-0.7290460205078125e2), SC_(0.1252804412841796875e3), SC_(0.3839300658689373815830761148374331937154e-46), + SC_(-0.7290460205078125e2), SC_(0.25554705810546875e3), SC_(0.2451728941031062427272665484306743376086e-107), + SC_(-0.7290460205078125e2), SC_(0.503011474609375e3), SC_(0.3804541047790449891831659262615119819112e-217), + SC_(-0.7290460205078125e2), SC_(0.10074598388671875e4), SC_(0.1609485041764832205733383613676089003386e-437), + SC_(-0.7290460205078125e2), SC_(0.1185395751953125e4), SC_(0.5286617461619307606407976695028744909355e-515), + SC_(-0.7290460205078125e2), SC_(0.353451806640625e4), SC_(0.4252727041870810007272294050962600690759e-1536), + SC_(-0.7290460205078125e2), SC_(0.80715478515625e4), SC_(0.7225421446583687935214716001980501582795e-3507), + SC_(-0.7290460205078125e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.617021607511078284203431245617539588877e-7050)), + SC_(-0.7290460205078125e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.4982778020141303245214047263053056397026e-13928)), + SC_(-0.7290460205078125e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.281689238316161921546343371679087454574e-15796)), + SC_(-0.62323604583740234375e2), SC_(0.24750102996826171875e2), SC_(0.6745183967776568226882524708487938056875e15), + SC_(-0.62323604583740234375e2), SC_(0.637722015380859375e2), SC_(0.6545311734942178902723924532558287624952e-16), + SC_(-0.62323604583740234375e2), SC_(0.1252804412841796875e3), SC_(0.1656532226161521639805764466363495194113e-48), + SC_(-0.62323604583740234375e2), SC_(0.25554705810546875e3), SC_(0.1547673376370412380297419400250693508513e-108), + SC_(-0.62323604583740234375e2), SC_(0.503011474609375e3), SC_(0.9227214789189674273358185346965399203582e-218), + SC_(-0.62323604583740234375e2), SC_(0.10074598388671875e4), SC_(0.7918944121135532385395798829085145544592e-438), + SC_(-0.62323604583740234375e2), SC_(0.1185395751953125e4), SC_(0.2892810675468518815348991889357281331268e-515), + SC_(-0.62323604583740234375e2), SC_(0.353451806640625e4), SC_(0.3473597010045323910900283230401929551928e-1536), + SC_(-0.62323604583740234375e2), SC_(0.80715478515625e4), SC_(0.6612598699249681198531835080793951126635e-3507), + SC_(-0.62323604583740234375e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5904134813790360951037473338098849774904e-7050)), + SC_(-0.62323604583740234375e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.4872840764596854116092324404256445242957e-13928)), + SC_(-0.62323604583740234375e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2762021177708928645685587252784589095343e-15796)), + SC_(-0.5579319000244140625e2), SC_(0.24750102996826171875e2), SC_(0.2000280553692923364816391845858003081304e11), + SC_(-0.5579319000244140625e2), SC_(0.637722015380859375e2), SC_(0.3011072877774196098095590001850230113398e-18), + SC_(-0.5579319000244140625e2), SC_(0.1252804412841796875e3), SC_(0.8546927999408637677019436633377190577324e-50), + SC_(-0.5579319000244140625e2), SC_(0.25554705810546875e3), SC_(0.3476662826409067664561159567913234387987e-109), + SC_(-0.5579319000244140625e2), SC_(0.503011474609375e3), SC_(0.4297054501709256270968786489029298481091e-218), + SC_(-0.5579319000244140625e2), SC_(0.10074598388671875e4), SC_(0.5402417605668363705190262971682628012869e-438), + SC_(-0.5579319000244140625e2), SC_(0.1185395751953125e4), SC_(0.2089958153703015756026346516755670166549e-515), + SC_(-0.5579319000244140625e2), SC_(0.353451806640625e4), SC_(0.3114579751849795632507912984614279046035e-1536), + SC_(-0.5579319000244140625e2), SC_(0.80715478515625e4), SC_(0.6304085854717380067005364169553964460527e-3507), + SC_(-0.5579319000244140625e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5765486071511390988466247681614919123209e-7050)), + SC_(-0.5579319000244140625e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.4814584754866145181310357527683878612407e-13928)), + SC_(-0.5579319000244140625e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2732885595625837213661882188978422888301e-15796)), + SC_(-0.4430035400390625e2), SC_(0.95070552825927734375e1), SC_(0.5693602607646284460254541471864922205948e23), + SC_(-0.4430035400390625e2), SC_(0.24750102996826171875e2), SC_(0.1242729664484783369574386233140179346878e4), + SC_(-0.4430035400390625e2), SC_(0.637722015380859375e2), SC_(0.7993412663367930219134100562570886747324e-22), + SC_(-0.4430035400390625e2), SC_(0.1252804412841796875e3), SC_(0.9881485422320279470670535583393602847552e-52), + SC_(-0.4430035400390625e2), SC_(0.25554705810546875e3), 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SC_(0.8731920655533864492797505716863887075778e-3507), + SC_(0.9150136566162109375e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6779648490610206356859880361410662474219e-7050)), + SC_(0.9150136566162109375e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5226073790697475866503341992543667357366e-13928)), + SC_(0.9150136566162109375e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2937821135047504313538653046960400698576e-15796)), + SC_(0.9297769927978515625e2), SC_(0.637722015380859375e2), SC_(0.2514159824708407029519182952938261461039e-2), + SC_(0.9297769927978515625e2), SC_(0.1252804412841796875e3), SC_(0.9571910315928452948496436653664936091399e-41), + SC_(0.9297769927978515625e2), SC_(0.25554705810546875e3), SC_(0.1464962376552961548773813405449012101125e-104), + SC_(0.9297769927978515625e2), SC_(0.503011474609375e3), SC_(0.102289622788229900557470609045948505245e-215), + SC_(0.9297769927978515625e2), SC_(0.10074598388671875e4), SC_(0.8379350403930188404501970121630099171596e-437), + SC_(0.9297769927978515625e2), SC_(0.1185395751953125e4), SC_(0.2149730062284109979164130574050367445585e-514), + SC_(0.9297769927978515625e2), SC_(0.353451806640625e4), SC_(0.6810640416069074949832134642840700061677e-1536), + SC_(0.9297769927978515625e2), SC_(0.80715478515625e4), SC_(0.8880475416546418613925474382495752636196e-3507), + SC_(0.9297769927978515625e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6836772391670514269657537156841197891453e-7050)), + SC_(0.9297769927978515625e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5248314285242353145777820274280636896797e-13928)), + SC_(0.9297769927978515625e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2948841983584072578394887812793739973464e-15796)), + SC_(0.935389862060546875e2), SC_(0.637722015380859375e2), SC_(0.4848538332206214872114685461457208526682e-2), + SC_(0.935389862060546875e2), SC_(0.1252804412841796875e3), SC_(0.1407357631531569552447817697356939114967e-40), + SC_(0.935389862060546875e2), SC_(0.25554705810546875e3), SC_(0.1789644517979726575647466756527040582949e-104), + SC_(0.935389862060546875e2), SC_(0.503011474609375e3), SC_(0.1134297740334902791076613679075967354327e-215), + SC_(0.935389862060546875e2), SC_(0.10074598388671875e4), SC_(0.8825347981316688657305249314794392766025e-437), + SC_(0.935389862060546875e2), SC_(0.1185395751953125e4), SC_(0.2246641303550294973758291904185796545436e-514), + SC_(0.935389862060546875e2), SC_(0.353451806640625e4), SC_(0.6912227290322635031334137954308461354241e-1536), + SC_(0.935389862060546875e2), SC_(0.80715478515625e4), SC_(0.893824833574208599253272536219631346374e-3507), + SC_(0.935389862060546875e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6858858035128739755837136827347333728619e-7050)), + SC_(0.935389862060546875e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5256888449089264657997112108384303458607e-13928)), + SC_(0.935389862060546875e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2953089268910496653492909777451085521704e-15796)), + SC_(0.937735595703125e2), SC_(0.637722015380859375e2), SC_(0.6385107666034147877046020721409920794724e-2), + SC_(0.937735595703125e2), SC_(0.1252804412841796875e3), SC_(0.1654344599410448916584423239120073465608e-40), + SC_(0.937735595703125e2), SC_(0.25554705810546875e3), SC_(0.1946478889917254423807361997687268337342e-104), + SC_(0.937735595703125e2), SC_(0.503011474609375e3), SC_(0.1184592506913226623646749833739806823329e-215), + SC_(0.937735595703125e2), SC_(0.10074598388671875e4), SC_(0.901953355688788108549058088701948794206e-437), + SC_(0.937735595703125e2), SC_(0.1185395751953125e4), SC_(0.2288605393501964892215062020117435573162e-514), + SC_(0.937735595703125e2), SC_(0.353451806640625e4), SC_(0.6955313792142442148137724010444647419301e-1536), + SC_(0.937735595703125e2), SC_(0.80715478515625e4), SC_(0.8962607682372740535877231935882633914515e-3507), + SC_(0.937735595703125e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6868148706801060740839033435132304040165e-7050)), + SC_(0.937735595703125e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.526049123050695967902464958981106803334e-13928)), + SC_(0.937735595703125e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.295487369292694824273684043549849890813e-15796)), + SC_(0.98576263427734375e2), SC_(0.637722015380859375e2), SC_(0.1990227834888151454373307286008775674612e1), + SC_(0.98576263427734375e2), SC_(0.1252804412841796875e3), SC_(0.4894424029224796393682491693146694616972e-39), + SC_(0.98576263427734375e2), SC_(0.25554705810546875e3), SC_(0.1136170375675789312265821779116321329958e-103), + SC_(0.98576263427734375e2), SC_(0.503011474609375e3), SC_(0.2948452830082071500214646231076147090724e-215), + SC_(0.98576263427734375e2), SC_(0.10074598388671875e4), SC_(0.1425279255280814495079497699158457698546e-436), + SC_(0.98576263427734375e2), SC_(0.1185395751953125e4), SC_(0.3377080491094056336148164895290399079231e-514), + SC_(0.98576263427734375e2), SC_(0.353451806640625e4), SC_(0.7926042309665472483589245698365986058321e-1536), + SC_(0.98576263427734375e2), SC_(0.80715478515625e4), SC_(0.9490411542227230445519501560068255751126e-3507), + SC_(0.98576263427734375e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.706642352671901713112325219552778166628e-7050)), + SC_(0.98576263427734375e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5336813326314223870941968492332354768707e-13928)), + SC_(0.98576263427734375e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2992641523816122158711374925214774085881e-15796)), + SC_(0.99292266845703125e2), SC_(0.637722015380859375e2), SC_(0.4765079470941391660151554471287672030993e1), + SC_(0.99292266845703125e2), SC_(0.1252804412841796875e3), SC_(0.8211475782164329588746473557436288828796e-39), + SC_(0.99292266845703125e2), SC_(0.25554705810546875e3), SC_(0.1488689678947819908654123225405437300755e-103), + SC_(0.99292266845703125e2), SC_(0.503011474609375e3), SC_(0.3390847523037554808868465852319867472935e-215), + SC_(0.99292266845703125e2), SC_(0.10074598388671875e4), SC_(0.1528876974454391292968555182897209684705e-436), + SC_(0.99292266845703125e2), SC_(0.1185395751953125e4), SC_(0.3584703400837929375694495921804618033184e-514), + SC_(0.99292266845703125e2), SC_(0.353451806640625e4), SC_(0.8086451033245101399967163404205099861285e-1536), + SC_(0.99292266845703125e2), SC_(0.80715478515625e4), SC_(0.9574060247932829286311891259694070669118e-3507), + SC_(0.99292266845703125e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.7097333247638353953239893327006028051669e-7050)), + SC_(0.99292266845703125e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5348615672629262632704311334792898428533e-13928)), + SC_(0.99292266845703125e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.299847616599843593158654409241939580736e-15796)) + }; +#undef SC_ + + diff --git a/test/bessel_k_int_data.ipp b/test/bessel_k_int_data.ipp new file mode 100644 index 000000000..3aa34db4e --- /dev/null +++ b/test/bessel_k_int_data.ipp @@ -0,0 +1,492 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 481> bessel_k_int_data = { + SC_(0), SC_(0.177219114266335964202880859375e-2), SC_(0.6451475930592273598846015135698055330078e1), + SC_(0), SC_(0.22177286446094512939453125e-2), SC_(0.6227212142001190939808570915268231760654e1), + SC_(0), SC_(0.7444499991834163665771484375e-2), SC_(0.5016294646816679195434588077252051358532e1), + SC_(0), SC_(0.1433600485324859619140625e-1), SC_(0.4361188048817122598222684820956136285199e1), + SC_(0), SC_(0.1760916970670223236083984375e-1), SC_(0.4155666670689396106825982497779831275659e1), + SC_(0), SC_(0.6152711808681488037109375e-1), SC_(0.2907904688572973437220285912023264651352e1), + SC_(0), SC_(0.11958599090576171875e0), SC_(0.2251245456228397094716239150833833783688e1), + SC_(0), SC_(0.15262925624847412109375e0), SC_(0.2013151217079277922721039040650374928823e1), + SC_(0), SC_(0.408089816570281982421875e0), SC_(0.1097070466164341232251948278975330916289e1), + SC_(0), SC_(0.6540834903717041015625e0), SC_(0.7111296101768869724219672824880816154124e0), + SC_(0), SC_(0.1097540378570556640625e1), SC_(0.3668587200933656003255821289886727335553e0), + SC_(0), SC_(0.30944411754608154296875e1), SC_(0.3115344887529544812621292520040581803004e-1), + SC_(0), SC_(0.51139926910400390625e1), SC_(0.325805941096065330441380826151925706171e-2), + SC_(0), SC_(0.95070552825927734375e1), SC_(0.2983575249299677934623174911041338567643e-4), + SC_(0), SC_(0.24750102996826171875e2), SC_(0.4469793219985647671692938809730755521561e-11), + SC_(0), SC_(0.637722015380859375e2), SC_(0.3154890666025357981487513910165521100024e-28), + SC_(0), SC_(0.1252804412841796875e3), SC_(0.4365986153732310357450484955539750321993e-55), + SC_(0), SC_(0.25554705810546875e3), SC_(0.8155212353606568575514680314443449984517e-112), + SC_(0), SC_(0.503011474609375e3), SC_(0.1959094651632950581341362431434333187503e-219), + SC_(0), SC_(0.10074598388671875e4), SC_(0.1153834312978712202246739136605238163053e-438), + SC_(0), SC_(0.1185395751953125e4), SC_(0.5626632279469502957817365401058836530616e-516), + SC_(0), SC_(0.353451806640625e4), SC_(0.2005335541692877275070776095045572408221e-1536), + SC_(0), SC_(0.80715478515625e4), SC_(0.5198552672839385593247348234265735246569e-3507), + SC_(0), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5238258665687646932029547633274667132227e-7050)), + SC_(0), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.4586477351514513511787402593637142120047e-13928)), + SC_(0), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.261839521735852199886433084148333502408e-15796)), + SC_(0.1e1), SC_(0.177219114266335964202880859375e-2), SC_(0.5642670589050394493876757991354791444425e3), + SC_(0.1e1), SC_(0.22177286446094512939453125e-2), SC_(0.4509043336519153776882032141395071321111e3), + SC_(0.1e1), SC_(0.7444499991834163665771484375e-2), SC_(0.134306823034307382114643500755390513023e3), + SC_(0.1e1), SC_(0.1433600485324859619140625e-1), SC_(0.6971959660478877278042038844910522107511e2), + SC_(0.1e1), SC_(0.1760916970670223236083984375e-1), SC_(0.5674760507791176149484792894248541539452e2), + SC_(0.1e1), SC_(0.6152711808681488037109375e-1), SC_(0.1614820987046735392831380358603921129883e2), + SC_(0.1e1), SC_(0.11958599090576171875e0), SC_(0.8197998310985025401124448473235927713019e1), + SC_(0.1e1), SC_(0.15262925624847412109375e0), SC_(0.6360645272530455596559051797225101283072e1), + SC_(0.1e1), SC_(0.408089816570281982421875e0), SC_(0.2132196083017461631334167216825680193136e1), + SC_(0.1e1), SC_(0.6540834903717041015625e0), SC_(0.1156576280544243110905012085298289192381e1), + SC_(0.1e1), SC_(0.1097540378570556640625e1), SC_(0.5118042111815067840711185047380239515098e0), + SC_(0.1e1), SC_(0.30944411754608154296875e1), SC_(0.3587084607310022256777513946093825420136e-1), + SC_(0.1e1), SC_(0.51139926910400390625e1), SC_(0.3563402139499414445927612094054750431128e-2), + SC_(0.1e1), SC_(0.95070552825927734375e1), SC_(0.3136737811772098452264479387949931309609e-4), + SC_(0.1e1), SC_(0.24750102996826171875e2), SC_(0.4559214298385623744840433425339909113277e-11), + SC_(0.1e1), SC_(0.637722015380859375e2), SC_(0.3179530807904064450989433716351000642288e-28), + SC_(0.1e1), SC_(0.1252804412841796875e3), SC_(0.4383376507619551733740470932900485417799e-55), + SC_(0.1e1), SC_(0.25554705810546875e3), SC_(0.8171153185119733731907215700781324087313e-112), + SC_(0.1e1), SC_(0.503011474609375e3), SC_(0.1961041051464076987061687841817509806692e-219), + SC_(0.1e1), SC_(0.10074598388671875e4), SC_(0.1154406816332980455168031108997781743075e-438), + SC_(0.1e1), SC_(0.1185395751953125e4), SC_(0.5629005093195648507075346585433996324305e-516), + SC_(0.1e1), SC_(0.353451806640625e4), SC_(0.2005619200413067947685927551685795058075e-1536), + SC_(0.1e1), SC_(0.80715478515625e4), 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SC_(0.9651821409596944246819658023453058852567e-106), + SC_(0.85e2), SC_(0.503011474609375e3), SC_(0.2515578850121789395121409417066593310199e-216), + SC_(0.85e2), SC_(0.10074598388671875e4), SC_(0.4146905619633520795687290702797802077516e-437), + SC_(0.85e2), SC_(0.1185395751953125e4), SC_(0.1182063426653318577358697854155415902088e-514), + SC_(0.85e2), SC_(0.353451806640625e4), SC_(0.5571590796064834896786127413958385484532e-1536), + SC_(0.85e2), SC_(0.80715478515625e4), SC_(0.8132823498430510533405211338608503625054e-3507), + SC_(0.85e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.6544164172431353956246831844735505709742e-7050)), + SC_(0.85e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5133398576643193156132512241768029727037e-13928)), + SC_(0.85e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2891837680670311977658996166762631480513e-15796)), + SC_(0.88e2), SC_(0.637722015380859375e2), SC_(0.8408884745537773166223428345981835442608e-5), + SC_(0.88e2), SC_(0.1252804412841796875e3), SC_(0.3427163660301187722220573043617848517407e-42), + SC_(0.88e2), SC_(0.25554705810546875e3), SC_(0.2611158848229128475717621937789423053511e-105), + SC_(0.88e2), SC_(0.503011474609375e3), SC_(0.4201277178177965425183058418537562017403e-216), + SC_(0.88e2), SC_(0.10074598388671875e4), SC_(0.5362866581825708379523263981965145477407e-437), + SC_(0.88e2), SC_(0.1185395751953125e4), SC_(0.1470923110114575284128795225669870357461e-514), + SC_(0.88e2), SC_(0.353451806640625e4), SC_(0.5995934685429881061512313674772217021537e-1536), + SC_(0.88e2), SC_(0.80715478515625e4), SC_(0.839852015080128730529999326058629197173e-3507), + SC_(0.88e2), SC_(0.1622925e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.664964032329384431282211022073156371507e-7050)), + SC_(0.88e2), SC_(0.3206622265625e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.5175109105174760942763088943597648303892e-13928)), + SC_(0.88e2), SC_(0.3636794921875e5), SC_(BOOST_MATH_SMALL_CONSTANT(0.2912545597593180629046239733714231938561e-15796)) + }; +#undef SC_ + + diff --git a/test/bessel_y01_data.ipp b/test/bessel_y01_data.ipp new file mode 100644 index 000000000..29f446187 --- /dev/null +++ b/test/bessel_y01_data.ipp @@ -0,0 +1,111 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> bessel_y01_data = { + SC_(0), SC_(0.23917420208454132080078125e0), SC_(-0.96144387845032600014206252125473909797e0), + SC_(0), SC_(0.1785583972930908203125e1), SC_(0.4741447443427281185149710186267060231689e0), + SC_(0), SC_(0.48770198822021484375e1), SC_(-0.2878028614684715596290259912770720755831e0), + SC_(0), SC_(0.54930877685546875e1), SC_(-0.3396368039207613416368370236707080035826e0), + SC_(0), SC_(0.5623225688934326171875e1), SC_(-0.3340373324911775573542553376291358673245e0), + SC_(0), SC_(0.6349340915679931640625e1), SC_(-0.2128871385683908276739449505491816562456e0), + SC_(0), SC_(0.677385044097900390625e1), SC_(-0.9426827045132919826659947856922431474509e-1), + SC_(0), SC_(0.7094316959381103515625e1), SC_(0.2479102692009034386110116691802019548306e-2), + SC_(0), SC_(0.788065433502197265625e1), SC_(0.2029626650223921424547972824606518098619e0), + SC_(0), SC_(0.941909885406494140625e1), SC_(0.1871389209333241942443487293304042083944e0), + SC_(0), SC_(0.105962162017822265625e2), SC_(-0.895681903138232043513826049370887073897e-1), + SC_(0), SC_(0.110517024993896484375e2), SC_(-0.1770624440732838746367860550588287464321e0), + SC_(0), SC_(0.139249114990234375e2), SC_(0.1142985183059252859759404038734709330922e0), + SC_(0), SC_(0.1485147190093994140625e2), SC_(0.2063286797703167877532498353892868456477e0), + SC_(0), SC_(0.15408351898193359375e2), SC_(0.1804637317868620678845766162727475608308e0), + SC_(0), SC_(0.180646991729736328125e2), SC_(-0.1876865372622732192971339060928812967449e0), + SC_(0), SC_(0.199369258880615234375e2), SC_(0.5206684116941857702281164878715312044063e-1), + SC_(0), SC_(0.210880641937255859375e2), SC_(0.1723989782023412731879808822208131224943e0), + SC_(0), SC_(0.242687816619873046875e2), SC_(-0.1613789161670016087114250427765172807239e0), + SC_(0), SC_(0.25183135986328125e2), SC_(-0.1071939400958389283049694534626391030773e0), + SC_(0), SC_(0.27344074249267578125e2), SC_(0.1508708516504150084978629879232740523815e0), + SC_(0), SC_(0.273610286712646484375e2), SC_(0.1511874592773052558132847555838893071071e0), + SC_(0), SC_(0.316179637908935546875e2), SC_(-0.7862384849343525606244284493490404212792e-1), + SC_(0), SC_(0.3198816680908203125e2), SC_(-0.3038016447495889126879755326091231324441e-1), + SC_(0), SC_(0.327870330810546875e2), SC_(0.7658058565089261314120406134639008633495e-1), + SC_(0), SC_(0.33936756134033203125e2), SC_(0.135185927745980170738744171379455779367e0), + SC_(0), SC_(0.340679779052734375e2), SC_(0.1308985104522637565167036775360209675704e0), + SC_(0), SC_(0.362919464111328125e2), SC_(-0.1073864718556727242939230205939491695933e0), + SC_(0), SC_(0.396103668212890625e2), SC_(0.1142550468868382394556084602403207743216e0), + SC_(0), SC_(0.3989643096923828125e2), SC_(0.124661273461738151669837813759338802014e0), + SC_(0), SC_(0.39905292510986328125e2), SC_(0.1248230809391841736060855994886626203322e0), + SC_(0), SC_(0.400140228271484375e2), SC_(0.1260052610908056072451525745074524623778e0), + SC_(0), SC_(0.4073618316650390625e2), SC_(0.9737382379702960925368875830625442357192e-1), + SC_(0), SC_(0.4175042724609375e2), SC_(-0.1494519243198415326119337512087694723669e-1), + SC_(0), SC_(0.424564666748046875e2), SC_(-0.9014506163034071267518605768672279547304e-1), + SC_(0), SC_(0.4392153167724609375e2), SC_(-0.9036719621072680503961718730174281767038e-1), + SC_(0), SC_(0.452895965576171875e2), SC_(0.5882036891280847548975142664645827422984e-1), + SC_(0), SC_(0.45668792724609375e2), SC_(0.9236225506165839536204051297244550413125e-1), + SC_(0), SC_(0.45786777496337890625e2), SC_(0.1002478087601361079531531328106555493414e0), + SC_(0), SC_(0.466996612548828125e2), SC_(0.1093312781568777333543706682350852686496e0), + SC_(0), SC_(0.47858348846435546875e2), SC_(0.6173117074433275797673442689748870252433e-2), + SC_(0), SC_(0.4787534332275390625e2), SC_(0.4214269749343140372639153683731024716149e-2), + SC_(0), SC_(0.47974620819091796875e2), SC_(-0.7221017162650168684644534187509493067458e-2), + SC_(0), SC_(0.48244426727294921875e2), SC_(-0.3749948283323452728759672710131060037256e-1), + SC_(0), SC_(0.48384746551513671875e2), SC_(-0.5224122081317449129185406414204132435727e-1), + SC_(0), SC_(0.48443389892578125e2), SC_(-0.5810149952964139807372900859379505792103e-1), + SC_(0), SC_(0.4852964019775390625e2), SC_(-0.6633946207791363484342690704116317279187e-1), + SC_(0), SC_(0.49055484771728515625e2), SC_(-0.1036808657798402644495161610234081710331e0), + SC_(0), SC_(0.4964406585693359375e2), SC_(-0.1117659301811653699040137406534766658576e0), + SC_(0), SC_(0.4982306671142578125e2), SC_(-0.1065466287843043964466710578381452573432e0), + SC_(0.1e1), SC_(0.23917420208454132080078125e0), SC_(-0.2816025505957848702890230692737435583999e1), + SC_(0.1e1), SC_(0.1785583972930908203125e1), SC_(-0.2323574880952473756698741233078133158232e0), + SC_(0.1e1), SC_(0.48770198822021484375e1), SC_(0.1887793753606189589996790522700420589945e0), + SC_(0.1e1), SC_(0.54930877685546875e1), SC_(-0.2143954097552594432326297820690591750424e-1), + SC_(0.1e1), SC_(0.5623225688934326171875e1), SC_(-0.6432984460938437015021689712475734307787e-1), + SC_(0.1e1), SC_(0.6349340915679931640625e1), SC_(-0.2511307972102585372917123452584402539094e0), + SC_(0.1e1), SC_(0.677385044097900390625e1), SC_(-0.2989785036869612052873436440028846450436e0), + SC_(0.1e1), SC_(0.7094316959381103515625e1), SC_(-0.2997377139745232636871782332312475969025e0), + SC_(0.1e1), SC_(0.788065433502197265625e1), SC_(-0.186133051469056393893281692679499181405e0), + SC_(0.1e1), SC_(0.941909885406494140625e1), SC_(0.190361227136102466719822742808004416221e0), + SC_(0.1e1), SC_(0.105962162017822265625e2), SC_(0.2240497166044468367366329478931466799755e0), + SC_(0.1e1), SC_(0.110517024993896484375e2), SC_(0.1540160818741671628552636923084057652304e0), + SC_(0.1e1), SC_(0.139249114990234375e2), SC_(-0.1766396467501490164858936378782126340342e0), + SC_(0.1e1), SC_(0.1485147190093994140625e2), SC_(-0.9506363681079245319129321117867079275859e-2), + SC_(0.1e1), SC_(0.15408351898193359375e2), SC_(0.99321403407708436538763939792420633861e-1), + SC_(0.1e1), SC_(0.180646991729736328125e2), SC_(-0.3995388258078618389192173389771024944513e-2), + SC_(0.1e1), SC_(0.199369258880615234375e2), SC_(-0.1696600513329557121425560056217261449047e0), + SC_(0.1e1), SC_(0.210880641937255859375e2), SC_(-0.1733957455990071497842225993373989557431e-1), + SC_(0.1e1), SC_(0.242687816619873046875e2), SC_(0.1021971395701211404109265949334393638881e-1), + SC_(0.1e1), SC_(0.25183135986328125e2), SC_(-0.119556229137516783528551234301878875503e0), + SC_(0.1e1), SC_(0.27344074249267578125e2), SC_(-0.1995985480706287524814653648262685912676e-1), + SC_(0.1e1), SC_(0.273610286712646484375e2), SC_(-0.173876060044410140856275457334661052251e-1), + SC_(0.1e1), SC_(0.316179637908935546875e2), SC_(-0.1193701231934014102558224854153342610094e0), + SC_(0.1e1), SC_(0.3198816680908203125e2), SC_(-0.1382462102065444071353719726581548003019e0), + SC_(0.1e1), SC_(0.327870330810546875e2), SC_(-0.1152502985662088783214562688328306893687e0), + SC_(0.1e1), SC_(0.33936756134033203125e2), SC_(0.2394218295328153690551974191920973663242e-1), + SC_(0.1e1), SC_(0.340679779052734375e2), SC_(0.4129906123027152853078275121624211732661e-1), + SC_(0.1e1), SC_(0.362919464111328125e2), SC_(0.7604014342236898104501488355770421840119e-1), + SC_(0.1e1), SC_(0.396103668212890625e2), SC_(-0.5348553383812884892017999878337862380908e-1), + SC_(0.1e1), SC_(0.3989643096923828125e2), SC_(-0.1881415396910400076456852195968705619502e-1), + SC_(0.1e1), SC_(0.39905292510986328125e2), SC_(-0.1770468345557173172933692614949148322983e-1), + SC_(0.1e1), SC_(0.400140228271484375e2), SC_(-0.4025288726928248087599225614592773179551e-2), + SC_(0.1e1), SC_(0.4073618316650390625e2), SC_(0.7959115261091509104088861641763392149822e-1), + SC_(0.1e1), SC_(0.4175042724609375e2), SC_(0.1224013166060462132230567141290330120648e0), + SC_(0.1e1), SC_(0.424564666748046875e2), SC_(0.8181451309209186481544883864207065020545e-1), + SC_(0.1e1), SC_(0.4392153167724609375e2), SC_(-0.8057815233036708376163307599491467159994e-1), + SC_(0.1e1), SC_(0.452895965576171875e2), SC_(-0.1022936127451748973670400377224684952561e0), + SC_(0.1e1), SC_(0.45668792724609375e2), SC_(-0.725345130575250866364960202825599272059e-1), + SC_(0.1e1), SC_(0.45786777496337890625e2), SC_(-0.6098613412165902384405909899253497503886e-1), + SC_(0.1e1), SC_(0.466996612548828125e2), SC_(0.4213746852981118912882681387493786761666e-1), + SC_(0.1e1), SC_(0.47858348846435546875e2), SC_(0.1152373194909642455028362253859850179066e0), + SC_(0.1e1), SC_(0.4787534332275390625e2), SC_(0.1152846618892812588531426336829335209133e0), + SC_(0.1e1), SC_(0.47974620819091796875e2), SC_(0.1148964835202570220707248379081032143507e0), + SC_(0.1e1), SC_(0.48244426727294921875e2), SC_(0.1081934550363820741605791245861795859361e0), + SC_(0.1e1), SC_(0.48384746551513671875e2), SC_(0.1015812774603423801064358240911218545139e0), + SC_(0.1e1), SC_(0.48443389892578125e2), SC_(0.9822383693317642997910135239495267124437e-1), + SC_(0.1e1), SC_(0.4852964019775390625e2), SC_(0.9268396308064766041470606107371977547449e-1), + SC_(0.1e1), SC_(0.49055484771728515625e2), SC_(0.4613847461932302214343079992899045052011e-1), + SC_(0.1e1), SC_(0.4964406585693359375e2), SC_(-0.1933115283084264304511911708126688592758e-1), + SC_(0.1e1), SC_(0.4982306671142578125e2), SC_(-0.3881735371825568460631754621811251910549e-1) + }; +#undef SC_ + + diff --git a/test/bessel_yn_data.ipp b/test/bessel_yn_data.ipp new file mode 100644 index 000000000..e8d964adf --- /dev/null +++ b/test/bessel_yn_data.ipp @@ -0,0 +1,311 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 300> bessel_yn_data = { + SC_(0.3e1), SC_(0.48770198822021484375e1), SC_(0.110763167474753768460862899966350902213e0), + SC_(0.3e1), SC_(0.6349340915679931640625e1), SC_(0.3354120577583840086404698897976135110485e0), + SC_(0.3e1), SC_(0.677385044097900390625e1), SC_(0.3025179709577162581120705977708622170952e0), + SC_(0.3e1), SC_(0.941909885406494140625e1), SC_(-0.2526681038602083570260072993947985762425e0), + SC_(0.3e1), SC_(0.110517024993896484375e2), SC_(-0.7984312482740648942854070462466960846308e-1), + SC_(0.3e1), SC_(0.139249114990234375e2), SC_(0.1365190825919215541308802073699101624338e0), + SC_(0.3e1), SC_(0.15408351898193359375e2), SC_(-0.1428229643813645478620540882069209010424e0), + SC_(0.3e1), SC_(0.27344074249267578125e2), SC_(-0.2323694273239485669670827936469439230487e-2), + SC_(0.3e1), 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SC_(-0.3695521231167434412849111458028999950705e16), + SC_(0.36e2), SC_(0.110517024993896484375e2), SC_(-0.1495013134817261901719756533597851889758e14), + SC_(0.36e2), SC_(0.139249114990234375e2), SC_(-0.6196420279439685191659884689826287903961e10), + SC_(0.36e2), SC_(0.15408351898193359375e2), SC_(-0.2244917516984348627389323607107640171731e9), + SC_(0.36e2), SC_(0.27344074249267578125e2), SC_(-0.1545427238791694297400129988367380768401e2), + SC_(0.36e2), SC_(0.273610286712646484375e2), SC_(-0.1523915770558155873570691291184852714414e2), + SC_(0.36e2), SC_(0.316179637908935546875e2), SC_(-0.9471823176252754612212600210916028343601e0), + SC_(0.36e2), SC_(0.4073618316650390625e2), SC_(0.1173425902348136586318833789495612852355e0), + SC_(0.36e2), SC_(0.4175042724609375e2), SC_(0.161213752895956137670111737354907566266e0), + SC_(0.36e2), SC_(0.452895965576171875e2), SC_(-0.9809890796436754979940541980047075943712e-2), + SC_(0.36e2), SC_(0.45668792724609375e2), SC_(-0.4400083682264260689426444674935174217496e-1), + SC_(0.36e2), SC_(0.48443389892578125e2), SC_(-0.1217719085278386136058589800678093840029e0), + SC_(0.39e2), SC_(0.48770198822021484375e1), SC_(-0.1554597256004980110812710730381202390693e30), + SC_(0.39e2), SC_(0.6349340915679931640625e1), SC_(-0.5898284655302538808806988321276025813845e25), + SC_(0.39e2), SC_(0.677385044097900390625e1), SC_(-0.4904509402336609269919728103147054195654e24), + SC_(0.39e2), SC_(0.941909885406494140625e1), SC_(-0.1701017598526523993821571611323638754576e19), + SC_(0.39e2), SC_(0.110517024993896484375e2), SC_(-0.417530693969750949899048280358754304865e16), + SC_(0.39e2), SC_(0.139249114990234375e2), SC_(-0.82735284556625303260320668828414334725e12), + SC_(0.39e2), SC_(0.15408351898193359375e2), SC_(-0.2150905699012014456845831243381060537536e11), + SC_(0.39e2), SC_(0.27344074249267578125e2), SC_(-0.173136202360576514562595059683668653722e3), + SC_(0.39e2), SC_(0.273610286712646484375e2), SC_(-0.170234226215207199774450660598470420203e3), + SC_(0.39e2), SC_(0.316179637908935546875e2), SC_(-0.476564200718666827425515092297538626754e1), + SC_(0.39e2), SC_(0.4073618316650390625e2), SC_(-0.1139728443291009808277788472915789866764e0), + SC_(0.39e2), SC_(0.4175042724609375e2), SC_(-0.3789392025172586855842365357167594570805e-1), + SC_(0.39e2), SC_(0.452895965576171875e2), SC_(0.1632271611911285678989399827341146194315e0), + SC_(0.39e2), SC_(0.45668792724609375e2), SC_(0.1627608022568010137669993716629446289949e0), + SC_(0.39e2), SC_(0.48443389892578125e2), SC_(-0.3909325122429817340070823484524675983864e-2), + SC_(0.42e2), SC_(0.48770198822021484375e1), SC_(-0.6778761113696976108831204681937591870923e33), + SC_(0.42e2), SC_(0.6349340915679931640625e1), SC_(-0.1156175851690544099284032996632355510762e29), + SC_(0.42e2), SC_(0.677385044097900390625e1), SC_(-0.7895552037472749766365607521724398751657e27), + SC_(0.42e2), SC_(0.941909885406494140625e1), SC_(-0.9970250213779731769482155702312393531646e21), + SC_(0.42e2), SC_(0.110517024993896484375e2), SC_(-0.1489524208745775017256038034837378483054e19), + SC_(0.42e2), SC_(0.139249114990234375e2), SC_(-0.1421193372601400943383360047532395451619e15), + SC_(0.42e2), SC_(0.15408351898193359375e2), SC_(-0.266364645856547563022563347741616424841e13), + SC_(0.42e2), SC_(0.27344074249267578125e2), SC_(-0.2743088520517812194850488183554456816175e4), + SC_(0.42e2), SC_(0.273610286712646484375e2), SC_(-0.2690083634003523361552911979208904087728e4), + SC_(0.42e2), SC_(0.316179637908935546875e2), SC_(-0.384805041736847573781666237608273412272e2), + SC_(0.42e2), SC_(0.4073618316650390625e2), SC_(-0.3041722094268398285198996796280539333307e0), + SC_(0.42e2), SC_(0.4175042724609375e2), SC_(-0.237942599990390419264117350357852632925e0), + SC_(0.42e2), SC_(0.452895965576171875e2), SC_(-0.1769165908357241638186400615615275561375e-2), + SC_(0.42e2), SC_(0.45668792724609375e2), SC_(0.2627264307139144752250072910959109882562e-1), + SC_(0.42e2), SC_(0.48443389892578125e2), SC_(0.1595445273747734203624873579243265164658e0), + SC_(0.45e2), SC_(0.48770198822021484375e1), SC_(-0.3678158809277406631158967826248579712809e37), + SC_(0.45e2), SC_(0.6349340915679931640625e1), SC_(-0.2823290277495845250473189621249822637552e32), + SC_(0.45e2), SC_(0.677385044097900390625e1), SC_(-0.1584048114824233231915994262683772557843e31), + SC_(0.45e2), SC_(0.941909885406494140625e1), SC_(-0.7304744289889455012949611900475401935963e24), + SC_(0.45e2), SC_(0.110517024993896484375e2), SC_(-0.6658312289631602826704537808573997071741e21), + SC_(0.45e2), SC_(0.139249114990234375e2), SC_(-0.3075861182582574432943275662208173510923e17), + SC_(0.45e2), SC_(0.15408351898193359375e2), SC_(-0.4170869187140386729800751855187493963783e15), + SC_(0.45e2), SC_(0.27344074249267578125e2), SC_(-0.5841747054602714111411632231058985786773e5), + SC_(0.45e2), SC_(0.273610286712646484375e2), SC_(-0.5714873510732163760028674196101817926392e5), + SC_(0.45e2), SC_(0.316179637908935546875e2), SC_(-0.4439504087675916287818144789832290059488e3), + SC_(0.45e2), SC_(0.4073618316650390625e2), SC_(-0.7185408908955997967519881533646995655741e0), + SC_(0.45e2), SC_(0.4175042724609375e2), SC_(-0.5010756124409158401140594717316266404372e0), + SC_(0.45e2), SC_(0.452895965576171875e2), SC_(-0.2012825578255357825420510083651093929494e0), + SC_(0.45e2), SC_(0.45668792724609375e2), SC_(-0.1794186322536855399767456087221221709347e0), + SC_(0.45e2), SC_(0.48443389892578125e2), SC_(0.4019015729778697779336545231834378773906e-2), + SC_(0.48e2), SC_(0.48770198822021484375e1), SC_(-0.2446594481694463075060840320557933670105e41), + SC_(0.48e2), SC_(0.6349340915679931640625e1), SC_(-0.8459207003255131733462304451406482680612e35), + SC_(0.48e2), SC_(0.677385044097900390625e1), SC_(-0.3900585059507033418338520667611796016876e34), + SC_(0.48e2), SC_(0.941909885406494140625e1), SC_(-0.6584509350557438141050929296978384665502e27), + SC_(0.48e2), SC_(0.110517024993896484375e2), SC_(-0.3668948448136800346846820718406706247001e24), + SC_(0.48e2), SC_(0.139249114990234375e2), SC_(-0.8242020060037745753714812787166372607277e19), + SC_(0.48e2), SC_(0.15408351898193359375e2), SC_(-0.8108471636196350279814807195501067088811e17), + SC_(0.48e2), SC_(0.27344074249267578125e2), SC_(-0.161478450826312060240470478671833452801e7), + SC_(0.48e2), SC_(0.273610286712646484375e2), SC_(-0.1576027872751121787147362803310570448531e7), + SC_(0.48e2), SC_(0.316179637908935546875e2), SC_(-0.6897332324403501516059328674303313990342e4), + SC_(0.48e2), SC_(0.4073618316650390625e2), SC_(-0.2873562437345553411051362683296702801187e1), + SC_(0.48e2), SC_(0.4175042724609375e2), SC_(-0.1645050914310854634110662323482130128629e1), + SC_(0.48e2), SC_(0.452895965576171875e2), SC_(-0.4095205157936721724910226616684093867583e0), + SC_(0.48e2), SC_(0.45668792724609375e2), SC_(-0.3702549409277610055202576821426084296534e0), + SC_(0.48e2), SC_(0.48443389892578125e2), SC_(-0.1889064785963676989110283778976852771445e0), + SC_(0.51e2), SC_(0.48770198822021484375e1), SC_(-0.1969182678091137840672922747889170648771e45), + SC_(0.51e2), SC_(0.6349340915679931640625e1), SC_(-0.3069134439111231226532450141215242288989e39), + SC_(0.51e2), SC_(0.677385044097900390625e1), SC_(-0.1163353956243705422475270788219762868015e38), + SC_(0.51e2), SC_(0.941909885406494140625e1), SC_(-0.7202983789334716962996909315059777818759e30), + SC_(0.51e2), SC_(0.110517024993896484375e2), SC_(-0.245737676689172511133503788575172294214e27), + SC_(0.51e2), SC_(0.139249114990234375e2), SC_(-0.2693876853731198001028217985845998477702e22), + SC_(0.51e2), SC_(0.15408351898193359375e2), SC_(-0.1927060716887504354505080102215582890868e20), + SC_(0.51e2), SC_(0.27344074249267578125e2), SC_(-0.5643892233771978378997816826534156972962e8), + SC_(0.51e2), SC_(0.273610286712646484375e2), SC_(-0.5496011251862539066107336526262783112098e8), + SC_(0.51e2), SC_(0.316179637908935546875e2), SC_(-0.1389344839181512847834347831384757681674e6), + SC_(0.51e2), SC_(0.4073618316650390625e2), SC_(-0.1776984353347768195784367735005036727279e2), + SC_(0.51e2), SC_(0.4175042724609375e2), SC_(-0.872915584752688114103030525708943114106e1), + SC_(0.51e2), SC_(0.452895965576171875e2), SC_(-0.1161540076556705952629084092382499590939e1), + SC_(0.51e2), SC_(0.45668792724609375e2), SC_(-0.9803416521580276513923752432914532429705e0), + SC_(0.51e2), SC_(0.48443389892578125e2), SC_(-0.3793943440135667994381484637963819220083e0), + SC_(0.54e2), SC_(0.48770198822021484375e1), SC_(-0.1895955328653395662512658715131596618242e49), + SC_(0.54e2), SC_(0.6349340915679931640625e1), SC_(-0.1332861075842053738784438239373281251896e43), + SC_(0.54e2), SC_(0.677385044097900390625e1), SC_(-0.415400028628792112351061646920744162805e41), + SC_(0.54e2), SC_(0.941909885406494140625e1), SC_(-0.9448773063492740921951882034107555889314e33), + SC_(0.54e2), SC_(0.110517024993896484375e2), SC_(-0.1976226953367884232990383015948873832395e30), + SC_(0.54e2), SC_(0.139249114990234375e2), SC_(-0.1060239929445304895052402195312333832751e25), + SC_(0.54e2), SC_(0.15408351898193359375e2), SC_(-0.5524852310470588596167631811840899911653e22), + SC_(0.54e2), SC_(0.27344074249267578125e2), SC_(-0.2443271932118273203257306735627770090503e10), + SC_(0.54e2), SC_(0.273610286712646484375e2), SC_(-0.2374028629035348856642105835047755802793e10), + SC_(0.54e2), SC_(0.316179637908935546875e2), SC_(-0.3530147161153490224199678627355098967203e7), + SC_(0.54e2), SC_(0.4073618316650390625e2), SC_(-0.1523687692433405507809303644341462788826e3), + SC_(0.54e2), SC_(0.4175042724609375e2), SC_(-0.6588811991238001972711661692129876612553e2), + SC_(0.54e2), SC_(0.452895965576171875e2), SC_(-0.5340581063654305505018449302519149157836e1), + SC_(0.54e2), SC_(0.45668792724609375e2), SC_(-0.4241843123154779988294497423588841413993e1), + SC_(0.54e2), SC_(0.48443389892578125e2), SC_(-0.101007717644958422192422706197265435244e1), + SC_(0.57e2), SC_(0.48770198822021484375e1), SC_(-0.2161626205739229156929028218866774547264e53), + SC_(0.57e2), SC_(0.6349340915679931640625e1), SC_(-0.6857816478363427954638793152049033022474e46), + SC_(0.57e2), SC_(0.677385044097900390625e1), SC_(-0.1757633522876923346308719323318346692964e45), + SC_(0.57e2), SC_(0.941909885406494140625e1), SC_(-0.1470724735456049669352892370483504273101e37), + SC_(0.57e2), SC_(0.110517024993896484375e2), SC_(-0.1887826470413712223495666019943375481003e33), + SC_(0.57e2), SC_(0.139249114990234375e2), SC_(-0.4968500213339560352225854720671631559582e27), + SC_(0.57e2), SC_(0.15408351898193359375e2), SC_(-0.1888817184319266555840067091337856309212e25), + SC_(0.57e2), SC_(0.27344074249267578125e2), SC_(-0.1288212844049340614131327714502525981742e12), + SC_(0.57e2), SC_(0.273610286712646484375e2), SC_(-0.1249013008694464348198848991016934457599e12), + SC_(0.57e2), SC_(0.316179637908935546875e2), SC_(-0.1107695891983348156343774480706314333398e9), + SC_(0.57e2), SC_(0.4073618316650390625e2), SC_(-0.1713142312580323786184747142248961806243e4), + SC_(0.57e2), SC_(0.4175042724609375e2), SC_(-0.6604590530228345805227426149899214659609e3), + SC_(0.57e2), SC_(0.452895965576171875e2), SC_(-0.3518710444917558080090981427376821716428e2), + SC_(0.57e2), SC_(0.45668792724609375e2), SC_(-0.2663864616630346108683365093711526997299e2), + SC_(0.57e2), SC_(0.48443389892578125e2), SC_(-0.4306298772312023516324511417973550342892e1), + SC_(0.6e2), SC_(0.48770198822021484375e1), SC_(-0.2892081353968073031088937298478587490074e57), + SC_(0.6e2), SC_(0.6349340915679931640625e1), SC_(-0.4142396646874563143741981167064851524547e50), + SC_(0.6e2), SC_(0.677385044097900390625e1), SC_(-0.8732096096999191430049139445386679139699e48), + SC_(0.6e2), SC_(0.941909885406494140625e1), SC_(-0.2690978379323628746253084206502905589883e40), + SC_(0.6e2), SC_(0.110517024993896484375e2), SC_(-0.2121795509181260451828069876653538790412e36), + SC_(0.6e2), SC_(0.139249114990234375e2), SC_(-0.2744926792364829807723799177062944239004e30), + SC_(0.6e2), SC_(0.15408351898193359375e2), SC_(-0.76223015502590825948030726266442383048e27), + SC_(0.6e2), SC_(0.27344074249267578125e2), SC_(-0.8156824115957781369979544049155027772995e13), + SC_(0.6e2), SC_(0.273610286712646484375e2), SC_(-0.7891899080153060843746196671232001179968e13), + SC_(0.6e2), SC_(0.316179637908935546875e2), SC_(-0.4219761955150242610522454447404911542549e10), + SC_(0.6e2), SC_(0.4073618316650390625e2), SC_(-0.2439731391973693955896903470108472114229e5), + SC_(0.6e2), SC_(0.4175042724609375e2), SC_(-0.8451812456206044962273070236632882957476e4), + SC_(0.6e2), SC_(0.452895965576171875e2), SC_(-0.3079126889383895719027479423955437627905e3), + SC_(0.6e2), SC_(0.45668792724609375e2), SC_(-0.2235394837400507231196792387743951216596e3), + SC_(0.6e2), SC_(0.48443389892578125e2), SC_(-0.2621645109951344510244589118146666833865e2) + }; +#undef SC_ + + diff --git a/test/bessel_yv_data.ipp b/test/bessel_yv_data.ipp new file mode 100644 index 000000000..73d3cf462 --- /dev/null +++ b/test/bessel_yv_data.ipp @@ -0,0 +1,443 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 432> bessel_yv_data = { + SC_(0.177219114266335964202880859375e-2), SC_(0.177219114266335964202880859375e-2), SC_(-0.4109981080896677237358777348659577384814e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.22177286446094512939453125e-2), SC_(-0.3967198166482792888587309535339999462266e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.7444499991834163665771484375e-2), SC_(-0.3196174808457488271204616814902805588745e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.1433600485324859619140625e-1), SC_(-0.277886401743182085883175930163269197664e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.1760916970670223236083984375e-1), SC_(-0.2647863781670063492437517931273354439201e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.6152711808681488037109375e-1), SC_(-0.1849304044017019356004789772749790770502e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.11958599090576171875e0), SC_(-0.1421209575863162389893134075657579819718e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.15262925624847412109375e0), SC_(-0.1262161021028632624881669027784829995884e1), + SC_(0.177219114266335964202880859375e-2), SC_(0.408089816570281982421875e0), SC_(-0.5944072046497630295250342068284430797962e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.6540834903717041015625e0), SC_(-0.2453277969150709954915717817184428440924e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.1097540378570556640625e1), SC_(0.1584374949163769422822208482661516201397e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.30944411754608154296875e1), SC_(0.3458497563774182491234075001630772667854e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.51139926910400390625e1), SC_(-0.3227704013798722509157489558679982720248e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.95070552825927734375e1), SC_(0.1703157314350648605712543095319781057303e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.24750102996826171875e2), SC_(-0.1480199026826370992952794039024870144586e0), + SC_(0.177219114266335964202880859375e-2), SC_(0.637722015380859375e2), SC_(0.1495108590261458231741268546248788758226e-1), + SC_(0.177219114266335964202880859375e-2), SC_(0.1252804412841796875e3), SC_(-0.6570327554025865783732578959902760519929e-1), + SC_(0.177219114266335964202880859375e-2), SC_(0.25554705810546875e3), SC_(-0.1424286729418854798832322396053032304441e-1), + SC_(0.177219114266335964202880859375e-2), SC_(0.503011474609375e3), SC_(-0.1488793551072535726513345519376745937703e-1), + SC_(0.177219114266335964202880859375e-2), SC_(0.10074598388671875e4), SC_(0.2459116676878273669207049701272585713077e-1), + SC_(0.177219114266335964202880859375e-2), SC_(0.1185395751953125e4), SC_(-0.5216296486026963890192536381882756251574e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.353451806640625e4), SC_(0.7150321510933910509861478052167676486744e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.80715478515625e4), SC_(-0.7217584852080900819222715241239866307669e-4), + SC_(0.177219114266335964202880859375e-2), SC_(0.1622925e5), SC_(-0.5289900963274037297853946955886625205773e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.3206622265625e5), SC_(0.3201736535368103742253839832309804984043e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.3636794921875e5), SC_(0.3530160743525716867126510667470988391367e-3), + SC_(0.22177286446094512939453125e-2), SC_(0.177219114266335964202880859375e-2), SC_(-0.4110719287432131383209663091223613660449e1), + SC_(0.22177286446094512939453125e-2), SC_(0.22177286446094512939453125e-2), SC_(-0.3967931677865958623227899442296989234961e1), + SC_(0.22177286446094512939453125e-2), SC_(0.7444499991834163665771484375e-2), SC_(-0.3196888768021427112833222816547069531807e1), + SC_(0.22177286446094512939453125e-2), SC_(0.1433600485324859619140625e-1), SC_(-0.2779570979763027030915138623734707116309e1), + SC_(0.22177286446094512939453125e-2), SC_(0.1760916970670223236083984375e-1), SC_(-0.2648568990595505812071805953119049709201e1), + SC_(0.22177286446094512939453125e-2), SC_(0.6152711808681488037109375e-1), SC_(-0.185000202041826350471111738892195254951e1), + SC_(0.22177286446094512939453125e-2), SC_(0.11958599090576171875e0), SC_(-0.1421904434134942202214431845401629010964e1), + SC_(0.22177286446094512939453125e-2), SC_(0.15262925624847412109375e0), SC_(-0.1262854140408887233259773811621052871948e1), + SC_(0.22177286446094512939453125e-2), SC_(0.408089816570281982421875e0), SC_(-0.5950761265617472237995217111777038762135e0), + SC_(0.22177286446094512939453125e-2), SC_(0.6540834903717041015625e0), SC_(-0.2459534859171060177043599079460653627876e0), + SC_(0.22177286446094512939453125e-2), SC_(0.1097540378570556640625e1), SC_(0.1579332201794321120135728809601595120035e0), + SC_(0.22177286446094512939453125e-2), SC_(0.30944411754608154296875e1), SC_(0.3460521593933925495064774929401352783066e0), + SC_(0.22177286446094512939453125e-2), SC_(0.51139926910400390625e1), SC_(-0.3226720162699480428923607985495542415611e0), + SC_(0.22177286446094512939453125e-2), SC_(0.95070552825927734375e1), SC_(0.1704518545616096824583473906088353974798e0), + SC_(0.22177286446094512939453125e-2), SC_(0.24750102996826171875e2), SC_(-0.1480630443547165728353407967940434766385e0), + SC_(0.22177286446094512939453125e-2), SC_(0.637722015380859375e2), SC_(0.1488194766429228836377915641199018452658e-1), + SC_(0.22177286446094512939453125e-2), SC_(0.1252804412841796875e3), SC_(-0.6572261100873119967302664606455129430061e-1), + SC_(0.22177286446094512939453125e-2), SC_(0.25554705810546875e3), SC_(-0.142093855992607823112630880080107228544e-1), + SC_(0.22177286446094512939453125e-2), SC_(0.503011474609375e3), SC_(-0.1491054423538811528779378781658535282739e-1), + SC_(0.22177286446094512939453125e-2), SC_(0.10074598388671875e4), SC_(0.2458751210966411371396701227739563430579e-1), + SC_(0.22177286446094512939453125e-2), SC_(0.1185395751953125e4), SC_(-0.5200492855806017431912763481277953331404e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.353451806640625e4), SC_(0.7158268136941132273892772334045520934666e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.80715478515625e4), SC_(-0.6596068471414521611228855010215014704418e-4), + SC_(0.22177286446094512939453125e-2), SC_(0.1622925e5), SC_(-0.5292246367479866864908050314944924753669e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.3206622265625e5), SC_(0.320390439737211080762980520435506250823e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.3636794921875e5), SC_(0.3500983370716348944297898811549471925567e-3), + SC_(0.7444499991834163665771484375e-2), SC_(0.177219114266335964202880859375e-2), SC_(-0.4120027412731051041206450410057429639349e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.22177286446094512939453125e-2), SC_(-0.3977105851156557119151823261612798218773e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.7444499991834163665771484375e-2), SC_(-0.3205504928732404892476271281075272444522e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.1433600485324859619140625e-1), SC_(-0.2787987583373352069958752481599007631499e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.1760916970670223236083984375e-1), SC_(-0.2656935742898321978445839607405640033284e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.6152711808681488037109375e-1), SC_(-0.1858171712468613875843642326777393322995e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.11958599090576171875e0), SC_(-0.1430015519366236809172334709693214910441e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.15262925624847412109375e0), SC_(-0.1270941805522165919290496753078963837638e1), + SC_(0.7444499991834163665771484375e-2), SC_(0.408089816570281982421875e0), SC_(-0.6028887952741650472210391450598434501064e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.6540834903717041015625e0), SC_(-0.253272254375116720240110301955438274904e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.1097540378570556640625e1), SC_(0.1520201824220985075413827678589759450294e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.30944411754608154296875e1), SC_(0.3484128787584500743767953634177680046837e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.51139926910400390625e1), SC_(-0.3215065328791237601271218312483710525199e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.95070552825927734375e1), SC_(0.1720422311734142500970235815393168124624e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.24750102996826171875e2), SC_(-0.1485637008291705383805876287336390101122e0), + SC_(0.7444499991834163665771484375e-2), SC_(0.637722015380859375e2), SC_(0.1407034822297315991341655297564501076178e-1), + SC_(0.7444499991834163665771484375e-2), SC_(0.1252804412841796875e3), SC_(-0.6594703326047541907061713679421759907013e-1), + SC_(0.7444499991834163665771484375e-2), SC_(0.25554705810546875e3), SC_(-0.1381608643410309470753787566141410262816e-1), + SC_(0.7444499991834163665771484375e-2), SC_(0.503011474609375e3), SC_(-0.151752265807192360259537475829210125324e-1), + SC_(0.7444499991834163665771484375e-2), SC_(0.10074598388671875e4), SC_(0.2454373914040585157738066701355246448403e-1), + SC_(0.7444499991834163665771484375e-2), SC_(0.1185395751953125e4), SC_(-0.5014906543268439324794963661123810184093e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.353451806640625e4), SC_(0.7251230144724707071573863762664925369871e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.80715478515625e4), SC_(0.6953369197393758981781818649242121542468e-5), + SC_(0.7444499991834163665771484375e-2), SC_(0.1622925e5), SC_(-0.5319567322960577952508664100308821679854e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.3206622265625e5), SC_(0.3229218945935765810090443669915499659257e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.3636794921875e5), SC_(0.3158568198916460292543522938944683153232e-3), + SC_(0.1433600485324859619140625e-1), SC_(0.177219114266335964202880859375e-2), SC_(-0.4134159952672901498743626308790551330992e1), + SC_(0.1433600485324859619140625e-1), SC_(0.22177286446094512939453125e-2), SC_(-0.3990837132176071416552743859631846467869e1), + SC_(0.1433600485324859619140625e-1), SC_(0.7444499991834163665771484375e-2), SC_(-0.3217563179354922543332500792794151411069e1), + SC_(0.1433600485324859619140625e-1), SC_(0.1433600485324859619140625e-1), SC_(-0.2799446740519488969651219625146622998737e1), + SC_(0.1433600485324859619140625e-1), SC_(0.1760916970670223236083984375e-1), SC_(-0.2668245252505030375352857746281865483493e1), + SC_(0.1433600485324859619140625e-1), SC_(0.6152711808681488037109375e-1), SC_(-0.186889810951487968438608309966422611667e1), + SC_(0.1433600485324859619140625e-1), SC_(0.11958599090576171875e0), SC_(-0.1440599803781774335574526240384283073629e1), + SC_(0.1433600485324859619140625e-1), SC_(0.15262925624847412109375e0), SC_(-0.1281485849537878284125998432302154093319e1), + SC_(0.1433600485324859619140625e-1), SC_(0.408089816570281982421875e0), SC_(-0.6130927188528948957629699386553428087544e0), + SC_(0.1433600485324859619140625e-1), SC_(0.6540834903717041015625e0), SC_(-0.2628616657161443753490102972580233083382e0), + SC_(0.1433600485324859619140625e-1), SC_(0.1097540378570556640625e1), SC_(0.1442321647711284365164473485125952896532e0), + SC_(0.1433600485324859619140625e-1), SC_(0.30944411754608154296875e1), SC_(0.3514866696796145177802316946038001144499e0), + SC_(0.1433600485324859619140625e-1), SC_(0.51139926910400390625e1), SC_(-0.3199381849177278131427345797992957474537e0), + SC_(0.1433600485324859619140625e-1), SC_(0.95070552825927734375e1), SC_(0.1741206187508240360674576027852156076872e0), + SC_(0.1433600485324859619140625e-1), SC_(0.24750102996826171875e2), SC_(-0.1492084150385547414276112923072727245566e0), + SC_(0.1433600485324859619140625e-1), SC_(0.637722015380859375e2), SC_(0.1299887757401767858160759862904849183224e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.1252804412841796875e3), SC_(-0.6623612901898277817530531932876652691197e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.25554705810546875e3), SC_(-0.1329610971315120839879080119446416949079e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.503011474609375e3), SC_(-0.1552264157837515558947215728593300366424e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.10074598388671875e4), SC_(0.2448349643616932259319127554623091231607e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.1185395751953125e4), SC_(-0.47696968979306790828599052079788410586e-2), + SC_(0.1433600485324859619140625e-1), SC_(0.353451806640625e4), SC_(0.7373052438334910802888978807399790026358e-2), + SC_(0.1433600485324859619140625e-1), SC_(0.80715478515625e4), SC_(0.103089084794957453201580199017916172336e-3), + SC_(0.1433600485324859619140625e-1), SC_(0.1622925e5), SC_(-0.5355041661602504411571715046096891970658e-2), + SC_(0.1433600485324859619140625e-1), SC_(0.3206622265625e5), SC_(0.3262263227226572779950829863684658963946e-2), + SC_(0.1433600485324859619140625e-1), SC_(0.3636794921875e5), SC_(0.2706771716021933545864407071325455578847e-3), + SC_(0.1760916970670223236083984375e-1), SC_(0.177219114266335964202880859375e-2), SC_(-0.414163021695610839626594818544969888135e1), + SC_(0.1760916970670223236083984375e-1), SC_(0.22177286446094512939453125e-2), SC_(-0.3998026070140048622843715154177888482864e1), + SC_(0.1760916970670223236083984375e-1), SC_(0.7444499991834163665771484375e-2), SC_(-0.3223578342484987127570834604684301589064e1), + SC_(0.1760916970670223236083984375e-1), SC_(0.1433600485324859619140625e-1), SC_(-0.2805041020636338065368011730266984060386e1), + SC_(0.1760916970670223236083984375e-1), 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SC_(-0.1499853482621027133664160846069426563295e-1), + SC_(0.95070552825927734375e1), SC_(0.10074598388671875e4), SC_(-0.2056061337856806920336623108105325621513e-1), + SC_(0.95070552825927734375e1), SC_(0.1185395751953125e4), SC_(0.200124531064386856030146552792672982826e-1), + SC_(0.95070552825927734375e1), SC_(0.353451806640625e4), SC_(0.3033471650974938272104310339983302036505e-2), + SC_(0.95070552825927734375e1), SC_(0.80715478515625e4), SC_(0.6313804457726326168433814890910007150901e-2), + SC_(0.95070552825927734375e1), SC_(0.1622925e5), SC_(0.1403165866529678983739736802708333117488e-2), + SC_(0.95070552825927734375e1), SC_(0.3206622265625e5), SC_(-0.1035230096656936586615653548264063815176e-3), + SC_(0.95070552825927734375e1), SC_(0.3636794921875e5), SC_(-0.3178411510645339458375062818266053238935e-2), + SC_(0.24750102996826171875e2), SC_(0.1097540378570556640625e1), SC_(-0.2538488265063196712152114613748075086351e30), + SC_(0.24750102996826171875e2), SC_(0.30944411754608154296875e1), SC_(-0.2001658856691900027586107047291395903374e19), + SC_(0.24750102996826171875e2), SC_(0.51139926910400390625e1), SC_(-0.9503574259305993133759318213869719739108e13), + SC_(0.24750102996826171875e2), SC_(0.95070552825927734375e1), SC_(-0.4122129578428727018796511082180354925791e7), + SC_(0.24750102996826171875e2), SC_(0.24750102996826171875e2), SC_(-0.2658990584952737858668063924548287218421e0), + SC_(0.24750102996826171875e2), SC_(0.637722015380859375e2), SC_(-0.6691180036576346233956293507614315544893e-1), + SC_(0.24750102996826171875e2), SC_(0.1252804412841796875e3), SC_(0.7533197321533906425069813365786808047345e-2), + SC_(0.24750102996826171875e2), SC_(0.25554705810546875e3), SC_(-0.1542283080029640948900332378050454496484e-1), + SC_(0.24750102996826171875e2), SC_(0.503011474609375e3), SC_(-0.2991988069469062154012100511573471727669e-1), + SC_(0.24750102996826171875e2), SC_(0.10074598388671875e4), SC_(0.1184200276216747266219053634744663039601e-1), + SC_(0.24750102996826171875e2), SC_(0.1185395751953125e4), SC_(0.147538779157775165689894252998980132164e-1), + SC_(0.24750102996826171875e2), SC_(0.353451806640625e4), SC_(0.1337803088680303126911599983254711741179e-1), + SC_(0.24750102996826171875e2), SC_(0.80715478515625e4), SC_(0.8029832288031143409523993739201196219101e-2), + SC_(0.24750102996826171875e2), SC_(0.1622925e5), SC_(-0.5198547432330736842863301364433701996388e-2), + SC_(0.24750102996826171875e2), SC_(0.3206622265625e5), SC_(0.4109381361370256630410240644020816524515e-2), + SC_(0.24750102996826171875e2), SC_(0.3636794921875e5), SC_(-0.3695089835104638523036034542189491664868e-2), + SC_(0.637722015380859375e2), SC_(0.24750102996826171875e2), SC_(-0.6267975543587496709349337927794592271998e18), + SC_(0.637722015380859375e2), SC_(0.637722015380859375e2), SC_(-0.1939298752945934292999130903839341072815e0), + SC_(0.637722015380859375e2), SC_(0.1252804412841796875e3), SC_(-0.7053508497894543897551296718023593343408e-2), + SC_(0.637722015380859375e2), SC_(0.25554705810546875e3), SC_(-0.3550962870822991127395561395887028412934e-1), + SC_(0.637722015380859375e2), SC_(0.503011474609375e3), SC_(-0.2648416216528672574838824635772159441549e-1), + SC_(0.637722015380859375e2), SC_(0.10074598388671875e4), SC_(-0.1420732900757196265534805768438896498951e-1), + SC_(0.637722015380859375e2), SC_(0.1185395751953125e4), SC_(-0.1724003388206597138922485614964423834268e-1), + SC_(0.637722015380859375e2), SC_(0.353451806640625e4), SC_(-0.4904153149047725659903093785508227452329e-2), + SC_(0.637722015380859375e2), SC_(0.80715478515625e4), SC_(-0.5165055881176750174649276537848635493731e-2), + SC_(0.637722015380859375e2), SC_(0.1622925e5), SC_(-0.3111673836710409913488025509896936578918e-2), + SC_(0.637722015380859375e2), SC_(0.3206622265625e5), SC_(0.1643282532322944952844854407146135350371e-2), + SC_(0.637722015380859375e2), SC_(0.3636794921875e5), SC_(0.2009528090577840712976850241949224122986e-2), + SC_(0.1252804412841796875e3), SC_(0.637722015380859375e2), SC_(-0.3797174624982978861579399751499143354312e23), + SC_(0.1252804412841796875e3), SC_(0.1252804412841796875e3), SC_(-0.154839290844967491628030470782884890493e0), + SC_(0.1252804412841796875e3), SC_(0.25554705810546875e3), SC_(0.5249253389690028543295568223911202330116e-1), + SC_(0.1252804412841796875e3), SC_(0.503011474609375e3), SC_(0.2263770495318548220431423611155582287596e-1), + SC_(0.1252804412841796875e3), SC_(0.10074598388671875e4), SC_(0.1928421153706604388682313816599232350549e-1), + SC_(0.1252804412841796875e3), SC_(0.1185395751953125e4), SC_(0.2303577167941297280388538081309628427215e-1), + SC_(0.1252804412841796875e3), SC_(0.353451806640625e4), SC_(0.4602232701140352366182281154234657598423e-2), + SC_(0.1252804412841796875e3), SC_(0.80715478515625e4), SC_(0.7605554856039814835669547074154003071661e-2), + SC_(0.1252804412841796875e3), SC_(0.1622925e5), SC_(-0.3591967493051195246860689634970817279493e-2), + SC_(0.1252804412841796875e3), SC_(0.3206622265625e5), SC_(0.2427101323753481633835014493220406978017e-2), + SC_(0.1252804412841796875e3), SC_(0.3636794921875e5), SC_(-0.4144432737996561571547765270607343786669e-2), + SC_(0.25554705810546875e3), SC_(0.25554705810546875e3), SC_(-0.1220899231301180796588329975642599451792e0), + SC_(0.25554705810546875e3), SC_(0.503011474609375e3), SC_(-0.2587460388550158251245313556927425626503e-1), + SC_(0.25554705810546875e3), SC_(0.10074598388671875e4), SC_(-0.2709588740650161660564368137667279157517e-2), + SC_(0.25554705810546875e3), SC_(0.1185395751953125e4), SC_(0.7388450600504248491679263004388188205535e-2), + SC_(0.25554705810546875e3), SC_(0.353451806640625e4), SC_(-0.4070364240355399016905231104224278184321e-3), + SC_(0.25554705810546875e3), SC_(0.80715478515625e4), SC_(0.8869456474018290057215067339235844280777e-2), + SC_(0.25554705810546875e3), SC_(0.1622925e5), SC_(0.6193548860770801980645313593061881324053e-2), + SC_(0.25554705810546875e3), SC_(0.3206622265625e5), SC_(-0.3573956395914134221575915050921377055356e-2), + SC_(0.25554705810546875e3), SC_(0.3636794921875e5), SC_(0.4150896139122617211447974747314758032795e-2), + SC_(0.503011474609375e3), SC_(0.503011474609375e3), SC_(-0.9741864482364844408259833924093581326514e-1), + SC_(0.503011474609375e3), SC_(0.10074598388671875e4), SC_(-0.1584672339561671902535695362033324616336e-1), + SC_(0.503011474609375e3), SC_(0.1185395751953125e4), SC_(0.5943568069873248119538400255697357733917e-2), + SC_(0.503011474609375e3), SC_(0.353451806640625e4), SC_(0.1015104946539387860338179999806441182473e-1), + SC_(0.503011474609375e3), SC_(0.80715478515625e4), SC_(0.8883753528547744135823658482106607009966e-2), + SC_(0.503011474609375e3), SC_(0.1622925e5), SC_(0.5522874844418632574706826538336184699582e-2), + SC_(0.503011474609375e3), SC_(0.3206622265625e5), SC_(0.4455587256750903432452764772837804809816e-2), + SC_(0.503011474609375e3), SC_(0.3636794921875e5), SC_(-0.3843435197254867050137588280055312232508e-2), + SC_(0.10074598388671875e4), SC_(0.10074598388671875e4), SC_(-0.7728430027806412142852417589285707550701e-1), + SC_(0.10074598388671875e4), SC_(0.1185395751953125e4), SC_(0.3043120394295723532238172036669867077737e-1), + SC_(0.10074598388671875e4), SC_(0.353451806640625e4), SC_(-0.4742168167846528568912593945029413563666e-2), + SC_(0.10074598388671875e4), SC_(0.80715478515625e4), SC_(-0.7420503792165366240145846571768477365506e-2), + SC_(0.10074598388671875e4), SC_(0.1622925e5), SC_(-0.1795717524473915811452628563407404683787e-2), + SC_(0.10074598388671875e4), SC_(0.3206622265625e5), SC_(0.7500534979641494608228235889183901707366e-3), + SC_(0.10074598388671875e4), SC_(0.3636794921875e5), SC_(0.3051253358553692403837737104117441878866e-2), + SC_(0.1185395751953125e4), SC_(0.10074598388671875e4), SC_(-0.7251757054382535422878260094266093685031e29), + SC_(0.1185395751953125e4), SC_(0.1185395751953125e4), SC_(-0.7320587935179758776740758943706909907242e-1), + SC_(0.1185395751953125e4), SC_(0.353451806640625e4), SC_(0.4795852538068195975343600795948509285902e-3), + SC_(0.1185395751953125e4), SC_(0.80715478515625e4), SC_(0.1749093092873678394701882014413670671251e-2), + SC_(0.1185395751953125e4), SC_(0.1622925e5), SC_(0.4162877531169379039027191426399181611799e-2), + SC_(0.1185395751953125e4), SC_(0.3206622265625e5), SC_(-0.3200560495047506921416083730251596135541e-3), + SC_(0.1185395751953125e4), SC_(0.3636794921875e5), SC_(-0.4176563093033758974497368789398642132897e-2) + }; +#undef SC_ + + diff --git a/test/beta_exp_data.ipp b/test/beta_exp_data.ipp new file mode 100644 index 000000000..2c787afc6 --- /dev/null +++ b/test/beta_exp_data.ipp @@ -0,0 +1,364 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 351> beta_exp_data = { { + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1155631.551635027016649268884796909927277) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1039549.452063747329381617654200841254652) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.216575062950141727924346923828125e-5), SC_(923467.3524924676425690820378921903570447) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(715366.9882608199489156088500706918884474) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.216575062950141727924346923828125e-5), SC_(599284.8886895402674421924477675861806454) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.72700195232755504548549652099609375e-5), SC_(275102.4248866129549503632384850636365051) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(649244.3230419389091055494874477610228104) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.216575062950141727924346923828125e-5), SC_(533162.2234706592346717218748633348329255) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(208979.7596677320047637517666971035319141) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.14000004739500582218170166015625e-4), SC_(142857.0944488511634633414470885024340382) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(635967.2966761120408405544283252854891051) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(519885.1971048323697502064131013612391462) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(195702.7333019051790657557774869592595619) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(129580.0680830243894812627999923649303964) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(116303.0417171976400618567245671670493297) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(594458.8435535046961781833034439822035691) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(478376.7439822250699480739314338811953296) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(154194.2801792984055346510144704788292969) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(88071.61496041830983457563965247905141874) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(74794.58859459188997822531683455967562115) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.60085076256655156612396240234375e-4), SC_(33286.13547199056171829186221434157408122) }, + { SC_(0.000116783194243907928466796875), SC_(0.1730655412757187150418758392333984375e-5), SC_(586378.651529572636659503110266369369984) }, + { SC_(0.000116783194243907928466796875), SC_(0.216575062950141727924346923828125e-5), SC_(470296.5519582930697306370777452683761698) }, + { SC_(0.000116783194243907928466796875), SC_(0.72700195232755504548549652099609375e-5), SC_(146114.0881553671010006794541334417041727) }, + { SC_(0.000116783194243907928466796875), SC_(0.14000004739500582218170166015625e-4), SC_(79991.42293648792255405447160736354988824) }, + { SC_(0.000116783194243907928466796875), SC_(0.17196454791701398789882659912109375e-4), 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SC_(0.001567578372048954697546190363817432566287) }, + { SC_(7.88237094879150390625), SC_(7.88237094879150390625), SC_(0.2304086783008952665744998296791021930007e-4) }, + { SC_(15.848876953125), SC_(0.1730655412757187150418758392333984375e-5), SC_(577812.467396494064942097256551467225537) }, + { SC_(15.848876953125), SC_(0.216575062950141727924346923828125e-5), SC_(461730.3678286549639904324126292681940118) }, + { SC_(15.848876953125), SC_(0.72700195232755504548549652099609375e-5), SC_(137547.9040660901316000472092040840673884) }, + { SC_(15.848876953125), SC_(0.14000004739500582218170166015625e-4), SC_(71425.23890042634784384697230625424169406) }, + { SC_(15.848876953125), SC_(0.17196454791701398789882659912109375e-4), SC_(58148.21255987503466037789783804288733992) }, + { SC_(15.848876953125), SC_(0.60085076256655156612396240234375e-4), SC_(16639.75977638419101692348021851942073003) }, + { SC_(15.848876953125), SC_(0.000116783194243907928466796875), SC_(8559.568200701221551218767963648849246027) }, + { SC_(15.848876953125), SC_(0.000149052008055150508880615234375), SC_(6705.760112274562377109693147086107759687) }, + { SC_(15.848876953125), SC_(0.0003985252114944159984588623046875), SC_(2505.945598047360836406987017984692903213) }, + { SC_(15.848876953125), SC_(0.00063875340856611728668212890625), SC_(1562.244937662732265726871770794485810834) }, + { SC_(15.848876953125), SC_(0.0010718167759478092193603515625), SC_(929.6935540045242455486443804204735207793) }, + { SC_(15.848876953125), SC_(0.00302191521041095256805419921875), SC_(327.6263775091254046984727283245152516425) }, + { SC_(15.848876953125), SC_(0.00499413348734378814697265625), SC_(196.9575542754211734070057984061189034343) }, + { SC_(15.848876953125), SC_(0.00928423367440700531005859375), SC_(104.4584212615345465801581259971762769937) }, + { SC_(15.848876953125), SC_(0.0241700224578380584716796875), SC_(38.21136981001168010710740679190694930632) }, + { SC_(15.848876953125), SC_(0.06227754056453704833984375), SC_(13.10618362350255476392166743793259749139) }, + { SC_(15.848876953125), SC_(0.12234418094158172607421875), SC_(5.514029391894522673178885977576812901954) }, + { SC_(15.848876953125), SC_(0.249557673931121826171875), SC_(1.833598229039232723593123386645917772312) }, + { SC_(15.848876953125), SC_(0.4912221431732177734375), SC_(0.467845046165576115379722912718972755625) }, + { SC_(15.848876953125), SC_(0.98384749889373779296875), SC_(0.06664123970861700317203157698863069745986) }, + { SC_(15.848876953125), SC_(1.1576130390167236328125), SC_(0.03776996613323409917542405876735205164805) }, + { SC_(15.848876953125), SC_(3.451677799224853515625), SC_(0.0001767056825017873997922509336879545386291) }, + { SC_(15.848876953125), SC_(7.88237094879150390625), SC_(0.310532280359287823834522315221229918373e-6) }, + { SC_(15.848876953125), SC_(15.848876953125), SC_(0.2576651451920685510042001174792567650789e-9) }, + { SC_(31.314670562744140625), SC_(0.1730655412757187150418758392333984375e-5), SC_(577811.7705852061307949254353294671322771) }, + { SC_(31.314670562744140625), SC_(0.216575062950141727924346923828125e-5), SC_(461729.6710184828160166319994100531668849) }, + { SC_(31.314670562744140625), SC_(0.72700195232755504548549652099609375e-5), SC_(137547.2072690075496939939676230577664587) }, + { SC_(31.314670562744140625), SC_(0.14000004739500582218170166015625e-4), SC_(71424.54212060195465862284514291276978183) }, + { SC_(31.314670562744140625), SC_(0.17196454791701398789882659912109375e-4), SC_(58147.51578824736212268726758935041995717) }, + { SC_(31.314670562744140625), SC_(0.60085076256655156612396240234375e-4), SC_(16639.06311472621275702017697078807585708) }, + { SC_(31.314670562744140625), SC_(0.000116783194243907928466796875), SC_(8558.871684391681177902610046087055681713) }, + { SC_(31.314670562744140625), SC_(0.000149052008055150508880615234375), SC_(6705.063678672548887519293397395332516276) }, + { SC_(31.314670562744140625), SC_(0.0003985252114944159984588623046875), SC_(2505.24980349352938047593785567583576373) }, + { SC_(31.314670562744140625), SC_(0.00063875340856611728668212890625), SC_(1561.549757854035533455898102964904417179) }, + { SC_(31.314670562744140625), SC_(0.0010718167759478092193603515625), SC_(928.999480869800191174333706227134595703) }, + { SC_(31.314670562744140625), SC_(0.00302191521041095256805419921875), SC_(326.9372633593889344749789623402894509337) }, + { SC_(31.314670562744140625), SC_(0.00499413348734378814697265625), SC_(196.2734150111948032449933491989589331273) }, + { SC_(31.314670562744140625), SC_(0.00928423367440700531005859375), SC_(103.7849653265487425512027398001941283229) }, + { SC_(31.314670562744140625), SC_(0.0241700224578380584716796875), SC_(37.57355943415404987582558557137728471079) }, + { SC_(31.314670562744140625), SC_(0.06227754056453704833984375), SC_(12.55038605336216829222092298247150246565) }, + { SC_(31.314670562744140625), SC_(0.12234418094158172607421875), SC_(5.06466450422007092993671829951452526177) }, + { SC_(31.314670562744140625), SC_(0.249557673931121826171875), SC_(1.542483843415710154392732807527862496098) }, + { SC_(31.314670562744140625), SC_(0.4912221431732177734375), SC_(0.3335279709424437595711652140518853567302) }, + { SC_(31.314670562744140625), SC_(0.98384749889373779296875), SC_(0.03409297661295516414885174102503442819277) }, + { SC_(31.314670562744140625), SC_(1.1576130390167236328125), SC_(0.01721838700844971854228152401760085523744) }, + { SC_(31.314670562744140625), SC_(3.451677799224853515625), SC_(0.1900362128922583001056640211044473004449e-4) }, + { SC_(31.314670562744140625), SC_(7.88237094879150390625), SC_(0.2885050414250820990910991383845819731228e-8) }, + { SC_(31.314670562744140625), SC_(15.848876953125), SC_(0.6530600415427592338710565326600917620279e-13) }, + { SC_(31.314670562744140625), SC_(31.314670562744140625), SC_(0.8915675179813854663775491877023168025702e-19) }, + { SC_(35.515575408935546875), SC_(0.1730655412757187150418758392333984375e-5), SC_(577811.6427939229375161086906177400275569) }, + { SC_(35.515575408935546875), SC_(0.216575062950141727924346923828125e-5), SC_(461729.5432274267192940591041545317110485) }, + { SC_(35.515575408935546875), SC_(0.72700195232755504548549652099609375e-5), SC_(137547.0794806155767312939551779369164394) }, + { SC_(35.515575408935546875), SC_(0.14000004739500582218170166015625e-4), SC_(71424.41433572253924394173477030901070617) }, + { SC_(35.515575408935546875), SC_(0.17196454791701398789882659912109375e-4), SC_(58147.38800503622156242701682984885441441) }, + { SC_(35.515575408935546875), SC_(0.60085076256655156612396240234375e-4), SC_(16638.93535389697145567616644195839808463) }, + { SC_(35.515575408935546875), SC_(0.000116783194243907928466796875), SC_(8558.7439531443773341365422276985698545) }, + { SC_(35.515575408935546875), SC_(0.000149052008055150508880615234375), SC_(6704.935964257969942256683796604190388483) }, + { SC_(35.515575408935546875), SC_(0.0003985252114944159984588623046875), SC_(2505.122219132224374561111132130667265447) }, + { SC_(35.515575408935546875), SC_(0.00063875340856611728668212890625), SC_(1561.422298589200014741982262412434636973) }, + { SC_(35.515575408935546875), SC_(0.0010718167759478092193603515625), SC_(928.8722467787222242808854452351889563633) }, + { SC_(35.515575408935546875), SC_(0.00302191521041095256805419921875), SC_(326.8110378414911903385288649321631400664) }, + { SC_(35.515575408935546875), SC_(0.00499413348734378814697265625), SC_(196.1482005962757034094399257872899036696) }, + { SC_(35.515575408935546875), SC_(0.00928423367440700531005859375), SC_(103.6619197982581616421872226071400592126) }, + { SC_(35.515575408935546875), SC_(0.0241700224578380584716796875), SC_(37.45772609209636964536574943912854380171) }, + { SC_(35.515575408935546875), SC_(0.06227754056453704833984375), SC_(12.450993041772750669407167680667607419) }, + { SC_(35.515575408935546875), SC_(0.12234418094158172607421875), SC_(4.986241033913772097760869076957541032543) }, + { SC_(35.515575408935546875), SC_(0.249557673931121826171875), SC_(1.494248013206312554952038234048230712297) }, + { SC_(35.515575408935546875), SC_(0.4912221431732177734375), SC_(0.3133802674518380533642801450941624241265) }, + { SC_(35.515575408935546875), SC_(0.98384749889373779296875), SC_(0.03012063183422189552497230190183310697765) }, + { SC_(35.515575408935546875), SC_(1.1576130390167236328125), SC_(0.01488854601694810064869001826934615844448) }, + { SC_(35.515575408935546875), SC_(3.451677799224853515625), SC_(0.1249353030883651639444207203869412646859e-4) }, + { SC_(35.515575408935546875), SC_(7.88237094879150390625), SC_(0.1169663515742618634451758579426537250388e-8) }, + { SC_(35.515575408935546875), SC_(15.848876953125), SC_(0.1250851178791656557811931908536049655477e-13) }, + { SC_(35.515575408935546875), SC_(31.314670562744140625), SC_(0.5365332615381537353158263927841062696094e-20) }, + { SC_(35.515575408935546875), SC_(35.515575408935546875), SC_(0.2474098388705327630350038282012894724267e-21) } + } }; +#undef SC_ + + + + diff --git a/test/beta_med_data.ipp b/test/beta_med_data.ipp new file mode 100644 index 000000000..60e869fb0 --- /dev/null +++ b/test/beta_med_data.ipp @@ -0,0 +1,1842 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1830> beta_med_data = { { + { SC_(0.4883005917072296142578125), SC_(0.4883005917072296142578125), SC_(3.245912809500479157065104747353807392371) }, + { SC_(3.5808107852935791015625), SC_(0.4883005917072296142578125), SC_(1.007653173802923954909901438393379243537) }, + { SC_(3.5808107852935791015625), SC_(3.5808107852935791015625), SC_(0.01354763979296020361276499134253466025558) }, + { SC_(9.76306438446044921875), SC_(0.4883005917072296142578125), SC_(0.6040068168955603700205507368871869179054) }, + { SC_(9.76306438446044921875), SC_(3.5808107852935791015625), SC_(0.0006768556385450819292424677664459339662542) }, + { SC_(9.76306438446044921875), SC_(9.76306438446044921875), SC_(0.1522016641487441392375569251478701938958e-5) }, + { SC_(10.9950771331787109375), SC_(0.4883005917072296142578125), SC_(0.5691374355039684505283786467923310741394) }, + { SC_(10.9950771331787109375), SC_(3.5808107852935791015625), SC_(0.0004623291188878679140702882645611609028529) }, + { SC_(10.9950771331787109375), SC_(9.76306438446044921875), SC_(0.6525428759744388549684590935535556520911e-6) }, + { SC_(10.9950771331787109375), SC_(10.9950771331787109375), SC_(0.2595645778453125149921775190075494972069e-6) }, + { SC_(11.2553272247314453125), SC_(0.4883005917072296142578125), SC_(0.5625252582720929656922435362566677067416) }, + { SC_(11.2553272247314453125), SC_(3.5808107852935791015625), SC_(0.0004286883639355158965650882249453525366545) }, + { SC_(11.2553272247314453125), SC_(9.76306438446044921875), SC_(0.5507764130648082456092816216454348231704e-6) }, + { SC_(11.2553272247314453125), SC_(10.9950771331787109375), SC_(0.215765220969841623426272969427339892725e-6) }, + { SC_(11.2553272247314453125), SC_(11.2553272247314453125), SC_(0.1787990823319435797792105752447442768941e-6) }, + { SC_(12.70741176605224609375), SC_(0.4883005917072296142578125), SC_(0.5294907274516666109456973690863173323156) }, + { SC_(12.70741176605224609375), SC_(3.5808107852935791015625), SC_(0.0002889727352835592358365667915148162805071) }, + { SC_(12.70741176605224609375), SC_(9.76306438446044921875), SC_(0.2252927431029049671249995618445578512183e-6) }, + { SC_(12.70741176605224609375), SC_(10.9950771331787109375), SC_(0.8131970313223237520922385412367530078239e-7) }, + { SC_(12.70741176605224609375), SC_(11.2553272247314453125), SC_(0.6627020372930043647557276756580896208093e-7) }, + { SC_(12.70741176605224609375), SC_(12.70741176605224609375), SC_(0.2245035027874900533178284692921746500427e-7) }, + { SC_(13.55634593963623046875), SC_(0.4883005917072296142578125), SC_(0.5127159488422374707026291329594215821195) }, + { SC_(13.55634593963623046875), SC_(3.5808107852935791015625), SC_(0.0002338050540339516890899323768977469084643) }, + { SC_(13.55634593963623046875), SC_(9.76306438446044921875), SC_(0.1385641428853223512568009007531028899044e-6) }, + { SC_(13.55634593963623046875), SC_(10.9950771331787109375), SC_(0.4779208342424526648121034774140947601376e-7) }, + { SC_(13.55634593963623046875), SC_(11.2553272247314453125), SC_(0.3858688985561532847951616400183530491078e-7) }, + { SC_(13.55634593963623046875), SC_(12.70741176605224609375), SC_(0.1243334579233090577547671408856256594842e-7) }, + { SC_(13.55634593963623046875), SC_(13.55634593963623046875), SC_(0.6695819170364036861651031065079190250677e-8) }, + { SC_(14.19721508026123046875), SC_(0.4883005917072296142578125), SC_(0.5010725503977163404563348242929462964601) }, + { SC_(14.19721508026123046875), SC_(3.5808107852935791015625), SC_(0.0002008402390975031689923237986077084533841) }, + { SC_(14.19721508026123046875), SC_(9.76306438446044921875), SC_(0.9753123839573875998586188687154385562149e-7) }, + { SC_(14.19721508026123046875), SC_(10.9950771331787109375), 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SC_(99.28820037841796875), SC_(96.76981353759765625), SC_(0.3484969370967557871486154959308680473194e-59) }, + { SC_(99.28820037841796875), SC_(96.88709259033203125), SC_(0.3207141270593581779254023249982845071696e-59) }, + { SC_(99.28820037841796875), SC_(97.0595703125), SC_(0.2838654062329006203358352082138793616487e-59) }, + { SC_(99.28820037841796875), SC_(98.1111602783203125), SC_(0.135331897584667014131258701945417599616e-59) }, + { SC_(99.28820037841796875), SC_(99.28820037841796875), SC_(0.5946341007968227163535653640336744451921e-60) }, + { SC_(99.64617156982421875), SC_(0.4883005917072296142578125), SC_(0.1920415401866252505960149155087364700429) }, + { SC_(99.64617156982421875), SC_(3.5808107852935791015625), SC_(0.2425007447493048943965527355796496259445e-6) }, + { SC_(99.64617156982421875), SC_(9.76306438446044921875), SC_(0.4338463329689989369521219399698331619339e-14) }, + { SC_(99.64617156982421875), SC_(10.9950771331787109375), SC_(0.2238793620730317918365780183630913591486e-15) }, + { SC_(99.64617156982421875), SC_(11.2553272247314453125), SC_(0.1217883523395548244503225558455778824065e-15) }, + { SC_(99.64617156982421875), SC_(12.70741176605224609375), SC_(0.4506937720795369966253108234128095796053e-17) }, + { SC_(99.64617156982421875), SC_(13.55634593963623046875), SC_(0.7053692076383500412728910873848177350615e-18) }, + { SC_(99.64617156982421875), SC_(14.19721508026123046875), SC_(0.1796747134889476173979330386276389059013e-18) }, + { SC_(99.64617156982421875), SC_(15.76973247528076171875), SC_(0.6990175412575383911153375435413556806998e-20) }, + { SC_(99.64617156982421875), SC_(17.126956939697265625), SC_(0.475219770637311911218530901770424242484e-21) }, + { SC_(99.64617156982421875), SC_(17.394779205322265625), SC_(0.2828061146103015181288433756774729344582e-21) }, + { SC_(99.64617156982421875), SC_(18.8463134765625), SC_(0.1805339957483893466812493111272198920639e-22) }, + { SC_(99.64617156982421875), SC_(21.200313568115234375), SC_(0.2551145044678329055180101519714681369991e-24) }, + { SC_(99.64617156982421875), SC_(22.111194610595703125), SC_(0.521827937109776436392706852004371656104e-25) }, + { SC_(99.64617156982421875), SC_(27.8570384979248046875), SC_(0.4585509723284941456360070341222362036382e-29) }, + { SC_(99.64617156982421875), SC_(29.709972381591796875), SC_(0.2797738978507146557808496231858841269347e-30) }, + { SC_(99.64617156982421875), SC_(30.198291778564453125), SC_(0.1359628995333712367120268567405627842371e-30) }, + { SC_(99.64617156982421875), SC_(30.8236217498779296875), SC_(0.5445278627462667801264942137517994423527e-31) }, + { SC_(99.64617156982421875), SC_(36.1357879638671875), SC_(0.3335676798398169433467181047773937919425e-34) }, + { SC_(99.64617156982421875), SC_(39.228778839111328125), SC_(0.5925480070878764501178860688204541780342e-36) }, + { SC_(99.64617156982421875), SC_(39.8798675537109375), SC_(0.2596346306023602973051248853685030558771e-36) }, + { SC_(99.64617156982421875), SC_(42.181911468505859375), SC_(0.1491806079618579894544070083102953058676e-37) }, + { SC_(99.64617156982421875), SC_(42.21454620361328125), SC_(0.1433543971257228329446310718021257196428e-37) }, + { SC_(99.64617156982421875), SC_(47.481121063232421875), SC_(0.2894219671371863633901114678428523677553e-40) }, + { SC_(99.64617156982421875), SC_(48.5427093505859375), SC_(0.8715859321707892083793914056580348529658e-41) }, + { SC_(99.64617156982421875), SC_(50.371234893798828125), SC_(0.1144304917248501740653089888841788532933e-41) }, + { SC_(99.64617156982421875), SC_(54.69268035888671875), SC_(0.1122326874270186317133861668699838855083e-43) }, + { SC_(99.64617156982421875), SC_(54.72658538818359375), SC_(0.108332632811810033121283078371423493224e-43) }, + { SC_(99.64617156982421875), SC_(63.23960113525390625), SC_(0.2262074110898120193155243766072235368719e-47) }, + { SC_(99.64617156982421875), SC_(63.979938507080078125), SC_(0.1121769230946183388130951853048359732083e-47) }, + { SC_(99.64617156982421875), SC_(65.55123138427734375), SC_(0.2576069444251818757386717313878324944765e-48) }, + { SC_(99.64617156982421875), SC_(65.57750701904296875), SC_(0.2513961622572047306221104339881351344355e-48) }, + { SC_(99.64617156982421875), SC_(67.8767242431640625), SC_(0.3044620609127321948461303153395712705542e-49) }, + { SC_(99.64617156982421875), SC_(68.13913726806640625), SC_(0.2399969117830299809884620846041547199087e-49) }, + { SC_(99.64617156982421875), SC_(72.586639404296875), SC_(0.4654774693873818497215598980564748298819e-51) }, + { SC_(99.64617156982421875), SC_(74.06732177734375), SC_(0.1298622741166349753386447918518841478102e-51) }, + { SC_(99.64617156982421875), SC_(74.31581878662109375), SC_(0.1049960916780689406311743991678991020359e-51) }, + { SC_(99.64617156982421875), SC_(75.77643585205078125), SC_(0.3039342449834348309579161767378331641473e-52) }, + { SC_(99.64617156982421875), SC_(79.222808837890625), SC_(0.1737297384572845553919699070496733332975e-53) }, + { SC_(99.64617156982421875), SC_(79.79488372802734375), SC_(0.1089347230301574574227379586143568671545e-53) }, + { SC_(99.64617156982421875), SC_(79.81259918212890625), SC_(0.1073754663619106685903160281830989629804e-53) }, + { SC_(99.64617156982421875), SC_(80.0300445556640625), SC_(0.8997668853899322225061748437651655966981e-54) }, + { SC_(99.64617156982421875), SC_(81.47422027587890625), SC_(0.2804407307548493204640087364124890413703e-54) }, + { SC_(99.64617156982421875), SC_(83.50250244140625), SC_(0.5586826121117789439615638077672711001932e-55) }, + { SC_(99.64617156982421875), SC_(84.9144439697265625), SC_(0.1846564298928271058587078878704235002733e-55) }, + { SC_(99.64617156982421875), SC_(87.8442840576171875), SC_(0.1933065218129760647098319832562443180994e-56) }, + { SC_(99.64617156982421875), SC_(90.58013916015625), SC_(0.246368912661760433645524568855257596525e-57) }, + { SC_(99.64617156982421875), SC_(91.3384552001953125), SC_(0.1402807696431983675171416011032570303907e-57) }, + { SC_(99.64617156982421875), SC_(91.57439422607421875), SC_(0.1178127749000117009577583032714136533809e-57) }, + { SC_(99.64617156982421875), SC_(93.3999786376953125), SC_(0.3085259587898341188615347001733307622013e-58) }, + { SC_(99.64617156982421875), SC_(95.71712493896484375), SC_(0.5784821309052389104753898006604136073638e-59) }, + { SC_(99.64617156982421875), SC_(95.75110626220703125), SC_(0.5645763032239863667263699872521759670044e-59) }, + { SC_(99.64617156982421875), SC_(95.94964599609375), SC_(0.489819130902001275392106966836686313721e-59) }, + { SC_(99.64617156982421875), SC_(96.48920440673828125), SC_(0.3333152772138028099176546108963123743121e-59) }, + { SC_(99.64617156982421875), SC_(96.76981353759765625), SC_(0.2730076878660219712815291918211112895904e-59) }, + { SC_(99.64617156982421875), SC_(96.88709259033203125), SC_(0.2511891524767099195545978116145973727058e-59) }, + { SC_(99.64617156982421875), SC_(97.0595703125), SC_(0.222258516470169272817490714164593170175e-59) }, + { SC_(99.64617156982421875), SC_(98.1111602783203125), SC_(0.1057582725557246879085406508118398211083e-59) }, + { SC_(99.64617156982421875), SC_(99.28820037841796875), SC_(0.4637012277372110272033377955412817752539e-60) }, + { SC_(99.64617156982421875), SC_(99.64617156982421875), SC_(0.3613651154665954292095783593077701467444e-60) } + } }; +#undef SC_ + + + diff --git a/test/beta_small_data.ipp b/test/beta_small_data.ipp new file mode 100644 index 000000000..e904ae2d7 --- /dev/null +++ b/test/beta_small_data.ipp @@ -0,0 +1,35 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 21> beta_small_data = { { + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1155631.551635027016649268884796909927277) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(1039549.452063747329381617654200841254652) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.216575062950141727924346923828125e-5), SC_(923467.3524924676425690820378921903570447) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(715366.9882608199489156088500706918884474) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.216575062950141727924346923828125e-5), SC_(599284.8886895402674421924477675861806454) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.72700195232755504548549652099609375e-5), SC_(275102.4248866129549503632384850636365051) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(649244.3230419389091055494874477610228104) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.216575062950141727924346923828125e-5), SC_(533162.2234706592346717218748633348329255) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(208979.7596677320047637517666971035319141) }, + { SC_(0.14000004739500582218170166015625e-4), SC_(0.14000004739500582218170166015625e-4), SC_(142857.0944488511634633414470885024340382) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(635967.2966761120408405544283252854891051) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(519885.1971048323697502064131013612391462) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(195702.7333019051790657557774869592595619) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(129580.0680830243894812627999923649303964) }, + { SC_(0.17196454791701398789882659912109375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(116303.0417171976400618567245671670493297) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.1730655412757187150418758392333984375e-5), SC_(594458.8435535046961781833034439822035691) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.216575062950141727924346923828125e-5), SC_(478376.7439822250699480739314338811953296) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.72700195232755504548549652099609375e-5), SC_(154194.2801792984055346510144704788292969) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.14000004739500582218170166015625e-4), SC_(88071.61496041830983457563965247905141874) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.17196454791701398789882659912109375e-4), SC_(74794.58859459188997822531683455967562115) }, + { SC_(0.60085076256655156612396240234375e-4), SC_(0.60085076256655156612396240234375e-4), SC_(33286.13547199056171829186221434157408122) } + } }; +#undef SC_ + + +// static_cast(BOOST_JOIN(x, L)) // changes type to T and adds suffix L to decimal digit string. + + diff --git a/test/binomial_data.ipp b/test/binomial_data.ipp new file mode 100644 index 000000000..2ad131388 --- /dev/null +++ b/test/binomial_data.ipp @@ -0,0 +1,169 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 159> binomial_data = { { + { SC_(0.15e2), SC_(0.15e2), SC_(0.1e1) }, + { SC_(0.19e2), SC_(0.15e2), SC_(0.3876e4) }, + { SC_(0.19e2), SC_(0.19e2), SC_(0.1e1) }, + { SC_(0.21e2), SC_(0.15e2), SC_(0.54264e5) }, + { SC_(0.21e2), SC_(0.19e2), SC_(0.21e3) }, + { SC_(0.21e2), SC_(0.21e2), SC_(0.1e1) }, + { SC_(0.29e2), SC_(0.15e2), SC_(0.7755876e8) }, + { SC_(0.29e2), SC_(0.19e2), SC_(0.2003001e8) }, + { SC_(0.29e2), SC_(0.21e2), SC_(0.4292145e7) }, + { SC_(0.29e2), SC_(0.29e2), SC_(0.1e1) }, + { SC_(0.33e2), SC_(0.15e2), SC_(0.103715832e10) }, + { SC_(0.33e2), SC_(0.19e2), SC_(0.8188092e9) }, + { SC_(0.33e2), SC_(0.21e2), SC_(0.35481732e9) }, + { SC_(0.33e2), SC_(0.29e2), SC_(0.4092e5) }, + { SC_(0.33e2), SC_(0.33e2), SC_(0.1e1) }, + { SC_(0.42e2), SC_(0.15e2), SC_(0.98672427616e11) }, + { SC_(0.42e2), SC_(0.19e2), SC_(0.4467753108e12) }, + { SC_(0.42e2), SC_(0.21e2), SC_(0.53825787444e12) }, + { SC_(0.42e2), SC_(0.29e2), SC_(0.2551873128e11) }, + { SC_(0.42e2), SC_(0.33e2), SC_(0.44589181e9) }, + { SC_(0.42e2), SC_(0.42e2), SC_(0.1e1) }, + { SC_(0.46e2), SC_(0.15e2), SC_(0.511738760544e12) }, + { SC_(0.46e2), SC_(0.19e2), SC_(0.415424667196e13) }, + { SC_(0.46e2), SC_(0.21e2), SC_(0.6943526580276e13) }, + { SC_(0.46e2), SC_(0.29e2), SC_(0.174969502686e13) }, + { SC_(0.46e2), SC_(0.33e2), SC_(0.10176623079e12) }, + { SC_(0.46e2), SC_(0.42e2), SC_(0.163185e6) }, + { SC_(0.46e2), SC_(0.46e2), SC_(0.1e1) }, + { SC_(0.82e2), SC_(0.15e2), SC_(0.996731056598616e16) }, + { SC_(0.82e2), SC_(0.19e2), SC_(0.19710382359693168e19) }, + { SC_(0.82e2), SC_(0.21e2), SC_(0.1833065559451464624e20) }, + { SC_(0.82e2), SC_(0.29e2), SC_(0.1257660762751009389168e23) }, + { SC_(0.82e2), SC_(0.33e2), SC_(0.899986590548788671513e23) }, + { SC_(0.82e2), SC_(0.42e2), SC_(0.41467066225715382349482e24) }, + { SC_(0.82e2), SC_(0.46e2), SC_(0.23223183395337370425708e24) }, + { SC_(0.82e2), SC_(0.82e2), SC_(0.1e1) }, + { SC_(0.95e2), SC_(0.15e2), SC_(0.110375347398090219e18) }, + { SC_(0.95e2), SC_(0.19e2), SC_(0.45038039715653129145e20) }, + { SC_(0.95e2), SC_(0.21e2), SC_(0.611230538998149609825e21) }, + { SC_(0.95e2), SC_(0.29e2), SC_(0.2146280142106099437545685e25) }, + { SC_(0.95e2), SC_(0.33e2), SC_(0.3780222443838484815806271e26) }, + { SC_(0.95e2), SC_(0.42e2), SC_(0.171987522327068244097964157e28) }, + { SC_(0.95e2), SC_(0.46e2), SC_(0.308620560869098008873280965e28) }, + { SC_(0.95e2), SC_(0.82e2), SC_(0.3489735464257595e16) }, + { SC_(0.95e2), SC_(0.95e2), SC_(0.1e1) }, + { SC_(0.122e3), SC_(0.15e2), SC_(0.6156937298378625264e19) }, + { SC_(0.122e3), SC_(0.19e2), SC_(0.819751088529043274604e22) }, + { SC_(0.122e3), SC_(0.21e2), SC_(0.205054879430622110547372e24) }, + { SC_(0.122e3), SC_(0.29e2), SC_(0.965499981454662046766678536e28) }, + { SC_(0.122e3), SC_(0.33e2), SC_(0.68890617994929806723272437813e30) }, + { SC_(0.122e3), SC_(0.42e2), SC_(0.982048893026815201335532502524191e33) }, + { SC_(0.122e3), SC_(0.46e2), SC_(0.9517963588769497111427223674615988e34) }, + { SC_(0.122e3), SC_(0.82e2), SC_(0.254605268562507644790693611765531e33) }, + { SC_(0.122e3), SC_(0.95e2), SC_(0.877923835320476575559398624e27) }, + { SC_(0.122e3), SC_(0.122e3), SC_(0.1e1) }, + { SC_(0.125e3), SC_(0.15e2), SC_(0.90648078331934398e19) }, + { SC_(0.125e3), SC_(0.19e2), SC_(0.1350175763944140060675e23) }, + { SC_(0.125e3), SC_(0.21e2), SC_(0.357796577445197116078875e24) }, + { SC_(0.125e3), SC_(0.29e2), SC_(0.21471697865846785089593512375e29) }, + { SC_(0.125e3), SC_(0.33e2), SC_(0.1743111472200107189546091504625e31) }, + { SC_(0.125e3), SC_(0.42e2), SC_(0.339619764433637564049548277312025e34) }, + { SC_(0.125e3), SC_(0.46e2), SC_(0.38244450869782214079034893241053e35) }, + { SC_(0.125e3), SC_(0.82e2), SC_(0.655545126697486460839825744579025e34) }, + { SC_(0.125e3), SC_(0.95e2), SC_(0.687094331707097122866992396e29) }, + { SC_(0.125e3), SC_(0.122e3), SC_(0.31775e6) }, + { SC_(0.125e3), SC_(0.125e3), SC_(0.1e1) }, + { SC_(0.135e3), SC_(0.15e2), SC_(0.307563739414613748e20) }, + { SC_(0.135e3), SC_(0.19e2), SC_(0.65183278299357679461e23) }, + { SC_(0.135e3), SC_(0.21e2), SC_(0.2070345077412932009547e25) }, + { SC_(0.135e3), SC_(0.29e2), SC_(0.2654598483954587033449048188e30) }, + { SC_(0.135e3), SC_(0.33e2), SC_(0.322268385697858320141605625441e32) }, + { SC_(0.135e3), SC_(0.42e2), SC_(0.165539796634100865734488051471458e36) }, + { SC_(0.135e3), SC_(0.46e2), SC_(0.2961867439565318449706072684149998e37) }, + { SC_(0.135e3), SC_(0.82e2), SC_(0.1323968752490674447090425029339391282e39) }, + { SC_(0.135e3), SC_(0.95e2), SC_(0.319215598884570762368184126129732e35) }, + { SC_(0.135e3), SC_(0.122e3), SC_(0.437531400061434e18) }, + { SC_(0.135e3), SC_(0.125e3), SC_(0.393812684240976e15) }, + { SC_(0.135e3), SC_(0.135e3), SC_(0.1e1) }, + { SC_(0.137e3), SC_(0.15e2), SC_(0.388194525997363728e20) }, + { SC_(0.137e3), SC_(0.19e2), SC_(0.87968625327656981292e23) }, + { SC_(0.137e3), SC_(0.21e2), SC_(0.28916543839848387707556e25) }, + { SC_(0.137e3), SC_(0.29e2), SC_(0.4280069137507949602563401336e30) }, + { SC_(0.137e3), SC_(0.33e2), SC_(0.560540007685072462740701644251e32) }, + { SC_(0.137e3), SC_(0.42e2), SC_(0.3453905364934565879468064249738192e36) }, + { SC_(0.137e3), SC_(0.46e2), SC_(0.67381580139170956477318127290699344e37) }, + { SC_(0.137e3), SC_(0.95e2), SC_(0.3453905364934565879468064249738192e36) }, + { SC_(0.137e3), SC_(0.122e3), SC_(0.388194525997363728e20) }, + { SC_(0.137e3), SC_(0.125e3), SC_(0.55587257066498976e17) }, + { SC_(0.137e3), SC_(0.135e3), SC_(0.9316e4) }, + { SC_(0.137e3), SC_(0.137e3), SC_(0.1e1) }, + { SC_(0.143e3), SC_(0.15e2), SC_(0.76435265515004093841e20) }, + { SC_(0.143e3), SC_(0.19e2), SC_(0.210374461432936964163e24) }, + { SC_(0.143e3), SC_(0.21e2), SC_(0.76395982994646537557478e25) }, + { SC_(0.143e3), SC_(0.29e2), SC_(0.17138496567725350060390313738e31) }, + { SC_(0.143e3), SC_(0.33e2), SC_(0.27947962468928852152878073112114e33) }, + { SC_(0.143e3), SC_(0.42e2), SC_(0.291038363317011408088847346619471914e37) }, + { SC_(0.143e3), SC_(0.46e2), SC_(0.728184459077800535203086713053471437e38) }, + { SC_(0.143e3), SC_(0.95e2), SC_(0.3005697554491346889987208985795179974e39) }, + { SC_(0.143e3), SC_(0.122e3), SC_(0.76395982994646537557478e25) }, + { SC_(0.143e3), SC_(0.125e3), SC_(0.31976918137806418552776e23) }, + { SC_(0.143e3), SC_(0.135e3), SC_(0.3553074951283e13) }, + { SC_(0.143e3), SC_(0.137e3), SC_(0.10679057389e11) }, + { SC_(0.143e3), SC_(0.143e3), SC_(0.1e1) }, + { SC_(0.144e3), SC_(0.15e2), SC_(0.85323087086516197776e20) }, + { SC_(0.144e3), SC_(0.19e2), SC_(0.242351379570743382715776e24) }, + { SC_(0.144e3), SC_(0.21e2), SC_(0.89439199603488629335584e25) }, + { SC_(0.144e3), SC_(0.29e2), SC_(0.214603783108908731190974363328e31) }, + { SC_(0.144e3), SC_(0.33e2), SC_(0.36256816175907700090220202956256e33) }, + { SC_(0.144e3), SC_(0.42e2), SC_(0.410877689388721987890137430521607408e37) }, + { SC_(0.144e3), SC_(0.46e2), SC_(0.1069985327624523235400453945711223336e39) }, + { SC_(0.144e3), SC_(0.122e3), SC_(0.500046434146777336739856e26) }, + { SC_(0.144e3), SC_(0.125e3), SC_(0.242351379570743382715776e24) }, + { SC_(0.144e3), SC_(0.135e3), SC_(0.56849199220528e14) }, + { SC_(0.144e3), SC_(0.137e3), SC_(0.219683466288e12) }, + { SC_(0.144e3), SC_(0.143e3), SC_(0.144e3) }, + { SC_(0.144e3), SC_(0.144e3), SC_(0.1e1) }, + { SC_(0.145e3), SC_(0.15e2), SC_(0.95168058673421912904e20) }, + { SC_(0.145e3), SC_(0.19e2), SC_(0.27889642887109357534752e24) }, + { SC_(0.145e3), SC_(0.21e2), SC_(0.10458616082666009075532e26) }, + { SC_(0.145e3), SC_(0.29e2), SC_(0.26825472888613591398871795416e31) }, + { SC_(0.145e3), SC_(0.33e2), SC_(0.4693962808488050458108865561301e33) }, + { SC_(0.145e3), SC_(0.42e2), SC_(0.57842004816858920625310609151100072e37) }, + { SC_(0.145e3), SC_(0.46e2), SC_(0.156715022732884716296026082957704428e39) }, + { SC_(0.145e3), SC_(0.122e3), SC_(0.315246665005577016640344e27) }, + { SC_(0.145e3), SC_(0.125e3), SC_(0.1757047501887889524689376e25) }, + { SC_(0.145e3), SC_(0.135e3), SC_(0.824313388697656e15) }, + { SC_(0.145e3), SC_(0.137e3), SC_(0.398176282647e13) }, + { SC_(0.145e3), SC_(0.143e3), SC_(0.1044e5) }, + { SC_(0.145e3), SC_(0.144e3), SC_(0.145e3) }, + { SC_(0.145e3), SC_(0.145e3), SC_(0.1e1) }, + { SC_(0.148e3), SC_(0.15e2), SC_(0.131439605927052712464e21) }, + { SC_(0.148e3), SC_(0.19e2), SC_(0.42244624913775365779698e24) }, + { SC_(0.148e3), SC_(0.21e2), SC_(0.16608172537529972375104128e26) }, + { SC_(0.148e3), SC_(0.29e2), SC_(0.51863815313548295889610045344e31) }, + { SC_(0.148e3), SC_(0.33e2), SC_(0.100644585365316213131406856592232e34) }, + { SC_(0.148e3), SC_(0.42e2), SC_(0.1587255130729102485141868921944628536e38) }, + { SC_(0.148e3), SC_(0.122e3), SC_(0.6418858594895863473128136624e29) }, + { SC_(0.148e3), SC_(0.125e3), SC_(0.525225250880542723214261376e27) }, + { SC_(0.148e3), SC_(0.135e3), SC_(0.1525832904625819216e19) }, + { SC_(0.148e3), SC_(0.137e3), SC_(0.12775329171405528e17) }, + { SC_(0.148e3), SC_(0.143e3), SC_(0.552689424e9) }, + { SC_(0.148e3), SC_(0.144e3), SC_(0.19190605e8) }, + { SC_(0.148e3), SC_(0.145e3), SC_(0.529396e6) }, + { SC_(0.148e3), SC_(0.148e3), SC_(0.1e1) }, + { SC_(0.149e3), SC_(0.15e2), SC_(0.146152994650230254904e21) }, + { SC_(0.149e3), SC_(0.19e2), SC_(0.484188393242502269321154e24) }, + { SC_(0.149e3), SC_(0.21e2), SC_(0.19332950844468483467894649e26) }, + { SC_(0.149e3), SC_(0.29e2), SC_(0.643975706809891340629324729688e31) }, + { SC_(0.149e3), SC_(0.33e2), SC_(0.129276234650276859970513979588298e34) }, + { SC_(0.149e3), SC_(0.42e2), SC_(0.2210289854940525890524658592240650952e38) }, + { SC_(0.149e3), SC_(0.122e3), SC_(0.35422590023684579907262679888e30) }, + { SC_(0.149e3), SC_(0.125e3), SC_(0.3260773432550036073288539376e28) }, + { SC_(0.149e3), SC_(0.135e3), SC_(0.16239221627803361656e20) }, + { SC_(0.149e3), SC_(0.137e3), SC_(0.158627003878285306e18) }, + { SC_(0.149e3), SC_(0.143e3), SC_(0.13725120696e11) }, + { SC_(0.149e3), SC_(0.144e3), SC_(0.571880029e9) }, + { SC_(0.149e3), SC_(0.145e3), SC_(0.19720001e8) }, + { SC_(0.149e3), SC_(0.148e3), SC_(0.149e3) }, + { SC_(0.149e3), SC_(0.149e3), SC_(0.1e1) } + } }; +#undef SC_ + diff --git a/test/binomial_large_data.ipp b/test/binomial_large_data.ipp new file mode 100644 index 000000000..7441d393d --- /dev/null +++ b/test/binomial_large_data.ipp @@ -0,0 +1,240 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 230> binomial_large_data = { { + { SC_(0.174e3), SC_(0.4e1), SC_(0.36890001e8) }, + { SC_(0.174e3), SC_(0.5e1), SC_(0.1254260034e10) }, + { SC_(0.174e3), SC_(0.6e1), SC_(0.35328324291e11) }, + { SC_(0.174e3), SC_(0.8e1), SC_(0.17699490469791e14) }, + { SC_(0.174e3), SC_(0.9e1), SC_(0.326457268665034e15) }, + { SC_(0.174e3), SC_(0.11e2), SC_(0.80308488091598364e17) }, + { SC_(0.174e3), SC_(0.13e2), SC_(0.13593756003504784614e20) }, + { SC_(0.174e3), SC_(0.22e2), SC_(0.4359346248249975839021623926e28) }, + { SC_(0.174e3), SC_(0.25e2), SC_(0.1087562120541667885404612090756e31) }, + { SC_(0.174e3), SC_(0.32e2), SC_(0.90596769259248050560796925567405729e35) }, + { SC_(0.174e3), SC_(0.33e2), SC_(0.389840643479188581201004952441564046e36) }, + { SC_(0.174e3), SC_(0.36e2), SC_(0.24968911018132735303785934846085665809e38) }, + { SC_(0.174e3), SC_(0.38e2), SC_(0.335748386706129085101975024324392317059e39) }, + { SC_(0.181e3), SC_(0.4e1), SC_(0.43252665e8) }, + { SC_(0.181e3), SC_(0.5e1), SC_(0.1531144341e10) }, + { SC_(0.181e3), SC_(0.6e1), SC_(0.44913567336e11) }, + { SC_(0.181e3), SC_(0.8e1), SC_(0.2442175223895e14) }, + { SC_(0.181e3), SC_(0.9e1), SC_(0.46944034859315e15) }, + { SC_(0.181e3), SC_(0.11e2), SC_(0.12551981393474298e18) }, + { SC_(0.181e3), SC_(0.13e2), SC_(0.2311656573298183215e20) }, + { SC_(0.181e3), SC_(0.22e2), SC_(0.109786047308135746805973924e29) }, + { SC_(0.181e3), SC_(0.25e2), SC_(0.3137775924888208973244704900292e31) }, + { SC_(0.181e3), SC_(0.32e2), SC_(0.362807370049633357494117222655465275e36) }, + { SC_(0.181e3), SC_(0.33e2), SC_(0.1638130246587738492927983823504979575e37) }, + { SC_(0.181e3), SC_(0.36e2), SC_(0.121459327734252437983487804513759407155e39) }, + { SC_(0.183e3), SC_(0.4e1), SC_(0.45212895e8) }, + { SC_(0.183e3), SC_(0.5e1), SC_(0.1618621641e10) }, + { SC_(0.183e3), SC_(0.6e1), SC_(0.48019108683e11) }, + { SC_(0.183e3), SC_(0.8e1), SC_(0.26712344173086e14) }, + { SC_(0.183e3), SC_(0.9e1), SC_(0.51940669225445e15) }, + { SC_(0.183e3), SC_(0.11e2), SC_(0.14213800227494049e18) }, + { SC_(0.183e3), SC_(0.13e2), SC_(0.2679848027506762623e20) }, + { SC_(0.183e3), SC_(0.22e2), SC_(0.14194619920981246828881085065e29) }, + { SC_(0.183e3), SC_(0.25e2), SC_(0.4212963192547234058811906047292e31) }, + { SC_(0.183e3), SC_(0.32e2), SC_(0.533495022820003911907243629923308011e36) }, + { SC_(0.183e3), SC_(0.33e2), SC_(0.2441143892297593657514963276315742717e37) }, + { SC_(0.183e3), SC_(0.36e2), SC_(0.188487763000513078905882248491998453765e39) }, + { SC_(0.197e3), SC_(0.4e1), SC_(0.60862165e8) }, + { SC_(0.197e3), SC_(0.5e1), SC_(0.2349279569e10) }, + { SC_(0.197e3), SC_(0.6e1), SC_(0.75176946208e11) }, + { SC_(0.197e3), SC_(0.8e1), SC_(0.4871734603372e14) }, + { SC_(0.197e3), SC_(0.9e1), SC_(0.102306426670812e16) }, + { SC_(0.197e3), SC_(0.11e2), SC_(0.326971339639915152e18) }, + { SC_(0.197e3), SC_(0.13e2), SC_(0.7212233203211205372e20) }, + { SC_(0.197e3), SC_(0.22e2), SC_(0.79183342998439646916544716576e29) }, + { SC_(0.197e3), SC_(0.25e2), SC_(0.30226519812632630000675368493832e32) }, + { SC_(0.197e3), SC_(0.32e2), SC_(0.7012229196864814144578953941342321778e37) }, + { SC_(0.197e3), SC_(0.33e2), SC_(0.3506114598432407072289476970671160889e38) }, + { SC_(0.205e3), SC_(0.4e1), SC_(0.71452955e8) }, + { SC_(0.205e3), SC_(0.5e1), SC_(0.2872408791e10) }, + { SC_(0.205e3), SC_(0.6e1), SC_(0.957469597e11) }, + { SC_(0.205e3), SC_(0.8e1), SC_(0.67368244751775e14) }, + { SC_(0.205e3), SC_(0.9e1), SC_(0.1474616024011075e16) }, + { SC_(0.205e3), SC_(0.11e2), SC_(0.51236204034275715e18) }, + { SC_(0.205e3), SC_(0.13e2), SC_(0.122973458426368674425e21) }, + { SC_(0.205e3), SC_(0.22e2), SC_(0.1996995395393768863820790084e30) }, + { SC_(0.205e3), SC_(0.25e2), SC_(0.87236660026494642796327227908148e32) }, + { SC_(0.205e3), SC_(0.32e2), SC_(0.2797639369858790056445408204110589505e38) }, + { SC_(0.205e3), SC_(0.33e2), SC_(0.14666412454108202417122897554882787405e39) }, + { SC_(0.219e3), SC_(0.4e1), SC_(0.93240126e8) }, + { SC_(0.219e3), SC_(0.5e1), SC_(0.4009325418e10) }, + { SC_(0.219e3), SC_(0.6e1), SC_(0.142999273242e12) }, + { SC_(0.219e3), SC_(0.8e1), SC_(0.115308485402067e15) }, + { SC_(0.219e3), SC_(0.9e1), SC_(0.2703343379981793e16) }, + { SC_(0.219e3), SC_(0.11e2), SC_(0.1078634008612735407e19) }, + { SC_(0.219e3), SC_(0.13e2), SC_(0.297702986377114972332e21) }, + { SC_(0.219e3), SC_(0.22e2), SC_(0.92286864572641294222383786261e30) }, + { SC_(0.219e3), SC_(0.25e2), SC_(0.503521145580703624811857345830723e33) }, + { SC_(0.219e3), SC_(0.32e2), SC_(0.275143042079930237307916608031577699934e39) }, + { SC_(0.227e3), SC_(0.4e1), SC_(0.1077342e9) }, + { SC_(0.227e3), SC_(0.5e1), SC_(0.480494532e10) }, + { SC_(0.227e3), SC_(0.6e1), SC_(0.17778297684e12) }, + { SC_(0.227e3), SC_(0.8e1), SC_(0.1543537202493e15) }, + { SC_(0.227e3), SC_(0.9e1), SC_(0.37559405260663e16) }, + { SC_(0.227e3), SC_(0.11e2), SC_(0.161525929569174898e19) }, + { SC_(0.227e3), SC_(0.13e2), SC_(0.4808502672559283502e21) }, + { SC_(0.227e3), SC_(0.22e2), SC_(0.211447113241809535208273441052e31) }, + { SC_(0.227e3), SC_(0.25e2), SC_(0.1300776673899864041137122497734284e34) }, + { SC_(0.286e3), SC_(0.4e1), SC_(0.272963405e9) }, + { SC_(0.286e3), SC_(0.5e1), SC_(0.15395136042e11) }, + { SC_(0.286e3), SC_(0.6e1), SC_(0.721005537967e12) }, + { SC_(0.286e3), SC_(0.8e1), SC_(0.1005802725463965e16) }, + { SC_(0.286e3), SC_(0.9e1), SC_(0.3106812863099803e17) }, + { SC_(0.286e3), SC_(0.11e2), SC_(0.21592914273609648996e20) }, + { SC_(0.286e3), SC_(0.13e2), SC_(0.1042965442638773751185e23) }, + { SC_(0.286e3), SC_(0.22e2), SC_(0.425965175292651465971486597897325e33) }, + { SC_(0.286e3), SC_(0.25e2), SC_(0.561508034879773671402684888107554506e36) }, + { SC_(0.308e3), SC_(0.4e1), SC_(0.367704645e9) }, + { SC_(0.308e3), SC_(0.5e1), SC_(0.22356442416e11) }, + { SC_(0.308e3), SC_(0.6e1), SC_(0.1129000342008e13) }, + { SC_(0.308e3), SC_(0.8e1), SC_(0.1832649805164486e16) }, + { SC_(0.308e3), SC_(0.9e1), SC_(0.610883268388162e17) }, + { SC_(0.308e3), SC_(0.11e2), SC_(0.4948265543629273684e20) }, + { SC_(0.308e3), SC_(0.13e2), SC_(0.2788537951740619923768e23) }, + { SC_(0.308e3), SC_(0.22e2), SC_(0.231085830083090590416670728502104e34) }, + { SC_(0.308e3), SC_(0.25e2), SC_(0.3876354280333804915261626106361337072e37) }, + { SC_(0.353e3), SC_(0.4e1), SC_(0.6360354e9) }, + { SC_(0.353e3), SC_(0.5e1), SC_(0.4439527092e11) }, + { SC_(0.353e3), SC_(0.6e1), SC_(0.257492571336e13) }, + { SC_(0.353e3), SC_(0.8e1), SC_(0.552054876781122e16) }, + { SC_(0.353e3), SC_(0.9e1), SC_(0.2116210360994301e18) }, + { SC_(0.353e3), SC_(0.11e2), SC_(0.22699626628585414872e21) }, + { SC_(0.353e3), SC_(0.13e2), SC_(0.16969717029992873417964e24) }, + { SC_(0.353e3), SC_(0.22e2), SC_(0.5130106833626650018875548966208696e35) }, + { SC_(0.353e3), SC_(0.25e2), SC_(0.133593446626656394879582879903973231364e39) }, + { SC_(0.358e3), SC_(0.4e1), SC_(0.673005095e9) }, + { SC_(0.358e3), SC_(0.5e1), SC_(0.47648760726e11) }, + { SC_(0.358e3), SC_(0.6e1), SC_(0.2803335422713e13) }, + { SC_(0.358e3), SC_(0.8e1), SC_(0.6184958895482796e16) }, + { SC_(0.358e3), SC_(0.9e1), SC_(0.2405261792687754e18) }, + { SC_(0.358e3), SC_(0.11e2), SC_(0.26556714113228463528e21) }, + { SC_(0.358e3), SC_(0.13e2), SC_(0.20438796217066896077556e24) }, + { SC_(0.358e3), SC_(0.22e2), SC_(0.7057427033366990358417620413368873e35) }, + { SC_(0.358e3), SC_(0.25e2), SC_(0.192263950446748795075597663873492257384e39) }, + { SC_(0.376e3), SC_(0.4e1), SC_(0.81957425e9) }, + { SC_(0.376e3), SC_(0.5e1), SC_(0.609763242e11) }, + { SC_(0.376e3), SC_(0.6e1), SC_(0.37703693797e13) }, + { SC_(0.376e3), SC_(0.8e1), SC_(0.9192295203757875e16) }, + { SC_(0.376e3), SC_(0.9e1), SC_(0.375862737220322e18) }, + { SC_(0.376e3), SC_(0.11e2), SC_(0.4589694053537099244e21) }, + { SC_(0.376e3), SC_(0.13e2), SC_(0.390888943559576285614e24) }, + { SC_(0.376e3), SC_(0.22e2), SC_(0.2144450623649450847150905469935555e36) }, + { SC_(0.376e3), SC_(0.372e3), SC_(0.81957425e9) }, + { SC_(0.376e3), SC_(0.373e3), SC_(0.8789e7) }, + { SC_(0.376e3), SC_(0.374e3), SC_(0.705e5) }, + { SC_(0.376e3), SC_(0.375e3), SC_(0.376e3) }, + { SC_(0.376e3), SC_(0.376e3), SC_(0.1e1) }, + { SC_(0.378e3), SC_(0.4e1), SC_(0.83722275e9) }, + { SC_(0.378e3), SC_(0.5e1), SC_(0.626242617e11) }, + { SC_(0.378e3), SC_(0.6e1), SC_(0.389314160235e13) }, + { SC_(0.378e3), SC_(0.8e1), SC_(0.9594647478991575e16) }, + { SC_(0.378e3), SC_(0.9e1), SC_(0.39444661858076475e18) }, + { SC_(0.378e3), SC_(0.11e2), SC_(0.4869335930029018812e21) }, + { SC_(0.378e3), SC_(0.13e2), SC_(0.4192685517906140159394e24) }, + { SC_(0.378e3), SC_(0.22e2), SC_(0.24180810300189004781143134586060785e36) }, + { SC_(0.378e3), SC_(0.372e3), SC_(0.389314160235e13) }, + { SC_(0.378e3), SC_(0.373e3), SC_(0.626242617e11) }, + { SC_(0.378e3), SC_(0.374e3), SC_(0.83722275e9) }, + { SC_(0.378e3), SC_(0.375e3), SC_(0.8930376e7) }, + { SC_(0.378e3), SC_(0.376e3), SC_(0.71253e5) }, + { SC_(0.378e3), SC_(0.378e3), SC_(0.1e1) }, + { SC_(0.389e3), SC_(0.4e1), SC_(0.939438501e9) }, + { SC_(0.389e3), SC_(0.5e1), SC_(0.72336764577e11) }, + { SC_(0.389e3), SC_(0.6e1), SC_(0.4629552932928e13) }, + { SC_(0.389e3), SC_(0.8e1), SC_(0.12095203060802928e17) }, + { SC_(0.389e3), SC_(0.9e1), SC_(0.512030262907323952e18) }, + { SC_(0.389e3), SC_(0.11e2), SC_(0.670387258762843596064e21) }, + { SC_(0.389e3), SC_(0.13e2), SC_(0.612398760879857625004464e24) }, + { SC_(0.389e3), SC_(0.22e2), SC_(0.4627435055579525047225070723215824e36) }, + { SC_(0.389e3), SC_(0.372e3), SC_(0.2108849811550582576133675754e30) }, + { SC_(0.389e3), SC_(0.373e3), SC_(0.96113798381661940467218466e28) }, + { SC_(0.389e3), SC_(0.374e3), SC_(0.4111820251621901196458544e27) }, + { SC_(0.389e3), SC_(0.375e3), SC_(0.16447281006487604785834176e26) }, + { SC_(0.389e3), SC_(0.376e3), SC_(0.612398760879857625004464e24) }, + { SC_(0.389e3), SC_(0.378e3), SC_(0.670387258762843596064e21) }, + { SC_(0.389e3), SC_(0.379e3), SC_(0.19457149990478310176e20) }, + { SC_(0.389e3), SC_(0.386e3), SC_(0.9735114e7) }, + { SC_(0.389e3), SC_(0.388e3), SC_(0.389e3) }, + { SC_(0.391e3), SC_(0.4e1), SC_(0.958984195e9) }, + { SC_(0.391e3), SC_(0.5e1), SC_(0.74225376693e11) }, + { SC_(0.391e3), SC_(0.6e1), SC_(0.4775165900583e13) }, + { SC_(0.391e3), SC_(0.8e1), SC_(0.1260643797753912e17) }, + { SC_(0.391e3), SC_(0.9e1), SC_(0.53647397171083144e18) }, + { SC_(0.391e3), SC_(0.11e2), SC_(0.709813589006707540368e21) }, + { SC_(0.391e3), SC_(0.13e2), SC_(0.65530354544067961515256e24) }, + { SC_(0.391e3), SC_(0.22e2), SC_(0.519645908172294225323547068003403e36) }, + { SC_(0.391e3), SC_(0.372e3), SC_(0.94028803439575537144626963663e32) }, + { SC_(0.391e3), SC_(0.373e3), SC_(0.4789670952686153366616386889e31) }, + { SC_(0.391e3), SC_(0.374e3), SC_(0.230518922856552835826457123e30) }, + { SC_(0.391e3), SC_(0.375e3), SC_(0.10450191169497061890799389576e29) }, + { SC_(0.391e3), SC_(0.376e3), SC_(0.444688985936045186842527216e27) }, + { SC_(0.391e3), SC_(0.378e3), SC_(0.65530354544067961515256e24) }, + { SC_(0.391e3), SC_(0.379e3), SC_(0.2247743031854573877832e23) }, + { SC_(0.391e3), SC_(0.386e3), SC_(0.74225376693e11) }, + { SC_(0.391e3), SC_(0.388e3), SC_(0.9886435e7) }, + { SC_(0.392e3), SC_(0.4e1), SC_(0.96887063e9) }, + { SC_(0.392e3), SC_(0.5e1), SC_(0.75184360888e11) }, + { SC_(0.392e3), SC_(0.6e1), SC_(0.4849391277276e13) }, + { SC_(0.392e3), SC_(0.8e1), SC_(0.12869072102071185e17) }, + { SC_(0.392e3), SC_(0.9e1), SC_(0.54908040968837056e18) }, + { SC_(0.392e3), SC_(0.11e2), SC_(0.730306894726061301376e21) }, + { SC_(0.392e3), SC_(0.13e2), SC_(0.67778097575922535393088e24) }, + { SC_(0.392e3), SC_(0.22e2), SC_(0.5505437729825387468292714882630648e36) }, + { SC_(0.392e3), SC_(0.372e3), SC_(0.18429645474156805280346884877948e34) }, + { SC_(0.392e3), SC_(0.373e3), SC_(0.98818474392261690511243350552e32) }, + { SC_(0.392e3), SC_(0.374e3), SC_(0.5020189875542706202442844012e31) }, + { SC_(0.392e3), SC_(0.375e3), SC_(0.240969114026049897717256512576e30) }, + { SC_(0.392e3), SC_(0.376e3), SC_(0.10894880155433107077641916792e29) }, + { SC_(0.392e3), SC_(0.378e3), SC_(0.1834849927233902922427168e26) }, + { SC_(0.392e3), SC_(0.379e3), SC_(0.67778097575922535393088e24) }, + { SC_(0.392e3), SC_(0.386e3), SC_(0.4849391277276e13) }, + { SC_(0.392e3), SC_(0.388e3), SC_(0.96887063e9) }, + { SC_(0.398e3), SC_(0.4e1), SC_(0.1029804105e10) }, + { SC_(0.398e3), SC_(0.5e1), SC_(0.81148563474e11) }, + { SC_(0.398e3), SC_(0.6e1), SC_(0.5315230907547e13) }, + { SC_(0.398e3), SC_(0.8e1), SC_(0.14547786993956139e17) }, + { SC_(0.398e3), SC_(0.9e1), SC_(0.63040410307143269e18) }, + { SC_(0.398e3), SC_(0.11e2), SC_(0.864983200770704352428e21) }, + { SC_(0.398e3), SC_(0.13e2), SC_(0.828287951907239471630766e24) }, + { SC_(0.398e3), SC_(0.22e2), SC_(0.776122173564640399865233597134960648e36) }, + { SC_(0.398e3), SC_(0.374e3), SC_(0.198248598682272276052532494920343209e39) }, + { SC_(0.398e3), SC_(0.375e3), SC_(0.12687910315665425667362079674901965376e38) }, + { SC_(0.398e3), SC_(0.376e3), SC_(0.776122173564640399865233597134960648e36) }, + { SC_(0.398e3), SC_(0.378e3), SC_(0.2516163840026833008699548944439896e34) }, + { SC_(0.398e3), SC_(0.379e3), SC_(0.13277909446051889227965957490448e33) }, + { SC_(0.398e3), SC_(0.386e3), SC_(0.27895708224855215365803e23) }, + { SC_(0.398e3), SC_(0.388e3), SC_(0.24522719609478731641e20) }, + { SC_(0.398e3), SC_(0.394e3), SC_(0.1029804105e10) }, + { SC_(0.398e3), SC_(0.395e3), SC_(0.10428396e8) }, + { SC_(0.398e3), SC_(0.397e3), SC_(0.398e3) }, + { SC_(0.398e3), SC_(0.398e3), SC_(0.1e1) }, + { SC_(0.399e3), SC_(0.4e1), SC_(0.1040232501e10) }, + { SC_(0.399e3), SC_(0.5e1), SC_(0.82178367579e11) }, + { SC_(0.399e3), SC_(0.6e1), SC_(0.5396379471021e13) }, + { SC_(0.399e3), SC_(0.8e1), SC_(0.14845439924778771e17) }, + { SC_(0.399e3), SC_(0.9e1), SC_(0.644951890065388829e18) }, + { SC_(0.399e3), SC_(0.11e2), SC_(0.889505920380183084069e21) }, + { SC_(0.399e3), SC_(0.13e2), SC_(0.856183660132094686996569e24) }, + { SC_(0.399e3), SC_(0.22e2), SC_(0.821413122685123394021825478134878776e36) }, + { SC_(0.399e3), SC_(0.375e3), SC_(0.210936508997937701719894574595245174376e39) }, + { SC_(0.399e3), SC_(0.376e3), SC_(0.13464032489230066067227313272036926024e38) }, + { SC_(0.399e3), SC_(0.378e3), SC_(0.47807112960509827165291429944358024e35) }, + { SC_(0.399e3), SC_(0.379e3), SC_(0.2648942934487351900979208519344376e34) }, + { SC_(0.399e3), SC_(0.386e3), SC_(0.856183660132094686996569e24) }, + { SC_(0.399e3), SC_(0.388e3), SC_(0.889505920380183084069e21) }, + { SC_(0.399e3), SC_(0.394e3), SC_(0.82178367579e11) }, + { SC_(0.399e3), SC_(0.395e3), SC_(0.1040232501e10) }, + { SC_(0.399e3), SC_(0.397e3), SC_(0.79401e5) }, + { SC_(0.399e3), SC_(0.398e3), SC_(0.399e3) }, + { SC_(0.399e3), SC_(0.399e3), SC_(0.1e1) } + } }; +#undef SC_ + diff --git a/test/binomial_quantile.ipp b/test/binomial_quantile.ipp new file mode 100644 index 000000000..b2a1c55ab --- /dev/null +++ b/test/binomial_quantile.ipp @@ -0,0 +1,4042 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 4032> binomial_quantile_data = {{ + { SC_(2), SC_(0.12698681652545928955078125), SC_(0.12698681652545928955078125), SC_(0), SC_(0.28467385230321224203411154382440248724380832183117) }, + { SC_(2), SC_(0.12698681652545928955078125), SC_(0.135477006435394287109375), SC_(0), SC_(0.25727865882740919932840773421531232758964168093055) }, + { SC_(2), SC_(0.12698681652545928955078125), SC_(0.22103404998779296875), SC_(0), SC_(0.035694410305790976496105074803969225386555435216338) }, + { SC_(2), SC_(0.12698681652545928955078125), SC_(0.814723670482635498046875), SC_(0.11873939407970113038114981488058971337236924303007), SC_(0) }, + { SC_(2), SC_(0.12698681652545928955078125), SC_(0.835008561611175537109375), SC_(0.17128223695871555977493584741313475336874139582588), SC_(0) }, + { SC_(2), SC_(0.12698681652545928955078125), 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SC_(51.192792213689340783250166225037111774388170370382) }, + { SC_(462), SC_(0.12698681652545928955078125), SC_(0.905791938304901123046875), SC_(67.665260544532309474320200503574452874539227729732), SC_(48.852266259282523771985777476676041562048252008723) }, + { SC_(462), SC_(0.12698681652545928955078125), SC_(0.9133758544921875), SC_(68.013388330367306251000688720114637847988999563299), SC_(48.535219685292496706664865041258838310315124444267) }, + { SC_(462), SC_(0.12698681652545928955078125), SC_(0.968867778778076171875), SC_(71.802540055421868170346684697572743253217029569524), SC_(45.150692977330678624468025972292740141400736443319) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.12698681652545928955078125), SC_(53.739795171222854539802098334928827183778178234774), SC_(70.514112790470495113367692848183326665903474182075) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.135477006435394287109375), SC_(54.021908304151288114318535953257101991256018663137), SC_(70.210228200197934082971211057716971043126861028263) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.22103404998779296875), SC_(56.387433823936690833244464396145242776633499077766), SC_(67.693520675373405004474010232492611258429808283056) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.814723670482635498046875), SC_(68.650737972587943016782286766131513932714835755564), SC_(55.481551020151500107128929845791794669646015535119) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.835008561611175537109375), SC_(69.247105190815425587192453376085891492117810757824), SC_(54.921007540176788098869374790341778693551504833492) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.905791938304901123046875), SC_(71.848183299959923271385148904301669635848906778566), SC_(52.510134373004159485314341368383765088809751258824) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.9133758544921875), SC_(72.205255990937807491501326387765126674760451036753), SC_(52.183431308470588357470614266264042994621119089194) }, + { SC_(462), SC_(0.135477006435394287109375), SC_(0.968867778778076171875), SC_(76.09021079023541008927650189975220767318323211873), SC_(48.693821115055950415866389131356912362436163992595) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.12698681652545928955078125), SC_(91.473890238455393922742272332918884569797011865451), SC_(111.81757327733425401312252686514606931564223516256) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.135477006435394287109375), SC_(91.820983099472539415515469227537429195455462137458), SC_(111.45382939810889865252685270988496922566292975343) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.22103404998779296875), SC_(94.724051968408192819921075770463769644815737250003), SC_(108.43512406972868279073171724673082414866727440413) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.814723670482635498046875), SC_(109.58460286018467659577340155079212172063994505968), SC_(93.613840186544673407397105108894081249299802991432) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.835008561611175537109375), SC_(110.29992385651988045247881502965037115365674382973), SC_(92.925920330649254578037662130581103523508857087667) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.905791938304901123046875), SC_(113.41254669803691235369926635444417693728356224189), SC_(89.958764785945043196999745612095002613403353240766) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.9133758544921875), SC_(113.8389369510853615165327351484589424506726480099), SC_(89.555598155398217252937943809693281297647221956114) }, + { SC_(462), SC_(0.22103404998779296875), SC_(0.968867778778076171875), SC_(118.46441618276542283999156428593171000508403703773), SC_(85.232366617435603858727915292447303108879448558759) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.12698681652545928955078125), SC_(366.34737420545874442238221115291633974255551979091), SC_(385.39412347147842949069932878055960066545995779033) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.135477006435394287109375), SC_(366.68950826720375537056067073635745094610807175144), SC_(385.07077828654809305974170276374350293065208267136) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.22103404998779296875), SC_(369.52678242708776628742959218778979998428825207247), SC_(382.36398368087413929821491434486112552834726684396) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.814723670482635498046875), SC_(383.39962627958838708488965344732287579600932141899), SC_(368.44683323291813379944568666580592072576455888311) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.835008561611175537109375), SC_(384.04103690353672291842299000701987459315148906768), SC_(367.77450432313584002403003049241276891691689337133) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.905791938304901123046875), SC_(386.80486880129491568765070057261717752595099504027), SC_(364.84652718705696461209218261146300999977406338047) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.9133758544921875), SC_(387.18006196770118088836897502089709358059882204361), SC_(364.44512754037228506136735492102500419868025189395) }, + { SC_(462), SC_(0.814723670482635498046875), SC_(0.968867778778076171875), SC_(391.1979372388894667313317488436941468297856338792), SC_(360.08615952790907327852939271832934339967258884367) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.12698681652545928955078125), SC_(376.14277892061017849266775901808505747312225934913), SC_(394.33789120042478975215523077546862132662147940277) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.135477006435394287109375), SC_(376.47062701059669499387868529275643470509113751059), SC_(394.03004590676159627740764045030348480565576709442) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.22103404998779296875), SC_(379.18809995995613471328554800138413075452476668576), SC_(391.45147951758687874169405809424639185125272456402) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.814723670482635498046875), SC_(392.43837928481028126646073601917700733527343373877), SC_(378.15403266219698379771114242082032368969242437767) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.835008561611175537109375), SC_(393.0494049218549429851402512675154900185236854482), SC_(377.51009193924422812717110873059467516599044508596) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.905791938304901123046875), SC_(395.68054520295444541561872495346098157216590762668), SC_(374.7042003492121975403116531525070739928849275944) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.9133758544921875), SC_(396.03750015577938107018070400739144079347599034163), SC_(374.31934482163543166773732423558235755036350427585) }, + { SC_(462), SC_(0.835008561611175537109375), SC_(0.968867778778076171875), SC_(399.85655233600266888086901662617538469166965290174), SC_(370.13713023266072785880996336774329538400032584841) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.12698681652545928955078125), SC_(410.77841779603348299009132062991806834191114132292), SC_(425.09192836850669467964774217200606567091489385596) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.135477006435394287109375), SC_(411.04050019870873412928204791009626631793165778548), SC_(424.85409990428803893880371527354629061091777136208) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.22103404998779296875), SC_(413.20743037050376397313177889683126115801350031291), SC_(422.85557132516048428686099311119423735829582532068) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.814723670482635498046875), SC_(423.62181315216437335232445710883865624489575861881), SC_(412.384010755884455878159061159658137846137080341) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.835008561611175537109375), SC_(424.09539327494980230925529034323151094008499154172), SC_(411.87052691167998064749135957293786099905640575261) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.905791938304901123046875), SC_(426.12723517699196060712792921408370314767680064867), SC_(409.62678643346206726187796520740152524187070973137) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.9133758544921875), SC_(426.4019321011556842693438692165731311536954876857), SC_(409.31825123167004990102031796718648244988746090145) }, + { SC_(462), SC_(0.905791938304901123046875), SC_(0.968867778778076171875), SC_(429.32593469528266849431422550896091856464531454781), SC_(405.95362577891838024646713447954699109277704571247) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.12698681652545928955078125), SC_(414.54749317241684103166151201044157289557560066664), SC_(428.32887976389006323958081567847829759072678046843) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.135477006435394287109375), SC_(414.80049552840271675364275050243742214362718825342), SC_(428.10058981763657174736653663422610344943505276056) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.22103404998779296875), SC_(416.89150989421192572249673797944131156939052148208), SC_(426.18115473504837209951197968533905334246040854066) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.814723670482635498046875), SC_(426.91729194717382243189697410571951485038139257899), SC_(416.09711705955833706875763226722903633636646325407) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.835008561611175537109375), SC_(427.37213017802529379127292358079248928283540824397), SC_(415.60162223352377968657825664449093045486831839776) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.905791938304901123046875), SC_(429.3223377515598904641695530576862301200922783231), SC_(413.4355089091988924882925074231621936947576819029) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.9133758544921875), SC_(429.5858401370714466977766917756196697426616416387), SC_(413.13752688833461862829157641357569678006403754388) }, + { SC_(462), SC_(0.9133758544921875), SC_(0.968867778778076171875), SC_(432.38818207988393867668921543142972096521280530344), SC_(409.88617366468320542592944593518289804952858533401) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.12698681652545928955078125), SC_(442.82393191141749173724816325072326820230463861198), SC_(451.31614712317827841869644152156836743662631399408) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.135477006435394287109375), SC_(442.98606274891854929687497244464980968557637066531), SC_(451.1821745050413112505324438707834719314419238511) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.22103404998779296875), SC_(444.31831042243092829560186140968343282457861353281), SC_(450.04528422783410389071851518631959672024544710996) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.814723670482635498046875), SC_(450.48346315283352091572919321540490887699874734837), SC_(443.81383106654679638874061205590535899657992881982) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.835008561611175537109375), SC_(450.75287287219138398255983761247801341317357905199), SC_(443.49813299555706441752047724189621951549274423869) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.905791938304901123046875), SC_(451.89591732812948291460872123812941435040707992501), SC_(442.10903364493891816867430327722153866406411695147) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.9133758544921875), SC_(452.0487864202684433692288504005674021335799201021), SC_(441.91683192409503729874122861358957872208941993018) }, + { SC_(462), SC_(0.968867778778076171875), SC_(0.968867778778076171875), SC_(453.64921455609407364749407038846093208870919124129), SC_(439.80319758393201038757522213012342652322688527575) } + }}; +#undef SC_ + diff --git a/test/cbrt_data.ipp b/test/cbrt_data.ipp new file mode 100644 index 000000000..12d5f5def --- /dev/null +++ b/test/cbrt_data.ipp @@ -0,0 +1,95 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 85> cbrt_data = { { + { SC_(0.266297021326439287136622624529991298914e-12), SC_(0.1888421455001568264216782004847867296461e-37) }, + { SC_(0.5920415525016708979677559909760020673275e-12), SC_(0.2075183790343685754527643464587844863336e-36) }, + { SC_(0.155163989296047688526414276566356420517e-11), SC_(0.3735707037930203997047439019589434984292e-35) }, + { SC_(0.326923297461201300961874949280172586441e-11), SC_(0.3494118359373226772153200620663814420149e-34) }, + { SC_(0.3753785910581841633870681107509881258011e-11), SC_(0.5289425440452335470097113858122815653748e-34) }, + { SC_(0.9579165585749116473834874341264367103577e-11), SC_(0.8789881934013975442181045034737061376284e-33) }, + { SC_(0.1858167439361402273334533674642443656921e-10), SC_(0.6415854952683870221087767100660047169015e-32) }, + { SC_(0.5449485307451595872407779097557067871094e-10), SC_(0.1618327663648593897188312867673509349544e-30) }, + { SC_(0.6089519166696533147842274047434329986572e-10), SC_(0.2258130336429980668979991908229127823735e-30) }, + { SC_(0.1337744776064297980155970435589551925659e-9), SC_(0.2393975994093713145816368117219309682822e-29) }, + { SC_(0.2554458866654840676346793770790100097656e-9), SC_(0.1666850852333084475602734478539689292864e-28) }, + { SC_(0.9285605062636648199259070679545402526855e-9), SC_(0.8006277238962806516943344887937136945761e-27) }, + { SC_(0.1698227447555211711005540564656257629395e-8), SC_(0.4897647988639492837638733591590200680017e-26) }, + { SC_(0.339355921141759608872234821319580078125e-8), SC_(0.3908105631910126517922937762400154556398e-25) }, + { SC_(0.6313728651008432279922999441623687744141e-8), SC_(0.2516852352568191806987306088580024017075e-24) }, + { SC_(0.8383264749056706932606175541877746582031e-8), SC_(0.5891685351226332190380177353564166513441e-24) }, + { SC_(0.1962631124285962869180366396903991699219e-7), SC_(0.7559899905537899689665020666334853594435e-23) }, + { SC_(0.5256384838503436185419559478759765625e-7), SC_(0.1452317136611010055610488165775172825142e-21) }, + { SC_(0.116242290459922514855861663818359375e-6), SC_(0.1570697224733838212725464129330562874032e-20) }, + { SC_(0.1776920584006802528165280818939208984375e-6), SC_(0.5610532144067232522265416539512374653302e-20) }, + { SC_(0.246631174150024889968335628509521484375e-6), SC_(0.1500181866107243920079950771800422025379e-19) }, + { SC_(0.7932688959044753573834896087646484375e-6), SC_(0.4991847137949555007936240315051986249607e-18) }, + { SC_(0.1372093493046122603118419647216796875e-5), SC_(0.2583158853420963423689975850731860849521e-17) }, + { SC_(0.214747751670074649155139923095703125e-5), SC_(0.9903435487644105337695875806797860570175e-17) }, + { SC_(0.527022712049074470996856689453125e-5), SC_(0.1463821071995824977534730930446568768253e-15) }, + { SC_(0.9233162927557714283466339111328125e-5), SC_(0.7871391209588633308733754268944773271037e-15) }, + { SC_(0.269396477960981428623199462890625e-4), SC_(0.1955130454371024476239796923077328444099e-13) }, + { SC_(0.3208058114978484809398651123046875e-4), SC_(0.3301616917426164412576388360054394990469e-13) }, + { SC_(0.10957030463032424449920654296875e-3), SC_(0.1315462909319166691700161404508507291713e-11) }, + { SC_(0.126518702018074691295623779296875e-3), SC_(0.2025182580848748191763613884783980516393e-11) }, + { SC_(0.28976381872780621051788330078125e-3), SC_(0.2432945998183714075166971697772880334043e-10) }, + { SC_(0.687857042066752910614013671875e-3), SC_(0.325457709339121122043443511174534212782e-9) }, + { SC_(0.145484809763729572296142578125e-2), SC_(0.3079306732417723714591019095094197606751e-8) }, + { SC_(0.2847635187208652496337890625e-2), SC_(0.2309154822558505456380682823926992940837e-7) }, + { SC_(0.56468211114406585693359375e-2), SC_(0.1800578620431549562186637161391699604382e-6) }, + { SC_(0.11621631681919097900390625e-1), SC_(0.1569644571431467249504618249826407884451e-5) }, + { SC_(0.257236398756504058837890625e-1), SC_(0.1702147780446687692243218673184664750751e-4) }, + { SC_(0.560617186129093170166015625e-1), SC_(0.1761972888887947511388016005088640519383e-3) }, + { SC_(0.106835305690765380859375e0), SC_(0.1219394946966688262159601656951790626948e-2) }, + { SC_(0.2401093542575836181640625e0), SC_(0.1384290502703277530255383713042680732253e-1) }, + { SC_(0.438671648502349853515625e0), SC_(0.8441482026963079214835649797480515710291e-1) }, + { SC_(0.903765499591827392578125e0), SC_(0.7381885006644863290177619358266650206879e0) }, + { SC_(0.12760250568389892578125e1), SC_(0.2077674969235041602851348126190789145085e1) }, + { SC_(0.3411548614501953125e1), SC_(0.3970586787024071934171232101107307244092e2) }, + { SC_(0.671881103515625e1), SC_(0.3033034012472492122469702735543251037598e3) }, + { SC_(0.802254772186279296875e1), SC_(0.5163413756551815056187668129261680860509e3) }, + { SC_(0.264815673828125e2), SC_(0.1857081908850696527224499732255935668945e5) }, + { SC_(0.54742523193359375e2), SC_(0.1640493194709731775162708800053223967552e6) }, + { SC_(0.744071502685546875e2), SC_(0.4119495333434053702710286870569689199328e6) }, + { SC_(0.210427001953125e3), SC_(0.9317607304574811554630286991596221923828e7) }, + { SC_(0.286463409423828125e3), SC_(0.2350755546524705072036454112094361335039e8) }, + { SC_(0.74548876953125e3), SC_(0.4143079969757992170052602887153625488281e9) }, + { SC_(0.15343248291015625e4), SC_(0.361203491025739912996687053237110376358e10) }, + { SC_(0.3632982421875e4), SC_(0.47950141115752447418868541717529296875e11) }, + { SC_(0.8027111328125e4), SC_(0.517223035506184733219444751739501953125e12) }, + { SC_(0.128921982421875e5), SC_(0.2142796483541738858568482100963592529297e13) }, + { SC_(0.2196087890625e5), SC_(0.10591297122360160745680332183837890625e14) }, + { SC_(0.614975859375e5), SC_(0.232580984311524292169094085693359375e15) }, + { SC_(0.103892109375e6), SC_(0.1121366795544843797206878662109375e16) }, + { SC_(0.2370011875e6), SC_(0.13312253103065122768310546875e17) }, + { SC_(0.32081496875e6), SC_(0.33018996548382043793914794921875e17) }, + { SC_(0.533606625e6), SC_(0.151937032114340403275390625e18) }, + { SC_(0.1836336625e7), SC_(0.6192369863782824692494140625e19) }, + { SC_(0.38194295e7), SC_(0.55717996837103384222375e20) }, + { SC_(0.52642505e7), SC_(0.145884664571511666937625e21) }, + { SC_(0.1527432e8), SC_(0.3563572958789165568e22) }, + { SC_(0.25265768e8), SC_(0.16128631219129563064832e23) }, + { SC_(0.6509808e8), SC_(0.275870040782346842112e24) }, + { SC_(0.114023104e9), SC_(0.1482444961322159867428864e25) }, + { SC_(0.189604896e9), SC_(0.6816299156208797501915136e25) }, + { SC_(0.507585472e9), SC_(0.130775849541591501155074048e27) }, + { SC_(0.764055936e9), SC_(0.446041700029835346607865856e27) }, + { SC_(0.2103773184e10), SC_(0.9311008970618745805030293504e28) }, + { SC_(0.3395078656e10), SC_(0.39133574711079901758452924416e29) }, + { SC_(0.6645239808e10), SC_(0.293448554175562916132775002112e30) }, + { SC_(0.9947638784e10), SC_(0.984373742549644225510117474304e30) }, + { SC_(0.19561418752e11), SC_(0.7485159350422880041966143275008e31) }, + { SC_(0.60532621312e11), SC_(0.221803524649682352197867382243328e33) }, + { SC_(0.78978883584e11), SC_(0.492643743012832976593364088520704e33) }, + { SC_(0.169071362048e12), SC_(0.4832926096640938859567595590254592e34) }, + { SC_(0.345661243392e12), SC_(0.41300191319644612608733712417292288e35) }, + { SC_(0.994912108544e12), SC_(0.984813853841992339417900620588253184e36) }, + { SC_(0.2023890092032e13), SC_(0.8290119158315231036253469850906656768e37) }, + { SC_(0.4372805189632e13), SC_(0.83614267463479414022236113855701843968e38) }, + { SC_(0.5516391612416e13), SC_(0.167866976532732026957407327191402807296e39) } + } }; +#undef SC_ + diff --git a/test/compile_test/dist_bernoulli_incl_test.cpp b/test/compile_test/dist_bernoulli_incl_test.cpp new file mode 100644 index 000000000..b959c61d4 --- /dev/null +++ b/test/compile_test/dist_bernoulli_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(bernoulli) +} + +template class boost::math::bernoulli_distribution >; +template class boost::math::bernoulli_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::bernoulli_distribution >; +#endif diff --git a/test/compile_test/dist_beta_incl_test.cpp b/test/compile_test/dist_beta_incl_test.cpp new file mode 100644 index 000000000..3c69c794e --- /dev/null +++ b/test/compile_test/dist_beta_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(beta) +} + +template class boost::math::beta_distribution >; +template class boost::math::beta_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::beta_distribution >; +#endif diff --git a/test/compile_test/dist_binomial_incl_test.cpp b/test/compile_test/dist_binomial_incl_test.cpp new file mode 100644 index 000000000..62e823517 --- /dev/null +++ b/test/compile_test/dist_binomial_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(binomial) +} + +template class boost::math::binomial_distribution >; +template class boost::math::binomial_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::binomial_distribution >; +#endif diff --git a/test/compile_test/dist_cauchy_incl_test.cpp b/test/compile_test/dist_cauchy_incl_test.cpp new file mode 100644 index 000000000..ff6186672 --- /dev/null +++ b/test/compile_test/dist_cauchy_incl_test.cpp @@ -0,0 +1,26 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(cauchy) +} + +template class boost::math::cauchy_distribution >; +template class boost::math::cauchy_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::cauchy_distribution >; +#endif diff --git a/test/compile_test/dist_chi_squared_incl_test.cpp b/test/compile_test/dist_chi_squared_incl_test.cpp new file mode 100644 index 000000000..8d4b42d03 --- /dev/null +++ b/test/compile_test/dist_chi_squared_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(chi_squared) +} + +template class boost::math::chi_squared_distribution >; +template class boost::math::chi_squared_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::chi_squared_distribution >; +#endif diff --git a/test/compile_test/dist_complement_incl_test.cpp b/test/compile_test/dist_complement_incl_test.cpp new file mode 100644 index 000000000..b61981f01 --- /dev/null +++ b/test/compile_test/dist_complement_incl_test.cpp @@ -0,0 +1,22 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + boost::math::complement(f, f); + boost::math::complement(f, f, d); + boost::math::complement(f, f, d, l); + boost::math::complement(f, f, d, l, i); +} diff --git a/test/compile_test/dist_exponential_incl_test.cpp b/test/compile_test/dist_exponential_incl_test.cpp new file mode 100644 index 000000000..a65db9364 --- /dev/null +++ b/test/compile_test/dist_exponential_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(exponential) +} + +template class boost::math::exponential_distribution >; +template class boost::math::exponential_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::exponential_distribution >; +#endif diff --git a/test/compile_test/dist_extreme_val_incl_test.cpp b/test/compile_test/dist_extreme_val_incl_test.cpp new file mode 100644 index 000000000..44c5ccf67 --- /dev/null +++ b/test/compile_test/dist_extreme_val_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(extreme_value) +} + +template class boost::math::extreme_value_distribution >; +template class boost::math::extreme_value_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::extreme_value_distribution >; +#endif diff --git a/test/compile_test/dist_fisher_f_incl_test.cpp b/test/compile_test/dist_fisher_f_incl_test.cpp new file mode 100644 index 000000000..d334a5c34 --- /dev/null +++ b/test/compile_test/dist_fisher_f_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(fisher_f) +} + +template class boost::math::fisher_f_distribution >; +template class boost::math::fisher_f_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::fisher_f_distribution >; +#endif diff --git a/test/compile_test/dist_gamma_incl_test.cpp b/test/compile_test/dist_gamma_incl_test.cpp new file mode 100644 index 000000000..22ea9a3f8 --- /dev/null +++ b/test/compile_test/dist_gamma_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(gamma) +} + +template class boost::math::gamma_distribution >; +template class boost::math::gamma_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::gamma_distribution >; +#endif diff --git a/test/compile_test/dist_lognormal_incl_test.cpp b/test/compile_test/dist_lognormal_incl_test.cpp new file mode 100644 index 000000000..cfa9e50f3 --- /dev/null +++ b/test/compile_test/dist_lognormal_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(lognormal) +} + +template class boost::math::lognormal_distribution >; +template class boost::math::lognormal_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::lognormal_distribution >; +#endif diff --git a/test/compile_test/dist_neg_binom_incl_test.cpp b/test/compile_test/dist_neg_binom_incl_test.cpp new file mode 100644 index 000000000..c366e5e94 --- /dev/null +++ b/test/compile_test/dist_neg_binom_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(negative_binomial) +} + +template class boost::math::negative_binomial_distribution >; +template class boost::math::negative_binomial_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::negative_binomial_distribution >; +#endif diff --git a/test/compile_test/dist_normal_incl_test.cpp b/test/compile_test/dist_normal_incl_test.cpp new file mode 100644 index 000000000..145b904b4 --- /dev/null +++ b/test/compile_test/dist_normal_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(normal) +} + +template class boost::math::normal_distribution >; +template class boost::math::normal_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::normal_distribution >; +#endif diff --git a/test/compile_test/dist_pareto_incl_test.cpp b/test/compile_test/dist_pareto_incl_test.cpp new file mode 100644 index 000000000..7bf0397c0 --- /dev/null +++ b/test/compile_test/dist_pareto_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(pareto) +} + +template class boost::math::pareto_distribution >; +template class boost::math::pareto_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::pareto_distribution >; +#endif diff --git a/test/compile_test/dist_poisson_incl_test.cpp b/test/compile_test/dist_poisson_incl_test.cpp new file mode 100644 index 000000000..40a3a7a0c --- /dev/null +++ b/test/compile_test/dist_poisson_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(poisson) +} + +template class boost::math::poisson_distribution >; +template class boost::math::poisson_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::poisson_distribution >; +#endif diff --git a/test/compile_test/dist_students_t_incl_test.cpp b/test/compile_test/dist_students_t_incl_test.cpp new file mode 100644 index 000000000..82fc203b1 --- /dev/null +++ b/test/compile_test/dist_students_t_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(students_t) +} + +template class boost::math::students_t_distribution >; +template class boost::math::students_t_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::students_t_distribution >; +#endif diff --git a/test/compile_test/dist_triangular_incl_test.cpp b/test/compile_test/dist_triangular_incl_test.cpp new file mode 100644 index 000000000..5272f6472 --- /dev/null +++ b/test/compile_test/dist_triangular_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(triangular) +} + +template class boost::math::triangular_distribution >; +template class boost::math::triangular_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::triangular_distribution >; +#endif diff --git a/test/compile_test/dist_uniform_incl_test.cpp b/test/compile_test/dist_uniform_incl_test.cpp new file mode 100644 index 000000000..73c219e38 --- /dev/null +++ b/test/compile_test/dist_uniform_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(uniform) +} + +template class boost::math::uniform_distribution >; +template class boost::math::uniform_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::uniform_distribution >; +#endif diff --git a/test/compile_test/dist_weibull_incl_test.cpp b/test/compile_test/dist_weibull_incl_test.cpp new file mode 100644 index 000000000..56d3cf531 --- /dev/null +++ b/test/compile_test/dist_weibull_incl_test.cpp @@ -0,0 +1,25 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + TEST_DIST_FUNC(weibull) +} + +template class boost::math::weibull_distribution >; +template class boost::math::weibull_distribution >; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +template class boost::math::weibull_distribution >; +#endif diff --git a/test/compile_test/distribution_concept_check.cpp b/test/compile_test/distribution_concept_check.cpp new file mode 100644 index 000000000..66c5f476a --- /dev/null +++ b/test/compile_test/distribution_concept_check.cpp @@ -0,0 +1,49 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false + +#include +#include + + +template +void instantiate(RealType) +{ + using namespace boost; + using namespace boost::math; + using namespace boost::math::concepts; + + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); +} + + +int main() +{ + instantiate(float(0)); + instantiate(double(0)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + instantiate((long double)(0)); +#endif +} + diff --git a/test/compile_test/generate.sh b/test/compile_test/generate.sh new file mode 100755 index 000000000..ac88405aa --- /dev/null +++ b/test/compile_test/generate.sh @@ -0,0 +1,46 @@ +for file in ../../../../boost/math/tools/*.hpp; do +cat > tools_$(basename $file .hpp)_inc_test.cpp << EOF; +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +EOF +done + +for file in ../../../../boost/math/distributions/*.hpp; do +cat > dist_$(basename $file .hpp)_incl_test.cpp << EOF; +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +EOF +done + +for file in ../../../../boost/math/special_functions/*.hpp; do +cat > sf_$(basename $file .hpp)_incl_test.cpp << EOF; +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +EOF +done + + diff --git a/test/compile_test/instantiate.hpp b/test/compile_test/instantiate.hpp new file mode 100644 index 000000000..e32caba5c --- /dev/null +++ b/test/compile_test/instantiate.hpp @@ -0,0 +1,742 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_LIBS_MATH_TEST_INSTANTIATE_HPP +#define BOOST_LIBS_MATH_TEST_INSTANTIATE_HPP + +#ifndef BOOST_MATH_ASSERT_UNDEFINED_POLICY +# define BOOST_MATH_ASSERT_UNDEFINED_POLICY false +#endif + +#include + +#include +#include + +typedef boost::math::policies::policy<> test_policy; + +namespace test{ + +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(test_policy) + +} + +namespace dist_test{ + +BOOST_MATH_DECLARE_DISTRIBUTIONS(double, test_policy) + +} + +template +void instantiate(RealType) +{ + using namespace boost; + using namespace boost::math; + using namespace boost::math::concepts; + + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + function_requires > >(); + + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + function_requires >(); + + int i; + RealType v1(0.5), v2(0.5), v3(0.5); + boost::math::tgamma(v1); + boost::math::tgamma1pm1(v1); + boost::math::lgamma(v1); + boost::math::lgamma(v1, &i); + boost::math::digamma(v1); + boost::math::tgamma_ratio(v1, v2); + boost::math::tgamma_delta_ratio(v1, v2); + boost::math::factorial(i); + boost::math::unchecked_factorial(i); + i = boost::math::max_factorial::value; + boost::math::double_factorial(i); + boost::math::rising_factorial(v1, i); + boost::math::falling_factorial(v1, i); + boost::math::tgamma(v1, v2); + boost::math::tgamma_lower(v1, v2); + boost::math::gamma_p(v1, v2); + boost::math::gamma_q(v1, v2); + boost::math::gamma_p_inv(v1, v2); + boost::math::gamma_q_inv(v1, v2); + boost::math::gamma_p_inva(v1, v2); + boost::math::gamma_q_inva(v1, v2); + boost::math::erf(v1); + boost::math::erfc(v1); + boost::math::erf_inv(v1); + boost::math::erfc_inv(v1); + boost::math::beta(v1, v2); + boost::math::beta(v1, v2, v3); + boost::math::betac(v1, v2, v3); + boost::math::ibeta(v1, v2, v3); + boost::math::ibetac(v1, v2, v3); + boost::math::ibeta_inv(v1, v2, v3); + boost::math::ibetac_inv(v1, v2, v3); + boost::math::ibeta_inva(v1, v2, v3); + boost::math::ibetac_inva(v1, v2, v3); + boost::math::ibeta_invb(v1, v2, v3); + boost::math::ibetac_invb(v1, v2, v3); + boost::math::gamma_p_derivative(v2, v3); + boost::math::ibeta_derivative(v1, v2, v3); + (boost::math::fpclassify)(v1); + (boost::math::isfinite)(v1); + (boost::math::isnormal)(v1); + (boost::math::isnan)(v1); + (boost::math::isinf)(v1); + boost::math::log1p(v1); + boost::math::expm1(v1); + boost::math::cbrt(v1); + boost::math::sqrt1pm1(v1); + boost::math::powm1(v1, v2); + boost::math::legendre_p(1, v1); + boost::math::legendre_p(1, 0, v1); + boost::math::legendre_q(1, v1); + boost::math::legendre_next(2, v1, v2, v3); + boost::math::legendre_next(2, 2, v1, v2, v3); + boost::math::laguerre(1, v1); + boost::math::laguerre(2, 1, v1); + boost::math::laguerre(2u, 1u, v1); + boost::math::laguerre_next(2, v1, v2, v3); + boost::math::laguerre_next(2, 1, v1, v2, v3); + boost::math::hermite(1, v1); + boost::math::hermite_next(2, v1, v2, v3); + boost::math::spherical_harmonic_r(2, 1, v1, v2); + boost::math::spherical_harmonic_i(2, 1, v1, v2); + boost::math::ellint_1(v1); + boost::math::ellint_1(v1, v2); + boost::math::ellint_2(v1); + boost::math::ellint_2(v1, v2); + boost::math::ellint_3(v1, v2); + boost::math::ellint_3(v1, v2, v3); + boost::math::ellint_rc(v1, v2); + boost::math::ellint_rd(v1, v2, v3); + boost::math::ellint_rf(v1, v2, v3); + boost::math::ellint_rj(v1, v2, v3, v1); + boost::math::hypot(v1, v2); + boost::math::sinc_pi(v1); + boost::math::sinhc_pi(v1); + boost::math::asinh(v1); + boost::math::acosh(v1); + boost::math::atanh(v1); + boost::math::sin_pi(v1); + boost::math::cos_pi(v1); + boost::math::cyl_neumann(v1, v2); + boost::math::cyl_neumann(i, v2); + boost::math::cyl_bessel_j(v1, v2); + boost::math::cyl_bessel_j(i, v2); + boost::math::cyl_bessel_i(v1, v2); + boost::math::cyl_bessel_i(i, v2); + boost::math::cyl_bessel_k(v1, v2); + boost::math::cyl_bessel_k(i, v2); + boost::math::sph_bessel(i, v2); + boost::math::sph_bessel(i, 1); + boost::math::sph_neumann(i, v2); + boost::math::sph_neumann(i, i); + // + // All over again, with a policy this time: + // + test_policy pol; + boost::math::tgamma(v1, pol); + boost::math::tgamma1pm1(v1, pol); + boost::math::lgamma(v1, pol); + boost::math::lgamma(v1, &i, pol); + boost::math::digamma(v1, pol); + boost::math::tgamma_ratio(v1, v2, pol); + boost::math::tgamma_delta_ratio(v1, v2, pol); + boost::math::factorial(i, pol); + boost::math::unchecked_factorial(i); + i = boost::math::max_factorial::value; + boost::math::double_factorial(i, pol); + boost::math::rising_factorial(v1, i, pol); + boost::math::falling_factorial(v1, i, pol); + boost::math::tgamma(v1, v2, pol); + boost::math::tgamma_lower(v1, v2, pol); + boost::math::gamma_p(v1, v2, pol); + boost::math::gamma_q(v1, v2, pol); + boost::math::gamma_p_inv(v1, v2, pol); + boost::math::gamma_q_inv(v1, v2, pol); + boost::math::gamma_p_inva(v1, v2, pol); + boost::math::gamma_q_inva(v1, v2, pol); + boost::math::erf(v1, pol); + boost::math::erfc(v1, pol); + boost::math::erf_inv(v1, pol); + boost::math::erfc_inv(v1, pol); + boost::math::beta(v1, v2, pol); + boost::math::beta(v1, v2, v3, pol); + boost::math::betac(v1, v2, v3, pol); + boost::math::ibeta(v1, v2, v3, pol); + boost::math::ibetac(v1, v2, v3, pol); + boost::math::ibeta_inv(v1, v2, v3, pol); + boost::math::ibetac_inv(v1, v2, v3, pol); + boost::math::ibeta_inva(v1, v2, v3, pol); + boost::math::ibetac_inva(v1, v2, v3, pol); + boost::math::ibeta_invb(v1, v2, v3, pol); + boost::math::ibetac_invb(v1, v2, v3, pol); + boost::math::gamma_p_derivative(v2, v3, pol); + boost::math::ibeta_derivative(v1, v2, v3, pol); + (boost::math::fpclassify)(v1); + (boost::math::isfinite)(v1); + (boost::math::isnormal)(v1); + (boost::math::isnan)(v1); + (boost::math::isinf)(v1); + boost::math::log1p(v1, pol); + boost::math::expm1(v1, pol); + boost::math::cbrt(v1, pol); + boost::math::sqrt1pm1(v1, pol); + boost::math::powm1(v1, v2, pol); + boost::math::legendre_p(1, v1, pol); + boost::math::legendre_p(1, 0, v1, pol); + boost::math::legendre_q(1, v1, pol); + boost::math::legendre_next(2, v1, v2, v3); + boost::math::legendre_next(2, 2, v1, v2, v3); + boost::math::laguerre(1, v1, pol); + boost::math::laguerre(2, 1, v1, pol); + boost::math::laguerre_next(2, v1, v2, v3); + boost::math::laguerre_next(2, 1, v1, v2, v3); + boost::math::hermite(1, v1, pol); + boost::math::hermite_next(2, v1, v2, v3); + boost::math::spherical_harmonic_r(2, 1, v1, v2, pol); + boost::math::spherical_harmonic_i(2, 1, v1, v2, pol); + boost::math::ellint_1(v1, pol); + boost::math::ellint_1(v1, v2, pol); + boost::math::ellint_2(v1, pol); + boost::math::ellint_2(v1, v2, pol); + boost::math::ellint_3(v1, v2, pol); + boost::math::ellint_3(v1, v2, v3, pol); + boost::math::ellint_rc(v1, v2, pol); + boost::math::ellint_rd(v1, v2, v3, pol); + boost::math::ellint_rf(v1, v2, v3, pol); + boost::math::ellint_rj(v1, v2, v3, v1, pol); + boost::math::hypot(v1, v2, pol); + boost::math::sinc_pi(v1, pol); + boost::math::sinhc_pi(v1, pol); + boost::math::asinh(v1, pol); + boost::math::acosh(v1, pol); + boost::math::atanh(v1, pol); + boost::math::sin_pi(v1, pol); + boost::math::cos_pi(v1, pol); + boost::math::cyl_neumann(v1, v2, pol); + boost::math::cyl_neumann(i, v2, pol); + boost::math::cyl_bessel_j(v1, v2, pol); + boost::math::cyl_bessel_j(i, v2, pol); + boost::math::cyl_bessel_i(v1, v2, pol); + boost::math::cyl_bessel_i(i, v2, pol); + boost::math::cyl_bessel_k(v1, v2, pol); + boost::math::cyl_bessel_k(i, v2, pol); + boost::math::sph_bessel(i, v2, pol); + boost::math::sph_bessel(i, 1, pol); + boost::math::sph_neumann(i, v2, pol); + boost::math::sph_neumann(i, i, pol); + // + // All over again with the versions in test:: + // + test::tgamma(v1); + test::tgamma1pm1(v1); + test::lgamma(v1); + test::lgamma(v1, &i); + test::digamma(v1); + test::tgamma_ratio(v1, v2); + test::tgamma_delta_ratio(v1, v2); + test::factorial(i); + test::unchecked_factorial(i); + i = test::max_factorial::value; + test::double_factorial(i); + test::rising_factorial(v1, i); + test::falling_factorial(v1, i); + test::tgamma(v1, v2); + test::tgamma_lower(v1, v2); + test::gamma_p(v1, v2); + test::gamma_q(v1, v2); + test::gamma_p_inv(v1, v2); + test::gamma_q_inv(v1, v2); + test::gamma_p_inva(v1, v2); + test::gamma_q_inva(v1, v2); + test::erf(v1); + test::erfc(v1); + test::erf_inv(v1); + test::erfc_inv(v1); + test::beta(v1, v2); + test::beta(v1, v2, v3); + test::betac(v1, v2, v3); + test::ibeta(v1, v2, v3); + test::ibetac(v1, v2, v3); + test::ibeta_inv(v1, v2, v3); + test::ibetac_inv(v1, v2, v3); + test::ibeta_inva(v1, v2, v3); + test::ibetac_inva(v1, v2, v3); + test::ibeta_invb(v1, v2, v3); + test::ibetac_invb(v1, v2, v3); + test::gamma_p_derivative(v2, v3); + test::ibeta_derivative(v1, v2, v3); + (test::fpclassify)(v1); + (test::isfinite)(v1); + (test::isnormal)(v1); + (test::isnan)(v1); + (test::isinf)(v1); + test::log1p(v1); + test::expm1(v1); + test::cbrt(v1); + test::sqrt1pm1(v1); + test::powm1(v1, v2); + test::legendre_p(1, v1); + test::legendre_p(1, 0, v1); + test::legendre_q(1, v1); + test::legendre_next(2, v1, v2, v3); + test::legendre_next(2, 2, v1, v2, v3); + test::laguerre(1, v1); + test::laguerre(2, 1, v1); + test::laguerre_next(2, v1, v2, v3); + test::laguerre_next(2, 1, v1, v2, v3); + test::hermite(1, v1); + test::hermite_next(2, v1, v2, v3); + test::spherical_harmonic_r(2, 1, v1, v2); + test::spherical_harmonic_i(2, 1, v1, v2); + test::ellint_1(v1); + test::ellint_1(v1, v2); + test::ellint_2(v1); + test::ellint_2(v1, v2); + test::ellint_3(v1, v2); + test::ellint_3(v1, v2, v3); + test::ellint_rc(v1, v2); + test::ellint_rd(v1, v2, v3); + test::ellint_rf(v1, v2, v3); + test::ellint_rj(v1, v2, v3, v1); + test::hypot(v1, v2); + test::sinc_pi(v1); + test::sinhc_pi(v1); + test::asinh(v1); + test::acosh(v1); + test::atanh(v1); + test::sin_pi(v1); + test::cos_pi(v1); + test::cyl_neumann(v1, v2); + test::cyl_neumann(i, v2); + test::cyl_bessel_j(v1, v2); + test::cyl_bessel_j(i, v2); + test::cyl_bessel_i(v1, v2); + test::cyl_bessel_i(i, v2); + test::cyl_bessel_k(v1, v2); + test::cyl_bessel_k(i, v2); + test::sph_bessel(i, v2); + test::sph_bessel(i, 1); + test::sph_neumann(i, v2); + test::sph_neumann(i, i); +} + +template +void instantiate_mixed(RealType) +{ + using namespace boost; + using namespace boost::math; + + int i = 1; + long l = 1; + short s = 1; + float fr = 0.5F; + double dr = 0.5; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + long double lr = 0.5L; +#else + double lr = 0.5L; +#endif + + boost::math::tgamma(i); + boost::math::tgamma1pm1(i); + boost::math::lgamma(i); + boost::math::lgamma(i, &i); + boost::math::digamma(i); + boost::math::tgamma_ratio(i, l); + boost::math::tgamma_ratio(fr, lr); + boost::math::tgamma_delta_ratio(i, s); + boost::math::tgamma_delta_ratio(fr, lr); + boost::math::rising_factorial(s, i); + boost::math::falling_factorial(s, i); + boost::math::tgamma(i, l); + boost::math::tgamma(fr, lr); + boost::math::tgamma_lower(i, s); + boost::math::tgamma_lower(fr, lr); + boost::math::gamma_p(i, s); + boost::math::gamma_p(fr, lr); + boost::math::gamma_q(i, s); + boost::math::gamma_q(fr, lr); + boost::math::gamma_p_inv(i, fr); + boost::math::gamma_q_inv(s, fr); + boost::math::gamma_p_inva(i, lr); + boost::math::gamma_q_inva(i, lr); + boost::math::erf(i); + boost::math::erfc(i); + boost::math::erf_inv(i); + boost::math::erfc_inv(i); + boost::math::beta(i, s); + boost::math::beta(fr, lr); + boost::math::beta(i, s, l); + boost::math::beta(fr, dr, lr); + boost::math::betac(l, i, s); + boost::math::betac(fr, dr, lr); + boost::math::ibeta(l, i, s); + boost::math::ibeta(fr, dr, lr); + boost::math::ibetac(l, i, s); + boost::math::ibetac(fr, dr, lr); + boost::math::ibeta_inv(l, s, i); + boost::math::ibeta_inv(fr, dr, lr); + boost::math::ibetac_inv(l, i, s); + boost::math::ibetac_inv(fr, dr, lr); + boost::math::ibeta_inva(l, i, s); + boost::math::ibeta_inva(fr, dr, lr); + boost::math::ibetac_inva(l, i, s); + boost::math::ibetac_inva(fr, dr, lr); + boost::math::ibeta_invb(l, i, s); + boost::math::ibeta_invb(fr, dr, lr); + boost::math::ibetac_invb(l, i, s); + boost::math::ibetac_invb(fr, dr, lr); + boost::math::gamma_p_derivative(i, l); + boost::math::gamma_p_derivative(fr, lr); + boost::math::ibeta_derivative(l, i, s); + boost::math::ibeta_derivative(fr, dr, lr); + (boost::math::fpclassify)(i); + (boost::math::isfinite)(s); + (boost::math::isnormal)(l); + (boost::math::isnan)(i); + (boost::math::isinf)(l); + boost::math::log1p(i); + boost::math::expm1(s); + boost::math::cbrt(l); + boost::math::sqrt1pm1(s); + boost::math::powm1(i, s); + boost::math::powm1(fr, lr); + //boost::math::legendre_p(1, i); + boost::math::legendre_p(1, 0, s); + boost::math::legendre_q(1, i); + boost::math::laguerre(1, i); + boost::math::laguerre(2, 1, i); + boost::math::laguerre(2u, 1u, s); + boost::math::hermite(1, s); + boost::math::spherical_harmonic_r(2, 1, s, i); + boost::math::spherical_harmonic_i(2, 1, fr, lr); + boost::math::ellint_1(i); + boost::math::ellint_1(i, s); + boost::math::ellint_1(fr, lr); + boost::math::ellint_2(i); + boost::math::ellint_2(i, l); + boost::math::ellint_2(fr, lr); + boost::math::ellint_3(i, l); + boost::math::ellint_3(fr, lr); + boost::math::ellint_3(s, l, i); + boost::math::ellint_3(fr, dr, lr); + boost::math::ellint_rc(i, s); + boost::math::ellint_rc(fr, lr); + boost::math::ellint_rd(s, i, l); + boost::math::ellint_rd(fr, lr, dr); + boost::math::ellint_rf(s, l, i); + boost::math::ellint_rf(fr, dr, lr); + boost::math::ellint_rj(i, i, s, l); + boost::math::ellint_rj(i, fr, dr, lr); + boost::math::hypot(i, s); + boost::math::hypot(fr, lr); + boost::math::sinc_pi(i); + boost::math::sinhc_pi(i); + boost::math::asinh(s); + boost::math::acosh(l); + boost::math::atanh(l); + boost::math::sin_pi(s); + boost::math::cos_pi(s); + boost::math::cyl_neumann(fr, dr); + boost::math::cyl_neumann(i, s); + boost::math::cyl_bessel_j(fr, lr); + boost::math::cyl_bessel_j(i, s); + boost::math::cyl_bessel_i(fr, lr); + boost::math::cyl_bessel_i(i, s); + boost::math::cyl_bessel_k(fr, lr); + boost::math::cyl_bessel_k(i, s); + boost::math::sph_bessel(i, fr); + boost::math::sph_bessel(i, 1); + boost::math::sph_neumann(i, lr); + boost::math::sph_neumann(i, i); + + boost::math::policies::policy<> pol; + + + boost::math::tgamma(i, pol); + boost::math::tgamma1pm1(i, pol); + boost::math::lgamma(i, pol); + boost::math::lgamma(i, &i, pol); + boost::math::digamma(i, pol); + boost::math::tgamma_ratio(i, l, pol); + boost::math::tgamma_ratio(fr, lr, pol); + boost::math::tgamma_delta_ratio(i, s, pol); + boost::math::tgamma_delta_ratio(fr, lr, pol); + boost::math::rising_factorial(s, i, pol); + boost::math::falling_factorial(s, i, pol); + boost::math::tgamma(i, l, pol); + boost::math::tgamma(fr, lr, pol); + boost::math::tgamma_lower(i, s, pol); + boost::math::tgamma_lower(fr, lr, pol); + boost::math::gamma_p(i, s, pol); + boost::math::gamma_p(fr, lr, pol); + boost::math::gamma_q(i, s, pol); + boost::math::gamma_q(fr, lr, pol); + boost::math::gamma_p_inv(i, fr, pol); + boost::math::gamma_q_inv(s, fr, pol); + boost::math::gamma_p_inva(i, lr, pol); + boost::math::gamma_q_inva(i, lr, pol); + boost::math::erf(i, pol); + boost::math::erfc(i, pol); + boost::math::erf_inv(i, pol); + boost::math::erfc_inv(i, pol); + boost::math::beta(i, s, pol); + boost::math::beta(fr, lr, pol); + boost::math::beta(i, s, l, pol); + boost::math::beta(fr, dr, lr, pol); + boost::math::betac(l, i, s, pol); + boost::math::betac(fr, dr, lr, pol); + boost::math::ibeta(l, i, s, pol); + boost::math::ibeta(fr, dr, lr, pol); + boost::math::ibetac(l, i, s, pol); + boost::math::ibetac(fr, dr, lr, pol); + boost::math::ibeta_inv(l, s, i, pol); + boost::math::ibeta_inv(fr, dr, lr, pol); + boost::math::ibetac_inv(l, i, s, pol); + boost::math::ibetac_inv(fr, dr, lr, pol); + boost::math::ibeta_inva(l, i, s, pol); + boost::math::ibeta_inva(fr, dr, lr, pol); + boost::math::ibetac_inva(l, i, s, pol); + boost::math::ibetac_inva(fr, dr, lr, pol); + boost::math::ibeta_invb(l, i, s, pol); + boost::math::ibeta_invb(fr, dr, lr, pol); + boost::math::ibetac_invb(l, i, s, pol); + boost::math::ibetac_invb(fr, dr, lr, pol); + boost::math::gamma_p_derivative(i, l, pol); + boost::math::gamma_p_derivative(fr, lr, pol); + boost::math::ibeta_derivative(l, i, s, pol); + boost::math::ibeta_derivative(fr, dr, lr, pol); + boost::math::log1p(i, pol); + boost::math::expm1(s, pol); + boost::math::cbrt(l, pol); + boost::math::sqrt1pm1(s, pol); + boost::math::powm1(i, s, pol); + boost::math::powm1(fr, lr, pol); + //boost::math::legendre_p(1, i, pol); + boost::math::legendre_p(1, 0, s, pol); + boost::math::legendre_q(1, i, pol); + boost::math::laguerre(1, i, pol); + boost::math::laguerre(2, 1, i, pol); + boost::math::laguerre(2u, 1u, s, pol); + boost::math::hermite(1, s, pol); + boost::math::spherical_harmonic_r(2, 1, s, i, pol); + boost::math::spherical_harmonic_i(2, 1, fr, lr, pol); + boost::math::ellint_1(i, pol); + boost::math::ellint_1(i, s, pol); + boost::math::ellint_1(fr, lr, pol); + boost::math::ellint_2(i, pol); + boost::math::ellint_2(i, l, pol); + boost::math::ellint_2(fr, lr, pol); + boost::math::ellint_3(i, l, pol); + boost::math::ellint_3(fr, lr, pol); + boost::math::ellint_3(s, l, i, pol); + boost::math::ellint_3(fr, dr, lr, pol); + boost::math::ellint_rc(i, s, pol); + boost::math::ellint_rc(fr, lr, pol); + boost::math::ellint_rd(s, i, l, pol); + boost::math::ellint_rd(fr, lr, dr, pol); + boost::math::ellint_rf(s, l, i, pol); + boost::math::ellint_rf(fr, dr, lr, pol); + boost::math::ellint_rj(i, i, s, l, pol); + boost::math::ellint_rj(i, fr, dr, lr, pol); + boost::math::hypot(i, s, pol); + boost::math::hypot(fr, lr, pol); + boost::math::sinc_pi(i, pol); + boost::math::sinhc_pi(i, pol); + boost::math::asinh(s, pol); + boost::math::acosh(l, pol); + boost::math::atanh(l, pol); + boost::math::sin_pi(s, pol); + boost::math::cos_pi(s, pol); + boost::math::cyl_neumann(fr, dr, pol); + boost::math::cyl_neumann(i, s, pol); + boost::math::cyl_bessel_j(fr, lr, pol); + boost::math::cyl_bessel_j(i, s, pol); + boost::math::cyl_bessel_i(fr, lr, pol); + boost::math::cyl_bessel_i(i, s, pol); + boost::math::cyl_bessel_k(fr, lr, pol); + boost::math::cyl_bessel_k(i, s, pol); + boost::math::sph_bessel(i, fr, pol); + boost::math::sph_bessel(i, 1, pol); + boost::math::sph_neumann(i, lr, pol); + boost::math::sph_neumann(i, i, pol); + + + test::tgamma(i); + test::tgamma1pm1(i); + test::lgamma(i); + test::lgamma(i, &i); + test::digamma(i); + test::tgamma_ratio(i, l); + test::tgamma_ratio(fr, lr); + test::tgamma_delta_ratio(i, s); + test::tgamma_delta_ratio(fr, lr); + test::rising_factorial(s, i); + test::falling_factorial(s, i); + test::tgamma(i, l); + test::tgamma(fr, lr); + test::tgamma_lower(i, s); + test::tgamma_lower(fr, lr); + test::gamma_p(i, s); + test::gamma_p(fr, lr); + test::gamma_q(i, s); + test::gamma_q(fr, lr); + test::gamma_p_inv(i, fr); + test::gamma_q_inv(s, fr); + test::gamma_p_inva(i, lr); + test::gamma_q_inva(i, lr); + test::erf(i); + test::erfc(i); + test::erf_inv(i); + test::erfc_inv(i); + test::beta(i, s); + test::beta(fr, lr); + test::beta(i, s, l); + test::beta(fr, dr, lr); + test::betac(l, i, s); + test::betac(fr, dr, lr); + test::ibeta(l, i, s); + test::ibeta(fr, dr, lr); + test::ibetac(l, i, s); + test::ibetac(fr, dr, lr); + test::ibeta_inv(l, s, i); + test::ibeta_inv(fr, dr, lr); + test::ibetac_inv(l, i, s); + test::ibetac_inv(fr, dr, lr); + test::ibeta_inva(l, i, s); + test::ibeta_inva(fr, dr, lr); + test::ibetac_inva(l, i, s); + test::ibetac_inva(fr, dr, lr); + test::ibeta_invb(l, i, s); + test::ibeta_invb(fr, dr, lr); + test::ibetac_invb(l, i, s); + test::ibetac_invb(fr, dr, lr); + test::gamma_p_derivative(i, l); + test::gamma_p_derivative(fr, lr); + test::ibeta_derivative(l, i, s); + test::ibeta_derivative(fr, dr, lr); + (test::fpclassify)(i); + (test::isfinite)(s); + (test::isnormal)(l); + (test::isnan)(i); + (test::isinf)(l); + test::log1p(i); + test::expm1(s); + test::cbrt(l); + test::sqrt1pm1(s); + test::powm1(i, s); + test::powm1(fr, lr); + //test::legendre_p(1, i); + test::legendre_p(1, 0, s); + test::legendre_q(1, i); + test::laguerre(1, i); + test::laguerre(2, 1, i); + test::laguerre(2u, 1u, s); + test::hermite(1, s); + test::spherical_harmonic_r(2, 1, s, i); + test::spherical_harmonic_i(2, 1, fr, lr); + test::ellint_1(i); + test::ellint_1(i, s); + test::ellint_1(fr, lr); + test::ellint_2(i); + test::ellint_2(i, l); + test::ellint_2(fr, lr); + test::ellint_3(i, l); + test::ellint_3(fr, lr); + test::ellint_3(s, l, i); + test::ellint_3(fr, dr, lr); + test::ellint_rc(i, s); + test::ellint_rc(fr, lr); + test::ellint_rd(s, i, l); + test::ellint_rd(fr, lr, dr); + test::ellint_rf(s, l, i); + test::ellint_rf(fr, dr, lr); + test::ellint_rj(i, i, s, l); + test::ellint_rj(i, fr, dr, lr); + test::hypot(i, s); + test::hypot(fr, lr); + test::sinc_pi(i); + test::sinhc_pi(i); + test::asinh(s); + test::acosh(l); + test::atanh(l); + test::sin_pi(s); + test::cos_pi(s); + test::cyl_neumann(fr, dr); + test::cyl_neumann(i, s); + test::cyl_bessel_j(fr, lr); + test::cyl_bessel_j(i, s); + test::cyl_bessel_i(fr, lr); + test::cyl_bessel_i(i, s); + test::cyl_bessel_k(fr, lr); + test::cyl_bessel_k(i, s); + test::sph_bessel(i, fr); + test::sph_bessel(i, 1); + test::sph_neumann(i, lr); + test::sph_neumann(i, i); +} + + +#endif // BOOST_LIBS_MATH_TEST_INSTANTIATE_HPP diff --git a/test/compile_test/sf_bessel_incl_test.cpp b/test/compile_test/sf_bessel_incl_test.cpp new file mode 100644 index 000000000..c1371b26a --- /dev/null +++ b/test/compile_test/sf_bessel_incl_test.cpp @@ -0,0 +1,53 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::cyl_bessel_j(f, f)); + check_result(boost::math::cyl_bessel_j(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cyl_bessel_j(l, l)); +#endif + + check_result(boost::math::cyl_neumann(f, f)); + check_result(boost::math::cyl_neumann(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cyl_neumann(l, l)); +#endif + + check_result(boost::math::cyl_bessel_i(f, f)); + check_result(boost::math::cyl_bessel_i(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cyl_bessel_i(l, l)); +#endif + + check_result(boost::math::cyl_bessel_k(f, f)); + check_result(boost::math::cyl_bessel_k(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cyl_bessel_k(l, l)); +#endif + + check_result(boost::math::sph_bessel(u, f)); + check_result(boost::math::sph_bessel(u, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sph_bessel(u, l)); +#endif + + check_result(boost::math::sph_neumann(u, f)); + check_result(boost::math::sph_neumann(u, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sph_neumann(u, l)); +#endif +} diff --git a/test/compile_test/sf_beta_incl_test.cpp b/test/compile_test/sf_beta_incl_test.cpp new file mode 100644 index 000000000..5b4c0ddac --- /dev/null +++ b/test/compile_test/sf_beta_incl_test.cpp @@ -0,0 +1,42 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::beta(f, f)); + check_result(boost::math::beta(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::beta(l, l)); +#endif + + check_result(boost::math::ibeta(f, f, f)); + check_result(boost::math::ibeta(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ibeta(l, l, l)); +#endif + + check_result(boost::math::ibeta_inv(f, f, f)); + check_result(boost::math::ibeta_inv(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ibeta_inv(l, l, l)); +#endif + + check_result(boost::math::ibeta_inva(f, f, f)); + check_result(boost::math::ibeta_inva(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ibeta_inva(l, l, l)); +#endif +} + diff --git a/test/compile_test/sf_binomial_incl_test.cpp b/test/compile_test/sf_binomial_incl_test.cpp new file mode 100644 index 000000000..ebdbdfa21 --- /dev/null +++ b/test/compile_test/sf_binomial_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::binomial_coefficient(u, u)); + check_result(boost::math::binomial_coefficient(u, u)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::binomial_coefficient(u, u)); +#endif +} diff --git a/test/compile_test/sf_cbrt_incl_test.cpp b/test/compile_test/sf_cbrt_incl_test.cpp new file mode 100644 index 000000000..6620cc0f8 --- /dev/null +++ b/test/compile_test/sf_cbrt_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::cbrt(f)); + check_result(boost::math::cbrt(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cbrt(l)); +#endif +} diff --git a/test/compile_test/sf_cos_pi_incl_test.cpp b/test/compile_test/sf_cos_pi_incl_test.cpp new file mode 100644 index 000000000..8053aaa4b --- /dev/null +++ b/test/compile_test/sf_cos_pi_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::cos_pi(f)); + check_result(boost::math::cos_pi(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::cos_pi(l)); +#endif +} diff --git a/test/compile_test/sf_digamma_incl_test.cpp b/test/compile_test/sf_digamma_incl_test.cpp new file mode 100644 index 000000000..12356d67d --- /dev/null +++ b/test/compile_test/sf_digamma_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::digamma(f)); + check_result(boost::math::digamma(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::digamma(l)); +#endif +} \ No newline at end of file diff --git a/test/compile_test/sf_ellint_1_incl_test.cpp b/test/compile_test/sf_ellint_1_incl_test.cpp new file mode 100644 index 000000000..74fb5df48 --- /dev/null +++ b/test/compile_test/sf_ellint_1_incl_test.cpp @@ -0,0 +1,29 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_1(f, f)); + check_result(boost::math::ellint_1(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_1(l, l)); +#endif + + check_result(boost::math::ellint_1(f)); + check_result(boost::math::ellint_1(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_1(l)); +#endif +} diff --git a/test/compile_test/sf_ellint_2_incl_test.cpp b/test/compile_test/sf_ellint_2_incl_test.cpp new file mode 100644 index 000000000..3ca5ff69b --- /dev/null +++ b/test/compile_test/sf_ellint_2_incl_test.cpp @@ -0,0 +1,29 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_2(f, f)); + check_result(boost::math::ellint_2(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_2(l, l)); +#endif + + check_result(boost::math::ellint_2(f)); + check_result(boost::math::ellint_2(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_2(l)); +#endif +} diff --git a/test/compile_test/sf_ellint_3_incl_test.cpp b/test/compile_test/sf_ellint_3_incl_test.cpp new file mode 100644 index 000000000..a46444b7d --- /dev/null +++ b/test/compile_test/sf_ellint_3_incl_test.cpp @@ -0,0 +1,29 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_3(f, f)); + check_result(boost::math::ellint_3(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_3(l, l)); +#endif + + check_result(boost::math::ellint_3(f, f, f)); + check_result(boost::math::ellint_3(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_3(l, l, l)); +#endif +} diff --git a/test/compile_test/sf_ellint_rc_incl_test.cpp b/test/compile_test/sf_ellint_rc_incl_test.cpp new file mode 100644 index 000000000..4e67aab04 --- /dev/null +++ b/test/compile_test/sf_ellint_rc_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_rc(f, f)); + check_result(boost::math::ellint_rc(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_rc(l, l)); +#endif +} diff --git a/test/compile_test/sf_ellint_rd_incl_test.cpp b/test/compile_test/sf_ellint_rd_incl_test.cpp new file mode 100644 index 000000000..b0be0fc07 --- /dev/null +++ b/test/compile_test/sf_ellint_rd_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_rd(f, f, f)); + check_result(boost::math::ellint_rd(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_rd(l, l, l)); +#endif +} diff --git a/test/compile_test/sf_ellint_rf_incl_test.cpp b/test/compile_test/sf_ellint_rf_incl_test.cpp new file mode 100644 index 000000000..6547cf184 --- /dev/null +++ b/test/compile_test/sf_ellint_rf_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_rf(f, f, f)); + check_result(boost::math::ellint_rf(d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_rf(l, l, l)); +#endif +} diff --git a/test/compile_test/sf_ellint_rj_incl_test.cpp b/test/compile_test/sf_ellint_rj_incl_test.cpp new file mode 100644 index 000000000..ee7e4d0c7 --- /dev/null +++ b/test/compile_test/sf_ellint_rj_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::ellint_rj(f, f, f, f)); + check_result(boost::math::ellint_rj(d, d, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::ellint_rj(l, l, l, l)); +#endif +} diff --git a/test/compile_test/sf_erf_incl_test.cpp b/test/compile_test/sf_erf_incl_test.cpp new file mode 100644 index 000000000..9931128f7 --- /dev/null +++ b/test/compile_test/sf_erf_incl_test.cpp @@ -0,0 +1,41 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::erf(f)); + check_result(boost::math::erf(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::erf(l)); +#endif + + check_result(boost::math::erfc(f)); + check_result(boost::math::erfc(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::erfc(l)); +#endif + + check_result(boost::math::erf_inv(f)); + check_result(boost::math::erf_inv(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::erf_inv(l)); +#endif + + check_result(boost::math::erfc_inv(f)); + check_result(boost::math::erfc_inv(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::erfc_inv(l)); +#endif +} diff --git a/test/compile_test/sf_expm1_incl_test.cpp b/test/compile_test/sf_expm1_incl_test.cpp new file mode 100644 index 000000000..dba9e9cca --- /dev/null +++ b/test/compile_test/sf_expm1_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::expm1(f)); + check_result(boost::math::expm1(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::expm1(l)); +#endif +} diff --git a/test/compile_test/sf_factorials_incl_test.cpp b/test/compile_test/sf_factorials_incl_test.cpp new file mode 100644 index 000000000..6d2a01817 --- /dev/null +++ b/test/compile_test/sf_factorials_incl_test.cpp @@ -0,0 +1,42 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::factorial(u)); + check_result(boost::math::factorial(u)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::factorial(u)); +#endif + + check_result(boost::math::double_factorial(u)); + check_result(boost::math::double_factorial(u)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::double_factorial(u)); +#endif + + check_result(boost::math::rising_factorial(f, i)); + check_result(boost::math::rising_factorial(d, i)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::rising_factorial(l, i)); +#endif + + check_result(boost::math::falling_factorial(f, u)); + check_result(boost::math::falling_factorial(d, u)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::falling_factorial(l, u)); +#endif +} + diff --git a/test/compile_test/sf_fpclassify_incl_test.cpp b/test/compile_test/sf_fpclassify_incl_test.cpp new file mode 100644 index 000000000..4515a901b --- /dev/null +++ b/test/compile_test/sf_fpclassify_incl_test.cpp @@ -0,0 +1,43 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::fpclassify BOOST_NO_MACRO_EXPAND(f)); + check_result(boost::math::fpclassify BOOST_NO_MACRO_EXPAND(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::fpclassify BOOST_NO_MACRO_EXPAND(l)); +#endif + + check_result(boost::math::isfinite BOOST_NO_MACRO_EXPAND(f)); + check_result(boost::math::isfinite BOOST_NO_MACRO_EXPAND(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::isfinite BOOST_NO_MACRO_EXPAND(l)); +#endif + + check_result(boost::math::isinf BOOST_NO_MACRO_EXPAND(f)); + check_result(boost::math::isinf BOOST_NO_MACRO_EXPAND(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::isinf BOOST_NO_MACRO_EXPAND(l)); +#endif + + check_result(boost::math::isnormal BOOST_NO_MACRO_EXPAND(f)); + check_result(boost::math::isnormal BOOST_NO_MACRO_EXPAND(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::isnormal BOOST_NO_MACRO_EXPAND(l)); +#endif +} + + diff --git a/test/compile_test/sf_gamma_incl_test.cpp b/test/compile_test/sf_gamma_incl_test.cpp new file mode 100644 index 000000000..fd0dabbf4 --- /dev/null +++ b/test/compile_test/sf_gamma_incl_test.cpp @@ -0,0 +1,83 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::tgamma(f)); + check_result(boost::math::tgamma(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::tgamma(l)); +#endif + + check_result(boost::math::lgamma(f)); + check_result(boost::math::lgamma(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::lgamma(l)); +#endif + + check_result(boost::math::gamma_p(f, f)); + check_result(boost::math::gamma_p(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_p(l, l)); +#endif + + check_result(boost::math::gamma_q(f, f)); + check_result(boost::math::gamma_q(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_q(l, l)); +#endif + + check_result(boost::math::gamma_p_inv(f, f)); + check_result(boost::math::gamma_p_inv(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_p_inv(l, l)); +#endif + + check_result(boost::math::gamma_q_inv(f, f)); + check_result(boost::math::gamma_q_inv(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_q_inv(l, l)); +#endif + + check_result(boost::math::gamma_p_inva(f, f)); + check_result(boost::math::gamma_p_inva(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_p_inva(l, l)); +#endif + + check_result(boost::math::gamma_q_inva(f, f)); + check_result(boost::math::gamma_q_inva(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_q_inva(l, l)); +#endif + + check_result(boost::math::gamma_p_derivative(f, f)); + check_result(boost::math::gamma_p_derivative(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::gamma_p_derivative(l, l)); +#endif + + check_result(boost::math::tgamma_ratio(f, f)); + check_result(boost::math::tgamma_ratio(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::tgamma_ratio(l, l)); +#endif + + check_result(boost::math::tgamma_delta_ratio(f, f)); + check_result(boost::math::tgamma_delta_ratio(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::tgamma_delta_ratio(l, l)); +#endif +} diff --git a/test/compile_test/sf_hermite_incl_test.cpp b/test/compile_test/sf_hermite_incl_test.cpp new file mode 100644 index 000000000..757897939 --- /dev/null +++ b/test/compile_test/sf_hermite_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::hermite(u, f)); + check_result(boost::math::hermite(u, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::hermite(u, l)); +#endif +} diff --git a/test/compile_test/sf_hypot_incl_test.cpp b/test/compile_test/sf_hypot_incl_test.cpp new file mode 100644 index 000000000..cb2c9d19a --- /dev/null +++ b/test/compile_test/sf_hypot_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::hypot(f, f)); + check_result(boost::math::hypot(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::hypot(l, l)); +#endif +} diff --git a/test/compile_test/sf_laguerre_incl_test.cpp b/test/compile_test/sf_laguerre_incl_test.cpp new file mode 100644 index 000000000..7683d1aab --- /dev/null +++ b/test/compile_test/sf_laguerre_incl_test.cpp @@ -0,0 +1,32 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::laguerre(u, f)); + check_result(boost::math::laguerre(u, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::laguerre(u, l)); +#endif + + typedef boost::math::policies::policy<> def_pol; + def_pol p; + + check_result(boost::math::laguerre(u, u, f, p)); + check_result(boost::math::laguerre(u, u, d, p)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::laguerre(u, u, l, p)); +#endif +} diff --git a/test/compile_test/sf_lanczos_incl_test.cpp b/test/compile_test/sf_lanczos_incl_test.cpp new file mode 100644 index 000000000..ef54883ba --- /dev/null +++ b/test/compile_test/sf_lanczos_incl_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/sf_legendre_incl_test.cpp b/test/compile_test/sf_legendre_incl_test.cpp new file mode 100644 index 000000000..8ba867249 --- /dev/null +++ b/test/compile_test/sf_legendre_incl_test.cpp @@ -0,0 +1,35 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::legendre_p(i, f)); + check_result(boost::math::legendre_p(i, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::legendre_p(i, l)); +#endif + + check_result(boost::math::legendre_p(i, i, f)); + check_result(boost::math::legendre_p(i, i, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::legendre_p(i, i, l)); +#endif + + check_result(boost::math::legendre_q(u, f)); + check_result(boost::math::legendre_q(u, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::legendre_q(u, l)); +#endif +} diff --git a/test/compile_test/sf_log1p_incl_test.cpp b/test/compile_test/sf_log1p_incl_test.cpp new file mode 100644 index 000000000..0d4737f6d --- /dev/null +++ b/test/compile_test/sf_log1p_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::log1p(f)); + check_result(boost::math::log1p(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::log1p(l)); +#endif +} diff --git a/test/compile_test/sf_math_fwd_incl_test.cpp b/test/compile_test/sf_math_fwd_incl_test.cpp new file mode 100644 index 000000000..6e790977d --- /dev/null +++ b/test/compile_test/sf_math_fwd_incl_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/sf_powm1_incl_test.cpp b/test/compile_test/sf_powm1_incl_test.cpp new file mode 100644 index 000000000..56f474709 --- /dev/null +++ b/test/compile_test/sf_powm1_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::powm1(f, f)); + check_result(boost::math::powm1(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::powm1(l, l)); +#endif +} diff --git a/test/compile_test/sf_sign_incl_test.cpp b/test/compile_test/sf_sign_incl_test.cpp new file mode 100644 index 000000000..161b943b1 --- /dev/null +++ b/test/compile_test/sf_sign_incl_test.cpp @@ -0,0 +1,35 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::sign(f)); + check_result(boost::math::sign(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sign(l)); +#endif + + check_result(boost::math::signbit(f)); + check_result(boost::math::signbit(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::signbit(l)); +#endif + + check_result(boost::math::copysign(f, f)); + check_result(boost::math::copysign(d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::copysign(l, l)); +#endif +} diff --git a/test/compile_test/sf_sin_pi_incl_test.cpp b/test/compile_test/sf_sin_pi_incl_test.cpp new file mode 100644 index 000000000..99e8c8853 --- /dev/null +++ b/test/compile_test/sf_sin_pi_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::sin_pi(f)); + check_result(boost::math::sin_pi(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sin_pi(l)); +#endif +} diff --git a/test/compile_test/sf_sinc_incl_test.cpp b/test/compile_test/sf_sinc_incl_test.cpp new file mode 100644 index 000000000..4c51cb329 --- /dev/null +++ b/test/compile_test/sf_sinc_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::sinc_pi(f)); + check_result(boost::math::sinc_pi(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sinc_pi(l)); +#endif +} diff --git a/test/compile_test/sf_sinhc_incl_test.cpp b/test/compile_test/sf_sinhc_incl_test.cpp new file mode 100644 index 000000000..7eeb6173a --- /dev/null +++ b/test/compile_test/sf_sinhc_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::sinhc_pi(f)); + check_result(boost::math::sinhc_pi(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sinhc_pi(l)); +#endif +} diff --git a/test/compile_test/sf_sph_harm_incl_test.cpp b/test/compile_test/sf_sph_harm_incl_test.cpp new file mode 100644 index 000000000..741d3b2d0 --- /dev/null +++ b/test/compile_test/sf_sph_harm_incl_test.cpp @@ -0,0 +1,43 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +inline void check_result_imp(std::complex, std::complex){} +inline void check_result_imp(std::complex, std::complex){} +inline void check_result_imp(std::complex, std::complex){} + +#include "test_compile_result.hpp" + + + +void check() +{ + check_result >(boost::math::spherical_harmonic(u, i, f, f)); + check_result >(boost::math::spherical_harmonic(u, i, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result >(boost::math::spherical_harmonic(u, i, l, l)); +#endif + + check_result(boost::math::spherical_harmonic_r(u, i, f, f)); + check_result(boost::math::spherical_harmonic_r(u, i, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::spherical_harmonic_r(u, i, l, l)); +#endif + + check_result(boost::math::spherical_harmonic_i(u, i, f, f)); + check_result(boost::math::spherical_harmonic_i(u, i, d, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::spherical_harmonic_i(u, i, l, l)); +#endif +} + + diff --git a/test/compile_test/sf_sqrt1pm1_incl_test.cpp b/test/compile_test/sf_sqrt1pm1_incl_test.cpp new file mode 100644 index 000000000..b26f6a2eb --- /dev/null +++ b/test/compile_test/sf_sqrt1pm1_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void check() +{ + check_result(boost::math::sqrt1pm1(f)); + check_result(boost::math::sqrt1pm1(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::sqrt1pm1(l)); +#endif +} diff --git a/test/compile_test/std_real_concept_check.cpp b/test/compile_test/std_real_concept_check.cpp new file mode 100644 index 000000000..10b407d44 --- /dev/null +++ b/test/compile_test/std_real_concept_check.cpp @@ -0,0 +1,200 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false + +#include +#include + +#include "instantiate.hpp" + +// +// The purpose of this test is to verify that our code compiles +// cleanly with a type whose std lib functions are in namespace +// std and can *not* be found by ADL. This verifies that we're +// not finding std lib functions that are in the global namespace +// for example calling ::pow(double) rather than std::pow(long double). +// This is a silent error that does the wrong thing at runtime, and +// of course we can't call std::pow() directly because we want +// the functions to be found by ADL when that's appropriate. +// +// Furthermore our code does different things internally depending +// on numeric_limits<>::digits, so there are some macros that can +// be defined that cause our concept-archetype to emulate various +// floating point types: +// +// EMULATE32: 32-bit float +// EMULATE64: 64-bit double +// EMULATE80: 80-bit long double +// EMULATE128: 128-bit long double +// +// In order to ensure total code coverage this file must be +// compiled with each of the above macros in turn, and then +// without any of the above as well! +// + +#define NULL_MACRO /**/ +#ifdef EMULATE32 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 24; + static const int digits10 = 6; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -125; + static const int min_exponent10 = -37; + static const int max_exponent = 128; + static const int max_exponent10 = 38; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE64 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 53; + static const int digits10 = 15; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -1021; + static const int min_exponent10 = -307; + static const int max_exponent = 1024; + static const int max_exponent10 = 308; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE80 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 64; + static const int digits10 = 18; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -16381; + static const int min_exponent10 = -4931; + static const int max_exponent = 16384; + static const int max_exponent10 = 4932; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE128 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 113; + static const int digits10 = 33; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -16381; + static const int min_exponent10 = -4931; + static const int max_exponent = 16384; + static const int max_exponent10 = 4932; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif + + + +int main() +{ +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + instantiate(boost::math::concepts::std_real_concept(0)); +#endif +} + diff --git a/test/compile_test/test_compile_result.hpp b/test/compile_test/test_compile_result.hpp new file mode 100644 index 000000000..9a3fd30ae --- /dev/null +++ b/test/compile_test/test_compile_result.hpp @@ -0,0 +1,131 @@ +// Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// +// Note this header must NOT include any other headers, for its +// use to be meaningful (because we use it in tests designed to +// detect missing includes). +// + +static const float f = 0; +static const double d = 0; +static const long double l = 0; +static const unsigned u = 0; +static const int i = 0; + +//template +//inline void check_result_imp(T, T){} + +inline void check_result_imp(float, float){} +inline void check_result_imp(double, double){} +inline void check_result_imp(long double, long double){} +inline void check_result_imp(int, int){} +inline void check_result_imp(bool, bool){} + +template +inline void check_result_imp(T1, T2) +{ + typedef int static_assertion[sizeof(T1) == 0xFFFF]; +} + +template +inline void check_result(T2) +{ + T1 a = T1(); + T2 b = T2(); + return check_result_imp(a, b); +} + + +template +struct DistributionConcept +{ + static void constraints() + { + typedef typename Distribution::value_type value_type; + + const Distribution& dist = DistributionConcept::get_object(); + + value_type x = 0; + // The result values are ignored in all these checks. + check_result(cdf(dist, x)); + check_result(cdf(complement(dist, x))); + check_result(pdf(dist, x)); + check_result(quantile(dist, x)); + check_result(quantile(complement(dist, x))); + check_result(mean(dist)); + check_result(mode(dist)); + check_result(standard_deviation(dist)); + check_result(variance(dist)); + check_result(hazard(dist, x)); + check_result(chf(dist, x)); + check_result(coefficient_of_variation(dist)); + check_result(skewness(dist)); + check_result(kurtosis(dist)); + check_result(kurtosis_excess(dist)); + check_result(median(dist)); + // + // we can't actually test that at std::pair is returned from these + // because that would mean including some std lib headers.... + // + range(dist); + support(dist); + + check_result(cdf(dist, f)); + check_result(cdf(complement(dist, f))); + check_result(pdf(dist, f)); + check_result(quantile(dist, f)); + check_result(quantile(complement(dist, f))); + check_result(hazard(dist, f)); + check_result(chf(dist, f)); + check_result(cdf(dist, d)); + check_result(cdf(complement(dist, d))); + check_result(pdf(dist, d)); + check_result(quantile(dist, d)); + check_result(quantile(complement(dist, d))); + check_result(hazard(dist, d)); + check_result(chf(dist, d)); + check_result(cdf(dist, l)); + check_result(cdf(complement(dist, l))); + check_result(pdf(dist, l)); + check_result(quantile(dist, l)); + check_result(quantile(complement(dist, l))); + check_result(hazard(dist, l)); + check_result(chf(dist, l)); + check_result(cdf(dist, i)); + check_result(cdf(complement(dist, i))); + check_result(pdf(dist, i)); + check_result(quantile(dist, i)); + check_result(quantile(complement(dist, i))); + check_result(hazard(dist, i)); + check_result(chf(dist, i)); + unsigned long li = 1; + check_result(cdf(dist, li)); + check_result(cdf(complement(dist, li))); + check_result(pdf(dist, li)); + check_result(quantile(dist, li)); + check_result(quantile(complement(dist, li))); + check_result(hazard(dist, li)); + check_result(chf(dist, li)); + } +private: + static Distribution* get_object_p(); + static Distribution& get_object() + { + // will never get called: + return *get_object_p(); + } +}; // struct DistributionConcept + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#define TEST_DIST_FUNC(dist)\ + DistributionConcept< boost::math::dist##_distribution >::constraints();\ + DistributionConcept< boost::math::dist##_distribution >::constraints();\ + DistributionConcept< boost::math::dist##_distribution >::constraints(); +#else +#define TEST_DIST_FUNC(dist)\ + DistributionConcept< boost::math::dist##_distribution >::constraints();\ + DistributionConcept< boost::math::dist##_distribution >::constraints(); +#endif diff --git a/test/compile_test/test_traits.cpp b/test/compile_test/test_traits.cpp new file mode 100644 index 000000000..6af0ef360 --- /dev/null +++ b/test/compile_test/test_traits.cpp @@ -0,0 +1,56 @@ +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include + +using namespace boost::math; + +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution >::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution >::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_distribution::value); + +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution >::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution >::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); +BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value == false); + diff --git a/test/compile_test/tools_config_inc_test.cpp b/test/compile_test/tools_config_inc_test.cpp new file mode 100644 index 000000000..d9f765940 --- /dev/null +++ b/test/compile_test/tools_config_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_fraction_inc_test.cpp b/test/compile_test/tools_fraction_inc_test.cpp new file mode 100644 index 000000000..e185137e0 --- /dev/null +++ b/test/compile_test/tools_fraction_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_minima_inc_test.cpp b/test/compile_test/tools_minima_inc_test.cpp new file mode 100644 index 000000000..367d7b63a --- /dev/null +++ b/test/compile_test/tools_minima_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_polynomial_inc_test.cpp b/test/compile_test/tools_polynomial_inc_test.cpp new file mode 100644 index 000000000..111d5be5d --- /dev/null +++ b/test/compile_test/tools_polynomial_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_precision_inc_test.cpp b/test/compile_test/tools_precision_inc_test.cpp new file mode 100644 index 000000000..84c59ef93 --- /dev/null +++ b/test/compile_test/tools_precision_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_rational_inc_test.cpp b/test/compile_test/tools_rational_inc_test.cpp new file mode 100644 index 000000000..46b5de44f --- /dev/null +++ b/test/compile_test/tools_rational_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_real_cast_inc_test.cpp b/test/compile_test/tools_real_cast_inc_test.cpp new file mode 100644 index 000000000..e46bdc94c --- /dev/null +++ b/test/compile_test/tools_real_cast_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_remez_inc_test.cpp b/test/compile_test/tools_remez_inc_test.cpp new file mode 100644 index 000000000..121b0b4da --- /dev/null +++ b/test/compile_test/tools_remez_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_roots_inc_test.cpp b/test/compile_test/tools_roots_inc_test.cpp new file mode 100644 index 000000000..6bd5a9ce7 --- /dev/null +++ b/test/compile_test/tools_roots_inc_test.cpp @@ -0,0 +1,38 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +inline void check_result_imp(std::pair, std::pair){} +inline void check_result_imp(std::pair, std::pair){} +inline void check_result_imp(std::pair, std::pair){} + +#include "test_compile_result.hpp" + +void check() +{ + typedef double (*F)(double); + typedef std::pair (*F2)(double); + typedef std::tr1::tuple (*F3)(double); + typedef boost::math::tools::eps_tolerance Tol; + Tol tol(u); + boost::uintmax_t max_iter = 0; + F f = 0; + F2 f2 = 0; + F3 f3 = 0; + + check_result >(boost::math::tools::bisect(f, d, d, tol, max_iter)); + check_result >(boost::math::tools::bisect(f, d, d, tol)); + check_result(boost::math::tools::newton_raphson_iterate(f2, d, d, d, i, max_iter)); + check_result(boost::math::tools::halley_iterate(f3, d, d, d, i, max_iter)); + check_result(boost::math::tools::schroeder_iterate(f3, d, d, d, i, max_iter)); +} + diff --git a/test/compile_test/tools_series_inc_test.cpp b/test/compile_test/tools_series_inc_test.cpp new file mode 100644 index 000000000..0a51905ee --- /dev/null +++ b/test/compile_test/tools_series_inc_test.cpp @@ -0,0 +1,35 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +struct Functor +{ + typedef double result_type; + double operator()(); +}; +#define U double + +Functor func; +boost::uintmax_t uim = 0; + +void check() +{ + check_result(boost::math::tools::sum_series(func, i)); + check_result(boost::math::tools::sum_series(func, i, uim)); + check_result(boost::math::tools::sum_series(func, i, d)); + check_result(boost::math::tools::sum_series(func, i, uim, d)); + check_result(boost::math::tools::kahan_sum_series(func, i)); + check_result(boost::math::tools::kahan_sum_series(func, i, uim)); +} + diff --git a/test/compile_test/tools_solve_inc_test.cpp b/test/compile_test/tools_solve_inc_test.cpp new file mode 100644 index 000000000..a0dcd0d3b --- /dev/null +++ b/test/compile_test/tools_solve_inc_test.cpp @@ -0,0 +1,10 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + diff --git a/test/compile_test/tools_stats_inc_test.cpp b/test/compile_test/tools_stats_inc_test.cpp new file mode 100644 index 000000000..6d5898b4d --- /dev/null +++ b/test/compile_test/tools_stats_inc_test.cpp @@ -0,0 +1,11 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +template class boost::math::tools::stats; diff --git a/test/compile_test/tools_test_data_inc_test.cpp b/test/compile_test/tools_test_data_inc_test.cpp new file mode 100644 index 000000000..d2daa5e99 --- /dev/null +++ b/test/compile_test/tools_test_data_inc_test.cpp @@ -0,0 +1,21 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +#define T double + +template boost::math::tools::parameter_info boost::math::tools::make_random_param(T start_range, T end_range, int n_points); +template boost::math::tools::parameter_info boost::math::tools::make_periodic_param(T start_range, T end_range, int n_points); +template boost::math::tools::parameter_info boost::math::tools::make_power_param(T basis, int start_exponent, int end_exponent); + +template class boost::math::tools::test_data; + +template bool boost::math::tools::get_user_parameter_info(boost::math::tools::parameter_info& info, const char* param_name); + + diff --git a/test/compile_test/tools_test_inc_test.cpp b/test/compile_test/tools_test_inc_test.cpp new file mode 100644 index 000000000..d6a724923 --- /dev/null +++ b/test/compile_test/tools_test_inc_test.cpp @@ -0,0 +1,36 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +inline void check_result_imp(boost::math::tools::test_result, boost::math::tools::test_result){} + +#include "test_compile_result.hpp" + + +void check() +{ + check_result(boost::math::tools::relative_error(f, f)); + + #define A boost::array, 2> + typedef double (*F1)(const boost::array&); + typedef F1 F2; + A a; + F1 f1 = 0; + F2 f2 = 0; + + check_result::value_type> > + (boost::math::tools::test(a, f1, f2)); + +} + diff --git a/test/compile_test/tools_toms748_inc_test.cpp b/test/compile_test/tools_toms748_inc_test.cpp new file mode 100644 index 000000000..3c321006f --- /dev/null +++ b/test/compile_test/tools_toms748_inc_test.cpp @@ -0,0 +1,18 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include + +#define T double +#define Tol boost::math::tools::eps_tolerance + +typedef T (*F)(T); + +template std::pair boost::math::tools::toms748_solve(F, const T&, const T&, const T&, const T&, Tol, boost::uintmax_t&); +template std::pair boost::math::tools::toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter); +template std::pair boost::math::tools::bracket_and_solve_root(F f, const T& guess, const T& factor, bool rising, Tol tol, boost::uintmax_t& max_iter); diff --git a/test/complex_test.cpp b/test/complex_test.cpp index bcb60c002..cf879c056 100644 --- a/test/complex_test.cpp +++ b/test/complex_test.cpp @@ -853,9 +853,11 @@ int test_main(int, char*[]) check_spots(float(0)); std::cout << "Running complex trig spot checks for type double." << std::endl; check_spots(double(0)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS std::cout << "Running complex trig spot checks for type long double." << std::endl; check_spots((long double)(0)); - +#endif + std::cout << "Running complex trig boundary and accuracy tests." << std::endl; test_boundaries(); return 0; diff --git a/test/digamma_data.ipp b/test/digamma_data.ipp new file mode 100644 index 000000000..b5ab09aca --- /dev/null +++ b/test/digamma_data.ipp @@ -0,0 +1,510 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> digamma_data = { { + { SC_(2.818432331085205078125), SC_(0.8484115700906551606307984398000472347785) }, + { SC_(4.6342258453369140625), SC_(1.421713669467331557347601964119226105014) }, + { SC_(4.783483982086181640625), SC_(1.457016504476551729585831562031964212238) }, + { SC_(8.0939121246337890625), SC_(2.02806725001046109435006913084666450899) }, + { SC_(11.90207004547119140625), SC_(2.434114987242530269375534072241650635459) }, + { SC_(13.53912639617919921875), SC_(2.568199379535107033473901142572267131293) }, + { SC_(15.40344142913818359375), SC_(2.70177959979566700235213147968567613712) }, + { SC_(16.5205841064453125), SC_(2.774036634025874647373138331993451055497) }, + { SC_(17.7738971710205078125), SC_(2.849336085709903318587899668469823726457) }, + { SC_(18.45084381103515625), 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SC_(6.767103998002709424523574394478208979411) }, + { SC_(870.186767578125), SC_(6.768133164641880669262131602717669262048) }, + { SC_(872.4288330078125), SC_(6.770707861861071404604235546152482005058) }, + { SC_(875.94281005859375), SC_(6.774729881353175605276607003284786011147) }, + { SC_(876.7574462890625), SC_(6.775659990483147584810245626024403172256) }, + { SC_(877.3638916015625), SC_(6.776351836800695164519969516994906905958) }, + { SC_(878.4306640625), SC_(6.777567674259764690685426386600703554342) }, + { SC_(885.99810791015625), SC_(6.786150373612182701581356639019071248638) }, + { SC_(887.72613525390625), SC_(6.788099946665576471808465592376033272437) }, + { SC_(890.90325927734375), SC_(6.791674512973996076620205962146827011757) }, + { SC_(900.0538330078125), SC_(6.80189895079299198904108771904549258351) }, + { SC_(900.1832275390625), SC_(6.802042783444049392389068654739673857143) }, + { SC_(902.71612548828125), SC_(6.804854149520866258225931342229077189181) }, + { SC_(904.88092041015625), SC_(6.807250694718158106286277547784604092869) }, + { SC_(905.79193115234375), SC_(6.808257518383144815884886145813559834871) }, + { SC_(907.36474609375), SC_(6.809993367381673942808480783672644328598) }, + { SC_(908.43359375), SC_(6.811171291835528522342757827992129791562) }, + { SC_(910.56494140625), SC_(6.81351601145607889196433654356525525846) }, + { SC_(910.6475830078125), SC_(6.813606815790404023323189518843380307986) }, + { SC_(912.5775146484375), SC_(6.815725030681610574873878555395906108156) }, + { SC_(913.33734130859375), SC_(6.816557756247443703552647371667225468139) }, + { SC_(913.3758544921875), SC_(6.816599945984404149476890515171707321892) }, + { SC_(915.73553466796875), SC_(6.819181496829614596845401444040303420405) }, + { SC_(917.117919921875), SC_(6.820690771903488486844573292669501047527) }, + { SC_(917.19366455078125), SC_(6.82077340337986035180672708919674786967) }, + { SC_(920.874755859375), SC_(6.824780979614736502636113976154470262398) }, + { SC_(923.379638671875), SC_(6.827498872615433477170026117448771541295) }, + { SC_(927.49285888671875), SC_(6.8319459105580415512096734861697582082) }, + { SC_(928.85418701171875), SC_(6.83341337535443392277100858245459000213) }, + { SC_(929.263671875), SC_(6.833854364953862271803203220770951435661) }, + { SC_(929.385986328125), SC_(6.833986052262011870912515832740972478803) }, + { SC_(933.99322509765625), SC_(6.838933753242727661252945811698284684698) }, + { SC_(934.01068115234375), SC_(6.838952452778342207434589177668657889371) }, + { SC_(939.0015869140625), SC_(6.844284594236099395567215793156428746645) }, + { SC_(940.07403564453125), SC_(6.845426666187627323729754250792575000864) }, + { SC_(942.05059814453125), SC_(6.847528135708265169237587238138487198853) }, + { SC_(944.7872314453125), SC_(6.850430437191786907351651758111614319783) }, + { SC_(948.925048828125), SC_(6.854802811834284555381489323758582235861) }, + { SC_(950.2220458984375), SC_(6.856169404714020378408540666432044534508) }, + { SC_(954.943359375), SC_(6.861128346520892642201796166599822038471) }, + { SC_(955.017578125), SC_(6.861206104783820446126170121986330983269) }, + { SC_(956.13458251953125), SC_(6.862375649747642715205235987065776445518) }, + { SC_(957.1669921875), SC_(6.863455405877975518619709756325785766734) }, + { SC_(957.5068359375), SC_(6.863810580012297609207700982417845904956) }, + { SC_(959.2913818359375), SC_(6.865673559355261880433729035303716037083) }, + { SC_(959.492431640625), SC_(6.865883228230126473126797544809231414745) }, + { SC_(959.74395751953125), SC_(6.866145475218019847811591867037562659094) }, + { SC_(961.8980712890625), SC_(6.868388594389408661650623078145486428762) }, + { SC_(964.88848876953125), SC_(6.871494254878780851989128297020995382715) }, + { SC_(964.96636962890625), SC_(6.871575008339416373031500751728687321642) }, + { SC_(967.6949462890625), SC_(6.874400118733980051857714900416741852838) }, + { SC_(968.640380859375), SC_(6.87537714276502884037846310503316214533) }, + { SC_(968.8677978515625), SC_(6.875612016002868563135875018985921894413) }, + { SC_(970.5927734375), SC_(6.877391753918391265533592997745089246348) }, + { SC_(977.00201416015625), SC_(6.883976856642686182085022703036491098977) }, + { SC_(979.74835205078125), SC_(6.886785333127100045440434410093146016624) }, + { SC_(979.9256591796875), SC_(6.886966381237166628662813047575594521666) }, + { SC_(981.10968017578125), SC_(6.888174544185669495233192817179284924608) }, + { SC_(981.72314453125), SC_(6.888799943368115034020659880636949694688) }, + { SC_(982.36053466796875), SC_(6.8894493197658087146525978621453836978) }, + { SC_(987.4595947265625), SC_(6.894629143910089536380903205901047293626) }, + { SC_(988.37939453125), SC_(6.895560662679555267312975090816879245672) }, + { SC_(988.5216064453125), SC_(6.895704609063605825635576817887832481836) }, + { SC_(990.1099853515625), SC_(6.897310953864133148550105914216515716675) }, + { SC_(992.88128662109375), SC_(6.900107437258989936275310085429403733899) }, + { SC_(993.53472900390625), SC_(6.900765679507473199060169339559463847836) }, + { SC_(994.0684814453125), SC_(6.901303031311155785795352189437299613696) }, + { SC_(996.13470458984375), SC_(6.903380469883631241011306751657568074638) }, + { SC_(996.4613037109375), SC_(6.903708447139272864766001846716835845992) } + } }; +#undef SC_ + diff --git a/test/digamma_neg_data.ipp b/test/digamma_neg_data.ipp new file mode 100644 index 000000000..d77f8215e --- /dev/null +++ b/test/digamma_neg_data.ipp @@ -0,0 +1,211 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 200> digamma_neg_data = { { + { SC_(-99.7181549072265625), SC_(2.03892909952497242038694056382195623059) }, + { SC_(-99.5216522216796875), SC_(4.3913617771941913117971009646096341326) }, + { SC_(-98.80979156494140625), SC_(-0.01795976054399933340057713377738267488686) }, + { SC_(-98.646087646484375), SC_(3.044214979883807667874683391024711390163) }, + { SC_(-98.2226104736328125), SC_(8.327061837969621994688065941368639650164) }, + { SC_(-96.81671905517578125), SC_(-0.2613967795671165924570265553692310141858) }, + { SC_(-96.555389404296875), SC_(4.023029392905834803729410679283679163172) }, + { SC_(-96.4288330078125), SC_(5.288313818176304987255615817663265383342) }, + { SC_(-96.35587310791015625), SC_(6.101607804168127653000043332720050192048) }, + { SC_(-95.3828582763671875), SC_(5.774458772879798661504785984613127024683) }, + { SC_(-94.87835693359375), SC_(-3.258775525008949657912384482006797161857) }, + { SC_(-94.60498809814453125), SC_(3.479509682356698077739870471765368709254) }, + { SC_(-93.2404632568359375), SC_(7.876249181059212250961342610258680843403) }, + { SC_(-92.41457366943359375), SC_(5.395652110469876120841383508575006667739) }, + { SC_(-90.2868194580078125), SC_(6.996060449908181672759658412431764721268) }, + { SC_(-90.2459564208984375), SC_(7.730510069525265980585453066651639424664) }, + { SC_(-89.013824462890625), SC_(76.7844527353150160809893161375899712728) }, + { SC_(-88.75354766845703125), SC_(1.279073328662160048752885420029490083884) }, + { SC_(-88.1002349853515625), SC_(14.12873575145206447240140060245136812233) }, + { SC_(-87.481719970703125), SC_(4.657749706875254767656016829355880725463) }, + { SC_(-87.41033935546875), SC_(5.385408972387998240975025273868320766616) }, + { SC_(-87.3013153076171875), SC_(6.737807742019520253440825060046509923858) }, + { SC_(-87.00937652587890625), SC_(111.0902104512757743601088229496385482381) }, + { SC_(-86.452301025390625), SC_(4.939690918031593302255270591348782158549) }, + { SC_(-86.1375579833984375), SC_(11.27311653802311845949475246171079728424) }, + { SC_(-85.81136322021484375), SC_(-0.2076024359899236820459284405564934682571) }, + { SC_(-85.08860015869140625), SC_(15.4432197184156901762673734012738165669) }, + { SC_(-85.0706024169921875), SC_(18.38013049788840734996810404641615698096) }, + { SC_(-84.55615997314453125), SC_(3.88321705564237834505242432670033691134) }, + { SC_(-84.2386932373046875), SC_(7.812678135978754145263274987740845778654) }, + { SC_(-84.194244384765625), SC_(8.931719867523162463837746400725488234199) }, + { SC_(-83.73882293701171875), SC_(1.505295935399338449868426369185929751229) }, + { SC_(-82.8813323974609375), SC_(-3.609402776980612588036128298328365683225) }, + { SC_(-82.6134796142578125), SC_(3.250221400281531800616333769504919375611) }, + { SC_(-81.31273651123046875), SC_(6.500209496762236219781012282892172792378) }, + { SC_(-81.16180419921875), SC_(10.04118609757139096209050165941519368701) }, + { SC_(-80.34047698974609375), SC_(6.113463877628090613614429157059502101004) }, + { SC_(-79.1931915283203125), SC_(8.902649813785723219406837941195708301933) }, + { SC_(-78.979095458984375), SC_(-43.39220275151493342151673584769713674459) }, + { SC_(-78.80756378173828125), SC_(-0.1741139524199893908631289936754328553595) }, + { SC_(-78.276214599609375), SC_(7.029126641974004636072711755974329319293) }, + { SC_(-77.8965911865234375), SC_(-4.965949067939776233732103672063117487233) }, + { SC_(-77.618804931640625), SC_(3.128016725588726483845498994960937165101) }, + { SC_(-76.9843902587890625), SC_(-59.66111542957708160367279993835116598861) }, + { SC_(-75.6475067138671875), SC_(2.762831843674791808646068344156198358292) }, + { SC_(-74.8916168212890625), SC_(-4.544467476010996269925717795920967036228) }, + { SC_(-74.57178497314453125), SC_(3.59770242115359475843120457380803078855) }, + { SC_(-74.49048614501953125), SC_(4.411294618813885078478516140128263239646) }, + { SC_(-74.249176025390625), SC_(7.472045154786435852081961669032387751756) }, + { SC_(-72.397491455078125), SC_(5.337270392731504876256252154330660996853) }, + { SC_(-72.30770111083984375), SC_(6.45618615562028783175344592523791857763) }, + { SC_(-72.150177001953125), SC_(10.44291983579332484719137950538141198138) }, + { SC_(-71.41609954833984375), SC_(5.123297394484169650507849264219211277233) }, + { SC_(-70.29705810546875), SC_(6.586989653308987758464004111780026164342) }, + { SC_(-70.0168304443359375), SC_(63.61661733083149880967450730403400964169) }, + { SC_(-69.808685302734375), SC_(-0.3289873714853887848889508326638941457588) }, + { SC_(-69.18329620361328125), SC_(9.082833286105283040745693219253094753599) }, + { SC_(-68.3449554443359375), SC_(5.895816498784197711310887233679346985484) }, + { SC_(-68.2900543212890625), SC_(6.667078439728477642775937857513723712022) }, + { SC_(-66.7551727294921875), SC_(0.9631089274457797371832913555201378277263) }, + { SC_(-66.2877349853515625), SC_(6.674402390903648107264524741943941318692) }, + { SC_(-65.96142578125), SC_(-21.60039097244717646250068532377671067773) }, + { SC_(-65.00162506103515625), SC_(619.5382407651047414044960295217348989637) }, + { SC_(-64.83405303955078125), SC_(-1.290399564904368875515657543138626740639) }, + { SC_(-64.72376251220703125), SC_(1.515712957371258935902920146011383158566) }, + { SC_(-64.36548614501953125), SC_(5.585036804770642096748234463405359373268) }, + { SC_(-63.870601654052734375), SC_(-3.132938413135016068514633000959251655011) }, + { SC_(-61.955417633056640625), SC_(-18.14906679867205714676685355624200206247) }, + { SC_(-61.84415435791015625), SC_(-1.762828050189299963414925510084105410157) }, + { SC_(-61.27040863037109375), SC_(6.885962246437210164247050669769241762116) }, + { SC_(-61.1430206298828125), SC_(10.63639337906714605332887925207272339472) }, + { SC_(-60.7772979736328125), SC_(0.3828568174285943765552196371752184023788) }, + { SC_(-60.767955780029296875), SC_(0.597630996899231770205669545427601282449) }, + { SC_(-60.50917816162109375), SC_(4.020425571512186535251807369058720729888) }, + { SC_(-60.126148223876953125), SC_(11.61249771821316227897371950731036451444) }, + { SC_(-59.5791473388671875), SC_(3.298014524278177804824357796228486139154) }, + { SC_(-59.12688446044921875), SC_(11.54738093463764209365528407465938463008) }, + { SC_(-58.733348846435546875), SC_(1.252516603233681404470535780337376850768) }, + { SC_(-57.823871612548828125), SC_(-1.020030805813282973065541328061417103161) }, + { SC_(-57.791233062744140625), SC_(-0.01721196139850082730515140522511875898686) }, + { SC_(-57.6834869384765625), SC_(2.021283718214246528807976825874750730944) }, + { SC_(-56.1255645751953125), SC_(11.58306242149007962260737588966574096145) }, + { SC_(-55.441379547119140625), SC_(4.6095090326877333856682583039277727222) }, + { SC_(-54.396717071533203125), SC_(5.062176320861478792112156635998489324447) }, + { SC_(-54.201084136962890625), SC_(8.295102791777500360299701370233860660785) }, + { SC_(-53.06093597412109375), SC_(20.19053903029668873838826742013474757079) }, + { SC_(-52.671115875244140625), SC_(2.10070123867348232297487204513445237558) }, + { SC_(-52.52413177490234375), SC_(3.73213352669881119442059765740051858078) }, + { SC_(-51.462436676025390625), SC_(4.323002262954298089562762598650242024201) }, + { SC_(-51.243106842041015625), SC_(7.226998934762993806818100988804419542686) }, + { SC_(-51.0235595703125), SC_(46.31011451882384360024092555028601375848) }, + { SC_(-50.94109344482421875), SC_(-12.8413475810657768775226051735008739843) }, + { SC_(-50.163593292236328125), SC_(9.490021355163248564818621481640556958139) }, + { SC_(-49.63372802734375), SC_(2.511313156813305996562305257857390789172) }, + { SC_(-49.404293060302734375), SC_(4.884246649262366888920411419024147958277) }, + { SC_(-47.262851715087890625), SC_(6.763893885944739117755918311467424143999) }, + { SC_(-46.920246124267578125), SC_(-8.416023355487745311972990967262253971971) }, + { SC_(-45.31185150146484375), SC_(5.932978383248106730387626359282358526882) }, + { SC_(-45.278446197509765625), SC_(6.448720313065481170040935613035156187478) }, + { SC_(-45.277942657470703125), SC_(6.457159638091017467737698925977740795224) }, + { SC_(-45.027637481689453125), SC_(39.91011698129930359222267887372752760268) }, + { SC_(-43.8442535400390625), SC_(-2.107933118887280485309161602735949216416) }, + { SC_(-43.217838287353515625), SC_(7.628256500265451720152832482834432955622) }, + { SC_(-43.11763763427734375), SC_(11.88558178274471415613706962118145261485) }, + { SC_(-42.62453460693359375), SC_(2.468202270232387819257535057032048703186) }, + { SC_(-41.904331207275390625), SC_(-6.388806142931646581376930809159704270731) }, + { SC_(-41.473590850830078125), SC_(3.998311840817845119612809226760453214553) }, + { SC_(-41.47322845458984375), SC_(4.001904991009103072678533254287758877523) }, + { SC_(-40.71761322021484375), SC_(1.159373041348507101344016731863232548419) }, + { SC_(-40.549640655517578125), SC_(3.220862320117017004190871786172571326259) }, + { SC_(-38.39553070068359375), SC_(4.73066601123284545617603996389975515062) }, + { SC_(-36.764072418212890625), SC_(0.185641778964637473051349775352193113594) }, + { SC_(-36.0236663818359375), SC_(45.77413334958292444754118070178767678465) }, + { SC_(-35.60390472412109375), SC_(2.522888342129267961559021180379112152327) }, + { SC_(-35.3686981201171875), SC_(4.954668236416411879662604875880175363163) }, + { SC_(-34.49019622802734375), SC_(3.651891900429621653936520779034983699363) }, + { SC_(-34.452213287353515625), SC_(4.029227315027716909301715786442880409125) }, + { SC_(-34.425933837890625), SC_(4.297752459205291355542460490934969321341) }, + { SC_(-33.6394500732421875), SC_(2.05879207452756962707410762095930722907) }, + { SC_(-32.12648773193359375), SC_(10.97049318285659411778704721045579811462) }, + { SC_(-32.02973175048828125), SC_(37.01840088144976405978561031600232101035) }, + { SC_(-32.018024444580078125), SC_(58.90273484616011864735901401561254400453) }, + { SC_(-31.864048004150390625), SC_(-3.425649819825478548122204473279221831269) }, + { SC_(-30.5171375274658203125), SC_(3.265278944239551980140922031510366995665) }, + { SC_(-30.4767131805419921875), SC_(3.663521748406629288062646856398131484428) }, + { SC_(-30.0923290252685546875), SC_(13.94615623704883065583726298948748620016) }, + { SC_(-29.4225749969482421875), SC_(4.178249593518503446782790798157935603259) }, + { SC_(-29.395389556884765625), SC_(4.46906479672600980848725781028158753644) }, + { SC_(-29.0635166168212890625), SC_(18.92098221867999526957587349227217750794) }, + { SC_(-28.929615020751953125), SC_(-10.59321162183541691346972253310776238772) }, + { SC_(-27.416103363037109375), SC_(4.17700594969969944426868830026842791532) }, + { SC_(-27.304531097412109375), SC_(5.54012214812515808522852628894283103891) }, + { SC_(-26.72013092041015625), SC_(0.7028843790088342764015208907756718567463) }, + { SC_(-25.9352741241455078125), SC_(-11.9614838597048426449728575053473182996) }, + { SC_(-25.6867523193359375), SC_(1.176826484915453835886205411786243733377) }, + { SC_(-24.8732929229736328125), SC_(-4.237136620209855403786484087613189060887) }, + { SC_(-24.6270904541015625), SC_(1.898484134239669451822687252748496404653) }, + { SC_(-24.53133392333984375), SC_(2.909938662308256016840011167255882888239) }, + { SC_(-24.27997589111328125), SC_(5.809456022692943160671997012588234680914) }, + { SC_(-24.225986480712890625), SC_(6.863255301509444395634121067991546441736) }, + { SC_(-23.8268795013427734375), SC_(-2.003570170516321537241642099057200674105) }, + { SC_(-23.6249980926513671875), SC_(1.882051906931976769141489221324624486755) }, + { SC_(-23.4483184814453125), SC_(3.690576780901990543590973501647480396476) }, + { SC_(-22.6082859039306640625), SC_(2.02829972990549794339388303789633983053) }, + { SC_(-22.1102294921875), SC_(11.82489297106174961055859856595077767589) }, + { SC_(-22.0832767486572265625), SC_(14.85021720005861723216276067526099288351) }, + { SC_(-20.779270172119140625), SC_(-0.7220286362185610999510531120120858367544) }, + { SC_(-20.6402416229248046875), SC_(1.570036502538595917265616936047416332984) }, + { SC_(-20.60250091552734375), SC_(2.001360446832507792499854867762750216251) }, + { SC_(-20.480007171630859375), SC_(3.241245764823419396963096620359809656269) }, + { SC_(-20.27201080322265625), SC_(5.768313318300353594949489118483966781535) }, + { SC_(-20.2071361541748046875), SC_(7.156819372589727235791276774661186960423) }, + { SC_(-20.1894168853759765625), SC_(7.670698039778866912498508209079922928436) }, + { SC_(-19.971954345703125), SC_(-32.54467844963252077717911577731405834236) }, + { SC_(-19.2469005584716796875), SC_(6.186480084023335387087272289090747328989) }, + { SC_(-19.1824493408203125), SC_(7.847002126080603757397510906504995585177) }, + { SC_(-19.0265483856201171875), SC_(40.55157678949956103482720787629324186475) }, + { SC_(-18.5715198516845703125), SC_(2.230314699128516174606595799951710607353) }, + { SC_(-18.5276336669921875), SC_(2.672586759609702793340088983364248246452) }, + { SC_(-17.9159221649169921875), SC_(-8.702502484177860216144583065634001249363) }, + { SC_(-17.8754024505615234375), SC_(-4.700545068020732462956282791568579773556) }, + { SC_(-17.8096714019775390625), SC_(-1.704910040367315348529245955199242558161) }, + { SC_(-17.654216766357421875), SC_(1.245525979337004465966586338242325823861) }, + { SC_(-16.9171390533447265625), SC_(-8.936978387990812220163335657856867859665) }, + { SC_(-16.4991436004638671875), SC_(2.841759414051282631909664692084257003501) }, + { SC_(-15.9282741546630859375), SC_(-10.90604642089629536174323573315025319791) }, + { SC_(-15.1532230377197265625), SC_(8.7652341843566231062515337784346464505) }, + { SC_(-15.0870685577392578125), SC_(13.94393218121241823357181745576517766981) }, + { SC_(-12.7571163177490234375), SC_(-0.7005285484888729861556721133426365160483) }, + { SC_(-12.32425594329833984375), SC_(4.486877647992933354241229013873681135288) }, + { SC_(-12.15693378448486328125), SC_(8.38572290746722802141392801697704201607) }, + { SC_(-10.90967655181884765625), SC_(-8.337781110433724243601733849861559188948) }, + { SC_(-9.42080593109130859375), SC_(3.093207549574642835823398326728433769791) }, + { SC_(-9.26352691650390625), SC_(5.164408378967561819557747661923746174341) }, + { SC_(-8.66241455078125), SC_(0.4573363030846697184753722937434504783991) }, + { SC_(-8.42644596099853515625), SC_(2.928692490941547323611393715468003596111) }, + { SC_(-8.28063488006591796875), SC_(4.761502420123261711232867499795716416469) }, + { SC_(-7.91252231597900390625), SC_(-9.011928670506620642435146780806294048492) }, + { SC_(-7.073635101318359375), SC_(15.36275862317279916670563199810883722471) }, + { SC_(-6.600677967071533203125), SC_(0.9328456790163266794082601288835098544981) }, + { SC_(-6.59893131256103515625), SC_(0.9516512759861559819594253210068193102636) }, + { SC_(-5.9925975799560546875), SC_(-133.1949549912806814878528595454864731658) }, + { SC_(-4.9777927398681640625), SC_(-43.25515309581336581959034619574585012987) }, + { SC_(-4.4982433319091796875), SC_(1.628080704179537179987368246302687190569) }, + { SC_(-4.28330326080322265625), SC_(4.11144188155776750659420254650051070116) }, + { SC_(-4.24931621551513671875), SC_(4.714952991490022604296929488005221371491) }, + { SC_(-4.0708599090576171875), SC_(15.40013806764590763994801524236096523633) }, + { SC_(-4.05075550079345703125), SC_(21.05232398567253746583677948874966443745) }, + { SC_(-4.0256023406982421875), SC_(40.48643215403911174179393965158605983731) }, + { SC_(-3.5111486911773681640625), SC_(1.281561880048097406180157051598176902444) }, + { SC_(-3.5033643245697021484375), SC_(1.356501581084328750160720447675663936527) }, + { SC_(-3.2305061817169189453125), SC_(4.871536212360443209892217244039906620205) }, + { SC_(-3.1132221221923828125), SC_(9.744280495465310545518498968361342433867) }, + { SC_(-2.9407203197479248046875), SC_(-15.4345615314581643435197891745342605011) }, + { SC_(-1.8890321254730224609375), SC_(-7.765572445501014244464453609042267110197) }, + { SC_(-1.2540400028228759765625), SC_(3.637637081727820317005755368640064000683) }, + { SC_(-1.1478424072265625), SC_(6.784401313345897445876900487443690652634) }, + { SC_(-0.7118701934814453125), SC_(-2.248647256037530489173954985789306442878) }, + { SC_(-0.593149662017822265625), SC_(-0.8263747473361177168793211424797478169291) }, + { SC_(-0.3538668155670166015625), SC_(1.443172070386911802413212733069332368731) } + } }; +#undef SC_ + + diff --git a/test/digamma_root_data.ipp b/test/digamma_root_data.ipp new file mode 100644 index 000000000..b0a3aa838 --- /dev/null +++ b/test/digamma_root_data.ipp @@ -0,0 +1,211 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 200> digamma_root_data = { { + { SC_(1.39999997615814208984375), SC_(-0.06138456903152256550686860248931989075643) }, + { SC_(1.4005000591278076171875), SC_(-0.06087192929516339380130503872301167293378) }, + { SC_(1.40100002288818359375), SC_(-0.06035965880979232302674791723368356347747) }, + { SC_(1.4014999866485595703125), SC_(-0.05984763511635311349827564040488766194063) }, + { SC_(1.401999950408935546875), SC_(-0.0593358579881174546738279942752396761834) }, + { SC_(1.40250003337860107421875), SC_(-0.05882420526073090100751004662830533539728) }, + { SC_(1.40299999713897705078125), SC_(-0.0583129206425757749570876729240439746001) }, + { SC_(1.40349996089935302734375), SC_(-0.05780188191118900126815738602890619172921) }, + { SC_(1.4040000438690185546875), SC_(-0.05729096707893393080771549505001408983822) }, + { SC_(1.40450000762939453125), SC_(-0.056780419503293858705549580743156540668) }, + { SC_(1.4049999713897705078125), SC_(-0.05627011713868186213939939894956276866569) }, + { SC_(1.40550005435943603515625), SC_(-0.05575993817369138452985739956960740694737) }, + { SC_(1.40600001811981201171875), SC_(-0.05525012561586888099905375034247029300063) }, + { SC_(1.40649998188018798828125), SC_(-0.05474055759601525571890189603476879643263) }, + { SC_(1.40699994564056396484375), SC_(-0.05423123389038862937136671914178719811082) }, + { SC_(1.4075000286102294921875), SC_(-0.05372203292179596750107248206039107172611) }, + { SC_(1.40799999237060546875), SC_(-0.05321319723270065414202667836306539722927) }, + { SC_(1.4084999561309814453125), SC_(-0.05270460518832646256646398238653334023619) }, + { SC_(1.40900003910064697265625), SC_(-0.05219613538658003557049855371629116390416) }, + { SC_(1.40950000286102294921875), SC_(-0.05168803002221822433984269085445887545932) }, + { SC_(1.40999996662139892578125), SC_(-0.05118016763572077571206202324130731620467) }, + { SC_(1.410500049591064453125), SC_(-0.05067242699961275111564488922723020894775) }, + { SC_(1.4110000133514404296875), SC_(-0.05016504996189404179318756205383368841508) }, + { SC_(1.41149997711181640625), SC_(-0.04965791523781849836663795623050040651275) }, + { SC_(1.41200006008148193359375), SC_(-0.04915090177405025431768617393574718923719) }, + { SC_(1.41250002384185791015625), SC_(-0.04864425107277807126795345488864174180773) }, + { SC_(1.41299998760223388671875), SC_(-0.04813784202354994896630230505781776120276) }, + { SC_(1.41349995136260986328125), SC_(-0.04763167440643069181645608937840354367671) }, + { SC_(1.414000034332275390625), SC_(-0.04712562739952290889837033293650476187953) }, + { SC_(1.4144999980926513671875), SC_(-0.04661994204540782942709091596942293402734) }, + { SC_(1.41499996185302734375), SC_(-0.04611449746527775349715574804732130222007) }, + { SC_(1.41550004482269287109375), SC_(-0.04560917301026328003252147788241984436241) }, + { SC_(1.41600000858306884765625), SC_(-0.0451042093793269736581833678634701785191) }, + { SC_(1.41649997234344482421875), SC_(-0.04459948586684279443839061743784846261709) }, + { SC_(1.4170000553131103515625), SC_(-0.04409488199649628730133723954943974407746) }, + { SC_(1.417500019073486328125), SC_(-0.0435906381245648444628637002942433530947) }, + { SC_(1.4179999828338623046875), SC_(-0.04308663371813044418132048918710682813622) }, + { SC_(1.41849994659423828125), SC_(-0.04258286856012920630307449298163993137931) }, + { SC_(1.41900002956390380859375), SC_(-0.0420792224035827733504570446125334946695) }, + { SC_(1.41949999332427978515625), SC_(-0.04157593514930978152490152156879729835277) }, + { SC_(1.41999995708465576171875), SC_(-0.04107288649393147397031813049746661830252) }, + { SC_(1.4205000400543212890625), SC_(-0.04056995636192793782845393152872471220703) }, + { SC_(1.421000003814697265625), SC_(-0.04006738431359655642636465412836862063583) }, + { SC_(1.4214999675750732421875), SC_(-0.03956505021716870619391607561977829898393) }, + { SC_(1.42200005054473876953125), SC_(-0.0390628341681160707895772877982347062534) }, + { SC_(1.42250001430511474609375), SC_(-0.03856097538713614221174291451172868720556) }, + { SC_(1.42299997806549072265625), SC_(-0.03805935391360299407483745199209552218662) }, + { SC_(1.42349994182586669921875), SC_(-0.03755796953327570443031191372677272436563) }, + { SC_(1.4240000247955322265625), SC_(-0.03705670256888721277125707475995514593767) }, + { SC_(1.424499988555908203125), SC_(-0.03655579178977111826937882476062071563405) }, + { SC_(1.4249999523162841796875), SC_(-0.0360551174627633043738925866730974379965) }, + { SC_(1.42550003528594970703125), SC_(-0.03555456008050613945891518063349475611711) }, + { SC_(1.42599999904632568359375), SC_(-0.03505435807486918460538076814912383388358) }, + { SC_(1.42649996280670166015625), SC_(-0.03455439188274749223938775275471736825604) }, + { SC_(1.4270000457763671875), SC_(-0.03405454216623210245980763665400513878778) }, + { SC_(1.4275000095367431640625), SC_(-0.03355504702063797243050550756083487574355) }, + { SC_(1.427999973297119140625), SC_(-0.03305578705245798813618865616700236433431) }, + { SC_(1.42850005626678466796875), SC_(-0.03255664309277291143229018160781330281834) }, + { SC_(1.42900002002716064453125), SC_(-0.03205785290124899051424545775403784997954) }, + { SC_(1.42949998378753662109375), SC_(-0.03155929725351741957783226833498005852699) }, + { SC_(1.42999994754791259765625), SC_(-0.03106097593893646101388889322754816060391) }, + { SC_(1.430500030517578125), SC_(-0.03056277001319123536849588910788667969385) }, + { SC_(1.4309999942779541015625), SC_(-0.03006491678982831278985395858099735606325) }, + { SC_(1.431499958038330078125), SC_(-0.02956729726928046744375226173805018083445) }, + { SC_(1.43200004100799560546875), SC_(-0.029069792675158646894282928074400495447) }, + { SC_(1.43250000476837158203125), SC_(-0.02857263998746077965589217391685781750225) }, + { SC_(1.43299996852874755859375), SC_(-0.02807572037469272194244519088403794652932) }, + { SC_(1.4335000514984130859375), SC_(-0.02757891522793825630459486913227966938778) }, + { SC_(1.4340000152587890625), SC_(-0.02708246119454089526602248424490897334451) }, + { SC_(1.4344999790191650390625), SC_(-0.02658623961062601916105249666629939819311) }, + { SC_(1.434999942779541015625), SC_(-0.02609025026826734736532129982459892559412) }, + { SC_(1.43550002574920654296875), SC_(-0.02559437478113976107312669529902398957573) }, + { SC_(1.43599998950958251953125), SC_(-0.0250988493544413555894730759899566827115) }, + { SC_(1.43649995326995849609375), SC_(-0.02460355554708331611314729590329885967853) }, + { SC_(1.4370000362396240234375), SC_(-0.02410837513916324870453956495027706651788) }, + { SC_(1.4375), SC_(-0.02361354400529789914172989421521805262163) }, + { SC_(1.4379999637603759765625), SC_(-0.02311894387096655710370815673046310749581) }, + { SC_(1.43850004673004150390625), SC_(-0.02262445668224241951726505286548155427726) }, + { SC_(1.43900001049041748046875), SC_(-0.02213031798404363285679840262837603735113) }, + { SC_(1.43949997425079345703125), SC_(-0.02163640966796990067907296550632756430219) }, + { SC_(1.440000057220458984375), SC_(-0.02114261384562461502572001614025897932072) }, + { SC_(1.4405000209808349609375), SC_(-0.02064916573310666785738104392419687958556) }, + { SC_(1.4409999847412109375), SC_(-0.02015594738769022700601666231128218163721) }, + { SC_(1.4414999485015869140625), SC_(-0.0196629586049117094653247721121789058042) }, + { SC_(1.44200003147125244140625), SC_(-0.0191700817163872207492164657910951176529) }, + { SC_(1.44249999523162841796875), SC_(-0.01867755150116066387436310533382643774849) }, + { SC_(1.44299995899200439453125), SC_(-0.01818525023671041840942493794786294366844) }, + { SC_(1.443500041961669921875), SC_(-0.01769306041915025068329883160626130368828) }, + { SC_(1.4440000057220458984375), SC_(-0.01720121650074344257459977191565623298452) }, + { SC_(1.444499969482421875), SC_(-0.01670960092360902193107619746814218063958) }, + { SC_(1.44500005245208740234375), SC_(-0.01621809634791875134279906903350577844016) }, + { SC_(1.44550001621246337890625), SC_(-0.015726936900023407129612092249722223488) }, + { SC_(1.44599997997283935546875), SC_(-0.01523600518624245606690212546401818985812) }, + { SC_(1.44649994373321533203125), SC_(-0.01474530100472522827513829838650753549117) }, + { SC_(1.447000026702880859375), SC_(-0.01425470723370217219431406682491211640879) }, + { SC_(1.4474999904632568359375), SC_(-0.01376445756632729291905843124879046109968) }, + { SC_(1.4479999542236328125), SC_(-0.01327443482716782005100353254837490055862) }, + { SC_(1.44850003719329833984375), SC_(-0.01278452205749292497551903285885539659936) }, + { SC_(1.44900000095367431640625), SC_(-0.01229495262655675620536025426734759387572) }, + { SC_(1.44949996471405029296875), SC_(-0.01180560952210644817133945874679139627505) }, + { SC_(1.4500000476837158203125), SC_(-0.01131637594801586656260394588481713457627) }, + { SC_(1.450500011444091796875), SC_(-0.01082748495049637469227173996198819829549) }, + { SC_(1.4509999752044677734375), SC_(-0.01033881968004044725477451816247304271862) }, + { SC_(1.45150005817413330078125), SC_(-0.009850263502694288097376018889413894603933) }, + { SC_(1.45200002193450927734375), SC_(-0.009362049142480268951706194627385896301701) }, + { SC_(1.45249998569488525390625), SC_(-0.008874059912203315333901183590153038939381) }, + { SC_(1.45299994945526123046875), SC_(-0.008386295613345236665485880245372577446211) }, + { SC_(1.4535000324249267578125), SC_(-0.007898639827508066919974235979462981148542) }, + { SC_(1.453999996185302734375), SC_(-0.007411324850459837999980330364290265721957) }, + { SC_(1.4544999599456787109375), SC_(-0.006924234210747293687116178727328347532758) }, + { SC_(1.45500004291534423828125), SC_(-0.006437251651142443774260894146878774165176) }, + { SC_(1.45550000667572021484375), SC_(-0.005950609147203835378592212115736517701905) }, + { SC_(1.45599997043609619140625), SC_(-0.005464190388787066551639512688711491505847) }, + { SC_(1.45650005340576171875), SC_(-0.00497787927940728167871412685817772062165) }, + { SC_(1.4570000171661376953125), SC_(-0.004491907475256527500473978270689662215921) }, + { SC_(1.457499980926513671875), SC_(-0.004006158827071890461464755144012668001093) }, + { SC_(1.4579999446868896484375), SC_(-0.003520633138850239630378016887483403867021) }, + { SC_(1.45850002765655517578125), SC_(-0.003035214527765513668034604216830692388851) }, + { SC_(1.45899999141693115234375), SC_(-0.002550134225501944697075651923125347621858) }, + { SC_(1.45949995517730712890625), SC_(-0.002065276296639069833107196292495616468625) }, + { SC_(1.46000003814697265625), SC_(-0.001580525018104361453955140711216969645092) }, + { SC_(1.4605000019073486328125), SC_(-0.001096111304170043846745287256190494732511) }, + { SC_(1.460999965667724609375), SC_(-0.0006119193793063274064283424010699749479896) }, + { SC_(1.46150004863739013671875), SC_(-0.0001278336797728010038687456086772293940197) }, + { SC_(1.46200001239776611328125), SC_(0.0003559151967379670943728162750581078035161) }, + { SC_(1.46249997615814208984375), SC_(0.0008394428662870266119150633759096763287959) }, + { SC_(1.4630000591278076171875), SC_(0.001322864733702479649625580945291659569672) }, + { SC_(1.46350002288818359375), SC_(0.001805950517042218036842926190553563525984) }, + { SC_(1.4639999866485595703125), SC_(0.002288815673318872744259324682705288271519) }, + { SC_(1.464499950408935546875), SC_(0.002771460395327381562525270989780009242693) }, + { SC_(1.46500003337860107421875), SC_(0.003253999876667223767472091225924873652768) }, + { SC_(1.46549999713897705078125), SC_(0.003736204255104145549500401970100669234964) }, + { SC_(1.46599996089935302734375), SC_(0.004218188776241473723916466805139222942859) }, + { SC_(1.4665000438690185546875), SC_(0.004700068475739702789490158181882191908657) }, + { SC_(1.46700000762939453125), SC_(0.005181613805192456365413438059635728009745) }, + { SC_(1.4674999713897705078125), SC_(0.005662939852129155621931915752726712941002) }, + { SC_(1.46800005435943603515625), SC_(0.006144161494685809209920008019685283315534) }, + { SC_(1.46850001811981201171875), SC_(0.006625049497465130483020027415596322710511) }, + { SC_(1.46899998188018798828125), SC_(0.007105718790337639298786852855563148822142) }, + { SC_(1.46949994564056396484375), SC_(0.007586169563676627501200448950668213028167) }, + { SC_(1.4700000286102294921875), SC_(0.008066516486231662885434708649134127678999) }, + { SC_(1.47049999237060546875), SC_(0.008546530738674327360836097967927454942833) }, + { SC_(1.4709999561309814453125), SC_(0.00902632704131026480217858186061492831619) }, + { SC_(1.47150003910064697265625), SC_(0.009506019906325596650154195641432409630907) }, + { SC_(1.47200000286102294921875), SC_(0.009985380825502499855688695923101130679945) }, + { SC_(1.47249996662139892578125), SC_(0.01046452436244992624581900727741018711986) }, + { SC_(1.473000049591064453125), SC_(0.01094356487319982808059040901671966801919) }, + { SC_(1.4735000133514404296875), SC_(0.01142227415983642922640932055783984854239) }, + { SC_(1.47399997711181640625), SC_(0.01190076662968176394047411758437134448755) }, + { SC_(1.47450006008148193359375), SC_(0.01237915648302134259978832581154852202161) }, + { SC_(1.47500002384185791015625), SC_(0.01285721583143592736486164760948599640939) }, + { SC_(1.47549998760223388671875), SC_(0.01333505892636885712587736290231507253031) }, + { SC_(1.47599995136260986328125), SC_(0.01381268595510390028319865656189032981603) }, + { SC_(1.476500034332275390625), SC_(0.01429021091090032875378729920511744436802) }, + { SC_(1.4769999980926513671875), SC_(0.01476740631674500824066000704296476101248) }, + { SC_(1.47749996185302734375), SC_(0.01524438621687947040773770812918073015903) }, + { SC_(1.47800004482269287109375), SC_(0.01572126444976191365504766913460853166159) }, + { SC_(1.47850000858306884765625), SC_(0.01619781384600798034797161476617322052959) }, + { SC_(1.47899997234344482421875), SC_(0.01667414829492702066558118850104769852234) }, + { SC_(1.4795000553131103515625), SC_(0.01715038148057877526958196636506777091166) }, + { SC_(1.480000019073486328125), SC_(0.01762628654041217983419823745186174695842) }, + { SC_(1.4804999828338623046875), SC_(0.01810197720920750584264288401814595315433) }, + { SC_(1.48099994659423828125), SC_(0.01857745367191608501032749919254492077255) }, + { SC_(1.48150002956390380859375), SC_(0.01905282940736078826428345835701187921344) }, + { SC_(1.48199999332427978515625), SC_(0.01952787796086124235035667894483836618612) }, + { SC_(1.48249995708465576171875), SC_(0.02000271286178744470899885416201568506841) }, + { SC_(1.4830000400543212890625), SC_(0.02047744743549734724595413353304339596842) }, + { SC_(1.483500003814697265625), SC_(0.02095185553229931162465762694540459626532) }, + { SC_(1.4839999675750732421875), SC_(0.02142605052796892258114323238639681979697) }, + { SC_(1.48450005054473876953125), SC_(0.02190014559479680012423194248738781677386) }, + { SC_(1.48500001430511474609375), SC_(0.02237391488729442162211914583054893402186) }, + { SC_(1.48549997806549072265625), SC_(0.02284747162804102549936129615381098290523) }, + { SC_(1.48600006103515625), SC_(0.02332092883665599429697089795154616828867) }, + { SC_(1.4865000247955322265625), SC_(0.02379406097107123019383531656762472857036) }, + { SC_(1.486999988555908203125), SC_(0.02426698110106619670748667123423481731356) }, + { SC_(1.4874999523162841796875), SC_(0.02473968940861605188874507129358473978676) }, + { SC_(1.48800003528594970703125), SC_(0.02521229871039613408736493282176431879518) }, + { SC_(1.48849999904632568359375), SC_(0.02568458386767842754083514014266512257621) }, + { SC_(1.48899996280670166015625), SC_(0.02615665774712770972822278823384134472911) }, + { SC_(1.4895000457763671875), SC_(0.02662863301366652729705485156514156921969) }, + { SC_(1.4900000095367431640625), SC_(0.02710028483017233972822145485920594582439) }, + { SC_(1.490499973297119140625), SC_(0.02757172591142969944611182410222535997026) }, + { SC_(1.49100005626678466796875), SC_(0.02804306877099953245333159632753441304838) }, + { SC_(1.49150002002716064453125), SC_(0.02851408887259174195648360999928964786226) }, + { SC_(1.49199998378753662109375), SC_(0.02898489877950229580989209771050696266479) }, + { SC_(1.49249994754791259765625), SC_(0.02945549867145914132386877573515702816588) }, + { SC_(1.493000030517578125), SC_(0.02992600086081766434888226036243716094018) }, + { SC_(1.4934999942779541015625), SC_(0.03039618121119222213671842099543294002594) }, + { SC_(1.493999958038330078125), SC_(0.03086615208450468685425685392189632212515) }, + { SC_(1.49450004100799560546875), SC_(0.03133602564265587447963502512063650812183) }, + { SC_(1.49500000476837158203125), SC_(0.0318055780482999943002748605694401059607) }, + { SC_(1.49549996852874755859375), SC_(0.03227492151277847133068637874480248065187) }, + { SC_(1.4960000514984130859375), SC_(0.03274416804792387851799905190703885999517) }, + { SC_(1.4965000152587890625), SC_(0.03321309411466675105852178534017282410089) }, + { SC_(1.4969999790191650390625), SC_(0.0336818117741548556801742840271873983606) }, + { SC_(1.49750006198883056640625), SC_(0.03415043288853741106672780407819814754177) }, + { SC_(1.49800002574920654296875), SC_(0.03461873421626057425769027729390348592731) }, + { SC_(1.49849998950958251953125), SC_(0.03508682766866380229344336380060751609967) }, + { SC_(1.49899995326995849609375), SC_(0.03555471342260717737839815920297809359884) }, + { SC_(1.4995000362396240234375), SC_(0.03602250314126230243963242841830489771559) } + } }; +#undef SC_ + + diff --git a/test/digamma_small_data.ipp b/test/digamma_small_data.ipp new file mode 100644 index 000000000..7377548db --- /dev/null +++ b/test/digamma_small_data.ipp @@ -0,0 +1,44 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 33> digamma_small_data = { { + { SC_(0.1690093176520690576580818742513656616211e-8), SC_(-591683355.0172646248558707395909205014789) }, + { SC_(0.2114990849122477811761200428009033203125e-8), SC_(-472815285.0570071002693788265718597340393) }, + { SC_(0.7099628440698779741069301962852478027344e-8), SC_(-140852442.1314070676912450420112690472767) }, + { SC_(0.136718796284185373224318027496337890625e-7), SC_(-73142832.95860873836229194739144163160831) }, + { SC_(0.1679341288252089725574478507041931152344e-7), SC_(-59547157.96538602413887273326920747232618) }, + { SC_(0.586768322818898013792932033538818359375e-7), SC_(-17042502.04007390142270552503694442817772) }, + { SC_(0.1140460881288163363933563232421875e-6), SC_(-8768385.503057815514505383587592316839882) }, + { SC_(0.1455586016163579188287258148193359375e-6), SC_(-6870085.813645861903628134604185829717845) }, + { SC_(0.38918477685001562349498271942138671875e-6), SC_(-2569474.152450422201931296992216788786447) }, + { SC_(0.623782625552848912775516510009765625e-6), SC_(-1603123.137920326054732670974252040029471) }, + { SC_(0.104669607026153244078159332275390625e-5), SC_(-955387.7506368215484908190151041836664367) }, + { SC_(0.2951089072666945867240428924560546875e-5), SC_(-338858.5294366592361422006155767583415332) }, + { SC_(0.4877083483734168112277984619140625e-5), SC_(-205041.1518328576044816818799248254570363) }, + { SC_(0.9066634447663091123104095458984375e-5), SC_(-110295.0867866920189113632829333848561728) }, + { SC_(0.2360353755648247897624969482421875e-4), SC_(-42367.10793974013680262660975431753380722) }, + { SC_(0.60817910707555711269378662109375e-4), SC_(-16443.10183188331524808941398664226054956) }, + { SC_(0.000119476739200763404369354248046875), SC_(-8370.407052088278326658500142471452559876) }, + { SC_(0.0002437086659483611583709716796875), SC_(-4103.836730163294582058528454940279573003) }, + { SC_(0.00047970912419259548187255859375), SC_(-2085.173007504666107151589993385624082421) }, + { SC_(0.000960788573138415813446044921875), SC_(-1041.387348674624002438653075235083900513) }, + { SC_(0.00113048148341476917266845703125), SC_(-885.1541983038739901476427277144444120955) }, + { SC_(0.0033707791008055210113525390625), SC_(-297.2390039349243537614805191705277299023) }, + { SC_(0.007697627879679203033447265625), SC_(-130.474775285766139271184373586979099581) }, + { SC_(0.0154774188995361328125), SC_(-65.1622965036480167454852163183367159499) }, + { SC_(0.0305807329714298248291015625), SC_(-33.22833448837103737169827250947288832931) }, + { SC_(0.0346831791102886199951171875), SC_(-29.35398677269140057359817914882516787732) }, + { SC_(0.09283597767353057861328125), SC_(-11.20575716299454988174075282116985360081) }, + { SC_(0.22476322948932647705078125), SC_(-4.707226986627979845318377740205580092891) }, + { SC_(0.4500701129436492919921875), SC_(-2.233123888225926352977853470362332872207) }, + { SC_(0.64851474761962890625), SC_(-1.375098345846253151137062991270963899568) }, + { SC_(1.14188635349273681640625), SC_(-0.3652988540135093192143191973421561168173) }, + { SC_(2.0095670223236083984375), SC_(0.4289360116443062870701048575669258219731) }, + { SC_(5.68704509735107421875), SC_(1.647702940606041030337068005600358865775) } + } }; +#undef SC_ + + diff --git a/test/ellint_e2_data.ipp b/test/ellint_e2_data.ipp new file mode 100644 index 000000000..9729c468f --- /dev/null +++ b/test/ellint_e2_data.ipp @@ -0,0 +1,534 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Each row of test data contains in order: +// +// k, phi, E(k, phi) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 520> ellint_e2_data = { + SC_(0.177219114266335964202880859375e-2), SC_(-0.804919183254241943359375e0), SC_(0.17721905416489978459933452744691987573693030723399e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.74602639675140380859375e0), SC_(0.17721906263793616232304803862551613935165980878975e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.72904598712921142578125e0), SC_(0.17721906496143189532808590572077500756745196250994e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.62323606014251708984375e0), SC_(0.17721907823458326176536911566374461949290266081915e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.5579319000244140625e0), SC_(0.17721908538996504890695781731051446713497872352528e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.44300353527069091796875e0), SC_(0.1772190960611723703532438177387763930589137687723e-2), + SC_(0.177219114266335964202880859375e-2), SC_(-0.38366591930389404296875e0), SC_(0.17721910061149527140815880383922381002724195459473e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.9376299381256103515625e-1), SC_(0.17721911345080004258530792787497396150296565903888e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.944411754608154296875e-1), SC_(0.17721911343895994071624688564628016634872782068401e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.264718532562255859375e0), SC_(0.17721910776580442894609156234461709334349303324512e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.62944734096527099609375e0), SC_(0.17721907751280664212690267651483481595911639313105e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.67001712322235107421875e0), SC_(0.17721907262237245229946682518231598438599636326697e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.81158387660980224609375e0), SC_(0.1772190531655048767532217861236813247332269413991e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.826751708984375e0), SC_(0.17721905086031437581845682018005163376219526346493e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.91501367092132568359375e0), SC_(0.17721903659952045902730471028534006712649156285481e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.92977702617645263671875e0), SC_(0.17721903407305898397789671421686956699622188608929e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.93538987636566162109375e0), SC_(0.17721903310191937273874074422793996165770412692148e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.93773555755615234375e0), SC_(0.17721903269433608619814655577621926258964232521032e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.98576259613037109375e0), SC_(0.17721902412478400771222500977665176615786698940728e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.99292266368865966796875e0), SC_(0.17721902281054485597950865679003652002775579957175e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.804919183254241943359375e0), SC_(0.22177274667911356404114891574256878749693907344194e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.74602639675140380859375e0), SC_(0.22177276328387303450482024510234767179030032679487e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.72904598712921142578125e0), SC_(0.2217727678372686367339028586754168902231177349864e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.62323606014251708984375e0), SC_(0.22177279384889650054782811230773624538568740820856e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.5579319000244140625e0), SC_(0.22177280787142304325383561406780984188050064485493e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.44300353527069091796875e0), SC_(0.22177282878397316166435787421159323409852657193959e-2), + SC_(0.22177286446094512939453125e-2), SC_(-0.38366591930389404296875e0), SC_(0.22177283770131950245199600240560529548081991406878e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.9376299381256103515625e-1), SC_(0.2217728628627254479372625258375135600386491716262e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.944411754608154296875e-1), SC_(0.2217728628395221979258299938047197957597496164853e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.264718532562255859375e0), SC_(0.22177285172174219060847672057602083161874684079641e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.62944734096527099609375e0), SC_(0.22177279243441823271547938764610382865435171389572e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.67001712322235107421875e0), SC_(0.22177278285054864598218183902335374842302009015249e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.81158387660980224609375e0), SC_(0.22177274472058160774429082218212375574848674857853e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.826751708984375e0), SC_(0.22177274020305878581113108501360596469572415665098e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.91501367092132568359375e0), SC_(0.22177271225592599152503205364411936997104330819552e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.92977702617645263671875e0), SC_(0.22177270730477403168508453406178297939782452362205e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.93538987636566162109375e0), SC_(0.22177270540161426246190034172682702788389529386352e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.93773555755615234375e0), SC_(0.22177270460286595928412185481890644744187621951273e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.98576259613037109375e0), SC_(0.22177268780896006483002343116598140190281467534408e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.99292266368865966796875e0), SC_(0.22177268523342180309535844744234619982692618957978e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.804919183254241943359375e0), SC_(0.74444554408367041960081599572734384580648039601292e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.74602639675140380859375e0), SC_(0.74444617216162272729935658343672599710823129917136e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.72904598712921142578125e0), SC_(0.7444463443944617202862815487638272084421484282213e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.62323606014251708984375e0), SC_(0.74444732828674762841832446009298546988014342845109e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.5579319000244140625e0), SC_(0.74444785868925647373677427756541386357216267403116e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.44300353527069091796875e0), SC_(0.7444486497059729002932025822805836773020204604622e-2), + SC_(0.7444499991834163665771484375e-2), SC_(-0.38366591930389404296875e0), SC_(0.7444489870039692628767038225671358877887084418318e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.9376299381256103515625e-1), SC_(0.74444993873102289959003988480526261034186232298732e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.944411754608154296875e-1), SC_(0.74444993785336377021615551123021090095012183365455e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.264718532562255859375e0), SC_(0.74444951732496163285321657924324274340742574731183e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.62944734096527099609375e0), SC_(0.74444727478402810869648210185177744717615995463001e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.67001712322235107421875e0), SC_(0.74444691227345931676783343972213308719772501191675e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.81158387660980224609375e0), SC_(0.74444547000179682301220146799104273617595274550494e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.826751708984375e0), SC_(0.74444529912552191587668074107636437041675982253389e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.91501367092132568359375e0), SC_(0.74444424201776730075960207569226490521597156912434e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.92977702617645263671875e0), SC_(0.74444405473886664282906533654411935648151688362809e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.93538987636566162109375e0), SC_(0.74444398275122133283989327309245789244705679180104e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.93773555755615234375e0), SC_(0.74444395253830019766374648366268259171697141665571e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.98576259613037109375e0), SC_(0.74444331730273938555860696278431505938817087276882e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.99292266368865966796875e0), SC_(0.74444321988199504330861521619247593689621297824599e-2), + SC_(0.1433600485324859619140625e-1), SC_(-0.804919183254241943359375e0), SC_(0.14335686705725696407060645727945600655565741414147e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.74602639675140380859375e0), SC_(0.14335731558581990142476732612631387140359085570156e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.72904598712921142578125e0), SC_(0.14335743858193055553888894193489743522839559243509e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.62323606014251708984375e0), SC_(0.14335814120326027832379597580040828122216388507481e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.5579319000244140625e0), SC_(0.14335851997467008238725204894793026916257019753693e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.44300353527069091796875e0), SC_(0.14335908485363563424533546850245012850090360873434e-1), + SC_(0.1433600485324859619140625e-1), SC_(-0.38366591930389404296875e0), SC_(0.14335932572318417439388238927592963489979339519294e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.9376299381256103515625e-1), SC_(0.14336000536285207953459760778071938813620926376329e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.944411754608154296875e-1), SC_(0.14336000473610715309472432563583032814457494094135e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.264718532562255859375e0), SC_(0.14335970443226789444113931927418674846075665585412e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.62944734096527099609375e0), SC_(0.14335810299578782495392921047255424568664286939309e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.67001712322235107421875e0), SC_(0.14335784411863985146631166860519228538944451465492e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.81158387660980224609375e0), SC_(0.14335681415313482527267826975008547916557671973967e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.826751708984375e0), SC_(0.14335669212508490437041994345359491275578416731562e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.91501367092132568359375e0), SC_(0.1433559372086014461936976066148820437936563510921e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.92977702617645263671875e0), SC_(0.14335580346584476426767024206037631424851951301064e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.93538987636566162109375e0), SC_(0.14335575205677374246197434850917518154812470284912e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.93773555755615234375e0), SC_(0.14335573048058908550764743332104831817119331680375e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.98576259613037109375e0), SC_(0.14335527683396662777675860041375777953747610473763e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.99292266368865966796875e0), SC_(0.14335520726182617060139852455489440586802090752734e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.804919183254241943359375e0), SC_(0.17608580108574407596974329704841169956949700562955e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.74602639675140380859375e0), SC_(0.17608663231619667706144284604540728747274691033403e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.72904598712921142578125e0), SC_(0.17608686025692226081166910626170820785576222384032e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.62323606014251708984375e0), SC_(0.17608816237609089248262489588718703344935310646926e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.5579319000244140625e0), SC_(0.17608886432433642683152813125927533432394416093985e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.44300353527069091796875e0), SC_(0.17608991116840504430604382009436726055680942592846e-1), + SC_(0.1760916970670223236083984375e-1), SC_(-0.38366591930389404296875e0), SC_(0.176090357551125362347926995427228626500868394589e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.9376299381256103515625e-1), SC_(0.17609161706489009824375122022213121549814666772137e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.944411754608154296875e-1), SC_(0.17609161590340381771438539655887170786578607686625e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.264718532562255859375e0), SC_(0.17609105937855647197851193299561879969853300958653e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.62944734096527099609375e0), 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SC_(-0.72904598712921142578125e0), SC_(0.42742682818966336795606721796292568150309618969066e3), + SC_(0.503011474609375e3), SC_(-0.62323606014251708984375e0), SC_(0.44990438001330262518694919031753384020576625696966e3), + SC_(0.503011474609375e3), SC_(-0.5579319000244140625e0), SC_(0.46124922502921161009662440273528750616727045937458e3), + SC_(0.503011474609375e3), SC_(-0.44300353527069091796875e0), SC_(0.47735745567472002811275473463900479664220214084528e3), + SC_(0.503011474609375e3), SC_(-0.38366591930389404296875e0), SC_(0.48396812043024017707709837542494532143253698991882e3), + SC_(0.503011474609375e3), SC_(0.9376299381256103515625e-1), SC_(0.50190480963296507338418291149366181766024849187853e3), + SC_(0.503011474609375e3), SC_(0.944411754608154296875e-1), SC_(0.50188871584285852712073221168824514015029692480846e3), + SC_(0.503011474609375e3), SC_(0.264718532562255859375e0), SC_(0.49408574205216958037913559664441409041763723627144e3), + SC_(0.503011474609375e3), SC_(0.62944734096527099609375e0), SC_(0.448732712753173223917291380485323902266910100027e3), + SC_(0.503011474609375e3), SC_(0.67001712322235107421875e0), SC_(0.44064980515604668179870304048719969769816683101636e3), + SC_(0.503011474609375e3), SC_(0.81158387660980224609375e0), SC_(0.40539941613509322668871139516155737996637247323477e3), + SC_(0.503011474609375e3), SC_(0.826751708984375e0), SC_(0.40080012733999218534312909286248833728961899399719e3), + SC_(0.503011474609375e3), SC_(0.91501367092132568359375e0), SC_(0.36919171399237342314526184351256549304483460904576e3), + SC_(0.503011474609375e3), SC_(0.92977702617645263671875e0), SC_(0.36279237787744593399714035595038205138016721213798e3), + SC_(0.503011474609375e3), SC_(0.93538987636566162109375e0), SC_(0.36023892853829175638985926444127462444440080265681e3), + SC_(0.503011474609375e3), SC_(0.93773555755615234375e0), SC_(0.35914990886887487679557974420087323548020127919695e3), + SC_(0.503011474609375e3), SC_(0.98576259613037109375e0), SC_(0.33252547245143178927979205996695688705563461577588e3), + SC_(0.503011474609375e3), SC_(0.99292266368865966796875e0), SC_(0.32718692844071197014439649471431843150228446330379e3), + SC_(0.10074598388671875e4), SC_(-0.804919183254241943359375e0), SC_(0.81566217335361231611771646986187800572268739295575e3), + SC_(0.10074598388671875e4), SC_(-0.74602639675140380859375e0), SC_(0.84755287064438863615038784712657137407744566356707e3), + SC_(0.10074598388671875e4), SC_(-0.72904598712921142578125e0), SC_(0.85590189104276483616202632098287025738931943507153e3), + SC_(0.10074598388671875e4), SC_(-0.62323606014251708984375e0), SC_(0.9009738212363659705854642096463530651141689117269e3), + SC_(0.10074598388671875e4), SC_(-0.5579319000244140625e0), SC_(0.92372203670774325404806275709264270484304517663929e3), + SC_(0.10074598388671875e4), SC_(-0.44300353527069091796875e0), SC_(0.95602114692228459835045237318476147078694497472822e3), + SC_(0.10074598388671875e4), SC_(-0.38366591930389404296875e0), SC_(0.96927625305559748266490795989045893214495805831554e3), + SC_(0.10074598388671875e4), SC_(0.9376299381256103515625e-1), SC_(0.10052408954500277585485462407617932594289289210602e4), + SC_(0.10074598388671875e4), SC_(0.944411754608154296875e-1), SC_(0.10052086262210379665209822645440559289291237665647e4), + SC_(0.10074598388671875e4), SC_(0.264718532562255859375e0), SC_(0.98956304231616558867754941531972183396056332288818e3), + SC_(0.10074598388671875e4), SC_(0.62944734096527099609375e0), SC_(0.89862442644009820675218558218153173632055069943683e3), + SC_(0.10074598388671875e4), SC_(0.67001712322235107421875e0), SC_(0.88241671832232838738799982334634993277048716392487e3), + SC_(0.10074598388671875e4), SC_(0.81158387660980224609375e0), SC_(0.81173128112196665137058131884505478065505214589458e3), + SC_(0.10074598388671875e4), SC_(0.826751708984375e0), SC_(0.80250834127587356001580304856418261116625810606362e3), + SC_(0.10074598388671875e4), SC_(0.91501367092132568359375e0), SC_(0.73912196799060491702381369566485537415592290789722e3), + SC_(0.10074598388671875e4), SC_(0.92977702617645263671875e0), SC_(0.7262884386240235721342337367344044620971138470138e3), + SC_(0.10074598388671875e4), SC_(0.93538987636566162109375e0), SC_(0.72116757185358740018078031894015653763422120624137e3), + SC_(0.10074598388671875e4), SC_(0.93773555755615234375e0), SC_(0.71898356440546660369364312811873716877046903540745e3), + SC_(0.10074598388671875e4), SC_(0.98576259613037109375e0), SC_(0.66558630336818747011593267003329852836650815741705e3), + SC_(0.10074598388671875e4), SC_(0.99292266368865966796875e0), SC_(0.6548786829121203110560807170365229948244134922981e3), + SC_(0.1185395751953125e4), SC_(-0.804919183254241943359375e0), SC_(0.95992720290264897479760554571647114770157610887996e3), + SC_(0.1185395751953125e4), SC_(-0.74602639675140380859375e0), SC_(0.99741366529362017349163653479322848619622124868772e3), + SC_(0.1185395751953125e4), SC_(-0.72904598712921142578125e0), SC_(0.10072278878934483167286843735137660835902486784055e4), + SC_(0.1185395751953125e4), SC_(-0.62323606014251708984375e0), SC_(0.10602110806127832608392589101187996342688647189112e4), + SC_(0.1185395751953125e4), SC_(-0.5579319000244140625e0), SC_(0.10869529954610434278852140581154818972542686423743e4), + SC_(0.1185395751953125e4), SC_(-0.44300353527069091796875e0), SC_(0.11249233882184205470868187837372292890877647671181e4), + SC_(0.1185395751953125e4), SC_(-0.38366591930389404296875e0), SC_(0.11405061649275106998721240053724233425466223964298e4), + SC_(0.1185395751953125e4), SC_(0.9376299381256103515625e-1), SC_(0.11827870772689532014486419251476519345293661688085e4), + SC_(0.1185395751953125e4), SC_(0.944411754608154296875e-1), SC_(0.11827491403552705968358829702474865775810482378628e4), + SC_(0.1185395751953125e4), SC_(0.264718532562255859375e0), SC_(0.11643556929058415420471543503560945498245581125436e4), + SC_(0.1185395751953125e4), SC_(0.62944734096527099609375e0), SC_(0.10574492529652112224851625390587579547328457881061e4), + SC_(0.1185395751953125e4), SC_(0.67001712322235107421875e0), SC_(0.1038396462932344155293121667046538419874167990145e4), + SC_(0.1185395751953125e4), SC_(0.81158387660980224609375e0), SC_(0.9553066699471665679840089770219957164232252629554e3), + SC_(0.1185395751953125e4), SC_(0.826751708984375e0), SC_(0.94446573985220084327197598058015000573173775032994e3), + SC_(0.1185395751953125e4), SC_(0.91501367092132568359375e0), SC_(0.86996356635663675306873109547180307652983494815843e3), + SC_(0.1185395751953125e4), SC_(0.92977702617645263671875e0), SC_(0.85488052815230820219522803855058047921692647892349e3), + SC_(0.1185395751953125e4), SC_(0.93538987636566162109375e0), SC_(0.84886217610355647741057825617131638665389927335333e3), + SC_(0.1185395751953125e4), SC_(0.93773555755615234375e0), SC_(0.84629542044995161715099588574641450818752524118888e3), + SC_(0.1185395751953125e4), SC_(0.98576259613037109375e0), SC_(0.78354531056875687291420826162420650224843893529921e3), + SC_(0.1185395751953125e4), SC_(0.99292266368865966796875e0), SC_(0.77096378731739644110407751882818742444499646261653e3) + }; +#undef SC_ + diff --git a/test/ellint_e_data.ipp b/test/ellint_e_data.ipp new file mode 100644 index 000000000..848a34be4 --- /dev/null +++ b/test/ellint_e_data.ipp @@ -0,0 +1,114 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Each row of test data contains in order: +// +// k, E(k) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> ellint_e_data = { + SC_(-0.990433037281036376953125e0), SC_(0.10274527688571687030049782227296053357094496206279e1), + SC_(-0.936334311962127685546875e0), SC_(0.11232874555917022066938321555637623562213676975972e1), + SC_(-0.931107819080352783203125e0), SC_(0.1130760221755419903369153157619439589360779303926e1), + SC_(-0.928576648235321044921875e0), SC_(0.11343097282371636644057123410674810478126636794066e1), + SC_(-0.92711746692657470703125e0), SC_(0.11363362574688624051162518398981635242155496915255e1), + SC_(-0.907657206058502197265625e0), SC_(0.11621185270599645325148449388373607697900140479452e1), + SC_(-0.89756715297698974609375e0), SC_(0.11746803765109846934010357339330910078574216811115e1), + SC_(-0.80573642253875732421875e0), SC_(0.12711544492641534140326897830960020400731161482408e1), + SC_(-0.804919183254241943359375e0), SC_(0.1271899578518333585916968732627512791380932403316e1), + SC_(-0.780276477336883544921875e0), SC_(0.12936183519259955902237796585053994588791953787341e1), + SC_(-0.775070965290069580078125e0), SC_(0.12980290901572191577411456044089492043970943779076e1), + SC_(-0.7496345043182373046875e0), SC_(0.13187607273206970520631617642065752154074679201729e1), + SC_(-0.74820673465728759765625e0), SC_(0.13198856849340247002745787630036800286336929689325e1), + SC_(-0.74602639675140380859375e0), SC_(0.13215959480936459650266993732312887701493706168114e1), + SC_(-0.72904598712921142578125e0), SC_(0.1334606672267782799453143920126631207219848911468e1), + SC_(-0.7162272930145263671875e0), SC_(0.13440793050791416705713681443513787752467694981452e1), + SC_(-0.701772034168243408203125e0), SC_(0.13544180205342472921449637366285993011101451511219e1), + SC_(-0.68477380275726318359375e0), SC_(0.13661308406125643621284019270477708148951227020526e1), + SC_(-0.657626628875732421875e0), SC_(0.13838947645626975911149829417421592164301145093627e1), + SC_(-0.652269661426544189453125e0), SC_(0.13872692599655636614951215534059306447757898114989e1), + SC_(-0.6262547969818115234375e0), SC_(0.14030728449607783731169275509403934977874920947839e1), + SC_(-0.62323606014251708984375e0), SC_(0.14048456682760981684618618684460963872624120364616e1), + SC_(-0.57958185672760009765625e0), SC_(0.14291386904685383227954283815576245068501134204508e1), + SC_(-0.576151371002197265625e0), SC_(0.14309448152636963724388781094697116200713399276732e1), + SC_(-0.5579319000244140625e0), SC_(0.14402965087704791609397947280292073414645780530405e1), + SC_(-0.446154057979583740234375e0), SC_(0.14894386503626442042107428737827539093660806612699e1), + SC_(-0.44300353527069091796875e0), SC_(0.14906320545924202935802686800501318065294882015735e1), + SC_(-0.40594112873077392578125e0), SC_(0.15039335802932226988591218103108193186540888213994e1), + SC_(-0.396173775196075439453125e0), SC_(0.15072169743011592499372913974729141564215181659934e1), + SC_(-0.38366591930389404296875e0), SC_(0.15112892448422190697630083347423467639448624581727e1), + SC_(-0.36689913272857666015625e0), SC_(0.15165180033802033267632053479981021035471425681416e1), + SC_(-0.365801036357879638671875e0), SC_(0.15168513413633160947387236074965552423019640205414e1), + SC_(-0.277411997318267822265625e0), SC_(0.15401245425176709136660196900048700278424091275112e1), + SC_(-0.236883103847503662109375e0), SC_(0.15485231276601733576918931368003657825692622988276e1), + SC_(-0.215545952320098876953125e0), SC_(0.15523894121876553934587790103948024781080753117537e1), + SC_(-0.202522933483123779296875e0), SC_(0.15545635291592032109785456524323662491832287557881e1), + SC_(-0.18253767490386962890625e0), SC_(0.15576286893796261035306290689920363346745725969978e1), + SC_(-0.156477451324462890625e0), SC_(0.15611364139446355493431072334668884516455428910749e1), + SC_(-0.1558246612548828125e0), SC_(0.15612172160025555618286865486857507482242942668515e1), + SC_(-0.12251126766204833984375e0), SC_(0.15648856105816674567926606395168937481071020785374e1), + SC_(-0.1088275909423828125e0), SC_(0.15661350376732836367309274463594953937245200605844e1), + SC_(-0.8402168750762939453125e-1), SC_(0.15680203305860144668144643526811660961420870159842e1), + SC_(-0.5048263072967529296875e-1), SC_(0.15697950560238758069422088665702687900822920595852e1), + SC_(-0.29248714447021484375e-1), SC_(0.15704603238122883636275639829068215175843565484093e1), + SC_(-0.2486217021942138671875e-1), SC_(0.1570553560549795220596952208506368318845323442966e1), + SC_(-0.2047121524810791015625e-1), SC_(0.15706317452006442392207295085961102343692070947481e1), + SC_(-0.18821895122528076171875e-1), SC_(0.15706571985088185541190068658162634466459956376283e1), + SC_(0.73254108428955078125e-2), SC_(0.15707752537045378829148682574129984681356739471373e1), + SC_(0.9376299381256103515625e-1), SC_(0.15673382012803800695565201533706558176677595081402e1), + SC_(0.944411754608154296875e-1), SC_(0.15672879111272468454088900653008917791981439593238e1), + SC_(0.264718532562255859375e0), SC_(0.15429050287765040115345161487328955754762740575513e1), + SC_(0.27952671051025390625e0), SC_(0.15396478942459482760618960796407428547408640525839e1), + SC_(0.29262602329254150390625e0), SC_(0.15366093920760526747388381843200711024138982306603e1), + SC_(0.3109557628631591796875e0), SC_(0.15321071618558340305908384979951601020818799999842e1), + SC_(0.31148135662078857421875e0), SC_(0.15319737203884180398267043979026904916868658518393e1), + SC_(0.32721102237701416015625e0), SC_(0.1527867114331003738198228458599913043176805971218e1), + SC_(0.3574702739715576171875e0), SC_(0.15193440688380299383044814920714973708033404406654e1), + SC_(0.362719058990478515625e0), SC_(0.15177809660808721467844113827968123756890771957761e1), + SC_(0.3896572589874267578125e0), SC_(0.15093570468318824817504413386396057198219992161732e1), + SC_(0.4120922088623046875e0), SC_(0.15018188427341099768691902236893549315641407909401e1), + SC_(0.41872966289520263671875e0), SC_(0.14994958178213154496366968552961386017760898588045e1), + SC_(0.45167791843414306640625e0), SC_(0.1487322085908655313535535255202869925685698085035e1), + SC_(0.48129451274871826171875e0), SC_(0.14754407459744073247384176419011374260929941397226e1), + SC_(0.4862649440765380859375e0), SC_(0.14733570928853164419583523432188100088746641413847e1), + SC_(0.50937330722808837890625e0), SC_(0.14633222267462579060234013375472616462719440305596e1), + SC_(0.5154802799224853515625e0), SC_(0.14605730543776299802897695844262830217129367088168e1), + SC_(0.52750003337860107421875e0), SC_(0.1455040905739579018637680781037806414611373457245e1), + SC_(0.53103363513946533203125e0), SC_(0.14533836180900849164859243315111429017381139244939e1), + SC_(0.58441460132598876953125e0), SC_(0.14265695055008370758871995090190851074198707889523e1), + SC_(0.5879499912261962890625e0), SC_(0.14246715204117826383644822612160764733188324494824e1), + SC_(0.59039986133575439453125e0), SC_(0.14233470653300894642788935610227647487095507501819e1), + SC_(0.59455978870391845703125e0), SC_(0.14210806804871026734537311084007732025147240364441e1), + SC_(0.59585726261138916015625e0), SC_(0.14203692855345104085630580991694310940176283484347e1), + SC_(0.5962116718292236328125e0), SC_(0.14201745910377376893163001298577620388569425640087e1), + SC_(0.6005609035491943359375e0), SC_(0.14177721978271857966949913233824083454479033587509e1), + SC_(0.6150619983673095703125e0), SC_(0.1409584186264459097822607166118300162769330850531e1), + SC_(0.62944734096527099609375e0), SC_(0.14011843899700989964492170135501929645905818746914e1), + SC_(0.64380657672882080078125e0), SC_(0.13925155171987190379716348493447778489398983557681e1), + SC_(0.6469156742095947265625e0), SC_(0.13906001623565653310360026300062173208737677732474e1), + SC_(0.67001712322235107421875e0), SC_(0.13759265375091920908763740804480465836032745124162e1), + SC_(0.6982586383819580078125e0), SC_(0.13568776608794218001220481427238400798591422244367e1), + SC_(0.74485766887664794921875e0), SC_(0.13225089163569380724816605958889295619432742743828e1), + SC_(0.75686132907867431640625e0), SC_(0.13130049536402767623871717650724612640189448665164e1), + SC_(0.81158387660980224609375e0), SC_(0.1265774005474254141411822958784348996686269275959e1), + SC_(0.826751708984375e0), SC_(0.1251401815196169298547115798872087290243528044558e1), + SC_(0.83147108554840087890625e0), SC_(0.12468018231406093242228210016848902253352539970597e1), + SC_(0.84174954891204833984375e0), SC_(0.12365605223473625325793146444569526624443067779814e1), + SC_(0.8679864406585693359375e0), SC_(0.12089081947735379320176271312129936213684657310011e1), + SC_(0.90044414997100830078125e0), SC_(0.11711493603723543295821235205304820794673798105369e1), + SC_(0.91433393955230712890625e0), SC_(0.11535184098186861192678956506006329721663192849427e1), + SC_(0.91501367092132568359375e0), SC_(0.11526291690338582866360141839353650400854743647163e1), + SC_(0.918984889984130859375e0), SC_(0.11473810486945145917917969281137061723789053926309e1), + SC_(0.92977702617645263671875e0), SC_(0.11326318918571500732674050143237850082646730427556e1), + SC_(0.93538987636566162109375e0), SC_(0.11246526120138518224375085002257139975074424198895e1), + SC_(0.93773555755615234375e0), SC_(0.11212495302257932905970525402254900901444230096575e1), + SC_(0.94118559360504150390625e0), SC_(0.11161664223439311396943776225269812587817500317889e1), + SC_(0.96221935749053955078125e0), SC_(0.10828018699366994143027254777963854661530573064246e1), + SC_(0.98576259613037109375e0), SC_(0.10380503716235214587344268369403753352254592216066e1), + SC_(0.9881370067596435546875e0), SC_(0.10327762472475272053177375153237179387280796303267e1), + SC_(0.99292266368865966796875e0), SC_(0.10213677605130466359178501196768639821158116338631e1) + }; +#undef SC_ + diff --git a/test/ellint_f_data.ipp b/test/ellint_f_data.ipp new file mode 100644 index 000000000..0482cd10a --- /dev/null +++ b/test/ellint_f_data.ipp @@ -0,0 +1,626 @@ +// Copyright (c) 2006 John Maddock +// +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// There are three values in each row: +// +// phi, k, ellint_1(k, phi) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 610> ellint_f_data = { + SC_(0.177219114266335964202880859375e-2), SC_(0.12698681652545928955078125e0), SC_(0.17721911576221833285284471505772299979929420100157e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.135477006435394287109375e0), SC_(0.17721911596893097495385621130075639520810631802367e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.22103404998779296875e0), SC_(0.17721911879842626333735727252300568988042608181058e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.308167040348052978515625e0), SC_(0.17721912307586304284414392153484736630244171106967e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.6323592662811279296875e0), SC_(0.17721915136072145749706721763844561838990574996748e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.814723670482635498046875e0), SC_(0.17721917584088468746420484903332375635064689539385e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.835008561611175537109375e0), SC_(0.17721917894520936923412246849940510596766888342397e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.905791938304901123046875e0), SC_(0.17721919037560789892218093103766859258846695998999e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.9133758544921875e0), SC_(0.17721919165542367899383364723376572314588337429883e-2), + SC_(0.177219114266335964202880859375e-2), SC_(0.968867778778076171875e0), SC_(0.1772192013446002825503764323931321827613955432354e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.12698681652545928955078125e0), SC_(0.22177286739245139979235205992077921703857874992614e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.135477006435394287109375e0), SC_(0.22177286779754970704335950293621400375492832466137e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.22103404998779296875e0), SC_(0.22177287334256025917867525959622978930360537074924e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.308167040348052978515625e0), SC_(0.22177288172512715350908153180610533224553721306023e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.6323592662811279296875e0), SC_(0.2217729371554693319729054703026208450026061763822e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.814723670482635498046875e0), SC_(0.22177298512970423745998954480900177381552708544037e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.835008561611175537109375e0), SC_(0.2217729912133089721595554486266261343813239076347e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.905791938304901123046875e0), SC_(0.22177301361368343651109035690490962300073942661987e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.9133758544921875e0), SC_(0.22177301612176360421071258583990800763547750726474e-2), + SC_(0.22177286446094512939453125e-2), SC_(0.968867778778076171875e0), SC_(0.22177303510983544844756451225638983381366612239496e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.12698681652545928955078125e0), SC_(0.74445011006718641819708303378218126160875363196343e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.135477006435394287109375e0), SC_(0.74445012538997345743072973657345052421185812852883e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.22103404998779296875e0), SC_(0.7444503351293811483247033675758072401037344482302e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.308167040348052978515625e0), SC_(0.74445065219959407401467925377833815036903664230965e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.6323592662811279296875e0), SC_(0.74445274886650082568519006341784743843976685124848e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.814723670482635498046875e0), SC_(0.74445456352769560796450156151400127382998272645198e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.835008561611175537109375e0), SC_(0.74445479364613446978911973710885424587373960669503e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.905791938304901123046875e0), SC_(0.74445564096573887453994590323758448047874147238395e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.9133758544921875e0), SC_(0.74445573583701508518948039844922043785925055729088e-2), + SC_(0.7444499991834163665771484375e-2), SC_(0.968867778778076171875e0), SC_(0.74445645408656423225725685299458948993261862803123e-2), + SC_(0.1433600485324859619140625e-1), SC_(0.12698681652545928955078125e0), SC_(0.14336012771572004313303282953555702877340432667875e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.135477006435394287109375e0), SC_(0.14336013865789252161967302705927923090033123892839e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.22103404998779296875e0), SC_(0.14336028843546510307044909433889108803989992630717e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.308167040348052978515625e0), SC_(0.14336051486049934782010291664500194812313169536713e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.6323592662811279296875e0), SC_(0.14336201216019080237489913190282411526700579073185e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.814723670482635498046875e0), SC_(0.14336330811999887083609618534824546245882513068397e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.835008561611175537109375e0), SC_(0.14336347246485787781200874226852611208775882622848e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.905791938304901123046875e0), SC_(0.14336407760582576201557263371323141017452648028165e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.9133758544921875e0), SC_(0.14336414536187317685947794701801663975831519240267e-1), + SC_(0.1433600485324859619140625e-1), SC_(0.968867778778076171875e0), SC_(0.14336465833209128166372255292238755432460966798543e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.12698681652545928955078125e0), SC_(0.17609184380978861185686844051402854987830396635258e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.135477006435394287109375e0), SC_(0.17609186408789500419846933977020874867517546176364e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.22103404998779296875e0), SC_(0.1760921416571103349085943082737303345872943430306e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.308167040348052978515625e0), SC_(0.17609256127163964831115570232150804486066285764182e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.6323592662811279296875e0), SC_(0.17609533613743185319361344589522300705915673688072e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.814723670482635498046875e0), SC_(0.17609773793502177357795159012740745279433408620657e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.835008561611175537109375e0), SC_(0.17609804251901222009248779176903833413486036667612e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.905791938304901123046875e0), SC_(0.17609916404852657704429322263138229104993266575624e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.9133758544921875e0), SC_(0.1760992896240552553398130726999861697860870407761e-1), + SC_(0.1760916970670223236083984375e-1), SC_(0.968867778778076171875e0), SC_(0.17610024034162638690605314405325566252936262309737e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.12698681652545928955078125e0), SC_(0.61527743617821017309837317251112211820225287909855e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.135477006435394287109375e0), SC_(0.61527830061250812048027946124763113656851093792434e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.22103404998779296875e0), SC_(0.61529013370063563692372946253072143453390904798089e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.308167040348052978515625e0), SC_(0.61530802448197340746490712692471232101050993961418e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.6323592662811279296875e0), SC_(0.61542639923962778105311852278000229689228218960744e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.814723670482635498046875e0), SC_(0.61552895025692965307933631422367268195916509703533e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.835008561611175537109375e0), SC_(0.61554196132063228221650243038365314737062757534577e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.905791938304901123046875e0), SC_(0.61558988200653669357386111420505273705513915221039e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.9133758544921875e0), SC_(0.61559524874653086436154791531898624131363535519786e-1), + SC_(0.6152711808681488037109375e-1), SC_(0.968867778778076171875e0), SC_(0.61563588723439851846457601010368343188236507772324e-1), + SC_(0.11958599090576171875e0), SC_(0.12698681652545928955078125e0), SC_(0.11959057453624120334328464580296093852234979832319e0), + SC_(0.11958599090576171875e0), SC_(0.135477006435394287109375e0), SC_(0.11959120801241224829404843422425331039825816399323e0), + SC_(0.11958599090576171875e0), SC_(0.22103404998779296875e0), SC_(0.11959988089267249852525096379530849147372849485837e0), + SC_(0.11958599090576171875e0), SC_(0.308167040348052978515625e0), SC_(0.11961299840115507344869578732891575108359266237168e0), + SC_(0.11958599090576171875e0), SC_(0.6323592662811279296875e0), SC_(0.11969993471932342512901186119137938273010491343839e0), + SC_(0.11958599090576171875e0), SC_(0.814723670482635498046875e0), SC_(0.11977545304297570964095700217118618012823329027572e0), + SC_(0.11958599090576171875e0), SC_(0.835008561611175537109375e0), SC_(0.11978504790549786604035595742049176901340492369488e0), + SC_(0.11958599090576171875e0), SC_(0.905791938304901123046875e0), SC_(0.11982041286438468953360393124738982966366389995088e0), + SC_(0.11958599090576171875e0), SC_(0.9133758544921875e0), SC_(0.11982437604947070134783045506622451496127174214852e0), + SC_(0.11958599090576171875e0), SC_(0.968867778778076171875e0), 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SC_(0.24750102996826171875e2), SC_(0.12698681652545928955078125e0), SC_(0.24852206550898319978190983067100514776615377727473e2), + SC_(0.24750102996826171875e2), SC_(0.135477006435394287109375e0), SC_(0.24866464070855511040663338301964479223598462797236e2), + SC_(0.24750102996826171875e2), SC_(0.22103404998779296875e0), SC_(0.25065363656732084822750387055815189187723429657877e2), + SC_(0.24750102996826171875e2), SC_(0.308167040348052978515625e0), SC_(0.25380073970395947922582583397566279723752708041733e2), + SC_(0.24750102996826171875e2), SC_(0.6323592662811279296875e0), SC_(0.2805260001123830228051193977609968789435237527768e2), + SC_(0.24750102996826171875e2), SC_(0.814723670482635498046875e0), SC_(0.32013812288161230383507254058691739835624677000322e2), + SC_(0.24750102996826171875e2), SC_(0.835008561611175537109375e0), SC_(0.32751766781102618452555944128390146420838932359409e2), + SC_(0.24750102996826171875e2), SC_(0.905791938304901123046875e0), SC_(0.36511896764707420590932811629213597607442192208936e2), + SC_(0.24750102996826171875e2), SC_(0.9133758544921875e0), SC_(0.37098090138580051103542727188386673865227695373655e2), + SC_(0.24750102996826171875e2), SC_(0.968867778778076171875e0), SC_(0.44576670431026372474999805087999761032786564151124e2), + SC_(0.637722015380859375e2), SC_(0.12698681652545928955078125e0), SC_(0.64029707052819358798026202945496034201051376675713e2), + SC_(0.637722015380859375e2), SC_(0.135477006435394287109375e0), SC_(0.64065662867329731510905167206470426348000579756421e2), + SC_(0.637722015380859375e2), SC_(0.22103404998779296875e0), SC_(0.64567221477870510803896006565094849896791667291395e2), + SC_(0.637722015380859375e2), SC_(0.308167040348052978515625e0), SC_(0.65360653914250445223499712404078058345986158088873e2), + SC_(0.637722015380859375e2), SC_(0.6323592662811279296875e0), SC_(0.72091068969413354768470555979894375250854883743041e2), + SC_(0.637722015380859375e2), SC_(0.814723670482635498046875e0), SC_(0.82046315600696025842317036872213782774465494655701e2), + SC_(0.637722015380859375e2), SC_(0.835008561611175537109375e0), SC_(0.83898717232428299594870923028741701777220145751954e2), + SC_(0.637722015380859375e2), SC_(0.905791938304901123046875e0), SC_(0.93328998463557747079788635557387681634136121872602e2), + SC_(0.637722015380859375e2), SC_(0.9133758544921875e0), SC_(0.94798090083448401328383552423060476950786086617102e2), + SC_(0.637722015380859375e2), SC_(0.968867778778076171875e0), SC_(0.11352388177887899862769270203556217867987795179232e3), + SC_(0.1252804412841796875e3), SC_(0.12698681652545928955078125e0), SC_(0.12579154382331624849995504971737598053587224775121e3), + SC_(0.1252804412841796875e3), SC_(0.135477006435394287109375e0), SC_(0.1258629123249332796639091198974708149109486889059e3), + SC_(0.1252804412841796875e3), SC_(0.22103404998779296875e0), SC_(0.12685851994596818963211475589675243092409541745734e3), + SC_(0.1252804412841796875e3), SC_(0.308167040348052978515625e0), SC_(0.12843375727132041082283110776238893197168320725174e3), + SC_(0.1252804412841796875e3), SC_(0.6323592662811279296875e0), SC_(0.14180779137123412498124168609891981092056098817107e3), + SC_(0.1252804412841796875e3), SC_(0.814723670482635498046875e0), SC_(0.16162410252299947507128901048395142821231251884648e3), + SC_(0.1252804412841796875e3), SC_(0.835008561611175537109375e0), SC_(0.1653152013141215856944195913668621513903319809522e3), + SC_(0.1252804412841796875e3), SC_(0.905791938304901123046875e0), SC_(0.18412078840911869664652903874125802301205742425445e3), + SC_(0.1252804412841796875e3), SC_(0.9133758544921875e0), SC_(0.18705231339830243358318239275545179927816644574167e3), + SC_(0.1252804412841796875e3), SC_(0.968867778778076171875e0), SC_(0.22444947582863925345693507386186458537730111006249e3), + SC_(0.25554705810546875e3), SC_(0.12698681652545928955078125e0), SC_(0.25658502090244978090877141970893960506917422719394e3), + SC_(0.25554705810546875e3), SC_(0.135477006435394287109375e0), SC_(0.25672995659370167636074162037295145926099041806219e3), + SC_(0.25554705810546875e3), SC_(0.22103404998779296875e0), SC_(0.25875180174660371624905238703656604914656092031662e3), + SC_(0.25554705810546875e3), SC_(0.308167040348052978515625e0), SC_(0.26195057844009785727683464893701821156596484750488e3), + SC_(0.25554705810546875e3), SC_(0.6323592662811279296875e0), SC_(0.2891011463129193460573977573707949150504396644584e3), + SC_(0.25554705810546875e3), SC_(0.814723670482635498046875e0), SC_(0.32930739590633854381605546287467793111283231579432e3), + SC_(0.25554705810546875e3), SC_(0.835008561611175537109375e0), SC_(0.33679375763609597539308592356025124507335101594018e3), + SC_(0.25554705810546875e3), SC_(0.905791938304901123046875e0), SC_(0.37492445861810085142108868488536553926419178382588e3), + SC_(0.25554705810546875e3), SC_(0.9133758544921875e0), SC_(0.38086699090058068271279985054050727201708494031721e3), + SC_(0.25554705810546875e3), SC_(0.968867778778076171875e0), SC_(0.45664903993544502840751750992103769324314215524713e3), + SC_(0.503011474609375e3), SC_(0.12698681652545928955078125e0), SC_(0.50505659203440279938830010253440085323070647638583e3), + SC_(0.503011474609375e3), SC_(0.135477006435394287109375e0), SC_(0.50534216388830445980335135399460817135115935936356e3), + SC_(0.503011474609375e3), SC_(0.22103404998779296875e0), SC_(0.50932593644875965893234599959022743691202259918901e3), + SC_(0.503011474609375e3), SC_(0.308167040348052978515625e0), SC_(0.51562892434896885100439228418671454505533824895074e3), + SC_(0.503011474609375e3), SC_(0.6323592662811279296875e0), SC_(0.5691388458382702939253765036826448847256790383481e3), + SC_(0.503011474609375e3), SC_(0.814723670482635498046875e0), SC_(0.64841647174915260562168094112340451126659760019193e3), + SC_(0.503011474609375e3), SC_(0.835008561611175537109375e0), SC_(0.66318246845203780675903083663338059471936088994611e3), + SC_(0.503011474609375e3), SC_(0.905791938304901123046875e0), SC_(0.73841077722591654304210596796019202721941795711926e3), + SC_(0.503011474609375e3), SC_(0.9133758544921875e0), SC_(0.75013755084726454965881401907276391709101666410419e3), + SC_(0.503011474609375e3), SC_(0.968867778778076171875e0), SC_(0.89973134282710925452142048450157816262200221102697e3), + SC_(0.10074598388671875e4), SC_(0.12698681652545928955078125e0), SC_(0.10115604555368949483687471890824497026532038567019e4), + SC_(0.10074598388671875e4), SC_(0.135477006435394287109375e0), SC_(0.1012133050497275830780347173002131083103600802088e4), + SC_(0.10074598388671875e4), SC_(0.22103404998779296875e0), SC_(0.1020120880922149604989950464445671598395259388161e4), + SC_(0.10074598388671875e4), SC_(0.308167040348052978515625e0), SC_(0.10327591029439108059157922326032636145569231829932e4), + SC_(0.10074598388671875e4), SC_(0.6323592662811279296875e0), SC_(0.11400597837237931811476044006089184883421460101765e4), + SC_(0.10074598388671875e4), SC_(0.814723670482635498046875e0), SC_(0.12990515604873895842003339764405472011586481003657e4), + SC_(0.10074598388671875e4), SC_(0.835008561611175537109375e0), SC_(0.13286671610382179011061935562304250712176020940221e4), + SC_(0.10074598388671875e4), SC_(0.905791938304901123046875e0), SC_(0.14795587929982445488758894910099973156560147854256e4), + SC_(0.10074598388671875e4), SC_(0.9133758544921875e0), SC_(0.15030813318756246891355137019045501942292731438192e4), + SC_(0.10074598388671875e4), SC_(0.968867778778076171875e0), SC_(0.18031683136020447255217816635153572296860929726591e4), + SC_(0.1185395751953125e4), SC_(0.12698681652545928955078125e0), SC_(0.11902165793870267987435741776753047875920265751213e4), + SC_(0.1185395751953125e4), SC_(0.135477006435394287109375e0), SC_(0.11908897391528240120679535097563994079186387409261e4), + SC_(0.1185395751953125e4), SC_(0.22103404998779296875e0), SC_(0.12002804010608050581829575008183192405062961123642e4), + SC_(0.1185395751953125e4), SC_(0.308167040348052978515625e0), SC_(0.12151378934996375824220394156626328228788742423087e4), + SC_(0.1185395751953125e4), SC_(0.6323592662811279296875e0), SC_(0.13412676685915296881822755838145300215000346890517e4), + SC_(0.1185395751953125e4), SC_(0.814723670482635498046875e0), SC_(0.15281201627814349911219322982937563560261834041215e4), + SC_(0.1185395751953125e4), SC_(0.835008561611175537109375e0), SC_(0.15629208273736966018850653753226433255433492941276e4), + SC_(0.1185395751953125e4), SC_(0.905791938304901123046875e0), SC_(0.17402117426998289313370860215974579190403457757437e4), + SC_(0.1185395751953125e4), SC_(0.9133758544921875e0), SC_(0.17678472099706618291007501715448169066167629554269e4), + SC_(0.1185395751953125e4), SC_(0.968867778778076171875e0), SC_(0.2120363403215687272189669275949606957433496693392e4) + }; +#undef SC_ + + diff --git a/test/ellint_k_data.ipp b/test/ellint_k_data.ipp new file mode 100644 index 000000000..0b8ea2255 --- /dev/null +++ b/test/ellint_k_data.ipp @@ -0,0 +1,110 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> ellint_k_data = { + SC_(-0.99042308330535888671875e0), SC_(0.3377711175347896212115917173531827081735908096628e1), + SC_(-0.936324596405029296875e0), SC_(0.24799928378892127263868582279024816073017669022263e1), + SC_(-0.931098163127899169921875e0), SC_(0.24445366497109921574142184681825951967450363552834e1), + SC_(-0.928566992282867431640625e0), SC_(0.24283992440860978354078883091691713688375646750641e1), + SC_(-0.927107870578765869140625e0), SC_(0.24193766718674199444433297470157473721017375354321e1), + SC_(-0.907647669315338134765625e0), SC_(0.23150567546618192429964125282245930582572797978818e1), + SC_(-0.89755761623382568359375e0), SC_(0.22701438887345149877132302334749426657376446371554e1), + SC_(-0.805727422237396240234375e0), SC_(0.20065967275502056522161145711205880615166307631054e1), + SC_(-0.804910182952880859375e0), SC_(0.20049607901713509576649915223436597681197065440791e1), + SC_(-0.78026759624481201171875e0), SC_(0.19592100556166276984041617991618803394021101627948e1), + SC_(-0.775062084197998046875e0), SC_(0.19503477481367618080344709104861737619417735663137e1), + SC_(-0.749625742435455322265625e0), SC_(0.19104400630324162677555937926672404286027386161554e1), + SC_(-0.748197972774505615234375e0), SC_(0.19083524493958065222471464174677965403570194813599e1), + SC_(-0.746017634868621826171875e0), SC_(0.19051932837924025803305082607208317930893376343866e1), + SC_(-0.729037344455718994140625e0), SC_(0.18817193270349712906579552152250851402884979195389e1), + SC_(-0.7162187099456787109375e0), SC_(0.18652211443514110895753017730892076794376353163691e1), + SC_(-0.70176351070404052734375e0), SC_(0.18477488779057501703815851124765277365377561449605e1), + SC_(-0.684765398502349853515625e0), SC_(0.18285893795597042965531449099985329446409496660036e1), + SC_(-0.657618343830108642578125e0), SC_(0.18007288418337930091030357471813257042005823390856e1), + SC_(-0.65226137638092041015625e0), SC_(0.17955901766989373531455842967467859457919280319805e1), + SC_(-0.626246631145477294921875e0), SC_(0.17721381024097650892501675317068276777213166539113e1), + SC_(-0.62322795391082763671875e0), SC_(0.17695682119254878963375370515480337143290717650699e1), + SC_(-0.579573929309844970703125e0), SC_(0.17355084423179730268283500692922635797002148145716e1), + SC_(-0.576143443584442138671875e0), SC_(0.17330585071273208935299581419876214110701077039073e1), + SC_(-0.557924091815948486328125e0), SC_(0.17205458226692737403200667710314658175352937313565e1), + SC_(-0.4461468160152435302734375e0), SC_(0.1659143428704545241870880533420140506147522333952e1), + SC_(-0.442996323108673095703125e0), SC_(0.16577357283229169972381488150848899831930773096426e1), + SC_(-0.4059340953826904296875e0), SC_(0.16422906110493949856538834271333672987146221978421e1), + SC_(-0.3961667716503143310546875e0), SC_(0.16385456546730544519392193048583469346592283426711e1), + SC_(-0.38365900516510009765625e0), SC_(0.16339371135272803257688801436358682987064021339502e1), + SC_(-0.366892278194427490234375e0), SC_(0.16280775355945302002930914045811739595274403877109e1), + SC_(-0.3657942116260528564453125e0), SC_(0.16277061660792581984612626919271028055659130917587e1), + SC_(-0.2774055898189544677734375e0), SC_(0.16023984430282800438657607498499405531471989727973e1), + SC_(-0.236876904964447021484375e0), SC_(0.15935547153447952975660160168795914210559129958763e1), + SC_(-0.215539872646331787109375e0), SC_(0.1589532821488995266987321628138860611930750500013e1), + SC_(-0.20251691341400146484375e0), SC_(0.15872846186561139876915281752585049624892405264006e1), + SC_(-0.18253175914287567138671875e0), SC_(0.15841312442860499386355917744088632471227360101517e1), + SC_(-0.15647165477275848388671875e0), SC_(0.15805456328241604119645385355233009390341499402715e1), + SC_(-0.155818879604339599609375e0), SC_(0.15804633259546865645050516399359359956031398166357e1), + SC_(-0.12250564992427825927734375e0), SC_(0.15767400871102419514183615836102806382524229776023e1), + SC_(-0.108822040259838104248046875e0), SC_(0.15754779970087063716770992170704593788694293192204e1), + SC_(-0.84016263484954833984375e-1), SC_(0.15735793449827667079522175566532496627258499829808e1), + SC_(-0.5047737061977386474609375e-1), SC_(0.15717983468978644730669790777144243904901266418816e1), + SC_(-0.2924356050789356231689453125e-1), SC_(0.15711323191301626906143245522892930046111479139043e1), + SC_(-0.2485703863203525543212890625e-1), SC_(0.15710390490728134946675992373238514306032071520432e1), + SC_(-0.2046610601246356964111328125e-1), SC_(0.15709608520853443438268600756375428033257896081621e1), + SC_(-0.1881679333746433258056640625e-1), SC_(0.15709353981303437885151982749950681277334084182671e1), + SC_(0.73303808458149433135986328125e-2), SC_(0.15708174289149913873348609319163921760640886342517e1), + SC_(0.93767531216144561767578125e-1), SC_(0.15742662557451933000168330868057640508121413022011e1), + SC_(0.94445712864398956298828125e-1), SC_(0.15743168849992435161106319232000048329301973530501e1), + SC_(0.26472222805023193359375e0), SC_(0.15994564178224078022860215956833860720177728374035e1), + SC_(0.2795303165912628173828125e0), SC_(0.16029072336719100270539575574782330640458373122935e1), + SC_(0.2926295697689056396484375e0), SC_(0.16061468472007804719214570477294979509859730905091e1), + SC_(0.31095921993255615234375e0), SC_(0.1610983807838740712761368067192691405393131953881e1), + SC_(0.311484813690185546875e0), SC_(0.16111278469143224539903626000312660918785264354958e1), + SC_(0.3272143900394439697265625e0), SC_(0.16155797975233104967915214233565515716306532094698e1), + SC_(0.35747349262237548828125e0), SC_(0.16249404264362997324992641484372408858215631632002e1), + SC_(0.3627222478389739990234375e0), SC_(0.16266751570835125835068247288118651474745966539618e1), + SC_(0.3896603286266326904296875e0), SC_(0.16361224926513534404184731359873689992038364400691e1), + SC_(0.4120951592922210693359375e0), SC_(0.16447205720518345410219553921960387345221364427466e1), + SC_(0.418732583522796630859375e0), SC_(0.16473983100889105980743072684285213201961112169823e1), + SC_(0.451680660247802734375e0), SC_(0.16616537578199064695178688197080983284328919711812e1), + SC_(0.4812971055507659912109375e0), SC_(0.16759409027443021223185102539476096588592566487743e1), + SC_(0.486267507076263427734375e0), SC_(0.16784859353297838419400330721985671192135840854661e1), + SC_(0.509375751018524169921875e0), SC_(0.16909133957115563796676413045863923443879243171298e1), + SC_(0.515482723712921142578125e0), SC_(0.16943684153133994180565117266182132481147687142707e1), + SC_(0.52750241756439208984375e0), SC_(0.17013882771778724726932964262059466393678608760062e1), + SC_(0.531035959720611572265625e0), SC_(0.17035089955832848907265521472430416089318818612814e1), + SC_(0.584416687488555908203125e0), SC_(0.17390197327988114565246996689229900966075086201035e1), + SC_(0.587952077388763427734375e0), SC_(0.17416228307854377082064181076439075579393688159215e1), + SC_(0.5904018878936767578125e0), SC_(0.17434466233796448758342732941787231810206075234501e1), + SC_(0.5945618152618408203125e0), SC_(0.17465816562648604997075231101739061981996142156151e1), + SC_(0.5958592891693115234375e0), SC_(0.17475694098213745790315441260074496951659216831348e1), + SC_(0.59621369838714599609375e0), SC_(0.17478400476807070940242210641459121619833657136515e1), + SC_(0.60056293010711669921875e0), SC_(0.17511905345983286033588753251945176786720874355107e1), + SC_(0.6150639057159423828125e0), SC_(0.17627654245195403785778656645694141888969985612575e1), + SC_(0.629449188709259033203125e0), SC_(0.17748975492324966654441702323788071861970996989102e1), + SC_(0.6438083648681640625e0), SC_(0.17877036110717852208569994416480820233927146151031e1), + SC_(0.64691746234893798828125e0), SC_(0.17905734108284599293495949910396246976486037925061e1), + SC_(0.67001879215240478515625e0), SC_(0.18130645541261783392588745817521015225200222756122e1), + SC_(0.698260128498077392578125e0), SC_(0.18436823516091780453599478590327018132893271380223e1), + SC_(0.744858920574188232421875e0), SC_(0.1903528375308813580432291088034833377552119409638e1), + SC_(0.75686252117156982421875e0), SC_(0.1921257878508255118587957424341444494608703851379e1), + SC_(0.81158483028411865234375e0), SC_(0.20185697260071845965032147456371482208660801062775e1), + SC_(0.826752603054046630859375e0), SC_(0.20517631918180497335416092658794507587336758082983e1), + SC_(0.831471920013427734375e0), SC_(0.20628007317688811564201523487885011440347375983853e1), + SC_(0.841750323772430419921875e0), SC_(0.20881524894194487687984856759006121206951763482194e1), + SC_(0.867987096309661865234375e0), SC_(0.21626553443645542835585774105841427525784980815445e1), + SC_(0.90044462680816650390625e0), SC_(0.22824738454300952253057616561926426370482782713694e1), + SC_(0.914334356784820556640625e0), SC_(0.23479275233217975943026959240752324894340069192044e1), + SC_(0.915014088153839111328125e0), SC_(0.23514264944956428937368978831156074286111428698483e1), + SC_(0.918985307216644287109375e0), SC_(0.23725035893866734721028942202359170270846633736916e1), + SC_(0.9297773838043212890625e0), SC_(0.2436037207911617888011813888250469085423814242563e1), + SC_(0.935390174388885498046875e0), SC_(0.24734276363166698887860324191541438396367793619322e1), + SC_(0.937735855579376220703125e0), SC_(0.24901081139855480473523004657581668979713789812992e1), + SC_(0.941185891628265380859375e0), SC_(0.25159148953544058237986278218012688105174747190388e1), + SC_(0.962219536304473876953125e0), SC_(0.27197444631906795382660830171523843323266857162631e1), + SC_(0.985762655735015869140625e0), SC_(0.31847978937935282927541645440535551111318175394252e1), + SC_(0.988137066364288330078125e0), SC_(0.32733547952035871434783118403258690613147460127497e1), + SC_(0.992922723293304443359375e0), SC_(0.35258676303484017878276559590157568137293088406988e1) + }; +#undef SC_ + diff --git a/test/ellint_pi2_data.ipp b/test/ellint_pi2_data.ipp new file mode 100644 index 000000000..573a4ed36 --- /dev/null +++ b/test/ellint_pi2_data.ipp @@ -0,0 +1,514 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Each row of data contains in order: +// +// n, k, PI(n, k) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> ellint_pi2_data = { + SC_(-0.871743316650390625e2), SC_(0.12698681652545928955078125e0), SC_(0.1674125184786922275852201144520277497358358213892e0), + SC_(-0.871743316650390625e2), SC_(0.135477006435394287109375e0), SC_(0.16743071541243053183466547814319460702264753849723e0), + SC_(-0.871743316650390625e2), SC_(0.22103404998779296875e0), SC_(0.16768328087467940131221559947745677007068360020936e0), + SC_(-0.871743316650390625e2), SC_(0.308167040348052978515625e0), SC_(0.16807808560619590184466399113136263382376022541855e0), + SC_(-0.871743316650390625e2), SC_(0.6323592662811279296875e0), SC_(0.17122026715243530525631538196923732076962513344995e0), + SC_(-0.871743316650390625e2), SC_(0.814723670482635498046875e0), SC_(0.17534141630242782956646324755267751416323066482835e0), + SC_(-0.871743316650390625e2), SC_(0.835008561611175537109375e0), SC_(0.17605554592184937069060075230541493581360045985831e0), + SC_(-0.871743316650390625e2), SC_(0.905791938304901123046875e0), SC_(0.17950085945970126738588840411224305832081644195523e0), + SC_(-0.871743316650390625e2), SC_(0.9133758544921875e0), SC_(0.18001354683126288823677625533339788309413921715434e0), + SC_(-0.871743316650390625e2), SC_(0.968867778778076171875e0), SC_(0.1861623643134889055406196973715380337805086459718e0), + SC_(-0.8631682586669921875e2), SC_(0.12698681652545928955078125e0), SC_(0.16823313926270090905448439997177323604952165614363e0), + SC_(-0.8631682586669921875e2), SC_(0.135477006435394287109375e0), SC_(0.16825150634432729564119302859989272874459890613116e0), + SC_(-0.8631682586669921875e2), SC_(0.22103404998779296875e0), SC_(0.1685064347403877889119812191305941553416022238658e0), + SC_(-0.8631682586669921875e2), SC_(0.308167040348052978515625e0), SC_(0.16890493817434378352408134048753732273764880942954e0), + SC_(-0.8631682586669921875e2), SC_(0.6323592662811279296875e0), SC_(0.17207676229429286979803490409285729827120332255809e0), + SC_(-0.8631682586669921875e2), SC_(0.814723670482635498046875e0), SC_(0.17623726816880267488571098227884869828610508117243e0), + SC_(-0.8631682586669921875e2), SC_(0.835008561611175537109375e0), SC_(0.17695826377184426719315773208991477000910029521482e0), + SC_(-0.8631682586669921875e2), SC_(0.905791938304901123046875e0), SC_(0.18043685472666137350386874269852689421871499307432e0), + SC_(-0.8631682586669921875e2), SC_(0.9133758544921875e0), SC_(0.18095451269215495642582809340040493458719660291695e0), + SC_(-0.8631682586669921875e2), SC_(0.968867778778076171875e0), SC_(0.18716320978769236210159810482536135896363176742343e0), + SC_(-0.7767555999755859375e2), SC_(0.12698681652545928955078125e0), SC_(0.17723805469497005138970658754479684034139004976308e0), + SC_(-0.7767555999755859375e2), SC_(0.135477006435394287109375e0), SC_(0.17725833547565335175365419889134424107345532565877e0), + SC_(-0.7767555999755859375e2), SC_(0.22103404998779296875e0), SC_(0.17753984165266644170812805305717146395671420094776e0), + SC_(-0.7767555999755859375e2), SC_(0.308167040348052978515625e0), SC_(0.17797995186642175489596380169052339842058010566152e0), + SC_(-0.7767555999755859375e2), SC_(0.6323592662811279296875e0), SC_(0.18148541433576540744902539513186881476311716728227e0), + SC_(-0.7767555999755859375e2), SC_(0.814723670482635498046875e0), SC_(0.18608934162792226467645945802477829030064897877359e0), + SC_(-0.7767555999755859375e2), SC_(0.835008561611175537109375e0), SC_(0.18688773786015825638183632881144496436170974248241e0), + SC_(-0.7767555999755859375e2), SC_(0.905791938304901123046875e0), SC_(0.19074161304026152406747562924366026789550786287803e0), + SC_(-0.7767555999755859375e2), SC_(0.9133758544921875e0), SC_(0.19131534459003262670423020772987735547496828534479e0), + SC_(-0.7767555999755859375e2), SC_(0.968867778778076171875e0), SC_(0.19819980751100172478383698040644955494927942439633e0), + SC_(-0.6887512969970703125e2), SC_(0.12698681652545928955078125e0), SC_(0.18807673277332863572439323149240003591140207896313e0), + SC_(-0.6887512969970703125e2), SC_(0.135477006435394287109375e0), SC_(0.18809942894445066895052818190744724738603008664113e0), + SC_(-0.6887512969970703125e2), SC_(0.22103404998779296875e0), SC_(0.18841448356519705070217464671784444233160761372693e0), + SC_(-0.6887512969970703125e2), SC_(0.308167040348052978515625e0), SC_(0.1889071243831848858362782229683818449975108831782e0), + SC_(-0.6887512969970703125e2), SC_(0.6323592662811279296875e0), SC_(0.19283429586558276570559152326779096606423284809356e0), + SC_(-0.6887512969970703125e2), SC_(0.814723670482635498046875e0), SC_(0.19799982175435638480350130578229601669520072124985e0), + SC_(-0.6887512969970703125e2), SC_(0.835008561611175537109375e0), SC_(0.19889635678180396557867565231805298960791065374116e0), + SC_(-0.6887512969970703125e2), SC_(0.905791938304901123046875e0), SC_(0.20322643729961523231468602423697484017124986208135e0), + SC_(-0.6887512969970703125e2), SC_(0.9133758544921875e0), SC_(0.20387136735607637387480442251720783691354881786227e0), + SC_(-0.6887512969970703125e2), SC_(0.968867778778076171875e0), SC_(0.21161453485391920919877033907030403099202567227684e0), + SC_(-0.361317138671875e2), SC_(0.12698681652545928955078125e0), SC_(0.25807395141922622129192766140410293065730739479983e0), + SC_(-0.361317138671875e2), SC_(0.135477006435394287109375e0), SC_(0.25811504785027394933935002408996823143204300773286e0), + SC_(-0.361317138671875e2), SC_(0.22103404998779296875e0), SC_(0.25868576719992802882962788582860568063380617782066e0), + SC_(-0.361317138671875e2), SC_(0.308167040348052978515625e0), SC_(0.25957908488371220675239650970681566250728634199314e0), + SC_(-0.361317138671875e2), SC_(0.6323592662811279296875e0), SC_(0.26673759528219383447347331664283586398767715711549e0), + SC_(-0.361317138671875e2), SC_(0.814723670482635498046875e0), SC_(0.27624188294753047128196814230568161804652916161863e0), + SC_(-0.361317138671875e2), SC_(0.835008561611175537109375e0), SC_(0.27790012081814387436020029758750215255750597263172e0), + SC_(-0.361317138671875e2), SC_(0.905791938304901123046875e0), SC_(0.2859382172923565009441543467819242355142627417165e0), + SC_(-0.361317138671875e2), SC_(0.9133758544921875e0), SC_(0.28713902465022766561269528868552426093781301498377e0), + SC_(-0.361317138671875e2), SC_(0.968867778778076171875e0), SC_(0.30160824142871788586221637018441354445836038223767e0), + SC_(-0.17712909698486328125e2), SC_(0.12698681652545928955078125e0), SC_(0.3636729206179323089584368881165294845044180186145e0), + SC_(-0.17712909698486328125e2), SC_(0.135477006435394287109375e0), SC_(0.36375007260253424844068666864895304741725434790239e0), + SC_(-0.17712909698486328125e2), SC_(0.22103404998779296875e0), SC_(0.36482213460485031933513677664348491484058391598753e0), + SC_(-0.17712909698486328125e2), SC_(0.308167040348052978515625e0), SC_(0.3665025099771102643902474028243205860953862778759e0), + SC_(-0.17712909698486328125e2), SC_(0.6323592662811279296875e0), SC_(0.38006573775283109160868693734996149636084699279112e0), + SC_(-0.17712909698486328125e2), SC_(0.814723670482635498046875e0), SC_(0.39831093780271045079340378355274398784600363650505e0), + SC_(-0.17712909698486328125e2), SC_(0.835008561611175537109375e0), SC_(0.40151794120384245966685931317645917709530730447343e0), + SC_(-0.17712909698486328125e2), SC_(0.905791938304901123046875e0), SC_(0.41714508520694688780251187692727469785205563133695e0), + SC_(-0.17712909698486328125e2), SC_(0.9133758544921875e0), SC_(0.41948979991844566329448196311636376356543866861521e0), + SC_(-0.17712909698486328125e2), SC_(0.968867778778076171875e0), 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SC_(0.957506835460662841796875e0), SC_(0.56128172471137549389344415739570835037807068562013e2), + SC_(0.992881298065185546875e0), SC_(0.964888513088226318359375e0), SC_(0.60675965176721448584035543697674959062919562854532e2), + SC_(0.992881298065185546875e0), SC_(0.967694938182830810546875e0), SC_(0.6276157259900044417409413119621359487406867750706e2), + SC_(0.992881298065185546875e0), SC_(0.968867778778076171875e0), SC_(0.63707286094632152287529888037895307035240586536621e2), + SC_(0.992881298065185546875e0), SC_(0.992881298065185546875e0), SC_(0.1120903461776478755543567802910659613184060305846e3), + SC_(0.992881298065185546875e0), SC_(0.996461331844329833984375e0), SC_(0.14251487713932698105862123589172439943765575331836e3), + SC_(0.996461331844329833984375e0), SC_(0.97540400922298431396484375e-1), SC_(0.26525228363305081889178508646675136190937635090909e2), + SC_(0.996461331844329833984375e0), SC_(0.12698681652545928955078125e0), SC_(0.26609183579980223193874498139758348544252737042937e2), + SC_(0.996461331844329833984375e0), SC_(0.135477006435394287109375e0), SC_(0.26637665635242177149342425025355691196208309182209e2), + SC_(0.996461331844329833984375e0), SC_(0.188381969928741455078125e0), SC_(0.26859854936512317034208182700195391532482709459344e2), + SC_(0.996461331844329833984375e0), SC_(0.22103404998779296875e0), SC_(0.27037248327624528229706950425004550527856127233531e2), + SC_(0.996461331844329833984375e0), SC_(0.278498232364654541015625e0), SC_(0.27430729276514554098633382905293142866236072404168e2), + SC_(0.996461331844329833984375e0), SC_(0.308167040348052978515625e0), SC_(0.27678118753104768841544241662525580388107866876282e2), + SC_(0.996461331844329833984375e0), SC_(0.546881496906280517578125e0), SC_(0.31213240738021743521644147864390542809845218224843e2), + SC_(0.996461331844329833984375e0), SC_(0.54722058773040771484375e0), SC_(0.31220997617474390630713051844594666758716918064286e2), + SC_(0.996461331844329833984375e0), SC_(0.6323592662811279296875e0), SC_(0.33569286497756543464285670293321380122426984476525e2), + SC_(0.996461331844329833984375e0), SC_(0.814723670482635498046875e0), SC_(0.4395386723690573604089341802484590769216982250272e2), + SC_(0.996461331844329833984375e0), SC_(0.835008561611175537109375e0), SC_(0.46132844280074174499202090082014892744405520616437e2), + SC_(0.996461331844329833984375e0), SC_(0.905791938304901123046875e0), SC_(0.58587652855298148334031858334558601606458070313074e2), + SC_(0.996461331844329833984375e0), SC_(0.9133758544921875e0), SC_(0.60748367915375862531255830821598900720628098154312e2), + SC_(0.996461331844329833984375e0), SC_(0.957506835460662841796875e0), SC_(0.82625470558383026944908893678685038814065230848324e2), + SC_(0.996461331844329833984375e0), SC_(0.964888513088226318359375e0), SC_(0.89667844444233284890246553088819296424506047104969e2), + SC_(0.996461331844329833984375e0), SC_(0.967694938182830810546875e0), SC_(0.92913130201351090779050947500643074594936139585892e2), + SC_(0.996461331844329833984375e0), SC_(0.968867778778076171875e0), SC_(0.94387923495063103062113556546657899915314134335375e2), + SC_(0.996461331844329833984375e0), SC_(0.992881298065185546875e0), SC_(0.17241273531195384234344983626787913729253040033677e3), + SC_(0.996461331844329833984375e0), SC_(0.996461331844329833984375e0), SC_(0.22388123241793230536309311882526020237726710081432e3) + }; +#undef SC_ + diff --git a/test/ellint_pi3_data.ipp b/test/ellint_pi3_data.ipp new file mode 100644 index 000000000..e4ff45fc3 --- /dev/null +++ b/test/ellint_pi3_data.ipp @@ -0,0 +1,414 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Each row of data contains in order: +// +// n, phi, k, PI(n, phi, k) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 400> ellint_pi3_data = { + SC_(0.97540400922298431396484375e-1), SC_(0.15321610867977142333984375e0), SC_(0.814723670482635498046875e0), SC_(0.15373203842748125370338546651780574888816846402842e0), + SC_(0.97540400922298431396484375e-1), SC_(0.1994704306125640869140625e0), SC_(0.135477006435394287109375e0), SC_(0.19975117232566753482238350513231879856319046077755e0), + SC_(0.97540400922298431396484375e-1), SC_(0.2128067910671234130859375e0), SC_(0.905791938304901123046875e0), SC_(0.21444962146028791552496083456124507126701014421723e0), + SC_(0.97540400922298431396484375e-1), SC_(0.295909702777862548828125e0), SC_(0.835008561611175537109375e0), SC_(0.2997980220798275355640498732995940042091456814389e0), + SC_(0.97540400922298431396484375e-1), SC_(0.3471994698047637939453125e0), SC_(0.12698681652545928955078125e0), SC_(0.3486476485633547682857103272510437125962758824058e0), + SC_(0.97540400922298431396484375e-1), SC_(0.4374639987945556640625e0), SC_(0.968867778778076171875e0), SC_(0.45394401267101670348318068398554096043112056011139e0), + SC_(0.97540400922298431396484375e-1), SC_(0.4840676784515380859375e0), SC_(0.9133758544921875e0), SC_(0.50429356834860180919136192213432534063086058053706e0), + SC_(0.97540400922298431396484375e-1), SC_(0.859039485454559326171875e0), SC_(0.22103404998779296875e0), SC_(0.88214398516284753446093152446515519356116616485632e0), + SC_(0.97540400922298431396484375e-1), SC_(0.859572112560272216796875e0), SC_(0.6323592662811279296875e0), SC_(0.92068616339339750141383142107393122316780108757901e0), + SC_(0.97540400922298431396484375e-1), SC_(0.993307650089263916015625e0), SC_(0.308167040348052978515625e0), SC_(0.10344161605846899329010060099405650089702846030678e1), + 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SC_(0.1994704306125640869140625e0), SC_(0.71846997737884521484375e0), SC_(0.20283934930431330261325692288358107410442895192458e0), + SC_(0.992881298065185546875e0), SC_(0.2128067910671234130859375e0), SC_(0.29632079601287841796875e0), SC_(0.21619844220527214635649026415392786440017796933018e0), + SC_(0.992881298065185546875e0), SC_(0.295909702777862548828125e0), SC_(0.2878049910068511962890625e0), SC_(0.3051647326285145064763788149974065042939649411249e0), + SC_(0.992881298065185546875e0), SC_(0.3471994698047637939453125e0), SC_(0.74469280242919921875e0), SC_(0.36593269361963920740660627814234613035125736530233e0), + SC_(0.992881298065185546875e0), SC_(0.4374639987945556640625e0), SC_(0.623435556888580322265625e0), SC_(0.47351038005245946606485569936799511854047412964646e0), + SC_(0.992881298065185546875e0), SC_(0.4840676784515380859375e0), SC_(0.188955008983612060546875e0), SC_(0.52619421119616314908557198345534215266757025628341e0), + SC_(0.992881298065185546875e0), SC_(0.859039485454559326171875e0), SC_(0.811873972415924072265625e0), SC_(0.12796147420322316976041013515903300942049403300717e1), + SC_(0.992881298065185546875e0), SC_(0.859572112560272216796875e0), SC_(0.6867754459381103515625e0), SC_(0.12394269966981472546595377372621649553539498692944e1), + SC_(0.992881298065185546875e0), SC_(0.993307650089263916015625e0), SC_(0.31250798702239990234375e0), SC_(0.15534506456507100564291157167033185308229885626081e1), + SC_(0.992881298065185546875e0), SC_(0.127976500988006591796875e1), SC_(0.18351115286350250244140625e0), SC_(0.32883134974911278663257207579294642984195789886813e1), + SC_(0.992881298065185546875e0), SC_(0.131162846088409423828125e1), SC_(0.3857105076313018798828125e0), SC_(0.3843930070475687905185740186007796682476313279295e1), + SC_(0.992881298065185546875e0), SC_(0.142281472682952880859375e1), SC_(0.3684845864772796630859375e0), SC_(0.64560281183013155713676475264767841267885661014251e1), + SC_(0.992881298065185546875e0), SC_(0.14347274303436279296875e1), SC_(0.3439297378063201904296875e0), SC_(0.68762448789590812867204741141193446919124297531292e1), + SC_(0.992881298065185546875e0), SC_(0.150404822826385498046875e1), SC_(0.62561857700347900390625e0), SC_(0.13204500086777579559121699778257730234656273348984e2), + SC_(0.992881298065185546875e0), SC_(0.15156433582305908203125e1), SC_(0.81576907634735107421875e0), SC_(0.18782686397271619660785945631793130083787364064634e2), + SC_(0.992881298065185546875e0), SC_(0.15200517177581787109375e1), SC_(0.7802274227142333984375e0), SC_(0.18302545936290427541117117730722002834553061125842e2), + SC_(0.992881298065185546875e0), SC_(0.1521893978118896484375e1), SC_(0.669285118579864501953125e0), SC_(0.16059155876792932443850017386340255003749606531065e2), + SC_(0.992881298065185546875e0), SC_(0.155961430072784423828125e1), SC_(0.811257660388946533203125e-1), SC_(0.17107262415330151383028878013437636789944479173721e2), + SC_(0.992881298065185546875e0), SC_(0.156523787975311279296875e1), SC_(0.379418849945068359375e0), SC_(0.19155786972659388496036522827203540528048007161044e2), + SC_(0.996461331844329833984375e0), SC_(0.15321610867977142333984375e0), SC_(0.929385960102081298828125e0), SC_(0.15494953789788259980228066836666586023123819724425e0), + SC_(0.996461331844329833984375e0), SC_(0.1994704306125640869140625e0), SC_(0.4584968984127044677734375e0), SC_(0.20243262749758402603750294407634188860785303898664e0), + SC_(0.996461331844329833984375e0), SC_(0.2128067910671234130859375e0), SC_(0.775712668895721435546875e0), SC_(0.21706303863374380621352712917030134681912729698606e0), + SC_(0.996461331844329833984375e0), SC_(0.295909702777862548828125e0), SC_(0.303613960742950439453125e0), SC_(0.30524071829572125811103009319660245769811854653147e0), + SC_(0.996461331844329833984375e0), SC_(0.3471994698047637939453125e0), SC_(0.4867916405200958251953125e0), SC_(0.36356073608058866564834344897710010972287253451556e0), + SC_(0.996461331844329833984375e0), SC_(0.4374639987945556640625e0), SC_(0.910564959049224853515625e0), SC_(0.48104301408957402493870044558559339292659904746407e0), + SC_(0.996461331844329833984375e0), SC_(0.4840676784515380859375e0), SC_(0.4358585774898529052734375e0), SC_(0.5296599894432982217860267176354900496432490384687e0), + SC_(0.996461331844329833984375e0), SC_(0.859039485454559326171875e0), SC_(0.488617718219757080078125e0), SC_(0.11959816962457412713215082894885440106679306405223e1), + SC_(0.996461331844329833984375e0), SC_(0.859572112560272216796875e0), SC_(0.4467837512493133544921875e0), SC_(0.11905815264303277217419460505969483267592093504852e1), + SC_(0.996461331844329833984375e0), SC_(0.993307650089263916015625e0), SC_(0.75578987598419189453125e0), SC_(0.17303829899385816376582825125863049753469531971278e1), + SC_(0.996461331844329833984375e0), SC_(0.127976500988006591796875e1), SC_(0.3063494861125946044921875e0), SC_(0.33959201984939021366299708945798000348464210899788e1), + SC_(0.996461331844329833984375e0), SC_(0.131162846088409423828125e1), SC_(0.108061917126178741455078125e0), SC_(0.37244311575010030859404652598599661337479280352413e1), + SC_(0.996461331844329833984375e0), SC_(0.142281472682952880859375e1), SC_(0.5085086822509765625e0), SC_(0.71622395151588347106386969682257760699123684023386e1), + SC_(0.996461331844329833984375e0), SC_(0.14347274303436279296875e1), SC_(0.39358985424041748046875e0), SC_(0.73636293852590201326616972102360781065664379097234e1), + SC_(0.996461331844329833984375e0), SC_(0.150404822826385498046875e1), SC_(0.510771572589874267578125e0), SC_(0.13953417098210141831895151114277855224069335314941e2), + SC_(0.996461331844329833984375e0), SC_(0.15156433582305908203125e1), SC_(0.870186746120452880859375e0), SC_(0.2556857706461314333232853600430574327145004073045e2), + SC_(0.996461331844329833984375e0), SC_(0.15200517177581787109375e1), SC_(0.817627727985382080078125e0), SC_(0.23625019083977016071256452688852096136119845707138e2), + SC_(0.996461331844329833984375e0), SC_(0.1521893978118896484375e1), SC_(0.36086070537567138671875e0), SC_(0.15792458525021165627427857010031529171479777626611e2), + SC_(0.996461331844329833984375e0), SC_(0.155961430072784423828125e1), SC_(0.79483139514923095703125e0), SC_(0.36995201253231850772712106938485439098332159356711e2), + SC_(0.996461331844329833984375e0), SC_(0.156523787975311279296875e1), SC_(0.917117893695831298828125e0), SC_(0.57989207945838172546694589268843963845498536168854e2) + }; +#undef SC_ + diff --git a/test/ellint_pi3_large_data.ipp b/test/ellint_pi3_large_data.ipp new file mode 100644 index 000000000..7e38f1a29 --- /dev/null +++ b/test/ellint_pi3_large_data.ipp @@ -0,0 +1,394 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Each row of data contains in order: +// +// n, phi, k, PI(n, phi, k) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 380> ellint_pi3_large_data = { + SC_(-0.882951507568359375e2), SC_(-0.80491924285888671875e1), SC_(0.814723670482635498046875e0), SC_(-0.87472421400728425336727040442604900083192977041785e0), + SC_(-0.882951507568359375e2), SC_(-0.7460263729095458984375e1), SC_(0.135477006435394287109375e0), SC_(-0.82718880423302271609361236805097048706326618138613e0), + SC_(-0.882951507568359375e2), SC_(-0.729045963287353515625e1), SC_(0.905791938304901123046875e0), SC_(-0.87747551276758882959544200152954525505618423971659e0), + SC_(-0.882951507568359375e2), SC_(-0.623236083984375e1), SC_(0.835008561611175537109375e0), SC_(-0.65215237953434450934522345100560800793657047297229e0), + SC_(-0.882951507568359375e2), SC_(-0.5579319000244140625e1), SC_(0.12698681652545928955078125e0), SC_(-0.5122762429234107157399593560830350036728566184486e0), + SC_(-0.882951507568359375e2), SC_(-0.443003559112548828125e1), SC_(0.968867778778076171875e0), SC_(-0.54332403292257364440317808333720203684254160082478e0), + SC_(-0.882951507568359375e2), SC_(-0.38366591930389404296875e1), SC_(0.9133758544921875e0), SC_(-0.51338947010848800530828929008203919356014977411471e0), + SC_(-0.882951507568359375e2), SC_(0.9376299381256103515625e0), SC_(0.22103404998779296875e0), SC_(0.15824345778373739469179000212986130442374553398e0), + SC_(-0.882951507568359375e2), SC_(0.944411754608154296875e0), SC_(0.6323592662811279296875e0), SC_(0.16010110842241419282972789586647601062389080124406e0), + SC_(-0.882951507568359375e2), SC_(0.264718532562255859375e1), SC_(0.308167040348052978515625e0), SC_(0.18812698999986759709990325160594334177917367010501e0), + SC_(-0.882951507568359375e2), SC_(0.62944736480712890625e1), SC_(0.97540400922298431396484375e-1), SC_(0.6764648718685968528811245633241126898305817526621e0), + SC_(-0.882951507568359375e2), SC_(0.670017147064208984375e1), SC_(0.54722058773040771484375e0), SC_(0.81778541776042463995319059339588552698088433898419e0), + SC_(-0.882951507568359375e2), SC_(0.81158390045166015625e1), SC_(0.278498232364654541015625e0), SC_(0.83745249382380518272911191808287678555946548573727e0), + SC_(-0.882951507568359375e2), SC_(0.826751708984375e1), SC_(0.188381969928741455078125e0), SC_(0.83757094918096990315704827205570637800372489102387e0), + SC_(-0.882951507568359375e2), SC_(0.91501369476318359375e1), SC_(0.546881496906280517578125e0), SC_(0.88536530789452037214536364734769886885031477691156e0), + SC_(-0.882951507568359375e2), SC_(0.929777050018310546875e1), SC_(0.992881298065185546875e0), SC_(0.10670104592156217618232174187269358103233999652509e1), + SC_(-0.882951507568359375e2), SC_(0.93538990020751953125e1), SC_(0.957506835460662841796875e0), SC_(0.10357277056795963385227029717982264964969263725296e1), + SC_(-0.882951507568359375e2), SC_(0.93773555755615234375e1), SC_(0.996461331844329833984375e0), SC_(0.11393287366513926492075339545953276894721889046337e1), + SC_(-0.882951507568359375e2), SC_(0.98576259613037109375e1), SC_(0.964888513088226318359375e0), SC_(0.12490646709828734017662810793035855992633603697925e1), + SC_(-0.882951507568359375e2), SC_(0.992922687530517578125e1), SC_(0.967694938182830810546875e0), SC_(0.12562075694306417376583251897842403492180529464883e1), + SC_(-0.8681658935546875e2), SC_(-0.80491924285888671875e1), SC_(0.15761308372020721435546875e0), SC_(-0.84140469773208335456919923830249117896698295285825e0), + SC_(-0.8681658935546875e2), SC_(-0.7460263729095458984375e1), SC_(0.725838959217071533203125e0), SC_(-0.85987671052370888277919669524246227679947360493402e0), + SC_(-0.8681658935546875e2), SC_(-0.729045963287353515625e1), SC_(0.970592796802520751953125e0), SC_(-0.91443922525100421734210147418874666650910614494338e0), + SC_(-0.8681658935546875e2), SC_(-0.623236083984375e1), SC_(0.981109678745269775390625e0), SC_(-0.7106272156738935312370492307790173553201815649551e0), + SC_(-0.8681658935546875e2), SC_(-0.5579319000244140625e1), SC_(0.957166969776153564453125e0), SC_(-0.58106009812366507614405013631862001052582083043834e0), + SC_(-0.8681658935546875e2), SC_(-0.443003559112548828125e1), SC_(0.109861753880977630615234375e0), SC_(-0.49983870530384115095256605303312645824352714073227e0), + SC_(-0.8681658935546875e2), SC_(-0.38366591930389404296875e1), SC_(0.4853756427764892578125e0), SC_(-0.49428566816516318921025209597689924685446558273393e0), + SC_(-0.8681658935546875e2), SC_(0.9376299381256103515625e0), SC_(0.79810583591461181640625e0), SC_(0.16264406175756977733142719519534386130654412249944e0), + SC_(-0.8681658935546875e2), SC_(0.944411754608154296875e0), SC_(0.80028045177459716796875e0), SC_(0.16282022749627678393354605008370427957677491731307e0), + SC_(-0.8681658935546875e2), SC_(0.264718532562255859375e1), SC_(0.297029435634613037109375e0), SC_(0.18977993070188281230059741069586097466083740104406e0), + SC_(-0.8681658935546875e2), SC_(0.62944736480712890625e1), SC_(0.14188633859157562255859375e0), SC_(0.68239209777614933756004515248637654641240799137633e0), + SC_(-0.8681658935546875e2), SC_(0.670017147064208984375e1), SC_(0.47834846191108226776123046875e-2), SC_(0.81288526517388767163902468894469430412802255190882e0), + SC_(-0.8681658935546875e2), SC_(0.81158390045166015625e1), SC_(0.4217612743377685546875e0), SC_(0.84924877776215008160678238935268625874487764674281e0), + SC_(-0.8681658935546875e2), SC_(0.826751708984375e1), SC_(0.1124645173549652099609375e0), SC_(0.8436484863509619135739901228550661577676789921708e0), + SC_(-0.8681658935546875e2), SC_(0.91501369476318359375e1), SC_(0.915735542774200439453125e0), 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SC_(0.40756534641409571950446138635247828926930790279057e1), + SC_(-0.422729969024658203125e1), SC_(0.98576259613037109375e1), SC_(0.111202754080295562744140625e0), SC_(0.4485670727454841708415959087291569852354460985191e1), + SC_(-0.422729969024658203125e1), SC_(0.992922687530517578125e1), SC_(0.16287212073802947998046875e0), SC_(0.45334366508455232455528579705650267157785917194655e1), + SC_(-0.396634578704833984375e1), SC_(-0.80491924285888671875e1), SC_(0.780252039432525634765625e0), SC_(-0.40943621028986137634975703353728608171096349083644e1), + SC_(-0.396634578704833984375e1), SC_(-0.7460263729095458984375e1), SC_(0.8773639202117919921875e0), SC_(-0.41529939833416188958459732301077246683216449362382e1), + SC_(-0.396634578704833984375e1), SC_(-0.729045963287353515625e1), SC_(0.3897388279438018798828125e0), SC_(-0.3480702220830587340721759704232773754695850228748e1), + SC_(-0.396634578704833984375e1), SC_(-0.623236083984375e1), SC_(0.210301876068115234375e0), SC_(-0.27885209025273530237198094751157992754542254286699e1), + SC_(-0.396634578704833984375e1), SC_(-0.5579319000244140625e1), SC_(0.241691291332244873046875e0), SC_(-0.23574190979731764925820378430269429039999026821228e1), + SC_(-0.396634578704833984375e1), SC_(-0.443003559112548828125e1), SC_(0.2727530300617218017578125e0), SC_(-0.20795595903717808369226602920751548446608046498173e1), + SC_(-0.396634578704833984375e1), SC_(-0.38366591930389404296875e1), SC_(0.4039121568202972412109375e0), SC_(-0.19363504684946027378474342759829701188128121771172e1), + SC_(-0.396634578704833984375e1), SC_(0.9376299381256103515625e0), SC_(0.4924419820308685302734375e0), SC_(0.57439960450437229566178730657936055087445971602828e0), + SC_(-0.396634578704833984375e1), SC_(0.944411754608154296875e0), SC_(0.964545309543609619140625e-1), SC_(0.56442474196093604965583566632417577278742858368547e0), + SC_(-0.396634578704833984375e1), SC_(0.264718532562255859375e1), SC_(0.22341994941234588623046875e0), SC_(0.10268979775772100508568940610675973674681502356369e1), + SC_(-0.396634578704833984375e1), SC_(0.62944736480712890625e1), SC_(0.1319732964038848876953125e0), SC_(0.28383877275791007314386299295586348532858393601241e1), + SC_(-0.396634578704833984375e1), SC_(0.670017147064208984375e1), SC_(0.49030148983001708984375e0), SC_(0.32907839164490880888445582345626377534487551184709e1), + SC_(-0.396634578704833984375e1), SC_(0.81158390045166015625e1), SC_(0.94205057621002197265625e0), SC_(0.48362019411346784721294532477720592781436962781609e1), + SC_(-0.396634578704833984375e1), SC_(0.826751708984375e1), SC_(0.954943358898162841796875e0), SC_(0.50567832895539918958405915199892399369403297399246e1), + SC_(-0.396634578704833984375e1), SC_(0.91501369476318359375e1), SC_(0.9561345577239990234375e0), SC_(0.55399195596034391177588570332150200763269427826113e1), + SC_(-0.396634578704833984375e1), SC_(0.929777050018310546875e1), SC_(0.651968657970428466796875e0), SC_(0.44651151939772538632794635991507283967810574236258e1), + SC_(-0.396634578704833984375e1), SC_(0.93538990020751953125e1), SC_(0.575208604335784912109375e0), SC_(0.44208286802520298236946850661246282685292115127784e1), + SC_(-0.396634578704833984375e1), SC_(0.93773555755615234375e1), SC_(0.75750362873077392578125e0), SC_(0.47354975575487988811361588081480294631498903701242e1), + SC_(-0.396634578704833984375e1), SC_(0.98576259613037109375e1), SC_(0.597795434296131134033203125e-1), SC_(0.45905273890268042154281499664878219445864193449935e1), + SC_(-0.396634578704833984375e1), SC_(0.992922687530517578125e1), SC_(0.435245692729949951171875e0), SC_(0.47676238150782921841655206628063738779088064504765e1), + SC_(-0.2233159542083740234375e1), SC_(-0.80491924285888671875e1), SC_(0.23477990925312042236328125e0), SC_(-0.44748533126920831422349545820794806553371277385033e1), + SC_(-0.2233159542083740234375e1), SC_(-0.7460263729095458984375e1), SC_(0.552881181240081787109375e0), SC_(-0.45036034718645244567658910223019378062296243109396e1), + SC_(-0.2233159542083740234375e1), SC_(-0.729045963287353515625e1), SC_(0.3531585633754730224609375e0), SC_(-0.42726238657165988625899083013084234770927337383475e1), + SC_(-0.2233159542083740234375e1), SC_(-0.623236083984375e1), SC_(0.53152568638324737548828125e-1), SC_(-0.34453886271845466532767992962242203359531735591743e1), + SC_(-0.2233159542083740234375e1), SC_(-0.5579319000244140625e1), SC_(0.82119405269622802734375e0), SC_(-0.36214083786208780467420759923323534444796335095604e1), + SC_(-0.2233159542083740234375e1), SC_(-0.443003559112548828125e1), SC_(0.3225107491016387939453125e0), SC_(-0.25784058164296168294688388728197558401803014682815e1), + SC_(-0.2233159542083740234375e1), SC_(-0.38366591930389404296875e1), SC_(0.15403441153466701507568359375e-1), SC_(-0.22937304274191202625404745929401752108326363938647e1), + SC_(-0.2233159542083740234375e1), SC_(0.9376299381256103515625e0), SC_(0.4049813747406005859375e0), SC_(0.66882671905380321365353696629501581726839659732368e0), + SC_(-0.2233159542083740234375e1), SC_(0.944411754608154296875e0), SC_(0.430237986147403717041015625e-1), SC_(0.66092945746372938652588605746346208982917240808408e0), + SC_(-0.2233159542083740234375e1), SC_(0.264718532562255859375e1), SC_(0.908433616161346435546875e0), SC_(0.18536900080764993605645667717175921713955588720638e1), + SC_(-0.2233159542083740234375e1), SC_(0.62944736480712890625e1), SC_(0.16899003088474273681640625e0), SC_(0.35237325007404202330837864635359809500592160070051e1), + SC_(-0.2233159542083740234375e1), SC_(0.670017147064208984375e1), SC_(0.80913722515106201171875e0), SC_(0.45426945752272245371383350900233494955058093762277e1), + SC_(-0.2233159542083740234375e1), SC_(0.81158390045166015625e1), SC_(0.649115502834320068359375e0), SC_(0.49038329428638199567178903183066367562827357258082e1), + SC_(-0.2233159542083740234375e1), SC_(0.826751708984375e1), SC_(0.25829589366912841796875e0), SC_(0.45593877951262054200620352500821353887097951116667e1), + SC_(-0.2233159542083740234375e1), SC_(0.91501369476318359375e1), SC_(0.731722414493560791015625e0), SC_(0.57004523382308203815337495168340937882687351079681e1), + SC_(-0.2233159542083740234375e1), SC_(0.929777050018310546875e1), SC_(0.12984652817249298095703125e0), SC_(0.51319275680071863858207399358794549369987990555794e1), + SC_(-0.2233159542083740234375e1), SC_(0.93538990020751953125e1), SC_(0.64774596691131591796875e0), SC_(0.56823138039205519976075282866343588596541504718097e1), + SC_(-0.2233159542083740234375e1), SC_(0.93773555755615234375e1), SC_(0.493326842784881591796875e0), SC_(0.5455958348692104517652116375856606963768856463546e1), + SC_(-0.2233159542083740234375e1), SC_(0.98576259613037109375e1), SC_(0.4509237110614776611328125e0), SC_(0.58425715884572127603208314559986328209865929040183e1), + SC_(-0.2233159542083740234375e1), SC_(0.992922687530517578125e1), SC_(0.3785004317760467529296875e0), SC_(0.58235654495321593144993439337365609617712620177635e1), + SC_(0.201027393341064453125e0), SC_(-0.80491924285888671875e1), SC_(0.54700887203216552734375e0), SC_(-0.99225565272299265057474332033452451298257411758289e1), + SC_(0.201027393341064453125e0), SC_(-0.7460263729095458984375e1), SC_(0.71846997737884521484375e0), SC_(-0.98733924062829495855268464439174693001319340908722e1), + SC_(0.201027393341064453125e0), SC_(-0.729045963287353515625e1), SC_(0.29632079601287841796875e0), SC_(-0.82838310087004006715984395376110573887564954006723e1), + SC_(0.201027393341064453125e0), SC_(-0.623236083984375e1), SC_(0.2878049910068511962890625e0), SC_(-0.71399649768593994233565781388330571566404664291638e1), + SC_(0.201027393341064453125e0), SC_(-0.5579319000244140625e1), SC_(0.74469280242919921875e0), SC_(-0.7852838275875078318898768277396954817312338949233e1), + SC_(0.201027393341064453125e0), SC_(-0.443003559112548828125e1), SC_(0.623435556888580322265625e0), SC_(-0.55353469466832669055948998690680459923127869705195e1), + SC_(0.201027393341064453125e0), SC_(-0.38366591930389404296875e1), SC_(0.188955008983612060546875e0), SC_(-0.42670151795740926550616070898142650427001824200856e1), + SC_(0.201027393341064453125e0), SC_(0.9376299381256103515625e0), SC_(0.811873972415924072265625e0), SC_(0.10948999881609283534865564650155039154841364221285e1), + SC_(0.201027393341064453125e0), SC_(0.944411754608154296875e0), SC_(0.6867754459381103515625e0), SC_(0.10678437743348588891519840198915283249591294529495e1), + SC_(0.201027393341064453125e0), SC_(0.264718532562255859375e1), SC_(0.31250798702239990234375e0), SC_(0.31064483566488608931281755219427099993121689654282e1), + SC_(0.201027393341064453125e0), SC_(0.62944736480712890625e1), SC_(0.18351115286350250244140625e0), SC_(0.71043510436913445771405941909028349861445575844728e1), + SC_(0.201027393341064453125e0), SC_(0.670017147064208984375e1), SC_(0.3857105076313018798828125e0), SC_(0.77553041204136176774038756662263769147319302501976e1), + SC_(0.201027393341064453125e0), SC_(0.81158390045166015625e1), SC_(0.3684845864772796630859375e0), SC_(0.94787138653519013648251654059066285072949428515173e1), + SC_(0.201027393341064453125e0), SC_(0.826751708984375e1), SC_(0.3439297378063201904296875e0), SC_(0.96231456672706330197751945234594804514605578493294e1), + SC_(0.201027393341064453125e0), SC_(0.91501369476318359375e1), SC_(0.62561857700347900390625e0), SC_(0.11697431340770649519046006010297689812160093617617e2), + SC_(0.201027393341064453125e0), SC_(0.929777050018310546875e1), SC_(0.81576907634735107421875e0), SC_(0.13689559632918850897185479014035065907489555706468e2), + SC_(0.201027393341064453125e0), SC_(0.93538990020751953125e1), SC_(0.7802274227142333984375e0), SC_(0.13252477031585661466687278470237361058228506710271e2), + SC_(0.201027393341064453125e0), SC_(0.93773555755615234375e1), SC_(0.669285118579864501953125e0), SC_(0.12219207125307825999387228477920190151422215626939e2), + SC_(0.201027393341064453125e0), SC_(0.98576259613037109375e1), SC_(0.811257660388946533203125e-1), SC_(0.11000670184663049479533855890007560504469595099336e2), + SC_(0.201027393341064453125e0), SC_(0.992922687530517578125e1), SC_(0.379418849945068359375e0), SC_(0.11497477476964669120779679338120232828710984170976e2) + }; +#undef SC_ + diff --git a/test/ellint_rc_data.ipp b/test/ellint_rc_data.ipp new file mode 100644 index 000000000..c03e26684 --- /dev/null +++ b/test/ellint_rc_data.ipp @@ -0,0 +1,216 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Test data for RC, each row contains in order: +// +// x, y, RC(x, y) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 201> ellint_rc_data = { + SC_(0.11698430441812742785454260394458960094979845402414e-30), SC_(0.1429457475085533184e20), SC_(0.41546482167023371731518451342517736095263780557582e-9), + SC_(0.60682196926171104626767727515008970841312337712241e-30), SC_(0.20031258624e11), SC_(0.11098537606275153066313383027431717728910787699004e-4), + SC_(0.75974287571611502022633594876763844865744160884209e-30), SC_(0.20657736771171195551744e23), SC_(0.10928951720653730370271800759007590309935803177545e-10), + SC_(0.10711124272997777587041261201942927950011067906889e-29), SC_(0.47946235e7), SC_(0.71736902777915132997000781571354752397421561836787e-3), + SC_(0.22148209406780902835423848339554340607932845033396e-29), SC_(0.19855378956352178823888896e26), SC_(0.35251758550831167800571476411635458611576128740787e-12), + SC_(0.41366469991877353664505423235416110752209856263632e-29), SC_(0.1080339869874636584075679427013641468012674864323e-29), SC_(0.73882898278016317196709226502699714371366335165311e15), + SC_(0.44767214645416419053919245787147718165594030426536e-29), SC_(0.26553176031232e14), SC_(0.30483276103299498395114390375939883603035900740311e-6), + SC_(0.60078474312334545656635060929362382794743178141472e-29), SC_(0.37131016039637643189053051173686981201171875e-8), SC_(0.25778133019186154518263488635604654249614963342631e5), + SC_(0.61834249098451553748337548320711315606925672313013e-29), SC_(0.15125532639232e16), SC_(0.40389133730438812928090167163598082343384172921425e-7), + SC_(0.15386248307178551019952166983379913769903023636025e-28), SC_(0.11083928e9), SC_(0.14920144492544889019454092007748150269559896140113e-3), + SC_(0.19696330564073048658818435820590984305819001943432e-28), 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SC_(0.130841693878173828125e2), SC_(0.516425012058796509045456896e28), SC_(0.21858292984467318233636908716192880532684611323597e-13), + SC_(0.2496727752685546875e2), SC_(0.755542572949963187056710012257099151611328125e-10), SC_(0.27928314162875666520962642485579357074135329858573e1), + SC_(0.4151866912841796875e2), SC_(0.84098145009994339684283693509730169685090217512879e-22), SC_(0.4341003402255665800060030872597376661848978347545e1), + SC_(0.10465634918212890625e3), SC_(0.60640822994173504412174224853515625e-5), SC_(0.88219974915068377829851076473240001539884786059211e0), + SC_(0.204871795654296875e3), SC_(0.351293559140227787146657792e27), SC_(0.83807866206725702601826883835498669002259349569411e-13), + SC_(0.2247014312744140625e3), SC_(0.275117858615052709715247104e27), SC_(0.94702291082189117328396733175326455545223390155305e-13), + SC_(0.322791595458984375e3), SC_(0.86205957643638764405225050069248027284629642963409e-15), SC_(0.11646880911415179034823341402428681949283112375143e1), + SC_(0.4562802734375e3), SC_(0.10197021348467909816516131089894163913918803398168e-29), SC_(0.17922488338074155572310350987171414204572553246592e1), + SC_(0.7562174072265625e3), SC_(0.1910448768e10), SC_(0.35923470132422725783693134841062611631507906827816e-4), + SC_(0.82290283203125e3), SC_(0.36223742228003175114281475543975830078125e-7), SC_(0.43980399212423650879963940207747478236239652368737e0), + SC_(0.10313978271484375e4), SC_(0.10318525717298732830018437311991874594241380691528e-14), SC_(0.6668515395137059692868787992249145942381978211602e0), + SC_(0.11206973876953125e4), SC_(0.159823103248039936e18), SC_(0.39291632590134749018316722859590852663653580203117e-8), + SC_(0.3524259033203125e4), SC_(0.38758790625e6), SC_(0.23805498764247481510606154825690304471222889948051e-2), + SC_(0.3542151123046875e4), SC_(0.23456829198204715014597354638681281358003616333008e-13), SC_(0.34396142776310129846777566611486054989278031943136e0), + SC_(0.69560673828125e4), SC_(0.747194576263427734375e1), SC_(0.49317089833424179564861860197880854610400944287808e-1), + SC_(0.7298279296875e4), SC_(0.135008262029312e15), SC_(0.13518785304121054341303872941843981481438748034264e-6), + SC_(0.897631640625e4), SC_(0.386145068359375e4), SC_(0.13761011816758051951614138230181415517505541771123e-1), + SC_(0.147901005859375e5), SC_(0.62520772668534154026944814518561754825703991045316e-23), SC_(0.2648413871016930409485455149414286637028184421264e0), + SC_(0.815536015625e5), SC_(0.230844914913177490234375e0), SC_(0.24794349449595078621345239278260580833606450246749e-1), + SC_(0.84901359375e5), SC_(0.18469835205078125e4), SC_(0.90274392712796214783133197618962325121969170157594e-2), + SC_(0.131195640625e6), SC_(0.399027378608056683915492826862951233901632974721e-21), SC_(0.8619841670060382832194238459033456814758956298182e-1), + SC_(0.41534478125e6), SC_(0.1918984889984130859375e1), SC_(0.1060665265394308433482579591641745485349613369435e-1), + SC_(0.6869200625e6), SC_(0.12406291034494643099606037139892578125e-5), SC_(0.17148853609122838287334270730377713622015582671517e-1), + SC_(0.9115189375e6), SC_(0.179409878467662494683889664e27), SC_(0.11727263974665223656008245016259368445874373950871e-12), + SC_(0.39914275e7), SC_(0.5617260831058956682682037353515625e-4), SC_(0.66003352582444822055431410346869855793086865122376e-2), + SC_(0.15278292e8), SC_(0.14619622e8), SC_(0.25961118278381005988486780674736203564957112818546e-3), + SC_(0.15583587e8), SC_(0.46741176605224609375e2), SC_(0.17863256980859237735262630967043753358312118433161e-2), + SC_(0.2359947e8), SC_(0.83892250504513065845701871993405683214864787598623e-28), SC_(0.85438729190979830430450108794113432317121617101695e-2), + SC_(0.44006012e8), SC_(0.41172763385266176e17), SC_(0.77411561887565592752972899889991015699435496559546e-8), + SC_(0.4426536e8), SC_(0.89961878334825112180062577083017316681434749625623e-18), SC_(0.45500003788487494954911364150613981149754477134657e-2), + SC_(0.14248488e9), SC_(0.61731988312258099732254031872e29), SC_(0.63221492521852077836928323850507110790467251568351e-14), + SC_(0.156271296e9), SC_(0.13287944602780044078826904296875e-3), SC_(0.11670991383498154005718703499716739301912113499221e-2), + SC_(0.339495296e9), SC_(0.27455182077952e14), SC_(0.29911423091309748142593311126423073053561438997539e-6), + SC_(0.355727936e9), SC_(0.49794778812580822691949943719295546451682199506905e-24), SC_(0.20422111659287768080880666674695585733215224225899e-2), + SC_(0.1626075136e10), SC_(0.482386627197265625e2), SC_(0.23211090322482879766302355319080524745211784432509e-3), + SC_(0.2606333696e10), SC_(0.12266837021255184297929829995155159849673509597778e-13), SC_(0.53963700104507610983907942976446519710715041862358e-3), + SC_(0.3957605632e10), SC_(0.55799657429356219751070966594852507114410400390625e-13), SC_(0.42920548607924953119431231450313296279672960982957e-3), + SC_(0.7740245504e10), SC_(0.589840515072e13), SC_(0.63227008905693272145832965758127025031892199413741e-6), + SC_(0.10259405824e11), SC_(0.35736448257653877024582131372021365223190514370799e-17), SC_(0.31894320284009100162338080720486603695808737357409e-3), + SC_(0.14041162752e11), SC_(0.16802999673239973138530304e26), SC_(0.38320084631624641337451768839752529254504071282248e-12), + SC_(0.82854125568e11), SC_(0.58830391620982291911135882706044579926359106014644e-24), SC_(0.14299276138191576425667241629897528275040512189828e-3), + SC_(0.11738591232e12), SC_(0.761073704058645716941100545227527618408203125e-10), SC_(0.7322171577268687983119525908536326486897490571614e-4), + SC_(0.191893782528e12), SC_(0.174726296875e6), SC_(0.17458388731589946205436634365599236215476886257278e-4), + SC_(0.446171086848e12), SC_(0.49623249953612003082525916397571563720703125e-9), SC_(0.37153579565929854820799756427432381213691623554542e-4), + SC_(0.572994879488e12), SC_(0.4479686882443445386870784e25), SC_(0.742157274353212118943917435113707955824870795782e-12), + SC_(0.92746285056e12), SC_(0.7325884342193603515625e1), SC_(0.13992334087709732045223614104220216290282058103286e-4), + SC_(0.13073324703744e14), SC_(0.48555353730121618377670656e27), SC_(0.71285511551986730987092942275962922466144519563445e-13), + SC_(0.1591604150272e14), SC_(0.351564170056957952e18), SC_(0.26379290397755114021116829042194283811143014246219e-8), + SC_(0.20741449842688e14), SC_(0.4743499375e6), SC_(0.20837274754438050716836877714486012090201227681745e-5), + SC_(0.31191365320704e14), SC_(0.2293932139873504638671875e0), SC_(0.30376223823640976472682511381218417507215286561237e-5), + SC_(0.51712110886912e14), SC_(0.156551992965379627797503279104e30), SC_(0.39700005931080030246901295737647079077294776223606e-14), + SC_(0.14830236860416e15), SC_(0.57842544071306897015280826246375056598481023684144e-17), SC_(0.30262887085625134071904723108221624416890211410925e-5), + SC_(0.3135611338752e15), SC_(0.2348838144e10), SC_(0.37238603812796720490349747070177231425073166090273e-6), + SC_(0.323813657018368e15), SC_(0.156708486328125e5), SC_(0.69847661515884606972303830302067390332677255420951e-6), + SC_(0.530327327997952e15), SC_(0.10366431646728266850195727608041629252966231433675e-18), SC_(0.17153262952128191564833782071342486326068147437365e-5), + SC_(0.727262651482112e15), SC_(0.1535589888e10), SC_(0.26799428982612804774219551247863136845480307410692e-6), + SC_(0.95759886188544e15), SC_(0.4659874708323741288040764629840850830078125e-8), SC_(0.8897375003298093960305950194491436152687587334261e-6), + SC_(0.1551973144854528e16), SC_(0.10139344880747031860591903938868807433237861914677e-19), SC_(0.10458468387194864418446638976290709697775378499213e-5), + SC_(0.183459460939776e16), SC_(0.58535432060362684618705258305616519358052915120161e-26), SC_(0.11315636043901720121388837656247249664177763590949e-5), + SC_(0.27152649027584e16), SC_(0.66861986169897136278450489044189453125e-6), SC_(0.49072897074683974426280640967267909946582802584589e-6), + SC_(0.3762704347037696e16), SC_(0.10164524532769642241629852708355797252185587220552e-24), SC_(0.77271884197062324056183842244132156703113101877894e-6), + SC_(0.6551729407524864e16), SC_(0.25762003497220575809478759765625e-3), SC_(0.28457658270165863549014595037995212294784586710095e-6), + SC_(0.9007602981666816e16), SC_(0.25765624e9), SC_(0.98811088491194357965387716654259888241690606624295e-7), + SC_(0.31614863423832064e17), SC_(0.64014885e7), SC_(0.66664506125039749834276834929108360333353634282303e-7), + SC_(0.45080659638616064e17), SC_(0.66192871423119660955992064e26), SC_(0.19306655679013338902506375891076552233340289260346e-12), + SC_(0.59877638017122304e17), SC_(0.53512100536178474827209150532780768116936087608337e-15), SC_(0.15362013975191677793880105907602135335521807551786e-6), + SC_(0.128395461743607808e18), SC_(0.206836903384823477636694016e27), SC_(0.10921919594473705616931168879025397857363831649357e-12), + SC_(0.140845240494850048e18), SC_(0.15238641357421875e4), SC_(0.44689988476580111818524171778021205499463465100743e-7), + SC_(0.245924553149120512e18), SC_(0.12432241406593107741605394588021265875441186132822e-25), SC_(0.10191370089476240496442184751961162333606213558986e-6), + SC_(0.10302131894484992e19), SC_(0.86229714468864e14), SC_(0.53078949279350852244495899285397248236587855360507e-8), + SC_(0.4173026783555223552e19), SC_(0.25325901731494375978251987705874090841683295149966e-25), SC_(0.25259325443772059313156731529761618841007461413987e-7), + SC_(0.38083625921602912256e20), SC_(0.14043506688e11), SC_(0.18721812477205878318873337191745626686057078380972e-8), + SC_(0.6036870791327383552e20), SC_(0.7191220191232e13), SC_(0.11151877475097208967247771680064286641451016014729e-8), + SC_(0.140771054246300745728e21), SC_(0.4364620208740234375e2), SC_(0.18544022482541756413624180462592298070565733208098e-8), + SC_(0.229345916576958775296e21), SC_(0.181080293376e13), SC_(0.66174853669110756842649598160820882183579693595045e-9), + SC_(0.486625867239520206848e21), SC_(0.2663514463837359985504355062296832912238642165903e-19), SC_(0.21326856610092306285181605511492710866111336973419e-8), + SC_(0.1241846768099137683456e22), SC_(0.16907547929525060340181807204941172504655995389999e-25), SC_(0.15507902235497089042906909015124525282864868106372e-8), + SC_(0.2085586588137791946752e23), SC_(0.1494682042368e13), SC_(0.85673852855596798336664141721336700864507390102108e-10), + SC_(0.140444835444342210428928e24), SC_(0.468889537015105749778432e24), SC_(0.17303641617108978771183397785345947512931840394585e-11), + SC_(0.277204771630374352584704e24), SC_(0.178649388253688812255859375e-1), SC_(0.56400743476075289901756964355291370430816192512748e-10), + SC_(0.292186842467552253181952e24), SC_(0.8000304500511383121995759616e28), SC_(0.17494456350937238152051998129209403413657779766045e-13), + SC_(0.404188410838900933656576e24), SC_(0.15307389070737408e17), SC_(0.14530172405497225203148631842861632557097281219397e-10), + SC_(0.432213410600102056165376e24), SC_(0.1959631583933896656101239768660304818581607833039e-18), SC_(0.75206449301552958256440392567562049430798996389352e-10), + SC_(0.48688847917066659823616e24), SC_(0.12212552573408426924125600686592708205013835254249e-21), SC_(0.76232145024480748870634232091696723820904729119412e-10), + SC_(0.1611108507323484384264192e25), SC_(0.11971012087915599989354498156046702206367626786232e-15), SC_(0.3694442785175160419076949350029585599009031148763e-10), + SC_(0.1626183676572159287754752e26), SC_(0.18501463382170069138510370976291596889495849609375e-11), SC_(0.10719260914536354712818827718038676299273171700718e-10), + SC_(0.18820064668587267094216704e26), SC_(0.302210201308383830109960399568080902099609375e-9), SC_(0.93936394693848721355885906623218887308774325852727e-11), + SC_(0.24925849334950778558742528e26), SC_(0.31317834636651731435069138009717243018286553235541e-24), SC_(0.11646010681447365669777654446715227593887198721003e-10), + SC_(0.9036469625899950631026688e26), SC_(0.690641880035400390625e1), SC_(0.31148497090217722784373888164259764652317411281318e-11), + SC_(0.7376334675354954747676196864e28), SC_(0.3832167254355459644063744e25), SC_(0.52109798218409315418287118331564136634721278594953e-13), + SC_(0.19445130267235161963321360384e29), SC_(0.27158976561908717618255837411567199524142779409885e-16), SC_(0.37530178368631060137545101809988924309836018928577e-12) + }; +#undef SC_ + + diff --git a/test/ellint_rd_data.ipp b/test/ellint_rd_data.ipp new file mode 100644 index 000000000..bb6b0bd86 --- /dev/null +++ b/test/ellint_rd_data.ipp @@ -0,0 +1,216 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Test data for RD, each row contains in order: +// +// x, y, z, RD(x, y, z) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 201> ellint_rd_data = { + SC_(0.60682196926171104626767727515008970841312337712241e-30), SC_(0.20031258624e11), SC_(0.10313978271484375e4), SC_(0.20551372495384815463856604017047221545095240191057e-7), + SC_(0.13810928266794127506964991575307193802925529863273e-29), SC_(0.2529551275074481964111328125e-3), SC_(0.7030536597341097149183042347431182861328125e-8), SC_(0.26827491422802771467722943731540343461246213838861e11), + SC_(0.44767214645416419053919245787147718165594030426536e-29), SC_(0.26553176031232e14), SC_(0.2085586588137791946752e23), SC_(0.10585138723496739834534066073267118675522211779171e-31), + SC_(0.60078474312334545656635060929362382794743178141472e-29), SC_(0.37131016039637643189053051173686981201171875e-8), SC_(0.32517390764125836699536241120728208287005145393778e-20), SC_(0.15139742233407639223792275679902886417433224652453e26), + SC_(0.61834249098451553748337548320711315606925672313013e-29), SC_(0.15125532639232e16), SC_(0.11666782739894188125617802143096923828125e-6), SC_(0.66117253645648417600031762235375783237535720054585e0), + SC_(0.80490202086679616440767211156605911864354431197377e-29), SC_(0.263248904192e12), SC_(0.297980882347216843954978816e27), SC_(0.10333434770883940055455846219131687005742728721005e-37), + SC_(0.15386248307178551019952166983379913769903023636025e-28), SC_(0.11083928e9), SC_(0.210290006361901760101318359375e-2), SC_(0.13550514102538012513935665942320904977981297554257e0), + SC_(0.27790564235134654243337661318427908127768828411199e-28), SC_(0.21656921296023725949679903214975529301966616912978e-20), SC_(0.16200692698475904762744903564453125e-4), SC_(0.85856403820113222790454310518632166416049096576362e9), + SC_(0.75787209056271355952485380019348769132814133643247e-28), SC_(0.3443290234375e5), SC_(0.87738095954812459262649215691458087947041111220869e-24), SC_(0.18256962837089036695688975681087926934231777165353e23), + SC_(0.83892250504513065845701871993405683214864787598623e-28), SC_(0.1116302655645995400846004486083984375e-5), SC_(0.5191992844812585795584e22), SC_(0.25853043071836854153543603031360043116599575432972e-30), + SC_(0.1287535325337014953720268054032123824574619620157e-27), SC_(0.13705567485615801022446745431989965211687137938007e-26), SC_(0.5078582763671875e0), SC_(0.25453721736248252436275234446728359554220261644493e3), + SC_(0.55263283825603576259688637477087526891102295303199e-27), SC_(0.14570690537170526341649579027404115549870766699314e-16), SC_(0.12941088853490731622934705547460304161866623869764e-26), SC_(0.36729174819285901249546490822632941293048355712711e36), + SC_(0.56127809703514701593677302775789618452802567406812e-27), SC_(0.9228809556838246663801328395493328571319580078125e-11), SC_(0.86422194270699415064029835775727406144142150878906e-12), SC_(0.10781910269701214744433782310389927085727282773814e19), + SC_(0.11649136528196886067987694424015483218044894522109e-26), SC_(0.36767340167168e14), SC_(0.4968523085117340087890625e0), SC_(0.99577870866168728614595399908281899687867831139901e-6), + SC_(0.25172518744793432127678756566548684645107602196601e-26), SC_(0.432213410600102056165376e24), SC_(0.1959631583933896656101239768660304818581607833039e-18), SC_(0.23283511871909436551998718906785080640182682915363e8), + SC_(0.28601321158264026057365024112938563807684451932578e-26), SC_(0.40558081566683637682324548023871102486737072467804e-15), SC_(0.208714828125e6), SC_(0.76237261396619453885730994908723652541092525168043e-6), + SC_(0.73955594308673193523198369636584653936223991005372e-26), SC_(0.15995684594580228399252064264146611094474792480469e-12), SC_(0.170420291286861873152e22), SC_(0.1687006236430131643441016712281856878753726523535e-29), + SC_(0.25325901731494375978251987705874090841683295149966e-25), SC_(0.2247014312744140625e3), SC_(0.275117858615052709715247104e27), SC_(0.18485690522734605390476411600907452658544362635922e-37), + SC_(0.34857664908449107348958795506280082952452713251912e-25), SC_(0.4151866912841796875e2), SC_(0.84098145009994339684283693509730169685090217512879e-22), SC_(0.54257544703354289137242893169458880766652554745316e22), + SC_(0.34919368622379363366113344149995795887854964367758e-25), SC_(0.253342638931968e15), SC_(0.7562174072265625e3), SC_(0.249241547815296223838694888499925239323224138633e-9), + SC_(0.43573388456807634241895548780677530339229715228289e-25), SC_(0.4951682e8), SC_(0.27826294978205525521980137859608542144911232096849e-25), SC_(0.68052511685869677076187126812000404571638179018885e22), + SC_(0.79820903020739723608895214436397557193018231780357e-25), SC_(0.64729720161775472443612539086288393264112528413534e-17), SC_(0.68450550546432e14), SC_(0.19125532541215326384896730405964498227631338694385e-18), + SC_(0.87440356429873925658498856170228582591246249688943e-25), SC_(0.897631640625e4), SC_(0.386145068359375e4), SC_(0.70431488067233168809120506088634745460689092194161e-5), + SC_(0.94440849851249091320281320378524601844016927998382e-25), SC_(0.339495296e9), SC_(0.27455182077952e14), SC_(0.12588696277491280442424745508482566976425259261561e-18), + SC_(0.14235380836999617679740726134029336127093459674064e-24), SC_(0.13978527726259656235898667313173260930711863658793e-24), SC_(0.2028485946357250213623046875e-1), SC_(0.27366411869676160015961055866442893198667542321359e5), + SC_(0.19953517994584643852448181812338140391502663839596e-24), SC_(0.20741449842688e14), SC_(0.4743499375e6), SC_(0.13886820075171731514726499953757541277555907027004e-11), + SC_(0.24754359063412526643006257824154938047573476511687e-24), SC_(0.542938704356835328e18), SC_(0.44974670432440498891960159215184658369128360699829e-26), SC_(0.10752772537029819829836280199737653161290081988757e18), + SC_(0.31317834636651731435069138009717243018286553235541e-24), SC_(0.446171086848e12), SC_(0.49623249953612003082525916397571563720703125e-9), SC_(0.90507658227268922734361508605808168072270615976688e4), + SC_(0.49794778812580822691949943719295546451682199506905e-24), SC_(0.1241846768099137683456e22), SC_(0.16907547929525060340181807204941172504655995389999e-25), SC_(0.78343915398198934519415679862427240727586271564883e15), + SC_(0.56225779987930131445265275562229206527363467582603e-24), SC_(0.59220483766176692878700381014733800100202643079683e-19), SC_(0.170295034922225631232e22), SC_(0.20048163240230817674563773499193885173665982026089e-29), + SC_(0.63171350201374448567926894124993295043246632758382e-24), SC_(0.34594001439795600424927311319582115491182941913878e-27), SC_(0.63795086741914598182229839666991088459324643622494e-23), SC_(0.30128244387255120033790929863955407018683622956366e36), + SC_(0.13769532587968157644195235343886302517048270122046e-23), SC_(0.2007315487162486533634364604949951171875e-6), SC_(0.484821384427736035149791860021650791168212890625e-10), SC_(0.13804448026385854623972866111304415486550268685946e15), + SC_(0.1809900535358029636909945199492450026643175498009e-23), SC_(0.4521496521192602813243865966796875e-5), SC_(0.8669365225699721122509799897670745849609375e-10), SC_(0.16273113459057256254769276696724703797987842259227e14), + SC_(0.23768503319901658006414734019219381138884883419848e-23), SC_(0.545301093941248e15), SC_(0.1237108193663516431115567684173583984375e-9), SC_(0.10384732803004085499895888503014146907429299054875e4), + SC_(0.30307608021550191165014175615417981032514993522398e-23), SC_(0.112407951746718026697635650634765625e-5), SC_(0.2082685879058432e16), SC_(0.78504137547976820009624215245629935490941195781392e-21), + SC_(0.62520772668534154026944814518561754825703991045316e-23), SC_(0.19445130267235161963321360384e29), SC_(0.27158976561908717618255837411567199524142779409885e-16), SC_(0.79176099627370786959440226480012961415298742955765e3), + SC_(0.13102694856934221302627057835335012189043046859638e-22), SC_(0.557648090759990083584e22), SC_(0.511986668338959560742296162061393260955810546875e-10), SC_(0.78466137092077764354738487755677479556614922112935e0), + SC_(0.22752379870925977520487843277428559665698237779452e-22), SC_(0.169495918280704e15), SC_(0.106314858496e12), SC_(0.21650121297190654998249828534683860885914030587266e-17), + SC_(0.25335998731998849030698736206528910198942927678445e-22), SC_(0.99404931640625e3), SC_(0.204871795654296875e3), SC_(0.42132166564785080959023378471888313620164092769291e-3), + SC_(0.34646067293379381351152691325339738961930358129848e-22), SC_(0.26973976499394821796760860105960760360263583912399e-23), SC_(0.238645943284598871514390339143574237823486328125e-10), SC_(0.35432548942737632785957707186626141058866961729541e18), + SC_(0.71863877935764437100797607417958061393203905709015e-22), SC_(0.174722980453023744e19), SC_(0.13073324703744e14), SC_(0.1736004266338139270794293208784155064732837114576e-21), + SC_(0.96362472102334232055222076765519517715929964651878e-21), SC_(0.229194643907248973846435546875e-3), SC_(0.354709317473833607436972670257091522216796875e-9), SC_(0.55865475580945127826780252757531774517543358031385e12), + SC_(0.27589042212186362242123678200494962808875243354123e-20), SC_(0.16342684e8), SC_(0.10564504593401150001064081607182743027806282043457e-12), SC_(0.70232843197284835279286086968072241001475849391442e10), + SC_(0.10233203223084091273258880352287070181205308472272e-19), SC_(0.85809811133878005628972118756792042404413223266602e-13), SC_(0.231948544e9), SC_(0.21282358830663000454676497251299241238551176927446e-10), + SC_(0.18673345450030310831705831391483241255002667458029e-19), SC_(0.16140768350376832622251404282443946390469477480956e-24), SC_(0.35751782069791464758326094786154009108614104661683e-22), SC_(0.57362515961254887975882907105995424791809261862104e33), + SC_(0.21109476364529939751075359117074570214356299402425e-19), SC_(0.8756801014807582267041677127782781970980266472715e-25), SC_(0.19232578752860259808180970650021732008816410797181e-22), SC_(0.10040849239749431240730130421363049484424180456536e34), + SC_(0.48988976909583188869227858043553514022505623870529e-19), SC_(0.20838384443777613341808319091796875e-4), SC_(0.15278292e8), SC_(0.70563639547351817695354788729725173202298905165509e-9), + SC_(0.82997499668329372719087076888666576479636205476709e-19), SC_(0.13155479315039595949803718770027355301939497866925e-23), SC_(0.1327809695148788465823841420387907419353723526001e-14), SC_(0.32382452643409713657086228689471450756020929884496e24), + SC_(0.16677509527778160624086499150275919589603290660307e-18), SC_(0.50082130538252574789989333416135111329551950562359e-29), SC_(0.27393523857242675896004658903787332135948417999316e-20), SC_(0.26379890286497467800536742716267974869467674676907e31), + SC_(0.30420727664434581210352358596193944606511649908498e-18), SC_(0.28656053473241627216339111328125e-4), SC_(0.94969911521617292598785838378394608128019171999767e-19), SC_(0.21152520435266024826733886337057191990575481859773e22), + SC_(0.75868695590633990953827528458397466692986199632287e-18), SC_(0.930611462144e12), SC_(0.18147922618313051311153033283257685840533790511131e-24), SC_(0.83768344778024722769204816337111831433373325848511e16), + SC_(0.79597584432992412060764095271814255738718202337623e-18), SC_(0.75974287571611502022633594876763844865744160884209e-30), SC_(0.20657736771171195551744e23), SC_(0.47403170903497016250215067728483228381506772219895e-31), + SC_(0.12784377253954858878954355810853016350847610738128e-17), SC_(0.687874899968e12), SC_(0.2496727752685546875e2), SC_(0.14487562897240575462420431915733354323986381404974e-6), + SC_(0.22468750405760049766485005051652734664457966573536e-17), SC_(0.20721947294077835977077484130859375e-4), SC_(0.33333534375e6), SC_(0.1891939096717128299814570834532131624242218004607e-6), + SC_(0.33297255999405977634352134242323728585688513703644e-17), SC_(0.31191365320704e14), SC_(0.2293932139873504638671875e0), SC_(0.23416581491449832804939246015659888605843645930834e-5), + SC_(0.35736448257653877024582131372021365223190514370799e-17), SC_(0.1626183676572159287754752e26), SC_(0.18501463382170069138510370976291596889495849609375e-11), SC_(0.40153851788485548350521938534047362493332593188358e0), + SC_(0.46871312598868006987253009842930850936681963503361e-17), SC_(0.3093168493023691324522496e26), SC_(0.16793731632787596740420928687663819571897812904515e-26), SC_(0.60797270634828688834833132685695209121001352766598e10), + SC_(0.57842544071306897015280826246375056598481023684144e-17), SC_(0.29870528793496387010009129880927503108978271484375e-12), SC_(0.2262512535863296e17), SC_(0.29646119268293442517889863887012770015886764481583e-22), + SC_(0.152616857880340486461855087352290638591512106359e-16), SC_(0.2803833618070566221119488e25), SC_(0.130162216009921394288539886474609375e-4), SC_(0.13764477533975189296577420989460030210624358668897e-6), + SC_(0.27022364375023134885640124780437076879024971276522e-16), SC_(0.27506543358959030582888269221280574638246675400223e-29), SC_(0.10302131894484992e19), SC_(0.11533202298793762115879808476821158493089208267767e-24), + SC_(0.51905646303429040628284185654450766378431580960751e-16), SC_(0.15583587e8), SC_(0.46741176605224609375e2), SC_(0.16258625638384341129980088810715602413277009050755e-4), + SC_(0.54374770505732477601443242143375300656771287322044e-16), SC_(0.4562802734375e3), SC_(0.10197021348467909816516131089894163913918803398168e-29), SC_(0.18861228040658158783485684274667884467513804993808e23), + SC_(0.60475637804894883005346617466102543403394520282745e-16), SC_(0.56145323516737172227849140410116390853545453865081e-18), SC_(0.64836257251954521052539348602294921875e-7), SC_(0.19426856083161503849129240968279840607620575171493e13), + SC_(0.17055543082513291604229710429763144929893314838409e-15), SC_(0.6943710148334503173828125e-1), SC_(0.61840874921367523613102701318666731822304427623749e-15), SC_(0.12070723069489973584177449356829135215953130632198e17), + SC_(0.31354745663454523796342954256033408455550670623779e-15), SC_(0.18973853599391077295877039432525634765625e-8), SC_(0.15741248433312655941040958396115456707775592803955e-13), SC_(0.38340329268483399676355291918194933462116017467071e19), + SC_(0.92657114094649205629505850367877428652718663215637e-15), SC_(0.32666126748015500425026402808725833892822265625e-10), SC_(0.62660384591791979530528422515089914668351411819458e-14), SC_(0.60469431091148756507917001500985998650828444618344e20), + SC_(0.10318525717298732830018437311991874594241380691528e-14), 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SC_(0.174148595112480631422976e24), SC_(0.187841462272e12), SC_(0.20361865097500372485155107117802705969273207578608e-24), + SC_(0.9007602981666816e16), SC_(0.25765624e9), SC_(0.1591604150272e14), SC_(0.19726797826086290726900599041820484051327569566171e-20), + SC_(0.18448819419086848e17), SC_(0.55460104261295954577309919031335994077380746603012e-16), SC_(0.154132149174272e16), SC_(0.13580314631955561794382801022694745071290705109491e-22), + SC_(0.41172763385266176e17), SC_(0.69560673828125e4), SC_(0.747194576263427734375e1), SC_(0.62793115296093377148750971856138022413722904539364e-10), + SC_(0.59877638017122304e17), SC_(0.53512100536178474827209150532780768116936087608337e-15), SC_(0.60865414403362585004788968196643994088311036341765e-26), SC_(0.67932276518492118905647587709216361710433902821356e13), + SC_(0.11384566732292096e18), SC_(0.58757351632535552e19), SC_(0.273440826416015625e3), SC_(0.2218197410675072553866425364806973969075261727874e-18), + SC_(0.128395461743607808e18), SC_(0.206836903384823477636694016e27), SC_(0.53310077419155277311801910400390625e-5), SC_(0.25213187167619695580929289478697105906035912959449e-18), + SC_(0.245924553149120512e18), SC_(0.12432241406593107741605394588021265875441186132822e-25), SC_(0.11698430441812742785454260394458960094979845402414e-30), SC_(0.15814359980865698013833243813020147773628412638471e21), + SC_(0.304309702915784704e18), SC_(0.530327327997952e15), SC_(0.10366431646728266850195727608041629252966231433675e-18), SC_(0.73346046074259210722247465233594194982314902683682e-6), + SC_(0.351564170056957952e18), SC_(0.39915753210320770987303899714279126384431587576396e-26), SC_(0.21695373623974084952946051325821291927687106275468e-28), SC_(0.16012914574357507234163722206932362300988672115721e20), + SC_(0.805980574786256896e18), SC_(0.7298279296875e4), SC_(0.135008262029312e15), SC_(0.24742374685179778771251059396546742229781662198914e-22), + SC_(0.1429457475085533184e20), SC_(0.14041162752e11), SC_(0.16802999673239973138530304e26), SC_(0.32121431604673931143271876891436876644565038320042e-36), + SC_(0.221162365880631296e20), SC_(0.4426536e8), SC_(0.89961878334825112180062577083017316681434749625623e-18), SC_(0.10108911246070441761891693525755853616606808949316e-3), + SC_(0.38083625921602912256e20), SC_(0.14043506688e11), SC_(0.13209346463532805303342509972708285204134881496429e-15), SC_(0.35692228203582006473843647627664557137685858827968e-6), + SC_(0.57692237126522896384e20), SC_(0.63429242800339125096797943115234375e-5), SC_(0.31824542466005171886425636864e29), SC_(0.55222293820695175519018810756922011735863676741191e-41), + SC_(0.6036870791327383552e20), SC_(0.7191220191232e13), SC_(0.68897207938789506442844867706298828125e-6), SC_(0.17346553888445511526875601632160113537907688737997e-12), + SC_(0.113674073964877447168e21), SC_(0.183615109375e6), SC_(0.404058591811917722225189208984375e-4), SC_(0.10330175955957541532693528313466256772187391201874e-9), + SC_(0.140771054246300745728e21), SC_(0.4364620208740234375e2), SC_(0.10711124272997777587041261201942927950011067906889e-29), SC_(0.36980588192743195268081186143904811424184826509079e5), + SC_(0.486625867239520206848e21), SC_(0.2663514463837359985504355062296832912238642165903e-19), SC_(0.4719122320543576863524228422930565122003386188676e-26), SC_(0.12125027886646699970512476707156771853870273067952e14), + SC_(0.1146824826980038344704e22), SC_(0.123134608e9), SC_(0.36464645756234167146109012677852867501115952109103e-20), SC_(0.13220462423972425935917229500420846663825040326569e-3), + SC_(0.317074383652647862272e22), SC_(0.89612589356621110444032e24), SC_(0.2342108211970048e16), SC_(0.11619224315234818923589971905969099280156494799642e-29), + SC_(0.32148523727519219712e22), SC_(0.60515264161661452673922225736467336588411333742066e-25), SC_(0.787468255232e13), SC_(0.67190469656717598102688231195715810897773628309487e-23), + SC_(0.13571258581209030066176e23), SC_(0.2301062643527984619140625e-1), SC_(0.14359374149799930542314996273489668965339660644531e-11), SC_(0.14166931188327778230186785722742287698171547666982e-3), + SC_(0.20821377315431516209152e23), SC_(0.31614863423832064e17), SC_(0.64014885e7), SC_(0.4621406894160877827384001215215505228565154941349e-22), + SC_(0.55817784918415767502848e23), SC_(0.229345916576958775296e21), SC_(0.181080293376e13), SC_(0.62303863493181421011759609358700084332302564282572e-27), + SC_(0.210179409744077661405184e24), SC_(0.775467603201754089842406683474462128114628768627e-26), SC_(0.1351436267417501696e19), SC_(0.48419735649202961080115073168670433174544124355214e-29), + SC_(0.277204771630374352584704e24), SC_(0.178649388253688812255859375e-1), SC_(0.1626075136e10), SC_(0.35041190711483516523923271898554740595024898229225e-20), + SC_(0.286204729096914532827136e24), SC_(0.18205785323743839398957788944244384765625e-7), SC_(0.6443661400749065109504e22), SC_(0.85231956001222048244789008910270023125100724496553e-33), + SC_(0.292186842467552253181952e24), SC_(0.8000304500511383121995759616e28), SC_(0.12346179962158203125e2), SC_(0.17659213899312742611429666455779433659781019844976e-25), + SC_(0.393077021751861389033472e24), SC_(0.85326592849126535957377352366076878413243909232699e-28), SC_(0.11113272845378330266896682587685063481330871582031e-11), SC_(0.43056670697313120117799317169619739766527357451448e1), + SC_(0.48688847917066659823616e24), SC_(0.12212552573408426924125600686592708205013835254249e-21), SC_(0.1269279595243605041993096779751049041351507185027e-17), SC_(0.33543623251839576830546760936703173774870805103267e7), + SC_(0.6959596827501176488460288e25), SC_(0.56396991515163736959758100092886750224006554485772e-29), SC_(0.164767579074269378772992e24), SC_(0.67539896107197358427396075668773079100885280216386e-35), + SC_(0.18820064668587267094216704e26), SC_(0.302210201308383830109960399568080902099609375e-9), SC_(0.33895318467300131244893458519885746926814107382597e-21), SC_(0.21606569536016446005127644096871253975571307187249e4), + SC_(0.66192871423119660955992064e26), SC_(0.65104581211365760324995297566040105840304400052279e-28), SC_(0.700861590985368820838630199432373046875e-9), SC_(0.5261182706092194133617741726329224767119336720926e-3), + SC_(0.9036469625899950631026688e26), SC_(0.690641880035400390625e1), SC_(0.1238969862461090087890625e0), SC_(0.3008677996342580895031853541445446250478520113248e-12), + SC_(0.351293559140227787146657792e27), SC_(0.92746285056e12), SC_(0.7325884342193603515625e1), SC_(0.61405412064015393151121436181259576895276598600887e-19), + SC_(0.48555353730121618377670656e27), SC_(0.41366469991877353664505423235416110752209856263632e-29), SC_(0.1080339869874636584075679427013641468012674864323e-29), SC_(0.42620833187540100646142696579397880259804927864477e17), + SC_(0.3196617981363007079661436928e28), SC_(0.357227936120807498809881508350372314453125e-8), SC_(0.190531306287766710738651454448699951171875e-7), SC_(0.19434029391873011348314670331341701583590531402094e-5), + SC_(0.4932620110638545248192561152e28), SC_(0.39087764918804168701171875e-1), SC_(0.37496308203098109226999139757157200603110425163322e-21), SC_(0.11157521852141686174291251786184006118424156415823e-1), + SC_(0.516425012058796509045456896e28), SC_(0.3000732163818651513042917353020530062720192427143e-28), SC_(0.28816120624542236328125e1), SC_(0.14487121655367107180122470700505306007568660707101e-13), + SC_(0.6724360036363488659828113408e28), SC_(0.39914275e7), SC_(0.5617260831058956682682037353515625e-4), SC_(0.24432489934783656632287527852861417375460919006681e-14), + SC_(0.7376334675354954747676196864e28), SC_(0.3832167254355459644063744e25), SC_(0.86986000872677090266227306614155168063007295131683e-15), SC_(0.60499813791534240550577796444003132563462392194689e-18), + SC_(0.1240879712420966629044649984e29), SC_(0.35030939907262229474824187104684165838808240778235e-22), SC_(0.23553908385443359223970771765266363217961043119431e-16), SC_(0.11419947735685798535870442055414653068651348519666e4), + SC_(0.13622372113626680630865559552e29), SC_(0.5065993732313771067901500902430633743733778828755e-18), SC_(0.1568245533434264871175400912761688232421875e-11), SC_(0.1638076044611111663076389515306947175644793927095e-1), + SC_(0.21413030105610117647127543808e29), SC_(0.56796892335439872e17), SC_(0.44545595303495604738852208133448318250202646595426e-19), SC_(0.40758436912717219001309081424162965116278950127384e-12), + SC_(0.240611655851932728105631744e29), SC_(0.15126033260477136128765155826087636266509091563425e-30), SC_(0.147685372829437255859375e1), SC_(0.13095601170876630580259197493117307804833462447144e-13), + SC_(0.61731988312258099732254031872e29), SC_(0.47155488899394500240213479534565823420775918905622e-28), SC_(0.4463543296e10), SC_(0.27051189723173151307626070067360099122375990020391e-23) + }; +#undef SC_ + + diff --git a/test/ellint_rf_data.ipp b/test/ellint_rf_data.ipp new file mode 100644 index 000000000..56d8b4964 --- /dev/null +++ b/test/ellint_rf_data.ipp @@ -0,0 +1,415 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Test data for RF, each row contains in order: +// +// x, y, z, RF(x, y, z) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 401> ellint_rf_data = { + SC_(0.60682196926171104626767727515008970841312337712241e-30), SC_(0.20031258624e11), SC_(0.10313978271484375e4), SC_(0.69081557947567857896297308369623231790841946303168e-4), + SC_(0.13810928266794127506964991575307193802925529863273e-29), SC_(0.2529551275074481964111328125e-3), SC_(0.7030536597341097149183042347431182861328125e-8), SC_(0.41696748464385961137940994051143282877208328256968e3), + SC_(0.14532352659780781796355564562931230420316401288775e-29), SC_(0.62687453931329231730011467674071804187172354186889e-30), SC_(0.26256847716031594952568184453411959111690521240234e-12), SC_(0.40492940437281068910143789937423107620726714449588e8), + SC_(0.21161119528926488937912315394561157356168786172334e-29), SC_(0.142681614079265273176133632659912109375e-6), SC_(0.1414351724088191986083984375e-1), SC_(0.60023495609336663619731475310462707680628111629781e2), + SC_(0.43727979838159528482554219689363356095561935561375e-29), SC_(0.7798122348544e13), SC_(0.8588943201947652980736e22), SC_(0.12728379993876013222161727164717897397898227135914e-9), + SC_(0.44767214645416419053919245787147718165594030426536e-29), SC_(0.26553176031232e14), SC_(0.2085586588137791946752e23), SC_(0.80511870764678968189176930995818394321307940480734e-10), + SC_(0.60078474312334545656635060929362382794743178141472e-29), SC_(0.37131016039637643189053051173686981201171875e-8), SC_(0.32517390764125836699536241120728208287005145393778e-20), SC_(0.25056280362165428100075135368904610528487634651412e6), + SC_(0.61834249098451553748337548320711315606925672313013e-29), SC_(0.15125532639232e16), SC_(0.11666782739894188125617802143096923828125e-6), SC_(0.69024104942927043057057398977626412595781610122053e-6), + SC_(0.80490202086679616440767211156605911864354431197377e-29), SC_(0.263248904192e12), SC_(0.297980882347216843954978816e27), SC_(0.10843189731083616299141336159195842305986739069878e-11), + SC_(0.15386248307178551019952166983379913769903023636025e-28), SC_(0.11083928e9), SC_(0.210290006361901760101318359375e-2), SC_(0.13041677890491263355196952840107362001464424225232e-2), + SC_(0.21260283480045472585820552908849842179312334994757e-28), SC_(0.117237625e7), SC_(0.28658838e8), SC_(0.56137169413466917815737398640168522533687819147944e-3), + SC_(0.23067565607429071342632211210292164463707216326199e-28), SC_(0.44169701635837554931640625e-1), SC_(0.14622103125e6), SC_(0.23255376087386493123620791606540418922162034060246e-1), + SC_(0.27790564235134654243337661318427908127768828411199e-28), SC_(0.21656921296023725949679903214975529301966616912978e-20), SC_(0.16200692698475904762744903564453125e-4), SC_(0.48848907341566731317828094161181483220489237197822e4), + SC_(0.3576658041416457175462348437019377323527317041928e-28), SC_(0.182191292416e12), SC_(0.355224928e9), SC_(0.1056145023109440334189908118197725710820599937924e-4), + SC_(0.45190074381881787947113833644192369803258850957835e-28), SC_(0.903221837196690095865856e24), SC_(0.197033285648384e15), SC_(0.1316234260403974959059386181123491622884455028604e-10), + SC_(0.75787209056271355952485380019348769132814133643247e-28), SC_(0.3443290234375e5), SC_(0.87738095954812459262649215691458087947041111220869e-24), SC_(0.18482788150207164039886687903794355893352157342876e0), + SC_(0.83892250504513065845701871993405683214864787598623e-28), SC_(0.1116302655645995400846004486083984375e-5), SC_(0.5191992844812585795584e22), SC_(0.46130757625625887687399713890619646292026008752597e-9), + SC_(0.10848934303015863927446282468623299997624646840919e-27), SC_(0.62221619626030569058053553009209074892000950850727e-30), SC_(0.343400804652490021666816e24), 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SC_(0.30069886354161477220500208874734175760615704540309e-11), + SC_(0.34176589422481911700258816e26), SC_(0.2594374745967797935009002685546875e-4), SC_(0.32639631015324299667668128677178174257278442382813e-11), SC_(0.61686726715099850869664269896236478457721173132798e-11), + SC_(0.66192871423119660955992064e26), SC_(0.65104581211365760324995297566040105840304400052279e-28), SC_(0.700861590985368820838630199432373046875e-9), SC_(0.51196495026574394146099382874482062017583142405701e-11), + SC_(0.8845430532586596087103488e26), SC_(0.19398925304412841796875e1), SC_(0.41011344364960677921772003173828125e-6), SC_(0.32883289253336466203877579763959792657158933119763e-11), + SC_(0.9036469625899950631026688e26), SC_(0.690641880035400390625e1), SC_(0.1238969862461090087890625e0), SC_(0.31745434338722854630828915646448539066116092136479e-11), + SC_(0.26160499139176195703177216e27), SC_(0.25551566240720070127823149164214555639773607254028e-14), SC_(0.2455513858048e13), SC_(0.10841986934216006931865443143259911799200916785229e-11), + SC_(0.351293559140227787146657792e27), SC_(0.92746285056e12), SC_(0.7325884342193603515625e1), SC_(0.96945165287587645794032553087381848752888109990602e-12), + SC_(0.48555353730121618377670656e27), SC_(0.41366469991877353664505423235416110752209856263632e-29), SC_(0.1080339869874636584075679427013641468012674864323e-29), SC_(0.29736856860648111150067984819515308053148330813485e-11), + SC_(0.576385747842019784963129344e27), SC_(0.42413690615195065447551314719021320343017578125e-10), SC_(0.429480374875002013368430198170244693756103515625e-10), SC_(0.18094440824622577258491088407553210906682687995255e-11), + SC_(0.861307183935277657013354496e27), SC_(0.13938332907855510711669921875e-1), SC_(0.68459855298386769923242872813539959368599538368623e-30), SC_(0.11766749224716932927375683670462877212205767926654e-11), + SC_(0.105505156605401800151924736e28), SC_(0.36789907058298876307844693656079471111297607421875e-11), SC_(0.10516135122406685860405589982143287428734135247055e-23), SC_(0.14057878532181048238381113039649629348002517816159e-11), + SC_(0.3196617981363007079661436928e28), SC_(0.357227936120807498809881508350372314453125e-8), SC_(0.190531306287766710738651454448699951171875e-7), SC_(0.73543482163433806936996915244342283274029549975336e-12), + SC_(0.4932620110638545248192561152e28), SC_(0.39087764918804168701171875e-1), SC_(0.37496308203098109226999139757157200603110425163322e-21), SC_(0.49677918885669315094834554588390832782238505953762e-12), + SC_(0.516425012058796509045456896e28), SC_(0.3000732163818651513042917353020530062720192427143e-28), SC_(0.28816120624542236328125e1), SC_(0.45590953716748809622767271344568834414324208839861e-12), + SC_(0.6724360036363488659828113408e28), SC_(0.39914275e7), SC_(0.5617260831058956682682037353515625e-4), SC_(0.3149211786161963517965091650354250566920868130489e-12), + SC_(0.7376334675354954747676196864e28), SC_(0.3832167254355459644063744e25), SC_(0.86986000872677090266227306614155168063007295131683e-15), SC_(0.60174664703890956134939928112815630373518374796924e-13), + SC_(0.1240879712420966629044649984e29), SC_(0.35030939907262229474824187104684165838808240778235e-22), SC_(0.23553908385443359223970771765266363217961043119431e-16), SC_(0.47464339774120723669000561802171052279059315153073e-12), + SC_(0.13622372113626680630865559552e29), SC_(0.5065993732313771067901500902430633743733778828755e-18), SC_(0.1568245533434264871175400912761688232421875e-11), SC_(0.40583512551936497170672012442547588122036553493224e-12), + SC_(0.21413030105610117647127543808e29), SC_(0.56796892335439872e17), SC_(0.44545595303495604738852208133448318250202646595426e-19), SC_(0.10055266250578662063485344934342287093203087991764e-12), + SC_(0.240611655851932728105631744e29), SC_(0.15126033260477136128765155826087636266509091563425e-30), SC_(0.147685372829437255859375e1), SC_(0.21832951144880765009500199074282822236333957933832e-12), + SC_(0.61731988312258099732254031872e29), SC_(0.47155488899394500240213479534565823420775918905622e-28), SC_(0.4463543296e10), SC_(0.94272971828494080864224866724511415301020102514649e-13), + SC_(0.74469943825056746033385570304e29), SC_(0.25381451890023692523876122624e29), SC_(0.353061568e9), SC_(0.73993207202984769111475156231178550093712585836288e-14), + SC_(0.117661009908837424815142862848e30), SC_(0.434518829345703125e3), SC_(0.429355800151824951171875e0), SC_(0.9266881807310148034392861576649650379030278607074e-13), + SC_(0.593089902380358100434877939712e30), SC_(0.6506956228718475954142519412926048971712589263916e-14), SC_(0.61562469482421875e2), SC_(0.43634363152950073747311554335524072760111106270713e-13), + SC_(0.1138966491359350198995104301056e31), SC_(0.16628866613248e14), SC_(0.83722254681922700089433337836521478725337885862245e-21), SC_(0.19460821722924338220942178264562269988472467417183e-13) + }; +#undef SC_ + diff --git a/test/ellint_rj_data.ipp b/test/ellint_rj_data.ipp new file mode 100644 index 000000000..5833fe475 --- /dev/null +++ b/test/ellint_rj_data.ipp @@ -0,0 +1,816 @@ +// Copyright (c) 2006 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Test data for RF, each row contains in order: +// +// x, y, z, p, RF(x, y, z, p) +// +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 801> ellint_rj_data = { + SC_(0.17778719640226356167347852945229440308832650104108e-30), SC_(0.140657017848070144e19), SC_(0.1004598712921142578125e2), SC_(-0.482979822802320768460049293935298919677734375e-9), SC_(-0.25179478604244669733973586875090503552089558764187e-9), + SC_(0.17971096343704528451751984491965084147481741634467e-30), SC_(0.89101844176293009628514463216220065078232437372208e-17), SC_(0.83764338663550372334498751058362691151424505491344e-28), SC_(0.25525904358368192101317869568e29), SC_(0.55264196567924337745014728862252651162552511364961e-18), + SC_(0.3374483874642825322028299383742755000492569610633e-30), SC_(0.46577155087788907954176e23), SC_(0.46983051300048828125e0), SC_(-0.4337557424349824941600672900676727294921875e-8), SC_(-0.29586494691174345632763961304004697052773087913144e-10), + SC_(0.34628584803331255547073953716003532083475191421235e-30), SC_(0.303861990424576e15), SC_(0.4400977e7), SC_(0.8120083456e10), SC_(0.94406118682435306969091503176397574982473232273954e-16), + SC_(0.52529654597568598687401224634973212611437919511155e-30), SC_(0.58548273201185353977823232e26), SC_(0.20248201901919065867629932142790494253858923912048e-14), SC_(-0.18734671312196414516097698394344282521492941247926e-27), SC_(0.33644488398603828475913952425044358424278852139661e8), + SC_(0.62221619626030569058053553009209074892000950850727e-30), SC_(0.343400804652490021666816e24), SC_(0.67300522327423095703125e0), SC_(-0.27625999903489739661921475999406538903713226318359e-12), SC_(-0.75889847310956721783906856673041055044273454295232e-11), + SC_(0.62687453931329231730011467674071804187172354186889e-30), SC_(0.26256847716031594952568184453411959111690521240234e-12), SC_(0.1063940752263547436662784e25), SC_(-0.136451337486336e15), SC_(-0.67037231980991720071044018087634320636670369191284e-24), + SC_(0.68459855298386769923242872813539959368599538368623e-30), SC_(0.3576658041416457175462348437019377323527317041928e-28), SC_(0.182191292416e12), SC_(-0.363291968e9), SC_(-0.83523497967218414236437029496187282518142685409801e-12), + SC_(0.79319377739814275716610132815898742724164603635128e-30), SC_(0.149536081295309486449696123600006103515625e-7), SC_(0.110968975e7), SC_(0.39225603641135470576542739402404764287313199133678e-22), SC_(0.58404081910170394697384547096361115025170471880191e13), + SC_(0.89048248588914662625862647642457234630028429798521e-30), SC_(0.19709305012440588607850390271996382609153184795395e-27), SC_(0.11493865037410894201880001018568858790897452128896e-30), SC_(-0.12642370874130226532372489600675180554389953613281e-11), SC_(-0.63945430527093703240104260994490070888470387853874e27), + SC_(0.91680349711592282765605796247359202706042027867974e-30), SC_(0.734585367784587264e18), SC_(0.1052182232419898336993917357499253478604839479385e-26), SC_(0.2412866171048483705123448575434206360597333418383e-23), SC_(0.65770139664720747088672501472576668594088104855097e16), + SC_(0.10737441240523986959769665275538259586246072191374e-29), SC_(0.17000503652400689169610359385842457413673400878906e-11), SC_(0.18877310806851385273422393249673449514610540367343e-24), SC_(-0.11655765639250777695895033508093171243654281372297e-28), SC_(0.43166762266635358600263434023771743134211282555865e33), + SC_(0.11714106834475308811660531464919355963339235126279e-29), SC_(0.24252292633056640625e2), SC_(0.367521736353615183872e22), SC_(-0.4790649536162908709687890753434018342726830042011e-21), SC_(0.22701967082264264796089141494292880179559115582815e-4), + SC_(0.15170719443124775223923085751802832491502232570833e-29), SC_(0.5680430348320442135445773601531982421875e-8), SC_(0.264763527397376e15), SC_(0.13735858601648942567408084869384765625e-5), SC_(0.46218215900161368547332976532325866494807961652724e0), + SC_(0.16414311592283418833973460759110207528131746805707e-29), SC_(0.2019777626058022108969416021903358834932440402099e-21), SC_(0.2589416226816e13), SC_(-0.8526492879801228497171905473805963993072509765625e-11), SC_(-0.28262895694754696636159676198309944171160827781306e7), + SC_(0.18514118780253326139548161460302589871739700953216e-29), SC_(0.386711545288562774658203125e-1), SC_(0.23784641790813854100861135520972311496734619140625e-12), SC_(-0.1573566764023808e16), SC_(-0.13857417194455984598251280402623707809865243033107e-12), + SC_(0.27003399803360749163257433233715641043057849874222e-29), SC_(0.4438962912167515363689288870188763384827534963506e-23), SC_(0.75457567458852864e17), SC_(-0.11436019672743120284552631460428528953343629837036e-13), SC_(-0.11008238028815716178378898314905171879133537621566e8), + SC_(0.27604985838665253543615615843832173935883006238474e-29), SC_(0.25638719488e11), SC_(0.441928942759635320832e22), SC_(0.880926163517870008945465087890625e-4), SC_(0.47167946924686267794681000601586718795332807589313e-13), + SC_(0.27727679237026680624547360688641283050751007274628e-29), SC_(0.574552528560161590576171875e-1), SC_(0.599083490669727325439453125e-1), SC_(0.326562033434624e15), SC_(0.59574348217068678836770193704601663452299198297862e-13), + SC_(0.30817991819236751066216748295709401603164490197841e-29), SC_(0.20504934709643312615865232140244245329085970297456e-17), SC_(0.73745083851314552802870354829278820574245401076041e-19), SC_(0.2719417170737870037555694580078125e-4), SC_(0.23633334716008312781302835692937431199979037634428e15), + SC_(0.40116172822941619726509594554900420026234064360422e-29), SC_(0.273342628705129491070976e24), SC_(0.13233334255616e14), SC_(-0.1860324374547417392022907733917236328125e-6), SC_(-0.41662640839102650251726859519844649467351936743988e-24), + SC_(0.41366469991877353664505423235416110752209856263632e-29), SC_(0.1080339869874636584075679427013641468012674864323e-29), SC_(0.31354745663454523796342954256033408455550670623779e-15), SC_(-0.55809874055512409540824592113494873046875e-7), SC_(-0.51464305863761418392864887639923614413758733102008e17), + SC_(0.42974403323689182985245685819973216614391367995564e-29), SC_(0.1226755567577137036837375827502683023340068757534e-15), SC_(0.73181544029618176e17), SC_(0.356656491756439208984375e0), SC_(0.57511100808304419237998619538080396492131651332701e-6), + SC_(0.43727979838159528482554219689363356095561935561375e-29), SC_(0.7798122348544e13), SC_(0.8588943201947652980736e22), SC_(-0.45831213261398661135397416775299529474035908904739e-23), SC_(0.52890106746769770489956114049541553965288744960747e-8), + SC_(0.44767214645416419053919245787147718165594030426536e-29), SC_(0.26553176031232e14), SC_(0.2085586588137791946752e23), SC_(0.790340763648e12), SC_(0.64331758941919291277431102504836936943662726638851e-23), + SC_(0.56396991515163736959758100092886750224006554485772e-29), SC_(0.164767579074269378772992e24), SC_(0.96442290460109492133966568871983326971530914306641e-14), SC_(-0.182207055936620593152e21), SC_(-0.16288586324488794352121434520748982365641475834901e-29), + SC_(0.71060984495908924431726061148159195715493970237986e-29), SC_(0.61842984e8), SC_(0.3555642664432525634765625e0), SC_(0.133759144e9), SC_(0.28834216749836541937606611695450951218706212147855e-10), + SC_(0.80490202086679616440767211156605911864354431197377e-29), SC_(0.263248904192e12), SC_(0.297980882347216843954978816e27), SC_(0.140897846221923828125e1), SC_(0.44824022549527823491543784090039049377556207542239e-18), + SC_(0.93229513968235881657927660254990574951111859851663e-29), SC_(0.366241894662380218505859375e-1), SC_(0.4340599750656e13), SC_(0.105478806095575292710985138176e30), SC_(0.24012124307552155152973805705552466780637562428371e-33), + SC_(0.98984072271443418184618106348812926220946298945177e-29), SC_(0.96973053952e11), SC_(0.1032632680728338890752e22), SC_(-0.34396671690046787261962890625e-2), SC_(-0.96271364111058320397683919234948561206704260148431e-21), + SC_(0.13796957375375282521258282185313922946375173684891e-28), SC_(0.603357184e9), SC_(0.56249721609616214201832207209008629433810710906982e-14), SC_(-0.58723545944948736e17), SC_(-0.58028697480534659570212041732835661494029465370265e-19), + SC_(0.14087878121271588812030448853437982084687159586205e-28), SC_(0.53508747383344354605481852928e29), SC_(0.62704322937005693765167776976277025369199691340327e-17), SC_(0.235499138513371136e18), SC_(0.22302880199358479320924395693872038333122707229792e-29), + SC_(0.19187669896268843882106199046212790333876075179759e-28), SC_(0.26183668812773328027354580171959241852164268493652e-13), SC_(0.5970707e7), SC_(0.3312188584361984e16), SC_(0.92017241034815941174045547113765004391798006057029e-17), + SC_(0.19696330564073048658818435820590984305819001943432e-28), SC_(0.32238440243425309843308409068705787565729120602853e-24), SC_(0.292186842467552253181952e24), SC_(-0.3806432217839128513100120064e28), SC_(-0.82513020195805722794885738700643372535526082646992e-37), + SC_(0.20092729076345475268161641680692456393259236283608e-28), SC_(0.5697601591236889362335205078125e-4), SC_(0.1162191569964638329110528e26), SC_(-0.16412856978298360348657569008246504688175448904985e-21), SC_(0.31839738515206016513063561009026484616216098410256e-2), + SC_(0.20426069933899679481125376462984716200236491481616e-28), SC_(0.3943929741312e13), SC_(0.23412235260009765625e2), SC_(0.10193842560436084643201024e26), SC_(0.21207853870411859766389282691659616682682417758342e-29), + SC_(0.22881844270846610652749196737333504046746971403743e-28), SC_(0.2179837304525471795325908264118197621428407728672e-16), SC_(0.86233940750171127054952112389596408092984347604215e-19), SC_(0.368157728e9), SC_(0.72529706319504487456480463098377420100634724772411e1), + SC_(0.27790564235134654243337661318427908127768828411199e-28), SC_(0.21656921296023725949679903214975529301966616912978e-20), SC_(0.16200692698475904762744903564453125e-4), SC_(0.1635257538336499151053522635820185529564874116204e-22), SC_(0.58938880546601329124832820085911559407364943754411e25), + SC_(0.38307908170359829668907275664710825900011835075561e-28), SC_(0.14482298173190688572731232852675020694732666015625e-10), SC_(0.28193880243627888738304e23), SC_(0.589604375e6), SC_(0.60047466992965579411135347476169935325115071814017e-15), + SC_(0.44431992244584514654678755926404518943295031559691e-28), SC_(0.48174394645500255940076463501836601110994706687052e-19), SC_(0.111553230384106831872e21), SC_(0.26075669569036418811408302358884491864634006858979e-29), SC_(0.41939587045610674859432459314508206490215441802069e15), + SC_(0.45190074381881787947113833644192369803258850957835e-28), SC_(0.903221837196690095865856e24), SC_(0.197033285648384e15), SC_(0.32623424794930855341590358875691890716552734375e-10), SC_(0.61845676096363676818434676933016847521984818056106e-13), + SC_(0.67043661646154515272860423358014245865689394675811e-28), SC_(0.27387009957888e14), SC_(0.11955467241431342253831366051651271417622290817867e-26), SC_(0.87310802944e11), SC_(0.28933511748345186860086749976012259395216961852377e-15), + SC_(0.72950579063556310785863719706888276620113808904403e-28), SC_(0.590493896484375e4), SC_(0.99392208896e11), SC_(-0.19527980804443359375e2), SC_(-0.16079545628777305110796074754669542322724883666308e-8), + SC_(0.94777901458212457182664258521518150736237789261152e-28), SC_(0.4428201e7), SC_(0.2530101420741656313673562390278253679619336276427e-31), SC_(0.26171386241912841796875e1), SC_(0.18205396527964645652297231995003573026647291521867e-1), + SC_(0.11153434660579423094016106021221904046901687488323e-27), SC_(0.750879536326465313322842121124267578125e-7), SC_(0.9310612177071408979145417106337845325469970703125e-12), SC_(0.31839368781345048800513961029245990630442975088954e-17), SC_(0.99763610438792567414792334839900304053722628109961e19), + SC_(0.11969815900999513257745596351318014772377229309672e-27), SC_(0.1668161242237629641861967998969556656319035425895e-21), SC_(0.2847195703125e5), SC_(-0.12113708519078755261729959946670476256258552893996e-18), SC_(-0.58469912606844089535572230231066119898125245972777e18), + SC_(0.14718026449967740426353456716434433271883469801847e-27), SC_(0.14995603510614981277719965857381914152826921874523e-28), SC_(0.1111803961616105472e19), SC_(0.508679587341021033353172242641448974609375e-8), SC_(0.12814521532382957625567079334028430774339053692899e2), + SC_(0.15284006704962108878625465914339150556459706251089e-27), SC_(0.1302969292737543582916259765625e-2), SC_(0.354480992801311143493632e24), SC_(0.37605104e9), SC_(0.18607859430748510805274307470148837042692731526225e-18), + SC_(0.16409516553343363785084154944027139734674650595365e-27), SC_(0.21342112026081242375155966654008687921617593019619e-20), SC_(0.78913960056218451777354516707418952137231826782227e-14), SC_(-0.1775841411699730088002979755401611328125e-6), SC_(-0.17015599078079798684750666220401593398087597887802e16), + SC_(0.18215559766104924690332214579658583558792903588607e-27), SC_(0.214253668673336505889892578125e-4), SC_(0.7301043754462531075178496e25), SC_(0.1068455829996046990704172685582307167351245880127e-12), SC_(0.11526214590948729311397784489864538809888995731212e-2), + SC_(0.20347543751535935665284243442334458135561822645851e-27), SC_(0.440987377729536e16), SC_(0.287391328125e5), SC_(-0.9624498046875e4), SC_(-0.12941824082139400213634919237534112886628905903587e-11), + SC_(0.21993065518257681430269719956396304729998221211967e-27), SC_(0.187684774264479346476449792e27), SC_(0.12580362244240480345597811588829353208161820898425e-23), SC_(0.22742198944091796875e2), 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SC_(0.6770315462223105083152407343178427212980572931536e-22), + SC_(0.32259278802051668555153276928e30), SC_(0.6638677978515625e3), SC_(0.11419584692467545095073688798947841860353946685791e-13), SC_(-0.914323027245700359344482421875e-4), SC_(-0.77167253067005495411786081658186819275011987939055e-17), + SC_(0.593089902380358100434877939712e30), SC_(0.6506956228718475954142519412926048971712589263916e-14), SC_(0.61562469482421875e2), SC_(0.2321270964224e13), SC_(0.21597485998566959444034291971333852159947917872266e-25), + SC_(0.653810657301073942950006751232e30), SC_(0.3684862263296e13), SC_(0.1658748045099008e17), SC_(-0.2043063195140096e16), SC_(-0.17992368419735800205714333900182443278381089486339e-30), + SC_(0.90054999923278052824918786048e30), SC_(0.5752800417419362304e19), SC_(0.23091611800228823082094034944e29), SC_(0.437946654856204986572265625e-1), SC_(0.20691242362670718543784273355620160692479713280658e-36), + SC_(0.1138966491359350198995104301056e31), SC_(0.16628866613248e14), SC_(0.83722254681922700089433337836521478725337885862245e-21), SC_(0.351686295552e12), SC_(0.16740484414913831836071532979123766364786418157375e-26), + SC_(0.1155650423248669340749283721216e31), SC_(0.55448366981994133391117331177699867339470074512064e-18), SC_(0.120595209390657634304e22), SC_(-0.31079530179668142456832e23), SC_(-0.20503537182647658707141982907149513184896420806907e-36) + }; +#undef SC_ + + diff --git a/test/erf_data.ipp b/test/erf_data.ipp new file mode 100644 index 000000000..d13707ad5 --- /dev/null +++ b/test/erf_data.ipp @@ -0,0 +1,510 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> erf_data = { { + { SC_(-7.954905033111572265625), SC_(-0.9999999999999999999999999999768236114552), SC_(1.999999999999999999999999999976823611455) }, + { SC_(-7.925852298736572265625), SC_(-0.9999999999999999999999999999631035087875), SC_(1.999999999999999999999999999963103508787) }, + { SC_(-7.923464298248291015625), SC_(-0.9999999999999999999999999999616689085769), SC_(1.999999999999999999999999999961668908577) }, + { SC_(-7.870497226715087890625), SC_(-0.999999999999999999999999999910929780403), SC_(1.999999999999999999999999999910929780403) }, + { SC_(-7.809566974639892578125), SC_(-0.9999999999999999999999999997666672740312), SC_(1.999999999999999999999999999766667274031) }, + { SC_(-7.78337383270263671875), SC_(-0.9999999999999999999999999996477997000614), SC_(1.999999999999999999999999999647799700061) }, + { SC_(-7.75354480743408203125), SC_(-0.9999999999999999999999999994380404842296), SC_(1.99999999999999999999999999943804048423) }, + { SC_(-7.735670566558837890625), SC_(-0.9999999999999999999999999992571019874276), SC_(1.999999999999999999999999999257101987428) }, + { SC_(-7.715617656707763671875), SC_(-0.9999999999999999999999999989846832415537), SC_(1.999999999999999999999999998984683241554) }, + { SC_(-7.7047863006591796875), SC_(-0.999999999999999999999999998798456908148), SC_(1.999999999999999999999999998798456908148) }, + { SC_(-7.53247547149658203125), SC_(-0.9999999999999999999999999830308685738632), SC_(1.999999999999999999999999983030868573863) }, + { SC_(-7.490674495697021484375), SC_(-0.9999999999999999999999999680283814121658), SC_(1.999999999999999999999999968028381412166) }, + { SC_(-7.4677028656005859375), SC_(-0.9999999999999999999999999547824309252063), SC_(1.999999999999999999999999954782430925206) }, + { SC_(-7.45092296600341796875), SC_(-0.9999999999999999999999999417916071317633), SC_(1.999999999999999999999999941791607131763) }, + { SC_(-7.448862552642822265625), SC_(-0.999999999999999999999999939960596846754), SC_(1.999999999999999999999999939960596846754) }, + { SC_(-7.428613185882568359375), SC_(-0.9999999999999999999999999186357837637381), SC_(1.999999999999999999999999918635783763738) }, + { SC_(-7.41693973541259765625), SC_(-0.9999999999999999999999999030902896226869), SC_(1.999999999999999999999999903090289622687) }, + { SC_(-7.352462291717529296875), SC_(-0.9999999999999999999999997466806221320354), SC_(1.999999999999999999999999746680622132035) }, + { SC_(-7.32713413238525390625), SC_(-0.9999999999999999999999996313499781458314), SC_(1.999999999999999999999999631349978145831) }, + { SC_(-7.311619281768798828125), SC_(-0.9999999999999999999999995363881484604713), SC_(1.999999999999999999999999536388148460471) }, + { SC_(-7.279047489166259765625), SC_(-0.9999999999999999999999992510468758087201), SC_(1.99999999999999999999999925104687580872) }, + { SC_(-7.261257648468017578125), SC_(-0.999999999999999999999999027617901396665), SC_(1.999999999999999999999999027617901396665) }, + { SC_(-7.232891559600830078125), SC_(-0.9999999999999999999999985274747006541921), SC_(1.999999999999999999999998527474700654192) }, + { SC_(-7.20552921295166015625), SC_(-0.9999999999999999999999978059072279635353), SC_(1.999999999999999999999997805907227963535) }, + { SC_(-7.18053722381591796875), SC_(-0.9999999999999999999999968458614521146074), SC_(1.999999999999999999999996845861452114607) }, + { SC_(-7.14955902099609375), SC_(-0.9999999999999999999999950624010083082766), SC_(1.999999999999999999999995062401008308277) }, + { SC_(-7.13679790496826171875), SC_(-0.9999999999999999999999940645544114571544), SC_(1.999999999999999999999994064554411457154) }, + { SC_(-7.043527126312255859375), SC_(-0.9999999999999999999999774337000610083234), SC_(1.999999999999999999999977433700061008323) }, + { SC_(-6.9184741973876953125), SC_(-0.9999999999999999999998683679500951423272), SC_(1.999999999999999999999868367950095142327) }, + { SC_(-6.7863311767578125), SC_(-0.9999999999999999999991795120414840889856), SC_(1.999999999999999999999179512041484088986) }, + { SC_(-6.784533023834228515625), SC_(-0.9999999999999999999991590255661336519112), SC_(1.999999999999999999999159025566133651911) }, + { SC_(-6.75908756256103515625), SC_(-0.9999999999999999999988086234364717344992), SC_(1.999999999999999999998808623436471734499) }, + { SC_(-6.7491912841796875), SC_(-0.9999999999999999999986362669470340916908), SC_(1.999999999999999999998636266947034091691) }, + { SC_(-6.701987743377685546875), SC_(-0.999999999999999999997409009041632256952), SC_(1.999999999999999999997409009041632256952) }, + { SC_(-6.658857822418212890625), SC_(-0.9999999999999999999953604766205454848685), SC_(1.999999999999999999995360476620545484869) }, + { SC_(-6.649026393890380859375), SC_(-0.9999999999999999999947043111854298902328), SC_(1.999999999999999999994704311185429890233) }, + { SC_(-6.63174724578857421875), SC_(-0.9999999999999999999933213242713558320894), SC_(1.999999999999999999993321324271355832089) }, + { SC_(-6.562829494476318359375), SC_(-0.9999999999999999999832486398825313868824), SC_(1.999999999999999999983248639882531386882) }, + { SC_(-6.474317073822021484375), SC_(-0.9999999999999999999461767620323245787117), SC_(1.999999999999999999946176762032324578712) }, + { SC_(-6.456727504730224609375), SC_(-0.9999999999999999999322505846330688168028), SC_(1.999999999999999999932250584633068816803) }, + { SC_(-6.44589138031005859375), SC_(-0.9999999999999999999219560784452381988458), SC_(1.999999999999999999921956078445238198846) }, + { SC_(-6.439353466033935546875), SC_(-0.9999999999999999999150123383981949772306), SC_(1.999999999999999999915012338398194977231) }, + { SC_(-6.388184070587158203125), SC_(-0.9999999999999999998348790206445813875893), SC_(1.999999999999999999834879020644581387589) }, + { SC_(-6.293555736541748046875), SC_(-0.9999999999999999994437019971919548391819), SC_(1.999999999999999999443701997191954839182) }, + { SC_(-6.2710094451904296875), SC_(-0.9999999999999999992589325173780715980195), SC_(1.99999999999999999925893251737807159802) }, + { SC_(-6.242211818695068359375), SC_(-0.9999999999999999989326493643877411691996), SC_(1.9999999999999999989326493643877411692) }, + { SC_(-6.22209262847900390625), SC_(-0.999999999999999998624109280849850203527), SC_(1.999999999999999998624109280849850203527) }, + { SC_(-6.220756053924560546875), SC_(-0.9999999999999999986007425210281758251292), SC_(1.999999999999999998600742521028175825129) }, + { SC_(-6.200567722320556640625), SC_(-0.9999999999999999981962301352174253788249), SC_(1.999999999999999998196230135217425378825) }, + { SC_(-6.188919544219970703125), SC_(-0.9999999999999999979123729253091532478511), SC_(1.999999999999999997912372925309153247851) }, + { SC_(-6.121317386627197265625), SC_(-0.9999999999999999951502004864042604960246), SC_(1.999999999999999995150200486404260496025) }, + { SC_(-6.0960369110107421875), SC_(-0.999999999999999993368492031514234286194), SC_(1.999999999999999993368492031514234286194) }, + { SC_(-6.08724498748779296875), SC_(-0.9999999999999999926083315666932664549236), SC_(1.999999999999999992608331566693266454924) }, + { SC_(-6.026896953582763671875), SC_(-0.9999999999999999844955556006137776564529), SC_(1.999999999999999984495555600613777656453) }, + { SC_(-5.9970760345458984375), SC_(-0.9999999999999999777013911828554459912853), SC_(1.999999999999999977701391182855445991285) }, + { SC_(-5.98565387725830078125), SC_(-0.9999999999999999743831687415390670278399), SC_(1.99999999999999997438316874153906702784) }, + { SC_(-5.96821117401123046875), SC_(-0.999999999999999968354398455085434720569), SC_(1.999999999999999968354398455085434720569) }, + { SC_(-5.92245578765869140625), SC_(-0.9999999999999999450648430800723663224774), SC_(1.999999999999999945064843080072366322477) }, + { SC_(-5.921500682830810546875), SC_(-0.9999999999999999444311537798288144041217), SC_(1.999999999999999944431153779828814404122) }, + { SC_(-5.888427257537841796875), SC_(-0.9999999999999999174281958873362911444154), SC_(1.999999999999999917428195887336291144415) }, + { SC_(-5.87206363677978515625), SC_(-0.9999999999999998996343433163177337705717), SC_(1.999999999999999899634343316317733770572) }, + { SC_(-5.860218048095703125), SC_(-0.9999999999999998844434106367807360220097), SC_(1.99999999999999988444341063678073602201) }, + { SC_(-5.83236789703369140625), SC_(-0.9999999999999998392204407202278816953224), SC_(1.999999999999999839220440720227881695322) }, + { SC_(-5.822903156280517578125), SC_(-0.9999999999999998201851297606348078114632), SC_(1.999999999999999820185129760634807811463) }, + { SC_(-5.7919788360595703125), SC_(-0.9999999999999997411398900200191738483065), SC_(1.999999999999999741139890020019173848306) }, + { SC_(-5.782009124755859375), SC_(-0.9999999999999997089916610994263590057672), SC_(1.999999999999999708991661099426359005767) }, + { SC_(-5.757698535919189453125), SC_(-0.9999999999999996131703510337485986984616), SC_(1.999999999999999613170351033748598698462) }, + { SC_(-5.7298183441162109375), SC_(-0.9999999999999994646206751994613491480939), SC_(1.999999999999999464620675199461349148094) }, + { SC_(-5.6807231903076171875), SC_(-0.9999999999999990546672380240389993944094), SC_(1.999999999999999054667238024038999394409) }, + { SC_(-5.67137622833251953125), SC_(-0.9999999999999989471628690116228142127541), SC_(1.999999999999998947162869011622814212754) }, + { SC_(-5.63473606109619140625), SC_(-0.9999999999999983967395492779530149899861), SC_(1.999999999999998396739549277953014989986) }, + { SC_(-5.614176273345947265625), SC_(-0.9999999999999979723800510435190721967745), SC_(1.999999999999997972380051043519072196774) }, + { SC_(-5.6112957000732421875), SC_(-0.9999999999999979047003987118507997439545), SC_(1.999999999999997904700398711850799743955) }, + { SC_(-5.56195163726806640625), SC_(-0.9999999999999963321711826091909555528649), SC_(1.999999999999996332171182609190955552865) }, + { SC_(-5.52898502349853515625), SC_(-0.999999999999994682538283828212635195642), SC_(1.999999999999994682538283828212635195642) }, + { SC_(-5.47819042205810546875), SC_(-0.9999999999999906156035801749079852647386), SC_(1.999999999999990615603580174907985264739) }, + { SC_(-5.4710788726806640625), SC_(-0.9999999999999898428944822493616106985002), SC_(1.9999999999999898428944822493616106985) }, + { SC_(-5.405083179473876953125), SC_(-0.9999999999999789329565303176610969074478), SC_(1.999999999999978932956530317661096907448) }, + { SC_(-5.402743816375732421875), SC_(-0.9999999999999783844591899263947698611185), SC_(1.999999999999978384459189926394769861119) }, + { SC_(-5.398212432861328125), SC_(-0.9999999999999772817558127269992185563875), SC_(1.999999999999977281755812726999218556388) }, + { SC_(-5.39404582977294921875), SC_(-0.9999999999999762190932060075070452678252), SC_(1.999999999999976219093206007507045267825) }, + { SC_(-5.349620342254638671875), SC_(-0.9999999999999613641252395001102021837028), SC_(1.999999999999961364125239500110202183703) }, + { SC_(-5.3191070556640625), SC_(-0.9999999999999462010476635137780577801192), SC_(1.999999999999946201047663513778057780119) }, + { SC_(-5.2961597442626953125), SC_(-0.9999999999999310743837318684932169157806), SC_(1.999999999999931074383731868493216915781) }, + { SC_(-5.2686710357666015625), SC_(-0.9999999999999073829182184635550018139178), SC_(1.999999999999907382918218463555001813918) }, + { SC_(-5.261013031005859375), SC_(-0.999999999999899463963517023463646064969), SC_(1.999999999999899463963517023463646064969) }, + { SC_(-5.218157291412353515625), SC_(-0.999999999999841220863178602997391812622), SC_(1.999999999999841220863178602997391812622) }, + { SC_(-5.09483242034912109375), SC_(-0.9999999999994203324744342031290577923874), SC_(1.999999999999420332474434203129057792387) }, + { SC_(-5.09044742584228515625), SC_(-0.9999999999993933525823679657367786914755), SC_(1.999999999999393352582367965736778691476) }, + { SC_(-5.0638217926025390625), SC_(-0.9999999999992009929816947679454761069971), SC_(1.999999999999200992981694767945476106997) }, + { SC_(-5.05747509002685546875), SC_(-0.9999999999991469518921467092111635858909), SC_(1.999999999999146951892146709211163585891) }, + { SC_(-5.04293918609619140625), SC_(-0.9999999999990093020187848982236516240641), SC_(1.999999999999009302018784898223651624064) }, + { SC_(-5.0100383758544921875), SC_(-0.9999999999986122073654265981749793751861), SC_(1.999999999998612207365426598174979375186) }, + { SC_(-4.98588848114013671875), SC_(-0.9999999999982250531272305424718207921526), SC_(1.999999999998225053127230542471820792153) }, + { SC_(-4.97671985626220703125), SC_(-0.999999999998051837006412113097549508585), SC_(1.999999999998051837006412113097549508585) }, + { SC_(-4.88807392120361328125), SC_(-0.9999999999952475475430397811200299233621), SC_(1.999999999995247547543039781120029923362) }, + { SC_(-4.883771419525146484375), SC_(-0.999999999995039278283384091602446410145), SC_(1.999999999995039278283384091602446410145) }, + { SC_(-4.85447597503662109375), SC_(-0.9999999999933632419317336295299664149692), SC_(1.999999999993363241931733629529966414969) }, + { SC_(-4.8071804046630859375), SC_(-0.9999999999894197712585954179441600183832), SC_(1.999999999989419771258595417944160018383) }, + { SC_(-4.8020343780517578125), SC_(-0.999999999988871990172530478547610656837), SC_(1.999999999988871990172530478547610656837) }, + { SC_(-4.67612361907958984375), SC_(-0.9999999999623486826561914023506448369931), SC_(1.999999999962348682656191402350644836993) }, + { SC_(-4.67091083526611328125), 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SC_(0.9999999999999999999999851346580378971058), SC_(0.1486534196210289422461831541803890066821e-22) }, + { SC_(7.11659526824951171875), SC_(0.9999999999999999999999920618538775044464), SC_(0.7938146122495553642615968497016709781571e-23) }, + { SC_(7.18280124664306640625), SC_(0.9999999999999999999999969477254268593726), SC_(0.3052274573140627399432973240733715500213e-23) }, + { SC_(7.20355319976806640625), SC_(0.9999999999999999999999977419301284355623), SC_(0.2258069871564437709331260570315365949665e-23) }, + { SC_(7.27909374237060546875), SC_(0.9999999999999999999999992515556916792261), SC_(0.7484443083207738734503483608935021035732e-24) }, + { SC_(7.28028106689453125), SC_(0.9999999999999999999999992645004451770347), SC_(0.7354995548229653146869970523781527988642e-24) }, + { SC_(7.298152923583984375), SC_(0.9999999999999999999999994345631928770567), SC_(0.5654368071229432894992332067407147179749e-24) }, + { SC_(7.31467151641845703125), SC_(0.9999999999999999999999995568116259996168), SC_(0.4431883740003831823890539135951501761503e-24) }, + { SC_(7.32010936737060546875), SC_(0.9999999999999999999999995910130381669911), SC_(0.4089869618330089157685151201949087925479e-24) }, + { SC_(7.34866237640380859375), SC_(0.9999999999999999999999997319906448357093), SC_(0.268009355164290673339087031887057715089e-24) }, + { SC_(7.351879119873046875), SC_(0.9999999999999999999999997444791197223142), SC_(0.2555208802776857586959896180952785091403e-24) }, + { SC_(7.35590362548828125), SC_(0.9999999999999999999999997592943074733691), SC_(0.2407056925266308623391590746789683246709e-24) }, + { SC_(7.390369415283203125), SC_(0.9999999999999999999999998558663763263787), SC_(0.1441336236736213391389793807586759175995e-24) }, + { SC_(7.43821620941162109375), SC_(0.9999999999999999999999999295502581038212), SC_(0.7044974189617879705218727765406635153304e-25) }, + { SC_(7.43946170806884765625), SC_(0.9999999999999999999999999308550578406467), SC_(0.6914494215935333790398074475111218230458e-25) }, + { SC_(7.48311901092529296875), SC_(0.999999999999999999999999964163218913179), SC_(0.3583678108682102561152351850893798856277e-25) }, + { SC_(7.49824619293212890625), SC_(0.9999999999999999999999999714868908856129), SC_(0.2851310911438713788012919629820992423858e-25) }, + { SC_(7.50188446044921875), SC_(0.9999999999999999999999999730141507157818), SC_(0.2698584928421822020035304290073321597406e-25) }, + { SC_(7.52948474884033203125), SC_(0.9999999999999999999999999822420651220879), SC_(0.1775793487791212989839150755243583008096e-25) }, + { SC_(7.6320323944091796875), SC_(0.9999999999999999999999999962984458271673), SC_(0.3701554172832712513006017529906907914499e-26) }, + { SC_(7.6759738922119140625), SC_(0.9999999999999999999999999981215459672183), SC_(0.1878454032781689292876320094611621328422e-26) }, + { SC_(7.67881011962890625), SC_(0.999999999999999999999999998202249704333), SC_(0.1797750295667047232713633441478606640198e-26) }, + { SC_(7.69775485992431640625), SC_(0.9999999999999999999999999986598159749852), SC_(0.1340184025014838921678286365792744470017e-26) }, + { SC_(7.70757007598876953125), SC_(0.9999999999999999999999999988493273258458), SC_(0.1150672674154231530685408186230565876878e-26) }, + { SC_(7.71776866912841796875), SC_(0.9999999999999999999999999990181051135537), SC_(0.9818948864462603043818495768683861311821e-27) }, + { SC_(7.79935359954833984375), SC_(0.9999999999999999999999999997259876414039), SC_(0.2740123585961319556857387807076098785701e-27) }, + { SC_(7.81407070159912109375), SC_(0.9999999999999999999999999997826446873284), SC_(0.2173553126715507615629930753265485721298e-27) }, + { SC_(7.81634521484375), SC_(0.9999999999999999999999999997902963473516), SC_(0.209703652648431358354620963217352346183e-27) }, + { SC_(7.84175968170166015625), SC_(0.9999999999999999999999999998595912638039), SC_(0.1404087361960659424532579514219630734144e-27) }, + { SC_(7.88610076904296875), SC_(0.9999999999999999999999999999304798650663), SC_(0.6952013493370366965787168537834557597352e-28) }, + { SC_(7.89655590057373046875), SC_(0.9999999999999999999999999999411317291543), SC_(0.5886827084565561878281328421149862522402e-28) }, + { SC_(7.9050960540771484375), SC_(0.9999999999999999999999999999486179207741), SC_(0.513820792259039401977061154556515432295e-28) }, + { SC_(7.93815517425537109375), SC_(0.9999999999999999999999999999696918627742), SC_(0.3030813722580726383815315462367670870671e-28) }, + { SC_(7.94338130950927734375), SC_(0.9999999999999999999999999999721239170155), SC_(0.2787608298449905464917225531595312562447e-28) } + } }; +#undef SC_ + diff --git a/test/erf_inv_data.ipp b/test/erf_inv_data.ipp new file mode 100644 index 000000000..b3ee380cc --- /dev/null +++ b/test/erf_inv_data.ipp @@ -0,0 +1,110 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> erf_inv_data = { { + { SC_(-0.990433037281036376953125), SC_(-1.832184533179510927322805923563700329767) }, + { SC_(-0.936334311962127685546875), SC_(-1.311339282092737086640055105484822812599) }, + { SC_(-0.931107819080352783203125), SC_(-1.286316461685184857889337272829270644576) }, + { SC_(-0.928576648235321044921875), SC_(-1.274755355308702535979544636706295190547) }, + { SC_(-0.92711746692657470703125), SC_(-1.268242390461126936128446743938528753699) }, + { SC_(-0.907657206058502197265625), SC_(-1.190178701802009872651528128114629153367) }, + { SC_(-0.89756715297698974609375), SC_(-1.154826929034364581020764782706068245251) }, + { SC_(-0.80573642253875732421875), SC_(-0.917873491512746130598510739795171676962) }, + { SC_(-0.804919183254241943359375), SC_(-0.9161942446913841771580833184012189161559) }, + { SC_(-0.780276477336883544921875), SC_(-0.867806431357551134410815525153998291827) }, + { SC_(-0.775070965290069580078125), SC_(-0.8580919924152284867224297625328485999768) }, + { SC_(-0.7496345043182373046875), SC_(-0.8127924290810407931222432768905514928867) }, + { SC_(-0.74820673465728759765625), SC_(-0.8103475955423936417307157186107256388648) }, + { SC_(-0.74602639675140380859375), SC_(-0.806632688757205819001462030577472890805) }, + { SC_(-0.72904598712921142578125), SC_(-0.7784313874823551106598826200232666844639) }, + { SC_(-0.7162272930145263671875), SC_(-0.7579355487144890063429377586715787838945) }, + { SC_(-0.701772034168243408203125), SC_(-0.7355614373173595299453301005770428516518) }, + { SC_(-0.68477380275726318359375), SC_(-0.7101588399949052872532446197423489724803) }, + { SC_(-0.657626628875732421875), SC_(-0.6713881533266128640126408255047881638389) }, + { SC_(-0.652269661426544189453125), SC_(-0.6639738263692763456669198307149427734317) }, + { SC_(-0.6262547969818115234375), SC_(-0.6289572925573171740836428308746584439674) }, + { SC_(-0.62323606014251708984375), SC_(-0.6249936843093662471116097431474787933967) }, + { SC_(-0.57958185672760009765625), SC_(-0.5697131213589784467617578394703976041604) }, + { SC_(-0.576151371002197265625), SC_(-0.5655172244109430153330195150336141500139) }, + { SC_(-0.5579319000244140625), SC_(-0.5435569422159360687847790186563654276103) }, + { SC_(-0.446154057979583740234375), SC_(-0.4186121208546731033057205459902879301199) }, + { SC_(-0.44300353527069091796875), SC_(-0.4152898953738801984047941692529271391195) }, + { SC_(-0.40594112873077392578125), SC_(-0.3768620801611051992528860948080812212023) }, + { SC_(-0.396173775196075439453125), SC_(-0.3669220210390825311547962776125822899061) }, + { SC_(-0.38366591930389404296875), SC_(-0.3542977152760563782151668726041057557165) }, + { SC_(-0.36689913272857666015625), SC_(-0.3375493847053488720470432821496358279516) }, + { SC_(-0.365801036357879638671875), SC_(-0.3364591774366710656954166264654945559873) }, + { SC_(-0.277411997318267822265625), SC_(-0.2510244067029671790889794981353227476998) }, + { SC_(-0.236883103847503662109375), SC_(-0.213115119330839975829499967157244997714) }, + { SC_(-0.215545952320098876953125), SC_(-0.1934073617841803428235669261097060281642) }, + { SC_(-0.202522933483123779296875), SC_(-0.1814532246720147926398927046057793150106) }, + { SC_(-0.18253767490386962890625), SC_(-0.1632073953550647568421821286058243218715) }, + { SC_(-0.156477451324462890625), SC_(-0.1395756320903277910768376053314442757507) }, + { SC_(-0.1558246612548828125), SC_(-0.1389857795955756484965030151195660030168) }, + { SC_(-0.12251126766204833984375), SC_(-0.109002961098867662134935094105847496074) }, + { SC_(-0.1088275909423828125), SC_(-0.09674694516640724629590870677194632943569) }, + { SC_(-0.08402168750762939453125), SC_(-0.07460044047654119877070700345343119035515) }, + { SC_(-0.05048263072967529296875), SC_(-0.04476895818328636312384562686715995170129) }, + { SC_(-0.029248714447021484375), SC_(-0.0259268064334840921104659134138093242797) }, + { SC_(-0.02486217021942138671875), SC_(-0.02203709146986755832638577823832075055744) }, + { SC_(-0.02047121524810791015625), SC_(-0.01814413302702029459097557481591610553903) }, + { SC_(-0.018821895122528076171875), SC_(-0.01668201759439857888105181293763417899072) }, + { SC_(0.0073254108428955078125), SC_(0.006492067534753215749601670217642082465642) }, + { SC_(0.09376299381256103515625), SC_(0.08328747254794857150987333986733043734817) }, + { SC_(0.0944411754608154296875), SC_(0.08389270963798942778622198997355058545872) }, + { SC_(0.264718532562255859375), SC_(0.2390787735821979528028028789569770109829) }, + { SC_(0.27952671051025390625), SC_(0.2530214201700340392837551955289041822603) }, + { SC_(0.29262602329254150390625), SC_(0.2654374523135675523971788948011709467352) }, + { SC_(0.3109557628631591796875), SC_(0.282950508503826367238408926581528085458) }, + { SC_(0.31148135662078857421875), SC_(0.2834552014554130441860525970673030809536) }, + { SC_(0.32721102237701416015625), SC_(0.2986277427848421570858990348074985028421) }, + { SC_(0.3574702739715576171875), SC_(0.3282140305634627092431945088114761850208) }, + { SC_(0.362719058990478515625), SC_(0.3334035993712283467959295804559099468454) }, + { SC_(0.3896572589874267578125), SC_(0.3603304982893212173104266596348905175268) }, + { SC_(0.4120922088623046875), SC_(0.3831602323665075533579267768785894144888) }, + { SC_(0.41872966289520263671875), SC_(0.3899906753567599452444107492361433402154) }, + { SC_(0.45167791843414306640625), SC_(0.4244594733907945411184647153213164209335) }, + { SC_(0.48129451274871826171875), SC_(0.4563258063707025027210352963461819167707) }, + { SC_(0.4862649440765380859375), SC_(0.4617640058971775089811390737537561779898) }, + { SC_(0.50937330722808837890625), SC_(0.4874174763856674076219106695373814892182) }, + { SC_(0.5154802799224853515625), SC_(0.4943041993872143888987628020569772222018) }, + { SC_(0.52750003337860107421875), SC_(0.5079978091910991117615000459548117088362) }, + { SC_(0.53103363513946533203125), SC_(0.5120597685873370942783226077302881881069) }, + { SC_(0.58441460132598876953125), SC_(0.5756584292527058478710392476034273328569) }, + { SC_(0.5879499912261962890625), SC_(0.5800336103175463592377907341030447077804) }, + { SC_(0.59039986133575439453125), SC_(0.5830784871670823806198622501806646778319) }, + { SC_(0.59455978870391845703125), SC_(0.588273673825686998734497652983815773459) }, + { SC_(0.59585726261138916015625), SC_(0.5899005483108011364541949539839185473259) }, + { SC_(0.5962116718292236328125), SC_(0.5903454775096607218832535637355431851718) }, + { SC_(0.6005609035491943359375), SC_(0.5958247243549040349587326482492767206448) }, + { SC_(0.6150619983673095703125), SC_(0.6143583249050861028039832921829036722514) }, + { SC_(0.62944734096527099609375), SC_(0.6331707263097125575937994856370309207836) }, + { SC_(0.64380657672882080078125), SC_(0.6524069265890823819975498133014027247554) }, + { SC_(0.6469156742095947265625), SC_(0.656635855345815020063728463464436343698) }, + { SC_(0.67001712322235107421875), SC_(0.6888269167957872563013714303376548038671) }, + { SC_(0.6982586383819580078125), SC_(0.7302336318927408409119676651737758401138) }, + { SC_(0.74485766887664794921875), SC_(0.8046505193013635090578266413458426260098) }, + { SC_(0.75686132907867431640625), SC_(0.8253191678260588578995203396384711816647) }, + { SC_(0.81158387660980224609375), SC_(0.9300427626888758122211127950646282789481) }, + { SC_(0.826751708984375), SC_(0.9629665092443368464606966822833571908852) }, + { SC_(0.83147108554840087890625), SC_(0.9736479209913771931387473923084901789046) }, + { SC_(0.84174954891204833984375), SC_(0.997713670556719074960678197806799852186) }, + { SC_(0.8679864406585693359375), SC_(1.065050516333636716777334376076374184102) }, + { SC_(0.90044414997100830078125), SC_(1.164612422633086435501625591693259387477) }, + { SC_(0.91433393955230712890625), SC_(1.215315881176612875682446995412738776976) }, + { SC_(0.91501367092132568359375), SC_(1.217962731073139868794942852653058932976) }, + { SC_(0.918984889984130859375), SC_(1.233778505900771488542027767896521427575) }, + { SC_(0.92977702617645263671875), SC_(1.28019542575660930623179572273596558907) }, + { SC_(0.93538987636566162109375), SC_(1.306695301483797253764522033930388453334) }, + { SC_(0.93773555755615234375), SC_(1.318335478463913327121670503572736587296) }, + { SC_(0.94118559360504150390625), SC_(1.33613349872692113073358883961598631154) }, + { SC_(0.96221935749053955078125), SC_(1.468821071545234761861756248744372345584) }, + { SC_(0.98576259613037109375), SC_(1.733272259459038694476413373595347034928) }, + { SC_(0.9881370067596435546875), SC_(1.77921769652839903464038407684397479173) }, + { SC_(0.99292266368865966796875), SC_(1.904368122482929779094714951471938518496) } + } }; +#undef SC_ + diff --git a/test/erf_large_data.ipp b/test/erf_large_data.ipp new file mode 100644 index 000000000..ef4ea8b3a --- /dev/null +++ b/test/erf_large_data.ipp @@ -0,0 +1,310 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 300> erf_large_data = { { + { SC_(8.2311115264892578125), SC_(0.9999999999999999999999999999997436415644), SC_(0.2563584356432915693836191701249115171878e-30) }, + { SC_(8.3800067901611328125), SC_(0.9999999999999999999999999999999787664373), SC_(0.212335626810981756102114466368867764939e-31) }, + { SC_(8.39224529266357421875), SC_(0.9999999999999999999999999999999827316301), SC_(0.1726836993826464997300336080711750877816e-31) }, + { SC_(8.66370105743408203125), SC_(0.9999999999999999999999999999999998367494), SC_(0.1632506054605993373182936619108145081694e-33) }, + { SC_(8.9759693145751953125), SC_(0.9999999999999999999999999999999999993611), SC_(0.6389109854135105643535885036658544645608e-36) }, + { SC_(9.1102085113525390625), SC_(0.9999999999999999999999999999999999999445), SC_(0.5554662272164108694298027082501260363284e-37) }, + { SC_(9.45745944976806640625), SC_(0.9999999999999999999999999999999999999999), SC_(0.8480477594433168680596946662186145070498e-40) }, + { SC_(10.61029338836669921875), SC_(1), SC_(0.6786472703144874611591306359565011820753e-50) }, + { SC_(10.8140201568603515625), SC_(1), SC_(0.8470069870191149998048954134093763207838e-52) }, + { SC_(10.82457828521728515625), SC_(1), SC_(0.6733586817565831939437647508350297170672e-52) }, + { SC_(10.9283580780029296875), SC_(1), SC_(0.6977670213613499261034444110870077646237e-53) }, + { SC_(10.98818302154541015625), SC_(1), SC_(0.1870385270398381154229086866501074415355e-53) }, + { SC_(11.31863117218017578125), SC_(1), SC_(0.1142544869245536263179923727458113771164e-56) }, + { SC_(11.69488239288330078125), SC_(1), SC_(0.1919907987324948057617154107998359393185e-60) }, + { SC_(11.78605365753173828125), SC_(1), SC_(0.2239752724414047638235296360419194553396e-61) }, + { SC_(12.1997470855712890625), SC_(1), SC_(0.1061491854672859805410277580039105082623e-65) }, + { SC_(12.4239101409912109375), SC_(1), SC_(0.4177140558891845034847722323657671701092e-68) }, + { SC_(13.54282093048095703125), SC_(1), SC_(0.9235451601555197945993611934762413642509e-81) }, + { SC_(14.22005176544189453125), SC_(1), SC_(0.6009313847910459672567608314935750835662e-89) }, + { SC_(14.22926807403564453125), SC_(1), SC_(0.4620339167397254212949689945484104460402e-89) }, + { SC_(14.359676361083984375), SC_(1), SC_(0.1100471801774583728133644248988374419641e-90) }, + { SC_(14.41039371490478515625), SC_(1), SC_(0.2548929323782609375991749296565683531577e-91) }, + { SC_(14.87335300445556640625), SC_(1), SC_(0.3198000452629152167848001696581299429735e-97) }, + { SC_(14.92373943328857421875), SC_(1), SC_(0.7101988862786309879894988884459001902733e-98) }, + { SC_(15.8191242218017578125), SC_(1), SC_(0.7438591168260150840237290305856485663443e-110) }, + { SC_(15.96480560302734375), SC_(1), SC_(0.7187861904421355163894524608883700422307e-112) }, + { SC_(15.99831295013427734375), SC_(1), SC_(0.2457896620902286177098546091913977024151e-112) }, + { SC_(16.7455272674560546875), SC_(1), SC_(0.5559992295828943910250640094814395579468e-123) }, + { SC_(17.008663177490234375), SC_(1), SC_(0.760237064211963037211163791564153667507e-127) }, + { SC_(17.2220897674560546875), SC_(1), SC_(0.5043169273534506801418127595237229711103e-130) }, + { SC_(17.757808685302734375), SC_(1), SC_(0.35565429074608249855471197796329792194e-138) }, + { SC_(17.802867889404296875), SC_(1), SC_(0.7145708921803004436285314631231014067581e-139) }, + { SC_(18.112152099609375), SC_(1), SC_(0.1053094819294519519788245447399920974245e-143) }, + { SC_(18.2649860382080078125), SC_(1), SC_(0.4020674492293458832133643654712400773494e-146) }, + { SC_(18.32352447509765625), SC_(1), SC_(0.4706809140073901333884996503518029500936e-147) }, + { SC_(18.4129180908203125), SC_(1), SC_(0.1755474923764976123532737166943228663005e-148) }, + { SC_(18.652309417724609375), SC_(1), SC_(0.2428091775485678872414439322587700295772e-152) }, + { SC_(18.9056758880615234375), SC_(1), SC_(0.1764835785229242910502507612159515936326e-156) }, + { SC_(19.1091136932373046875), SC_(1), SC_(0.7645206854732087495539968665185944859194e-160) }, + { SC_(19.1576213836669921875), SC_(1), SC_(0.1191626992794708978523133508616313098621e-160) }, + { SC_(19.31610870361328125), SC_(1), SC_(0.2657165206820982194424196529775105931442e-163) }, + { SC_(19.3672046661376953125), SC_(1), SC_(0.3671679040400985756060408739652361271147e-164) }, + { SC_(19.6346797943115234375), SC_(1), SC_(0.1067452532475426004914816798590685683097e-168) }, + { SC_(19.8862934112548828125), SC_(1), SC_(0.5060597273643268882175716699743699135461e-173) }, + { SC_(19.93419647216796875), SC_(1), SC_(0.7494327004502860500754429706601818608063e-174) }, + { SC_(20.2273464202880859375), SC_(1), SC_(0.5692529890798582547111055729709760376723e-179) }, + { SC_(20.242107391357421875), SC_(1), SC_(0.3130080167182599318145495989493729041355e-179) }, + { SC_(20.4949970245361328125), SC_(1), SC_(0.1037715327566096248596862690543722677311e-183) }, + { SC_(20.6639499664306640625), SC_(1), SC_(0.9828104968491392552509099594974133192176e-187) }, + { SC_(20.9242725372314453125), SC_(1), SC_(0.1928501126858728847552898327055431396084e-191) }, + { SC_(20.9607219696044921875), SC_(1), SC_(0.4182492638018175069922633264256087874033e-192) }, + { SC_(21.2989482879638671875), SC_(1), SC_(0.2552602767811562690060253249228934667833e-198) }, + { SC_(21.3341617584228515625), SC_(1), SC_(0.5679087385569991824361542895704294427451e-199) }, + { SC_(21.5831966400146484375), SC_(1), SC_(0.1280987506428470296014925421711821162698e-203) }, + { SC_(22.0373077392578125), SC_(1), SC_(0.3131661581007378806102060325912155502701e-212) }, + { SC_(22.2569446563720703125), SC_(1), SC_(0.1846614406013614928732974356509343390705e-216) }, + { SC_(22.9114551544189453125), SC_(1), SC_(0.2598259766741800523533570657530873506244e-229) }, + { SC_(23.0804386138916015625), SC_(1), SC_(0.108696918487531328017390687052866837422e-232) }, + { SC_(23.3235530853271484375), SC_(1), SC_(0.1355781333962075845012366321672254343728e-237) }, + { SC_(23.4473209381103515625), SC_(1), SC_(0.4129348514781646974836324488402049303682e-240) }, + { SC_(24.12081146240234375), SC_(1), SC_(0.4900590012091086604540120205982908317771e-254) }, + { SC_(24.3631992340087890625), SC_(1), SC_(0.382044899410099463972224844951303224334e-259) }, + { SC_(25.061580657958984375), SC_(1), SC_(0.379460392354870154626118964688104627376e-274) }, + { SC_(25.23714447021484375), SC_(1), SC_(0.5508622514289808668363975738198678197993e-278) }, + { SC_(25.3777942657470703125), SC_(1), SC_(0.4435055276360357438046878641441812378362e-281) }, + { SC_(25.813503265380859375), SC_(1), SC_(0.8969991006618404791166973289604328222239e-291) }, + { SC_(26.1247920989990234375), SC_(1), SC_(0.8433396822097075603573541828301520902564e-298) }, + { SC_(26.3525791168212890625), SC_(1), SC_(0.5380559555748413036380180439113786214843e-303) }, + { SC_(26.776111602783203125), SC_(1), SC_(0.8944111506115654515313274252719979616663e-313) }, + { SC_(26.8727970123291015625), SC_(1), SC_(0.4980346568584009068204868294049186460128e-315) }, + { SC_(27.6731243133544921875), SC_(1), SC_(0.5315985423107748521341095763480509306686e-334) }, + { SC_(27.6761074066162109375), SC_(1), SC_(0.4506400985060962000709584088621043872847e-334) }, + { SC_(27.96904754638671875), SC_(1), SC_(0.3715003445893695695072259525562685194324e-341) }, + { SC_(28.58887481689453125), SC_(1), SC_(0.2166518476138800933879816106005670654441e-356) }, + { SC_(28.742389678955078125), SC_(1), SC_(0.3244339622815072602163752228132919409512e-360) }, + { SC_(28.851139068603515625), SC_(1), SC_(0.6157273676049535809152624774082572414474e-363) }, + { SC_(28.9178009033203125), SC_(1), SC_(0.1305949729571576658796058283659336601413e-364) }, + { SC_(29.1156768798828125), SC_(1), SC_(0.1335911177834992334477211342269950231736e-369) }, + { SC_(29.3093738555908203125), SC_(1), SC_(0.1614718326165092385338896025176950706834e-374) }, + { SC_(29.5636444091796875), SC_(1), SC_(0.504780346276441675830815513402480460203e-381) }, + { SC_(29.631839752197265625), SC_(1), SC_(0.8890342529017924395163297463196226030207e-383) }, + { SC_(30.6340579986572265625), SC_(1), SC_(0.5049904008337056546410563604215235735623e-409) }, + { SC_(30.707683563232421875), SC_(1), SC_(0.5505904741434267060539220169867548888751e-411) }, + { SC_(30.8368549346923828125), SC_(1), SC_(0.1934000588075928658952471412237852868705e-414) }, + { SC_(31.179210662841796875), SC_(1), SC_(0.1150586385665917046799828457578881631925e-423) }, + { SC_(31.4387989044189453125), SC_(1), SC_(0.9951971607786623315295164065875436156955e-431) }, + { SC_(32.356414794921875), SC_(1), SC_(0.3647911693171115642349382796707730440408e-456) }, + { SC_(32.586200714111328125), SC_(1), SC_(0.1196854549366123742924784427293234502016e-462) }, + { SC_(32.75687408447265625), SC_(1), SC_(0.1707585732299349124939643284536648783473e-467) }, + { SC_(33.2696990966796875), SC_(1), SC_(0.3314340732586949021659796359200734182646e-482) }, + { SC_(33.519634246826171875), SC_(1), SC_(0.1851275204503437416978123025595612676848e-489) }, + { SC_(33.957134246826171875), SC_(1), SC_(0.276057351872287608387770565675906984022e-502) }, + { SC_(34.00215911865234375), SC_(1), SC_(0.1292839762753917184628820650991754160974e-503) }, + { SC_(35.2607574462890625), SC_(1), SC_(0.1723862426577913396207111099832970757108e-541) }, + { SC_(35.6440582275390625), SC_(1), SC_(0.2682942190174394220973041800979694793591e-553) }, + { SC_(35.898014068603515625), SC_(1), SC_(0.3427988522143486471852246306766064547166e-561) }, + { SC_(35.91162872314453125), SC_(1), SC_(0.128908321457486470920213918176837870702e-561) }, + { SC_(36.391139984130859375), SC_(1), SC_(0.1115677489279136454855597068297808612163e-576) }, + { SC_(36.69866943359375), SC_(1), SC_(0.1914841862056771258044968899947647500071e-586) }, + { SC_(36.778095245361328125), SC_(1), SC_(0.5580499602695800066261393252867262445483e-589) }, + { SC_(36.836078643798828125), SC_(1), SC_(0.7802735251877915942902082027097296754913e-591) }, + { SC_(36.926517486572265625), SC_(1), SC_(0.9862852517813094777393024048350874991345e-594) }, + { SC_(37.220302581787109375), SC_(1), SC_(0.3390210353535371981540883592991293306856e-603) }, + { SC_(37.62610626220703125), SC_(1), SC_(0.2161308855123068080619903224122596349173e-616) }, + { SC_(37.78128814697265625), SC_(1), SC_(0.1781872758239509973907000853718925676836e-621) }, + { SC_(39.196559906005859375), SC_(1), SC_(0.8334676760927633157297582502293215474801e-669) }, + { SC_(39.287792205810546875), SC_(1), SC_(0.6459479389166880954444837324458596842583e-672) }, + { SC_(39.3513031005859375), SC_(1), SC_(0.4369573452597126768250060965811055881436e-674) }, + { SC_(39.75826263427734375), SC_(1), SC_(0.4509483787208385166749854593762529916563e-688) }, + { SC_(39.86272430419921875), SC_(1), SC_(0.1098531610189679641881639506181744668225e-691) }, + { SC_(40.162616729736328125), SC_(1), SC_(0.4120302102977277191187506721340552814976e-702) }, + { SC_(40.170276641845703125), SC_(1), SC_(0.2226415732825465240782514517928537101293e-702) }, + { SC_(40.382472991943359375), SC_(1), SC_(0.8354572576931370443665770929403858031245e-710) }, + { SC_(40.696559906005859375), SC_(1), SC_(0.7225661474676652832663905782266313267167e-721) }, + { SC_(40.782176971435546875), SC_(1), SC_(0.673503764299797652470711537205502876345e-724) }, + { SC_(40.9482574462890625), SC_(1), SC_(0.8541553669484881480206884088881859451573e-730) }, + { SC_(41.145099639892578125), SC_(1), SC_(0.8156452609298646477772692892407567909292e-737) }, + { SC_(41.515956878662109375), SC_(1), SC_(0.3927474520057692651749738005168570105415e-750) }, + { SC_(41.838653564453125), SC_(1), SC_(0.8109391651273807861676399802649936141133e-762) }, + { SC_(42.584423065185546875), SC_(1), SC_(0.3614692055111044536160065200200925139014e-789) }, + { SC_(42.6111907958984375), SC_(1), SC_(0.3693115729392217140417024607374602650062e-790) }, + { SC_(42.6995391845703125), SC_(1), SC_(0.1964210029215776788610761375896332961826e-793) }, + { SC_(43.375934600830078125), SC_(1), SC_(0.1002957731076485676078330474511548731955e-818) }, + { SC_(43.761638641357421875), SC_(1), SC_(0.2518258316895643034341457105016197161517e-833) }, + { SC_(43.97068023681640625), SC_(1), SC_(0.2717687606713113972202367030042368252574e-841) }, + { SC_(43.977039337158203125), SC_(1), SC_(0.1553279441110419250099473836953698361303e-841) }, + { SC_(44.299617767333984375), SC_(1), SC_(0.662284188550331482050024128495832271562e-854) }, + { SC_(44.5380706787109375), SC_(1), SC_(0.4157106895546200361762199057481171702463e-863) }, + { SC_(44.7019195556640625), SC_(1), SC_(0.1849263581704522121134821381775513596569e-869) }, + { SC_(44.944408416748046875), SC_(1), SC_(0.6665835026232982550759602414166037648457e-879) }, + { SC_(45.39469146728515625), SC_(1), SC_(0.1423064152931413223321009373154837793895e-896) }, + { SC_(45.555110931396484375), SC_(1), SC_(0.6535595984749616464428461737502955522054e-903) }, + { SC_(45.922885894775390625), SC_(1), SC_(0.1587405585152606350596738044081514147077e-917) }, + { SC_(46.490032196044921875), SC_(1), SC_(0.2711887374954078187100515917478258475121e-940) }, + { SC_(46.700458526611328125), SC_(1), SC_(0.8220768482426773312423439218857395793096e-949) }, + { SC_(46.80968475341796875), SC_(1), SC_(0.300689768353676045411682389398910728652e-953) }, + { SC_(46.93021392822265625), SC_(1), SC_(0.3716830576798476782128277462132351948199e-958) }, + { SC_(47.80080413818359375), SC_(1), SC_(0.5560372820563731575239827056999357124531e-994) }, + { SC_(47.98065185546875), SC_(1), SC_(0.1829295366299049625367873244831792588204e-1001) }, + { SC_(48.160678863525390625), SC_(1), SC_(0.5544641890677600654790600067979757964649e-1009) }, + { SC_(48.228302001953125), SC_(1), SC_(0.8174825675757120829683639765764518487113e-1012) }, + { SC_(48.86585235595703125), SC_(1), SC_(0.1054147185420251960062940386161051253868e-1038) }, + { SC_(48.88062286376953125), SC_(1), SC_(0.2487429897424450019093445470126394812466e-1039) }, + { SC_(49.2615814208984375), SC_(1), SC_(0.1428662569339560177970591891033928074615e-1055) }, + { SC_(49.300342559814453125), SC_(1), SC_(0.3129115225209756198596637394420488168835e-1057) }, + { SC_(49.3912200927734375), SC_(1), SC_(0.3976507934443091883911509821945069748171e-1061) }, + { SC_(49.450878143310546875), SC_(1), SC_(0.1091595819414626077382205537328240830492e-1063) }, + { SC_(49.4884796142578125), SC_(1), SC_(0.2642659241845869227101266198364933762561e-1065) }, + { SC_(50.086460113525390625), SC_(1), SC_(0.3607878684331161959098857667661007215536e-1091) }, + { SC_(51.2444610595703125), SC_(1), SC_(0.3860548279363456071149897668546024070402e-1142) }, + { SC_(51.339717864990234375), SC_(1), SC_(0.2197789266327651540257336225175053590062e-1146) }, + { SC_(51.447040557861328125), SC_(1), SC_(0.354996685112096717011426990868690176895e-1151) }, + { SC_(51.52539825439453125), SC_(1), SC_(0.1110143282538327027879325761520351634781e-1154) }, + { SC_(51.782512664794921875), SC_(1), SC_(0.3217426516835935075931472670615112969831e-1166) }, + { SC_(52.144077301025390625), SC_(1), SC_(0.1532358547576965427522636443372608038782e-1182) }, + { SC_(52.820537567138671875), SC_(1), SC_(0.2202669107160275614166035168539425499729e-1213) }, + { SC_(52.8442840576171875), SC_(1), SC_(0.1790754358721238111167473370381879109436e-1214) }, + { SC_(52.871673583984375), SC_(1), SC_(0.9892522319813664639479190973985140674034e-1216) }, + { SC_(52.872089385986328125), SC_(1), SC_(0.9466912463849392521290423205053783831664e-1216) }, + { SC_(53.07733917236328125), SC_(1), SC_(0.3390908799279318246606852431264844551837e-1225) }, + { SC_(53.088535308837890625), SC_(1), SC_(0.1032764828225424213243462240852153489629e-1225) }, + { SC_(53.778041839599609375), SC_(1), SC_(0.1017002820918486891770554932818686415531e-1257) }, + { SC_(54.0477142333984375), SC_(1), SC_(0.2381751287333891734284837295419122638673e-1270) }, + { SC_(54.561374664306640625), SC_(1), SC_(0.139407502357667351124920158334676469294e-1294) }, + { SC_(54.6435394287109375), SC_(1), SC_(0.1765212687763302697244562949068373410255e-1298) }, + { SC_(55.047882080078125), SC_(1), SC_(0.9580050759351008955877580245822791128101e-1318) }, + { SC_(55.53577423095703125), SC_(1), SC_(0.3516341067901357169932495766343569323525e-1341) }, + { SC_(55.63845062255859375), SC_(1), SC_(0.3871072763485538347396913418646976027199e-1346) }, + { SC_(55.9916534423828125), SC_(1), SC_(0.289528812850324808187999318666603687985e-1363) }, + { SC_(55.991954803466796875), SC_(1), SC_(0.2799194648396030874187262104833020570154e-1363) }, + { SC_(56.6115570068359375), SC_(1), SC_(0.138610665425682272373334040969461433112e-1393) }, + { SC_(56.749294281005859375), SC_(1), SC_(0.2289081242224445766088406090902886148302e-1400) }, + { SC_(57.362518310546875), SC_(1), SC_(0.9220583307012274413074526609845212209396e-1431) }, + { SC_(57.5469818115234375), SC_(1), SC_(0.5725247795522802723716771177781160453318e-1440) }, + { SC_(58.515666961669921875), SC_(1), SC_(0.8387169924209196731842556332538391198272e-1489) }, + { SC_(58.8695220947265625), SC_(1), SC_(0.7612989272837028720721864253441810819973e-1507) }, + { SC_(59.008518218994140625), SC_(1), SC_(0.5818271285898335632169611265925278206909e-1514) }, + { SC_(59.44551849365234375), SC_(1), SC_(0.1907993812890002855694347258421797523393e-1536) }, + { SC_(59.853458404541015625), SC_(1), SC_(0.1386369169907035409186787094207373016361e-1557) }, + { SC_(60.46059417724609375), SC_(1), SC_(0.2591809034926289626577001708586017024727e-1589) }, + { SC_(60.804798126220703125), SC_(1), SC_(0.1921652566797868246807189497430043640543e-1607) }, + { SC_(60.9976654052734375), SC_(1), SC_(0.1202209001001935503022907284085976927471e-1617) }, + { SC_(61.63448333740234375), SC_(1), SC_(0.1443858912194400214601650177255810967138e-1651) }, + { SC_(61.718036651611328125), SC_(1), SC_(0.4818094804505875544021186960960583940739e-1656) }, + { SC_(61.749187469482421875), SC_(1), SC_(0.1028760000725124920333886117069810603088e-1657) }, + { SC_(61.7707366943359375), SC_(1), SC_(0.7180844057437257539436486908675040497688e-1659) }, + { SC_(62.129795074462890625), SC_(1), SC_(0.3411714823101512343643373517899997588934e-1678) }, + { SC_(62.415653228759765625), SC_(1), SC_(0.1172420264374634693794507306774825437228e-1693) }, + { SC_(63.656280517578125), SC_(1), SC_(0.1359202639129155863922014558394013590937e-1761) }, + { SC_(63.720615386962890625), SC_(1), SC_(0.3748863560151785919817454048142060747672e-1765) }, + { SC_(63.735622406005859375), SC_(1), SC_(0.5534848650018337136859100029355185261672e-1766) }, + { SC_(63.745220184326171875), SC_(1), SC_(0.1628046476475599666226420695090084595775e-1766) }, + { SC_(63.783538818359375), SC_(1), SC_(0.1227798051543007555849422132322321585877e-1768) }, + { SC_(63.87148284912109375), SC_(1), SC_(0.1632762187944705003794943454328122376037e-1773) }, + { SC_(64.515594482421875), SC_(1), SC_(0.1969332977981314849672526645022956975392e-1809) }, + { SC_(64.97594451904296875), SC_(1), SC_(0.2525310788037616742275863982689138714012e-1835) }, + { SC_(65.00909423828125), SC_(1), SC_(0.3394161846767033496272256639147831568993e-1837) }, + { SC_(65.32428741455078125), SC_(1), SC_(0.4872272181442093260036099253187279262469e-1855) }, + { SC_(65.8734893798828125), SC_(1), SC_(0.2462682463022898342795255163765824211512e-1886) }, + { SC_(65.895782470703125), SC_(1), SC_(0.1304674583839710679197974600800374601509e-1887) }, + { SC_(66.16791534423828125), SC_(1), SC_(0.3203747642032728309060331250397804757235e-1903) }, + { SC_(66.27771759033203125), SC_(1), SC_(0.1545497931044982506797819622759819541777e-1909) }, + { SC_(66.7202606201171875), SC_(1), SC_(0.4214693722661996667087731543216442503726e-1935) }, + { SC_(67.0804595947265625), SC_(1), SC_(0.4916509034440560388255363947533037583151e-1956) }, + { SC_(67.51879119873046875), SC_(1), SC_(0.1163704950246886518963746466814171576513e-1981) }, + { SC_(67.61028289794921875), SC_(1), SC_(0.4965812608096806464498341746692252789426e-1987) }, + { SC_(68.0894927978515625), SC_(1), SC_(0.2827017688141873952389885023460208246821e-2015) }, + { SC_(68.7330780029296875), SC_(1), SC_(0.1601703792022999521045114304499339964142e-2053) }, + { SC_(68.936859130859375), SC_(1), SC_(0.1045603435996106684348791239772064284538e-2065) }, + { SC_(69.3484344482421875), SC_(1), SC_(0.1990640753018030347229172891391208563692e-2090) }, + { SC_(69.4951629638671875), SC_(1), SC_(0.2821572392833612189407917583240153887473e-2099) }, + { SC_(69.6038970947265625), SC_(1), SC_(0.7606563437817722315286512269995042517896e-2106) }, + { SC_(69.8057861328125), SC_(1), SC_(0.4535113025492646420574078398459972783487e-2118) }, + { SC_(69.884307861328125), SC_(1), SC_(0.7806399428418380776194369203828958230443e-2123) }, + { SC_(70.090423583984375), SC_(1), SC_(0.2297973354469257356841567122165370429556e-2135) }, + { SC_(70.1346893310546875), SC_(1), SC_(0.4627340318338513849654024935557448249929e-2138) }, + { SC_(70.4619598388671875), SC_(1), SC_(0.4786946947734960725308155565435423870322e-2158) }, + { SC_(70.51854705810546875), SC_(1), SC_(0.1640745397943748179034151831339364278733e-2161) }, + { SC_(70.62750244140625), SC_(1), SC_(0.3431785975089560474469843416846358607766e-2168) }, + { SC_(70.77237701416015625), SC_(1), SC_(0.4345108777332364055904822498668881910844e-2177) }, + { SC_(71.46120452880859375), SC_(1), SC_(0.1213533316440842841172645481517647701339e-2219) }, + { SC_(71.54265594482421875), SC_(1), SC_(0.1059138463142173993761026331012035279463e-2224) }, + { SC_(71.8696136474609375), SC_(1), SC_(0.4560525021368102193256510775118687007107e-2245) }, + { SC_(71.89171600341796875), SC_(1), SC_(0.1900755308625425801959286367021480241643e-2246) }, + { SC_(72.96099853515625), SC_(1), SC_(0.1012288958237199779078623773782261227323e-2313) }, + { SC_(73.07500457763671875), SC_(1), SC_(0.5943790030417452214269807905148252301626e-2321) }, + { SC_(73.10594940185546875), SC_(1), SC_(0.6446581210317819300626283182106697992806e-2323) }, + { SC_(73.1313323974609375), SC_(1), SC_(0.1574358163400440570765391931676616457296e-2324) }, + { SC_(73.20639801025390625), SC_(1), SC_(0.2666631324666853271144728061565555980879e-2329) }, + { SC_(73.376953125), SC_(1), SC_(0.3692800481745944696527608913821150780197e-2340) }, + { SC_(73.430145263671875), SC_(1), SC_(0.1498451636616422925337939153598716367853e-2343) }, + { SC_(73.44467926025390625), SC_(1), SC_(0.1772057595578452829782717746213217866987e-2344) }, + { SC_(73.60561370849609375), SC_(1), SC_(0.9327165682410597575108877219484860075423e-2355) }, + { SC_(73.62299346923828125), SC_(1), SC_(0.7217309662507557984084259593722358671817e-2356) }, + { SC_(73.77313995361328125), SC_(1), SC_(0.1762441226759232140980568288616069877708e-2365) }, + { SC_(74.2175445556640625), SC_(1), SC_(0.4796668342626380739003100430828430758387e-2394) }, + { SC_(74.27039337158203125), SC_(1), SC_(0.1873022555292045249630763229328993092174e-2397) }, + { SC_(74.39823150634765625), SC_(1), SC_(0.1041846943079459800746159253005438318555e-2405) }, + { SC_(74.77135467529296875), SC_(1), SC_(0.6972625788619583784396293640501867635862e-2430) }, + { SC_(74.807342529296875), SC_(1), SC_(0.320164469944205443450167902957404361688e-2432) }, + { SC_(75.01886749267578125), SC_(1), SC_(0.5501656655265385463409067358984172337294e-2446) }, + { SC_(75.3089447021484375), SC_(1), SC_(0.6319468915000774061299279774718180641493e-2465) }, + { SC_(75.34217071533203125), SC_(1), SC_(0.4232650489503930895532015931831887522867e-2467) }, + { SC_(75.3960723876953125), SC_(1), SC_(0.1252103871082113384792176510455204354046e-2470) }, + { SC_(75.45360565185546875), SC_(1), SC_(0.2128736422560450047027718674310538561411e-2474) }, + { SC_(75.5235443115234375), SC_(1), SC_(0.5520057148754538352100449496414500231534e-2479) }, + { SC_(75.7169952392578125), SC_(1), SC_(0.1082452065927623584648516750055819887681e-2491) }, + { SC_(76.1279449462890625), SC_(1), SC_(0.8546877316832444989935075245988734481682e-2519) }, + { SC_(76.470703125), SC_(1), SC_(0.1638058714171406332549731275799690588823e-2541) }, + { SC_(76.938812255859375), SC_(1), SC_(0.1056702363179452330608841676726809135679e-2572) }, + { SC_(77.5743560791015625), SC_(1), SC_(0.2358904373500824632540699420486265405702e-2615) }, + { SC_(77.62860107421875), SC_(1), SC_(0.5201021650937707207333604556951161883551e-2619) }, + { SC_(77.94854736328125), SC_(1), SC_(0.1249443652740786709208083871965551047597e-2640) }, + { SC_(78.06497955322265625), SC_(1), SC_(0.1611056496774920949293225513029265281761e-2648) }, + { SC_(78.1817626953125), SC_(1), SC_(0.1913800020341339537443259258588625217145e-2656) }, + { SC_(78.7396087646484375), SC_(1), SC_(0.1826214635685593066183737049160168091905e-2694) }, + { SC_(79.23296356201171875), SC_(1), SC_(0.2578910365673578199346966028090255709301e-2728) }, + { SC_(79.28195953369140625), SC_(1), SC_(0.1091891461191934352451814384442219402849e-2731) }, + { SC_(79.53916168212890625), SC_(1), SC_(0.1977970219864078402856631374181827871126e-2749) }, + { SC_(79.89411163330078125), SC_(1), SC_(0.5214347553208988003055180927464712623108e-2774) }, + { SC_(80.03131103515625), SC_(1), SC_(0.1539244869801023794218894091743149755693e-2783) }, + { SC_(81.0540618896484375), SC_(1), SC_(0.4282424762803665933789596667679893825473e-2855) }, + { SC_(82.27494049072265625), SC_(1), SC_(0.1058661602699748027888078315752637932386e-2941) }, + { SC_(82.40390777587890625), SC_(1), SC_(0.6316127985700229083559360532999075433841e-2951) }, + { SC_(82.67310333251953125), SC_(1), SC_(0.3161239679691280797883924212187842089968e-2970) }, + { SC_(82.83135223388671875), SC_(1), SC_(0.1331877990458538379342065811181569680933e-2981) }, + { SC_(82.8936614990234375), SC_(1), SC_(0.4360382415173864210795784221294622630074e-2986) }, + { SC_(82.896820068359375), SC_(1), SC_(0.2582766043573584963841163098015327607626e-2986) }, + { SC_(83.0903167724609375), SC_(1), SC_(0.290013476198442869046820654577037968453e-3000) }, + { SC_(83.20987701416015625), SC_(1), SC_(0.6710637821460133939254155513062400075894e-3009) }, + { SC_(83.5117340087890625), SC_(1), SC_(0.930787700439063954052855467933526662801e-3031) }, + { SC_(84.054412841796875), SC_(1), SC_(0.2976066148553277399257428533933691648135e-3070) }, + { SC_(84.19962310791015625), SC_(1), SC_(0.7279769443639285050812101610482378417998e-3081) }, + { SC_(84.58744049072265625), SC_(1), SC_(0.2702912324282395041651536403393819660776e-3109) }, + { SC_(84.58887481689453125), SC_(1), SC_(0.2120514736394887571017949592511629200089e-3109) }, + { SC_(85.08606719970703125), SC_(1), SC_(0.4856698297979953595767807992015152776328e-3146) }, + { SC_(85.918212890625), SC_(1), SC_(0.761735048565653930137023626638747177308e-3208) }, + { SC_(86.31143951416015625), SC_(1), SC_(0.2931592864000316245471343335876130048243e-3237) }, + { SC_(86.48769378662109375), SC_(1), SC_(0.1734166482219001810675374526329398742724e-3250) }, + { SC_(86.51556396484375), SC_(1), SC_(0.1396184688760208541263525791606580135888e-3252) }, + { SC_(86.661895751953125), SC_(1), SC_(0.137591974249259342744647335702120081949e-3263) }, + { SC_(86.67838287353515625), SC_(1), SC_(0.7894924978682044731133080698270675959047e-3265) }, + { SC_(86.699005126953125), SC_(1), SC_(0.2210314800408904240090566206375166498669e-3266) }, + { SC_(86.875640869140625), SC_(1), SC_(0.1067393543048809336522125654614257792709e-3279) }, + { SC_(87.12085723876953125), SC_(1), SC_(0.3141588749441732230579799060940491765735e-3298) }, + { SC_(87.1272430419921875), SC_(1), SC_(0.1032456874487975042510718930756313489085e-3298) }, + { SC_(87.350982666015625), SC_(1), SC_(0.1145259102841530225822158676931062199611e-3315) }, + { SC_(87.4471588134765625), SC_(1), SC_(0.5719031848650119145301950377270835213481e-3323) }, + { SC_(87.5886077880859375), SC_(1), SC_(0.100944910275613191627060617308017176992e-3333) }, + { SC_(88.114166259765625), SC_(1), SC_(0.790369642782630853434920273428079682513e-3374) }, + { SC_(88.35390472412109375), SC_(1), SC_(0.3336629458423611349557710086599109785783e-3392) }, + { SC_(88.45099639892578125), SC_(1), SC_(0.1168446033182320025804519666787072055362e-3399) }, + { SC_(88.55356597900390625), SC_(1), SC_(0.1521829146954713302263784465478849666514e-3407) }, + { SC_(88.97168731689453125), SC_(1), SC_(0.8788241912132849825702219058327694185942e-3440) }, + { SC_(89.04711151123046875), SC_(1), SC_(0.129507473240771950228590714535613326805e-3445) }, + { SC_(89.05876922607421875), SC_(1), SC_(0.1623712240263443145358629211111315306735e-3446) }, + { SC_(89.41626739501953125), SC_(1), SC_(0.3153753084460525420724595238130726065242e-3474) }, + { SC_(89.51361846923828125), SC_(1), SC_(0.8577730193641580987437405819558340102121e-3482) }, + { SC_(89.68304443359375), SC_(1), SC_(0.5586301300368715269248196410139332100555e-3495) }, + { SC_(89.70983123779296875), SC_(1), SC_(0.4571434971968977143179977283713339395878e-3497) } + } }; +#undef SC_ + diff --git a/test/erf_small_data.ipp b/test/erf_small_data.ipp new file mode 100644 index 000000000..f05d8ee44 --- /dev/null +++ b/test/erf_small_data.ipp @@ -0,0 +1,162 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 150> erf_small_data = { { + { SC_(0), SC_(0), SC_(1) }, + { SC_(0.140129846432481707092372958328991613128e-44), SC_(0.1581195994027057927040695988659415347357e-44), SC_(1) }, + { SC_(0.2802596928649634141847459166579832262561e-44), SC_(0.3162391988054115854081391977318830694715e-44), SC_(1) }, + { SC_(0.4203895392974451212771188749869748393841e-44), SC_(0.4743587982081173781122087965978246042072e-44), SC_(1) }, + { SC_(0.8407790785948902425542377499739496787682e-44), SC_(0.9487175964162347562244175931956492084145e-44), SC_(1) }, + { SC_(0.1541428310757298778016102541618907744408e-43), SC_(0.1739315593429763719744765587525356882093e-43), SC_(1) }, + { SC_(0.3222986467947079263124578041566807101945e-43), SC_(0.3636750786262233232193600773916655298922e-43), SC_(1) }, + { SC_(0.7426881860921530475895766791436555495785e-43), 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SC_(0.1037359531608217366738244891166687011719e-7), SC_(0.1170534884254671319566985248995045101925e-7), SC_(0.9999999882946511574532868043301475100495) }, + { SC_(0.2613260008388351707253605127334594726563e-7), SC_(0.2948748151669259830348980474563029807727e-7), SC_(0.9999999705125184833074016965101952543697) }, + { SC_(0.4653804097642932902090251445770263671875e-7), SC_(0.525125559152401230823420797240431154212e-7), SC_(0.9999999474874440847598769176579202759569) }, + { SC_(0.8228098380413939594291150569915771484375e-7), SC_(0.9284414797271396176853241560056526014051e-7), SC_(0.9999999071558520272860382314675843994347) }, + { SC_(0.1440129295815495424903929233551025390625e-6), SC_(0.1625011895322124534932667008476984221693e-6), SC_(0.9999998374988104677875465067332991523016) }, + { SC_(0.373797803376874071545898914337158203125e-6), SC_(0.4217856540365096975937056642610100404098e-6), SC_(0.99999957821434596349030240629433573899) }, + { SC_(0.72830744102247990667819976806640625e-6), SC_(0.8218069436902647194923144120911901417405e-6), SC_(0.9999991781930563097352805076855879088099) }, + { SC_(0.10260146154905669391155242919921875e-5), SC_(0.1157733517254662264894221065154775581876e-5), SC_(0.9999988422664827453377351057789348452244) }, + { SC_(0.2678315240700612775981426239013671875e-5), SC_(0.3022155120513748374723247514313931180799e-5), SC_(0.9999969778448794862516252767524856860688) }, + { SC_(0.40205004552262835204601287841796875e-5), SC_(0.4536648954950918836538342339896039137139e-5), SC_(0.9999954633510450490811634616576601039609) }, + { SC_(0.10320758519810624420642852783203125e-4), SC_(0.1164572890196433412287423508862147085112e-4), SC_(0.9999883542710980356658771257649113785291) }, + { SC_(0.233581158681772649288177490234375e-4), SC_(0.2635681132346089850308780034445740445022e-4), SC_(0.9999736431886765391014969121996555425955) }, + { SC_(0.4860912667936645448207855224609375e-4), SC_(0.5484952583250342934869652426491962534587e-4), SC_(0.9999451504741674965706513034757350803747) }, + { SC_(0.0001085917465388774871826171875), SC_(0.0001225326640313436930517738593548252912385), SC_(0.9998774673359686563069482261406451747088) }, + { SC_(0.000165569479577243328094482421875), SC_(0.0001868251497546456322442598271644064384812), SC_(0.9998131748502453543677557401728355935615) }, + { SC_(0.0004721705918200314044952392578125), SC_(0.0005327874195307728646012144512357625925006), SC_(0.9994672125804692271353987855487642374075) }, + { SC_(0.00095945619978010654449462890625), SC_(0.001082630055365222472802272152269070171599), SC_(0.9989173699446347775271977278477309298284) }, + { SC_(0.0011034240014851093292236328125), SC_(0.001245080150435437151082626328085359462604), SC_(0.9987549198495645628489173736719146405374) }, + { SC_(0.00225476245395839214324951171875), SC_(0.002544222668224971136268866790058243266433), SC_(0.9974557773317750288637311332099417567336) }, + { SC_(0.006128217093646526336669921875), SC_(0.0069148659371010946376763193961158375309), SC_(0.9930851340628989053623236806038841624691) }, + { SC_(0.01089772023260593414306640625), SC_(0.01229627370763712993609092392080824128196), SC_(0.987703726292362870063909076079191758718) }, + { SC_(0.0229592286050319671630859375), SC_(0.02590216393432758991206927422057375200344), SC_(0.9740978360656724100879307257794262479966) }, + { SC_(0.043352998793125152587890625), SC_(0.04888799071126766325637634356499925915047), SC_(0.9511120092887323367436236564350007408495) }, + { SC_(0.06324388086795806884765625), SC_(0.07126804593959971882312270799434219648871), SC_(0.9287319540604002811768772920056578035113) }, + { SC_(0.2158693373203277587890625), SC_(0.2398511631288111077158786048968679107576), SC_(0.7601488368711888922841213951031320892424) }, + { SC_(0.334280669689178466796875), SC_(0.3636043520990431448970776140491250415292), SC_(0.6363956479009568551029223859508749584708) } + } }; +#undef SC_ + + + diff --git a/test/erfc_inv_big_data.ipp b/test/erfc_inv_big_data.ipp new file mode 100644 index 000000000..60725380c --- /dev/null +++ b/test/erfc_inv_big_data.ipp @@ -0,0 +1,211 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 200> erfc_inv_big_data = { { + { SC_(0.5460825444184401149953083908674803067585e-4312), SC_(99.62016927389407649911084501709563799849) }, + { SC_(0.2645475979732877501542006305722535728925e-4291), SC_(99.38083815930157675670279543643240126733) }, + { SC_(0.4647218972549971591905525747707221131215e-4281), SC_(99.26209205835381203650501813598769577392) }, + { SC_(0.1525121257807440558802279626797226161005e-4243), SC_(98.82602533559047574603223381461761500878) }, + { SC_(0.1981708500756372844759510276075956268897e-4234), SC_(98.71980149740490464004477100962342642155) }, + { SC_(0.3841754068409692466399340503654464602243e-4180), SC_(98.08467819532462847486581213069198608289) }, + { SC_(0.5335004017651487579647125338582580235046e-4091), SC_(97.03275964437690000965320992419441429924) }, + { SC_(0.1806383895924070461510470704435189756123e-4076), SC_(96.86022098874718527180735862424264805433) }, + { SC_(0.1321513657594592521699131916159148731226e-4066), SC_(96.7429083707271579347385166737183074303) }, + { SC_(0.1054920495502343332517378247415093412624e-4043), SC_(96.46999015250599733880053488460821308812) }, + { SC_(0.47363892632169499253148937212672347289e-4043), SC_(96.46220643869959630846141605535921442023) }, + { SC_(0.4750819439081903839348296656606873483722e-4000), SC_(95.94763372873564880985964950938277645413) }, + { SC_(0.5934696803923580795229088781443547003391e-3998), SC_(95.92247396394370347463100485254034797278) }, + { SC_(0.2809661603171360146097006130839773212914e-3996), SC_(95.90236599111862577117586443056337455754) }, + { SC_(0.201344506190345013258215914381814651279e-3981), SC_(95.72387427307012892119049758800169512382) }, + { SC_(0.1349680361063247504563298949422573179728e-3978), SC_(95.68987765258188263846875190956134957764) }, + { SC_(0.9554154264888789862328696045397641686626e-3961), SC_(95.47488611856686267812468521780070611561) }, + { SC_(0.4565370419999699854685454921616341777152e-3921), SC_(94.99523139627283958425843803677334678489) }, + { SC_(0.9326136640408284179797117117017987832624e-3838), SC_(93.98018858040542889098559401088499617323) }, + { SC_(0.2085705673165100591822581196874037577838e-3788), SC_(93.37371566951938600144821032880068555931) }, + { SC_(0.28906230618616666278153940715368369251e-3788), SC_(93.37196812654735398207358428222836165935) }, + { SC_(0.540152632181355759951616925660164590124e-3750), SC_(92.89890240230691810677222482470759338646) }, + { SC_(0.1373243759745268803582425259844487470076e-3682), SC_(92.05981131540500971264194940023399842876) }, + { SC_(0.3331647658738942966983713783383120828716e-3653), SC_(91.69161143487247689253830777510103757803) }, + { SC_(0.134842155208242065478707324403653741427e-3641), SC_(91.54576292987357613504022634708802175022) }, + { SC_(0.7492191291954483620899871520373510239375e-3623), SC_(91.30973569873630381671516956816451328373) }, + { SC_(0.2551662007977519576358324440130450442778e-3575), SC_(90.70847518912252606205543434018735665529) }, + { SC_(0.6190686739111601149790106116510494551314e-3563), SC_(90.55115617451523669098548678613430984841) }, + { SC_(0.9276632609820886396580236856359763026966e-3447), SC_(89.06191184952516547664083098302741022803) }, + { SC_(0.6433072193358217453796447947689324967478e-3351), SC_(87.8143275613841604460220844122740180476) }, + { SC_(0.3653541017426498234750653436978737170327e-3338), SC_(87.64696333406539636312319479272468979115) }, + { SC_(0.2789596742720445243557533363147985407484e-3305), SC_(87.21398755358298377575097222058344254911) }, + { SC_(0.443825473802222651705813573190067261172e-3131), SC_(84.88340554604607048910870088690204110131) }, + { SC_(0.3345784652046112831225855034049282590715e-3126), SC_(84.81723263217786918755009758623351862559) }, + { SC_(0.2414445336524691790354556146640969706061e-3069), SC_(84.0419598414264046587129844579410948894) }, + { SC_(0.8478603490304246265311990779170856959861e-3028), SC_(83.47092766845828325139370216002957076776) }, + { SC_(0.14740427247445948990191590432902058536e-2997), SC_(83.05281564078241081289202980345540186102) }, + { SC_(0.1329036517699801662619770631013011165738e-2944), SC_(82.31552494008283774947792893056404158871) }, + { SC_(0.1706141285384843345987169410441595650031e-2933), SC_(82.16002264325748303060260330185689090281) }, + { SC_(0.8373844481146949976640588846233709740831e-2929), SC_(82.09426838736262780476565867158939922013) }, + { SC_(0.6495088549453856516509184167801252067894e-2926), SC_(82.05373670564597872871969963977451713693) }, + { SC_(0.1833720938105362893360888392913141441369e-2882), SC_(81.44184518033949369735162216055497263014) }, + { SC_(0.2333520177487467820415573739119469955902e-2879), SC_(81.39794775935971788819430665383756478382) }, + { SC_(0.2894427653993793412567889456980821259226e-2847), SC_(80.94277839157518781341184937011412770395) }, + { SC_(0.2604952067990663450155633194749904382233e-2836), SC_(80.78683208931494064074748747620852519014) }, + { SC_(0.8678138849264725191436450361584810016324e-2832), SC_(80.72235938139247087444254555436435678849) }, + { SC_(0.3682667903200544588455731527142145038363e-2781), SC_(79.99708530932539906643334265914340895581) }, + { SC_(0.4588121885193651068401508885826145226945e-2766), SC_(79.77955734877589476980263452200168409691) }, + { SC_(0.7764413237799866858364453900343995777469e-2765), SC_(79.76182876388495108788138224124277407448) }, + { SC_(0.2423421525461451159440550098196593570056e-2760), SC_(79.69693586691073000502029265856910229446) }, + { SC_(0.5878986254507577903335876253702952707265e-2746), SC_(79.48887849840574283826340114327272440985) }, + { SC_(0.1673845625470905295897015228985949168928e-2737), SC_(79.36634262823360407978363436458177566234) }, + { SC_(0.6698391998725201197546429856824581586977e-2735), SC_(79.32858818851906731315743566169843426962) }, + { SC_(0.2515130355594925802647014162660517938496e-2725), SC_(79.18952171986121353050089556124781639244) }, + { SC_(0.2461647325252598254899739145066165462755e-2636), SC_(77.88509727460664014637457387303638162253) }, + { SC_(0.1641105794585019759166735829835878355641e-2634), SC_(77.85813389734403078489439365068353770847) }, + { SC_(0.6351595345143555864526860293158388570963e-2601), SC_(77.35991719682138597284770268761692627695) }, + { SC_(0.9223028410198721360701049566280075593579e-2545), SC_(76.51960497818744876859269133142591483997) }, + { SC_(0.5002814464212067387766735395165041257035e-2533), SC_(76.34286436673947487064698531398136817799) }, + { SC_(0.1185824625931586847282481488476410873355e-2519), SC_(76.14091514413348192509930444973601480465) }, + { SC_(0.257414900548022702014576953401369926289e-2493), SC_(75.74167889782762540410788473501883567486) }, + { SC_(0.9183566141599824732209230606531307948203e-2490), SC_(75.68766731518135433104337024242062602941) }, + { SC_(0.2932809029313999449803475825227510422988e-2473), SC_(75.43622352043665347320282000269125370548) }, + { SC_(0.5498724163832736366794865810754062391466e-2467), SC_(75.34043418668345319181733147217173430757) }, + { SC_(0.6795395146348654066574869189625572073492e-2451), SC_(75.09414895042043847993638837712888889371) }, + { SC_(0.4232772623968377298803745728444220344828e-2398), SC_(74.2804044020029126990405093970664224902) }, + { SC_(0.437705010214781194544970904918846009988e-2382), SC_(74.0317968899859238324605680410762845653) }, + { SC_(0.7328784181257135130009299062120147357384e-2332), SC_(73.24665809155865681149475856856425835323) }, + { SC_(0.8195784402152374810490855567845040936044e-2295), SC_(72.6620497559170838922485865519610574788) }, + { SC_(0.8987013844942659551846558843047951168834e-2288), SC_(72.55042905061235174343554742739711560742) }, + { SC_(0.2390567310984670056453397739020313330694e-2193), SC_(71.03636046035780611691848769848245798184) }, + { SC_(0.4332167143233490736155663100970351222456e-2185), SC_(70.90240600026913612396368503494705740935) }, + { SC_(0.108890416531056942890760795215821213854e-2164), SC_(70.57040795030909610470208299858512551923) }, + { SC_(0.5060627104045047637085866910481125331788e-2163), SC_(70.54320633525883330072442815574897293795) }, + { SC_(0.3691223561779676390769470042383461854014e-2122), SC_(69.87319260661719659265781125754219708719) }, + { SC_(0.6969547608629390600217634449465487521466e-2099), SC_(69.48865744111861740557954052160177140024) }, + { SC_(0.1920331015960815466646483904103550820861e-2096), SC_(69.44822094478638613766926468731012767531) }, + { SC_(0.6083975528963974425124099594841117907605e-2016), SC_(68.1007710158724367714118546612371304706) }, + { SC_(0.7700980458334894771243720474212758259757e-2007), SC_(67.94673150215833308050516745181191739631) }, + { SC_(0.3784507382437993534366757035181644553005e-2006), SC_(67.93501557417646729265187361704446762791) }, + { SC_(0.1933058527290172128745148456282632896472e-2000), SC_(67.8382198195958948309248955546035843938) }, + { SC_(0.3073129169339213242656468129490128120275e-1912), SC_(66.32462247762686573271005635222158246919) }, + { SC_(0.3563704808140633534254743351059329459844e-1867), SC_(65.53779859486217889665515029788703171428) }, + { SC_(0.1117328435404434848431164536157383381954e-1865), SC_(65.5115114749327337441199600755950731015) }, + { SC_(0.162291367493531393505178202172746460504e-1863), SC_(65.47350810130320090152901346075787089108) }, + { SC_(0.7291877778593992272357923862974438803504e-1814), SC_(64.59461716634387479104306278695990989692) }, + { SC_(0.1103697274312455454535522513124420450513e-1800), SC_(64.35930484119184106703256655579503018943) }, + { SC_(0.2787184985694988691274106077027481674296e-1777), SC_(63.93935283526837024038868862230807595467) }, + { SC_(0.2242899394948684173448369886336215330859e-1736), SC_(63.19860571879709151914226858360392775225) }, + { SC_(0.6373703493315477830264234158158397230918e-1648), SC_(61.5663708938977044791256528616543682363) }, + { SC_(0.1130761791970139872085460502062402383311e-1534), SC_(59.4111793960172436477860564668325556624) }, + { SC_(0.5186477168932786698797072956323906176672e-1526), SC_(59.24311964529112251267296011750890265511) }, + { SC_(0.2371756563156473396673371175154689282089e-1487), SC_(58.48710687097943040207443293838886041518) }, + { SC_(0.2474530000251364788339443399087025927602e-1487), SC_(58.48674428195209156566913277116595315737) }, + { SC_(0.1620952363661926127361288574589236587446e-1465), SC_(58.05577331395656272747602508275617390989) }, + { SC_(0.2122625807107458215797391334648973459688e-1429), SC_(57.33517766371867417445655972866006304427) }, + { SC_(0.6243197332907990298222556346398382173979e-1405), SC_(56.8418011386019473870793880887159457241) }, + { SC_(0.5516957459853521159388169416018722269647e-1400), SC_(56.74154440911648902947449827969373477014) }, + { SC_(0.2926235914834555267234269989321844626118e-1369), SC_(56.11477468249401615221533062929223131088) }, + { SC_(0.3794627523015867743608205312364027747225e-1312), SC_(54.93070214218908343008136754660768021631) }, + { SC_(0.318880483330771111942926508195676807607e-1300), SC_(54.68024803482526161939511993227336637443) }, + { SC_(0.3369469880227599686172744089074258905398e-1300), SC_(54.67974419399797522074814528351877319048) }, + { SC_(0.1302713766629726626489873195118034634277e-1298), SC_(54.64631892482891526341853652123547792494) }, + { SC_(0.2469170628653646881926626888059718255132e-1223), SC_(53.03693750856997083336291247307325166587) }, + { SC_(0.1371607091728437802604087975394383012252e-1207), SC_(52.69411504628090115129531887668248361258) }, + { SC_(0.1731629169156886034424126010604259325511e-1111), SC_(50.55126123375417963154846480444273825353) }, + { SC_(0.1988578488030194560097782052960341565225e-1101), SC_(50.32166883510982608447898399736737553384) }, + { SC_(0.2274776484360344132543141099781042532126e-1078), SC_(49.79143524779148474319978052812994435046) }, + { SC_(0.5627666511768513324968956052244126556016e-1045), SC_(49.01338231598138050658354300270473769631) }, + { SC_(0.1825674918577823738426304992234296317652e-1041), SC_(48.93085667837639283695386917651720233174) }, + { SC_(0.3802925527627705362359485795331572740116e-1032), SC_(48.71115012911996883239788333878826905934) }, + { SC_(0.704333512302902280929638896129066471028e-1023), SC_(48.49165994415978768781022851843247093366) }, + { SC_(0.1313561715183354223096034930474262234265e-978), SC_(47.429170762115701641491822625509511735) }, + { SC_(0.1479971733547495818402245675295016370485e-972), SC_(47.28207432819548183826826245137119654147) }, + { SC_(0.1347434440250964238000754812764303681229e-956), SC_(46.89195339639151403964904162145015071721) }, + { SC_(0.1121116975652110010815188524143927325837e-910), SC_(45.75090030899601472255890798805286514432) }, + { SC_(0.6537208325713531384544436742840979538836e-903), SC_(45.55510822465697895367331173242574718725) }, + { SC_(0.5260776239804675672628897143432967891746e-862), SC_(44.50957576907949796448736564849939576426) }, + { SC_(0.9950661094879265377710346605980409852453e-836), SC_(43.8248018686713805421278904903492358615) }, + { SC_(0.2062512012152859485600938747081581263563e-777), SC_(42.26547837659956767422521162018458102221) }, + { SC_(0.1860519246978454733361447959343549701547e-773), SC_(42.15763210320445259646388169147606822404) }, + { SC_(0.292259268934242545722820123169012540872e-772), SC_(42.12496313029953419941505368125427033208) }, + { SC_(0.2655070971534510004697091453590968778995e-739), SC_(41.21462174108933535284728637920523260807) }, + { SC_(0.2901429667218951401071997769907995675312e-725), SC_(40.82070209799865627866597450674844815644) }, + { SC_(0.2691466393457144658729027931985645462202e-709), SC_(40.36799005611971249179085189213378251021) }, + { SC_(0.3197098867400748771350059689777583564847e-690), SC_(39.82043439287531202279866114834532850691) }, + { SC_(0.509206568051986013963235738541488132177e-677), SC_(39.43700823628539555409237521877515418161) }, + { SC_(0.3582760294415859512291360611861492812312e-668), SC_(39.17795922357671427001830102855480201337) }, + { SC_(0.7456401788156278694200104791854742334257e-668), SC_(39.16860717559069454303739314459059279729) }, + { SC_(0.1879442582827104576451480885999452191966e-655), SC_(38.80249559566898835529327655718231853047) }, + { SC_(0.3684861623635251188317018685920580219909e-651), SC_(38.67497066678513891028432141977632830295) }, + { SC_(0.5300532049641075447010094275260357817648e-632), SC_(38.10059865129674828052108905587617617209) }, + { SC_(0.254919090870011244639387672109478129648e-628), SC_(37.98921207590422610976320034158408446716) }, + { SC_(0.1261526994355170599813296346837863228294e-566), SC_(36.07180567207263624533709617168647173026) }, + { SC_(0.3348102113183452824057762060804616114175e-551), SC_(35.57631428343251267029769551692709310927) }, + { SC_(0.3615878633639651283997811017708582782249e-540), SC_(35.21759342981655987654847385348063502236) }, + { SC_(0.8556456214239050595803534051499578042028e-537), SC_(35.10716335050071391896400132832542452084) }, + { SC_(0.109310372545754088817907678327638735624e-531), SC_(34.93937406241154814427167189866546786437) }, + { SC_(0.6542713735351179620273602253478996592612e-503), SC_(33.97831683163889299935986132301256667123) }, + { SC_(0.2612280308120653787093215442892321688194e-493), SC_(33.65156678099809689208816819469703821624) }, + { SC_(0.1022554993887925308488446602367781896336e-479), SC_(33.18348488984914247206612191269120081123) }, + { SC_(0.2031774862883333653515762991534010669034e-436), SC_(31.64637875978842562095759897711662201891) }, + { SC_(0.6588827942394778508739583209849917530674e-435), SC_(31.59139083855039387728200786192864837174) }, + { SC_(0.3663469071718277378831360961951773982499e-412), SC_(30.75175430540050080137890276895073229866) }, + { SC_(0.1361305298482799765369196276048348425522e-395), SC_(30.12535428053783863012447180563559906292) }, + { SC_(0.3744171560141971093473580527502555697533e-390), SC_(29.91687192152261750809710487517869880922) }, + { SC_(0.6874405809297488106605573621300258853644e-383), SC_(29.63617622599272249872842949718868018609) }, + { SC_(0.1459704391344155418984387882925473942446e-354), SC_(28.51518995595697899285114107951299228827) }, + { SC_(0.143123157725799928952051937462499680388e-336), SC_(27.77976460222037023739857761278401004959) }, + { SC_(0.9025996658203895649285047801834901806925e-333), SC_(27.62194204731470015756869063464069373815) }, + { SC_(0.1293287994487535182921055345118992149121e-330), SC_(27.53198333089873848069767951797741434292) }, + { SC_(0.1040364789689237257780791904879173249974e-287), SC_(25.67656975866313564221220720210665888424) }, + { SC_(0.2450719147172718682892603854216257656211e-282), SC_(25.43473899116627259523415573506406682191) }, + { SC_(0.1540635014300999959290867576988859893076e-280), SC_(25.35326739382267978364899192426764093558) }, + { SC_(0.1611739209855438499046222515484436760851e-273), SC_(25.03273169525729329624232590800413859205) }, + { SC_(0.2781078901633922872827030068281452871724e-257), SC_(24.27512000889987237167959986465966890736) }, + { SC_(0.8837820681846044382839818307987545000847e-230), SC_(22.93495456229819340317355998004446065241) }, + { SC_(0.2845511846316072211634522373099423142668e-217), SC_(22.29887672117465208331735160799934601323) }, + { SC_(0.662011186385949636511655220013184768187e-212), SC_(22.02033498263166080103335807537366542004) }, + { SC_(0.1113036584088835972703336205837326051999e-204), SC_(21.63966022176108818726656867477709317659) }, + { SC_(0.891278753475701910699424577762349195273e-195), SC_(21.10677883438545215904960900943812596651) }, + { SC_(0.1243375606125932867518669565282002251187e-191), SC_(20.93474611896499049089384377937974962862) }, + { SC_(0.963481566993839758726177317091390488753e-188), SC_(20.72000418866581389344212197908962674052) }, + { SC_(0.1401265332225531798315780528328964001988e-167), SC_(19.56909048260682534539216053987194710384) }, + { SC_(0.7560947579095629715718115752684784187336e-154), SC_(18.74494560773490096872968649468667611647) }, + { SC_(0.2179858592534951368829782528927445425064e-151), SC_(18.59346814776643084908600446605555382182) }, + { SC_(0.5208405693419352209961246987938629746382e-131), SC_(17.28776816936911830963159975455855394785) }, + { SC_(0.5376896865772724579278898126622880802843e-127), SC_(17.01882418428649428645068852272615582504) }, + { SC_(0.1449951754876351411423239477096839160613e-114), SC_(16.15763261758311680195806763260744183267) }, + { SC_(0.3189212836315126918184908007667112228414e-108), SC_(15.70012576642621709878560027430710336585) }, + { SC_(0.1297369886602648502231884634897739722895e-107), SC_(15.65546665388935777226018284593032382368) }, + { SC_(0.2602821327589283546034644516083144727197e-103), SC_(15.33647743152077165227354683273671003413) }, + { SC_(0.1590035594081776210615338926776093483999e-96), SC_(14.81946110818452708436758045284808001241) }, + { SC_(0.2720908074515874075065365442027770927174e-96), SC_(14.80136596875698966601900922593357151598) }, + { SC_(0.3706750818860597777102228512408751197629e-87), SC_(14.07473190298865115531360325271322276549) }, + { SC_(0.3489539664837006993042769801460156282803e-83), SC_(13.74669364199157990777768669458685265472) }, + { SC_(0.1947620269151566018960312643171165602404e-79), SC_(13.43009987431500967622734624289673694188) }, + { SC_(0.5129832297533069185312436951335270626736e-73), SC_(12.86957639029561313119037589037764922829) }, + { SC_(0.9272206234111032518003166213125699095688e-70), SC_(12.57574048509944911276422466059543556873) }, + { SC_(0.1460449453933976163637836336160553873706e-68), SC_(12.46599637375040876166599138465603014844) }, + { SC_(0.8768204308820258161647995326423963318019e-68), SC_(12.39412855962299302779918858089973570208) }, + { SC_(0.3176564512338672251415481009303329209995e-61), SC_(11.77127316747622053532997731227084711173) }, + { SC_(0.1172937774279091349820752991146331290881e-54), SC_(11.11297358832626209702547835299261897395) }, + { SC_(0.3822700066028246575792241970441049035743e-52), SC_(10.85058828114725350216779162566256390403) }, + { SC_(0.4793602326185941396320885530096873382082e-41), SC_(9.607343723542133339487829511083057069273) }, + { SC_(0.106201511009181306109371950926034372112e-40), SC_(9.566077632118694585964030456246300276422) }, + { SC_(0.1026229395621872573137267432392849554594e-24), SC_(7.413111396559360630306454710337113823294) }, + { SC_(0.1433834013674021626308791056277965507134e-19), SC_(6.574536501026477531602186421374903506051) }, + { SC_(0.2287484604245551701383736767135709586086e-12), SC_(5.183672646329402197657171322814515766894) }, + { SC_(0.4054382118091501052221361988694253843776e-11), SC_(4.903978376537783549083010816442468785255) }, + { SC_(0.5517250582467272934968166422809728429772e-9), SC_(4.386634719241854622420456791837035899963) }, + { SC_(0.1708920553446964553659243704866375034024e-5), SC_(3.383585107532389209677350898502297644278) }, + { SC_(0.2892930668406028955896934416060055101499e-5), SC_(3.308042643013470022433136596357284983766) }, + { SC_(0.7029998339557698614154475319387477445787e-5), SC_(3.17687395056466498458108882620439294841) }, + { SC_(0.3973417837559780922949941034264323413971e-4), SC_(2.90551585834893682957467771364140403613) }, + { SC_(0.04246334897368917726269942070072005435577), SC_(1.434684541881674882915735690265484394337) }, + { SC_(0.159920364968560744996532371753339418774), SC_(0.9937250495458864822841753804433979428041) }, + { SC_(0.2425390104583346613758947085681683120129), SC_(0.826370276571483139432927467391341431105) }, + { SC_(0.7954739362140872788414606986417965117653), SC_(0.1832884885283639734589195991287031281603) }, + { SC_(0.9236374866673857278562449757419727802699), SC_(0.06777816075457032123722158048171767182127) } + } }; +#undef SC_ + + diff --git a/test/erfc_inv_data.ipp b/test/erfc_inv_data.ipp new file mode 100644 index 000000000..13ae441bd --- /dev/null +++ b/test/erfc_inv_data.ipp @@ -0,0 +1,110 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 100> erfc_inv_data = { { + { SC_(0.00956696830689907073974609375), SC_(1.832184391051582711731256541599359331735) }, + { SC_(0.063665688037872314453125), SC_(1.311339282092737086640055105484822812599) }, + { SC_(0.068892158567905426025390625), SC_(1.286316565305373898738127895195338854718) }, + { SC_(0.071423359215259552001953125), SC_(1.274755321776344058215704428086211324587) }, + { SC_(0.0728825032711029052734375), SC_(1.268242522387371561738393687518023984868) }, + { SC_(0.09234277904033660888671875), SC_(1.190178756246535875259766567441510867604) }, + { SC_(0.102432854473590850830078125), SC_(1.154826903977823078146497880706118113588) }, + { SC_(0.1942635476589202880859375), SC_(0.9178735528443878579995511780412810469667) }, + { SC_(0.19508080184459686279296875), SC_(0.9161942752629032646404767631869277618212) }, + { SC_(0.21972350776195526123046875), SC_(0.8678064594007161661713535461829067693456) }, + { SC_(0.224929034709930419921875), SC_(0.8580919924152284867224297625328485999768) }, + { SC_(0.2503655254840850830078125), SC_(0.8127923779477598070926819995714374417663) }, + { SC_(0.25179326534271240234375), SC_(0.8103475955423936417307157186107256388648) }, + { SC_(0.2539736330509185791015625), SC_(0.8066326381314558738773191719462921998277) }, + { SC_(0.27095401287078857421875), SC_(0.7784313874823551106598826200232666844639) }, + { SC_(0.2837726771831512451171875), SC_(0.7579355956263440770864088550908621329508) }, + { SC_(0.298227965831756591796875), SC_(0.7355614373173595299453301005770428516518) }, + { SC_(0.3152261674404144287109375), SC_(0.7101588837290742217667270852502075077668) }, + { SC_(0.342373371124267578125), SC_(0.6713881533266128640126408255047881638389) }, + { SC_(0.347730338573455810546875), SC_(0.6639738263692763456669198307149427734317) }, + { SC_(0.3737452030181884765625), SC_(0.6289572925573171740836428308746584439674) }, + { SC_(0.37676393985748291015625), SC_(0.6249936843093662471116097431474787933967) }, + { SC_(0.42041814327239990234375), SC_(0.5697131213589784467617578394703976041604) }, + { SC_(0.4238486588001251220703125), SC_(0.5655171880456876504494070613171955224472) }, + { SC_(0.4420680999755859375), SC_(0.5435569422159360687847790186563654276103) }, + { SC_(0.553845942020416259765625), SC_(0.4186121208546731033057205459902879301199) }, + { SC_(0.55699646472930908203125), SC_(0.4152898953738801984047941692529271391195) }, + { SC_(0.59405887126922607421875), SC_(0.3768620801611051992528860948080812212023) }, + { SC_(0.603826224803924560546875), SC_(0.3669220210390825311547962776125822899061) }, + { SC_(0.61633408069610595703125), SC_(0.3542977152760563782151668726041057557165) }, + { SC_(0.63310086727142333984375), SC_(0.3375493847053488720470432821496358279516) }, + { SC_(0.634198963642120361328125), SC_(0.3364591774366710656954166264654945559873) }, + { SC_(0.722588002681732177734375), SC_(0.2510244067029671790889794981353227476998) }, + { SC_(0.763116896152496337890625), SC_(0.213115119330839975829499967157244997714) }, + { SC_(0.784454047679901123046875), SC_(0.1934073617841803428235669261097060281642) }, + { SC_(0.797477066516876220703125), SC_(0.1814532246720147926398927046057793150106) }, + { SC_(0.81746232509613037109375), SC_(0.1632073953550647568421821286058243218715) }, + { SC_(0.843522548675537109375), SC_(0.1395756320903277910768376053314442757507) }, + { SC_(0.8441753387451171875), SC_(0.1389857795955756484965030151195660030168) }, + { SC_(0.87748873233795166015625), SC_(0.109002961098867662134935094105847496074) }, + { SC_(0.8911724090576171875), SC_(0.09674694516640724629590870677194632943569) }, + { SC_(0.91597831249237060546875), SC_(0.07460044047654119877070700345343119035515) }, + { SC_(0.94951736927032470703125), SC_(0.04476895818328636312384562686715995170129) }, + { SC_(0.970751285552978515625), SC_(0.0259268064334840921104659134138093242797) }, + { SC_(0.97513782978057861328125), SC_(0.02203709146986755832638577823832075055744) }, + { SC_(0.97952878475189208984375), SC_(0.01814413302702029459097557481591610553903) }, + { SC_(0.981178104877471923828125), SC_(0.01668201759439857888105181293763417899072) }, + { SC_(1.0073254108428955078125), SC_(-0.006492067534753215749601670217642082465642) }, + { SC_(1.09376299381256103515625), SC_(-0.08328747254794857150987333986733043734817) }, + { SC_(1.0944411754608154296875), SC_(-0.08389270963798942778622198997355058545872) }, + { SC_(1.264718532562255859375), SC_(-0.2390787735821979528028028789569770109829) }, + { SC_(1.27952671051025390625), SC_(-0.2530214201700340392837551955289041822603) }, + { SC_(1.29262602329254150390625), SC_(-0.2654374523135675523971788948011709467352) }, + { SC_(1.3109557628631591796875), SC_(-0.282950508503826367238408926581528085458) }, + { SC_(1.31148135662078857421875), SC_(-0.2834552014554130441860525970673030809536) }, + { SC_(1.32721102237701416015625), SC_(-0.2986277427848421570858990348074985028421) }, + { SC_(1.3574702739715576171875), SC_(-0.3282140305634627092431945088114761850208) }, + { SC_(1.362719058990478515625), SC_(-0.3334035993712283467959295804559099468454) }, + { SC_(1.3896572589874267578125), SC_(-0.3603304982893212173104266596348905175268) }, + { SC_(1.4120922088623046875), SC_(-0.3831602323665075533579267768785894144888) }, + { SC_(1.41872966289520263671875), SC_(-0.3899906753567599452444107492361433402154) }, + { SC_(1.45167791843414306640625), SC_(-0.4244594733907945411184647153213164209335) }, + { SC_(1.48129451274871826171875), SC_(-0.4563258063707025027210352963461819167707) }, + { SC_(1.4862649440765380859375), SC_(-0.4617640058971775089811390737537561779898) }, + { SC_(1.50937330722808837890625), SC_(-0.4874174763856674076219106695373814892182) }, + { SC_(1.5154802799224853515625), SC_(-0.4943041993872143888987628020569772222018) }, + { SC_(1.52750003337860107421875), SC_(-0.5079978091910991117615000459548117088362) }, + { SC_(1.53103363513946533203125), SC_(-0.5120597685873370942783226077302881881069) }, + { SC_(1.58441460132598876953125), SC_(-0.5756584292527058478710392476034273328569) }, + { SC_(1.5879499912261962890625), SC_(-0.5800336103175463592377907341030447077804) }, + { SC_(1.59039986133575439453125), SC_(-0.5830784871670823806198622501806646778319) }, + { SC_(1.59455978870391845703125), SC_(-0.588273673825686998734497652983815773459) }, + { SC_(1.59585726261138916015625), SC_(-0.5899005483108011364541949539839185473259) }, + { SC_(1.5962116718292236328125), SC_(-0.5903454775096607218832535637355431851718) }, + { SC_(1.6005609035491943359375), SC_(-0.5958247243549040349587326482492767206448) }, + { SC_(1.6150619983673095703125), SC_(-0.6143583249050861028039832921829036722514) }, + { SC_(1.62944734096527099609375), SC_(-0.6331707263097125575937994856370309207836) }, + { SC_(1.64380657672882080078125), SC_(-0.6524069265890823819975498133014027247554) }, + { SC_(1.6469156742095947265625), SC_(-0.656635855345815020063728463464436343698) }, + { SC_(1.67001712322235107421875), SC_(-0.6888269167957872563013714303376548038671) }, + { SC_(1.6982586383819580078125), SC_(-0.7302336318927408409119676651737758401138) }, + { SC_(1.74485766887664794921875), SC_(-0.8046505193013635090578266413458426260098) }, + { SC_(1.75686132907867431640625), SC_(-0.8253191678260588578995203396384711816647) }, + { SC_(1.81158387660980224609375), SC_(-0.9300427626888758122211127950646282789481) }, + { SC_(1.826751708984375), SC_(-0.9629665092443368464606966822833571908852) }, + { SC_(1.83147108554840087890625), SC_(-0.9736479209913771931387473923084901789046) }, + { SC_(1.84174954891204833984375), SC_(-0.997713670556719074960678197806799852186) }, + { SC_(1.8679864406585693359375), SC_(-1.065050516333636716777334376076374184102) }, + { SC_(1.90044414997100830078125), SC_(-1.164612422633086435501625591693259387477) }, + { SC_(1.91433393955230712890625), SC_(-1.215315881176612875682446995412738776976) }, + { SC_(1.91501367092132568359375), SC_(-1.217962731073139868794942852653058932976) }, + { SC_(1.918984889984130859375), SC_(-1.233778505900771488542027767896521427575) }, + { SC_(1.92977702617645263671875), SC_(-1.28019542575660930623179572273596558907) }, + { SC_(1.93538987636566162109375), SC_(-1.306695301483797253764522033930388453334) }, + { SC_(1.93773555755615234375), SC_(-1.318335478463913327121670503572736587296) }, + { SC_(1.94118559360504150390625), SC_(-1.33613349872692113073358883961598631154) }, + { SC_(1.96221935749053955078125), SC_(-1.468821071545234761861756248744372345584) }, + { SC_(1.98576259613037109375), SC_(-1.733272259459038694476413373595347034928) }, + { SC_(1.9881370067596435546875), SC_(-1.77921769652839903464038407684397479173) }, + { SC_(1.99292266368865966796875), SC_(-1.904368122482929779094714951471938518496) } + } }; +#undef SC_ + diff --git a/test/functor.hpp b/test/functor.hpp new file mode 100644 index 000000000..cde300100 --- /dev/null +++ b/test/functor.hpp @@ -0,0 +1,209 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#ifndef BOOST_MATH_TEST_FUNCTOR_HPP +#define BOOST_MATH_TEST_FUNCTOR_HPP + +struct extract_result_type +{ + extract_result_type(unsigned i) : m_location(i){} + + template + typename S::value_type operator()(const S& row) + { + return row[m_location]; + } +private: + unsigned m_location; +}; + +inline extract_result_type extract_result(unsigned i) +{ + return extract_result_type(i); +} + +template +struct row_binder1 +{ + row_binder1(F _f, unsigned i) : f(_f), m_i(i) {} + + template + typename S::value_type operator()(const S& row) + { + return f(row[m_i]); + } + +private: + F f; + unsigned m_i; +}; + +template +inline row_binder1 bind_func(F f, unsigned i) +{ + return row_binder1(f, i); +} + +template +struct row_binder2 +{ + row_binder2(F _f, unsigned i, unsigned j) : f(_f), m_i(i), m_j(j) {} + + template + typename S::value_type operator()(const S& row) + { + return f(row[m_i], row[m_j]); + } + +private: + F f; + unsigned m_i, m_j; +}; + +template +inline row_binder2 bind_func(F f, unsigned i, unsigned j) +{ + return row_binder2(f, i, j); +} + +template +struct row_binder3 +{ + row_binder3(F _f, unsigned i, unsigned j, unsigned k) : f(_f), m_i(i), m_j(j), m_k(k) {} + + template + typename S::value_type operator()(const S& row) + { + return f(row[m_i], row[m_j], row[m_k]); + } + +private: + F f; + unsigned m_i, m_j, m_k; +}; + +template +inline row_binder3 bind_func(F f, unsigned i, unsigned j, unsigned k) +{ + return row_binder3(f, i, j, k); +} + +template +struct row_binder4 +{ + row_binder4(F _f, unsigned i, unsigned j, unsigned k, unsigned l) : f(_f), m_i(i), m_j(j), m_k(k), m_l(l) {} + + template + typename S::value_type operator()(const S& row) + { + return f(row[m_i], row[m_j], row[m_k], row[m_l]); + } + +private: + F f; + unsigned m_i, m_j, m_k, m_l; +}; + +template +inline row_binder4 bind_func(F f, unsigned i, unsigned j, unsigned k, unsigned l) +{ + return row_binder4(f, i, j, k, l); +} + +template +struct row_binder2_i1 +{ + row_binder2_i1(F _f, unsigned i, unsigned j) : f(_f), m_i(i), m_j(j) {} + + template + typename S::value_type operator()(const S& row) + { + return f(boost::math::tools::real_cast(row[m_i]), row[m_j]); + } + +private: + F f; + unsigned m_i, m_j; +}; + +template +inline row_binder2_i1 bind_func_int1(F f, unsigned i, unsigned j) +{ + return row_binder2_i1(f, i, j); +} + +template +struct row_binder3_i2 +{ + row_binder3_i2(F _f, unsigned i, unsigned j, unsigned k) : f(_f), m_i(i), m_j(j), m_k(k) {} + + template + typename S::value_type operator()(const S& row) + { + return f( + boost::math::tools::real_cast(row[m_i]), + boost::math::tools::real_cast(row[m_j]), + row[m_k]); + } + +private: + F f; + unsigned m_i, m_j, m_k; +}; + +template +inline row_binder3_i2 bind_func_int2(F f, unsigned i, unsigned j, unsigned k) +{ + return row_binder3_i2(f, i, j, k); +} + +template +struct row_binder4_i2 +{ + row_binder4_i2(F _f, unsigned i, unsigned j, unsigned k, unsigned l) : f(_f), m_i(i), m_j(j), m_k(k), m_l(l) {} + + template + typename S::value_type operator()(const S& row) + { + return f( + boost::math::tools::real_cast(row[m_i]), + boost::math::tools::real_cast(row[m_j]), + row[m_k], + row[m_l]); + } + +private: + F f; + unsigned m_i, m_j, m_k, m_l; +}; + +template +inline row_binder4_i2 bind_func_int2(F f, unsigned i, unsigned j, unsigned k, unsigned l) +{ + return row_binder4_i2(f, i, j, k, l); +} + +template +struct negate_type +{ + negate_type(F f) : m_f(f){} + + template + typename S::value_type operator()(const S& row) + { + return -m_f(row); + } +private: + F m_f; +}; + +template +inline negate_type negate(F f) +{ + return negate_type(f); +} + +#endif diff --git a/test/gamma_inv_big_data.ipp b/test/gamma_inv_big_data.ipp new file mode 100644 index 000000000..89975a2a8 --- /dev/null +++ b/test/gamma_inv_big_data.ipp @@ -0,0 +1,141 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 130> gamma_inv_big_data = { { + { SC_(464.56927490234375), SC_(0.12698681652545928955078125), SC_(440.0905015614498663381793089656310373835539712118699915057655756381750702144551901256933419148259778), SC_(489.2489328115381710888005650788332818303123263238882523231037266766796974176786885492715539575179927) }, + { SC_(464.56927490234375), SC_(0.135477006435394287109375), SC_(440.9201626373476098986751145295281211597851207555116002836559277456581502883881279408150408804624974), SC_(488.3596775997958518494466317847702951287228545295142304855338082929544530999908165788368449119633373) }, + { SC_(464.56927490234375), SC_(0.22103404998779296875), SC_(447.8707097559117500373730010123474103179731538915785876172851019700931527209673015311979439047788638), SC_(480.9951697728710868281134950146803822897647194680614404153327850687509247330919989892805468881282719) }, + { SC_(464.56927490234375), SC_(0.308167040348052978515625), SC_(453.5243991344879957369314553806138325182261928645340021307574697842383156443940903587657239443012288), SC_(475.1149279136111828287528971449692848991760379145030491172523136909085503134739001275580865573262267) }, + { SC_(464.56927490234375), SC_(0.6323592662811279296875), SC_(471.5586320968249281038445929212575977142103853999994962593917575305139999808448747718468071828960383), SC_(456.9895432265540105242329036050454287207764722373079606619854048979024366594658280592117689264157992) }, + { SC_(464.56927490234375), SC_(0.814723670482635498046875), SC_(483.7962479352174443104604814361337114707603182758678093812861642605333000597647127303417718754975292), SC_(445.2102204107761028803290781726535172344996283727995749912027305150811353997318714622791119089755303) }, + { SC_(464.56927490234375), SC_(0.835008561611175537109375), SC_(485.5413501990585182267303678139012739812082864500829920491986081765032489089832623030329596860813566), SC_(443.563211692283275278200948788175382734228099875239143691921387947329995022394758603528566488084557) }, + { SC_(464.56927490234375), SC_(0.905791938304901123046875), SC_(493.1530014001506637843558286915347874415133013116095848032725512370251487272101090417362802627532), SC_(436.4721607137502194476515600944759096109462004602962052350793459815125792956858001222848555476140453) }, + { SC_(464.56927490234375), SC_(0.9133758544921875), SC_(494.1980223559677419266737342424067342426878050583575377942941770747042395000546795305409031642061286), SC_(435.5102279299145333712817106435136572491533987684414157140918464149027877988350441152775081891128202) }, + { SC_(464.56927490234375), SC_(0.968867778778076171875), SC_(505.5712348542740722018328982886099757924348908530379539931536263900192339710371141619690805609322114), SC_(425.2177557338328763751700654579137573643963906759571233052948938600914520447536651895743912216230868) }, + { SC_(581.3642578125), SC_(0.12698681652545928955078125), SC_(553.9669793782038068678935676734904494531472719532817972687153514964534286624955148869921984604413705), SC_(608.9624188837483057145747405401351852945713877500076249931706854984537278152556192058495450356661075) }, + { SC_(581.3642578125), SC_(0.135477006435394287109375), SC_(554.8986621587098884100882390919729772471677865482900127899068620712298280708065488221682075356651129), SC_(607.9711405772914939464180467890095695334450990881469482983020058157368245651758798593316096874550468) }, + { SC_(581.3642578125), SC_(0.22103404998779296875), SC_(562.6989508379479360762737530017662823397707943257475597443912537342208024566957230695521899983347868), SC_(599.7568829483629048923260578901093044774886065740947188058513300551937073309051987160205195739811767) }, + { SC_(581.3642578125), SC_(0.308167040348052978515625), SC_(569.0374062500558452762422305261339987947322576241495557758885978701225529419403450771958235312394718), SC_(593.191871592557187889891303913053953793280772650663480941873292317612622398375064237989516603392702) }, + { SC_(581.3642578125), SC_(0.6323592662811279296875), SC_(589.2186806341801070520302845910583423903524143672474575926325599791162545901801711272745264959036703), SC_(572.9194442994055998344224745611620692130093494754880686659071973997039947893559220289586718667113235) }, + { SC_(581.3642578125), SC_(0.814723670482635498046875), SC_(602.8821834284482585691264752294355355054448218400550509491940689285405704202939688051624635513663289), SC_(559.7142416966026142699409579688200021128645733168138617853078743374086688135798773103892188217158995) }, + { SC_(581.3642578125), SC_(0.835008561611175537109375), SC_(604.828643910421477172904091611031783808541641171718978680317644118217631106164554044832916318314536), 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SC_(106814.1655385261881930404229732982932315967031144349630174728684365233416506163298048686969756243235), SC_(107141.9289142929117445526105750926681991067546071431542916760994950589383960650401663422672269967697) }, + { SC_(106978.296875), SC_(0.6323592662811279296875), SC_(107088.5883939339366789132635543218784222126363076723929464152640936921360514979892904256754782836202), SC_(106867.4149012959514842871740820819912876035326559814362503394815886893461205537905345858197524921503) }, + { SC_(106978.296875), SC_(0.814723670482635498046875), SC_(107271.1062601546124727026601294184511892667252827484197615804221472642319275593094402184268802800732), SC_(106685.355363598826654695221788222612153635895098179662024013576731553208328075853329795574072474843) }, + { SC_(106978.296875), SC_(0.835008561611175537109375), SC_(107296.8992994913869402418466674420311219379899864632641552560933440701747303396252073712935466163884), SC_(106659.6604273573002063815229431263566237748637743622261182212981422323668756139498467915581435044571) }, + { SC_(106978.296875), SC_(0.905791938304901123046875), SC_(107408.734998923265541925220213930532489899427477954826595105190870582743619595466816264015557152955), SC_(106548.3453903465392932839969654246405965260624999938958828309703296188141318404673733944550378196684) }, + { SC_(106978.296875), SC_(0.9133758544921875), SC_(107424.0055670202970824638449232697216412857488407216160829379996654120213895137065538290553524438463), SC_(106533.1579221417583551950937315667626037830032018763190969794071719898382331654652699608236673323027) }, + { SC_(106978.296875), SC_(0.968867778778076171875), SC_(107588.9235613222152829931679225549014754124395269284113443203255103948124031662805528602664547683086), SC_(106369.3208655605404481804038740303298105183580397635384112856884253986898629793630975440978403633314) }, + { SC_(171464.0625), SC_(0.12698681652545928955078125), SC_(170991.7987738160400661711532241368523815758482376794424363282331416985687517598530113021351567322652), SC_(171936.5271011605645630168791502834268148052500147715712053943581753043298682506567262206437629713908) }, + { SC_(171464.0625), SC_(0.135477006435394287109375), SC_(171008.2836354564209492823743402973811390337608777526702016261413773534660200796449809982051665077003), SC_(171919.9826384883939702551089975861816589350398843659033957710496659891252213316559001815815327400668) }, + { SC_(171464.0625), SC_(0.22103404998779296875), SC_(171145.6191662746123960347552525693670128191028466757466580336222001947839794403773558314059941595646), SC_(171782.2331060803849520335453303183326526337638884076039027498474047799995742641818132862456747537524) }, + { SC_(171464.0625), SC_(0.308167040348052978515625), SC_(171256.3361047097958387452441541471127499385772793150774116115411855101108847691601832573807619567347), SC_(171671.2895979873130236903578632050660794939876719454087826341290089368450153075267593819684360323877) }, + { SC_(171464.0625), SC_(0.6323592662811279296875), SC_(171603.7718023434324974514302206535409055277466973477498540819203050822468613895559823545353022871572), SC_(171323.7627427548550174468619344221281973249811264358000342338320713853780519662237808191551395326248) }, + { SC_(171464.0625), SC_(0.814723670482635498046875), SC_(171834.7812554227987093134480125253002251121865641876340406728469955856798309411540595789975061266601), SC_(171093.2116182572157852580015242631784731817460262145817086170604362162178153569504512508225251352553) }, + { SC_(171464.0625), SC_(0.835008561611175537109375), SC_(171867.4225853696747287532499924362369112417451458992299610929008452549979384474959400833956401103073), SC_(171060.6683914212385500445147373534713684868992048983316972221307056329253829244630268336756229649048) }, + { SC_(171464.0625), SC_(0.905791938304901123046875), SC_(172008.9390039799149245319369194588737700358279596931623900607969038096782602911984385367690545112371), SC_(170919.6726353340561334782062787174007300952740547857469982263053520346768626380151551392105674125141) }, + { SC_(171464.0625), SC_(0.9133758544921875), SC_(172028.2607163503237074804240420988332805635207751072405443517988230279811664015979896111754298047853), SC_(170900.4340228751077815276338342892187005146619881755020781374282366220361628632145020859022759543646) }, + { SC_(171464.0625), SC_(0.968867778778076171875), SC_(172236.905515185348613520159589244969601354143298301884427671166903159884473708788358619633391766896), SC_(170692.8701620843835232223758792946766066205588069757104160071945587119499747986058865001180462595334) }, + { SC_(287713.625), SC_(0.12698681652545928955078125), SC_(287101.8390891334881074097768409639811422105174055074247663991980671146735409214228746030537490827894), SC_(288325.6117858325819400290984010245379181857090949256951183079858217963457877563186041586190041828876) }, + { SC_(287713.625), SC_(0.135477006435394287109375), SC_(287123.2018824342060405509691117625160296096920082481743462259847740285682319170194185122522020998578), SC_(288304.1893914925111228677903149683935820829058722939396270910064082845368520684242652738579958079628) }, + { SC_(287713.625), SC_(0.22103404998779296875), SC_(287301.1633297324049890437619176952336530543502592319387571077608781438650607731795664155648991042492), SC_(288125.8139425596597585955824696111531518443099937996959008431783465244705148305958926033655366242125) }, + { SC_(287713.625), SC_(0.308167040348052978515625), SC_(287444.6161875962380911288819205876894144456711595429288695779716233686378459491093953007953541560695), SC_(287982.1345150191015439663126413778115578427703552946723363867416179375439869486600021290202033928835) }, + { SC_(287713.625), SC_(0.6323592662811279296875), SC_(287894.6874489179014071783111281890207171400441598448189286641946463264879755117085964912831960132818), SC_(287531.9720960921756986707071354480752766558762675841713700852373467579201294942464262449709073610385) }, + { SC_(287713.625), SC_(0.814723670482635498046875), SC_(288193.8624685657253120713231978071896495272422042999916280573686361169253355069989533169539117598378), SC_(287233.2554050650734668559921717066438674227127230795422265297442527832155003247585279503101394869551) }, + { SC_(287713.625), SC_(0.835008561611175537109375), SC_(288236.1305748677261264763139679877739018298385456598709163321801212462074437452189204330390376852663), SC_(287191.0854018844619610439948437360860177127073148068576003551456242291546450349862447341532794138293) }, + { SC_(287713.625), SC_(0.905791938304901123046875), SC_(288419.3697666997195573795607481350142848162288944628671823192696454698912657762168311546144078778697), SC_(287008.3668726438555694570198503384645300805316567228857999141205296735618456088406804345883798515772) }, + { SC_(287713.625), SC_(0.9133758544921875), SC_(288444.3862532392817919544710751213857359531657572985942506273978341080423727813156579392633735805302), SC_(286983.4334860286298928127732334989709654886825090910274253943356470926729788108952795775884980653786) }, + { SC_(287713.625), SC_(0.968867778778076171875), SC_(288714.4987040958494150791768379727138187987696180207795301950993211293148528792180512566346062773377), SC_(286714.4019734332685576845212232609186614672218264529859295616546913400761609253508670570311556417372) }, + { SC_(811189.1875), SC_(0.12698681652545928955078125), SC_(810161.8590312371865843168933980041081328325710070299088178861493265134389880308605518391017994747884), SC_(812216.7168437188563979284647249628251019755537428376273766031544801981839282189973470746075052877665) }, + { SC_(811189.1875), SC_(0.135477006435394287109375), SC_(810197.7498979269406216229333830071350035973327816952880714391211131397799458403161724415876017573062), SC_(812180.7663759825506891792362570130089364226324810883040952150567099852012280358577738543974186957542) }, + { SC_(811189.1875), SC_(0.22103404998779296875), SC_(810496.7085818541638251177278868436507795786808493806841003579619404032620103009446072000163980337239), SC_(811881.3936903780003325280400757211189636198717447205229093798838652404976779035708631850160770953495) }, + { SC_(811189.1875), SC_(0.308167040348052978515625), SC_(810737.6596810209441188397088671254688261035908114807153029865438117474573459893950971804983662488815), SC_(811640.2160215165658363785038485826735353144913795793280443540697836953543164349103546800608981662806) }, + { SC_(811189.1875), SC_(0.6323592662811279296875), SC_(811493.4130930334681784464934025270672115225675384515035359787878148824734579705400730129452548515192), SC_(810884.3714518926483987655962044487674578474311600687306107087577719289978417856154576706566527627155) }, + { SC_(811189.1875), SC_(0.814723670482635498046875), SC_(811995.6073944837810507278595067896837851547513348796820987055379061808847590525648652941945063796637), SC_(810382.6354791001731534931504185529321689341432235194142157415772145748880181384490815012739271443493) }, + { SC_(811189.1875), SC_(0.835008561611175537109375), SC_(812066.5471821809268774383895112728062329337775853423876890314159367663267493240015649616128193898023), SC_(810311.7937945344017345682607983665585726031514332575059314866957462286550595648052571524209586357645) }, + { SC_(811189.1875), SC_(0.905791938304901123046875), SC_(812374.0504188456832837387624199860361643216611730814901346962998291782800792848840022126928346732626), SC_(810004.8112205260695868832902010461963334632154865047659274031951119044731278686388069124903893504951) }, + { SC_(811189.1875), SC_(0.9133758544921875), SC_(812416.0278043274019558628676997090748402578371572470019350929977553824684311176493949507102425008448), SC_(809962.9169349809432409688756513667346638313258479402030261893546740734029634599290918129699775355502) }, + { SC_(811189.1875), SC_(0.968867778778076171875), SC_(812869.2110457723077173766634087729640503055024244625964488641332722990514021020625997363526699219015), SC_(809510.8146320036991512143859204546175464755726797868383946222879006660812841804439525397298317135306) }, + { SC_(1340602.5), SC_(0.12698681652545928955078125), SC_(1339281.789217310227603638956713067406749799202177619440960783591278712272782687984485318465325634698), SC_(1341923.411657643639007333777049618448197159121016565415209623831539285788844610953752838000410332646) }, + { SC_(1340602.5), SC_(0.135477006435394287109375), SC_(1339327.937228622487435434723558076320553619035659265800295103113312125271806499402498542109463722967), SC_(1341877.204045283265012388313242013019815575145051478003059780209276879235388989175125836420473319923) }, + { SC_(1340602.5), SC_(0.22103404998779296875), SC_(1339712.322663835647115463627093526039417214514450079134571740611687493566900248789753047878427302874), SC_(1341492.404608383515658278630627107716714934192755525237250501053389901217595022258339024841963265735) }, + { SC_(1340602.5), SC_(0.308167040348052978515625), SC_(1340022.109695761389373850715324652137775669433408208271739480026258057424153514144279448404087341583), SC_(1341182.391006759227701894203217959369703988360879936035471868679603927201823099528589868185221044072) }, + { SC_(1340602.5), SC_(0.6323592662811279296875), SC_(1340993.681516583716822573407774790571084098092620222888109700458969964450847155204734529531816156073), SC_(1340210.728028324176179123620867989801125006955369657205955659737293424313505301822663996693364643671) }, + { SC_(1340602.5), SC_(0.814723670482635498046875), SC_(1341639.21195865400971691711247042743964641649950184497692636094105194253010661428811607626202482777), SC_(1339565.655914919776901964481359375383609125899415767739147337874327439960785358915178860477236871205) }, + { SC_(1340602.5), SC_(0.835008561611175537109375), SC_(1341730.394593718344730348431912807931874774159152781393115335369100867655446425097778085615493303823), SC_(1339474.571382988983554873353395399477444166684037486807114439003468351504601181586784778488843724471) }, + { SC_(1340602.5), SC_(0.905791938304901123046875), SC_(1342125.631024741747482365874818097900648331829089766546145385089458404019214871024479928583367510744), SC_(1339079.85561463612134401235478017313574989598767112679109272711123705107621660725882816550403989336) }, + { SC_(1340602.5), SC_(0.9133758544921875), SC_(1342179.58318466684205310449571855171979481228479144098285572172412554068281390306320099547841289301), SC_(1339025.98655465027920550229578505314111660672260003165765196039783266546290600690587108716340600317) }, + { SC_(1340602.5), SC_(0.968867778778076171875), SC_(1342762.018513537047540360796759927612282441879039821617202574156172598071372824957647151677236387446), SC_(1338444.632164292546392718217459403862675869956866545071666772833839055979245806235185884461105636674) } + } }; +#undef SC_ + + diff --git a/test/gamma_inv_data.ipp b/test/gamma_inv_data.ipp new file mode 100644 index 000000000..e397a4619 --- /dev/null +++ b/test/gamma_inv_data.ipp @@ -0,0 +1,211 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 200> gamma_inv_data = { { + { SC_(9.754039764404296875), SC_(0.12698681652545928955078125), SC_(6.349849983781954486964960115093039567468747967664820851534065876063014180392060424555352423162443804), SC_(13.35954665556050630769710564071384909890705633863167415813992116806093647564994418684947673739937248) }, + { SC_(9.754039764404296875), SC_(0.135477006435394287109375), SC_(6.443655518427200469944002789678415067225918847146256390238048987798948404483563975737891105237977816), SC_(13.20648307403762788665455912344161337928185855631083250140430065533748994841668643195553261756810763) }, + { SC_(9.754039764404296875), SC_(0.22103404998779296875), SC_(7.262299107203547511919478506673940833772294421637239641242141768466064388735007729524335022945474356), 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SC_(0.135477006435394287109375), SC_(88.40714494536321930498314541057705121558781002694191911704447280462237009318667590621848989831217393), SC_(110.3104697481858991610699642415404049205540549382020546666969615332025792346398477535378399730503648) }, + { SC_(99.2881317138671875), SC_(0.22103404998779296875), SC_(91.50599625629075695333925558823944000033068037616404547594790241831618127835838880946155946667230488), SC_(106.7978091882579180957227826327399587317254598205887947452130155885496447480794670408964903808852888) }, + { SC_(99.2881317138671875), SC_(0.308167040348052978515625), SC_(94.05542966722570497040226539391420130180692037513060042769613095280747364242278963119073910822820164), SC_(104.0218869621341187749106671918013336314305862052064089048569771555329392581844795819010636668843943) }, + { SC_(99.2881317138671875), SC_(0.6323592662811279296875), SC_(102.3556282071769826944006172794969028033974462498658650696807922850126796685689047547579596993931716), SC_(95.63055847544334211087138998067211381097918211020892885108659046928523862524001913745220040268349381) }, + { SC_(99.2881317138671875), SC_(0.814723670482635498046875), SC_(108.1291543562602928024304751748147804453998345888983839614623715132630298762313311048580828345727242), SC_(90.31519355853058782089312287195256334067426270863324117470164569056437169804425894072388785174414288) }, + { SC_(99.2881317138671875), SC_(0.835008561611175537109375), SC_(108.9615144259886825743527192622351022097444435851716881383594981271001329308716835208190947446502812), SC_(89.58089159449409507162289598004762894727083770234022756966805353276045874719787836940516270710021009) }, + { SC_(99.2881317138671875), SC_(0.905791938304901123046875), SC_(112.6179577986881681054950590064902019242389712272583112203724181108477339819399828823424603515837391), SC_(86.44481769564459753753709972148206867820207960268756150487253069049902935247326722392696033050460213) }, + { SC_(99.2881317138671875), SC_(0.9133758544921875), SC_(113.1232225034369918026921896528672269476075103218287973199007233652864695087604791431815686564542933), SC_(86.02259769219673888125653444633780832251026932468113275214161763610722920488400091593471887527263333) }, + { SC_(99.2881317138671875), SC_(0.968867778778076171875), SC_(118.6722260958635779664708398764065569812316717572450903065068638638609506808950759768663333735117402), SC_(81.55360245504441686527910864220010764542026336630740606229239169661041788764418800722537023927646871) }, + { SC_(99.6461334228515625), SC_(0.12698681652545928955078125), SC_(88.37737888107073636627500424906781041431096054016259174725751830523083014165218482437664265598787634), SC_(111.1158076628131535311143363083607562906226185676374371847096774914864724460972364755857498710894997) }, + { SC_(99.6461334228515625), SC_(0.135477006435394287109375), SC_(88.74535620460806340987854361816128428024320481905389016348646200088835179377114089479753531259500676), SC_(110.6882616289176406115134461249367870803764967094106173548894808946298474490315397099426966057522927) }, + { SC_(99.6461334228515625), SC_(0.22103404998779296875), SC_(91.85017604570320860123048741031650279971434577969683060829287638362971682925038461612569617055793725), SC_(107.1696318444364396459981501446757609223861217010281426345016448193737147409852464617600305678038153) }, + { SC_(99.6461334228515625), SC_(0.308167040348052978515625), SC_(94.40442088829506749763117819247805230794301031739748106412466854828283371649057563467379541863694276), SC_(104.388897894676876174661084670063039489707876655517563928756296867121616845615313628471178466624892) }, + { SC_(99.6461334228515625), SC_(0.6323592662811279296875), SC_(102.7197092844094654604597714681631790586173807125641768128671855549332709686069679473684528779727183), SC_(95.98247945192366927890502346818486620765311827085464016312581482129126586465008442638253208379782958) }, + { SC_(99.6461334228515625), SC_(0.814723670482635498046875), SC_(108.5032548424476463598183504899336466748975635468938732532626605779594870328299994789652482831953253), SC_(90.6570957304451106144185164419238349454797081258790911488865971436148449646391904615295180665722972) }, + { SC_(99.6461334228515625), SC_(0.835008561611175537109375), SC_(109.3370293976377603308634730796822871722180822156146079704298307111056783571551863417375442399229845), SC_(89.92137944342759462480459816579961426775458462692578398945994611785848960085110828841885491701675714) }, + { SC_(99.6461334228515625), SC_(0.905791938304901123046875), SC_(112.999601070860905668295141701157176040361351555369172846468529901685867407811198602410255587767563), SC_(86.7791783017191311959905806085341928361560825596664875827422581491589972224048336103416333886978357) }, + { SC_(99.6461334228515625), SC_(0.9133758544921875), SC_(113.5057019266017757553572395745632780679072776470909404929295494401706417401173397780562513398354094), SC_(86.35612234649621027962836929188910335213017738233582627676662911520231216242529540127044730529945437) }, + { SC_(99.6461334228515625), SC_(0.968867778778076171875), SC_(119.0637249422376961671341002862243780120617645758579051054825546123558214039984323476889351592427586), SC_(81.87811103947525043911330324465564757254090604246213044539733449282580273670542563644266281418367064) } + } }; +#undef SC_ + + diff --git a/test/gamma_inv_small_data.ipp b/test/gamma_inv_small_data.ipp new file mode 100644 index 000000000..936577b9d --- /dev/null +++ b/test/gamma_inv_small_data.ipp @@ -0,0 +1,239 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 229> gamma_inv_small_data = { { + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.12698681652545928955078125), SC_(BOOST_MATH_SMALL_CONSTANT(0.2239623606222809074122747811596115646210220735131141509259977248899758059576948436798908594057794725e-517862)), SC_(BOOST_MATH_SMALL_CONSTANT(0.4348301951174619607003912855228982264838968134589390827069898370149065135278987288014463439625604227e-34079)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.135477006435394287109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.9832661142546970309065948494116195914044751609094617663675341704778529043412686011701793912754530322e-501622)), SC_(BOOST_MATH_SMALL_CONSTANT(0.174464879621346471044494182889773112103066192989857880445657763562407515813032064473382568887549155e-36531)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.22103404998779296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.8367188988033804556441828789142666841098768711906267440588049611090664598162301930590308375972597743e-378782)), SC_(BOOST_MATH_SMALL_CONSTANT(0.258455732678645501224573885451125893770282157149004436646252270592694652815341234774299743101856032e-62682)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.308167040348052978515625), SC_(BOOST_MATH_SMALL_CONSTANT(0.2205873777306860224249147129964654825654772808335654855092638386176586877302351181235652317545193535e-295387)), SC_(BOOST_MATH_SMALL_CONSTANT(0.8841860184607764624716199689352601821223421516721282102648409952215997935928956722103353938434066324e-92450)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.6323592662811279296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.3493006236342872627718268547235558974110453236408370553242260146873678814721124761086879547194341511e-115006)), SC_(BOOST_MATH_SMALL_CONSTANT(0.4608598583105815395151633406100700185774288566360162761619249236808047706604292396779780216945145655e-251105)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.814723670482635498046875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1307814977375223234061697346935750965506413900088172878810937417241710162327371948752530053786611785e-51419)), SC_(BOOST_MATH_SMALL_CONSTANT(0.3622247004413029118815942894731774856416774764001901351075227450895011592470141353244995696694896319e-423065)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.835008561611175537109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.3407200009988954922968231040073535782592944660979975072820794043769535120578823528671413904276314187e-45248)), SC_(BOOST_MATH_SMALL_CONSTANT(0.40372680297867266879468280760885460454290870234025769400910020388606341293084660523346882864736631e-452163)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.905791938304901123046875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1289547938840298804104533289571370292449761595410863583261482629787772675014213604864800473713128265e-24829)), SC_(BOOST_MATH_SMALL_CONSTANT(0.4473385981314550569653428208559396173592281928696895222750064028650962596615670418178431916147100131e-592788)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.9133758544921875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2660916718277898648353162270511365962890563186837505224306478516822433113175485818467727525519960066e-22737)), SC_(BOOST_MATH_SMALL_CONSTANT(0.6065564736497368801762295026989180367578487822196046971460097223432141011146248705024295896736469553e-613849)) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.968867778778076171875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1473132778697774093465699897329346789995005439716750867355563556776005696738800270428366009192934656e-7936)), SC_(BOOST_MATH_SMALL_CONSTANT(0.6008673144885279892931053509531579477623416731984800692761855880946172764550286223383877167245766902e-870647)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.12698681652545928955078125), SC_(BOOST_MATH_SMALL_CONSTANT(0.8500545195325853360129687030255002945655259106744411868936199400386357779413785394252606937409149347e-413825)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1168004659330719336985231445704960139768804061229080049955849277439667205757483213992057220049540351e-27232)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.135477006435394287109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.7202712693841376368216354323694296792211539722228687704587193232171864702453427173668794173358037833e-400847)), SC_(BOOST_MATH_SMALL_CONSTANT(0.2252614264634486253201447701279415032436991420138140648287940022937659078016357382571415445200213866e-29192)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.22103404998779296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.3000327882810984063485886505392286395644270610646887344028999837107973933925695988567563167728045096e-302685)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1498490062697145666757563645833685565995577423177516513148297134617720423214544808430852418820345691e-50089)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.308167040348052978515625), SC_(BOOST_MATH_SMALL_CONSTANT(0.1307004433406758598014428627573055623164994782163856086219980606033295840179348304515381612579150121e-236044)), SC_(BOOST_MATH_SMALL_CONSTANT(0.8660761056125063203743178336674650301770491595301193399539391174485981887085468404902606561784465577e-73877)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.6323592662811279296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1183316844216403526657583856991948601012391672622156410900987978451851712537691445331530312832797228e-91901)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1591103740030003499118723418460479045052885251014777131066205491018123438721843692240137750119143378e-200658)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.814723670482635498046875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1674134325233505112606127764095613275142520744956844047709789044616079191515075547796473275770297974e-41089)), SC_(BOOST_MATH_SMALL_CONSTANT(0.3612384181299129469928622773435995792869558303198558837492185251476669534034831500723829680749590518e-338072)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.835008561611175537109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.6513834432351054603935790801928696636769937839154029815627322893900997971153735962011923847861105547e-36158)), SC_(BOOST_MATH_SMALL_CONSTANT(0.2131860779789139256037424176809792095632204680337931857718865949785434825481960848149849797507583928e-361324)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.905791938304901123046875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2178005192333624215784803979717837598562757410620706503281900091107825680350931337396209745019443998e-19841)), SC_(BOOST_MATH_SMALL_CONSTANT(0.4588085686858952503042372234625485480201105499533793863948812241019952913466033029898154440318470087e-473698)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.9133758544921875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2044546663391562477705354540370760821409485274148331480980359986548683615601729039239239614213065235e-18169)), SC_(BOOST_MATH_SMALL_CONSTANT(0.7629190706250225994074616748192014275571599576934851198947217095948168969106405661485602778753139218e-490528)) }, + { SC_(0.216575062950141727924346923828125e-5), SC_(0.968867778778076171875), SC_(BOOST_MATH_SMALL_CONSTANT(0.4096487433832820170639333743831954761351590021205697452562807451168473548459292965564190974788844749e-6342)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1297397706877886168379273776356177633807822967666079798581567618567115144239687805362946264983224597e-695735)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.12698681652545928955078125), SC_(BOOST_MATH_SMALL_CONSTANT(0.460289138417023240211835453578728773675893958998560741266616339802513531273727174963248143404409346e-123279)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1227830593913913689043483463822718345251662053270570059659747620207531684387617971000150489495390622e-8112)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.135477006435394287109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.6439991263768059081770908658752929598087302065287439367522303278837306676039083594400840449710127147e-119413)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1936099411280065786600982265792717722767945024296565235561154800531592756609798398701501116557434413e-8696)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.22103404998779296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2064880008938546016135832883384716687050715182089334885089172611730709160883330305789994119189833204e-90170)), SC_(BOOST_MATH_SMALL_CONSTANT(0.9636789658116208361662045675832611481515953065813422246552622845538828381107902733604814371406262203e-14922)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.308167040348052978515625), SC_(BOOST_MATH_SMALL_CONSTANT(0.4672763627314931787409452529965189883928299396526058431027743531008131781333308125789010166445570443e-70318)), SC_(BOOST_MATH_SMALL_CONSTANT(0.5333958645484846264899444976530540348886306684971665853490471154195979393073606296468477524956455995e-22008)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.6323592662811279296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1232079542911688319553284787645084489874316487664902104337227730213965329139279051931070447347538552e-27377)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1745508475051527213009559427098382072109145551947732820968960362083588524860039683359355038443208669e-59776)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.814723670482635498046875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1296871584563229453031476552376127081733495432987283917208707733104304941994062901047954719016264728e-12240)), SC_(BOOST_MATH_SMALL_CONSTANT(0.3119684049016554066402196968135164105103042293007478934952055073327703314807997817770702753165916888e-100712)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.835008561611175537109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.1744156521766280393141688111513263176846896834358940986864024481897475752003777342235119371016119832e-10771)), SC_(BOOST_MATH_SMALL_CONSTANT(0.414282922235332770074768291902894089464647802773088955269313913489638472173573774702189380297337679e-107639)) }, + { SC_(0.72700195232755504548549652099609375e-5), SC_(0.905791938304901123046875), 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SC_(0.9133758544921875), SC_(11.92028503924933449320581621786613441617387709458224965565099431099659181938764039147894571145926575), SC_(4.411739446802953797688885844731597182665766011787872803903470630812608976887737596597075028566679532) }, + { SC_(7.88237094879150390625), SC_(0.968867778778076171875), SC_(13.87112145371288914260504318784118251728461784612328590191541928786695318093130503135200110507250744), SC_(3.529662237308642030515257624627728599863992204849868392777953155280407948755004639499026009930636495) } + } }; +#undef SC_ + diff --git a/test/handle_test_result.hpp b/test/handle_test_result.hpp new file mode 100644 index 000000000..b1f797f8f --- /dev/null +++ b/test/handle_test_result.hpp @@ -0,0 +1,198 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_HANDLE_TEST_RESULT +#define BOOST_MATH_HANDLE_TEST_RESULT + +#include +#include +#include +#include +#include + +// +// Every client of this header has to define this function, +// and initialise the table of expected results: +// +void expected_results(); + +typedef std::pair > expected_data_type; +typedef std::list list_type; + +inline list_type& + get_expected_data() +{ + static list_type data; + return data; +} + +inline void add_expected_result( + const char* compiler, + const char* library, + const char* platform, + const char* type_name, + const char* test_name, + const char* group_name, + boost::uintmax_t max_peek_error, + boost::uintmax_t max_mean_error) +{ + std::string re("(?:"); + re += compiler; + re += ")"; + re += "\\|"; + re += "(?:"; + re += library; + re += ")"; + re += "\\|"; + re += "(?:"; + re += platform; + re += ")"; + re += "\\|"; + re += "(?:"; + re += type_name; + re += ")"; + re += "\\|"; + re += "(?:"; + re += test_name; + re += ")"; + re += "\\|"; + re += "(?:"; + re += group_name; + re += ")"; + get_expected_data().push_back( + std::make_pair(boost::regex(re, boost::regex::perl | boost::regex::icase), + std::make_pair(max_peek_error, max_mean_error))); +} + +inline std::string build_test_name(const char* type_name, const char* test_name, const char* group_name) +{ + std::string result(BOOST_COMPILER); + result += "|"; + result += BOOST_STDLIB; + result += "|"; + result += BOOST_PLATFORM; + result += "|"; + result += type_name; + result += "|"; + result += group_name; + result += "|"; + result += test_name; + return result; +} + +inline const std::pair& + get_max_errors(const char* type_name, const char* test_name, const char* group_name) +{ + static const std::pair defaults(1, 1); + std::string name = build_test_name(type_name, test_name, group_name); + list_type& l = get_expected_data(); + list_type::const_iterator a(l.begin()), b(l.end()); + while(a != b) + { + if(regex_match(name, a->first)) + { +#if 0 + std::cout << name << std::endl; + std::cout << a->first.str() << std::endl; +#endif + return a->second; + } + ++a; + } + return defaults; +} + +template +void handle_test_result(const boost::math::tools::test_result& result, + const Seq& worst, int row, + const char* type_name, + const char* test_name, + const char* group_name) +{ + using namespace std; // To aid selection of the right pow. + T eps = boost::math::tools::epsilon(); + std::cout << std::setprecision(4); + + T max_error_found = (result.max)()/eps; + T mean_error_found = result.rms()/eps; + // + // Begin by printing the main tag line with the results: + // + std::cout << test_name << "<" << type_name << "> Max = " << max_error_found + << " RMS Mean=" << mean_error_found; + // + // If the max error is non-zero, give the row of the table that + // produced the worst error: + // + if((result.max)() != 0) + { + std::cout << "\n worst case at row: " + << row << "\n { "; + for(unsigned i = 0; i < worst.size(); ++i) + { + if(i) + std::cout << ", "; +#if defined(__SGI_STL_PORT) + std::cout << boost::math::tools::real_cast(worst[i]); +#else + std::cout << worst[i]; +#endif + } + std::cout << " }"; + } + std::cout << std::endl; + // + // Now verify that the results are within our expected bounds: + // + std::pair const& bounds = get_max_errors(type_name, test_name, group_name); + if(bounds.first < max_error_found) + { + std::cerr << "Peak error greater than expected value of " << bounds.first << std::endl; + BOOST_CHECK(bounds.first >= max_error_found); + } + if(bounds.second < mean_error_found) + { + std::cerr << "Mean error greater than expected value of " << bounds.second << std::endl; + BOOST_CHECK(bounds.second >= mean_error_found); + } + std::cout << std::endl; +} + +template +void print_test_result(const boost::math::tools::test_result& result, + const Seq& worst, int row, const char* name, const char* test) +{ + using namespace std; // To aid selection of the right pow. + T eps = boost::math::tools::epsilon(); + std::cout << std::setprecision(4); + + T max_error_found = (result.max)()/eps; + T mean_error_found = result.rms()/eps; + // + // Begin by printing the main tag line with the results: + // + std::cout << test << "(" << name << ") Max = " << max_error_found + << " RMS Mean=" << mean_error_found; + // + // If the max error is non-zero, give the row of the table that + // produced the worst error: + // + if((result.max)() != 0) + { + std::cout << "\n worst case at row: " + << row << "\n { "; + for(unsigned i = 0; i < worst.size(); ++i) + { + if(i) + std::cout << ", "; + std::cout << worst[i]; + } + std::cout << " }"; + } + std::cout << std::endl; +} + +#endif // BOOST_MATH_HANDLE_TEST_RESULT + diff --git a/test/hermite.ipp b/test/hermite.ipp new file mode 100644 index 000000000..a42413232 --- /dev/null +++ b/test/hermite.ipp @@ -0,0 +1,430 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 420> hermite = { + SC_(0.8e1), SC_(-0.804919189453125e3), SC_(0.45107507538695517471998224862706929168983312035236e26), + SC_(0.8e1), SC_(-0.7460263671875e3), SC_(0.24561928260207418635049717146784087504133748838575e26), + SC_(0.8e1), SC_(-0.72904595947265625e3), SC_(0.20429972623973894937590136235800300689033201111301e26), + SC_(0.8e1), SC_(-0.623236083984375e3), SC_(0.5827000192786290015079053630015377969828881322477e25), + SC_(0.8e1), SC_(-0.557931884765625e3), SC_(0.24036425469465274704663172230600209480309473246511e25), + SC_(0.8e1), SC_(-0.4430035400390625e3), SC_(0.3797226972301982762616920977628718931846209669113e24), + SC_(0.8e1), SC_(-0.383665924072265625e3), SC_(0.12017786536547968144293561686294350272042177165962e24), + SC_(0.8e1), SC_(0.9376299285888671875e2), SC_(0.15268620781186000127986962828387837477310566397968e19), + SC_(0.8e1), SC_(0.944411773681640625e2), SC_(0.1617518154852919675909314869798309833011143307635e19), + SC_(0.8e1), SC_(0.264718536376953125e3), SC_(0.61720298828736019075790614156102805167876087211952e22), + SC_(0.8e1), SC_(0.62944732666015625e3), SC_(0.63081173846769146140441083274224793875281883735246e25), + SC_(0.8e1), SC_(0.67001715087890625e3), SC_(0.10397137311210792549517430287956468909599308482363e26), + SC_(0.8e1), SC_(0.8115838623046875e3), SC_(0.48183457131767562763518406277181444679084105960509e26), + SC_(0.8e1), SC_(0.826751708984375e3), SC_(0.55876845531601243780899074396489049059605490708721e26), + SC_(0.8e1), SC_(0.915013671875e3), SC_(0.12579210047506473419699887821873790018411396965251e27), + SC_(0.8e1), SC_(0.92977703857421875e3), SC_(0.14297610657915447015410284337196030534976367717228e27), + SC_(0.8e1), SC_(0.935389892578125e3), SC_(0.15002871812830612180035463350811490994647498745898e27), + SC_(0.8e1), SC_(0.93773553466796875e3), SC_(0.15306505051932713956696576348429859356902240309652e27), + SC_(0.8e1), SC_(0.9857625732421875e3), SC_(0.2282508005179494438672517847212865156472291867269e27), + SC_(0.8e1), SC_(0.99292266845703125e3), SC_(0.24185618845978155462978780829055436870571431239936e27), + SC_(0.9e1), SC_(-0.804919189453125e3), SC_(-0.72615348491801522758107817154280636761854800448258e29), + SC_(0.9e1), SC_(-0.7460263671875e3), SC_(-0.36647428831123627780613206119656435975372881269689e29), + SC_(0.9e1), SC_(-0.72904595947265625e3), SC_(-0.29788553802717105399309090578189082490491064795881e29), + SC_(0.9e1), SC_(-0.623236083984375e3), SC_(-0.72631187656840434630386580888842027480654740183583e28), + SC_(0.9e1), SC_(-0.557931884765625e3), SC_(-0.26821031676217242168707855784900940170157970241264e28), + SC_(0.9e1), SC_(-0.4430035400390625e3), SC_(-0.33643014085005895416146465835311020646005187659932e27), + SC_(0.9e1), SC_(-0.383665924072265625e3), SC_(-0.92213797591750116970480652180852706466705815198925e26), + SC_(0.9e1), SC_(0.9376299285888671875e2), SC_(0.28619599017630076419048218930351610672808471971911e21), + SC_(0.9e1), SC_(0.944411773681640625e2), SC_(0.30538356606793385774776392717760624059016600770529e21), + SC_(0.9e1), SC_(0.264718536376953125e3), SC_(0.32675149012792009950839328597362709432321349932704e25), + SC_(0.9e1), SC_(0.62944732666015625e3), SC_(0.79411750739676620994218584143838991056650844543993e28), + SC_(0.9e1), SC_(0.67001715087890625e3), SC_(0.13932396494395279118751516299633800114577839547729e29), + SC_(0.9e1), SC_(0.8115838623046875e3), SC_(0.78209357516588915667750630886227046953282003885816e29), + SC_(0.9e1), SC_(0.826751708984375e3), SC_(0.9239201438100206122484942895031603534508035207534e29), + SC_(0.9e1), SC_(0.915013671875e3), SC_(0.23020188368730794526744784417674030233447879296414e30), + SC_(0.9e1), SC_(0.92977703857421875e3), SC_(0.26587057172217180147109231631091242505401497841279e30), + SC_(0.9e1), SC_(0.935389892578125e3), SC_(0.28066940992909396296554569662089642014388342016689e30), + SC_(0.9e1), SC_(0.93773553466796875e3), SC_(0.2870677681432185184539265574407337379961356759468e30), + SC_(0.9e1), SC_(0.9857625732421875e3), SC_(0.4500003405401322163412586784924678900107709701962e30), + SC_(0.9e1), SC_(0.99292266845703125e3), SC_(0.48028703540906417924253986480733367446328052415498e30), + SC_(0.1e2), SC_(-0.804919189453125e3), SC_(0.11689816296461847274622300874825539661285328959095e33), + SC_(0.1e2), SC_(-0.7460263671875e3), SC_(0.54679454280582535200198215910160226619047153251183e32), + SC_(0.1e2), SC_(-0.72904595947265625e3), SC_(0.43434081837302238559608884831319578537002297874261e32), + SC_(0.1e2), SC_(-0.623236083984375e3), SC_(0.90531705080732310437321262346769305582516409498699e31), + SC_(0.1e2), SC_(-0.557931884765625e3), SC_(0.29928184853282382212205807781073913971873952893684e31), + SC_(0.1e2), SC_(-0.4430035400390625e3), SC_(0.29807265173628291298939129580674147768362711153657e30), + SC_(0.1e2), SC_(-0.383665924072265625e3), SC_(0.70756420528926763608091741538465186246363823788564e29), + SC_(0.1e2), SC_(0.9376299285888671875e2), SC_(0.53641701648878869328402154187886529569000541972093e23), + SC_(0.1e2), SC_(0.944411773681640625e2), SC_(0.57652451729901027022147781233362208672090049328676e23), + SC_(0.1e2), SC_(0.264718536376953125e3), SC_(0.17298324279751374105449100095978732528697456059849e28), + SC_(0.1e2), SC_(0.62944732666015625e3), SC_(0.99969892955855027682901459214621192921416240033293e31), + SC_(0.1e2), SC_(0.67001715087890625e3), SC_(0.18669702059708370695739115280703586121887901490207e32), + SC_(0.1e2), SC_(0.8115838623046875e3), SC_(0.12694603758113437790859235888888381018069026780075e33), + SC_(0.1e2), SC_(0.826751708984375e3), SC_(0.15276950578878524322853942670808003516571872263785e33), + SC_(0.1e2), SC_(0.915013671875e3), SC_(0.42127347747272206350084241564061602373253483547743e33), + SC_(0.1e2), SC_(0.92977703857421875e3), SC_(0.49439813206983222292955141736901537001605526567434e33), + SC_(0.1e2), SC_(0.935389892578125e3), SC_(0.52506795789015555507271006606207920041267992294258e33), + SC_(0.1e2), SC_(0.93773553466796875e3), SC_(0.5383845389205336603774821335902360432595407417418e33), + SC_(0.1e2), SC_(0.9857625732421875e3), SC_(0.88718287878699347854064053369680200599130466268723e33), + SC_(0.1e2), SC_(0.99292266845703125e3), SC_(0.95377141623597704470908019348004898122546466610927e33), + SC_(0.11e2), SC_(-0.804919189453125e3), SC_(-0.18818569685711019771084718279949279407835364975145e36), + SC_(0.11e2), SC_(-0.7460263671875e3), SC_(-0.81583896324899347640769823826628802342599459007862e35), + SC_(0.11e2), SC_(-0.72904595947265625e3), SC_(-0.63330287962703711111790462420108580900426293742353e35), + SC_(0.11e2), SC_(-0.623236083984375e3), SC_(-0.11284379807813476542599136955709023783640757794916e35), + SC_(0.11e2), SC_(-0.557931884765625e3), SC_(-0.33395240744978214895073421867397301877129142707028e34), + SC_(0.11e2), SC_(-0.4430035400390625e3), SC_(-0.26408775121319087852943852625927639373812809780785e33), + SC_(0.11e2), SC_(-0.383665924072265625e3), SC_(-0.54291810656601189878288220477937246911188498521391e32), + SC_(0.11e2), SC_(0.9376299285888671875e2), SC_(0.10053489057481196687076296855379756797368225593143e26), + SC_(0.11e2), SC_(0.944411773681640625e2), SC_(0.10883423167744841177736957439483334412283561854401e26), + SC_(0.11e2), SC_(0.264718536376953125e3), SC_(0.91577206672391353431978833974641874893100570036399e30), + SC_(0.11e2), SC_(0.62944732666015625e3), SC_(0.12584997550011507219817405571418069606411457160603e35), + SC_(0.11e2), SC_(0.67001715087890625e3), SC_(0.25017762515677812475654023661074854439693249674989e35), + SC_(0.11e2), SC_(0.8115838623046875e3), SC_(0.20605314678159576340073614574689144722841355692387e36), + SC_(0.11e2), SC_(0.826751708984375e3), SC_(0.25260305214286551864922726137009996295874170160231e36), + SC_(0.11e2), SC_(0.915013671875e3), SC_(0.77093737893405727479156012282524646171889099584724e36), + SC_(0.11e2), SC_(0.92977703857421875e3), SC_(0.91935474481359373814770652069446021505623126342922e36), + SC_(0.11e2), SC_(0.935389892578125e3), SC_(0.98228090806597754947655769132113085309480225491242e36), + SC_(0.11e2), SC_(0.93773553466796875e3), SC_(0.10097188855678660617518870586147923895925111174325e37), + SC_(0.11e2), SC_(0.9857625732421875e3), SC_(0.17490943550521460249490073361246584037466880560676e37), + SC_(0.11e2), SC_(0.99292266845703125e3), SC_(0.18940329136734281935082032957044303460975085779168e37), + SC_(0.12e2), SC_(-0.804919189453125e3), SC_(0.30294598540220804462044524153306777663509970628739e39), + SC_(0.12e2), SC_(-0.7460263671875e3), SC_(0.12172627264453841222828991650975850694842130932071e39), + SC_(0.12e2), SC_(-0.72904595947265625e3), SC_(0.92340425553097458791203508279141174969141820492299e38), + SC_(0.12e2), SC_(-0.623236083984375e3), SC_(0.14065466193476872965593557347378160930420069772452e38), + SC_(0.12e2), SC_(-0.557931884765625e3), SC_(0.37263880802028208129069576191917543233991535476065e37), + SC_(0.12e2), SC_(-0.4430035400390625e3), SC_(0.23397705973845936510794729346708943708034608906903e36), + SC_(0.12e2), SC_(-0.383665924072265625e3), SC_(0.41658278768991111298468308002249529032249010160273e35), + SC_(0.12e2), SC_(0.9376299285888671875e2), SC_(0.1884110327970735092254408128261723357492772304494e28), + SC_(0.12e2), SC_(0.944411773681640625e2), SC_(0.20544182415774952250997210456061363935901042940234e28), + SC_(0.12e2), SC_(0.264718536376953125e3), SC_(0.48480562600268824564698370021815193972032016354896e33), + SC_(0.12e2), SC_(0.62944732666015625e3), SC_(0.15842966193994215674176186033328108010563621242497e38), + SC_(0.12e2), SC_(0.67001715087890625e3), SC_(0.3352424919079377855427890512067629280453469707889e38), + SC_(0.12e2), SC_(0.8115838623046875e3), SC_(0.33445602459725757173093631890694210384349814179179e39), + SC_(0.12e2), SC_(0.826751708984375e3), SC_(0.4176766491784391607460244444090637220753635269242e39), + SC_(0.12e2), SC_(0.915013671875e3), SC_(0.14108272157517756411117655879591654342259860325597e40), + SC_(0.12e2), SC_(0.92977703857421875e3), SC_(0.17095789873049740244186014591319864325495552151212e40), + SC_(0.12e2), SC_(0.935389892578125e3), SC_(0.18376197146596820542876788527941003168360783639003e40), + SC_(0.12e2), SC_(0.93773553466796875e3), SC_(0.1893686713584800621262558335100385915682586408288e40), + SC_(0.12e2), SC_(0.9857625732421875e3), SC_(0.34483639865358422938227666676511723324380657223673e40), + SC_(0.12e2), SC_(0.99292266845703125e3), SC_(0.37612354466089752738688053508788073512368471136832e40), + SC_(0.14e2), SC_(-0.804919189453125e3), SC_(0.78509349018738235160733032424141654678585448892725e45), + SC_(0.14e2), SC_(-0.7460263671875e3), SC_(0.27098354215438353455074292442751387993705409786796e45), + 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SC_(0.26497695928277797274126269348797127034206387419765e201), + SC_(0.68e2), SC_(-0.383665924072265625e3), SC_(0.14979523193992880308778135191580896285724901609253e197), + SC_(0.68e2), SC_(0.9376299285888671875e2), SC_(0.32486210198364986836850518198819815917013154840805e155), + SC_(0.68e2), SC_(0.944411773681640625e2), SC_(0.53130717007794880022412938335190211916976004706358e155), + SC_(0.68e2), SC_(0.264718536376953125e3), SC_(0.16304722685276499354792918086631605049027512979648e186), + SC_(0.68e2), SC_(0.62944732666015625e3), SC_(0.62810152575355789173710546297818953525375445861782e211), + SC_(0.68e2), SC_(0.67001715087890625e3), SC_(0.43931930018461310588579182708322391145380014657504e213), + SC_(0.68e2), SC_(0.8115838623046875e3), SC_(0.20135390598861294342717207643575046227436307960836e219), + SC_(0.68e2), SC_(0.826751708984375e3), SC_(0.70928840597252373257415593809153908579367912600248e219), + SC_(0.68e2), SC_(0.915013671875e3), SC_(0.70230300036076744196859829023426726976063282233909e222), + SC_(0.68e2), SC_(0.92977703857421875e3), SC_(0.2085580812121490632054276902743441722437005132514e223), + SC_(0.68e2), SC_(0.935389892578125e3), SC_(0.31403357047962017949167805427442757459524063192189e223), + SC_(0.68e2), SC_(0.93773553466796875e3), SC_(0.37234244977852621303836114362570239474544522838737e223), + SC_(0.68e2), SC_(0.9857625732421875e3), SC_(0.11118565108245626237683701654760798812770117180759e225), + SC_(0.68e2), SC_(0.99292266845703125e3), SC_(0.18188081940210288468029442085277750466942046125126e225), + SC_(0.69e2), SC_(-0.804919189453125e3), SC_(-0.1850081162399338654500028296297913252183226453783e222), + SC_(0.69e2), SC_(-0.7460263671875e3), SC_(-0.97767059274089595678968982337826854278140139586519e219), + SC_(0.69e2), SC_(-0.72904595947265625e3), SC_(-0.19961804161143040928554705782175334155063670416805e219), + SC_(0.69e2), SC_(-0.623236083984375e3), SC_(-0.39882904216104119883607723276479909045844467958239e214), + SC_(0.69e2), SC_(-0.557931884765625e3), SC_(-0.19210316224226509857981603548527956113322666380888e211), + SC_(0.69e2), SC_(-0.4430035400390625e3), SC_(-0.23473078169771562662136461616843138538409966193101e204), + SC_(0.69e2), SC_(-0.383665924072265625e3), SC_(-0.11491609678618516149736741904351620655468528386497e200), + SC_(0.69e2), SC_(0.9376299285888671875e2), SC_(0.60683580626155093065325198848663480816576016092558e157), + SC_(0.69e2), SC_(0.944411773681640625e2), SC_(0.99970547300737778074794585861930502873180772144862e157), + SC_(0.69e2), SC_(0.264718536376953125e3), SC_(0.86281343445478761898014189584430395982291880632205e188), + SC_(0.69e2), SC_(0.62944732666015625e3), SC_(0.79064579216384384919861590442204758056478775930199e214), + SC_(0.69e2), SC_(0.67001715087890625e3), SC_(0.58865834185481932389898790561772677853802379658932e216), + SC_(0.69e2), SC_(0.8115838623046875e3), SC_(0.32681428977081852902063701892528533163973331914428e222), + SC_(0.69e2), SC_(0.826751708984375e3), SC_(0.11727524620517137714481833672797036884220596207432e223), + SC_(0.69e2), SC_(0.915013671875e3), SC_(0.1285181499939382408174721290089724705670081991561e226), + SC_(0.69e2), SC_(0.92977703857421875e3), SC_(0.38780977658427262634355280276612655888568370486454e226), + SC_(0.69e2), SC_(0.935389892578125e3), SC_(0.58746482535419337038893161712202335100520637411516e226), + SC_(0.69e2), SC_(0.93773553466796875e3), SC_(0.69829049096108586661576816358605949519674563343367e226), + SC_(0.69e2), SC_(0.9857625732421875e3), SC_(0.21919763695005329034056321076749362840532324690225e228), + SC_(0.69e2), SC_(0.99292266845703125e3), SC_(0.36117472060917448197562379204160182702204274274195e228) + }; +#undef SC_ + diff --git a/test/hypot_test.cpp b/test/hypot_test.cpp index e01f46bd7..40b2da5f1 100644 --- a/test/hypot_test.cpp +++ b/test/hypot_test.cpp @@ -3,6 +3,7 @@ // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error #include #include #include @@ -40,8 +41,13 @@ const float boundaries[] = { void do_test_boundaries(float x, float y) { float expected = static_cast((boost::math::hypot)( +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS static_cast(x), static_cast(y))); +#else + static_cast(x), + static_cast(y))); +#endif float found = (boost::math::hypot)(x, y); BOOST_CHECK_CLOSE(expected, found, tolerance); } @@ -115,6 +121,7 @@ void test_spots() int test_main(int, char* []) { + BOOST_MATH_CONTROL_FP; test_boundaries(); test_spots(); return 0; diff --git a/test/ibeta_data.ipp b/test/ibeta_data.ipp new file mode 100644 index 000000000..65f5e1394 --- /dev/null +++ b/test/ibeta_data.ipp @@ -0,0 +1,511 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> ibeta_data = { { + { SC_(0.115105740725994110107421875), SC_(27.2666988372802734375), SC_(0.913345992565155029296875), SC_(5.624704191155661855917495760118550518449), SC_(0.4314397017151442751805333868688125395477e-30), SC_(0.9999999999999999999999999999999232955748), SC_(0.7670442516666816870156810539487652961219e-31) }, + { SC_(0.4634225666522979736328125), SC_(3.4317314624786376953125), SC_(0.24176712334156036376953125), SC_(0.9303555938707241359170267278058957882937), SC_(0.1886421251845313169553337074104181129545), SC_(0.8314186687138222136543152962911694480678), SC_(0.1685813312861777863456847037088305519322) }, + { SC_(0.671531736850738525390625), SC_(23.0631923675537109375), SC_(0.908442795276641845703125), SC_(0.1643292146089890181565750406388380802073), SC_(0.5052411887003343404176234528376913340103e-25), SC_(0.9999999999999999999999996925432949323565), SC_(0.3074567050676435237921994231096931905187e-24) }, + { SC_(1.190207004547119140625), SC_(72.69547271728515625), SC_(0.19963125884532928466796875), SC_(0.00559660970823449190019737902600132825008), SC_(0.9534862442725159579596623739736284684983e-9), SC_(0.9999998296314855714196326655015050202504), SC_(0.170368514428580367334498494979749619704e-6) }, + { SC_(1.54034411907196044921875), SC_(40.498138427734375), SC_(0.3462987840175628662109375), SC_(0.002938208699935392440146870722865374716805), SC_(0.4756199818667749603935341902256155409499e-9), SC_(0.9999998381258945674369009890750742808727), SC_(0.1618741054325630990109249257191272986861e-6) }, + { SC_(1.54871237277984619140625), SC_(81.61170196533203125), SC_(0.678767263889312744140625), SC_(0.0009678001476208569653032183047916640238319), SC_(0.559750588331056488014869497117437903989e-42), SC_(0.9999999999999999999999999999999999999994), SC_(0.5783741506002983362466276981258467292731e-39) }, + { SC_(2.251259326934814453125), SC_(4.89832019805908203125), SC_(0.79967319965362548828125), SC_(0.02442974530836840837734643304711539009959), SC_(0.6178004958231200706879909723189065523108e-4), SC_(0.9974774927784456570206007053676991220858), SC_(0.002522507221554342979399294632300877914206) }, + { SC_(2.8674151897430419921875), SC_(59.60579681396484375), SC_(0.575251102447509765625), SC_(0.1379388062890870134560096364710423528806e-4), SC_(0.4175062181357307894530268550925291936655e-24), SC_(0.9999999999999999999697325046252222304163), SC_(0.3026749537477776958367907987526182267324e-19) }, + { SC_(2.922028064727783203125), SC_(26.592067718505859375), SC_(0.4733414947986602783203125), SC_(0.000115508766121651474024307828538813831237), SC_(0.3796145990560169217472146784816274010906e-9), SC_(0.9999967135537481783877208030551367243803), SC_(0.3286446251821612279196944863275619723855e-5) }, + { SC_(3.0540943145751953125), SC_(42.39783477783203125), SC_(0.546926796436309814453125), SC_(0.2096012544527507305106729688625599112948e-4), SC_(0.1877791782437026107953033098682180353671e-16), SC_(0.9999999999991041123359027203750870846506), SC_(0.8958876640972796249129153494479842438984e-12) }, + { SC_(3.183284282684326171875), SC_(31.6550426483154296875), SC_(0.077649272978305816650390625), SC_(0.1607872346465483285868910326819359030108e-4), SC_(0.1977878368269834064074165442158816223284e-4), SC_(0.4484060589761797174626811812282881431733), SC_(0.5515939410238202825373188187717118568267) }, + { SC_(3.2600824832916259765625), SC_(6.254324436187744140625), SC_(0.743158161640167236328125), SC_(0.003877690538964813730515800489545485233393), SC_(0.1849121735270227071877446919569946133643e-4), SC_(0.9952540157237994764714787832333762033991), SC_(0.004745984276200523528521216766623796600881) }, + { SC_(3.360383510589599609375), SC_(52.89206695556640625), SC_(0.81474220752716064453125), SC_(0.4293999091288321831907350544825170972342e-5), SC_(0.219834444785251736057188895022180823046e-40), SC_(0.9999999999999999999999999999999999948804), SC_(0.5119573621504776597210994477995646618901e-35) }, + { SC_(3.4446079730987548828125), SC_(66.3605499267578125), SC_(0.096544884145259857177734375), SC_(0.1474102736157956762304488717716147895171e-5), SC_(0.8307589573110103561152668922877179107493e-7), SC_(0.9466497330301024630866270516938583276551), SC_(0.05335026696989753691337294830614167234487) }, + { SC_(3.57116794586181640625), SC_(36.129398345947265625), SC_(0.046266771852970123291015625), SC_(0.1377307832566030389205420277524302194542e-5), SC_(0.7311615977995211503865956731396528899981e-5), SC_(0.1585130520872948273412729049865238748685), SC_(0.8414869479127051726587270950134761251315) }, + { SC_(3.5762732028961181640625), SC_(80.64510345458984375), SC_(0.92886126041412353515625), SC_(0.5195049488417132478527686360715001708256e-6), SC_(0.2753515092058996575690963489155893940797e-94), SC_(1), SC_(0.530026729908584311094876547172139505977e-88) }, + { SC_(3.7738864421844482421875), SC_(88.39685821533203125), SC_(0.50323975086212158203125), SC_(0.1936358080461588466938455069456644840687e-6), SC_(0.2398889407764917341676367096721572497995e-29), SC_(0.9999999999999999999999876113337095529831), SC_(0.1238866629044701687424663630369236796346e-22) }, + { SC_(4.24311351776123046875), SC_(87.180511474609375), SC_(0.594544112682342529296875), SC_(0.4437344750551262251613273178405984994839e-7), SC_(0.1440454807297923187241336158457814894158e-36), SC_(0.9999999999999999999999999999967537910884), SC_(0.3246208911577068569804118543592340399042e-29) }, + { SC_(4.302380084991455078125), SC_(90.84336090087890625), SC_(0.3500487506389617919921875), SC_(0.3088652285368942607072200051700648121738e-7), SC_(0.3690261013058404601783388874804478116026e-20), SC_(0.9999999999998805219664725983124099139968), SC_(0.1194780335274016875900860032189749548454e-12) }, + { SC_(4.61713886260986328125), SC_(14.9113979339599609375), SC_(0.081217654049396514892578125), SC_(0.7777055873291569797054048770916368511548e-6), SC_(0.3050394533002200879235444490511467363605e-4), SC_(0.02486139844038035518263312395393841283781), SC_(0.9751386015596196448173668760460615871622) }, + { SC_(4.965442657470703125), SC_(82.34554290771484375), SC_(0.3013162314891815185546875), SC_(0.6232388896113233687607109871883895899506e-8), SC_(0.1757129987256299654131721919929978230207e-16), SC_(0.9999999971806477233823653568856711213164), SC_(0.2819352276617634643114328878683629953111e-8) }, + { SC_(5.26769924163818359375), SC_(94.65772247314453125), SC_(0.695263326168060302734375), SC_(0.1253627465271932390249428025335036505934e-8), SC_(0.3222623306430773933281977469822481222259e-51), SC_(1), SC_(0.2570638722989156911038000229513107483269e-42) }, + { SC_(5.39501190185546875), SC_(35.276241302490234375), SC_(0.184897840023040771484375), SC_(0.1209301166284995442719813406613075674954e-6), SC_(0.2280352502477099546779864717478435018705e-7), SC_(0.8413487283667377937115292338183332209645), SC_(0.1586512716332622062884707661816667790355) }, + { SC_(5.961885929107666015625), SC_(95.24720001220703125), SC_(0.587086021900177001953125), SC_(0.1538572075523752828118796821074440813526e-9), SC_(0.2000258675867804464202225099585025891472e-39), SC_(0.9999999999999999999999999999986999252699), SC_(0.1300074730127210093509800162113431715462e-29) }, + { SC_(5.977954387664794921875), SC_(43.524570465087890625), SC_(0.3404516875743865966796875), SC_(0.1330232014752472065680412448138951627377e-7), SC_(0.1838187070983182316780060409965932424395e-11), SC_(0.9998618336444325666209910880648121429145), SC_(0.0001381663555674333790089119351878570854794) }, + { SC_(6.04711818695068359375), SC_(27.766407012939453125), SC_(0.65411365032196044921875), SC_(0.1448258716618381548156837823010682065087e-6), SC_(0.7331873767991696645616237465028962091657e-15), SC_(0.9999999949374558241960009728719676377959), SC_(0.5062544175803999027128032362204089305663e-8) }, + { SC_(6.7992763519287109375), SC_(90.44748687744140625), SC_(0.887737333774566650390625), SC_(0.1998993534488701791116031926385820610152e-10), SC_(0.6971997317630963088710683423020133066068e-88), SC_(1), SC_(0.3487753810776704370476639121868597367948e-77) }, + { SC_(6.88061046600341796875), SC_(29.616764068603515625), SC_(0.815787494182586669921875), SC_(0.2281625605381182233289702097867621117898e-7), SC_(0.1856504037414286050377032409208117834665e-23), SC_(0.9999999999999999186323981885657766556028), SC_(0.8136760181143422334439720929849176822682e-16) }, + { SC_(7.14454555511474609375), SC_(93.46026611328125), SC_(0.5949366092681884765625), SC_(0.6261688000720345538602778281565999642712e-11), SC_(0.9596403591635324364962526501236935948176e-40), SC_(0.9999999999999999999999999999846744143264), 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SC_(0.9999999999999999999999999998901537636949), SC_(0.1098462363050708135613420006073096079804e-27) }, + { SC_(91.71936798095703125), SC_(69.523284912109375), SC_(0.170790970325469970703125), SC_(0.137298772709890694092701913934115684283e-77), SC_(0.5344906320448489479576222507564953931518e-48), SC_(0.2568777907006796621013521352120924429406e-29), SC_(0.999999999999999999999999999997431222093) }, + { SC_(91.742431640625), SC_(80.176300048828125), SC_(0.950227081775665283203125), SC_(0.1001107102701496758482172879825461687447e-51), SC_(0.4363674030870381660841751091381232725007e-108), SC_(1), SC_(0.4358848338099856653448549117193567190292e-56) }, + { SC_(92.033203125), SC_(39.246906280517578125), SC_(0.69485914707183837890625), SC_(0.3429603695399085196479912729784230311888e-35), SC_(0.4568842822110191193956928476545785495226e-35), SC_(0.4287837254261052363447145924559889366464), SC_(0.5712162745738947636552854075440110633536) }, + { SC_(92.3379669189453125), SC_(64.45505523681640625), SC_(0.4359149932861328125), SC_(0.1865276169340167763205291709996432353182e-50), SC_(0.3113144092675435028788977675902943909337e-46), SC_(0.5991256300190186593748026830039317039068e-4), SC_(0.9999400874369980981340625197316996068296) }, + { SC_(92.88541412353515625), SC_(24.4138278961181640625), SC_(0.47481119632720947265625), SC_(0.3517087569118679257544224321978885705116e-38), SC_(0.5033547361837899810418297479760310574297e-26), SC_(0.6987294081667341495372864749025001962669e-12), SC_(0.9999999999993012705918332658504627135251) }, + { SC_(92.92636871337890625), SC_(21.723785400390625), SC_(0.15452297031879425048828125), SC_(0.1496355919808997710160646497363599642223e-78), SC_(0.4033357831201168472873957975134700875523e-24), SC_(0.3709950821207872136467177497502667334427e-54), SC_(1) }, + { SC_(92.9385986328125), SC_(45.849689483642578125), SC_(0.3805077970027923583984375), SC_(0.7123009475318467090796828077561395619685e-50), SC_(0.260666357059090116628536966680822451934e-38), SC_(0.2732615576349308826639148876145050880397e-11), SC_(0.9999999999972673844236506911733608511239) }, + { SC_(93.399322509765625), SC_(68.135955810546875), SC_(0.0497494600713253021240234375), SC_(0.6905215639710705512278514815975036528166e-125), SC_(0.6864215712687726843530898730264241411204e-48), SC_(0.1005972995129974563314022676955487947004e-76), SC_(1) }, + { SC_(93.40106964111328125), SC_(96.49663543701171875), SC_(0.18903611600399017333984375), SC_(0.7657063231082140070082353793406337177991e-78), SC_(0.2555891770317885383063245635780211930521e-57), SC_(0.2995847993254348083418454071685611149657e-20), SC_(0.9999999999999999999970041520067456519166) }, + { SC_(93.90015411376953125), SC_(24.007595062255859375), SC_(0.403971731662750244140625), SC_(0.9335734346720279535279023803571936252236e-44), SC_(0.7613169380702204900368864669107621241774e-26), SC_(0.1226261216568281958863292968272309752221e-17), SC_(0.9999999999999999987737387834317180411367) }, + { SC_(94.20505523681640625), SC_(95.4943389892578125), SC_(0.3371889293193817138671875), SC_(0.9297654369865549462825173884928826911307e-63), SC_(0.2873180569011825880858826754896885391403e-57), SC_(0.3236004163085036181879114898511069822143e-5), SC_(0.9999967639958369149638181208851014889302) }, + { SC_(94.2736968994140625), SC_(13.12218475341796875), SC_(0.709393918514251708984375), SC_(0.4136446298056877569444391910150569799517e-22), SC_(0.3591337944167607654097996956175751590464e-17), SC_(0.1151770933278035243241152703989893222014e-4), SC_(0.9999884822906672196475675884729601010678) }, + { SC_(94.36226654052734375), SC_(75.62816619873046875), SC_(0.86930525302886962890625), SC_(0.7332140206931858197543993390917094119579e-51), SC_(0.494193161847161404627682054188893296014e-74), SC_(0.999999999999999999999993259905731481408), SC_(0.674009426851859198700510182276484925711e-23) }, + { SC_(94.47872161865234375), SC_(4.794428348541259765625), SC_(0.3019829094409942626953125), SC_(0.2037612994500144366883878655315176352494e-51), SC_(0.5445271727836805564325394307425243332438e-8), SC_(0.3741985884898362402462482734954103002849e-43), SC_(1) }, + { SC_(94.5174102783203125), SC_(2.595887660980224609375), SC_(0.9688708782196044921875), SC_(0.337761821443508199381120070196992983155e-5), SC_(0.7003104354886139691687572520659797605122e-5), SC_(0.3253740952886229454250383398376993551864), SC_(0.6746259047113770545749616601623006448136) }, + { SC_(95.02220916748046875), SC_(49.05890655517578125), SC_(0.095445640385150909423828125), SC_(0.1014296591921831167485134416924426771213e-100), SC_(0.3255049410331931492209518274123644776711e-40), SC_(0.3116071260553918747274711046692546109596e-60), SC_(1) }, + { SC_(95.16304779052734375), SC_(87.8050537109375), SC_(0.68924558162689208984375), SC_(0.3591386390171570330217034323842412396658e-55), SC_(0.3500715414450358091351879835688815561025e-61), SC_(0.9999990252477395079575524827693321958401), SC_(0.974752260492042447517230667804159929911e-6) }, + { SC_(95.61345672607421875), SC_(65.196868896484375), SC_(0.3377856314182281494140625), SC_(0.4333859722014505980501156640750924994333e-58), SC_(0.2842452980133411144067320657040665393916e-47), SC_(0.1524690030842662316266754200421873949762e-10), SC_(0.9999999999847530996915733768373324579958) }, + { SC_(95.71669769287109375), SC_(10.986175537109375), SC_(0.0333652384579181671142578125), SC_(0.3372800085083612728245814732509161788213e-143), SC_(0.3482559491895011198520376372913856125954e-15), SC_(0.9684831208004220966663378958012853318006e-128), SC_(1) }, + { SC_(95.7506866455078125), SC_(99.6461334228515625), SC_(0.01787211932241916656494140625), SC_(0.7925981877666200622765224435998813190314e-170), SC_(0.5647605029015430138123291624587681412693e-59), SC_(0.140342354625460930114931308563781999924e-110), SC_(1) }, + { SC_(95.7693939208984375), SC_(72.4906005859375), SC_(0.872441589832305908203125), SC_(0.4388142633049923644259492345034689476008e-50), SC_(0.6107180919399320674461648364002632421015e-72), SC_(0.9999999999999999999998608253780676540823), SC_(0.1391746219323459176926850046732349646854e-21) }, + { SC_(95.92913818359375), SC_(58.095668792724609375), SC_(0.1387105882167816162109375), SC_(0.1152963776753726723714207279128040302867e-87), SC_(0.1964676344164222838212765957522296319386e-44), SC_(0.5868466733355003420468381699190122888759e-43), SC_(1) }, + { SC_(95.94924163818359375), SC_(50.36627197265625), SC_(0.043119497597217559814453125), SC_(0.1205105244843612170118997464853132481334e-133), SC_(0.5379464386572334970803452281919284791023e-41), SC_(0.2240195599866172144586170468512280290734e-92), SC_(1) }, + { SC_(95.974395751953125), SC_(57.37546539306640625), SC_(0.12707412242889404296875), SC_(0.5509848073830600725513709899102562059558e-91), SC_(0.3907686547756237468365485949936283442849e-44), SC_(0.1410002569677527400672535793471909277884e-46), SC_(1) }, + { SC_(96.18981170654296875), SC_(97.99256134033203125), SC_(0.240151941776275634765625), SC_(0.1047338624529782694403256402509401725883e-72), SC_(0.1274994550921882776529206680489908409938e-58), SC_(0.8214455691379210750681642067899947422341e-14), SC_(0.9999999999999917855443086207892493183579) }, + { SC_(96.3088531494140625), SC_(60.5928192138671875), SC_(0.488668859004974365234375), SC_(0.1173424000075725323832188274859866785982e-48), SC_(0.1455980145613272932867048313858796718915e-45), SC_(0.0008052850602336237442086193512349341441502), SC_(0.9991947149397663762557913806487650658558) }, + { SC_(96.48885345458984375), SC_(96.76949310302734375), SC_(0.0185489989817142486572265625), SC_(0.1435982258016881740011473579074188068313e-169), SC_(0.2405150906117843685469869117960864428404e-58), SC_(0.5970445573141612300096758741661990216964e-111), SC_(1) }, + { SC_(96.86492919921875), SC_(33.23390960693359375), SC_(0.822622716426849365234375), SC_(0.3896126830362454261608108680217826494112e-32), SC_(0.575732196536069935996623203779523859796e-34), SC_(0.985438141759546703124794958382314271643), SC_(0.01456185824045329687520504161768572835698) }, + { SC_(97.0592803955078125), SC_(98.11096954345703125), SC_(0.0319296605885028839111328125), SC_(0.2998869565906142596550640980837126732485e-148), SC_(0.6376859616143078939902779400501204314318e-59), SC_(0.4702737313386163666185665116771431706157e-89), SC_(1) }, + { SC_(97.297454833984375), SC_(14.22172069549560546875), SC_(0.62565600872039794921875), SC_(0.4586358283116612515366441141888728005777e-27), SC_(0.2345703889146973155066568361856950228888e-18), SC_(0.1955216212655529321825058592196298245196e-8), SC_(0.9999999980447837873444706781749414078037) }, + { SC_(97.86806488037109375), SC_(23.2400360107421875), SC_(0.580998599529266357421875), SC_(0.4847838580994435860838596547982930354105e-33), SC_(0.1112477977342289130957389321924258614326e-25), SC_(0.4357693786731819507126485869030923714513e-7), SC_(0.9999999564230621326818049287351413096908) }, + { SC_(97.9748382568359375), SC_(35.46381378173828125), SC_(0.43892610073089599609375), SC_(0.2873257135274528993245007467042681059185e-45), SC_(0.1374390079964617837539348395028823804413e-33), SC_(0.2090568883720763685344749440226251580624e-11), SC_(0.9999999999979094311162792363146552505598) }, + { SC_(98.16379547119140625), SC_(49.7869415283203125), SC_(0.7558143138885498046875), SC_(0.3971958028052333965608830276024843615237e-41), SC_(0.2499072385121668210487453408177231300208e-43), SC_(0.9937475495928450856212412081950470642903), SC_(0.006252450407154914378758791804952935709679) }, + { SC_(98.2663421630859375), SC_(89.943603515625), SC_(0.79222810268402099609375), SC_(0.9703020130843336605117450383730907257821e-57), SC_(0.9342902223591938348587142832364306200003e-73), SC_(0.9999999999999999037114001866973304919699), SC_(0.962885998133026695080301263760365136479e-16) }, + { SC_(98.3052520751953125), SC_(55.438030242919921875), SC_(0.711244642734527587890625), SC_(0.9148851548605620864233078020635181477222e-44), SC_(0.2676495491351328940765645110959640891094e-45), SC_(0.9715765392732393467357708912202385044325), SC_(0.02842346072676065326422910877976149556749) }, + { SC_(98.4063720703125), SC_(29.338825225830078125), SC_(0.679551780223846435546875), SC_(0.7234863966571889292957198162735201822678e-32), SC_(0.6644618653733999798962457728260649872896e-30), SC_(0.01077102753452599131081375931128823317142), SC_(0.9892289724654740086891862406887117668286) }, + { SC_(98.79819488525390625), SC_(76.92431640625), SC_(0.498594343662261962890625), SC_(0.8547721404370727048220658289647343126557e-54), SC_(0.1805898574966365262303134358305105727825e-52), SC_(0.04519314170159976658173488989109380979713), SC_(0.9548068582984002334182651101089061902029) }, + { SC_(99.61347198486328125), SC_(44.7584381103515625), SC_(0.2373598515987396240234375), SC_(0.4979063210905853961279025180666656173251e-69), SC_(0.6862752454273526015268742998992848087559e-39), SC_(0.7255198616125700349403582088841519836769e-30), SC_(0.9999999999999999999999999999992744801384) }, + { SC_(99.90804290771484375), SC_(67.421295166015625), SC_(0.73282539844512939453125), SC_(0.4022690295295743797802135225234299639637e-49), SC_(0.3062978751814600527884142222488817508074e-53), SC_(0.9999238632526556439070749307531342152962), SC_(0.7613674734435609292506924686578470377178e-4) } + } }; +#undef SC_ + + diff --git a/test/ibeta_int_data.ipp b/test/ibeta_int_data.ipp new file mode 100644 index 000000000..73266d83c --- /dev/null +++ b/test/ibeta_int_data.ipp @@ -0,0 +1,1010 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1000> ibeta_int_data = { { + { SC_(1), SC_(1), SC_(0.12707412242889404296875), SC_(0.12707412242889404296875), SC_(0.87292587757110595703125), SC_(0.12707412242889404296875), SC_(0.87292587757110595703125) }, + { SC_(1), SC_(1), SC_(0.1355634629726409912109375), SC_(0.1355634629726409912109375), SC_(0.8644365370273590087890625), SC_(0.1355634629726409912109375), SC_(0.8644365370273590087890625) }, + { SC_(1), SC_(1), SC_(0.221111953258514404296875), SC_(0.221111953258514404296875), SC_(0.778888046741485595703125), SC_(0.221111953258514404296875), SC_(0.778888046741485595703125) }, + { SC_(1), SC_(1), SC_(0.3082362115383148193359375), SC_(0.3082362115383148193359375), SC_(0.6917637884616851806640625), SC_(0.3082362115383148193359375), SC_(0.6917637884616851806640625) }, + { SC_(1), SC_(1), SC_(0.632396042346954345703125), SC_(0.632396042346954345703125), SC_(0.367603957653045654296875), SC_(0.632396042346954345703125), SC_(0.367603957653045654296875) }, + { SC_(1), SC_(1), SC_(0.81474220752716064453125), SC_(0.81474220752716064453125), SC_(0.18525779247283935546875), SC_(0.81474220752716064453125), SC_(0.18525779247283935546875) }, + { SC_(1), SC_(1), SC_(0.8350250720977783203125), SC_(0.8350250720977783203125), SC_(0.1649749279022216796875), SC_(0.8350250720977783203125), SC_(0.1649749279022216796875) }, + { SC_(1), SC_(1), SC_(0.905801355838775634765625), SC_(0.905801355838775634765625), SC_(0.094198644161224365234375), SC_(0.905801355838775634765625), SC_(0.094198644161224365234375) }, + { SC_(1), SC_(1), SC_(0.913384497165679931640625), SC_(0.913384497165679931640625), SC_(0.086615502834320068359375), SC_(0.913384497165679931640625), SC_(0.086615502834320068359375) }, + { SC_(1), SC_(1), SC_(0.9688708782196044921875), SC_(0.9688708782196044921875), 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SC_(0.1435884301788758945509351628456887462036e-12) }, + { SC_(37), SC_(29), SC_(0.913384497165679931640625), SC_(0.1375136208127587270061499750741568866244e-19), SC_(0.2309185010347207989575339864017076569671e-33), SC_(0.9999999999999832075906612087444484899632), SC_(0.1679240933879125555151003684677899740315e-13) }, + { SC_(37), SC_(29), SC_(0.9688708782196044921875), SC_(0.1375136208127610361911603199804487667413e-19), SC_(0.2301697695222936221483666697794390147364e-45), SC_(0.9999999999999999999999999832620384684879), SC_(0.1673796153151210231687675483695080784991e-25) }, + { SC_(37), SC_(33), SC_(0.12707412242889404296875), SC_(0.2815266604188693653913302322995077909319e-36), SC_(0.5720050994072747663256989409539299862616e-21), SC_(0.4921750884923820105209519233224389835091e-15), SC_(0.9999999999999995078249115076179894790481) }, + { SC_(37), SC_(33), SC_(0.1355634629726409912109375), SC_(0.2277707779163919100908933095549890064306e-35), SC_(0.5720050994072727701445801959041944686588e-21), SC_(0.3981971107467630507203470725973361693706e-14), SC_(0.9999999999999960180288925323694927965293) }, + { SC_(37), SC_(33), SC_(0.221111953258514404296875), SC_(0.6699377276839075532455324311258095644127e-29), SC_(0.5720050927078977710132838273679710663338e-21), SC_(0.1171209362255882974546065024517529113122e-7), SC_(0.9999999882879063774411702545393497548247) }, + { SC_(37), SC_(33), SC_(0.3082362115383148193359375), SC_(0.3941556123526471052080162297773181229352e-25), SC_(0.5719656838460397831418385582003176457796e-21), SC_(0.6890770952235920510157252810387715473455e-4), SC_(0.9999310922904776407948984274718961228453) }, + { SC_(37), SC_(33), SC_(0.632396042346954345703125), SC_(0.5497340418871139522096404153992420609406e-21), SC_(0.2227105752016109564271894442405331665125e-22), SC_(0.9610649318629521274528570243755901534052), SC_(0.03893506813704787254714297562440984659483) }, + { SC_(37), SC_(33), SC_(0.81474220752716064453125), SC_(0.5720050823395897515434476704627684761846e-21), SC_(0.1706768529630891168936052690140727778083e-28), SC_(0.9999999701616553523826219507757071998362), SC_(0.2983834464761737804922429280016380616641e-7) }, + { SC_(37), SC_(33), SC_(0.8350250720977783203125), SC_(0.5720050985407469884566837838946864514369e-21), SC_(0.8665280593956755759286089261549931701012e-30), SC_(0.9999999984851043106196220087456203622238), SC_(0.1514895689380377991254379637776220948978e-8) }, + { SC_(37), SC_(33), SC_(0.905801355838775634765625), SC_(0.572005099407274913436774974821917171636e-21), SC_(0.1344155843850013782059558794526739210542e-36), SC_(0.9999999999999997650098145553493760303857), SC_(0.2349901854446506239696142929776751238858e-15) }, + { SC_(37), SC_(33), SC_(0.913384497165679931640625), SC_(0.5720050994072750365977061126186163585042e-21), SC_(0.1125465324720467901908766696294229578058e-37), SC_(0.999999999999999980324208195229357614054), SC_(0.1967579180477064238594600964257347683977e-16) }, + { SC_(37), SC_(33), SC_(0.9688708782196044921875), SC_(0.5720050994072750478523593598231063318166e-21), SC_(0.1890457752154400724263869930444876559631e-51), SC_(0.9999999999999999999999999999996695033394), SC_(0.3304966606265113541490993068263355840044e-30) }, + { SC_(37), SC_(37), SC_(0.12707412242889404296875), SC_(0.1663315126472082518689392220455877254431e-36), SC_(0.3095647894727759768982379894849795807496e-22), SC_(0.5373075953841159720995259363454571859457e-14), SC_(0.9999999999999946269240461588402790047406) }, + { SC_(37), SC_(37), SC_(0.1355634629726409912109375), SC_(0.1296111998868566405720542382682176670195e-35), SC_(0.309564789472764679093375775903441064718e-22), SC_(0.4186884435649110798542051137448125301801e-13), SC_(0.9999999999999581311556435088920145794886) }, + { SC_(37), SC_(37), SC_(0.221111953258514404296875), SC_(0.2562741045594853507301922042614016365325e-29), SC_(0.3095647638453671842648293885482778440017e-22), SC_(0.8278528866152636518115304178551121854591e-7), SC_(0.9999999172147113384736348188469582144888) }, + { SC_(37), SC_(37), SC_(0.3082362115383148193359375), SC_(0.9697027380656677021427830653945592127811e-26), SC_(0.3094678191989710734431501832609588142206e-22), SC_(0.0003132471040124351596576730870327977659869), SC_(0.999686752895987564840342326912967202234) }, + { SC_(37), SC_(37), SC_(0.632396042346954345703125), SC_(0.3063451207632721012988627372821913143173e-22), SC_(0.3219668709505538914501724285306955824562e-24), SC_(0.9895993704097000715411905826153733915169), SC_(0.01040062959029992845880941738462660848308) }, + { SC_(37), SC_(37), SC_(0.81474220752716064453125), SC_(0.3095647892990086195192122018743245560235e-22), SC_(0.1737690206941522596931737141183088624137e-31), SC_(0.9999999994386667133878509901348317251675), SC_(0.5613332866121490098651682748324536630957e-9) }, + { SC_(37), SC_(37), SC_(0.8350250720977783203125), SC_(0.3095647894672021976547480411236536892802e-22), SC_(0.557544255861642044384458086164795234761e-33), SC_(0.9999999999819894162765991478047915369078), SC_(0.180105837234008521952084630921920026118e-10) }, + { SC_(37), SC_(37), SC_(0.905801355838775634765625), SC_(0.3095647894727776401201650427507053565973e-22), SC_(0.9319941881679291354451743800364440933825e-41), SC_(0.9999999999999999996989340455175098675848), SC_(0.3010659544824901324152261922870847584496e-18) }, + { SC_(37), SC_(37), SC_(0.913384497165679931640625), SC_(0.3095647894727776402077793811087210433219e-22), SC_(0.5585080458777226819971826512604809980239e-42), SC_(0.9999999999999999999819582825673125691735), SC_(0.1804171743268743082647407545766726774271e-19) }, + { SC_(37), SC_(37), SC_(0.9688708782196044921875), SC_(0.3095647894727776402133644615674982685644e-22), SC_(0.1577418664709856221006413731479117116045e-57), SC_(0.9999999999999999999999999999999999949044), SC_(0.5095601044926236696504373430890795334844e-35) } + } }; +#undef SC_ + diff --git a/test/ibeta_inv_data.ipp b/test/ibeta_inv_data.ipp new file mode 100644 index 000000000..771ad8bdd --- /dev/null +++ b/test/ibeta_inv_data.ipp @@ -0,0 +1,1220 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1210> ibeta_inv_data = { { + { SC_(0.104760829344741068780422210693359375e-4), SC_(39078.1875), SC_(0.913384497165679931640625), SC_(0.2135769916611873809373928693612157637794870746322109915739746618652569766882593624385400901395922558e-3760), SC_(BOOST_MATH_SMALL_CONSTANT(0.6169146818683135737866366973021696361959736535710675458813428774517757535576031054239811928697394346e-101417)) }, + { SC_(0.1127331415773369371891021728515625e-4), SC_(0.0226620174944400787353515625), SC_(0.1355634629726409912109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.6398069040886718270700942839013650119743974939592699138527850527236181568779785317811717302449254101e-76964)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1123304116325227195797533167182832111272986902383876280357481893374380043283848201429836361152904889e-5592)) }, + { SC_(0.113778432933031581342220306396484375e-4), SC_(0.03654421865940093994140625), SC_(0.9688708782196044921875), SC_(0.5825356514402150924555439351704966386176716144780670249830283141506929468296452211723778443894234951e-1195), SC_(BOOST_MATH_SMALL_CONSTANT(0.1299246431009640780314778465681510321819328250560790012308930348335051591055920757982605267188242976e-132423)) }, + { SC_(0.1142846667789854109287261962890625e-4), SC_(0.00244517601095139980316162109375), SC_(0.1355634629726409912109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.9626654068785645417511885142439608103434581112273323673426712424844243107090177710187642425011765594e-75761)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1968131210354581442057523237500619235039616488884597039029510000400233015927645935497663749543679489e-5358)) }, + { SC_(0.1184685606858693063259124755859375e-4), SC_(0.015964560210704803466796875), SC_(0.3082362115383148193359375), SC_(BOOST_MATH_SMALL_CONSTANT(0.3589991782488314563613963926946558354830549116688735009134661806895352582322279412078679966549495e-43116)), SC_(BOOST_MATH_SMALL_CONSTANT(0.8498116607235052405320447197085177573554285064950351753253894145078534997511294854886147550142192844e-13482)) }, + { SC_(0.1215800511999987065792083740234375e-4), SC_(24110.49609375), SC_(0.1355634629726409912109375), SC_(BOOST_MATH_SMALL_CONSTANT(0.6446009470761694712696178802124602151206198100070272397820841954953457589393814080011513726764543968e-71386)), SC_(BOOST_MATH_SMALL_CONSTANT(0.4411620506592737532109016496457382048657857194340645526246628280311812592288988016544405559990034441e-5208)) }, + { SC_(0.130341722979210317134857177734375e-4), SC_(26168.341796875), SC_(0.12707412242889404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2166416563492047698893468112939141389204329099488114879770220649554966170201180232315257069497901648e-68742)), SC_(0.1077379305584195394942430687801727576986179591753422104711707268006164084179781893335299072659829737e-4532) }, + { SC_(0.13885271982871927320957183837890625e-4), SC_(0.04976274073123931884765625), SC_(0.632396042346954345703125), SC_(BOOST_MATH_SMALL_CONSTANT(0.1505435304898114171757823372486217209606557096566030133214604079062750941794084777046798389177683298e-14323)), SC_(BOOST_MATH_SMALL_CONSTANT(0.8191336340381130196534918861139224810035912334983040442057176348873687528604185195239248980481620309e-31292)) }, + { SC_(0.139016165121574886143207550048828125e-4), SC_(0.3188882328686304390430450439453125e-4), SC_(0.81474220752716064453125), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.4280498729519778987331529352631079454822421273284878121766790306889766290450686026189682387674704682e-41368)) }, + { SC_(0.14759907571715302765369415283203125e-4), SC_(2.8241312503814697265625), SC_(0.632396042346954345703125), SC_(BOOST_MATH_SMALL_CONSTANT(0.149627207977763220942721958324439773028753443201215406980939699041850526798527355589362837986915531e-13483)), SC_(BOOST_MATH_SMALL_CONSTANT(0.2558857410093259430689102879079456979338542212672378310349005537672813392923619357148699354237694984e-29446)) }, + { SC_(0.150794721776037476956844329833984375e-4), SC_(0.4875471131526865065097808837890625e-4), SC_(0.221111953258514404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1196552230711909114279654256938815998740725267957310722981422063213023453615282876024769302334980208e-35700)), SC_(1) }, + { SC_(0.1519624856882728636264801025390625e-4), SC_(16177.537109375), SC_(0.81474220752716064453125), SC_(BOOST_MATH_SMALL_CONSTANT(0.1452425209503955537338422359333948812006899768260140476005442384340571893909020382026092251217563149e-5859)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1118345883008806780173762266422459055560035027762640953226900168492818873939725415273834608038325661e-48188)) }, + { SC_(0.1554497794131748378276824951171875e-4), SC_(40.46924591064453125), SC_(0.905801355838775634765625), SC_(0.1263629472761081761336139611957096532344967812079201565462877028056022440086149370452532349289977537e-2765), SC_(BOOST_MATH_SMALL_CONSTANT(0.9919220074175430612528899218323716721906730140324380689847652631919111677660506550414306011565356188e-66001)) }, + { SC_(0.15675454051233828067779541015625e-4), SC_(0.000101913392427377402782440185546875), SC_(0.81474220752716064453125), SC_(0.2853716542422836893914328033448649871440784183324642438029547731008406102487895766538278835997616547e-1712), SC_(BOOST_MATH_SMALL_CONSTANT(0.2298245136733929898089558753739834100156855492642462724846011273903684294484329372722305239196215964e-42747)) }, + { SC_(0.15971760149113833904266357421875e-4), SC_(19.206241607666015625), SC_(0.913384497165679931640625), SC_(0.9582815834618172404720774906601516866545834853534306710079796796969465944037138120013655983754855937e-2465), SC_(BOOST_MATH_SMALL_CONSTANT(0.6326194310940251351062236483308143964837777396539364617390517845304796920345734401134182052813379253e-66519)) }, + { SC_(0.16304524251609109342098236083984375e-4), SC_(0.00039033559733070433139801025390625), SC_(0.12707412242889404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2749781704753392658488528370082054060663950436757594599541089373221441866088075915394337505609699445e-53860)), SC_(0.9828097758952314597050411618472387868890317665219757564769723884179197098195673563971975573178259902e-2530) }, + { SC_(0.16487292668898589909076690673828125e-4), SC_(470997.15625), SC_(0.12707412242889404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.4541512737936566032722209936653561647042510033334139786119604257285077373847313505998057776219766456e-54347)), SC_(0.1549128670548823853403220618952571853257756905438794574182300339725200919598045497915626657567121817e-3585) }, + { SC_(0.165044693858362734317779541015625e-4), SC_(3.57584381103515625), SC_(0.3082362115383148193359375), SC_(BOOST_MATH_SMALL_CONSTANT(0.794238046161941112343749823902683082201010853905108950104923153872086335073632221934630976272829339e-30969)), SC_(BOOST_MATH_SMALL_CONSTANT(0.2292466372763638638323475388751441681337812313997406078153894115038015753694828319985703460438800914e-9697)) }, + { SC_(0.166259487741626799106597900390625e-4), SC_(147818.875), SC_(0.632396042346954345703125), SC_(BOOST_MATH_SMALL_CONSTANT(0.4840219552903389147605728190427790980761548760276021823940610272825144416373810916979400186204380198e-11975)), SC_(BOOST_MATH_SMALL_CONSTANT(0.3355691678737767782891887487339395952689357895271229870171944057769814627603827220615993999615612822e-26146)) }, + { SC_(0.16847467122715897858142852783203125e-4), SC_(0.002207652665674686431884765625), SC_(0.9688708782196044921875), SC_(0.5906847920417500109372659805901565510642317169243157405152707420759307862602125145324167267749837092e-619), 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SC_(0.9999392952515020529866132904758764722582237171318891514012201689689695587113485590174396303118997795) }, + { SC_(499024.09375), SC_(0.3617647588253021240234375), SC_(0.221111953258514404296875), SC_(0.9999989677510921954271873521993859181834023852192245015532772531114512443689750755333329075681050925), SC_(0.9999999774076213040954883510429323955048191846373634062085510975273422302416340711527143505910773273) } + } }; +#undef SC_ + diff --git a/test/ibeta_inva_data.ipp b/test/ibeta_inva_data.ipp new file mode 100644 index 000000000..05a6bd211 --- /dev/null +++ b/test/ibeta_inva_data.ipp @@ -0,0 +1,1110 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1100> ibeta_inva_data = { { + { SC_(0.101913392427377402782440185546875e-4), SC_(0.3082362115383148193359375), SC_(0.1355634629726409912109375), SC_(0.6498233713152427462579302903941895526823485861809005407005756658138299437345439685916818979079350952e-4), SC_(0.1598220360006443400909761969445329244603859759388130816641624923434214075962489978723834186220509792e-5), SC_(0.1598250823507423266003361933591577902014719110010440833489018043541995851576985396009647692344594598e-5), SC_(0.6499023539055942142435327607856798948132754954188633069136984730655318067141056849740949064680308213e-4) }, + { SC_(0.1127331415773369371891021728515625e-4), SC_(0.8350250720977783203125), SC_(0.221111953258514404296875), SC_(0.3971461302443922423613753292972913694448775540766244520233346130148330830638824141461619735572445997e-4), SC_(0.3200361118082439458200937030865347915510238320387461167764804252841304593011303390960681217527273982e-5), SC_(0.3200210889570744035937464686925692217326089474042347373846591880760057034684133216576634448402269381e-5), SC_(0.3970804661143976120973144952058224085647966940391220340079248912116191892887943249201895859632245494e-4) }, + { SC_(0.113778432933031581342220306396484375e-4), SC_(0.905801355838775634765625), SC_(0.913384497165679931640625), SC_(0.1078981970343159393325856300516135956140969598440175633458632674725240047510167692675710778553847126e-5), SC_(0.0001200181935703441995121934466228033111754654801609287621930678605774967528649020200620769888673021826), SC_(0.0001199468547533371496891932800417754246696194915133685876058567801242730706768275318621189458708399548), SC_(0.1078921129179782267266584693661729948023960221141040813461658037144034851960955828005953048025067861e-5) }, + { SC_(0.1142846667789854109287261962890625e-4), SC_(0.632396042346954345703125), SC_(0.221111953258514404296875), SC_(0.4025899796795053338948240609285498211472682534957119282666669306536310006800286817936757572860013938e-4), SC_(0.3244356766903463719306913859577321911415693698681174694348315965945146265497858043549761761998515535e-5), SC_(0.3244305116814100828743022380284672509844786339827561909255756253408557410865952781989509881881736442e-5), SC_(0.402567403617356621571805857579531660366440062441234650794612972668478828468673367856838957420528343e-4) }, + { SC_(0.1145765054388903081417083740234375e-4), SC_(0.913384497165679931640625), SC_(0.905801355838775634765625), SC_(0.1191571750545867392537313798907431542344344095879200999308344375910813077398702631095528359741186503e-5), SC_(0.0001102067845592199693917010458194599552572103205410984424051016635846792835186985601306422395955097474), SC_(0.0001101436549179224812644413611401269090813984984476836579744033601646714018004549966518797920528872344), SC_(0.1191500741959144922072835433076419034115688201288730002114876689721036040236828451248448892509304819e-5) }, + { SC_(0.1184685606858693063259124755859375e-4), SC_(0.8350250720977783203125), SC_(0.8350250720977783203125), SC_(0.2340623363191603898370775227750571590200502357875341269550358518828489422976009146847049767670005029e-5), SC_(0.5997016652255219747886060771729383753308007448753449600990778600678820278172428956506474372626790804e-4), SC_(0.5995620164847073584050553181224642561226948778199420970812017399765536897356590939928613936314653308e-4), SC_(0.234051566405018175204269278558069354736428629628287051287220082664267809248151547790462782021755034e-5) }, + { SC_(0.12048656572005711495876312255859375e-4), SC_(0.913384497165679931640625), SC_(0.81474220752716064453125), SC_(0.2739744250951670275113298279765201756581440367980696202896543991551475858455303285016122111531160698e-5), SC_(0.5299670821761171764372900056596763966402488146854288536412386951147420756371449036647596214642744172e-4), SC_(0.5298047274004390572318749610229599999656628731752066034988441477464604972189618031020718116326084136e-4), SC_(0.2739553374006434941292646031005528692950850835558781259804553220755863034131702899478155309841849826e-5) }, + { SC_(0.13885271982871927320957183837890625e-4), SC_(0.8350250720977783203125), SC_(0.913384497165679931640625), SC_(0.1316761416595919370130125478806042690058228402723744932513555575390264383428267099674470556162378215e-5), SC_(0.0001464621138187770636368288091715451405788452187762438244322570896511463096988914144097451646517393708), SC_(0.0001463859922463820577136477421662789627417973019629595681808688398863611174765967906151751169713754156), SC_(0.1316696495791707946163884336944625169957551861831120145642154917636213029136344378330816865851253778e-5) }, + { SC_(0.139016165121574886143207550048828125e-4), SC_(0.1355634629726409912109375), SC_(0.12707412242889404296875), SC_(0.9547673292113491262034069793405709241018001446440966950759890317490878173052656894523453165695624926e-4), SC_(0.2023635278368965262888447203152206885757241398427865002786378633252671147811944223712212058587588226e-5), SC_(0.2023754690333906934784743578875515210120065850224750194606142612432969845718622103368626791471648456e-5), SC_(0.9551543863643827044064851822439522331538353359186806068477116827790284692824136317548192394272389844e-4) }, + { SC_(0.143944707815535366535186767578125e-4), SC_(0.221111953258514404296875), SC_(0.81474220752716064453125), SC_(0.3272972153593272792294315950670347359831261283602618957436021027450675786174856995623862264110326696e-5), SC_(0.632990141303189719133353279896837410679405196953125322609425221598638065093292166662050411772898045e-4), SC_(0.6331140087531343209152630672946940443241052431215532098504247271212384333561961538316669124336267509e-4), SC_(0.3273117781877539280798489020583032342225811653678924427425569001005416493844068702682445156689086415e-5) }, + { SC_(0.14498580640065483748912811279296875e-4), SC_(0.3082362115383148193359375), SC_(0.3082362115383148193359375), SC_(0.3253742212386968781612656316620215840333726254656645874658994387373581238578087455806150295721983642e-4), SC_(0.6460170263304768607166472283634580938080587045554068917515889569677695568527422292732443965595733546e-5), SC_(0.6460389168964756781185792834191710616285016843656138451140030019016010603469220070557969454976470937e-5), SC_(0.3253989653200702274919661149423446286389104607980329666927131807017854240337258963755040377916930538e-4) }, + { SC_(0.150073101394809782505035400390625e-4), SC_(0.905801355838775634765625), SC_(0.3082362115383148193359375), SC_(0.3368409504251561290464334473768649337939776230437629072347636411186828106568821467963048055712919614e-4), SC_(0.6687287839256708751603744004615321146312284138895378752133829995735298947731901687375615081162665865e-5), SC_(0.6686631150723913246638218241012310033772813288325333543118501948241656165404065320414326799591238772e-5), SC_(0.3367667217741028633923677261223016046989181571572493614821457869552722619328990219221837086365139157e-4) }, + { SC_(0.150358655446325428783893585205078125e-4), SC_(0.913384497165679931640625), SC_(0.1355634629726409912109375), SC_(0.9590303934205324631565257968984418682325537068446175465533121391485224886714546031448674308598768754e-4), SC_(0.2358064976440925144147073399479359701859791978119588953669585750075394212334040246114720302574396224e-5), SC_(0.2357871747378977311589396746549307159106310325274719313965661500873802598080130111071124153255736699e-5), SC_(0.9585294284458174387238014758587531939522744711214339460244053261369492265424734665782500243897903548e-4) }, + { SC_(0.150794721776037476956844329833984375e-4), SC_(0.12707412242889404296875), SC_(0.632396042346954345703125), SC_(0.87651069540304628423081918011260974905033687777361892246598630128014121055416273931319481920790515e-5), SC_(0.259394507865675995744762355153851792156958953609180785329566058577092563181989551106130502682548618e-4), SC_(0.2594355203373817883698461757521940058816181951323031229743551991870650806867128742063568243515966938e-4), SC_(0.8765912509183101407045558460058661316022689816530705908193566381502601877100638232366285667845801709e-5) }, + { SC_(0.15419080227729864418506622314453125e-4), SC_(0.12707412242889404296875), SC_(0.1355634629726409912109375), SC_(0.9830005405914737446060892363685712552152782311310204370882529948507189082695976623585226251217629048e-4), SC_(0.2417982090725863219268402865263656889157071672332778168735691070344962656880642022791761256750260214e-5), SC_(0.2418148323498376774634162885245152923095533505622569142619373610219681319883028827406881417407724923e-5), SC_(0.9834315167538787468882315193663229818124191644081977735590761688303094176815343087199599992309813292e-4) }, + { SC_(0.15675454051233828067779541015625e-4), SC_(0.3082362115383148193359375), SC_(0.12707412242889404296875), SC_(0.0001076705852659451793853865643565345351084475810373544400849947491064482183831269674483630442891227277), SC_(0.2281884068014367922411896063517274399583263910072808042169062857538812657905414478059107182988935904e-5), SC_(0.2281950317358702456902944257528144442000952715480241963747411206463352849834574570250349766756506247e-5), SC_(0.0001076920590890716618082136667711265533451107466155867551262184847803119057614262957145136457318713641) }, + { SC_(0.15964558770065195858478546142578125e-4), SC_(0.8350250720977783203125), SC_(0.905801355838775634765625), SC_(0.1660278783926417628556777160282997881556780127471291513930816808973189010497802355733599800163014022e-5), SC_(0.0001535552230132099996415866651208432663272262371647920595961863118674016207241551750339476731435767165), SC_(0.0001534708524536018031472228456674893374691254721961681088171006888239037200588582248419618310759235827), SC_(0.1660183881247630944760800805383653749744450292432029092339200275943746932481995662921241116809458963e-5) }, + { SC_(0.16304524251609109342098236083984375e-4), SC_(0.221111953258514404296875), SC_(0.12707412242889404296875), SC_(0.0001119845795811380561590323350505600731783209229321913051109174418673719036122152799154205160023158248), SC_(0.2373436779571213434043230033533137782270002877418358817665203764904913082412060259067142704435236263e-5), SC_(0.237354842387939075315235443146445014249627692147638075678416887622432328868767915597108680961090348e-5), SC_(0.000112020767308378375840566420540839970686221727315573048682688015419168849250873387024663623743467003) }, + { SC_(0.16847467122715897858142852783203125e-4), SC_(0.81474220752716064453125), SC_(0.913384497165679931640625), SC_(0.1597675138409293700309435180784469739232620351573060252420737905448171241490589056171813734376102797e-5), SC_(0.0001777124009802608826555105463099742495952656042301109910545990701930609772354857022376451861750454172), SC_(0.0001776100511814565558783790404856355430494800495259308451582193538709656878082298741615995907450789185), SC_(0.1597587846781442995028519903465322758894812154787584197845211972613431887705001337520744293208672331e-5) }, + { SC_(0.1695460741757415235042572021484375e-4), SC_(0.81474220752716064453125), SC_(0.905801355838775634765625), SC_(0.1763240139192367764350577078574126894495994363862021385705968362952943728559008233251209625751048006e-5), SC_(0.0001630766663055383146731232183002331937355493105587993916536858283999468056902346280484683877950440511), SC_(0.0001629897543666882871856905079787663075585451963966648415370701491698237494062073781966706404486930145), SC_(0.1763142377412093970664774302592003086305568525455565300103669618366872409088556579141664191554965497e-5) }, + { SC_(0.1747490387060679495334625244140625e-4), SC_(0.913384497165679931640625), SC_(0.913384497165679931640625), SC_(0.1657205490431766653288219519839018968888546479296707296937718099328611521158562162673563617224824575e-5), SC_(0.0001843653172192470436245570152447584219034861068531324705290813556424203611303058551178626221584331899), SC_(0.0001841901877263695417504790356833842525593500782750209287429911099919852050415191801904617209210017375), SC_(0.1657056122824094680842979074245439365062200089255551812950422064402396188284924313740737229005812151e-5) }, + { SC_(0.17535277947899885475635528564453125e-4), SC_(0.1355634629726409912109375), SC_(0.3082362115383148193359375), SC_(0.3934966619007810949666151809439796625761525755030458189033923939964942952549970487522772045241080464e-4), SC_(0.7813004844990431736875950082152823956499576004090834911169966325316944097732757790249052538329811151e-5), SC_(0.7813738690758194900025805668521577972307954518890831563516466130616372853353627433972933369976640684e-5), SC_(0.3935796118240089581784673294734863779612840643384986837483979048055208451716421353815051705178379411e-4) }, + { SC_(0.18408363757771439850330352783203125e-4), SC_(0.913384497165679931640625), SC_(0.3082362115383148193359375), SC_(0.413190619940077812148654535015486566874115279053224890650250082320279230593940677934502728871627653e-4), SC_(0.8202915653135443071254499739337398381484679516674675735681744216705620879155339990070901487941605462e-5), SC_(0.8201887319734426941626810875350578124969960624841708092206613535895147568423858458335946233424988906e-5), SC_(0.4130743831720140403888528190085862518549793340166449318755313043679828999244864499097442313882705138e-4) }, + { SC_(0.18476972400094382464885711669921875e-4), SC_(0.632396042346954345703125), 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SC_(3075673.343007852414765473783587634909885248447311793172436329302049106864266224074483562352620124036) }, + { SC_(448846.15625), SC_(0.9688708782196044921875), SC_(0.905801355838775634765625), SC_(13942148.14815611659064722112747818128627692558477209845530274937530403703010314738619713308434710383), SC_(13997877.13137363238891165544798253008428157956288687495976169844140682358130508342293587223794208822), SC_(14582.20509558332364458183492591473653375236960659867288176623116326823028248461128186433315865346464), SC_(14261.25976832042826122812485513044218145263768128533306080932256002127144716956873137761320234726799) }, + { SC_(467754.28125), SC_(0.9688708782196044921875), SC_(0.1355634629726409912109375), SC_(14582307.27749050806862514534441144029293736814556627825608323590943974657212073974512771840178001964), SC_(14534709.85379777204144202250848456557271434486494359952219887901749931232339169351948582272785520112), SC_(14892.0869980069314749260353980304174964844718061074818482931435407581107499244785056336160375827793), SC_(15166.20253163388900385334372720029740009082048553955730836477017740142502713368416445237970556885013) }, + { SC_(473141.4375), SC_(0.81474220752716064453125), SC_(0.221111953258514404296875), SC_(2083396.177078763261310247756210796709050986068912986314606144732424519707190749275957193786929212863), SC_(2078245.423957970390429173580046940301627576838628045485422158201767188767913815871568210851554479025), SC_(107305.0550947709258110643640197054815587384906300543600549141575195933780408202908438136093287591433), SC_(107863.5317465997746516385261209267501517088116598822296362271288937537024663332026184699882279661038) }, + { SC_(475931.78125), SC_(0.1355634629726409912109375), SC_(0.221111953258514404296875), SC_(74863.22159423657178639135843064319186867849712135432679140745009910166430025378444104631149358051552), SC_(74411.62359846101533194367027997650975048798856694024723181729993473406725909453910573442714644750971), SC_(3031199.27574924233997859053704277970442582820164378907041604417813349570600051850022942680116940976), SC_(3038471.007954823489719352635939274579304822773919251533092003024823266976921005941089319124285500496) }, + { SC_(480505.75), SC_(0.913384497165679931640625), SC_(0.905801355838775634765625), SC_(5057009.137883346991297280272316655884843512474132092967659893246907363070814150415616505779213755808), SC_(5077130.008669136984101475927154584113851934847615921791753015711169641012846352004783223552006266754), SC_(45860.39578823761216780930027528343441271409993632576700986556608110936794777922189190471234767214715), SC_(45272.82619384517959483726844004670545212496499292353295676138585412022031957970747357936964957150843) }, + { SC_(481037.90625), SC_(0.3082362115383148193359375), SC_(0.1355634629726409912109375), SC_(214954.0730415459324272322098059261658344195921911048215227103731874995082410590810383910277671282622), SC_(213728.9433236639020931731891795221242722212174046095998104798208675703012870270153304541107798144009), SC_(1077517.795801841403487141321214750115179387272101781637533476788045360859794269288938283235224199637), SC_(1081636.810871830922880032963090014830142863695032235420818616239563778235210349598024972698342430219) }, + { SC_(485980.90625), SC_(0.1355634629726409912109375), SC_(0.81474220752716064453125), SC_(75947.50287755040088992558685298601037560620447249457062259186705941456229777606961809318551961112457), SC_(76479.30117347133042069077426985013802863207922409730304956347781449509840371152867686698312667407036), SC_(3103196.634482552348526388073690399281667050881158209209257160529514613456386016945448500178200416261), SC_(3094633.498847298449800834614348937260696174622946733336073730690861985736659159836937933593087681597) }, + { SC_(489096.34375), SC_(0.81474220752716064453125), SC_(0.632396042346954345703125), SC_(2149835.420432935738571504824669405535112026843128171823218720145168010286191521063636486128819640379), SC_(2152140.267058834375526429568585141743365314468040624002321395548815791537567896033314059219767174233), SC_(111336.990938262038284792110970236277584339209153903880265239957333478239591138095337997321244983344), SC_(111087.0851343228376540732483260823293660124553023808030297399680696356784984406728076454354233386814) }, + { SC_(490224.65625), SC_(0.8350250720977783203125), SC_(0.9688708782196044921875), SC_(2474059.781112585100225923474277388945644841132414541483235018154724374112161333898186161710511319547), SC_(2488521.187172101754693537105176727049011115863706614369301342646676606307103207412127159070864309253), SC_(97489.16520339287822299971952377074411630970076546169823169145986944645114749814051528899672166177443), SC_(96219.21150955986998802274923363974940278386581612184471409847771598317921003666209856647350389402225) }, + { SC_(492616.1875), SC_(0.81474220752716064453125), SC_(0.9688708782196044921875), SC_(2160097.330587066231445009615179450496605337977400732745848956148755035070499897025280043407966765649), SC_(2172849.060248862788069481343937878596794903922837153025629325486461452111332884974573767684987699383), SC_(112704.513279915547893306409021394612149255526474732330209037454571458216341356862770454420824870536), SC_(111321.8919149126350976334581215168831576517722380867094186255880986306392649715612321977436708934672) }, + { SC_(493067.15625), SC_(0.1355634629726409912109375), SC_(0.9688708782196044921875), SC_(76767.64922907893070937377260795058987611362823891825524085896234884328550113407276897167228036558757), SC_(77882.89867960827431192138690174915177255321134308258910288268085009341055734572331426861469361693711), SC_(3153086.514367219226967064126240311772170736384107525381665234442715424152190338059638973443194990823), SC_(3135128.509847232411063129274533589230908159139891303513987728335691375301577612946664126754477170654) }, + { SC_(497097), SC_(0.12707412242889404296875), SC_(0.1355634629726409912109375), SC_(72681.11415387065674957690132016868169191955575858086107661197432136383823238646114474079457781392721), SC_(72047.41954020240893325737182119598226364424389984420063381611270985375436608408806330762383061821211), SC_(3409065.827055807531678961859073645657230358185950103860885952543964343334592856668738573506077026751), SC_(3420475.169677388105656745142209467687699219296898877898752790493449083963329175908034795593158414778) }, + { SC_(497481.8125), SC_(0.3082362115383148193359375), SC_(0.12707412242889404296875), SC_(222314.1253130015627660072636473322752712891751839119915721545837501534862620518004663188386017641378), SC_(221023.1037330142642378955151286775664560624517786400554218612145985519782410506686001446831861587568), SC_(1114311.595597452012343496149547782092708841542094172969811686130629128619077973244337814566303316361), SC_(1118652.146040181182739176928249543706836426766938346914879802716806902165572254306413904435006003401) } + } }; +#undef SC_ + diff --git a/test/ibeta_large_data.ipp b/test/ibeta_large_data.ipp new file mode 100644 index 000000000..d9a1b4ef5 --- /dev/null +++ b/test/ibeta_large_data.ipp @@ -0,0 +1,1222 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1210> ibeta_large_data = { { + { SC_(0.104760829344741068780422210693359375e-4), SC_(39078.1875), SC_(0.913384497165679931640625), SC_(95444.37547576888548779405522478045372688), SC_(BOOST_MATH_SMALL_CONSTANT(0.4076397251031275963346153642645211144346e-41521)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.4270966445860582673748568606264586002504e-41526)) }, + { SC_(0.1127331415773369371891021728515625e-4), SC_(0.0226620174944400787353515625), SC_(0.1355634629726409912109375), SC_(88703.20318198098901713372585734808194292), SC_(45.9460483635769831377505786833842050448), SC_(0.9994822930837981780736860587426162687504), SC_(0.0005177069162018219263139412573837312496096) }, + { SC_(0.113778432933031581342220306396484375e-4), SC_(0.03654421865940093994140625), SC_(0.9688708782196044921875), SC_(87893.29210911967129223056449206866305889), SC_(24.13233514927218107101077084629346084654), SC_(0.999725511349976252733137998478545702411), SC_(0.0002744886500237472668620015214542975889712) }, + { SC_(0.1142846667789854109287261962890625e-4), SC_(0.00244517601095139980316162109375), SC_(0.1355634629726409912109375), SC_(87498.94901523223439182946063290895919618), SC_(410.8174698800966982396187764410683048418), SC_(0.9953268278792474174353255397883295739773), SC_(0.004673172120752582564674460211670426022665) }, + { SC_(0.1184685606858693063259124755859375e-4), SC_(0.015964560210704803466796875), SC_(0.3082362115383148193359375), SC_(84409.76586511709213323237371074024560834), SC_(63.42758275377907514602214367744158071269), SC_(0.9992491395179357013663738064942314292026), SC_(0.000750860482064298633626193505768570797446) }, + { SC_(0.1215800511999987065792083740234375e-4), SC_(24110.49609375), SC_(0.1355634629726409912109375), SC_(82239.66859351932879196086451784571067159), SC_(0.1228963536503215267489197519054078145969e-1528), SC_(1), SC_(0.1494368298804234679967895435418141236754e-1533) }, + { SC_(0.130341722979210317134857177734375e-4), SC_(26168.341796875), SC_(0.12707412242889404296875), SC_(76710.65536325700740870385917451316226344), SC_(0.8987481009726700678494101665597031705645e-1548), SC_(1), SC_(0.1171607903382812021950729983554623721701e-1552) }, + { SC_(0.13885271982871927320957183837890625e-4), SC_(0.04976274073123931884765625), SC_(0.632396042346954345703125), SC_(72019.23463150248088958543324182892626471), SC_(19.53662131349699149968802706905083676178), SC_(0.9997288040735046086759949196286322002492), SC_(0.0002711959264953913240050803713677997508421) }, + { SC_(0.139016165121574886143207550048828125e-4), SC_(0.3188882328686304390430450439453125e-4), SC_(0.81474220752716064453125), SC_(71935.56143195648321209894597982190925135), SC_(31357.46843093207518473308985368320533006), SC_(0.6964222225589077802406713530999497187926), SC_(0.3035777774410922197593286469000502812074) }, + { SC_(0.14759907571715302765369415283203125e-4), SC_(2.8241312503814697265625), SC_(0.632396042346954345703125), SC_(67749.64468000029269520566988398084664291), SC_(0.02906706447901268252429885336709779665611), SC_(0.9999995709637719063997441118286889404404), SC_(0.4290362280936002558881713110595595793096e-6) }, + { SC_(0.150794721776037476956844329833984375e-4), SC_(0.4875471131526865065097808837890625e-4), SC_(0.221111953258514404296875), SC_(66314.05927892349341973416366818795267336), SC_(20512.09739510432080805623001488366337009), SC_(0.763756704421306309320206420222803196485), SC_(0.236243295578693690679793579777196803515) }, + { SC_(0.1519624856882728636264801025390625e-4), SC_(16177.537109375), SC_(0.81474220752716064453125), SC_(65795.44709267135893551075697872372424105), SC_(BOOST_MATH_SMALL_CONSTANT(0.2027537374480409874707652773038782413935e-11849)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.3081577014933374612265415972232266786926e-11854)) }, + { SC_(0.1554497794131748378276824951171875e-4), SC_(40.46924591064453125), SC_(0.905801355838775634765625), SC_(64325.19244254056356455961240584532088293), SC_(0.822463871000651749828830546797984530393e-43), SC_(1), SC_(0.1278603047686689558334345413308146407234e-47) }, + { SC_(0.15675454051233828067779541015625e-4), SC_(0.000101913392427377402782440185546875), SC_(0.81474220752716064453125), SC_(63795.48621858577102490380365796969496633), SC_(9810.772055381842015487811747997830665087), SC_(0.8667128001689005026050394523916069667826), SC_(0.1332871998310994973949605476083930332174) }, + { SC_(0.15971760149113833904266357421875e-4), SC_(19.206241607666015625), SC_(0.913384497165679931640625), SC_(62607.00088023805345936256566188949995708), SC_(0.2234001688321881800710378923972526401497e-21), SC_(0.9999999999999999999999999964317062678097), SC_(0.3568293732190334165134476285535997648472e-26) }, + { SC_(0.16304524251609109342098236083984375e-4), SC_(0.00039033559733070433139801025390625), SC_(0.12707412242889404296875), SC_(61330.74255992408673684618143646855164824), SC_(2563.824473817131717490401647850652347432), SC_(0.9598741396516039259205408266092071862094), SC_(0.04012586034839607407945917339079281379055) }, + { SC_(0.16487292668898589909076690673828125e-4), SC_(470997.15625), SC_(0.12707412242889404296875), SC_(60639.13353449960017258664254772876309268), SC_(BOOST_MATH_SMALL_CONSTANT(0.5384739990831595984009356761636892828401e-27804)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.887997515295636614071071990663640203201e-27809)) }, + { SC_(0.165044693858362734317779541015625e-4), SC_(3.57584381103515625), SC_(0.3082362115383148193359375), SC_(60587.77026541341888225085862457381274413), SC_(0.1732225036748861524733225741802535443988), SC_(0.9999971409740337294746579825837771947623), SC_(0.2859025966270525342017416222805237665672e-5) }, + { SC_(0.166259487741626799106597900390625e-4), SC_(147818.875), SC_(0.632396042346954345703125), SC_(60134.46375982525806101037989862809783137), SC_(BOOST_MATH_SMALL_CONSTANT(0.1037988440880944042360986327562736779854e-64249)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.1726112408728927999761612396367365655052e-64254)) }, + { SC_(0.16847467122715897858142852783203125e-4), SC_(0.002207652665674686431884765625), SC_(0.9688708782196044921875), SC_(59359.52475707738611546412137293027748567), SC_(449.5447623669572666846242510916839845304), SC_(0.9924836690157701392419526809332561139692), SC_(0.007516330984229860758047319066743886030786) }, + { SC_(0.1747490387060679495334625244140625e-4), SC_(0.26349246501922607421875), SC_(0.632396042346954345703125), SC_(57225.14946260528992955268005621257633375), SC_(3.201269200695144242963625771809339374153), SC_(0.9999440614807213732173180138063205712752), SC_(0.5593851927862678268198619367942872477437e-4) }, + { SC_(0.17863809262053109705448150634765625e-4), SC_(439.38714599609375), SC_(0.8350250720977783203125), SC_(55972.44090864893014330138933206516204433), SC_(0.3791074904984294581987640686637822179263e-346), SC_(1), SC_(0.6773109843774015299501716456463481210658e-351) }, + { SC_(0.1895247623906470835208892822265625e-4), SC_(1.07109558582305908203125), SC_(0.9688708782196044921875), SC_(52763.41949943324228341686952963434540238), SC_(0.02308304962964954759921229436647956617731), SC_(0.9999995625181280852782419366072564332654), SC_(0.4374818719147217580633927435667345739681e-6) }, + { SC_(0.19740999050554819405078887939453125e-4), SC_(105.41565704345703125), SC_(0.3082362115383148193359375), SC_(50650.76748327468387622526146707985788856), SC_(0.4058462091652395705622130232256784578791e-18), SC_(0.999999999999999999999991987363087849489), SC_(0.8012636912150510965168560289672323749562e-23) }, + { SC_(0.2004335328820161521434783935546875e-4), SC_(482.00701904296875), SC_(0.905801355838775634765625), SC_(49885.0975467399541005594058055485920959), SC_(0.6952488973381382600133321305803386175123e-497), SC_(1), SC_(0.1393700587007418889673088312510010012273e-501) }, + { SC_(0.2017128645093180239200592041015625e-4), SC_(232.9792938232421875), SC_(0.1355634629726409912109375), SC_(49569.39447695058637088307802861825336459), SC_(0.5613839078401083563926364753233791797764e-16), SC_(0.9999999999999999999988674787865299668824), SC_(0.1132521213470033117556444752410328436421e-20) }, + { SC_(0.20203804524498991668224334716796875e-4), SC_(42.8336334228515625), SC_(0.1355634629726409912109375), SC_(49491.30543730291452299570686539488489706), SC_(0.0002970093232930177329636992017779607619212), SC_(0.9999999939987575820392696584870441824195), SC_(0.6001242417960730341512955817580536150083e-8) }, + { SC_(0.20300331016187556087970733642578125e-4), SC_(42.38938140869140625), SC_(0.9688708782196044921875), SC_(49255.96842848213676438936988840982061108), SC_(0.3254102070311191847990477740588760715854e-65), SC_(1), SC_(0.660651322902325825501909533025783367306e-70) }, + { SC_(0.2051885167020373046398162841796875e-4), SC_(236087.890625), SC_(0.8350250720977783203125), SC_(48722.72335464117531549642945589803667543), SC_(BOOST_MATH_SMALL_CONSTANT(0.3623657681223884685762672034466978337415e-184763)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.7437305289460397760237525357326064309497e-184768)) }, + { SC_(0.21041001673438586294651031494140625e-4), SC_(3.8230969905853271484375), SC_(0.12707412242889404296875), SC_(47523.85310963552184634237668034540815544), SC_(0.6195869439847247414271234613685101142876), SC_(0.9999869627813034247834524610650576290292), SC_(0.1303721869657521654753893494237097084781e-4) }, + { SC_(0.2107691761921159923076629638671875e-4), SC_(0.04489715397357940673828125), SC_(0.221111953258514404296875), SC_(47443.99666807844164422517130961809149441), SC_(23.47266252209839487239051712128153957277), SC_(0.999505500022370764381415753210237199735), SC_(0.0004944999776292356185842467897628002650075) }, + { SC_(0.21139929231139831244945526123046875e-4), SC_(0.17535277947899885475635528564453125e-4), SC_(0.8350250720977783203125), SC_(47305.46963235175970278465413513213767625), SC_(57026.27394012006075825157468938510447443), SC_(0.4534139659948462776784686703995923371492), SC_(0.5465860340051537223215313296004076628508) }, + { SC_(0.21433561414596624672412872314453125e-4), SC_(3719.279052734375), SC_(0.81474220752716064453125), SC_(46647.00459679377138587098836772819692673), SC_(0.1495926352972579290785339523472754005755e-2726), SC_(1), SC_(0.3206907637270668567387926794334274553413e-2731) }, + { SC_(0.22448601157520897686481475830078125e-4), SC_(445071.28125), SC_(0.221111953258514404296875), SC_(44532.62436928473062336268384761066982164), SC_(BOOST_MATH_SMALL_CONSTANT(0.4607684427809900763431582243857051777023e-48306)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.1034676148793947213721892942120911058743e-48310)) }, + { SC_(0.22683978386339731514453887939453125e-4), SC_(0.0405500046908855438232421875), SC_(0.905801355838775634765625), SC_(44086.07924003136859651634907996836330649), SC_(22.49465708555873548404357783254789649371), SC_(0.9994900162236478687022169258555052598576), SC_(0.0005099837763521312977830741444947401424393) }, + { SC_(0.234849067055620253086090087890625e-4), SC_(25542.79296875), SC_(0.9688708782196044921875), SC_(42569.81566430276714554195940795833966451), SC_(BOOST_MATH_SMALL_CONSTANT(0.7581371473885469585126327225518771230987e-38493)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.1780926545153655561405185897694774794731e-38497)) }, + { SC_(0.23615430109202861785888671875e-4), SC_(4.50702953338623046875), SC_(0.905801355838775634765625), SC_(42343.22752013782255946235697838204438231), SC_(0.5715334129764534783145383504186846510886e-5), SC_(0.999999999865023653988705273815604260127), SC_(0.1349763460112947261843957398729530965949e-9) }, + { SC_(0.2399934965069405734539031982421875e-4), SC_(462.945892333984375), SC_(0.81474220752716064453125), SC_(41661.08259105714016534966079504412432568), SC_(0.2775658605653054188832508127259377541357e-341), SC_(1), SC_(0.6662473543711679396846266818523256261124e-346) }, + { SC_(0.24066381229204125702381134033203125e-4), SC_(1270.3814697265625), SC_(0.12707412242889404296875), SC_(41544.01619584179617307996501816686679587), SC_(0.6432823532988754950678493440100093053861e-77), SC_(1), SC_(0.1548435640565879148514421887738552981949e-81) }, + { SC_(0.25217834263457916676998138427734375e-4), SC_(1832.2723388671875), SC_(0.913384497165679931640625), SC_(39646.38644935289616324338485234769483149), SC_(0.1452763128621285161831040219171995838252e-1949), SC_(1), SC_(0.3664301488048974504586999693295043470729e-1954) }, + { SC_(0.252623358392156660556793212890625e-4), SC_(0.18408362567424774169921875), SC_(0.1355634629726409912109375), SC_(39582.74118681315029489567111244553726972), SC_(7.044686151999700105392458377191074685903), SC_(0.9998220579880223516008255071920565810242), SC_(0.0001779420119776483991744928079434189758066) }, + { SC_(0.2568908166722394526004791259765625e-4), SC_(0.0026883506216108798980712890625), SC_(0.905801355838775634765625), SC_(38929.29587722904553152445726009402934876), SC_(369.7184745062910124389774018949948166521), SC_(0.9905921692794321886093269352728967872691), SC_(0.00940783072056781139067306472710321273093) }, + { SC_(0.26870451620197854936122894287109375e-4), SC_(0.000144985810038633644580841064453125), SC_(0.8350250720977783203125), SC_(37217.22199709670655355686997165747036372), SC_(6895.605189165562504177708220015695722412), SC_(0.8436825379600015811441576566235048540017), SC_(0.1563174620399984188558423433764951459983) }, + { SC_(0.27549775040824897587299346923828125e-4), SC_(0.1503586471080780029296875), SC_(0.12707412242889404296875), SC_(36295.98898214209116290907229901584968788), SC_(8.375418806411123975304744110348984534574), SC_(0.999769299946587350130083574806330185546), SC_(0.0002307000534126498699164251936698144539948) }, + { SC_(0.2782344745355658233165740966796875e-4), SC_(0.295027554035186767578125), SC_(0.8350250720977783203125), SC_(35941.82178311282183858216166056846305608), SC_(2.074584015871362718585422877982321716874), SC_(0.9999422827176363493743336035161848668839), SC_(0.5771728236365062566639648381513311607213e-4) }, + { SC_(0.28775624741683714091777801513671875e-4), SC_(25491.8359375), SC_(0.905801355838775634765625), SC_(34740.91309313126314329366380972784559115), SC_(BOOST_MATH_SMALL_CONSTANT(0.1416279658206675284644779676116805674026e-26157)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.4076690944794690667194646507475445069595e-26162)) }, + { SC_(0.2893155397032387554645538330078125e-4), SC_(494.983856201171875), SC_(0.9688708782196044921875), SC_(34557.55760263256678063136756722525969856), 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SC_(BOOST_MATH_SMALL_CONSTANT(0.544865187939051001712434628531605526675e-105651)) }, + { SC_(489093.59375), SC_(101.60869598388671875), SC_(0.221111953258514404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.2679049247465053509778519007374647739724e-320562)), SC_(0.1239584973345346670544577370734255574289e-418), SC_(BOOST_MATH_SMALL_CONSTANT(0.2161246953675901063139086740469757800594e-320143)), SC_(1) }, + { SC_(489183.34375), SC_(118808.328125), SC_(0.905801355838775634765625), SC_(BOOST_MATH_SMALL_CONSTANT(0.1458359410123570793275225181353576621217e-130434)), SC_(BOOST_MATH_SMALL_CONSTANT(0.259453747115697820698795192063284012701e-142915)), SC_(1), SC_(BOOST_MATH_SMALL_CONSTANT(0.1779079596666185278653733063531830995884e-12480)) }, + { SC_(491651.625), SC_(32.955524444580078125), SC_(0.3082362115383148193359375), SC_(BOOST_MATH_SMALL_CONSTANT(0.1033057257566332368173504705063859426151e-251301)), SC_(0.603837473523874333668117011509794982175e-152), SC_(BOOST_MATH_SMALL_CONSTANT(0.1710820051524158477748585452714475118872e-251149)), SC_(1) }, + { SC_(499024.09375), SC_(0.3617647588253021240234375), SC_(0.221111953258514404296875), SC_(BOOST_MATH_SMALL_CONSTANT(0.1198688160746637782814955712359261486421e-327059)), SC_(0.0213597393543653960374826366383814420494), SC_(BOOST_MATH_SMALL_CONSTANT(0.5611904437877215617130419280865220458283e-327058)), SC_(1) } + } }; +#undef SC_ + + + diff --git a/test/ibeta_small_data.ipp b/test/ibeta_small_data.ipp new file mode 100644 index 000000000..abfb11061 --- /dev/null +++ b/test/ibeta_small_data.ipp @@ -0,0 +1,512 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 500> ibeta_small_data = { { + { SC_(0.011510574258863925933837890625), SC_(2.726669788360595703125), SC_(0.913345992565155029296875), SC_(85.50804647765112454526520766832090120908), SC_(0.0004969774007283958673037601318581948437309), SC_(0.9999941879795790272650838125485136788379), SC_(0.5812020420972734916187451486321162137375e-5) }, + { SC_(0.0463422574102878570556640625), SC_(0.34317314624786376953125), SC_(0.24176712334156036376953125), SC_(20.36367089471426800179696837574809289039), SC_(3.630773740238788693031723031598949317136), SC_(0.8486827348798151491124959025982318049325), SC_(0.1513172651201848508875040974017681950675) }, + { SC_(0.06715317070484161376953125), SC_(2.30631923675537109375), SC_(0.908442795276641845703125), SC_(13.78727207173260438424514878253152024011), SC_(0.001859217475218142037637565734833320618317), SC_(0.9998651679038930266805148691555636600556), SC_(0.0001348320961069733194851308444363399443688) }, + { SC_(0.1190207004547119140625), SC_(7.26954746246337890625), SC_(0.19963125884532928466796875), SC_(6.225047194692518378537141946282961080975), SC_(0.08420413075357450486624447742592649170405), SC_(0.9866538632858121421788318729611834919717), SC_(0.01334613671418785782116812703881650802825) }, + { SC_(0.1540344059467315673828125), SC_(4.049813747406005859375), SC_(0.3462987840175628662109375), SC_(4.872033861103043731550905432482222659763), SC_(0.08561968850485947325255754361164029251743), SC_(0.9827297959310547093553196782752749843992), SC_(0.01727020406894529064468032172472501560077) }, + { SC_(0.15487124025821685791015625), SC_(8.16117000579833984375), SC_(0.678767263889312744140625), SC_(4.38038087876606402301557100951561214911), SC_(0.1540537895441550779731794373253219559374e-4), SC_(0.999996483108386751271915947803000461492), SC_(0.3516891613248728084052196999538507955892e-5) }, + { SC_(0.2251259386539459228515625), SC_(0.4898320138454437255859375), SC_(0.79967319965362548828125), SC_(4.764013990849157580013322088552892632018), SC_(0.9820281524501242644370508756885101403883), SC_(0.8290948573018540323461920074910160298235), SC_(0.1709051426981459676538079925089839701765) }, + { SC_(0.2867415249347686767578125), SC_(5.96057987213134765625), SC_(0.575251102447509765625), SC_(1.911839728052784613470170990261300207059), SC_(0.001412230939458232329324385634994224021816), SC_(0.9992618688130328170087283879882410324961), SC_(0.0007381311869671829912716120117589675039269) }, + { SC_(0.2922028005123138427734375), SC_(2.659206867218017578125), SC_(0.4733414947986602783203125), SC_(2.306780982820498015018593188560459962841), SC_(0.09785807660771959252183084474688989446611), SC_(0.9593044635018992897241131636398162350677), SC_(0.04069553649810071027588683636018376493234) }, + { SC_(0.30540943145751953125), SC_(4.23978328704833984375), SC_(0.546926796436309814453125), SC_(1.925766552541500211935789109529683925689), SC_(0.01136621113828240461645649770280082638802), SC_(0.9941324563027414190664847981329029657831), SC_(0.005867543697258580933515201867097034216903) }, + { SC_(0.318328440189361572265625), SC_(3.1655042171478271484375), SC_(0.077649272978305816650390625), SC_(1.337499870967964297403057650377070193563), SC_(0.6794195418585711764434318012650276929832), SC_(0.6631399660602084538462156083737590843824), SC_(0.3368600339397915461537843916262409156176) }, + { SC_(0.326008260250091552734375), SC_(0.625432431697845458984375), SC_(0.743158161640167236328125), SC_(3.078454179457483361692247221422692039972), SC_(0.7360974885092882497469458885357603771633), SC_(0.8070290947450602569822924525981547293294), SC_(0.1929709052549397430177075474018452706706) }, + { SC_(0.3360383510589599609375), SC_(5.28920650482177734375), SC_(0.81474220752716064453125), SC_(1.550409503660757234080944256429985207784), SC_(0.2836427788609597588510100729645041742761e-4), SC_(0.9999817056339550018928818220116909157891), SC_(0.1829436604499810711817798830908421094105e-4) }, + { SC_(0.344460785388946533203125), SC_(6.63605499267578125), SC_(0.096544884145259857177734375), SC_(1.138099220831291561221229605806786314538), SC_(0.2341007981779031558780678617792900307434), SC_(0.8293974676177770014617748040947131716032), SC_(0.1706025323822229985382251959052868283968) }, + { SC_(0.3571167886257171630859375), SC_(3.6129398345947265625), SC_(0.046266771852970123291015625), SC_(0.9053135817270937856051949826437502245961), SC_(0.7221758562362922538389449085733137575244), SC_(0.5562638752728180895224241888333329380732), SC_(0.4437361247271819104775758111666670619268) }, + { SC_(0.357627332210540771484375), SC_(8.064510345458984375), SC_(0.92886126041412353515625), SC_(1.197177044295353240338824861021333896969), SC_(0.7152754878374974083055684082879234042397e-10), SC_(0.9999999999402531571079558956505173243843), SC_(0.597468428920441043494826756157346017527e-10) }, + { SC_(0.377388656139373779296875), SC_(8.8396854400634765625), SC_(0.50323975086212158203125), SC_(1.048244561726343473675201855489355128708), SC_(0.0003378999437661089518583588179790252882915), SC_(0.999677755487891791262427317664954237941), SC_(0.0003222445121082087375726823350457620590389) }, + { SC_(0.424311339855194091796875), SC_(8.71805095672607421875), SC_(0.594544112682342529296875), SC_(0.8450634150576302527396320143587314116559), SC_(0.5692485563137910015348490929019764830298e-4), SC_(0.9999326428995458545641785392037209582211), SC_(0.6735710045414543582146079627904177891161e-4) }, + { SC_(0.4302380084991455078125), SC_(9.08433628082275390625), SC_(0.3500487506389617919921875), SC_(0.8041671576172428143402874467426430937693), SC_(0.003657242502533209987243300892131746946366), SC_(0.995472725877070621917637160657188675184), SC_(0.004527274122929378082362839342811324816028) }, + { SC_(0.4617138803005218505859375), SC_(1.49113976955413818359375), SC_(0.081217654049396514892578125), SC_(0.6708426431384324314892486199862307536228), SC_(1.06148098502757264044386168615001572695), SC_(0.3872501836441769856742304061740122471689), SC_(0.6127498163558230143257695938259877528311) }, + { SC_(0.4965442717075347900390625), SC_(8.234554290771484375), SC_(0.3013162314891815185546875), SC_(0.6255445999459755311064947246119255484269), SC_(0.01047121638037352884799746710410909796172), SC_(0.9835362327294694289867309219284499434766), SC_(0.01646376727053057101326907807155005652337) }, + { SC_(0.526769936084747314453125), SC_(9.4657726287841796875), SC_(0.695263326168060302734375), SC_(0.5223867123087661970619213259031225900887), SC_(0.1604184643861934563272584858102025314868e-5), SC_(0.9999969291337627338160603638685334704082), SC_(0.3070866237266183939636131466529591820232e-5) }, + { SC_(0.539501190185546875), SC_(3.5276241302490234375), SC_(0.184897840023040771484375), SC_(0.6337086405553985369329386662133114583969), SC_(0.2298885748176341425212030686754049306522), SC_(0.7338011624802040649313396945278096189402), SC_(0.2661988375197959350686603054721903810598) }, + { SC_(0.596188604831695556640625), SC_(9.52472019195556640625), SC_(0.587086021900177001953125), SC_(0.3957027656575689988900171488472459553927), SC_(0.2784518837222712883966491998692226153708e-4), SC_(0.9999296360008322244237915105214427770265), SC_(0.7036399916777557620848947855722297349452e-4) }, + { SC_(0.597795426845550537109375), SC_(4.3524570465087890625), SC_(0.3404516875743865966796875), SC_(0.5857283108172326731547619033650850279516), SC_(0.05176933170314495414753269319008084049281), SC_(0.9187929048671107105645293523464974421304), SC_(0.08120709513288928943547064765350255786955) }, + { SC_(0.604711830615997314453125), SC_(2.7766406536102294921875), SC_(0.65411365032196044921875), SC_(0.8104703643388840470345296228912560465778), SC_(0.02126375318260348616345995572262831556693), SC_(0.9744344343526894636219031995625457203905), SC_(0.02556556564731053637809680043745427960951) }, + { SC_(0.679927647113800048828125), SC_(9.0447483062744140625), SC_(0.887737333774566650390625), SC_(0.3013615312130418242114530511558325766293), SC_(0.2937963547130558679504530467684762405074e-9), SC_(0.999999999025103326081320534896178383616), SC_(0.9748966739186794651038216163839749164796e-9) }, + { SC_(0.688061058521270751953125), SC_(2.9616763591766357421875), SC_(0.815787494182586669921875), SC_(0.6441996486596996003790817490051121624011), SC_(0.0023593844359375506996753278475536050391), SC_(0.9963508599908640401822148000540199843775), SC_(0.003649140009135959817785199945980015622505) }, + { SC_(0.71445453166961669921875), SC_(9.34602642059326171875), SC_(0.5949366092681884765625), SC_(0.2611868075149835081076007360078062823636), SC_(0.2618210260815823472350439860777132136683e-4), SC_(0.9998997672257934489288388082946320821168), SC_(0.0001002327742065510711611917053679178831703) }, + { SC_(0.75854289531707763671875), SC_(4.042085170745849609375), SC_(0.18398940563201904296875), SC_(0.2869641794222026502350604842071731344923), SC_(0.1433368221355461874324336015227538920014), SC_(0.6668917301687721681209003462437186369765), SC_(0.3331082698312278318790996537562813630235) }, + { SC_(0.759666860103607177734375), SC_(7.62421321868896484375), SC_(0.285910427570343017578125), SC_(0.2494067464807902016116972999567207456019), SC_(0.01285822954049965889923791220383840028034), SC_(0.9509723725387721326945071222846981919121), SC_(0.04902762746122786730549287771530180808794) }, + { SC_(0.78175532817840576171875), SC_(8.5445098876953125), SC_(0.2395785152912139892578125), SC_(0.2092181979177349561806385620026173078435), SC_(0.01454489283267034607676299413579126814072), SC_(0.9349986953438432477441465286773755595707), SC_(0.06500130465615675225585347132262444042931) }, + { SC_(0.811257660388946533203125), SC_(3.7941887378692626953125), SC_(0.379480898380279541015625), SC_(0.3489982897256404699956459179424488728876), SC_(0.04918171477157487449577833255024935191165), SC_(0.876483715364670390428440227316937452589), SC_(0.123516284635329609571559772683062547411) }, + { SC_(0.834698200225830078125), SC_(9.0694408416748046875), SC_(0.644353687763214111328125), SC_(0.1803151611837914682987734789219117172738), SC_(0.9959010328687650223890520801118033780347e-5), SC_(0.9999447719190872199805310662295119611992), SC_(0.5522808091278001946893377048803880084592e-4) }, + { SC_(0.83821380138397216796875), SC_(9.12577533721923828125), SC_(0.22599919140338897705078125), SC_(0.1645323980962702319386216945486588491031), SC_(0.01289064037523516697593443936007900173636), SC_(0.9273451718204823928666892977505184246911), SC_(0.07265482817951760713331070224948157530893) }, + { SC_(0.84435856342315673828125), SC_(3.462334156036376953125), SC_(0.2511587440967559814453125), SC_(0.2764659412617420491570801392360458860717), SC_(0.1225038182191884368747109382285656512776), SC_(0.6929496150821833440766199995023216044932), SC_(0.3070503849178166559233800004976783955068) }, + { SC_(0.8551578521728515625), SC_(9.93534755706787109375), SC_(0.47097623348236083984375), SC_(0.156230204191103226500775887222958181338), SC_(0.0001981074086433488391675923367618586808585), SC_(0.9987335578411774548179603749802068600465), SC_(0.001266442158822545182039625019793139953536) }, + { SC_(0.908232867717742919921875), SC_(0.678848326206207275390625), SC_(0.84073317050933837890625), SC_(1.160861004416774987619512191580855025408), SC_(0.4258943431472485997275346962213866620687), SC_(0.7315941970505542617319292810425302688679), SC_(0.2684058029494457382680707189574697311321) }, + { SC_(0.9111347198486328125), SC_(0.862172305583953857421875), SC_(0.95751106739044189453125), SC_(1.185899552667666342813996371641383797478), SC_(0.07629806988531935777484778370164513501807), SC_(0.9395514073850060300319313133177312277542), SC_(0.06044859261499396996806868668226877224576) }, + { 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SC_(0.007521320523051118214918717491848671514352), SC_(0.3488847647374800274842030525271354542582e-6), SC_(0.9999996511152352625199725157969474728645) }, + { SC_(9.2938594818115234375), SC_(4.584969043731689453125), SC_(0.3805077970027923583984375), SC_(0.3033084082238698622351283936762757320358e-5), SC_(0.0002160623300405638004025369269425911334952), SC_(0.01384366758374349929295072479433910689838), SC_(0.9861563324162565007070492752056608931016) }, + { SC_(9.33993244171142578125), SC_(6.81359577178955078125), SC_(0.0497494600713253021240234375), SC_(0.5518070299877559210026646747715538340941e-13), SC_(0.2146633044365963053975520213612198005479e-4), SC_(0.2570569897903910302588882280913495968099e-8), SC_(0.9999999974294301020960896974111177190865) }, + { SC_(9.340106964111328125), SC_(9.6496639251708984375), SC_(0.18903611600399017333984375), SC_(0.3760693208748964009075562773732041031133e-8), SC_(0.2241360584188258329348724623108553890864e-5), SC_(0.001675051252959087450179950156748128851185), SC_(0.9983249487470409125498200498432518711488) }, + { SC_(9.39001560211181640625), SC_(2.4007594585418701171875), SC_(0.403971731662750244140625), SC_(0.1134959668917263176160589414159781875943e-4), SC_(0.004844797940116370573271103607794275185958), SC_(0.002337160599663038722280937028362435559564), SC_(0.9976628394003369612777190629716375644404) }, + { SC_(9.420505523681640625), SC_(9.54943370819091796875), SC_(0.3371889293193817138671875), SC_(0.1805691602626601023406871171319958471924e-6), SC_(0.2091846165246050032401702854147298202011e-5), SC_(0.07946133712253384615168111931846197902557), SC_(0.9205386628774661538483188806815380209744) }, + { SC_(9.4273700714111328125), SC_(1.3122184276580810546875), SC_(0.709393918514251708984375), SC_(0.003017013444722590640311000913969701534262), SC_(0.04315436163162597522990240737111380539958), SC_(0.06534380749400867203205742252815333005062), SC_(0.9346561925059913279679425774718466699494) }, + { SC_(9.43622684478759765625), SC_(7.562816619873046875), SC_(0.86930525302886962890625), SC_(0.1050377778686616769098759960271509592204e-4), SC_(0.9810847461408665570279905884259728037155e-8), SC_(0.9990668412278015532617620281113608196916), SC_(0.0009331587721984467382379718886391803084328) }, + { SC_(9.447872161865234375), SC_(0.4794428348541259765625), SC_(0.3019829094409942626953125), SC_(0.1527301820162022411297571755969196094484e-5), SC_(0.6378057092170519230917855314560001309605), SC_(0.2394613501875548352256068326033443059895e-5), SC_(0.9999976053864981244516477439316739665569) }, + { SC_(9.45174121856689453125), SC_(0.259588778018951416015625), SC_(0.9688708782196044921875), SC_(0.4793292977141484299090680804809562384231), SC_(1.485529678525234390368678419714687359757), SC_(0.2439509926720312131315430238386720081336), SC_(0.7560490073279687868684569761613279918664) }, + { SC_(9.50222110748291015625), SC_(4.90589046478271484375), SC_(0.095445640385150909423828125), SC_(0.1494828487479906048651881175052573767788e-10), SC_(0.0001378928182363681434821788452535606188609), SC_(0.1084050891519629129496674016152519495829e-6), SC_(0.9999998915949108480370870503325983847481) }, + { SC_(9.51630496978759765625), SC_(8.78050518035888671875), SC_(0.68924558162689208984375), SC_(0.3487950539831946130214243133355575362614e-5), SC_(0.2597595086647305428948201602139996658804e-6), SC_(0.9306884723462189452892185573366006247018), SC_(0.06931152765378105471078144266339937529823) }, + { SC_(9.5613460540771484375), SC_(6.51968669891357421875), SC_(0.3377856314182281494140625), SC_(0.4444839214175533505542773743694858127194e-6), SC_(0.2448798812291126478463498053849728750408e-4), SC_(0.01782751107179748780824836030020235500629), SC_(0.9821724889282025121917516396997976449937) }, + { SC_(9.57166957855224609375), SC_(1.0986175537109375), SC_(0.0333652384579181671142578125), SC_(0.7641149432459472271709082287842355172746e-15), SC_(0.07915132376380966467712577074419741010585), SC_(0.9653849195574867821285790668820816696202e-14), SC_(0.9999999999999903461508044251321787142093) }, + { SC_(9.57506847381591796875), SC_(9.9646129608154296875), SC_(0.01787211932241916656494140625), SC_(0.1658996384153681417135737467814313678843e-17), SC_(0.1513424851355072886884114448435894461463e-5), SC_(0.1096186825970546076551374980438478385039e-11), SC_(0.9999999999989038131740294539234486250196) }, + { SC_(9.57693958282470703125), SC_(7.249060153961181640625), SC_(0.872441589832305908203125), SC_(0.1265918745248280535982657188870855265927e-4), SC_(0.1650447533802566668031298868233442384058e-7), SC_(0.9986979428474589734503870049545106253976), SC_(0.001302057152541026549612995045489374602406) }, + { SC_(9.59291362762451171875), SC_(5.809566974639892578125), SC_(0.1387105882167816162109375), SC_(0.3226608546341382422172814363970325173078e-9), SC_(0.4951645101829454492779247772056654797778e-4), SC_(0.6516193012187428032807345415266845925099e-5), SC_(0.9999934838069878125719671926545847331541) }, + { SC_(9.59492397308349609375), SC_(5.036627292633056640625), SC_(0.043119497597217559814453125), SC_(0.70463813192480350035637684220091276433e-14), SC_(0.0001140624644588322714943677495101932553999), SC_(0.6177651300315304735247529704567042174447e-10), SC_(0.9999999999382234869968469526475247029543) }, + { SC_(9.59743976593017578125), SC_(5.737546443939208984375), SC_(0.12707412242889404296875), SC_(0.1472749033768804794567481194100647733112e-9), SC_(0.5323124639940113790246026061688731976865e-4), SC_(0.276669260771152772688627996297166358751e-5), SC_(0.9999972333073922884722731137200370283364) }, + { SC_(9.61898136138916015625), SC_(9.79925632476806640625), SC_(0.240151941776275634765625), SC_(0.1353843923794317977412052106059477829317e-7), SC_(0.1632783113888411433729758988565696586839e-5), SC_(0.008223447729414564260765034318619123062768), SC_(0.9917765522705854357392349656813808769372) }, + { SC_(9.63088512420654296875), SC_(6.059281826019287109375), SC_(0.488668859004974365234375), SC_(0.581055997045127220418244921269735439886e-5), SC_(0.3187439986268775277133352875803521062324e-4), SC_(0.1541877713596934710853644077040271007484), SC_(0.8458122286403065289146355922959728992516) }, + { SC_(9.6488857269287109375), SC_(9.67694950103759765625), SC_(0.0185489989817142486572265625), SC_(0.1749481595836606883382303122964370689815e-17), SC_(0.1758002864820450185505025032329301561348e-5), SC_(0.995152869681782569163934165768269529895e-12), SC_(0.9999999999990048471303182174308360658342) }, + { SC_(9.686492919921875), SC_(3.323390960693359375), SC_(0.822622716426849365234375), SC_(0.0007256304807823739090101393096545824752395), SC_(0.0002824860673401657300814152223119963953949), SC_(0.7197882845329221918507242489433549020479), SC_(0.2802117154670778081492757510566450979521) }, + { SC_(9.70592784881591796875), SC_(9.81109714508056640625), SC_(0.0319296605885028839111328125), SC_(0.2412648809123635167598641163183182877987e-15), SC_(0.153247481975863672328821013292784593199e-5), SC_(0.1574348092142740770669069066716607378887e-9), SC_(0.9999999998425651907857259229330930933283) }, + { SC_(9.7297458648681640625), SC_(1.422172069549560546875), SC_(0.62565600872039794921875), SC_(0.0007515285211317483408827317273891061880333), SC_(0.03308064211512736093153684853289103287133), SC_(0.02221342902327139455287559010633701337676), SC_(0.9777865709767286054471244098936629866232) }, + { SC_(9.7868061065673828125), SC_(2.324003696441650390625), SC_(0.580998599529266357421875), SC_(0.0001869421048606273257218647038338269425064), SC_(0.004902502132333050539451707571566560474365), SC_(0.03673133964106605423746931249168049356184), SC_(0.9632686603589339457625306875083195064382) }, + { SC_(9.7974834442138671875), SC_(3.546381473541259765625), SC_(0.43892610073089599609375), SC_(0.8846383934988994958104111659144072787642e-5), SC_(0.0006942444162306305791078085627402992944978), SC_(0.01258213581077315616174240184063815608328), SC_(0.9874178641892268438382575981593618439167) }, + { SC_(9.816379547119140625), SC_(4.97869396209716796875), SC_(0.7558143138885498046875), SC_(0.8471586497414725426535462974925383298225e-4), SC_(0.2600787706776670771042095868990896095329e-4), SC_(0.7651102050188879474911089577484436075733), SC_(0.2348897949811120525088910422515563924267) }, + { SC_(9.82663440704345703125), SC_(8.9943599700927734375), SC_(0.79222810268402099609375), SC_(0.2564909668495030018187514949876471506381e-5), SC_(0.1328272157019951695723866392922830834089e-7), SC_(0.9948480487253848974511529374575346247921), SC_(0.005151951274615102548847062542465375207862) }, + { SC_(9.83052539825439453125), SC_(5.54380321502685546875), SC_(0.711244642734527587890625), SC_(0.4142099063694028106257790388260333526588e-4), SC_(0.1706128131291109079313412403482492307802e-4), SC_(0.7082657573983930918019463517369821910095), SC_(0.2917342426016069081980536482630178089905) }, + { SC_(9.84063720703125), SC_(2.9338824748992919921875), SC_(0.679551780223846435546875), SC_(0.0003620677940926280979183654032292271566579), SC_(0.001398063860838349825529157603027032578726), SC_(0.2057049500122911434857054575556857430709), SC_(0.7942950499877088565142945424443142569291) }, + { SC_(9.8798198699951171875), SC_(7.69243144989013671875), SC_(0.498594343662261962890625), SC_(0.2102276377997196459781769309658523199559e-5), SC_(0.5090672027747954852585150635916233053816e-5), SC_(0.2922690751289229878585770869253518082993), SC_(0.7077309248710770121414229130746481917007) }, + { SC_(9.9613475799560546875), SC_(4.475843906402587890625), SC_(0.2373598515987396240234375), SC_(0.2596148971864002645551220232138375643592e-7), SC_(0.0001912089032647389537199491735362492125718), SC_(0.0001357570950881480228485230092737049890207), SC_(0.999864242904911851977151476990726295011) }, + { SC_(9.9908046722412109375), SC_(6.74212932586669921875), SC_(0.73282539844512939453125), SC_(0.1400972426630799368521044086815964034595e-4), SC_(0.1993917002921952527694349144803786583093e-5), SC_(0.8754085417575787330833090864097798601476), SC_(0.1245914582424212669166909135902201398524) } + } }; +#undef SC_ + + + diff --git a/test/igamma_big_data.ipp b/test/igamma_big_data.ipp new file mode 100644 index 000000000..99f6ec939 --- /dev/null +++ b/test/igamma_big_data.ipp @@ -0,0 +1,298 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 287> igamma_big_data = { { + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655441178896580822765827178955078125e-7), SC_(-1), SC_(0.2993117912029903240422582053899047098779533956541385552546168739467202263345519375546082579259835689e-4), SC_(577797.90391619920649973095027584836128926282155274216943338008435801586707772349755359609709258809), SC_(0.9999700688208797009675957741794610095290122046604345861444745383126053279773665448062442806862583006) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.8653277063785935752093791961669921875e-6), SC_(13.38277582367555562294055416847373915500263659346494217875993623821221196887112446988546507684731302), SC_(0.2316099655383339755405731669847576162961123204586203558223557251327214945205980959748823352959805302e-4), SC_(577801.8158305834504756449335090514614341009677858339594077124115869397920785194231187741120056354752), SC_(0.9999768390034461666024459426833015242383703887679541379644177644274867278505479401904023114002150846) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.15575898260067333467304706573486328125e-5), SC_(12.79500378214091254793715786456975103024623413471734292810881005720669047076428987935745392946724363), SC_(0.2214376467251173939070843604623148303831679646741287024258011317172427658231597089388494422353631048e-4), SC_(577802.4036026249851187199369053553654222257241882927070069630627131207976000175299533646400167828553), SC_(0.9999778562353274882606092915639537685169616832035325871297574198868282757234176840291059234224082336) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.1730655412757187150418758392333984375e-5), SC_(12.68964583907062077871829961486723468135682689673266287978904878277523088091119954467857396528045035), SC_(0.2196142619591161296471772336172777007255716660501426039935550192381060291894820144394071807910637318e-4), SC_(577802.5089605680554104891557636050679385746135955306916870113824743952290596073830436993188967470421), SC_(0.9999780385738040883870352822766382722299274428333949857396006444980761893970810517985558692139197711) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.19037209995076409541070461273193359375e-5), SC_(12.59433798893655043954589862357055470940362305301481664366705173244103638298812628401438143132385277), SC_(0.2179648098442542012919725801196235526752581619134538802152189012490537997981862095991665699048812574e-4), SC_(577802.6042684181894808283281645963646185465667993744095332475044714455632541053061169599830892809987), SC_(0.9999782035190155745798708027419880376447324741838086546119784781098750946200201813790398946910551472) }, + { SC_(0.1730655412757187150418758392333984375e-5), SC_(0.346131082551437430083751678466796875e-5), 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SC_(0.1026762455478787231848021279669200901989201648818956533640028599511278074166334792418412467632444988), SC_(1) }, + { SC_(788352.3125), SC_(1), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), SC_(0.4666440397038730095030131088923330001395921365337505093937538015695853474965515882892883491065022498e-6), SC_(BOOST_MATH_SMALL_CONSTANT(0.1433908471829622508376220400714432285652733465995812102127318484064704211691023381454450301792300023e-4306319)) }, + { SC_(788352.3125), SC_(394176.15625), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1), SC_(0.2006257520633973004844236201403980302780110902909388684784270381180202010418543872474966575049005566), SC_(BOOST_MATH_SMALL_CONSTANT(0.6164848172783960800240480360461358265727377175617248578986815365384881763390067655148178076072402911e-66132)) }, + { SC_(788352.3125), SC_(788350.3125), SC_(0.162961216861866486196123635890099179907765436181771479621640418745608625572387287597608545912315394), SC_(0.5007488568506734531584407973208886646226494394047062415392632213315407911442369980288546641886893887), SC_(0.1624738083656850062172671442384129180894308808036804291026031361346630666772952933350920757512800554), SC_(0.4992511431493265468415592026791113353773505605952937584607367786684592088557630019711453358113106113) }, + { SC_(788352.3125), SC_(788352.3125), SC_(0.1626687717822932899593035312904247710866671194963517534486901004447793852560311314241268483442598936), SC_(0.4998502286855928443434206403315169955579486533591805378314370596112904256380207126657714482687295139), SC_(0.1627662534452582024540872488380873269105291974891001552755534544354923069936514495085737733193355558), SC_(0.5001497713144071556565793596684830044420513466408194621685629403887095743619792873342285517312704861) }, + { SC_(788352.3125), SC_(788354.3125), SC_(0.1623763274446328985015387393773169322760017933096679464665287143383813538555079409462603516012733209), SC_(0.4989516028002693214791004285569604766689865869988938477436476252444986636812350914588652370815612146), SC_(0.1630586977829185939118520407511951657211945236757839622577148405418903383941746399864402700623221286), SC_(0.5010483971997306785208995714430395233310134130011061522563523747555013363187649085411347629184387854) }, + { SC_(788352.3125), SC_(1576704.625), SC_(0.62966881391456604663265189232296651965212021691993178179931323658375951801607008550560163062833006), SC_(BOOST_MATH_SMALL_CONSTANT(0.1934852628337369167484390536774609470572595166386051921886173923840391906771189648683653522231175771e-105062)), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1) }, + { SC_(788352.3125), SC_(78835232), SC_(BOOST_MATH_SMALL_CONSTANT(0.4164740077948943840500786922849307561907212018086545639004814350805550398839532201156819761152062039e-28012316)), SC_(BOOST_MATH_SMALL_CONSTANT(0.1279745496059271390694582975257723346837265207507390129414161314216004095755396957400977650588924393e-32318629)), SC_(0.3254350252275514924133907801285120979971963169854519087242435548802716922496825809327006216635954494), SC_(1) }, + { SC_(1736170), SC_(1), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), SC_(0.2118914928047443977144200433110895948573532788053230331876929560838004923241637943455403176192920957e-6), SC_(BOOST_MATH_SMALL_CONSTANT(0.1678638954845296242754981240024346426980228514280855063671268705695352948287648087179192945942541815e-10078987)) }, + { SC_(1736170), SC_(868085), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1), SC_(0.1448940961583454163407547513975813077281954384418820421315934004081116003090269523192520567562712675), SC_(BOOST_MATH_SMALL_CONSTANT(0.1147874654706537747553860317681134464810452807999399860509102977323900833593912850990760752412482674e-145637)) }, + { SC_(1736170), SC_(1736168), SC_(0.6317777290022475594523330311129906002358832226132013915031845078518775760353610272181933223363237277), SC_(0.500504618033028326360191641057343755999405997498206839871815176619438983969420893139030447209664989), SC_(0.6305037889687944450359199539867683474985226650229812801702613966546458595016893892268912941388886983), SC_(0.499495381966971673639808358942656244000594002501793160128184823380561016030579106860969552790335011) }, + { SC_(1736170), SC_(1736170), SC_(0.6310133650036994505936467834743236622047002119530026486269595591827934240774463130132409395432514033), SC_(0.49989907641044580817879562048675209725080620507540160516376663057508236554638837205467803575236936), SC_(0.6312681529673425538946062016254352855297056756831800230464863453237300114596041034318436769319610226), SC_(0.50010092358955419182120437951324790274919379492459839483623336942491763445361162794532196424763064) }, + { SC_(1736170), SC_(1736172), SC_(0.63024900188566778658967343804876836928228054774250423301619921925975649543556103050219224039778483), SC_(0.4992935354854227744062593568501916692180042004614849112072827608836331168589189268904245367712978671), SC_(0.632032516085374217898579547050990578452125339893678438657246685246766940101489385942892376077427596), SC_(0.5007064645145772255937406431498083307819957995385150887927172391163668831410810731095754632287021329) }, + { SC_(1736170), SC_(3472340), SC_(0.6014445720839096678691921265326588051367345601002845203175453094832083718879547519600219908634900144), SC_(BOOST_MATH_SMALL_CONSTANT(0.4764741965410820548147263974496875110171884751124591798801059384427379413001614590138374358389181743e-231373)), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1) }, + { SC_(1736170), SC_(173616992), SC_(BOOST_MATH_SMALL_CONSTANT(0.1028718049106474382663595835099645962497203244264180111334446299544683261400046764833876858973945412e-61095576)), SC_(BOOST_MATH_SMALL_CONSTANT(0.8149672117199407490039815813486797316083480325648451121983304934903758539219857770713382104685664004e-71174558)), SC_(0.1262281517971042004488252985099758947734405887636182671673445904506523435537050416445084616475212426), SC_(1) } + } }; +#undef SC_ + + diff --git a/test/igamma_int_data.ipp b/test/igamma_int_data.ipp new file mode 100644 index 000000000..77e20f289 --- /dev/null +++ b/test/igamma_int_data.ipp @@ -0,0 +1,151 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 140> igamma_int_data = { { + { SC_(0.5), SC_(0.004999999888241291046142578125), SC_(1.631267845368323485191815380032984903074365412587098308286220239190324188420710121791871591309590004), SC_(0.9203443263332001265162236927545643793578786527087679769237787431564844748135321134774112708209554194), SC_(0.1411860055371925421063521033081602797231840435352888199275875506625870961703220596656577919527041757), SC_(0.07965567366679987348377630724543562064212134729123202307622125684351552518646788652258872917904458057) }, + { SC_(0.5), SC_(0.25), SC_(0.8498918380799311297867616098602389766300312781938048348528626579829435160004444680322048858624092311), SC_(0.479500122186953462317253346108035471263548424242036299941194274352806478283146429109461900411394659), SC_(0.9225620128255848975114058734809062061675181779285822933609451318699677685905877134253244973998849491), SC_(0.520499877813046537682746653891964528736451575757963700058805725647193521716853570890538099588605341) }, + { SC_(0.5), SC_(0.449999988079071044921875), SC_(0.6075647752751784703583581789426278456752564880945645704809989500539948786183333019925969429005007538), SC_(0.3427817175407890768272360273834711985342418909050114189064883223090985450970247707985828865357086527), SC_(1.164889075630337556939809304398517337122292968027822557732808839798916405972698879464932440361793426), SC_(0.6572182824592109231727639726165288014657581090949885810935116776909014549029752292014171134642913473) }, + { SC_(0.5), SC_(0.5), SC_(0.5624182315944071242794949573020430690267675647965080152711827351225288269777202721053138351861008813), SC_(0.3173105078629141028295349087359241550441740665467912180252110995140171160255903409776584970430004241), SC_(1.210035619311108903018672526039102113770781891325879112942625054730382457613311909352215548076193299), SC_(0.6826894921370858971704650912640758449558259334532087819747889004859828839744096590223415029569995759) }, + { SC_(0.5), SC_(0.550000011920928955078125), SC_(0.5215730804923022694296157870452972118167645274078781347659207691038036243173236920652888890778334457), SC_(0.2942660990726723860419838415374453483460616872211918064199672606812661428450855377898215276021760526), SC_(1.250880770413213757868551696295847970980784928714508993447887020749107660273708489392240494184460734), SC_(0.7057339009273276139580161584625546516539383127788081935800327393187338571549144622101784723978239474) }, + { SC_(0.5), SC_(1), SC_(0.2788055852806619764992326110774391720885500824971744701584987919357006294674865183689945658651387992), SC_(0.1572992070502851306587793649173907407039330020336970915400621021652827459039891587248999441840165472), SC_(1.493648265624854050798934872263706010708999373625212658055308997917210655123545663088534817397155381), SC_(0.8427007929497148693412206350826092592960669979663029084599378978347172540960108412751000558159834528) }, + { SC_(0.5), SC_(100), SC_(0.3701747860408278920253566448133907572136210252033768764800615129976497004743801458060294976199881689e-44), SC_(0.2088487583762544757000786294957788611560818119321163727012213713938174695833440290513463628030792355e-44), SC_(1.772453850905516027298167483341145182797549452420639267805528869599344836457124609321319131228525415), SC_(0.999999999999999999999999999999999999999999997911512416237455242999213705042211388439181880678836273) }, + { SC_(1), SC_(0.00999999977648258209228515625), SC_(0.9900498339704614360382138392834551161113407838357779440886813669674202814166581683103367609018741592), SC_(0.9900498339704614360382138392834551161113407838357779440886813669674202814166581683014245491001600443), SC_(0.009950166029538563961786160716544883888659216164222055911318633032579718583341831698248133897676852056), SC_(0.009950166029538563961786160716544883888659216164222055911318633032579718583341831698575450899839955738) }, + { SC_(1), SC_(0.5), SC_(0.6065306597126334236037995349911804534419181354871869556828921587350565194137484240072325063075404673), SC_(0.606530659712633423603799534991180453441918135487186955682892158735056519413748423985704188802918523), SC_(0.393469340287366576396200465008819546558081864512813044317107841264943480586251576001352388492010544), SC_(0.393469340287366576396200465008819546558081864512813044317107841264943480586251576014295811197081477) }, + { SC_(1), SC_(0.89999997615814208984375), SC_(0.4065696694339752855534404747740105862953642332665029835031979841616081291898465027056760538362422369), SC_(0.4065696694339752855534404747740105862953642332665029835031979841616081291898465026775698931385242319), SC_(0.5934303305660247144465595252259894137046357667334970164968020158383918708101534973029088409633087743), SC_(0.5934303305660247144465595252259894137046357667334970164968020158383918708101534973224301068614757681) }, + { SC_(1), SC_(1), SC_(0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033087215393207514013), SC_(0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033055633330193769539), SC_(0.63212055882855767840447622983853913255418886896823216549216319830253850425510019669986335547879961), SC_(0.6321205588285576784044762298385391325541888689682321654921631983025385042551001966944366669806230461) }, + { SC_(1), SC_(1.10000002384185791015625), SC_(0.3328710757618145679671571848455601132593649668322841328963855852147267744456382506080158515376624924), SC_(0.332871075761814567967157184845560113259364966832284132896385585214726774445638250618965856020172342), SC_(0.6671289242381854320328428151544398867406350331677158671036144147852732255543617494005690432618885189), SC_(0.667128924238185432032842815154439886740635033167715867103614414785273225554361749381034143979827658) }, + { SC_(1), SC_(2), SC_(0.1353352832366126918939994949724844034076315459095758814681588726540733741014876899370981224906570488), SC_(0.1353352832366126918939994949724844034076315459095758814681588726540733741014876899415500621934116259), SC_(0.8646647167633873081060005050275155965923684540904241185318411273459266258985123100714867723088939625), SC_(0.8646647167633873081060005050275155965923684540904241185318411273459266258985123100584499378065883741) }, + { SC_(1), SC_(100), SC_(0.3720075976020835962959695803863118337358892292376781967120613876663290475895815718157118778642281497e-43), SC_(0.37200759760208359629596958038631183373588922923767819671206138766632904758958157182794930297528019e-43), SC_(0.9999999999999999999999999999999999999999999627992402397916403704030419613688166264196619710317313401), SC_(0.9999999999999999999999999999999999999999999627992402397916403704030419613688166264110770762321803288) }, + { SC_(4.5), SC_(0.04500000178813934326171875), SC_(11.63172821024549441698223162601989743958836118115278238773312379208547617681782066799483071466068875), SC_(0.9999999839815762404539939126624023829295483903490073208372721857918957147514500440457971141443476684), SC_(0.1863219545121619924834063678225205571246503831411699898288242541283108286160415992612505831218724607e-6), SC_(0.1601842375954600608733759761707045160965099267916272781420810428524854995595420288585565233163730242e-7) }, + { SC_(4.5), SC_(2.25), SC_(10.18403214286566234526019345802929635910911247967051220522868706375276166311515909530042932735839335), SC_(0.8755390252983378432498497196860962665082171215981371572250371700817082375787028909649723325825612257), 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SC_(32252126974542171289411048249083781405372.53474714482154916805178576650614938828011202647624996849196), SC_(0.522014276547313947601528692598500029895500785401096881691266864819900195072518250950429228686742359) }, + { SC_(36.5), SC_(38.5), SC_(21742960397427655971999858476245787542154.15150586635725616093880001666211564993165974656956851031785), SC_(0.3519189835392620395302152269324892873693310528827889471335875731231959108019664428015002605063426072), SC_(40041033687682249313617362599307398090717.31443404234592002014059498309205144499003814879162837098991), SC_(0.6480810164607379604697847730675107126306689471172110528664124268768040891980335571984997394936573928) }, + { SC_(36.5), SC_(73), SC_(52949442154506109118983369217871874.38246594734925702394423038249326654537455487682886341892236797151), SC_(0.8570090512692680556802347857159858050844535163506777789989580690865292168793799100092698899450718342e-6), SC_(61783941135667750779508102092183967760997.08347396135391915713516461726090054954714301853233346238539), SC_(0.9999991429909487307319443197652142840141949155464836493222210010419309134707831206200899907301100549) }, + { SC_(36.5), SC_(3650), SC_(0.1953355084312233771020954281999372828097757306846922312364706684936133958688600435101169722435920175e-1458), SC_(0.3161587581439635603576946151249429130551615750835979291506861400626078334974808393076908597655919673e-1499), SC_(61783994085109905285617221075553185632871.46593990870317618107939499975416709492169789536119688130775), SC_(1) }, + { SC_(37), SC_(0.37000000476837158203125), SC_(371993326789901217467999448150835199999999.999999999999999998010144152088536780620259902094609630282), SC_(0.9999999999999999999999999999999999999999999999999999999999946508291802582432073734328927504122445964), SC_(0.1989855848803199113330559282105407777739456877542387398548067882388920416527083270608322077134689013e-17), SC_(0.5349170819741756792626567107249587755403553332314009452247515306615350728798447484985751790393880222e-59) }, + { SC_(37), SC_(18.5), SC_(371956712793231153197597830186294584210422.2758111030818172279500018253795285911205314038120175905779), SC_(0.9999015735121217818090760205581543280418411120156879903724853447672046973321848456420679481585530018), SC_(36613996670064270401617964540615789577.72418889691818277204999817551220730283028778038799981744356341), SC_(0.9842648787821819092397944184567195815888798431200962751465523279530266781515435793205184144699823239e-4) }, + { SC_(37), SC_(35), SC_(226989024014514715184435975122876165449017.4354380189351676839148067141921226508640903916939172052318), SC_(0.6101964945804424491761955895760276454618789066404902936457902256016569955365846250467251759013351627), SC_(145004302775386502283563473027959034550982.5645619810648323160851932866996132430867287925061002027896), SC_(0.3898035054195575508238044104239723545381210933595097063542097743983430044634153749532748240986648373) }, + { SC_(37), SC_(37), SC_(177863034503958843994685514205494501614349.5019963234044377047596609195056110031786919237288259549572), SC_(0.4781350139767814391926669215604954535637725270056258313717131858978224542335868299177336178494837268), SC_(194130292285942373473313933945340698385650.4980036765955622952403390813861248907721272604711914530642), SC_(0.5218649860232185608073330784395045464362274729943741686282868141021775457664131700822663821505162732) }, + { SC_(37), SC_(39), SC_(131256580134582475503504585688775961443809.3151145960527074790695530203993802515906461401417853004337), SC_(0.3528465988012605299562448021483701523550172331761938728513557987821171322564921159639473486646029879), SC_(240736746655318741964494862462059238556190.6848854039472925209304469804923556423601730440582321075878), SC_(0.6471534011987394700437551978516298476449827668238061271486442012178828677435078840360526513353970121) }, + { SC_(37), SC_(74), SC_(271778651018441097347065883466473848.2746511977815237362658432276453536467157425861343116997640371484), SC_(0.730600877611816548123508890538883154677474348252631497380790705322862094368509049491763524535733931e-6), SC_(371993055011250199026902101084951733526151.7253488022184762637341567732463822472350765980657057082574), SC_(0.9999992693991223881834518764911094611168453225256517473685026192092946771379056314909505082364754643) }, + { SC_(37), SC_(3700), SC_(0.3714703196437381532843643639864485147019841838742418471961281810239780186658714011436027292802695334e-1478), SC_(0.9985940415902717138423033467831033580547327587141446387368733042084326864839438382850644573580819965e-1520), SC_(371993326789901217467999448150835200000000.0000000000000000000000000008917358939508191842000174080214), SC_(1) } + } }; +#undef SC_ + + diff --git a/test/igamma_inva_data.ipp b/test/igamma_inva_data.ipp new file mode 100644 index 000000000..573821934 --- /dev/null +++ b/test/igamma_inva_data.ipp @@ -0,0 +1,445 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 435> igamma_inva_data = { { + { SC_(0.11342023313045501708984375), SC_(0.097540400922298431396484375), SC_(1.035869900800721563193351335409353423054156223778352824799732510014114331303190861923913851617789576), SC_(0.05862165929221091309602935851268320728966552926925608108837955463403857541085640683871851098187935567) }, + { SC_(0.11342023313045501708984375), SC_(0.12698681652545928955078125), SC_(0.93513162632210132756150227031041961003749580131891848591112697892138664595709032559736553036690404), SC_(0.07702073629515159209895492482246819546302457039495283952439305125976814739590655422243170396591695024) }, + { SC_(0.11342023313045501708984375), SC_(0.135477006435394287109375), SC_(0.9100380607375349140135424653157145054283763683426464165277887838730942363760587929004439481331867906), SC_(0.08239454114833290164904666725589763295213501324448854257672874472236795673343348101283994380015493955) }, + { SC_(0.11342023313045501708984375), SC_(0.188381969928741455078125), SC_(0.779595202624673387363485521164023245809684162801375448392802218438362825559076033610794317185133861), SC_(0.1166391941338030559840976122781824609204768300357456413264936214065498992433137767918843993082512688) }, + { SC_(0.11342023313045501708984375), SC_(0.22103404998779296875), SC_(0.7145928353898281531229899766663421465043332388718242321020543488710755276457340056824233507290026167), SC_(0.1384900078344770795221486812547034082011104407568501332022902743558707268824034463784809050893959221) }, + { SC_(0.11342023313045501708984375), SC_(0.278498232364654541015625), SC_(0.6182586795562308310975349170899684675397900921120847475062459321020936519806102953807288680346855742), SC_(0.1784721480822330050905288284953586748392689370078276538971887861788093058242214704511862158537019216) }, + { SC_(0.11342023313045501708984375), SC_(0.308167040348052978515625), SC_(0.5750762878067446972629162563822329468974137308881623861798146424843915189358227607746127802679658861), SC_(0.1999745971366197606214810194646538480932240109343778328885211338343706229230203595508410364650077464) }, + { SC_(0.11342023313045501708984375), SC_(0.546881496906280517578125), SC_(0.3158581972518262198680262971873178145282238457421313069361242252141961771513906035921435181902024621), SC_(0.40399522554055015965171997281891601803562228187095717362147123937459020167864162427846789506528334) }, + { SC_(0.11342023313045501708984375), SC_(0.54722058773040771484375), SC_(0.3155616219617029735803748550238394725944210138946535670470388364147063820887200160768983610508607508), SC_(0.4043391538070912752469413697358973847567093976055568141867354582154449982775159287792880341381161789) }, + { SC_(0.11342023313045501708984375), SC_(0.6323592662811279296875), SC_(0.2451306925552628674700116671932204741230337234325446757090933316372456823776133088401164345457104668), SC_(0.4981721948503688022008818739022596218839268441108020243647633715157177066183221315388893154605681197) }, + { SC_(0.11342023313045501708984375), SC_(0.814723670482635498046875), SC_(0.1145907253421634340428550841021180213245634728137200905215078642375729374894281996476238899008128165), SC_(0.7862852295173094376574861816166162829803656701458377721112754273299144168379358322286031004477952406) }, + { SC_(0.11342023313045501708984375), SC_(0.835008561611175537109375), SC_(0.101331527591007847744573524795847576083929747580314104040226733069549334001280984610814567479857817), SC_(0.8326058923439134249687721563121076618130208457176003407255229510096659223637127918548740068611902389) }, + { SC_(0.11342023313045501708984375), SC_(0.905791938304901123046875), SC_(0.05656193657838154866774148682961754372226868790952633895057593923638262083267913677844800561062284631), SC_(1.04896840759462485702081614165933409290459738742990942007871315448214243965980816410908535790073786) }, + { SC_(0.11342023313045501708984375), SC_(0.9133758544921875), SC_(0.05189064964852823459637928419581919698509481017420310002496395476818562158762986899770261198573844241), SC_(1.080435768098731745020005560573961257374011084999542196321375433066538881295544756909074244422125722) }, + { SC_(0.11342023313045501708984375), SC_(0.968867778778076171875), SC_(0.01835818396061160048337643098614074549706602146731102647725732066970461614194476008520575667903820769), SC_(1.448797407656090219493165071703219939123058592597047518890784499076538070587802667567859080671028321) }, + { SC_(0.1419346332550048828125), SC_(0.097540400922298431396484375), SC_(1.124578830537963702051998473796288168586685310061027732355203576629632283960817297660062046074915143), SC_(0.0659284721147043282303849180640934562566897224895804233539249128398260667430925608435729540214754515) }, + { SC_(0.1419346332550048828125), SC_(0.12698681652545928955078125), SC_(1.017033751189331824504325403924341720905739138565526684761461549984097760120918333327463981882765665), SC_(0.08649278031454385525728666007963182163925805103763420954040262025741180677454847460411841098791601186) }, + { SC_(0.1419346332550048828125), SC_(0.135477006435394287109375), SC_(0.9902142921858343274435115921427109074354611809918110403382037220186217516256053009853608288056704689), SC_(0.09248877472325344088393739866931009589190859884078310814802341635680304235539767062268472977333731618) }, + { SC_(0.1419346332550048828125), SC_(0.188381969928741455078125), SC_(0.8505785998195794526287491308010932266793339719426224792262622633066161829593612618292878964211114605), SC_(0.1305982554060110903644352571539294106230392294124679024676030185902887691963181187121592257303657878) }, + { SC_(0.1419346332550048828125), SC_(0.22103404998779296875), SC_(0.7808395661964723025065274478432724703812506573362937686319290770429457825258588979051496462804184209), SC_(0.1548317951205897711220384965795747080251080507849479059158897171375778172596890940331713045635641511) }, + { SC_(0.1419346332550048828125), SC_(0.278498232364654541015625), SC_(0.6772609184694136395829214678207043149699253868567694609654501808726365693498576674966232500529840506), SC_(0.1990250470315257234650602086315659437473047214824637904232710507788991505540315624327049715768915247) }, + { SC_(0.1419346332550048828125), SC_(0.308167040348052978515625), SC_(0.630732340145806754814474818656317258739611054139979482135367574881529997048038535582075675545383329), SC_(0.2227203252084107833339010674280717585720589607238234483333954941372899120632213762241351353111801409) }, + { SC_(0.1419346332550048828125), SC_(0.546881496906280517578125), SC_(0.349708717751391355159479214660277171464283691195026088708202769671562811924679353305737371727474314), SC_(0.4456482378008532807988050947059477473750827663770176898902670239364470277606341238138830195865284648) }, + { SC_(0.1419346332550048828125), SC_(0.54722058773040771484375), SC_(0.34938505235700051109422458264248273843369824263538926665971457103150418551006634428413713838463443), SC_(0.4460217175733750342583186470185268231571709276780851712061443842332352649527336346223436093968900516) }, + { SC_(0.1419346332550048828125), SC_(0.6323592662811279296875), SC_(0.2723365665725819880697084193348315672163942836223061717484924369936181601343161297604977832581277765), SC_(0.5476941234441064195196321208511198969055308360244560161002793861902290381370045763117008954093133652) }, + { SC_(0.1419346332550048828125), SC_(0.814723670482635498046875), SC_(0.1283232223230380098811206610224084691036303771630733595757042570301959629906377767938930312924094034), SC_(0.8577498466883962411368177728739427398074845373836199107327704588709507726781916595107240718092847017) }, + { SC_(0.1419346332550048828125), SC_(0.835008561611175537109375), SC_(0.1135836623459303051429194205019356086721666257185933943800095114966831710249674856235457280143561524), SC_(0.9073723077768857915584037443594473495284668640552456222581651191944986071977435720707982921349172822) }, + { SC_(0.1419346332550048828125), SC_(0.905791938304901123046875), SC_(0.06362286490652974750551515900848775590349209643142950504238254147486041451220153135727937086746593346), SC_(1.138548958417806819877672656752528834516707875999494557854245128596101999842871365821587081467468729) }, + { SC_(0.1419346332550048828125), SC_(0.9133758544921875), SC_(0.05839122910435104787959801317712985297073242431714212769707923902518190300188050313568727292734281841), SC_(1.172098433700713966567441591625443115999983390102098023915355276771621952667238157429389664117756093) }, + { SC_(0.1419346332550048828125), SC_(0.968867778778076171875), SC_(0.02071934836253989804136051808125936032827483325843529223729374263952186251046878164288294079384678798), SC_(1.56377644811475640972171606288309488019030758199907780303791244112105872633625554623267734499110393) }, + { SC_(0.476447999477386474609375), SC_(0.097540400922298431396484375), SC_(1.906240501146735627405728538719050272742196757761857063398095587230594456801036092842821773110568176), SC_(0.1557234989288209275828472080090140421199447184166737519995512435500240874319270398254513654236859738) }, + { SC_(0.476447999477386474609375), SC_(0.12698681652545928955078125), SC_(1.746889660226439769707796999615493342006338076754771883758583845080036360375775016213480275141899151), SC_(0.2002596077596651342317325888043366373380954823329671027872122725030008232437680807213017399722523492) }, + { SC_(0.476447999477386474609375), SC_(0.135477006435394287109375), SC_(1.706868853024925966866869647473148895344239016501717692391243187680702442620890834745165798505101534), 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SC_(2325471.727242497392773449639911571066022914913370149675655043817114137532777832647461474174075784684), SC_(2329627.057627175636482679321890332587124140179464134521091120057293262780055485755607840175871399635) }, + { SC_(2327548.75), SC_(0.968867778778076171875), SC_(2324705.264985226344294745203011389326002772462961055280128731391064922559949044503112872046703368467), SC_(2330394.060353776798876224782809597216579299224955719757290503934606630340754068086724147607390669768) }, + { SC_(6230117), SC_(0.097540400922298431396484375), SC_(6233351.692472792061056579423995328493857103363396239930285849771856937399508188067562089564731873418), SC_(6226883.533801842955995418263348267065006324244587298450820602128533786716023362663081376081505278596) }, + { SC_(6230117), SC_(0.12698681652545928955078125), SC_(6232964.887316261881701097347977065367130687058335028215195760161155574674778388777016592959892786101), SC_(6227270.213121212666669646893421478332353213715522094655483973849209986323367490884986293265596032224) }, + { SC_(6230117), SC_(0.135477006435394287109375), SC_(6232865.324662868111190527453037563536761042866213055644551828614498129899296431946798065713700170309), SC_(6227369.745974080667874198600668355855866443350107162047433655601935476580330379827328115355241930538) }, + { SC_(6230117), SC_(0.188381969928741455078125), SC_(6232323.63227185115778658441373123602508832270222793713035148463068189665774304541385361276622984635), SC_(6227911.294806215678453456704655164982523620764595109639499042366654806988561384785145329706145958711) }, + { SC_(6230117), SC_(0.22103404998779296875), SC_(6232036.136536250933679987772671460235733726477628676496527697942146593792117713490188974877896657093), SC_(6228198.727099844400404830340828350312187148760053110689306546858635128471089524390627975414927556548) }, + { SC_(6230117), SC_(0.278498232364654541015625), SC_(6231583.324968307181681406950097442146625275486837869234235041148491080696490646684242945262614873862), SC_(6228651.456675485708645793762391965898293897905472473126963140667980274570043834720563802098830776494) }, + { SC_(6230117), SC_(0.308167040348052978515625), SC_(6231368.01268140366726819175665186638268466543674045388244783605009365981150484906350062278363002165), SC_(6228866.737669838129552659768389036176940614109203793210530779899481841299310894708215176090916781291) }, + { SC_(6230117), SC_(0.546881496906280517578125), SC_(6229823.338725792076249396465479515707976738291090698907050753153528364353246948669279119423361512767), SC_(6230411.332565403676924567021771163447589886484281784991639253912767759502501563110172430307851857965) }, + { SC_(6230117), SC_(0.54722058773040771484375), SC_(6229821.20232969720644266921283636481256709952913709169187035576735749241917052776040711574219186358), SC_(6230413.469028954311951916289817041956497186222746860523204205914506151906118372519765070461530163706) }, + { SC_(6230117), SC_(0.6323592662811279296875), SC_(6229273.426873074414556366664038917730342926119497520622955950028248233012730923731157371779835274121), SC_(6230961.277899359562011793446984848521160452797500133271897124953410729884385903965899957485493616226) }, + { SC_(6230117), SC_(0.814723670482635498046875), SC_(6227882.434048976058402554738706177641614293741446680191726074222637100597948821666067849039597462162), SC_(6232352.49988779969296455486124268083236030976120352890683487451263766450087335752456747789340459989) }, + { SC_(6230117), SC_(0.835008561611175537109375), SC_(6227685.997474726247688087235059841375491699004535827525396018185766370072460519898747215701210901658), SC_(6232548.985513618649457852034579253993054240803224395784331070556455526838927599609658335554516685135) }, + { SC_(6230117), SC_(0.905791938304901123046875), SC_(6226834.658470939251130269088475617573565344834275615564082295817604878528766465312582574980965566951), SC_(6233400.584848756385324680328194932048486178740428764315512985319315075751357006490348772335643597714) }, + { SC_(6230117), SC_(0.9133758544921875), SC_(6226718.46152837504987605910618350148896811668349713499970654969426128470753421783619578121107382213), SC_(6233516.823341293127817703348946693534220889012539925157887472970468357261827764647617813444939050571) }, + { SC_(6230117), SC_(0.968867778778076171875), SC_(6225464.31285320598579378900341589648562621739072429909204840552840676406014931804970799580191477189), SC_(6234771.512485767533460305424549342618110995053933756316716261789635898723192636506248708591368854061) }, + { SC_(15083605), SC_(0.097540400922298431396484375), SC_(15088637.78129839240544084967639490379844477975678197902399407486600207945266721185803386193838048788), SC_(15078573.44497624163292763488000992712939959763710279840335208548585316282432025007551150520834892593) }, + { SC_(15083605), SC_(0.12698681652545928955078125), SC_(15088035.95454486752580740261889753816930358428372354823140840080143163186851089423134281705030456187), SC_(15079175.14589260744374955642608743874599125416522689478871576618133114925591835305586515102168450869) }, + { SC_(15083605), SC_(0.135477006435394287109375), SC_(15087881.04516515058008645681821244786240897066985410758979447193614902902100508841528459831467151797), SC_(15079330.02547179892254772110264023468178754075598873509528620157408889322324875656941277086187068723) }, + { SC_(15083605), SC_(0.188381969928741455078125), SC_(15087038.22164129576331239830876873353215959212349671143829660806147341521657730913830483880795924604), SC_(15080172.70543677309686154480478630061895108326075818775376184189438877404905927698452585456460439436) }, + { SC_(15083605), SC_(0.22103404998779296875), SC_(15086590.90118294501418553538586701765236949743021493354695067962707784877870875603880811865737188093), SC_(15080619.96245315283600996122158277029601757754302700450619766582009047870716487351388585974357884923) }, + { SC_(15083605), SC_(0.278498232364654541015625), SC_(15085886.35747723304468589943584481796024609839822718371399281266195551388857413330568267228846771924), SC_(15081324.42416656292294392879954060790691623450955961557020548135058993368970804049112726766191292319) }, + { SC_(15083605), SC_(0.308167040348052978515625), SC_(15085551.34420515641571775001974011059238876676606782093094374386448901839125525873664701387018457173), SC_(15081659.4061460886503203161240084124632660964064929967501551529928113396007364677184216561021748007) }, + { SC_(15083605), SC_(0.546881496906280517578125), SC_(15083147.88182872830072612604647044724558765934450113173248706039646401749665137590731770281561564279), SC_(15084062.78946247115173608372375734029812941014549468133844220227152081657865687456247081611122282205) }, + { SC_(15083605), SC_(0.54722058773040771484375), SC_(15083144.55761682266607161006212263153448991348641093068040602030513183087566712109971802285493643745), SC_(15084066.11374183255127774349397874273282083208958473269307345912470558386422663180275755856718286894) }, + { SC_(15083605), SC_(0.6323592662811279296875), SC_(15082292.21976072050789874456495350894784125555141581904825165790430182222017429560913482757052410598), SC_(15084918.4850117169954112119454349395093713303826023318028934780952962765491698393599514678303984078) }, + { SC_(15083605), SC_(0.814723670482635498046875), SC_(15080127.79672086471855145296910274760322807117874269748523561539117756164468929742101440033412619451), SC_(15087083.13721591300051187437457165132686653041492870555194851269781626386717551279263922333330837915) }, + { SC_(15083605), SC_(0.835008561611175537109375), SC_(15079822.13137600794879441883791234166660844799568532069150956980805393588464890726312515528384620475), SC_(15087388.85161233849662625412502891217716929149866356777620050803844415678951962999244150794846728711) }, + { SC_(15083605), SC_(0.905791938304901123046875), SC_(15078497.39111859196942643525234208299210155465881136002686264100877073259043613787120296293082019187), SC_(15088713.85220110248343127416083317298417971060035075742320769629655969166242700390981290477097864291) }, + { SC_(15083605), SC_(0.9133758544921875), SC_(15078316.57925461811008046272040141036219718278649375130100601062440179434244992344435579799277359019), SC_(15088894.70561504836921552772384009007949176914160796656104057265657981769338965029262519632953830189) }, + { SC_(15083605), SC_(0.968867778778076171875), SC_(15076364.99664491882324687003477038147511063884443803257690739420141880711939968486790583272510921072), SC_(15090846.82869404432549959125993359239420390163029355290721639099401542481263745861685706736887672797) }, + { SC_(30203694), SC_(0.097540400922298431396484375), SC_(30210815.47566831752036352507646724966902397170938923180599788499483624078712473926944711066637024068), SC_(30196573.75060631617324365602586584702812846344145665137998045911407907692772654641163877762705456397) }, + { SC_(30203694), SC_(0.12698681652545928955078125), SC_(30209963.8757157795480995601711819490850855177202359573561810789532756328719973708836474351335854889), SC_(30197425.22472169556982870125185614985218844456494507799910460728140621558219262938938751348734740307) }, + { SC_(30203694), SC_(0.135477006435394287109375), SC_(30209744.67452031337899415070813774353602917889889616197547160928602825672821125303891065223640309171), SC_(30197644.39611663637853198149681272295131038344158276965149015781362885264090975193062615327536605494) }, + { SC_(30203694), SC_(0.188381969928741455078125), SC_(30208552.05131349505367548465974563992383024423755241876800896941592304014554064168696807336242707805), SC_(30198836.87576457451947027519100177908268502372878129639212126829680623490205917339927211144716523626) }, + { SC_(30203694), SC_(0.22103404998779296875), SC_(30207919.07540045596968115923357225804129946123666237495696182304373656661881886764705270512529825631), SC_(30199469.78823564276686545281804196696004068558549613806405242745080779095956522018947522727857563533) }, + { SC_(30203694), SC_(0.278498232364654541015625), SC_(30206922.11482883621349902576400149622677934703180000493881690157818678904256430739792356870432623573), SC_(30200466.66681496083817322221637013993723066065566124378008761476684957400999798723604743234095626697) }, + { SC_(30203694), SC_(0.308167040348052978515625), SC_(30206448.05460290491450971241953253008423903107052255039907494646751223838472473892280165946724898133), SC_(30200940.69574834130317660059010636635891931231336377577854780281206532038547042794469057516760843273) }, + { SC_(30203694), SC_(0.546881496906280517578125), SC_(30203047.00717058636669137357304329205959160149968637834771106489786699129221742385725439984711498261), SC_(30204341.66412061438892037858238817499573385966690259844837484508562445579642440156011876568390745967) }, + { SC_(30203694), SC_(0.54722058773040771484375), SC_(30203042.30316911510841423755943355619862052255525737384753926509535075864473371673718417804236785854), SC_(30204346.36818954141196718386901594529173091315276193703386689325465835489738438729376667618975581351) }, + { SC_(30203694), SC_(0.6323592662811279296875), SC_(30201836.17980432310579115059624887819436825142777050805037400727522822838623497800967776780877762698), SC_(30205552.52496811563988534014867005476936232886423535822792563048853981792883735017378794452129469264) }, + { SC_(30203694), SC_(0.814723670482635498046875), SC_(30198773.32543295466384303637189930446471572734223223892225111721858159661322380144370334171399592945), SC_(30208615.60850382374838123766556759581715316364613287151987225419255301976073762617401758260722361711) }, + { SC_(30203694), SC_(0.835008561611175537109375), SC_(30198340.77785915372227694900099056895390988180231312503107906339171260486697177517187178707615377744), SC_(30209048.20512919326855490888268848804451625500482635894067819189284827655903055503571740161890393015) }, + { SC_(30203694), SC_(0.905791938304901123046875), SC_(30196466.12565226888150073567973527983647592469381424487408323665472782970590030369753565377529416458), SC_(30210923.11766742515441063993124028259470221758493890083864245553374244107973324231229306719090975719) }, + { SC_(30203694), SC_(0.9133758544921875), SC_(30196210.25585998280064237944520762297790392031265667224175460272447471873119884901896113469469345057), SC_(30211179.02900968308035831437060149239618022354135751179509266594271502924955014841375878474172002289) }, + { SC_(30203694), SC_(0.968867778778076171875), SC_(30193448.52065679169569881618335847186614165117290739943799101539336869203467308211264064437967727749), SC_(30213941.30468216779982532499490538896952642738252393109170337539122327630305298668623088994994675055) } + } }; +#undef SC_ + diff --git a/test/igamma_med_data.ipp b/test/igamma_med_data.ipp new file mode 100644 index 000000000..c4968dd8f --- /dev/null +++ b/test/igamma_med_data.ipp @@ -0,0 +1,712 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 700> igamma_med_data = { { + { SC_(0.9759566783905029296875), SC_(0.009759566746652126312255859375), SC_(1.003339192007827076679495082852814844336347745360235976200689788561799007730110137330271807878843422), SC_(0.9890348932311204142405476428315735322132825375129739181325170152149560168983641202727073881868421137), SC_(0.01112369385656880593053787042266942213006008594885039559843996330162391632861423778173725952365124566), SC_(0.01096510676887958575945235716842646778671746248702608186748298478504398310163587972729261181315788632) }, + { SC_(0.9759566783905029296875), SC_(0.48797833919525146484375), SC_(0.6107075742899761923771199703161020169732161418698525715736270035568544567698769445946517198087313906), SC_(0.60200090392622800556746880021527940973985765617427715855663599678855269649431562996389062378697249), SC_(0.4037553115744196902329129829593822494931916894392338002255027483065684672888474305173573475937632775), SC_(0.39799909607377199443253119978472059026014234382572284144336400321144730350568437003610937621302751) }, + { SC_(0.9759566783905029296875), SC_(0.8783609867095947265625), SC_(0.4103188243195825736649265764483608742565650301514019080978258727846258380934756822777647401521337079), SC_(0.4044690348331089562847173039266064850324957624072768224521011340506874965505832795400716939018472498), SC_(0.6041440615448133089451063768271233922098428011576844637013038790787970859652486928342443272503609601), SC_(0.5955309651668910437152826960733935149675042375927231775478988659493125034494167204599283060981527502) }, + { SC_(0.9759566783905029296875), SC_(0.9759566783905029296875), SC_(0.3716156455565117999679764359314206771522167659503088545148165288456984447337491300360711644068227913), 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SC_(0.2525536615874421367012932901172485860763539011272499932099422653784982044920184787105406746658177969e155), SC_(0.4080174886867464334217982943121703482051615723869957786212134749545282348319943206732155228552149625), SC_(0.3664238788123985643999928981588916872333748665168733268688265826065195876451843472891838939035823178e155), SC_(0.5919825113132535665782017056878296517948384276130042213787865250454717651680056793267844771447850375) }, + { SC_(99.40981292724609375), SC_(198.8196258544921875), SC_(0.1371819884134727259345696989796477150536656151060715801081402111868202863481570951651604961881278798e141), SC_(0.221626762620267814217739611891735278077275735293365054216933274355798919338201866535892704439999524e-14), SC_(0.6189775403998393292814020535488809276127389711669727834226177872692167107350909577968610869984484631e155), SC_(0.9999999999999977837323737973218578226038810826472192272426470663494578306672564420108066179813346411) }, + { SC_(99.40981292724609375), SC_(9940.9814453125), SC_(0.1194537579446871083493681851309120071876520743662587465911793406448948030112600502063890936876032829e-3923), SC_(0.1929856095707828218066317814300956090965211451849739733633670381338185644181153034168965828385051012e-4078), SC_(0.6189775403998407011012861882761402733097287676441233200787688479850177921372028259997245685694001147e155), SC_(1) }, + { SC_(99.6479034423828125), SC_(0.996479034423828125), SC_(0.1848618324328595326256540103756933748634832912163837099714564642354930615866313317180770173409626267e156), SC_(1), SC_(0.002632953621713545975680045402254573025958139457581454220024431257252884916204475441566275959699470982), SC_(0.1424281901278791824660099798410885257699748667114052276410912168756372599058364270087640660099318927e-157) }, + { SC_(99.6479034423828125), SC_(49.82395172119140625), SC_(0.1848618323694318986984968527056010536947721607403919764188529025226665743038734891777793577541211654e156), SC_(0.9999999996568916736763733473241955787224548174526161914240157740209114383969351598000863202596909474), SC_(0.6342763392715715767009232116871113047599173355260356171282648728275784254029765958684146128176885942e146), SC_(0.3431083263236266526758044212775451825473838085759842259790885616030648401999136797403090526350547118e-9) }, + { SC_(99.6479034423828125), SC_(97.6479034423828125), SC_(0.104781143510080106941519063115430082168881879110811778245289071826214589716454123584899890589776219e156), SC_(0.5668078809514984818732949250616079106146259355202233070637680474174180977553905187864349475271238675), SC_(0.8008068892277942568413494726026329269460141210557193172616739240927847187017720813317712675118640764e155), SC_(0.4331921190485015181267050749383920893853740644797766929362319525825819022446094812135650524728761325) }, + { SC_(99.6479034423828125), SC_(99.6479034423828125), SC_(0.8996813603114861425627742217275710021870970554377666868921695606836039683900496541598731785012627279e155), SC_(0.4866777249101670793371012779886767047310287873694062698229024013112932567958949261113829499504654837), SC_(0.9489369640171091836937658820293627464477358567260704128223950816713266474762636630208969949083635387e155), SC_(0.5133222750898329206628987220113232952689712126305937301770975986887067432041050738886170500495345163) }, + { SC_(99.6479034423828125), SC_(101.6479034423828125), SC_(0.7544659013596046014875316445704815685260451593376435295742314736115164671432240751181442400434537482e155), SC_(0.4081242144094947811955420069383153567301560395689175929907159931951304506349246950413365552505599538), SC_(0.1094152422968990724769008459186452180108787752826193570140333168743414148723089242062625933366172518e156), SC_(0.5918757855905052188044579930616846432698439604310824070092840068048695493650753049586634447494400462) }, + { SC_(99.6479034423828125), SC_(199.295806884765625), SC_(0.3804006384186390812353329092798841093741132631690706197243637647942568433128668478754826060017677696e141), SC_(0.2057756506101917034700047476092601152101401495116289370474041581162582830156798739943260060771748412e-14), SC_(0.1848618324328591522250155917366121395305740113322743358581932951648733372228665374612337044741147512e156), SC_(0.9999999999999979422434938980829652999525239073988478985985048837106295259584188374171698432012600567) }, + { SC_(99.6479034423828125), SC_(9964.7900390625), SC_(0.6187187441149196707967788811980241646983120637234991013972314808151116439559127711068191156963192852e-3933), SC_(0.3346925300762848381679977351230198692364721942446516591127645420968259313810798749854292537402422257e-4088), SC_(0.1848618324328595326256540103756933748634832912163837099714564642354930615866313317180770173409626267e156), SC_(1) } + } }; +#undef SC_ + + + diff --git a/test/igamma_small_data.ipp b/test/igamma_small_data.ipp new file mode 100644 index 000000000..9a915b80b --- /dev/null +++ b/test/igamma_small_data.ipp @@ -0,0 +1,264 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 252> igamma_small_data = { { + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.165048164480104120332981665342231281101703643798828125e-13), SC_(31.15790848492937617968754572910879626038987427321665501824062445965678033809696860327421033588997389), SC_(0.5142555520027874508595595722328287870710653491133268223717400674935981376413784737279829921742056631e-10), SC_(605883754914.8750136902267543620760601500284208480879474455026885251060577270869941538181411625849707), SC_(0.9999999999485744447997212549140440427767171212928934650886673177628259932506401862358621522444918976) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.825240808847993445596102901617996394634246826171875e-12), SC_(27.24588549611682625963801859326817650865189920983978484217386169027034380426371292313397641222047523), SC_(0.4496883316923336685099678252541612612557662391990416220973173325702496188138998642060577791819560088e-10), SC_(605883754918.7870366790393042821255872858690405998259225088795587011728204964734306876513968427252046), SC_(0.9999999999550311668307666331490032174745838738744233760800958377902682667429750381186100131967398492) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.1485433412558301302652807862614281475543975830078125e-11), SC_(26.65809886043765184861833815671322602877461879268583897860494287446522932950747972178867249013415618), SC_(0.4399870213191659715445515622143744278264487110204265430288897607131180016977120873571283497579633446e-10), SC_(605883754919.3748233147184786931452677224239910797032029260335045647417393122785451624076300440705085), SC_(0.9999999999560012978680834028455448437785625572173551288979573456971110239286881998302287908816411206) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(26.55273831558914007164077488340563437622573207289701433865490306535179619489488183113436762977413663), SC_(0.4382480648941352919111303365625055162099824756683096022952805756000873913572998244587797811350613329e-10), SC_(605883754919.4801838595669904701228309957315827322520896458223292046917791213919782970202279347248134), SC_(0.9999999999561751935105864708088869663437494483790017524331690397704719424399912608642700171714774703) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.1815529822833672479731603743857704102993011474609375e-11), SC_(26.45742811190195049533727267568504724879726922010500532361189289376431739839451074960681764629918148), SC_(0.4366749875024881266012524590033039215380435741991291470061983416947539967123967894565937878826126117e-10), SC_(605883754919.5754940632541800464263332034521698596805524986143382197347892929794570935205990162523633), SC_(0.9999999999563325012497511873398747540996696078461956425800870852993801658305246003287603206716974201) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(0.33009632353919737823844116064719855785369873046875e-11), SC_(25.85959113506148614558473903473463000631861077966174981641096804406377885881031868583310284349737406), SC_(0.4268077980959374477897019960137392551284102019659605405168797702263707908186064406194329606569899375e-10), SC_(605883754920.1733310400946443961788668444025871021592109390575937269357141426799956331047910800260781), SC_(0.9999999999573192201904062552210298003986260744871589798034039459483120229773629209181393555554219737) }, + { SC_(0.165048161769598689119220580323599278926849365234375e-11), SC_(100), SC_(0.3683597761710089903097254101781941823648659605630519043409070893481216676279802460792521302677220332e-45), SC_(0.6079710392694377664924030954068840443869972626811800754595244480141084992747696378316144427416336236e-57), SC_(605883754946.0329221751561305417636058791372171084778217183509837671756731964340490869383638804970089), SC_(0.9999999999999999999999999999999999999999999999999999999993920289607305622335075969045931159556130027) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.206542072399111542591043644279125146567821502685546875e-13), SC_(30.93364168989949088662225783062562821275952768601909775986922127020315858008889124877888658573404498), SC_(0.6389098545336816619892930503520603759746754856217578076876091016189435240618029150566633773613501762e-10), SC_(484162851275.7955760243922734768637389008721827255248363671120740129064287230363410221766509320371893), SC_(0.9999999999361090145466318338010706949647939624025324514378242192312390898381056475938197081115165694) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.103271037554808486902402364648878574371337890625e-11), SC_(27.02161867158789878931620977320803696878717217632829425949582007780935030750254752153846820592898784), SC_(0.5581101193257189977881105127089252324891172434544119082354068522575555605654075175919360728270166827e-10), SC_(484162851279.7075990427038655741697869582897739694971918768028775132798299154301492947629946592776076), SC_(0.9999999999441889880674281002211889487291074767510882756545588091764593147742444439434592478580586656) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.185887871935463966366341992397792637348175048828125e-11), SC_(26.43383198338951447575879438498253053353949275687755657420480627486005125327251472959382229622556174), SC_(0.5459698510954741536654673991582696808426815259889022512892158090257346409981992469903104624210780402e-10), SC_(484162851280.2953857309022498877272023465152804047448712962536151985708437183794483489930274512222535), SC_(0.999999999945403014890452584633453260084173031915731847401109774871078419097426535900180074918231649) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(26.32847149106800163239094870586891526845820120393763229343432134045290266659697616892501543756547086), SC_(0.5437937136236059464756234080125646641077276944217179074523941203826804633593467352842472989896410538e-10), SC_(484162851280.4007462232237627310950480256288956698261628491935394793413286527865969356685660118910604), SC_(0.9999999999456206286376394053524376591987435335892272305578282092547605879617319536640653260888398335) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.227196278283769981243267466197721660137176513671875e-11), SC_(26.23316133035754938556791779149687928744103895259996107320150619846356673193156719633839660209741738), SC_(0.5418251577863868103129995208921398586460271143353281036151920862215441160993128135190632372203888294e-10), SC_(484162851280.4960563839342149779180789400009316508433251005312106995741437947759328703339749844776792), SC_(0.9999999999458174842213613189687000479107860141353972885664671896384807913778455883900687182653599296) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(0.4130841502192339476096094585955142974853515625e-11), SC_(25.63532431054814484821636003256695231324162132983547239417046128926165869713281292469158572207801148), SC_(0.5294773079214937290303064802551754036436189663559802177554527074961441840149493021783730734829548128e-10), SC_(484162851281.0938934037436195152696366989308586250427427232956993786051887039778409051327292561244901), SC_(0.9999999999470522692078506270969693519744824596356381033644019782244547292503855815985050693994395465) }, + { SC_(0.20654207510961697380480472929775714874267578125e-11), SC_(100), SC_(0.3683597761717143751552636437264047570124229557729804467214558990981423263608011608473991574457783741e-45), SC_(0.7608179255751063148390974562103337563986691710549682719520828630510110088500246233042380098159938087e-57), SC_(484162851306.7292177142917643634859967314978109382843640527628119966039356180842359626956245681566386), SC_(0.9999999999999999999999999999999999999999999999999999999992391820744248936851609025437896662436013308) }, + { SC_(0.6933230899119902090887990198098123073577880859375e-11), SC_(0.693323098043506502730082274865708313882350921630859375e-13), SC_(29.72264969834645019434398391335764181503761657632206675371880634359092033544076863168741139909711396), SC_(0.206073993293117147372414902249210966803119325754493450865229927096149608694728859685200779850164835e-9), SC_(144232900121.1921430521268665012736046760265611169427879527355317858388116749258141965866398426913323), SC_(0.9999999997939260067068828526275850977507890331968806742455065491347700729038503913052711399307582724) }, + { SC_(0.6933230899119902090887990198098123073577880859375e-11), SC_(0.34666154495599510454439950990490615367889404296875e-11), SC_(25.81062670541879230446808925169387675802578353778167356850954822607902657677034794738407159230904849), SC_(0.1789510346003750462291870271302182045706001584982927847421887098868836775584657333644555074456521954e-9), SC_(144232900125.1041660450545243911494993376903261739546209912759249710480697924377079552570605269946721), SC_(0.9999999998210489653996249537708129728697817954293998415017072152578112901131163224415342662517363926) }, + { SC_(0.6933230899119902090887990198098123073577880859375e-11), SC_(0.62399078959440856806395458988845348358154296875e-11), 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SC_(0.9783640828550343335348322355579639593718670787893691525957788365683450289514105688535329977996498619) }, + { SC_(0.005553754977881908416748046875), SC_(100), SC_(0.3779230356323651469251400251436801657115323311828038629573248866991682540425440079590315860323569243e-45), SC_(0.2105577913040578184612113756477261211410811213425891696964581142045415939710425561995401419156912192e-47), SC_(179.4866071171035817405728387011929869119560612303966919638574491215705499715813417359332742369180329), SC_(0.9999999999999999999999999999999999999999999999978944220869594218153878862435227387885891887865741083) }, + { SC_(0.008691129274666309356689453125), SC_(0.869112918735481798648834228515625e-4), SC_(8.412150535244441203168046070317310253724567623691078331486778211100219408535590313077554856570699598), SC_(0.0734742369214086089210153054947260329971557695201341707775521940117156621771297470370953689874816908), SC_(106.0790083759050634625337598716446870920526426681515083014260468453871574230843464967785306090985135), SC_(0.9265257630785913910789846945052739670028442304798658292224478059882843378228702529622462611547659734) }, + { SC_(0.008691129274666309356689453125), SC_(0.0043455646373331546783447265625), SC_(4.747483533337902183946818699971831287857052577276252258033190690531537548113688962993636655174478423), SC_(0.04146594006461382313537482617248709417766142416202542486142474476114080262861893941422971907404955593), SC_(109.7436753778116024817549872419901660579201577145663343748796343659558392835062478468624488104947346), SC_(0.9585340599353861768646251738275129058223385758379745751385752552388591973713810605853961293779614401) }, + { SC_(0.008691129274666309356689453125), SC_(0.00782201625406742095947265625), SC_(4.188714405951040634652072336322477039569342050275559543021997361729942472593051923062465693734697627), SC_(0.03658548350621272832275084129392436578573561492653168803027125335941528070349701044338510087997499593), SC_(110.3024445051984640310497336056395203062078682415670270898908276947574343590268848867936197719345154), SC_(0.9634145164937872716772491587060756342142643850734683119697287466405847192965029895562840836832982795) }, + { SC_(0.008691129274666309356689453125), SC_(0.008691129274666309356689453125), SC_(4.08848737420065935929134320387738037551705209754263277179328266373220927403385787131830593805064886), SC_(0.03571007065596669193904108016015709813865155263375873910595316749759359782915444208415216634230658664), SC_(110.4026715369488453064104627380846169702601581942999538611195423927551675575860789385377795276185642), SC_(0.9642899293440333080609589198398429018613484473662412608940468325024064021708455579155247914668146388) }, + { SC_(0.008691129274666309356689453125), SC_(0.00956024229526519775390625), SC_(3.99782079814824918307908619479527349158197798771392975095659098640536858272574657135383729506757053), SC_(0.03491816168312825922906539505187828450927190406957704260083957374549678225908496848420561100299384268), SC_(110.4933381130012554826227197471667238541952323041286568819562340700820082488941902385022481706016425), SC_(0.9650818383168717407709346049481217154907280959304229573991604262545032177409150315154783785776319255) }, + { SC_(0.008691129274666309356689453125), SC_(0.01738225854933261871337890625), SC_(3.429652593561693644638834896451401252091493865356407433916434404687048079702486661999920641307007191), SC_(0.02995561077535484202820021234487128647994519436125259940646933255344394936860990914312183685175046186), SC_(111.0615063175878110210629710455105960936857164264861791989963906518003287519174501478561648243622059), SC_(0.9700443892246451579717997876551287135200548056387474005935306674465560506313900908566062177985487162) }, + { SC_(0.008691129274666309356689453125), SC_(100), SC_(0.3834347555474465586395291403125251405405389593008695231904802071464782586691835903731927721307539629e-45), SC_(0.3349033752422838748916689259240733539203045003131547237515235304204672805448804838430882271615821179e-47), SC_(114.4911589111495046657018059419619973457772102914591518773653784978478476913074116703328547030652432), SC_(0.9999999999999999999999999999999999999999999999966509662475771612510833107407592664607969549968684528) }, + { SC_(0.0299333669245243072509765625), SC_(0.00029933368205092847347259521484375), SC_(6.65558792122502145790203122705781927392537710568382982650902879917995730340091807239175568813213068), SC_(0.2025490756976798423479118675089777352736957583605687315155725757125233270513154513279880668403039601), SC_(26.20354954113990720677295171562168818585244384318224849305166810303853836693243368037159645376977416), SC_(0.7974509243023201576520881324910222647263042416394312684844274242874766729486845486700376252368081425) }, + { SC_(0.0299333669245243072509765625), SC_(0.01496668346226215362548828125), SC_(3.412781457563949627493849132800205293732370924443084983071137019169168281694119440771156646106558311), SC_(0.1038609568334763548457972777288529241469923900253962807775257423932593724995787612069914513230733471), SC_(29.44635600480097903718113380987930216604545002442299333648955988304932738863923231199219549579534653), SC_(0.8961390431665236451542027222711470758530076099746037192224742576067406275004212387919853803026229454) }, + { SC_(0.0299333669245243072509765625), SC_(0.02694003097712993621826171875), SC_(2.900428060083442521700414212209539293181960970813005023783465175158447954051342500859962243168594771), SC_(0.08826853910591643763067442789157912471708232570630547469480133709625147782109251555169415725060617567), SC_(29.95870940228148614297456873046996816659585997805307329577723172706004771628200925190338989873331007), SC_(0.9117314608940835623693255721084208752829176742936945253051986629037485221789074844474329514837168865) }, + { SC_(0.0299333669245243072509765625), SC_(0.0299333669245243072509765625), SC_(2.80837448661732195170761906499999493821123607368254428592631444869917451230049026839722143354330746), SC_(0.08546707867283131075432095045307988463257577828543753893332063453617336634663328499785869123628549113), SC_(30.05076297574760671296736387767951252156658487518353403363438245351932115803286148436613070835859738), SC_(0.9145329213271686892456790495469201153674242217145624610666793654638266336533667150012954175041413206) }, + { SC_(0.0299333669245243072509765625), SC_(0.0329267047345638275146484375), SC_(2.725101221535109535021413719100843696433626664532974402284107300109171958371467977939595309468299124), SC_(0.08293282879553038976697110966809591185042939605421337827594738688004093044577294705637033103753406651), SC_(30.13403624082981912965356922357866376334419428433310391727658960210932371196188377482375683243360572), SC_(0.9170671712044696102330288903319040881495706039457866217240526131199590695542270529428082023784467733) }, + { SC_(0.0299333669245243072509765625), SC_(0.059866733849048614501953125), SC_(2.204476228650003963881192397589198776933448409172425381454446680594408859247730852369919972696722456), SC_(0.06708868214130365722661165568506456062512493800386324405219217526484257111958289695748025178580326486), SC_(30.65466123371492470079379054509030868284437253969365293810625022162408681108562090039343216920518239), SC_(0.9329113178586963427733883443149354393748750619961367559478078247351574288804171030418509848600074949) }, + { SC_(0.0299333669245243072509765625), SC_(100), SC_(0.4229279999195938340595688371433305871889682449285871973650802503052523480463068726888829167170666972e-45), SC_(0.1287094040140258076521551344643561104581741100997942473491238945795244936062181105143232448905621274e-46), SC_(32.85913746236492866467498294267950745977782094844315031964110306815892683319002116589180337092808888), SC_(0.9999999999999999999999999999999999999999999999871290595985974192347844865535643889541825889900205753) }, + { SC_(0.05124260485172271728515625), SC_(0.00051242602057754993438720703125), SC_(5.750398977913244900274962421353297368031325639433868586950794911944578215691256892664192413362591034), SC_(0.3028722400658670309060582521260906731221817754690304799285207885428553696560930593067285904368322388), SC_(13.23582100930869053226053270179389011404206015970023183245494980462996606312399917775634092248377199), SC_(0.6971277599341329690939417478739093268778182245309695200714792114571446303439069406900635548967258876) }, + { SC_(0.05124260485172271728515625), SC_(0.025621302425861358642578125), SC_(2.832141251899738927184198830918552560705390731127443324742929027769956627452021708549562866974237375), SC_(0.1491682522274481432092518761039267269452211637367753155145843941704919377471471128043007783037985871), SC_(16.15407873532219650535129629222863492136799506800665709466281568880458765136323436187097046887212565), SC_(0.85083174777255185679074812389607327305477883626322468448541560582950806225285288719409849230013501) }, + { SC_(0.05124260485172271728515625), SC_(0.0461183451116085052490234375), SC_(2.354535041475650840188296544626824459348071689176520405240621833882890783690260790374595349243249817), SC_(0.1240128389463671496190159389825520574179898719077050768343103933526658807883438828524452225211835791), SC_(16.63168494574628459234719857852036302272531410995758001416512288269165349512499528004593798660311321), SC_(0.8759871610536328503809840610174479425820101280922949231656896066473341192116561171462170724905049019) }, + { SC_(0.05124260485172271728515625), SC_(0.05124260485172271728515625), SC_(2.268581955935316913153430290814781029572894726877321715668417204937011887527880456166238463358814554), SC_(0.119485708975357549703407111440249731302572941365773848592259365032731048905891169977364697529085816), SC_(16.71763803128661851938206483233240645250049107225677870373732751163753239128737561425429487248754847), SC_(0.8805142910246424502965928885597502686974270586342261514077406349672689510941088300213449330467672867) }, + { SC_(0.05124260485172271728515625), SC_(0.0563668645918369293212890625), SC_(2.190826928493114755175474738392958609573869639135473828240129903375769251681938201165488344589073467), SC_(0.1153903689079543131568507907786239460633868239209515305709967719710419887432512190607714956691512904), SC_(16.79539305872882067736002038475422887249951615999862659116561481319877502713331786925504499125728956), SC_(0.8846096310920456868431492092213760539366131760790484694290032280289580112567487809379809556858218957) }, + { SC_(0.05124260485172271728515625), SC_(0.1024852097034454345703125), SC_(1.705832980948029655675174791358196762256920326164848611959849532446454449849713723349992966594024533), SC_(0.08984584514959194964111225566819986886942747214355191893796722859826575799935451129133496979377311521), SC_(17.2803870062739057768603203317889907198164654729692518074458951841280898289655423470705403692523385), SC_(0.9101541548504080503588877443318001311305725278564480810620327714017342420006454887076845744989248092) }, + { SC_(0.05124260485172271728515625), SC_(100), SC_(0.4666332762470370840214357140870669083484141793110941527015291021554978221892371620740912810024635374e-45), SC_(0.2457747126921997105292850810706506718048727997191850109419737442424277356633748861407006331354595514e-46), SC_(18.98621998722193543253549512314718748207338579866746714315870763255310856472818916227141762310922602), SC_(0.9999999999999999999999999999999999999999999999754225287307800289470714918929349328195127200280814989) } + } }; +#undef SC_ + + + diff --git a/test/laguerre2.ipp b/test/laguerre2.ipp new file mode 100644 index 000000000..ccbb26fe5 --- /dev/null +++ b/test/laguerre2.ipp @@ -0,0 +1,290 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 280> laguerre2 = { + SC_(0.5e1), SC_(0.9754039764404296875e2), SC_(-0.56218428868911115998451316426215010600803852349048e8), + SC_(0.5e1), SC_(0.12698681640625e3), SC_(-0.2243354877625806499089339248835065869040287604245e9), + SC_(0.5e1), SC_(0.1354770050048828125e3), SC_(-0.31418973293934559300911242611564538290439848722478e9), + SC_(0.5e1), SC_(0.1883819732666015625e3), SC_(-0.17256344438562512861154328485542977773202418102338e10), + SC_(0.5e1), SC_(0.2210340576171875e3), SC_(-0.39170586692926680364193521563384606561329196831404e10), + SC_(0.5e1), SC_(0.27849822998046875e3), SC_(-0.12743797194803555297871516510157023166259117561797e11), + SC_(0.5e1), SC_(0.30816705322265625e3), SC_(-0.21330001890239721242857731991140245635183835676285e11), + SC_(0.5e1), SC_(0.5468814697265625e3), SC_(-0.38928352555423736374909603306210170085670252415611e12), + SC_(0.5e1), SC_(0.5472205810546875e3), SC_(-0.39050320967058708332723880470821630124002299400132e12), + SC_(0.5e1), SC_(0.6323592529296875e3), SC_(-0.8097387851220286833483200510030936982898650914701e12), + SC_(0.5e1), SC_(0.81472369384765625e3), SC_(-0.29004794312410106580901353153584987792805041014836e13), + SC_(0.5e1), SC_(0.835008544921875e3), SC_(-0.32824640180206380842810590985413965806061220575884e13), + SC_(0.5e1), SC_(0.90579193115234375e3), SC_(-0.49421323240476384503475902217971923134518824327159e13), + SC_(0.5e1), SC_(0.9133758544921875e3), SC_(-0.51537136874727263493269087352302138644536549927366e13), + SC_(0.5e1), SC_(0.9575068359375e3), SC_(-0.65333402285178696166233025823588675962128036189824e13), + SC_(0.5e1), SC_(0.96488848876953125e3), SC_(-0.67904577991104770306176706554532355404413293600378e13), + SC_(0.5e1), SC_(0.9676949462890625e3), SC_(-0.68903095684447650874743766361869788199054585412876e13), + 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SC_(0.10365782377701183452195461556076353497800893592323e45), + SC_(0.26e2), SC_(0.5472205810546875e3), SC_(0.10543127968944428085136728714429047403339769641033e45), + SC_(0.26e2), SC_(0.6323592529296875e3), SC_(0.54336316553928368949063555184249670518669259521127e46), + SC_(0.26e2), SC_(0.81472369384765625e3), SC_(0.51097552617218427263241448611405994401811535704536e49), + SC_(0.26e2), SC_(0.835008544921875e3), SC_(0.98954520453823509774054797214466944211629878869384e49), + SC_(0.26e2), SC_(0.90579193115234375e3), SC_(0.87786375743747677334359888017430029312432056322196e50), + SC_(0.26e2), SC_(0.9133758544921875e3), SC_(0.10975609743646233178198950358966597197707924620818e51), + SC_(0.26e2), SC_(0.9575068359375e3), SC_(0.38807135479289092313349072980539966631345258500198e51), + SC_(0.26e2), SC_(0.96488848876953125e3), SC_(0.47654821142237731628816607894656922038832987256136e51), + SC_(0.26e2), SC_(0.9676949462890625e3), SC_(0.51503194237863964904639112627180500133770486086125e51), + SC_(0.26e2), SC_(0.9688677978515625e3), SC_(0.53198500654007856666562730301281355378772116339286e51), + SC_(0.26e2), SC_(0.99288128662109375e3), SC_(0.10235016527486692862890189415524722222034128311139e52), + SC_(0.26e2), SC_(0.9964613037109375e3), SC_(0.11268082142512355561240466068867830761377375700468e52), + SC_(0.32e2), SC_(0.9754039764404296875e2), SC_(0.57896844151857557407003122100491148696303115075994e20), + SC_(0.32e2), SC_(0.12698681640625e3), SC_(0.42245682831545366496930901995710141383069638790405e27), + SC_(0.32e2), SC_(0.1354770050048828125e3), SC_(0.12500869474828256329426453946772458815922358559551e29), + SC_(0.32e2), SC_(0.1883819732666015625e3), SC_(0.28266779160071220855863136752236809118289513550655e35), + SC_(0.32e2), SC_(0.2210340576171875e3), SC_(0.16043258633437377313296386512455676335765088879499e38), + SC_(0.32e2), SC_(0.27849822998046875e3), SC_(0.98152083075443552200456383499687746658509404557554e41), + SC_(0.32e2), SC_(0.30816705322265625e3), SC_(0.39660303068870304789061389710467952730805922359184e43), + SC_(0.32e2), SC_(0.5468814697265625e3), SC_(0.2124691778655363299602608193872213697408649234399e52), + SC_(0.32e2), SC_(0.5472205810546875e3), SC_(0.21701146117737990401627817320898898731158045140974e52), + SC_(0.32e2), SC_(0.6323592529296875e3), SC_(0.29453956822267682637622660585710995711330784866261e54), + SC_(0.32e2), SC_(0.81472369384765625e3), SC_(0.14579187010342463619209867543395472012389261100613e58), + SC_(0.32e2), SC_(0.835008544921875e3), SC_(0.33104915961295333458609667123641141490590693664242e58), + SC_(0.32e2), SC_(0.90579193115234375e3), SC_(0.49617254391613681021336977691940022261460402301085e59), + SC_(0.32e2), SC_(0.9133758544921875e3), SC_(0.65447509655180124047279400990868215055245034097822e59), + SC_(0.32e2), SC_(0.9575068359375e3), SC_(0.31314186002835766887651202030265122705420348145384e60), + SC_(0.32e2), SC_(0.96488848876953125e3), SC_(0.40390183129399277888276794404622531827627609452775e60), + SC_(0.32e2), SC_(0.9676949462890625e3), SC_(0.44470349220352623223005295792036557179541953880227e60), + SC_(0.32e2), SC_(0.9688677978515625e3), SC_(0.46291347880797126364720392445817548485904681626203e60), + SC_(0.32e2), SC_(0.99288128662109375e3), SC_(0.10414215853249607927566783846023113314868365327492e61), + SC_(0.32e2), SC_(0.9964613037109375e3), SC_(0.11731839293063017507522289233426225751615807834469e61), + SC_(0.33e2), SC_(0.9754039764404296875e2), SC_(-0.15519766552964914960282162902087252393890944756729e21), + SC_(0.33e2), SC_(0.12698681640625e3), SC_(-0.49834394329409786429107636406724247060325054488444e27), + SC_(0.33e2), SC_(0.1354770050048828125e3), SC_(-0.19840581047253598115611184022145906036118542457808e29), + SC_(0.33e2), SC_(0.1883819732666015625e3), SC_(-0.98212428153399951310107146404282928611923578181199e35), + SC_(0.33e2), SC_(0.2210340576171875e3), SC_(-0.72578044049144687694845377262857641629094641588691e38), + SC_(0.33e2), SC_(0.27849822998046875e3), SC_(-0.62057397100619303564513894784492557621908539158042e42), + SC_(0.33e2), SC_(0.30816705322265625e3), SC_(-0.28714296782385167119124999018599500804762523101335e44), + SC_(0.33e2), SC_(0.5468814697265625e3), SC_(-0.30888931981090919336348667046948287489524269522361e53), + SC_(0.33e2), SC_(0.5472205810546875e3), SC_(-0.31571691597216716720754649436373448511713945451381e53), + SC_(0.33e2), SC_(0.6323592529296875e3), SC_(-0.50478315840421718669131419020872977517323737909924e55), + SC_(0.33e2), SC_(0.81472369384765625e3), SC_(-0.33062025262205293767486203308022299641518265442287e59), + SC_(0.33e2), SC_(0.835008544921875e3), SC_(-0.77112383731383286471230156867910260520520657245124e59), + SC_(0.33e2), SC_(0.90579193115234375e3), SC_(-0.1262346064599467935961005524047195599692654321956e61), + SC_(0.33e2), SC_(0.9133758544921875e3), SC_(-0.16801567279045565630405655696613192336457136904877e61), + SC_(0.33e2), SC_(0.9575068359375e3), SC_(-0.84582529186776869237021206078881700754491798000552e61), + SC_(0.33e2), SC_(0.96488848876953125e3), SC_(-0.11000226620410340743835989544374720411357868296468e62), + SC_(0.33e2), SC_(0.9676949462890625e3), SC_(-0.12149322908840894133499880791452022966588461024746e62), + SC_(0.33e2), SC_(0.9688677978515625e3), SC_(-0.12663293722507282266407117288015925037770799060973e62), + SC_(0.33e2), SC_(0.99288128662109375e3), SC_(-0.29247499878319565468778047006263649127299274582216e62), + SC_(0.33e2), SC_(0.9964613037109375e3), SC_(-0.33075364818279888462182974219279083273759877224615e62), + SC_(0.36e2), SC_(0.9754039764404296875e2), SC_(-0.14202381258875603292784398243371786895083677644968e21), + SC_(0.36e2), SC_(0.12698681640625e3), SC_(-0.99689260408784512857796674097683890787134923850651e23), + SC_(0.36e2), SC_(0.1354770050048828125e3), SC_(0.33085802053384704162061389949004801316875589574167e29), + SC_(0.36e2), SC_(0.1883819732666015625e3), SC_(0.29691002837109064218075098689453177656242661416649e37), + SC_(0.36e2), SC_(0.2210340576171875e3), SC_(0.50722869577926074483465513588442579657337298353026e40), + SC_(0.36e2), SC_(0.27849822998046875e3), SC_(0.12276018084343264808614960453898425578975135870535e45), + SC_(0.36e2), SC_(0.30816705322265625e3), SC_(0.86176792520194608969749051518980054897501919421191e46), + SC_(0.36e2), SC_(0.5468814697265625e3), SC_(0.77496906300517296899212867022166471009005365376408e56), + SC_(0.36e2), SC_(0.5472205810546875e3), SC_(0.79380346709534996703125888877181202810222361959297e56), + SC_(0.36e2), SC_(0.6323592529296875e3), SC_(0.20837628323063930783612574671644907853428390635937e59), + SC_(0.36e2), SC_(0.81472369384765625e3), SC_(0.31805284832302734783735872178451062267986916005535e63), + SC_(0.36e2), SC_(0.835008544921875e3), SC_(0.80427224577995045297081384061933456722295034711368e63), + SC_(0.36e2), SC_(0.90579193115234375e3), SC_(0.17180125938368536870586199334103942944296763636498e65), + SC_(0.36e2), SC_(0.9133758544921875e3), SC_(0.2349585543781729049082862094571853647181110727758e65), + SC_(0.36e2), SC_(0.9575068359375e3), SC_(0.13788032216346989091599957999852901194859847097388e66), + SC_(0.36e2), SC_(0.96488848876953125e3), SC_(0.18383754088256510557951005527578044897458936835611e66), + SC_(0.36e2), SC_(0.9676949462890625e3), SC_(0.20496108373672385020966386552222912655862108987654e66), + SC_(0.36e2), SC_(0.9688677978515625e3), SC_(0.21447176171446511024838576925006696393416059852805e66), + SC_(0.36e2), SC_(0.99288128662109375e3), SC_(0.53619191227443818889184097649341805455543103147757e66), + SC_(0.36e2), SC_(0.9964613037109375e3), SC_(0.6134634193224111642857607073780448563124536964454e66), + SC_(0.38e2), SC_(0.9754039764404296875e2), SC_(0.13026109621262939918139850088931337407072881569374e21), + SC_(0.38e2), SC_(0.12698681640625e3), SC_(-0.39045802697684580087249595226480504736900650572604e27), + SC_(0.38e2), SC_(0.1354770050048828125e3), SC_(0.4703084156408515882792903403585270483597230661564e28), + SC_(0.38e2), SC_(0.1883819732666015625e3), SC_(0.21811950666425589665298269088166544127028329061958e38), + SC_(0.38e2), SC_(0.2210340576171875e3), SC_(0.68226335876612312689262066243026383595759661918497e41), + SC_(0.38e2), SC_(0.27849822998046875e3), SC_(0.3417431296914681619225852746415756851618849310483e46), + SC_(0.38e2), SC_(0.30816705322265625e3), SC_(0.31970819524145633109548465679514234447503951190819e48), + SC_(0.38e2), SC_(0.5468814697265625e3), SC_(0.12178742579145686663348414006307088453127654992376e59), + SC_(0.38e2), SC_(0.5472205810546875e3), SC_(0.12492841620300639426815182320836485753685517201536e59), + SC_(0.38e2), SC_(0.6323592529296875e3), SC_(0.45810661530573839216333825165973798523285084973924e61), + SC_(0.38e2), SC_(0.81472369384765625e3), SC_(0.12351347140304796554466586331231292139128275112892e66), + SC_(0.38e2), SC_(0.835008544921875e3), SC_(0.32975853153845151581573454625890910497305801446349e66), + SC_(0.38e2), SC_(0.90579193115234375e3), SC_(0.84216312274962467518837779101216307869158288654706e67), + SC_(0.38e2), SC_(0.9133758544921875e3), SC_(0.11729374185740406136241147249355309364674121773484e68), + SC_(0.38e2), SC_(0.9575068359375e3), SC_(0.76287452793236059887210998415733522854114359820263e68), + SC_(0.38e2), SC_(0.96488848876953125e3), SC_(0.10342759141411866835170522110986503338234205828331e69), + SC_(0.38e2), SC_(0.9676949462890625e3), SC_(0.11604186211493955868813819686880547173842867350967e69), + SC_(0.38e2), SC_(0.9688677978515625e3), SC_(0.12174646477307642518592858476271949696633470067733e69), + SC_(0.38e2), SC_(0.99288128662109375e3), SC_(0.32098315809650975303906367704092470560267558301581e69), + SC_(0.38e2), SC_(0.9964613037109375e3), SC_(0.37011679718261969297940214190611219199030995011675e69), + SC_(0.39e2), SC_(0.9754039764404296875e2), SC_(-0.81788056310459660065120382039447835033044197049489e20), + SC_(0.39e2), SC_(0.12698681640625e3), SC_(0.22229715039776080683928576767759683825519547792559e27), + SC_(0.39e2), SC_(0.1354770050048828125e3), SC_(0.15550461280029801190156888344119462891700106953436e29), + SC_(0.39e2), SC_(0.1883819732666015625e3), SC_(-0.54226895058411233344275595074274527247091920347531e38), + SC_(0.39e2), SC_(0.2210340576171875e3), SC_(-0.23342183174200163814210714804874246930233709812346e42), + SC_(0.39e2), SC_(0.27849822998046875e3), SC_(-0.17013038203469478431364477259901167967979013129897e47), + SC_(0.39e2), SC_(0.30816705322265625e3), SC_(-0.18429253515276943444879261921619357066598024943888e49), + SC_(0.39e2), SC_(0.5468814697265625e3), SC_(-0.14577093593047894021012074885872239431277128765197e60), + SC_(0.39e2), SC_(0.5472205810546875e3), SC_(-0.14963982369811780116397608759812127691336981298577e60), + SC_(0.39e2), SC_(0.6323592529296875e3), SC_(-0.6492862403071900096406830281442380096038500168606e62), + SC_(0.39e2), SC_(0.81472369384765625e3), SC_(-0.23301820931113238390488179765094297019115309966495e67), + SC_(0.39e2), SC_(0.835008544921875e3), SC_(-0.6393122118320767120277111175042508309950280050084e67), + SC_(0.39e2), SC_(0.90579193115234375e3), SC_(-0.1785926484809201389439108598459489796533771170743e69), + SC_(0.39e2), SC_(0.9133758544921875e3), SC_(-0.25102368049271316850934544583200188325859894245069e69), + SC_(0.39e2), SC_(0.9575068359375e3), SC_(-0.17191430467481203804500407886563167282958450012617e70), + SC_(0.39e2), SC_(0.96488848876953125e3), SC_(-0.2350360204862166230883463530178770075607651772555e70), + SC_(0.39e2), SC_(0.9676949462890625e3), SC_(-0.26453813089044823183666856904573749479668606778886e70), + SC_(0.39e2), SC_(0.9688677978515625e3), SC_(-0.27790958611107454497958236687689216620181150284596e70), + SC_(0.39e2), SC_(0.99288128662109375e3), SC_(-0.75250430660256030378942490914723092683335523198301e70), + SC_(0.39e2), SC_(0.9964613037109375e3), SC_(-0.87109521955089946525219892186343098358449543636311e70) + }; +#undef SC_ + diff --git a/test/laguerre3.ipp b/test/laguerre3.ipp new file mode 100644 index 000000000..3567163f7 --- /dev/null +++ b/test/laguerre3.ipp @@ -0,0 +1,2250 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 2240> laguerre3 = { + SC_(0.6e1), SC_(0.4e1), SC_(0.9754039764404296875e2), SC_(0.61248773400035441372705568899743424188675775638604e9), + SC_(0.6e1), SC_(0.4e1), SC_(0.12698681640625e3), SC_(0.35204789737752362425049635886299775785128782532338e10), + SC_(0.6e1), SC_(0.4e1), SC_(0.1354770050048828125e3), SC_(0.53680020529340456542399945558001901900007952546989e10), + SC_(0.6e1), SC_(0.4e1), SC_(0.1883819732666015625e3), SC_(0.44534031167752568381323984620072514172994661540171e11), + SC_(0.6e1), SC_(0.4e1), SC_(0.2210340576171875e3), SC_(0.12226434763229329677261432059244153549746269983841e12), + SC_(0.6e1), SC_(0.4e1), SC_(0.27849822998046875e3), SC_(0.51928144111428742292292453631991614690587389994904e12), + SC_(0.6e1), SC_(0.4e1), SC_(0.30816705322265625e3), SC_(0.97428426563622634057559347373791034771828164883086e12), + SC_(0.6e1), SC_(0.4e1), SC_(0.5468814697265625e3), SC_(0.3324381244018248505151003408444078258062512743881e14), + SC_(0.6e1), SC_(0.4e1), SC_(0.5472205810546875e3), SC_(0.33370020703520225304974334354074929274249826530701e14), + SC_(0.6e1), SC_(0.4e1), SC_(0.6323592529296875e3), SC_(0.80676208197265431223389966116130786752172758946929e14), + SC_(0.6e1), SC_(0.4e1), SC_(0.81472369384765625e3), SC_(0.37709198164073899574491439717306334644653651087643e15), + SC_(0.6e1), SC_(0.4e1), SC_(0.835008544921875e3), SC_(0.43784677681230003590939175326930086254310296977861e15), + SC_(0.6e1), SC_(0.4e1), SC_(0.90579193115234375e3), SC_(0.71751164134703529160531275753374390004620108776236e15), + SC_(0.6e1), SC_(0.4e1), SC_(0.9133758544921875e3), SC_(0.75474163633107959381273333540455867778846765466731e15), + SC_(0.6e1), SC_(0.4e1), SC_(0.9575068359375e3), SC_(0.10048213277050885981803871378237693755448856069989e16), + SC_(0.6e1), SC_(0.4e1), SC_(0.96488848876953125e3), SC_(0.10527174037983159903287634170928938343127358723874e16), + SC_(0.6e1), SC_(0.4e1), SC_(0.9676949462890625e3), SC_(0.10714192604719650298629709048594099989758377336641e16), + SC_(0.6e1), SC_(0.4e1), SC_(0.9688677978515625e3), SC_(0.10793165886036598413491640043052411777219226808173e16), + SC_(0.6e1), SC_(0.4e1), SC_(0.99288128662109375e3), SC_(0.12520056327602887353590139034934048397213872282597e16), + SC_(0.6e1), SC_(0.4e1), SC_(0.9964613037109375e3), SC_(0.12796190491413846860531891431736127176350842769258e16), + SC_(0.6e1), SC_(0.5e1), SC_(0.9754039764404296875e2), SC_(0.57021748735352796844833665641714981145537658965832e9), + SC_(0.6e1), SC_(0.5e1), SC_(0.12698681640625e3), SC_(0.33392337350031500089312750393952812869969520242424e10), + SC_(0.6e1), SC_(0.5e1), SC_(0.1354770050048828125e3), SC_(0.51104706607057706503593300411908865340079809564711e10), + SC_(0.6e1), SC_(0.5e1), SC_(0.1883819732666015625e3), SC_(0.43033684039351042149204372971415695875947452322601e11), + SC_(0.6e1), SC_(0.5e1), SC_(0.2210340576171875e3), SC_(0.11878421144708082332834762334306478479136415095637e12), + SC_(0.6e1), SC_(0.5e1), SC_(0.27849822998046875e3), SC_(0.50766913019696586493282931975810537104713641422173e12), + SC_(0.6e1), SC_(0.5e1), SC_(0.30816705322265625e3), SC_(0.95466760718538857591082062972549747072346150612994e12), + SC_(0.6e1), SC_(0.5e1), SC_(0.5468814697265625e3), SC_(0.32872226240917190297091739750525454366627699186064e14), + SC_(0.6e1), SC_(0.5e1), SC_(0.5472205810546875e3), SC_(0.32997259328655255330496759553150199138853700279145e14), + SC_(0.6e1), SC_(0.5e1), SC_(0.6323592529296875e3), SC_(0.79898329260889892967192058996859678757129709781078e14), + SC_(0.6e1), SC_(0.5e1), SC_(0.81472369384765625e3), SC_(0.37428017493482775405894670005457409154697608991328e15), + SC_(0.6e1), SC_(0.5e1), SC_(0.835008544921875e3), SC_(0.43466223450876443481509940956447737669130708637296e15), + SC_(0.6e1), SC_(0.5e1), SC_(0.90579193115234375e3), SC_(0.71270545904261412829731553082318032163659205189309e15), + SC_(0.6e1), SC_(0.5e1), SC_(0.9133758544921875e3), SC_(0.7497285196890962326633452898612092751671091947701e15), + SC_(0.6e1), SC_(0.5e1), SC_(0.9575068359375e3), SC_(0.99845803303929955008572627409241608030868175838179e15), + SC_(0.6e1), SC_(0.5e1), SC_(0.96488848876953125e3), SC_(0.10461023358117816588476498791118019449318399295782e16), + SC_(0.6e1), SC_(0.5e1), SC_(0.9676949462890625e3), SC_(0.10647064052554109781969489136618692170398077176558e16), + SC_(0.6e1), SC_(0.5e1), SC_(0.9688677978515625e3), SC_(0.10725625259217407089116034019717976945427232766523e16), + SC_(0.6e1), SC_(0.5e1), SC_(0.99288128662109375e3), SC_(0.12443623656417308867522424203310475916181187862679e16), + SC_(0.6e1), SC_(0.5e1), SC_(0.9964613037109375e3), SC_(0.12718355604200258530774511416682851768226221644704e16), + SC_(0.6e1), SC_(0.7e1), SC_(0.9754039764404296875e2), SC_(0.49284811793209280365434949754966958001727610684493e9), + SC_(0.6e1), SC_(0.7e1), SC_(0.12698681640625e3), SC_(0.29997862506208045020727302705263463492969194184528e10), + SC_(0.6e1), SC_(0.7e1), SC_(0.1354770050048828125e3), SC_(0.46259460134591536879819144489901451018431480056059e10), + SC_(0.6e1), SC_(0.7e1), SC_(0.1883819732666015625e3), SC_(0.40158369321443255951219796747513198534532980552585e11), + SC_(0.6e1), SC_(0.7e1), SC_(0.2210340576171875e3), SC_(0.1120699625624019830448773936385820118244175747615e12), + SC_(0.6e1), SC_(0.7e1), SC_(0.27849822998046875e3), SC_(0.48509038473703058950212476663392650648272020840751e12), + SC_(0.6e1), SC_(0.7e1), SC_(0.30816705322265625e3), SC_(0.91641718510448442971341501529316096562199222581998e12), + SC_(0.6e1), SC_(0.7e1), SC_(0.5468814697265625e3), SC_(0.32139411296589331991485384644984994696859399675058e14), + SC_(0.6e1), SC_(0.7e1), SC_(0.5472205810546875e3), SC_(0.32262120245377330138811808763450924797742804132045e14), + SC_(0.6e1), SC_(0.7e1), SC_(0.6323592529296875e3), SC_(0.78361281470358220056716850107433788659136608638212e14), + SC_(0.6e1), SC_(0.7e1), SC_(0.81472369384765625e3), SC_(0.36870888982652886020361269968737789868814568588064e15), + SC_(0.6e1), SC_(0.7e1), SC_(0.835008544921875e3), SC_(0.42835095990188177652844730504615356961881860142468e15), + SC_(0.6e1), SC_(0.7e1), SC_(0.90579193115234375e3), SC_(0.70317345799024105504252323584691230510601850060279e15), + SC_(0.6e1), SC_(0.7e1), SC_(0.9133758544921875e3), SC_(0.73978540734723097232993500405412015992828695917165e15), + SC_(0.6e1), SC_(0.7e1), SC_(0.9575068359375e3), SC_(0.98583204388899485014640108498828265136751380914859e15), + SC_(0.6e1), SC_(0.7e1), SC_(0.96488848876953125e3), SC_(0.10329759730285883611535736100827742332871039794527e16), + SC_(0.6e1), SC_(0.7e1), SC_(0.9676949462890625e3), SC_(0.10513856938469798930399543075676194173463962645542e16), + SC_(0.6e1), SC_(0.7e1), SC_(0.9688677978515625e3), SC_(0.10591599150778794161004483492934728758631873253144e16), + SC_(0.6e1), SC_(0.7e1), SC_(0.99288128662109375e3), SC_(0.1229192323834801613580240381635673120588670474927e16), + SC_(0.6e1), SC_(0.7e1), SC_(0.9964613037109375e3), SC_(0.12563867825730434806513584802886506865386445819117e16), + SC_(0.6e1), SC_(0.13e2), SC_(0.9754039764404296875e2), SC_(0.31056176609328262744354290307490499447749128627984e9), + SC_(0.6e1), SC_(0.13e2), SC_(0.12698681640625e3), SC_(0.21474641645375196945003617651061372691638804186296e10), + SC_(0.6e1), SC_(0.13e2), SC_(0.1354770050048828125e3), SC_(0.33939912866617433675302309449585922660229492695138e10), + SC_(0.6e1), SC_(0.13e2), SC_(0.1883819732666015625e3), SC_(0.3246939393485798363604662321181037517515150375214e11), + SC_(0.6e1), SC_(0.13e2), SC_(0.2210340576171875e3), SC_(0.93785476831971153918912095143821834557237292167987e11), + SC_(0.6e1), SC_(0.13e2), SC_(0.27849822998046875e3), SC_(0.42229348699358054630103206097276707070997137917402e12), + SC_(0.6e1), SC_(0.13e2), SC_(0.30816705322265625e3), SC_(0.80921665035606831983918570980305083995391250863279e12), + SC_(0.6e1), SC_(0.13e2), SC_(0.5468814697265625e3), SC_(0.30021987362168608006705510091464548855616102605746e14), + SC_(0.6e1), SC_(0.13e2), SC_(0.5472205810546875e3), SC_(0.3013793009404698889289850495408042782537109837479e14), + SC_(0.6e1), SC_(0.13e2), SC_(0.6323592529296875e3), SC_(0.7389695059502189306373674846273437884319461768123e14), + SC_(0.6e1), SC_(0.13e2), SC_(0.81472369384765625e3), SC_(0.35240744843758068568366054624709488158274087060876e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.835008544921875e3), SC_(0.4098729207588772848089021141281301420706627999216e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.90579193115234375e3), SC_(0.67521178818255342139438661537971495337615596039104e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.9133758544921875e3), SC_(0.71061224267344889512925363465089197035709820934262e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.9575068359375e3), SC_(0.94874872695964208516621641733125659722361544545777e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.96488848876953125e3), SC_(0.99441667971961652225223077796484088222447277866909e15), + SC_(0.6e1), SC_(0.13e2), SC_(0.9676949462890625e3), SC_(0.10122530692858831068332258430998293296495283930143e16), + SC_(0.6e1), SC_(0.13e2), SC_(0.9688677978515625e3), SC_(0.10197856775415755500167732055815127952881513946199e16), + SC_(0.6e1), SC_(0.13e2), SC_(0.99288128662109375e3), SC_(0.11846028047307665541922047585702480764211063113749e16), + SC_(0.6e1), SC_(0.13e2), SC_(0.9964613037109375e3), SC_(0.12109745881583554125132338218009211405499508666071e16), + SC_(0.6e1), SC_(0.16e2), SC_(0.9754039764404296875e2), SC_(0.24278455726195980361923424496753165147487010730985e9), + SC_(0.6e1), SC_(0.16e2), SC_(0.12698681640625e3), SC_(0.18030561300133652370839361014673853459811752535873e10), + SC_(0.6e1), SC_(0.16e2), SC_(0.1354770050048828125e3), SC_(0.28881912653305133981890911698470926640429114963515e10), + SC_(0.6e1), SC_(0.16e2), SC_(0.1883819732666015625e3), SC_(0.29108812862427597218212064605049495961212544330731e11), + SC_(0.6e1), SC_(0.16e2), SC_(0.2210340576171875e3), SC_(0.85616143635967071526809182235517018079343589729518e11), + SC_(0.6e1), SC_(0.16e2), SC_(0.27849822998046875e3), SC_(0.39352212446993374561133734251485121814802348879652e12), + SC_(0.6e1), SC_(0.16e2), SC_(0.30816705322265625e3), SC_(0.75965534404424419844204496845212470192096727671235e12), + SC_(0.6e1), SC_(0.16e2), SC_(0.5468814697265625e3), SC_(0.29007606682763749078137760938450840888887892431046e14), + SC_(0.6e1), SC_(0.16e2), SC_(0.5472205810546875e3), SC_(0.29120279915635749780172021234758221516201513893146e14), + SC_(0.6e1), SC_(0.16e2), SC_(0.6323592529296875e3), SC_(0.71745423807489525509070867847267145602821278253669e14), + SC_(0.6e1), SC_(0.16e2), SC_(0.81472369384765625e3), SC_(0.34448448721794017468717487738286623037893967559024e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.835008544921875e3), SC_(0.4008857272032547098139448124146495500684039156006e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.90579193115234375e3), SC_(0.66158192958719979428580597488440490193264309489379e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.9133758544921875e3), SC_(0.69638877078679263739469509705373976108324301755808e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.9575068359375e3), SC_(0.9306471468871133044489467037648480350793545046173e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.96488848876953125e3), SC_(0.97559109159816676915953242609482016333608597005721e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.9676949462890625e3), SC_(0.99314621712374015523569716516073804333679788291835e15), + SC_(0.6e1), SC_(0.16e2), SC_(0.9688677978515625e3), SC_(0.10005602904979136235528290138828979990104151199654e16), + SC_(0.6e1), SC_(0.16e2), SC_(0.99288128662109375e3), SC_(0.11628181660003715807927735007640600139772036681539e16), + SC_(0.6e1), SC_(0.16e2), SC_(0.9964613037109375e3), SC_(0.11887861388958864158184058615107773425485087664827e16), + SC_(0.6e1), SC_(0.17e2), SC_(0.9754039764404296875e2), SC_(0.22309335653387736494310721981505397269088117661973e9), + SC_(0.6e1), SC_(0.17e2), SC_(0.12698681640625e3), SC_(0.16989221636139193381325228565875327380060701000354e10), + SC_(0.6e1), SC_(0.17e2), SC_(0.1354770050048828125e3), SC_(0.27341067762482514123133136489132498264704946000748e10), + SC_(0.6e1), SC_(0.17e2), SC_(0.1883819732666015625e3), SC_(0.28054797583055755151994997428941789365119484852733e11), + SC_(0.6e1), SC_(0.17e2), SC_(0.2210340576171875e3), SC_(0.83027843509612007538993540360122495875672501473246e11), + SC_(0.6e1), SC_(0.17e2), SC_(0.27849822998046875e3), SC_(0.38430118332897729090795123137235693236025494217705e12), + SC_(0.6e1), SC_(0.17e2), SC_(0.30816705322265625e3), SC_(0.74370607174932254937454808736955812958708844808005e12), + SC_(0.6e1), SC_(0.17e2), SC_(0.5468814697265625e3), SC_(0.28675882807252638302604168226582949972059845232418e14), + SC_(0.6e1), SC_(0.17e2), SC_(0.5472205810546875e3), 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SC_(-0.3229794917111444064930798736319578366982249750048e79), + SC_(0.49e2), SC_(0.19e2), SC_(0.90579193115234375e3), SC_(-0.24997713010410319875351696557857678569354844836328e81), + SC_(0.49e2), SC_(0.19e2), SC_(0.9133758544921875e3), SC_(-0.38959675198826857041536767909021901669084098876338e81), + SC_(0.49e2), SC_(0.19e2), SC_(0.9575068359375e3), SC_(-0.47698754627870214398837894364370617412641440207875e82), + SC_(0.49e2), SC_(0.19e2), SC_(0.96488848876953125e3), SC_(-0.7163777266035546050195813883475085233727823154041e82), + SC_(0.49e2), SC_(0.19e2), SC_(0.9676949462890625e3), SC_(-0.83543571310763374889834849414881575556735386136858e82), + SC_(0.49e2), SC_(0.19e2), SC_(0.9688677978515625e3), SC_(-0.89074864582832301584239019351245730273845614840483e82), + SC_(0.49e2), SC_(0.19e2), SC_(0.99288128662109375e3), SC_(-0.32501886194029633537695219597823561980219500935259e83), + SC_(0.49e2), SC_(0.19e2), SC_(0.9964613037109375e3), SC_(-0.39305507139881250084440920416798458859478419817795e83) + }; +#undef SC_ + diff --git a/test/legendre_p.ipp b/test/legendre_p.ipp new file mode 100644 index 000000000..0a21b9279 --- /dev/null +++ b/test/legendre_p.ipp @@ -0,0 +1,150 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 140> legendre_p = { + SC_(3), SC_(-0.804919183254241943359375), SC_(-0.09637879251279735399302410605920296560178428535437), SC_(-0.84585603807674271376114723023136351467791030262194), + SC_(3), SC_(-0.74602639675140380859375), SC_(0.081027074619746344180737168075984167003866787126753), SC_(-0.80282652754865351221792807765781454966620965987346), + SC_(3), SC_(-0.72904598712921142578125), SC_(0.12483445078630286253952851342405305778981983166886), SC_(-0.77778604740049228448138273219369426999256253154477), + SC_(3), SC_(-0.62323606014251708984375), SC_(0.32965574890576083785628027375894100181596968468511), SC_(-0.54513199810739721932916528089324249381856320981233), + SC_(3), SC_(-0.5579319000244140625), SC_(0.40270407973501426077134190961714921286329627037048), SC_(-0.36518655840013507868281802164420656404202160032323), + SC_(3), SC_(-0.44300353527069091796875), SC_(0.44715433191447753543307951822408097264371917844983), SC_(-0.036791936427532285617966259083782977179826877960429), + SC_(3), SC_(-0.38366591930389404296875), SC_(0.43431026413595020613207998332397363761003816762241), SC_(0.12305391050245773781086426664843252339855233834581), + SC_(3), SC_(0.09376299381256103515625), SC_(-0.13858369755095318860018104847528497280961801152444), SC_(0.63165561565784611978985918320060987131739507929363), + SC_(3), SC_(0.0944411754608154296875), SC_(-0.13955592906053357908142302859499928047171124489978), SC_(0.63114960637026901412475946814198954544987429570206), + SC_(3), SC_(0.264718532562255859375), SC_(-0.35070182432271000038054645231433426033618161454797), SC_(0.39637508312826953308089998658220280933868101174524), + SC_(3), SC_(0.62944734096527099609375), SC_(-0.32069719648529449984920462269136209876307930244366), SC_(-0.561319593950734471650860089348537659182887471015), + SC_(3), SC_(0.67001712322235107421875), SC_(-0.25306053375130974366335052634993663112084050226258), SC_(-0.66081564793406011705012545543655510789740391137235), + SC_(3), SC_(0.81158387660980224609375), SC_(0.11903579598709513016345544955329471825677956076106), SC_(-0.84529721625683760880474150426292608647145977571808), + SC_(3), SC_(0.826751708984375), SC_(0.1726224256643575927228084765374660491943359375), SC_(-0.83881729781391507585288129748948101183986537673708), + SC_(3), SC_(0.91501367092132568359375), SC_(0.54271752467888026988712457448701953488523486157646), SC_(-0.58117864722536578272928121147096403274716994164107), + SC_(3), SC_(0.92977702617645263671875), SC_(0.61478093203605979210896718576903619002393952541752), SC_(-0.47601316933958185359002896997987802108555441085956), + SC_(3), SC_(0.93538987636566162109375), SC_(0.6429734865180513243156944550547871042311953715398), SC_(-0.42776266738524467923677588882284251961316602328383), + SC_(3), SC_(0.93773555755615234375), SC_(0.65488632484399120345028300071987814590102061629295), SC_(-0.40599139678289761995855284290619999445381396586062), + SC_(3), SC_(0.98576259613037109375), SC_(0.91608863936432686136610015203984858089825138449669), SC_(0.49911743300628611193386173368321463301956284745397), + SC_(3), SC_(0.99292266368865966796875), SC_(0.95791076106518834501582130387317692843396343960194), SC_(0.90345674705561627727354266945334737890516023119515), + SC_(5), SC_(-0.804919183254241943359375), SC_(0.39312973382980111956333250391790337182567613112521), SC_(-0.30797722800923861903533473474484308050729533072248), + SC_(5), SC_(-0.74602639675140380859375), SC_(0.4144532363564166856565791456062465926118819061177), SC_(0.036750999302184486102911681342167235245694654813727), + SC_(5), SC_(-0.72904598712921142578125), SC_(0.40170549167165668611417028849058086074027575851596), SC_(0.12520396031165822675600338672833850727710544023424), + SC_(5), SC_(-0.62323606014251708984375), SC_(0.20914641668663499200234579439954172182527145448352), SC_(0.50490192751447088632949650396821036768725468002192), + SC_(5), SC_(-0.5579319000244140625), SC_(0.047804048918209109846499488120279081448233535491368), SC_(0.58010824986594190037517814500067698165903587991196), + SC_(5), SC_(-0.44300353527069091796875), SC_(-0.20426834340719068505951661242682923196540890996203), SC_(0.46263002281764960331577712774593562735119375851165), + SC_(5), SC_(-0.38366591930389404296875), SC_(-0.29067957811088193540744987867754401309946084909752), SC_(0.31516732667905959819121026003137539434799697742685), + SC_(5), SC_(0.09376299381256103515625), SC_(0.16864990731338440263027533820227754755159887863031), SC_(-0.46423435769108284256474941359029876833900529649787), + SC_(5), SC_(0.0944411754608154296875), SC_(0.1697659485252767697264924108060913297868060247664), SC_(-0.46324927263674125093814131181689301223878566421762), + SC_(5), SC_(0.264718532562255859375), SC_(0.34426832813003273714672924371824224133559143244821), SC_(-0.049432441822728069543372084272506540578906568566679), + SC_(5), SC_(0.62944734096527099609375), SC_(-0.22382279528405448977447983371074041669623686569369), SC_(0.49147664711760436466007581944059038967809611892101), + SC_(5), SC_(0.67001712322235107421875), SC_(-0.31223652976417549895070288708720094161868379195206), SC_(0.37610337461310797969209584800253042022904925554525), + SC_(5), SC_(0.81158387660980224609375), SC_(-0.38292680966875416452976755202641274083447045975907), SC_(-0.34885118284757894374218709400505373551960888245662), + SC_(5), SC_(0.826751708984375), SC_(-0.35269763131671469331857533366092200943081591546502), SC_(-0.44136440965471238150156372311112451085353528866709), + SC_(5), SC_(0.91501367092132568359375), SC_(0.063446713438062204473932078136911612759332109107889), SC_(-0.82663633769127728796241230589411701186971587004582), + SC_(5), SC_(0.92977702617645263671875), SC_(0.18225073809630641977015280395310276646936069311641), SC_(-0.82168035297079603201814519993667772968868461426771), + SC_(5), SC_(0.93538987636566162109375), SC_(0.23180510604150429189260138185795569842018206442858), SC_(-0.80887163427971286589959630152315983255456115675709), + SC_(5), SC_(0.93773555755615234375), SC_(0.25325565314935514475658826960850521331006389130604), SC_(-0.80136139259062971113064196979296442577453518790514), + SC_(5), SC_(0.98576259613037109375), SC_(0.79688047965898906667942422168075865228912496237335), SC_(-0.050056541330186097160719119175881895101011997635397), + SC_(5), SC_(0.99292266368865966796875), SC_(0.89644489554848727542361918961404754824639399670787), SC_(0.37903738567797794270627705612855765856516256104433), + SC_(6), SC_(-0.804919183254241943359375), SC_(-0.39836758461472829348998598547346973134856336403357), SC_(-0.11186770789341157073263552304164952167231422725269), + SC_(6), SC_(-0.74602639675140380859375), SC_(-0.26942943612721316815918357350921061418816560550185), SC_(-0.42602747957292468844799426387152791743497741383254), + SC_(6), SC_(-0.72904598712921142578125), SC_(-0.21840146170346573698680050916460454542235139698046), SC_(-0.48296923649820365053381714059193346150823509339657), + SC_(6), SC_(-0.62323606014251708984375), SC_(0.11229562713036701432807908627179645376710821780817), SC_(-0.52579618536742217260788491630929608589881417853436), + SC_(6), SC_(-0.5579319000244140625), SC_(0.25809538238812630583446736370855025636974655379774), SC_(-0.35443454005327430793711175663169328286248363280561), + SC_(6), SC_(-0.44300353527069091796875), SC_(0.32627027269443981686050692334106934126675080884247), SC_(0.076978364356597061911602221500765420329999192593694), + SC_(6), SC_(-0.38366591930389404296875), SC_(0.27296210523874472670236080623125046782530388804834), SC_(0.27741174731966691723650037805968774957839220262642), + SC_(6), SC_(0.09376299381256103515625), SC_(-0.25631763331851885315854599502995062449465614809422), SC_(-0.28268746516151787527692774690914081177303614109476), + SC_(6), SC_(0.0944411754608154296875), SC_(-0.25552408862019782534903797380377055156453603599383), SC_(-0.28448177559590539095516583083575884593430785034601), + SC_(6), SC_(0.264718532562255859375), SC_(0.055663225234594337836693345273561919051283660765644), SC_(-0.49211101331909515135704356361137060677480116821877), + SC_(6), SC_(0.62944734096527099609375), SC_(0.095034738211053556569114686960941777812118998024525), SC_(0.53595663488501203764878594691541436002403012434814), + SC_(6), SC_(0.67001712322235107421875), SC_(-0.027907528859134634498084497388775217092645960712684), SC_(0.56739928086411463246848716706273921826782270933536), + SC_(6), SC_(0.81158387660980224609375), SC_(-0.40564261599347283927948120545903068249295719183982), SC_(0.065698612969198511696728446106187822049301308842024), + SC_(6), SC_(0.826751708984375), SC_(-0.41441668785915980661759850483046331378288487166901), SC_(-0.046051089316127435599875211778830659378778305951565), + SC_(6), SC_(0.91501367092132568359375), SC_(-0.14534417319305000517260528484257988550512306946976), SC_(-0.73321199968869349837797508987098833755655170036581), + SC_(6), SC_(0.92977702617645263671875), SC_(-0.02497873763140766287065827948618680160671583843944), SC_(-0.80187400193518611909857727430796462753119035734657), + SC_(6), SC_(0.93538987636566162109375), SC_(0.028201141383986142054022312024873160606967286580461), SC_(-0.81740119198478364508466660088619509288367234232148), + SC_(6), SC_(0.93773555755615234375), SC_(0.051703886374808225952970501830818053254301264204132), SC_(-0.82169997971024778036115328485043201232954643879117), + SC_(6), SC_(0.98576259613037109375), SC_(0.72170038823497209265542323838206061870527088205587), SC_(-0.25448294588059600387767244854330786413565245709897), + SC_(6), SC_(0.99292266368865966796875), SC_(0.85656129838850274253099953133227725425228299308345), SC_(0.17625410031563921455149796282056956604031098075843), + SC_(13), SC_(-0.804919183254241943359375), SC_(-0.021346989675489582899777513316783512919027984178499), SC_(-0.44112202059370695044753367609499513618897172902607), + SC_(13), SC_(-0.74602639675140380859375), SC_(0.24666838562906154791363844279971485004009388494544), SC_(-0.1559921230869429475133589681910169153699159544855), + SC_(13), SC_(-0.72904598712921142578125), SC_(0.26205171362935296430727406212496048008100313584788), SC_(-0.017605179298092333976782550762974395271667547123874), + SC_(13), SC_(-0.62323606014251708984375), SC_(-0.080372069801645230864998901272264760659582933477599), SC_(0.36425485923732018458592174575151844471405378913141), + SC_(13), SC_(-0.5579319000244140625), SC_(-0.23583528015736737224719659667717786026703268284614), SC_(0.053245393722305081751995109192359134354871591534969), + SC_(13), SC_(-0.44300353527069091796875), SC_(0.018972604977717555259851272576261414873385597702604), SC_(-0.35886939657929715282594368713239077257966834584912), + SC_(13), SC_(-0.38366591930389404296875), SC_(0.18550250811451569348004068552647395938057310465493), SC_(-0.20246148054258569990560069049122417793152050446365), + SC_(13), SC_(0.09376299381256103515625), SC_(0.20769838303968474575312727060982309147206670120161), SC_(-0.10173158142023370803559517312227984933062321078173), + SC_(13), SC_(0.0944411754608154296875), SC_(0.20829227701017635616317556146048162430074934166827), SC_(-0.098728123460385300129782227650749493941872811511327), + SC_(13), SC_(0.264718532562255859375), SC_(-0.1016381458266690823353213981274025318520849829061), SC_(0.30835386562966326517046404619145380164126531151263), + SC_(13), SC_(0.62944734096527099609375), SC_(0.055159037199559177575178459223260433973722342159236), SC_(0.37691576754738027132316322394241161656121459482015), + SC_(13), SC_(0.67001712322235107421875), SC_(-0.11980080975332609846766208424077118471193674213212), SC_(0.34804362893711095691292965673736967284289808313288), + SC_(13), SC_(0.81158387660980224609375), SC_(0.064420961086402494165295142893303224035523617990131), SC_(-0.43419593067780672264639624363068831709493843634964), + SC_(13), SC_(0.826751708984375), SC_(0.16013654257672660076220863480719945813853670795106), SC_(-0.37830883499601794980891054166266352530711732686068), + SC_(13), SC_(0.91501367092132568359375), SC_(0.02981406506695430902920205723465070991144692911286), SC_(0.53391573874244763290223203488170268282792221646375), + SC_(13), SC_(0.92977702617645263671875), SC_(-0.14928783504228836625545729908131879929441545864731), SC_(0.5094824048642415323846575901763459962675439032328), + SC_(13), SC_(0.93538987636566162109375), SC_(-0.21821472006169780300413719938049820079476182672815), SC_(0.45809358526457816919278185678834785451752393960047), + SC_(13), SC_(0.93773555755615234375), SC_(-0.24593495540807619378595982256918019147674926458319), SC_(0.42889513803137504816802334074350455620384537313833), + SC_(13), SC_(0.98576259613037109375), SC_(0.06582920955682109946957491732104452005420774878568), SC_(-0.81713360182404631829162888606556475045158637145556), + SC_(13), SC_(0.99292266368865966796875), SC_(0.45168281361591264659940935116533646174194486856019), SC_(-0.66523405291864568469611605694139032227348875420525), + SC_(16), SC_(-0.804919183254241943359375), SC_(-0.24617676998426440746802697580367697602069689994431), SC_(-0.10342588249052076335368259863624425129111161555058), + SC_(16), SC_(-0.74602639675140380859375), SC_(0.05589693997183205349782501045594495596630477937041), SC_(-0.36756591277520722601404123599925254242454264467136), + SC_(16), SC_(-0.72904598712921142578125), SC_(0.1435839632236998433207972126531002284333052188772), SC_(-0.29678713314015209268198753833877314533807120416601), + SC_(16), SC_(-0.62323606014251708984375), SC_(0.025006544465484963551355899284888512107603542872025), SC_(0.34655424303519465823894024987098609170947386899599), + SC_(16), SC_(-0.5579319000244140625), SC_(-0.20273668414164894060160750060929827332311158866899), SC_(0.1149894928335745044316704348175343096216902152789), + SC_(16), SC_(-0.44300353527069091796875), SC_(0.0568430344346968196836808967847074379571404518767), SC_(-0.31329734713414163463145524081279862392937646591181), + SC_(16), SC_(-0.38366591930389404296875), SC_(0.19955350157037941974526184597940130951592808285846), SC_(-0.069121706148999889753860133645349635646774147838909), + SC_(16), SC_(0.09376299381256103515625), SC_(0.004077511401713283600036273485948150451706308548591), SC_(0.3090887073999325166929620341518083398087059069866), + SC_(16), SC_(0.0944411754608154296875), SC_(0.0018646455756569841632128890021862805086238621865179), SC_(0.30915113128562592599142879370578903469780546339442), + SC_(16), SC_(0.264718532562255859375), SC_(-0.057138811421491531852629180943328986609990232285623), SC_(-0.30102711196776165823913047161703041527490961215075), + SC_(16), SC_(0.62944734096527099609375), SC_(0.053905813652862864225047719177357021912552840167578), SC_(-0.3394893321631230893291882500714616497801870033187), + SC_(16), SC_(0.67001712322235107421875), SC_(0.20571755182888692558409436652095184688145604215735), SC_(-0.15400029036097541197972382773740560927016105513573), + SC_(16), SC_(0.81158387660980224609375), SC_(-0.25610293236665229776934216307688324622589786937523), SC_(0.029961309199690485279175505247898935243066233951638), + SC_(16), SC_(0.826751708984375), SC_(-0.24467257950234667771221825741251222771947492803914), SC_(-0.14584909786233540893014977656569646339205583145946), + SC_(16), SC_(0.91501367092132568359375), SC_(0.30042521371658467516725427482154698462078838756), SC_(0.11242410274334292000251659391599358954645603738835), + SC_(16), SC_(0.92977702617645263671875), SC_(0.20912498055485121957345853172873004479220934035096), SC_(0.38711457227561848903449378495312209659537067985191), + SC_(16), SC_(0.93538987636566162109375), SC_(0.14228176834914109691746234218230423568131480850111), SC_(0.46724475576348077611133336497614776597835295379528), + SC_(16), SC_(0.93773555755615234375), SC_(0.10947822546659067656059296835577708662292349876443), SC_(0.49345958632627785174194039683464075153298036877136), + SC_(16), SC_(0.98576259613037109375), SC_(-0.18053759743174070857427031356500616942177764243751), SC_(-0.69129540587532453519248426712601771804500778047449), + SC_(16), SC_(0.99292266368865966796875), SC_(0.24467850628321941403216962899355195806033451938156), SC_(-0.79611153201220615596111924787398204555433320171123), + SC_(18), SC_(-0.804919183254241943359375), SC_(-0.0091062060443244939758547861277241601517458288228401), SC_(-0.37780874716766649794749661161502596284251171176729), + SC_(18), SC_(-0.74602639675140380859375), SC_(0.22554068680311218202775373160884926839138620763174), SC_(0.043450468136242147653982143862844811198611491554443), + SC_(18), SC_(-0.72904598712921142578125), SC_(0.18653718888369370131458265584018026946159296567964), SC_(0.19518736469516489835563573974741507967360934238079), + SC_(18), SC_(-0.62323606014251708984375), SC_(-0.20837325214812717413067018822585361068165851336255), SC_(-0.037088866497216472583736911887498188987603773369659), + SC_(18), SC_(-0.5579319000244140625), SC_(0.0083537300257362812848909087265145663947906734657682), SC_(-0.31950967163170480424900959261902943297087436887194), + SC_(18), SC_(-0.44300353527069091796875), SC_(0.11695183881480336805185423703698692829975597728824), SC_(0.24681367379374692383023201907678584193776277803732), + SC_(18), SC_(-0.38366591930389404296875), SC_(-0.10358901613970675221183097893317277350294739582856), SC_(0.25578907869474625948919953810862798903271057530686), + SC_(18), SC_(0.09376299381256103515625), SC_(0.030899020693203570750329014297680833991336820156792), SC_(-0.28791754441150052849520877694768651589010597918611), + SC_(18), SC_(0.0944411754608154296875), SC_(0.033208344966164570458319052396398896477890882959009), SC_(-0.28729203288102887394022408730410045130704945170246), + SC_(18), SC_(0.264718532562255859375), SC_(-0.045964012675459912647004385671790829004749499845969), SC_(0.28775273984847125983134846044367147353322574723096), + SC_(18), SC_(0.62944734096527099609375), SC_(-0.21025192720175270695719746239016794207250758550671), SC_(-0.011451910307176188739521846496709015868005938376706), + SC_(18), SC_(0.67001712322235107421875), SC_(-0.11209473575977880537554630008831322938651681961203), SC_(-0.28861574077574071045967706163764266303939167641235), + SC_(18), SC_(0.81158387660980224609375), SC_(-0.059414308841952830929071045427642264332649709039605), SC_(0.36941533931174362399141104265754795528484218175544), + SC_(18), SC_(0.826751708984375), SC_(-0.1661861116384618036402153021294856784649918532195), SC_(0.28742497280291907514593608858531889117823779493292), + SC_(18), SC_(0.91501367092132568359375), SC_(0.24104179066621669826861428047478740284125071072384), SC_(-0.25816581242891276150855737931173321477113736399573), + SC_(18), SC_(0.92977702617645263671875), SC_(0.30345956430814040115575308179633295683878952009221), SC_(0.053205379543234927660568467761641814470230916738039), + SC_(18), SC_(0.93538987636566162109375), SC_(0.28695924419323148602457516358630434274549630594269), SC_(0.19040502613280754358590157800975680518526525190133), + SC_(18), SC_(0.93773555755615234375), SC_(0.27214637032176476176235696410333280489913435468021), SC_(0.24695230662862708114901251701483136230530918002153), + SC_(18), SC_(0.98576259613037109375), SC_(-0.30043454768104230746662772278855454488514250456508), SC_(-0.52586497321495004128692505709794988132542865806662), + SC_(18), SC_(0.99292266368865966796875), SC_(0.10905904618186494808823511070262036756283922080124), SC_(-0.8190043052645567950966181297246945012178431035647), + SC_(19), SC_(-0.804919183254241943359375), SC_(-0.13197080377202665736616914483295430386103226798339), SC_(0.30439185079461838727386102511417679445130143697504), + SC_(19), SC_(-0.74602639675140380859375), SC_(-0.145888472515752488322967820082056444353328274887), SC_(-0.26146280837274387072049326070853702574943967213499), + SC_(19), SC_(-0.72904598712921142578125), SC_(-0.049543273739376098938164118341034405465324305394334), SC_(-0.33399130950651187664252780722066231231086715215037), + SC_(19), SC_(-0.62323606014251708984375), SC_(0.108462474462736719241499115773313996144190665045), SC_(0.27188868340165133975799203281538322711757069710654), + SC_(19), SC_(-0.5579319000244140625), SC_(-0.16898700597587011902855624616392747901805702881528), SC_(0.16296957620279172219978419815087191821651266886829), + SC_(19), SC_(-0.44300353527069091796875), SC_(0.086774244261237041187502205186727910747248943821998), SC_(-0.2669005399736008170995423913834449884708993279043), + SC_(19), SC_(-0.38366591930389404296875), SC_(0.18518233719073980253783272267881493406790394149709), SC_(0.050815807727239433167794934837935315089529954541468), + SC_(19), SC_(0.09376299381256103515625), SC_(-0.1749285947092888721974786955725138685456169025118), SC_(-0.073354024188019371331713593919507214584741353774141), + SC_(19), SC_(0.0944411754608154296875), SC_(-0.17429825143908666454510603569046385180196182465626), SC_(-0.077001126265999192631865199712139132518527301044888), + SC_(19), SC_(0.264718532562255859375), SC_(0.16022500206625985912825819052448183125933330717671), SC_(0.14198924834209712722889554522828334583526568794205), + SC_(19), SC_(0.62944734096527099609375), SC_(-0.13438326361094362774586020859943409104199788819994), SC_(0.24300673266449070282404274854258133237165626165118), + SC_(19), SC_(0.67001712322235107421875), SC_(-0.20602713406941943301157630033028134054188569383329), SC_(-0.060940372673870113382880029493240963759634545830116), + SC_(19), SC_(0.81158387660980224609375), SC_(0.086971455844324875234132438927179131969983584970598), SC_(0.34508503887069582338167006750623249283260807122053), + SC_(19), SC_(0.826751708984375), SC_(-0.033442914793680897952010218712581529745899360903155), SC_(0.37454337915916963566398826067574338174597268932648), + SC_(19), SC_(0.91501367092132568359375), SC_(0.15007656103598576739971044779394255546718356845339), SC_(-0.37908868831917397336990106081043384870642885722423), + SC_(19), SC_(0.92977702617645263671875), SC_(0.28693501649909716776829384064013974960722420152263), SC_(-0.12313038336912701300360856578205807776474791522058), + SC_(19), SC_(0.93538987636566162109375), SC_(0.30324802829239922164147482899163792077902516015462), SC_(0.017801725881771420248228319415115282604840122400748), + SC_(19), SC_(0.93773555755615234375), SC_(0.30185089350015097041813236342216465473811492066808), SC_(0.080512539757557010183574032982523614885735562454396), + SC_(19), SC_(0.98576259613037109375), SC_(-0.34396150797274909426987918450335894287593307514199), SC_(-0.42686745322594445556165789826564473675847159812306), + SC_(19), SC_(0.99292266368865966796875), SC_(0.043969381393564338898972458003559718667557179801894), SC_(-0.81279540498362619452519349374691829897388634698573) + }; +#undef SC_ + diff --git a/test/legendre_p_large.ipp b/test/legendre_p_large.ipp new file mode 100644 index 000000000..a1f7d1ac8 --- /dev/null +++ b/test/legendre_p_large.ipp @@ -0,0 +1,170 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 160> legendre_p_large = { + SC_(29), SC_(-0.74602639675140380859375), SC_(0.050915219643735786802064817454102557266509665552523), SC_(-0.27118035040452065163236941090242943684321195237749), + SC_(29), SC_(-0.72904598712921142578125), SC_(0.15209960929167220423613043592541930303920942697128), SC_(-0.1438359066051312703697159687668902900032679225431), + SC_(29), SC_(-0.5579319000244140625), SC_(0.15849246733249484229246386045081847903407720368835), SC_(-0.046562152771403674797644638451970346262085750814402), + SC_(29), SC_(-0.38366591930389404296875), SC_(0.12421123432704035296982084866318407821031736589989), SC_(-0.13993608234219292039527623183264127314107515942999), + SC_(29), SC_(0.264718532562255859375), SC_(0.14939214703729469665461134129487953904270632801499), SC_(0.011880798886655750194841085329195617037793542295307), + SC_(29), SC_(0.62944734096527099609375), SC_(0.15752641351603055527713997332857496041323176430518), SC_(-0.085333543185746265108134728242042619782666873361817), + SC_(29), SC_(0.67001712322235107421875), SC_(0.054837169775284662482177068305626352949947763721645), SC_(0.25355488589896471253311476119270056443895635234837), + SC_(29), SC_(0.81158387660980224609375), SC_(0.063381360227496634467950288363046198618583970049264), SC_(0.28493842219370104406874711812243725560916796813782), + SC_(29), SC_(0.826751708984375), SC_(-0.08420018636937566660354955907203801287585967235842), SC_(0.27769559592859630765679556893873497642068025145103), + SC_(29), SC_(0.93773555755615234375), SC_(-0.24176816577505910452921016056528287088649954762181), SC_(0.094296671036158090707906492415425184420130303729612), + SC_(32), SC_(-0.74602639675140380859375), SC_(-0.10624862108385367062544557604098705758947684718283), SC_(-0.21143967375447832725357384431573889445833865564692), + SC_(32), SC_(-0.72904598712921142578125), SC_(0.025162565093345926014241859993355739041989277225088), SC_(-0.26274148479323222549160374105375114037174210402628), + SC_(32), SC_(-0.5579319000244140625), SC_(0.14211456817063838568644446010057604214542220046016), SC_(-0.09163254161603810457705450004999666401809984055068), + SC_(32), SC_(-0.38366591930389404296875), SC_(0.14171288019304087574298677776314705938680717437038), SC_(-0.052711866447932948820821929976771090258669811818703), + SC_(32), SC_(0.264718532562255859375), SC_(-0.10746376970464863575465113222711354062288152228845), SC_(0.14703447108655800801097998317667773187484429086529), + SC_(32), SC_(0.62944734096527099609375), SC_(-0.15721886386953844900014008976876572130375292635363), SC_(-0.03463115417113285817814346783115703627607698101613), + SC_(32), SC_(0.67001712322235107421875), SC_(0.048628774250235041116639532793364743422360425715375), SC_(-0.24343097490553436355382554969125452470269482841518), + SC_(32), SC_(0.81158387660980224609375), SC_(0.14709429758897851782623104453504062279048901484918), SC_(-0.17120780885496551232352932901235943774904473019944), + SC_(32), SC_(0.826751708984375), SC_(0.18197608635110136442722424209924919380946511973609), SC_(0.064598428569533734464113024874557402334768980022172), + SC_(32), SC_(0.93773555755615234375), SC_(-0.061512016664104396409494145001588438528631649237376), SC_(0.36010369834698605740012216441227373597825633684005), + SC_(33), SC_(-0.74602639675140380859375), SC_(-0.010238220971698350467857407217583972140726845648941), SC_(0.26483415891807978747956511959896313413438051711664), + SC_(33), SC_(-0.72904598712921142578125), SC_(-0.1308485763851203410864936505227876528881511136519), SC_(0.16199934006548616731309863119775152390583901324519), + SC_(33), SC_(-0.5579319000244140625), SC_(-0.12577548566890528018101278253700971908249029962743), 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SC_(-0.1417288093884680768867130033428123726827027042855), + SC_(115), SC_(-0.72904598712921142578125), SC_(0.0097131893606856202751128572690136150201795716231646), SC_(0.14012961153023905234915467648238152308202483249179), + SC_(115), SC_(-0.5579319000244140625), SC_(-0.055599872031625140506136394171510860309410123809349), SC_(0.093594483130628843819694780782575925754784924670163), + SC_(115), SC_(-0.38366591930389404296875), SC_(0.077050446497529102420650609981634503271775399693683), SC_(0.0088565512206407663055683543736671275288183109816667), + SC_(115), SC_(0.264718532562255859375), SC_(0.034359839114246272577599724967370326173387403660347), SC_(0.10578248128418573373870536860675213912264059240692), + SC_(115), SC_(0.62944734096527099609375), SC_(0.0082643990325571512244908953318239675941855106132265), SC_(-0.13165640504236233394765218021604504786233485133267), + SC_(115), SC_(0.67001712322235107421875), SC_(-0.0015721055642368064825337799606692523719130974520218), SC_(-0.1353286900783831392195565600906463227612561145681), + SC_(115), SC_(0.81158387660980224609375), SC_(-0.054210930576897856499034808655933306801074683806062), SC_(-0.12659507936588546926051398902517626516964689218928), + SC_(115), SC_(0.826751708984375), SC_(0.06212883372643213611372963364539976692767234086995), SC_(0.12103709534949250997508001328034108589273615963615), + SC_(115), SC_(0.93773555755615234375), SC_(-0.09980919675793132053530446510639647979845319955803), SC_(-0.12070840338940873219064713828143611372512223844091), + SC_(116), SC_(-0.74602639675140380859375), SC_(-0.068487247444151474237653315277483909414503521300912), SC_(0.093133560384715286625946002729620416337163521029867), + SC_(116), SC_(-0.72904598712921142578125), SC_(0.053747789206590672037151315696697693783833792642693), SC_(-0.11211897478819628686868719378051029417346515093252), + SC_(116), SC_(-0.5579319000244140625), SC_(0.080122819627702864075844102189789799443576781457988), SC_(0.020173351559352067990742477146036299831908728103273), + SC_(116), SC_(-0.38366591930389404296875), SC_(-0.024249827411491278686372542928360990815946534826784), SC_(-0.11467110049800068759284312552536360348652573880772), + SC_(116), SC_(0.264718532562255859375), SC_(0.073718014154393579573780297435185676405997097861731), SC_(-0.023941183188976671650101777466534611583995107650774), + SC_(116), SC_(0.62944734096527099609375), SC_(-0.059668625982518363289929597563274657874779402078409), SC_(-0.09255759087664323717343674706577199358384636772319), + SC_(116), SC_(0.67001712322235107421875), SC_(-0.064729455762043452505626574517572044801030359192088), SC_(-0.088456402518418445483355798933259689021877395507254), + SC_(116), SC_(0.81158387660980224609375), SC_(-0.09069070444539916571162153776802786573166522696411), SC_(-0.0527625562114936996735580314666853717965865324919), + SC_(116), SC_(0.826751708984375), SC_(0.094306540731834905126868750171910154657206855132072), SC_(0.044969460070177494430473922626932676591613349156668), + SC_(116), SC_(0.93773555755615234375), SC_(-0.11977051432565183689355134402005152657631906753244), SC_(-0.058477756477300972556925093606746906197796783801524), + SC_(119), SC_(-0.74602639675140380859375), SC_(0.0087647065593993755222972658179029408444330186700542), SC_(0.13981912827212672744690925773170472610548387878822), + SC_(119), SC_(-0.72904598712921142578125), SC_(-0.020501605529893412888893201088044338252329402923723), SC_(-0.13478486824495203061528622211548575586334551817436), + SC_(119), SC_(-0.5579319000244140625), SC_(0.080036079293206370171512732389414344286704185353321), SC_(-0.0057776707755366365863380008852867161540142918927643), + SC_(119), SC_(-0.38366591930389404296875), SC_(0.0052218893934731418741815240011222420959400585517152), SC_(-0.11902359008249444427689952801477258632653373728298), + SC_(119), SC_(0.264718532562255859375), SC_(-0.041958445146548813480092404170567661553263062970998), SC_(0.096369084287714671301426340819694101521306372531871), + SC_(119), SC_(0.62944734096527099609375), SC_(0.026043742963778131871545929796683083958834802939707), SC_(0.12346064721462223505658160979882243719998039566685), + SC_(119), SC_(0.67001712322235107421875), SC_(0.018730148385420890536233247762177938725955754094692), SC_(0.12977342706190187482418503847938799787319981967217), + SC_(119), SC_(0.81158387660980224609375), SC_(-0.0051210369574630302606435588094752910907524904265636), SC_(0.14977897540214287229711857221722487285805056065051), + SC_(119), SC_(0.826751708984375), SC_(0.0070994904638281005505473502488606481760552210237098), SC_(-0.15244878741668898076438519389186244649902911784835), + SC_(119), SC_(0.93773555755615234375), SC_(-0.089511411656126705336943924861664813522610049042024), SC_(0.13442707735952316190476066030580938662981623871763) + }; +#undef SC_ + diff --git a/test/log1p_expm1_data.ipp b/test/log1p_expm1_data.ipp new file mode 100644 index 000000000..a4cd8da04 --- /dev/null +++ b/test/log1p_expm1_data.ipp @@ -0,0 +1,90 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 80> log1p_expm1_data = { { + { SC_(-0.69330310821533203125e0), SC_(-0.1181895342296499380302723361817935835636e1), SC_(-0.5000779577496508480606742934033661111325e0) }, + { SC_(-0.650003612041473388671875e0), SC_(-0.1049832444670425873798449427248829256278e1), SC_(-0.477956108886575099597621504254337139212e0) }, + { SC_(-0.5634434223175048828125e0), SC_(-0.8288372954181591063099417140530721209296e0), SC_(-0.4307544676154126107123951950891833745657e0) }, + { SC_(-0.546193540096282958984375e0), SC_(-0.7900844716039173375586798387434829539395e0), SC_(-0.4208498682390169422105137923243733478616e0) }, + { SC_(-0.542549669742584228515625e0), SC_(-0.7820869679038185958797390511513816016257e0), SC_(-0.4187356706519199578716316143188015118402e0) }, + { SC_(-0.5222184658050537109375e0), SC_(-0.7386016923991178813343233368825674656574e0), SC_(-0.4067969136237976293436891521638796489665e0) }, + { SC_(-0.510332167148590087890625e0), SC_(-0.7140280098902130849529032452448458253718e0), SC_(-0.3997038529677125427667940816216463479375e0) }, + { SC_(-0.50135910511016845703125e0), SC_(-0.6958690918220109127943821843553816517932e0), SC_(-0.3942931192785328340465460021958309251731e0) }, + { SC_(-0.479341685771942138671875e0), SC_(-0.6526612791947966365578043302467499708984e0), SC_(-0.3808091201700817035685503987715593769588e0) }, + { SC_(-0.4362652301788330078125e0), SC_(-0.5731714043684679946812364099230074705554e0), SC_(-0.3535537538578334381043477595638178633635e0) }, + { SC_(-0.4033059179782867431640625e0), SC_(-0.5163507223485766856253687547437478575359e0), SC_(-0.3318923180936536087108170796894639170118e0) }, + { SC_(-0.3905523121356964111328125e0), SC_(-0.4952021616945213559322173628246423264147e0), SC_(-0.3233169689815286035820337522007644642142e0) }, + { SC_(-0.3101024627685546875e0), SC_(-0.3712121891835896680955152331639073308925e0), SC_(-0.2666281909314749260551669118328805822824e0) }, + { SC_(-0.2841587960720062255859375e0), SC_(-0.3342969188422780419049667231102322921458e0), SC_(-0.247352882189668478949423010818860745509e0) }, + { SC_(-0.268566131591796875e0), SC_(-0.3127484680657795755742977783527708289938e0), SC_(-0.2355251348015818742687440901815837983695e0) }, + { SC_(-0.19418840110301971435546875e0), SC_(-0.215905312065666655084617543044814851896e0), SC_(-0.1764972591721576227824907304661137322776e0) }, + { SC_(-0.14176605641841888427734375e0), SC_(-0.1528785551425763265780363302506293842933e0), SC_(-0.1321757453457543023426366655287893957346e0) }, + { SC_(-0.109534211456775665283203125e0), SC_(-0.1160105952462048156067755365883457898243e0), SC_(-0.1037484982319069072752933517880363955799e0) }, + { SC_(-0.2047410048544406890869140625e-1), SC_(-0.2068660038044094868521052319477265955827e-1), SC_(-0.2026592921724753704129022027337835687888e-1) }, + { SC_(0.1690093176520690576580818742513656616211e-8), SC_(0.1690093175092483105529122131518271037775e-8), SC_(0.16900931779488980500463190560092746436e-8) }, + { SC_(0.2114990849122477811761200428009033203125e-8), SC_(0.2114990846885884668978873262661703032735e-8), SC_(0.2114990851359070959273901626052243209537e-8) }, + { SC_(0.7099628440698779741069301962852478027344e-8), SC_(0.7099628415496417862364745346932718974223e-8), SC_(0.7099628465901141798701264063185361883662e-8) }, + { SC_(0.136718796284185373224318027496337890625e-7), SC_(0.1367187953495839188729947299064627330362e-7), SC_(0.1367187972187868403533999531964021065056e-7) }, + { SC_(0.1679341288252089725574478507041931152344e-7), SC_(0.1679341274151154071302093036120830402912e-7), SC_(0.1679341302353025616649699444201095417115e-7) }, + { SC_(0.586768322818898013792932033538818359375e-7), SC_(0.5867683056040454540164807679535005383825e-7), SC_(0.5867683400337515836824145241465956859382e-7) }, + { SC_(0.1140460881288163363933563232421875e-6), SC_(0.1140460816255617220976462903776019835232e-6), SC_(0.1140460946320716923598363683695123099251e-6) }, + { SC_(0.1455586016163579188287258148193359375e-6), SC_(0.1455585910227056945720521239406972268757e-6), SC_(0.1455586122100116850826593912609916845078e-6) }, + { SC_(0.38918477685001562349498271942138671875e-6), SC_(0.3891847011176400068545868550436683472421e-6), SC_(0.3891848525824207140259509650302544482884e-6) }, + { SC_(0.623782625552848912775516510009765625e-6), SC_(0.6237824310005478475318960793412787601785e-6), SC_(0.623782820105271336383228077398355607783e-6) }, + { SC_(0.104669607026153244078159332275390625e-5), SC_(0.1046695522475582933881286527214809459507e-5), SC_(0.1046696618048055313232628353357778906342e-5) }, + { SC_(0.2951089072666945867240428924560546875e-5), SC_(0.2951084718212155380639451770777340673208e-5), SC_(0.2951093427134586747271699561094046027716e-5) }, + { SC_(0.4877083483734168112277984619140625e-5), SC_(0.4877071590801182991613986686611799404767e-5), SC_(0.487709537672515613069814424540824650525e-5) }, + { SC_(0.9066634447663091123104095458984375e-5), SC_(0.9066593345981423108474522054350755440144e-5), SC_(0.9066675549717413905283549214899456610946e-5) }, + { SC_(0.2360353755648247897624969482421875e-4), SC_(0.2360325899737320048013584393963240356542e-4), SC_(0.236038161221667766684232806678163246188e-4) }, + { SC_(0.60817910707555711269378662109375e-4), SC_(0.6081606137340567462263175491115417378507e-4), SC_(0.6081976015418009724388783415958425809014e-4) }, + { SC_(0.119476739200763404369354248046875e-3), SC_(0.1194696024236052968763590872811035649465e-3), SC_(0.1194838768306258449523761123420162837989e-3) }, + { SC_(0.2437086659483611583709716796875e-3), SC_(0.2436789738154874267053728955508067082563e-3), SC_(0.2437383653179059006664060856490910639369e-3) }, + { SC_(0.47970912419259548187255859375e-3), SC_(0.4795941005544676642733520039561612799338e-3), SC_(0.4798242030152303995117671825532525519771e-3) }, + { SC_(0.960788573138415813446044921875e-3), SC_(0.9603273112237537137213831663159454349065e-3), SC_(0.9612502783347382477096533911074491600195e-3) }, + { SC_(0.113048148341476917266845703125e-2), SC_(0.112984297039538753162021050860817616211e-2), SC_(0.1131120718465376073328378699867756887911e-2) }, + { SC_(0.33707791008055210113525390625e-2), SC_(0.3365110759179022039875145678841719780479e-2), SC_(0.3376466565278741394700008076507908493073e-2) }, + { SC_(0.512778759002685546875e-2), SC_(0.5114685258802700303545874906083848033916e-2), SC_(0.5140957193498527263160102812012697907234e-2) }, + { SC_(0.7697627879679203033447265625e-2), SC_(0.7668152306886386648889951166330157951323e-2), SC_(0.7727330782215801339522867700900079915854e-2) }, + { SC_(0.154774188995361328125e-1), SC_(0.1535886535535876489145011045382406354476e-1), SC_(0.1559781448309931312470733745915557934013e-1) }, + { SC_(0.305807329714298248291015625e-1), SC_(0.3012246177452263928099228685195389251529e-1), SC_(0.3105312667138388983748287762435717112768e-1) }, + { SC_(0.346831791102886199951171875e-1), SC_(0.3409527271716101078966670703672108048896e-1), SC_(0.3529165481200088712490838501531684667149e-1) }, + { SC_(0.65634094178676605224609375e-1), SC_(0.6357001557994644965537090025497140391848e-1), SC_(0.6783591829384049159063415288088650400045e-1) }, + { SC_(0.6610882282257080078125e-1), SC_(0.6401540573272318085520396443445644947172e-1), SC_(0.6834297093791793596420077701374399081879e-1) }, + { SC_(0.9283597767353057861328125e-1), SC_(0.8877613176091369848730538080276748892544e-1), SC_(0.9728174180292914206798492307510621372797e-1) }, + { SC_(0.1853029727935791015625e0), SC_(0.1699984151517873986675151014366108622057e0), SC_(0.2035830378085867638230987836336350617284e0) }, + { SC_(0.195668697357177734375e0), SC_(0.1787056082557073958655865192859050644124e0), SC_(0.2161239335120153310995245962871550390859e0) }, + { SC_(0.218036949634552001953125e0), SC_(0.1972405051449346869157040783806052110702e0), SC_(0.2436330185556740271801313404522599276529e0) }, + { SC_(0.22476322948932647705078125e0), SC_(0.2027475432657783430097849833923015479242e0), SC_(0.2520262382028310304455135247279894706452e0) }, + { SC_(0.250229179859161376953125e0), SC_(0.2233268783961024315029665215194302402664e0), SC_(0.2843197231751691135236590716143900016849e0) }, + { SC_(0.253903329372406005859375e0), SC_(0.2262613494244882135168356599690803021814e0), SC_(0.2890471852440059179173797372107595021134e0) }, + { SC_(0.3161745369434356689453125e0), SC_(0.2747294509656754848973082105230889771257e0), SC_(0.3718696766716845552650417052219688730639e0) }, + { SC_(0.409090220928192138671875e0), SC_(0.3429442627531700751753321547062422000484e0), SC_(0.5054475372328648428694815984023805796901e0) }, + { SC_(0.41710007190704345703125e0), SC_(0.3486125805910782371810230966107318794672e0), SC_(0.5175543698964877990407235081110698982288e0) }, + { SC_(0.4173481762409210205078125e0), SC_(0.3487876441750920349995178382655357863143e0), SC_(0.5179309284235235985742545616245146385983e0) }, + { SC_(0.42039263248443603515625e0), SC_(0.3509333351432084595068912946293452186787e0), SC_(0.522559244493788135782100429719474184743e0) }, + { SC_(0.4406131207942962646484375e0), SC_(0.3650688012994099484177768785497336743499e0), SC_(0.5536595074987176176278695373720601072416e0) }, + { SC_(0.4500701129436492919921875e0), SC_(0.3716119090177188338876380950054248074284e0), SC_(0.5684221483289186932443765450666370231476e0) }, + { SC_(0.4690119922161102294921875e0), SC_(0.3845900606819792767555511238898325590012e0), SC_(0.5984141671678306664208145374987554621041e0) }, + { SC_(0.488781034946441650390625e0), SC_(0.3979576877817231446668587804772904526162e0), SC_(0.6303276957438828775174515106733850816524e0) }, + { SC_(0.52980291843414306640625e0), SC_(0.4251389156264805389990504406429844143619e0), SC_(0.6985975133709526468615664928080778563331e0) }, + { SC_(0.56810867786407470703125e0), SC_(0.4498702293886911216071551868503271338352e0), SC_(0.7649258494577443195036874224587751059599e0) }, + { SC_(0.57872617244720458984375e0), SC_(0.4566183018986474768197353974513078612477e0), SC_(0.7837647742172442075416378527651943088102e0) }, + { SC_(0.582029759883880615234375e0), SC_(0.4587086807259736626531803258754840111707e0), SC_(0.7896673415707786528734865994546559029663e0) }, + { SC_(0.607590496540069580078125e0), SC_(0.474736471992995596813856979079879927429e0), SC_(0.836002211172822800500580737750918478977e0) }, + { SC_(0.640033721923828125e0), SC_(0.4947168037733846594212322562172572984505e0), SC_(0.8965448333670257147748892521303229061693e0) }, + { SC_(0.640509545803070068359375e0), SC_(0.4950068922396821400142385044888446891867e0), SC_(0.8974474694176578931586020775970152155848e0) }, + { SC_(0.643289387226104736328125e0), SC_(0.4966999569758336894986723845165627033027e0), SC_(0.9027294105692233274199303070362746252986e0) }, + { SC_(0.64851474761962890625e0), SC_(0.4998747295737266888041967257469236722589e0), SC_(0.9126978792095101927158614842576974964632e0) }, + { SC_(0.650843918323516845703125e0), SC_(0.5012866228091318430397939496252397876475e0), SC_(0.9171580713331758702589264747154474769605e0) }, + { SC_(0.65477287769317626953125e0), SC_(0.5036637653893261974582119443232408393674e0), SC_(0.9247053242168904876365979871823094489023e0) }, + { SC_(0.65641486644744873046875e0), SC_(0.504655547804425337585747233639878779038e0), SC_(0.927868264760303541518755094801625468889e0) }, + { SC_(0.6588299274444580078125e0), SC_(0.5061124908504770714998094003973873746865e0), SC_(0.932529810907327764547265337859703076731e0) }, + { SC_(0.673553526401519775390625e0), SC_(0.5149492258613715166147831918156453824087e0), SC_(0.9611941077924732903931571711562687857294e0) }, + { SC_(0.69003379344940185546875e0), SC_(0.5247485248589687068613254590642295922543e0), SC_(0.9937829089064982966773652105234409046975e0) }, + { SC_(0.69504582881927490234375e0), SC_(0.5277097781183924897416903820415806504912e0), SC_(0.1003800903666412193968978978979482061619e1) } + } }; +#undef SC_ + diff --git a/test/log1p_expm1_test.cpp b/test/log1p_expm1_test.cpp index 9c3e23d6d..bbb823114 100644 --- a/test/log1p_expm1_test.cpp +++ b/test/log1p_expm1_test.cpp @@ -3,556 +3,119 @@ // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include #include #include #include #include +#include +#include "functor.hpp" -#include - -#ifdef BOOST_NO_STDC_NAMESPACE -namespace std{ using ::sqrt; using ::exp; } -#endif +#include "handle_test_result.hpp" // -// These test values were generated at 256-bit precision with NTL, -// we'll use these as a comparison for our own algorithms: +// DESCRIPTION: +// ~~~~~~~~~~~~ // -long double data[][3] = { - { 0.7071067811865475244008443621048490392848L, 0.5347999967395703705239932642507040249904L, 1.028114981647472451108126112746351175174L, }, - { -0.7071067811865475244008443621048490392848L, -1.227947177299515679941225385708880593066L, -0.5069313086047602121540426193848225247642L, }, - { 0.5L, 0.405465108108164381978013115464349136572L, 0.6487212707001281468486507878141635716538L, }, - { -0.5L, -0.6931471805599453094172321214581765680755L, -0.3934693402873665763962004650088195465581L, }, - { 0.3535533905932737622004221810524245196424L, 0.3027332756136080026565008168738916532429L, 0.4241190194809816004138284781974755046316L, }, - { -0.3535533905932737622004221810524245196424L, -0.4362646682381306258028444378052416278323L, -0.2978114986734404037618125202537819364955L, }, - { 0.25L, 0.2231435513142097557662950903098345033746L, 0.2840254166877414840734205680624364583363L, }, - { -0.25L, -0.2876820724517809274392190059938274315035L, -0.2211992169285951317548297330216793527032L, }, - { 0.1767766952966368811002110905262122598212L, 0.1627790866561079067978577664745092478057L, 0.1933645794479495947956947190066751660075L, }, - { -0.1767766952966368811002110905262122598212L, -0.1945277849706882079548540492230348777333L, -0.1620331144212442112773768997051282549751L, }, - { 0.125L, 0.1177830356563834545387941094705217050685L, 0.1331484530668263168290072278117938725655L, }, - { -0.125L, -0.1335313926245226231463436209313499745894L, -0.117503097415404597135107856770949263778L, }, - { 0.0883883476483184405501055452631061299106L, 0.08469802192359444824992033285217108168932L, 0.09241227540153062617063135584671815396877L, }, - { -0.0883883476483184405501055452631061299106L, -0.09254119938462034112310437534311466334391L, -0.08459468781377732889540834505145255847266L, }, - { 0.0625L, 0.0606246218164348425806061320404202632862L, 0.06449445891785942956339059464288967310073L, }, - { -0.0625L, -0.06453852113757117167292391568399292812891L, -0.06058693718652421388028917537769491547532L, }, - { 0.0441941738241592202750527726315530649553L, 0.0432454624194302551610854640871481968471L, 0.04518528280947901327000081260336002430674L, }, - { -0.0441941738241592202750527726315530649553L, -0.04520049725523360571871295632901500898476L, -0.04323183989734343213613354147207468639883L, }, - { 0.03125L, 0.0307716586667536883710282075967721640917L, 0.0317434074991026709387478152815071441945L, }, - { -0.03125L, -0.03174869831458030115699628274852562992756L, -0.03076676552365591815189080675364721639527L, }, - { 0.02209708691207961013752638631577653247765L, 0.02185648424813694627588486704954425304883L, 0.02234303578078870447841893477675632033383L, }, - { -0.02209708691207961013752638631577653247765L, -0.02234488474624582074086983060944116403229L, -0.02185473466225040063461756538971286083712L, }, - { 0.015625L, 0.01550418653596525415085404604244683587787L, 0.01574770858668574745853507208235174890672L, }, - { -0.015625L, -0.01574835696813916860754951146082826952093L, -0.01550356299459159401301117030297963029214L, }, - { 0.01104854345603980506876319315788826623883L, 0.01098795417351052684893633616419361769653L, 0.01110980401773807767277088699717081730887L, }, - { -0.01104854345603980506876319315788826623883L, -0.01111003193719750909051920695450559324905L, -0.0109877324634695920927755509825266486829L, }, - { 0.0078125L, 0.007782140442054948947462900061136763678126L, 0.007843097206447977693453559760123579193392L, }, - { -0.0078125L, -0.007843177461025892873184042490943581654592L, -0.00778206173975648789406277388638216992911L, }, - { 0.005524271728019902534381596578944133119413L, 0.005509068902949034570040757911345564446908L, 0.005539558653829430569890102084787436958606L, }, - { -0.005524271728019902534381596578944133119413L, -0.00553958694674479599776921231398085523465L, -0.005509040998094137069364339639027839483272L, }, - { 0.00390625L, 0.003898640415657323013937343095842907010724L, 0.003913889338347573443609603903460281898814L, }, - { -0.00390625L, -0.003913899321136329092317783643572664842706L, -0.003898630529882509937135338054830955713597L, }, - { 0.002762135864009951267190798289472066559706L, 0.002758328176699244507992500461404507081665L, 0.002765954075939611809020032526494603236975L, }, - { -0.002762135864009951267190798289472066559706L, -0.002765957600334472995309858879390478447378L, -0.002758324676557722165983108723111282889863L, }, - { 0.001953125L, 0.00195122013126174943967404953184153850035L, 0.001955033591002812046518898047477215867356L, }, - { -0.001953125L, -0.001955034835803350557627492241866812137664L, -0.001951218892524527289957340917326404744062L, }, - { 0.001381067932004975633595399144736033279853L, 0.001380115134839422362840066413196608682497L, 0.001382022045502678201141870910789095246896L, }, - { -0.001381067932004975633595399144736033279853L, -0.001382022485291226579353866252268409285565L, -0.001380114696566700781276188886360328730002L, }, - { 0.0009765625L, 0.0009760859730554588959608249080171866726118L, 0.0009770394924165352428452926116065064658516L, }, - { -0.0009765625L, -0.000977039647826612785968075151753465835865L, -0.0009760858180243377652882103896705696807979L, }, - { 0.0006905339660024878167976995723680166399266L, 0.0006902956571239922163465924178424595489941L, 0.0006907724394697735895620855250217767118257L, }, - { -0.0006905339660024878167976995723680166399266L, -0.0006907724943958822152082937884673220890625L, -0.0006902956022926226579739822473659270196051L, }, - { 0.00048828125L, 0.0004881620795013511885370496926454098503177L, 0.0004884004786944731261736238071633537881055L, }, - { -0.00048828125L, -0.0004884004981088744649849635598969109834608L, -0.0004881620601106346120642438969970786650047L, }, - { 0.0003452669830012439083988497861840083199633L, 0.0003452073920725940347605594851408457654352L, 0.0003453265945064500836804861194113777618491L, }, - { -0.0003452669830012439083988497861840083199633L, -0.0003453266013692502439328491689895266497932L, -0.0003452073852157150645328468804957180492811L, }, - { 0.000244140625L, 0.0002441108275273627091604790858234536994612L, 0.0002441704297478549370052339241357737575196L, }, - { -0.000244140625L, -0.0002441704321739144566954654183814336430665L, -0.0002441108251027834869064123317332192226443L, }, - { 0.0001726334915006219541994248930920041599816L, 0.0001726185920541657488047052564217192486562L, 0.0001726483935193326386701093624723297495011L, }, - { -0.0001726334915006219541994248930920041599816L, -0.0001726483943769975333358785791023362275292L, -0.0001726185911968709284892274915373687749247L, }, - { 0.0001220703125L, 0.0001220628625256773716230553671622032006662L, 0.0001220777633837710765035196704053169651876L, }, - { -0.0001220703125L, -0.0001220777636869822415828707903119755525239L, -0.0001220628622225587251301833935672790920603L, }, - { 0.8631674575031097709971244654600207999082e-4L, 0.8631302067436859538170634567406786112601e-4L, 0.8632047114779673053115205017424336944456e-4L, }, - { -0.8631674575031097709971244654600207999082e-4L, -0.8632047125499327478558483802377432851254e-4L, -0.8631302056719518077375662982669623892259e-4L, }, - { 0.6103515625e-4L, 0.6103329368063852491315878964896399244278e-4L, 0.610370189330454217791205985471093033817e-4L, }, - { -0.6103515625e-4L, -0.6103701897094392572114242980681797646601e-4L, -0.6103329364274580338274063706796778431152e-4L, }, - { 0.4315837287515548854985622327300103999541e-4L, 0.4315744157937625036775176424278410274829e-4L, 0.4315930421112837090401086653281340081162e-4L, }, - { -0.4315837287515548854985622327300103999541e-4L, -0.4315930422452721604826114516606004744788e-4L, -0.4315744156597885082640026911399617719788e-4L, }, - { 0.30517578125e-4L, 0.3051711247318637856906951416899468124305e-4L, 0.3051804379102429545128481254124317032922e-4L, }, - { -0.30517578125e-4L, -0.3051804379576142772845440263529078774196e-4L, -0.3051711246844960769262435293040988383306e-4L, }, - { 0.2157918643757774427492811163650051999771e-4L, 0.2157895361028356677465179910857365631679e-4L, 0.2157941926989617246887680913732626816569e-4L, }, - { -0.2157918643757774427492811163650051999771e-4L, -0.2157941927157098293412689381708235824685e-4L, -0.2157895360860884665958278826510182396057e-4L, }, - { 0.152587890625e-4L, 0.1525867264836239740575732513488881988266e-4L, 0.1525890547841394814004262248066173018701e-4L, }, - { -0.152587890625e-4L, -0.152589054790060783804405477297578319824e-4L, -0.1525867264777028975290466417266610381266e-4L, }, - { 0.1078959321878887213746405581825025999885e-4L, 0.1078953501154664660426995857578938351957e-4L, 0.1078965142665913183071896911252005879737e-4L, }, - { -0.1078959321878887213746405581825025999885e-4L, -0.1078965142686848031534799026109936625941e-4L, -0.1078953501133730376652725297740371278837e-4L, }, - { 0.762939453125e-5L, 0.7629365427567572155885296849132278805619e-5L, 0.7629423635154471743184330338220061308624e-5L, }, - { -0.762939453125e-5L, -0.7629423635228487317358417985971365713551e-5L, -0.7629365427493557993432788023738794879446e-5L, }, - { 0.5394796609394436068732027909125129999427e-5L, 0.5394782057531543905412357533557238337288e-5L, 0.5394811161335832678521641682790705055058e-5L, }, - 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{ 0.8312000267129179839093605233132203581414e-38L, 0.831200026712917983909360523313220358138e-38L, 0.8312000267129179839093605233132203581449e-38L, }, - { -0.8312000267129179839093605233132203581414e-38L, -0.8312000267129179839093605233132203581449e-38L, -0.831200026712917983909360523313220358138e-38L, }, - { 0.5877471754111437539843682686111228389093e-38L, 0.5877471754111437539843682686111228389076e-38L, 0.5877471754111437539843682686111228389111e-38L, }, - { -0.5877471754111437539843682686111228389093e-38L, -0.5877471754111437539843682686111228389111e-38L, -0.5877471754111437539843682686111228389076e-38L, }, - { 0.4156000133564589919546802616566101790707e-38L, 0.4156000133564589919546802616566101790699e-38L, 0.4156000133564589919546802616566101790716e-38L, }, - { -0.4156000133564589919546802616566101790707e-38L, -0.4156000133564589919546802616566101790716e-38L, -0.4156000133564589919546802616566101790699e-38L, }, - { 0.2938735877055718769921841343055614194547e-38L, 0.2938735877055718769921841343055614194542e-38L, 0.2938735877055718769921841343055614194551e-38L, }, - { -0.2938735877055718769921841343055614194547e-38L, -0.2938735877055718769921841343055614194551e-38L, -0.2938735877055718769921841343055614194542e-38L, }, -}; +// This file tests the functions log1p and expm1. The accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", // test function + 4, // Max Peek error + 3); // Max mean error + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} template -void test(T) +void do_test(const T& data, const char* type_name, const char* test_name) { - static const T two = 2; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = &boost::math::log1p; +#else + pg funcp = &boost::math::log1p; +#endif + + boost::math::tools::test_result result; + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; // - // Fudge factor is 3 epsilon, should be less than that, but some - // platforms have poor long double support (some of the test values - // actually test the accuracy of std::log and std::exp, and it's - // usually these, rather than our series expansion that fails otherwise). - // - static const T factor = std::pow(two, 1-std::numeric_limits::digits) * 300; - for(unsigned i = 0; i < sizeof(data)/sizeof(data[0]); ++i) - { - T input_value = static_cast(data[i][0]); - T expected_log1p = static_cast(data[i][1]); - T expected_expm1 = static_cast(data[i][2]); - BOOST_CHECK_CLOSE(boost::math::log1p(input_value), expected_log1p, factor); - BOOST_CHECK_CLOSE(boost::math::expm1(input_value), expected_expm1, factor); - BOOST_CHECK_CLOSE(boost::math::expm1(-input_value), -boost::math::expm1(input_value)/static_cast(std::exp(input_value)), 2*factor); - } + // test log1p against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::log1p; +#else + funcp = &boost::math::log1p; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::log1p", "log1p and expm1"); + std::cout << std::endl; + // + // test expm1 against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::expm1; +#else + funcp = boost::math::expm1; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::expm1", "log1p and expm1"); + std::cout << std::endl; +} + +template +void test(T, const char* type_name) +{ +# include "log1p_expm1_data.ipp" + + do_test(log1p_expm1_data, type_name, "expm1 and log1p"); + // // C99 Appendix F special cases: static const T zero = 0; @@ -571,18 +134,20 @@ void test(T) int test_main(int, char* []) { - std::cout << "Running float tests" << std::endl; - test(float(0)); - std::cout << "Running double tests" << std::endl; - test(double(0)); + expected_results(); + BOOST_MATH_CONTROL_FP; + test(float(0), "float"); + test(double(0), "double"); // // The long double version of these tests fails on some platforms // due to poor std lib support (not enough digits returned from // std::log and std::exp): // -#if !defined(__CYGWIN__) && !defined(__FreeBSD__) && !(defined(__GNUC__) && defined(__sun)) - std::cout << "Running long double tests" << std::endl; - test((long double)(0)); +#if !defined(BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS) + test((long double)(0), "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test((boost::math::concepts::real_concept)(0), "real_concept"); +#endif #else std::cout << "The long double tests have been disabled on this platform " "either because the long double overloads of the usual math functions are " diff --git a/test/negative_binomial_quantile.ipp b/test/negative_binomial_quantile.ipp new file mode 100644 index 000000000..e20981f94 --- /dev/null +++ b/test/negative_binomial_quantile.ipp @@ -0,0 +1,802 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 792> negative_binomial_quantile_data = {{ + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.097540400922298431396484375), SC_(11.568381290037563253305975817351444024377036234904), SC_(49.67581419477884086070549390307050513757197652133) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.12698681652545928955078125), SC_(12.977136041273636067294825573363051160267257422387), SC_(46.43808301937089644496373095130068129633820080023) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.135477006435394287109375), SC_(13.355446548799362499196093574014767375702882517351), SC_(45.626103029359612367556766004407934151298781150048) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.188381969928741455078125), SC_(15.52940284708738793480132106889874510563268887437), SC_(41.360131052208165673316312211602173007116266843573) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.22103404998779296875), SC_(16.764430009390827897971671993143269955166674809408), SC_(39.199062325902508267047473072514261193252696080063) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.278498232364654541015625), SC_(18.827330684864743271502385688632742201801058689866), SC_(35.937947666898633946380128330076651597919337173838) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.308167040348052978515625), SC_(19.859834305054294539273728119716020535354199181236), SC_(34.447725391983382465551236140743414030393753548228) }, + { SC_(4.285762786865234375), SC_(0.12698681652545928955078125), SC_(0.546881496906280517578125), SC_(28.286804773212901440345106101213208895684167630136), SC_(24.86330569705991784268291712045956516535854891059) }, + { 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SC_(0.905791938304901123046875), SC_(0.12698681652545928955078125), SC_(21.489496587266890076517016454951601933868240710191), SC_(34.21419151204538642582501073593898455181225393677) }, + { SC_(272.01507568359375), SC_(0.905791938304901123046875), SC_(0.135477006435394287109375), SC_(21.693684376277839805051109736192556966243931278661), SC_(33.9738887895006144506312349546109034434543780342) }, + { SC_(272.01507568359375), SC_(0.905791938304901123046875), SC_(0.814723670482635498046875), SC_(32.746018517690654246567264816843784711545696390601), SC_(22.755946481269646466611582848763719170337310131891) }, + { SC_(272.01507568359375), SC_(0.905791938304901123046875), SC_(0.835008561611175537109375), SC_(33.214512480133603302329410116305283062246842134753), SC_(22.346866227457822973178099650153865406868569125644) }, + { SC_(272.01507568359375), SC_(0.905791938304901123046875), SC_(0.905791938304901123046875), SC_(35.273093826701032451443765605959004625391699421517), SC_(20.603814129178545613790080798274981448289703629005) }, + { SC_(272.01507568359375), SC_(0.905791938304901123046875), SC_(0.968867778778076171875), SC_(38.682163065413445271801733134293205558716062791966), SC_(17.901424501093686966740632922099917275901391863203) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.12698681652545928955078125), SC_(4.8860982307362946848490148677021209303963709988433), SC_(11.700973803207663097096356923398241331317568902751) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.135477006435394287109375), SC_(4.9887446454577528986006399833823374787746289587588), SC_(11.566258084594318426060043263571263743668198571926) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.814723670482635498046875), SC_(10.880759501342490984683807982974895981350858245686), SC_(5.5273280527774480805413749184523466447940455089765) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.835008561611175537109375), SC_(11.141738049641109754624772567404173364587779651847), SC_(5.3190303970916925333893252428450932703036838979127) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.905791938304901123046875), SC_(12.296689997181380679405006454997551301575272020897), SC_(4.4443542423997783257006693663771333961825154928459) }, + { SC_(272.01507568359375), SC_(0.968867778778076171875), SC_(0.968867778778076171875), SC_(14.235964125170580443450965323386985634646137027791), SC_(3.1360623190240134481707711472606441020054701402056) } + }}; +#undef SC_ + diff --git a/test/poisson_quantile.ipp b/test/poisson_quantile.ipp new file mode 100644 index 000000000..4ed508817 --- /dev/null +++ b/test/poisson_quantile.ipp @@ -0,0 +1,632 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 619> poisson_quantile_data = {{ + { SC_(2.539736270904541015625), SC_(0.097540400922298431396484375), SC_(0.1236392659323415267286721455855935332272165776019), SC_(4.1794244675777288954971650240102219690023733491107) }, + { SC_(2.539736270904541015625), SC_(0.12698681652545928955078125), SC_(0.29692152360861802647109703200927257466043783201072), SC_(3.8759279753758017243263560996428606489658831749751) }, + { SC_(2.539736270904541015625), SC_(0.135477006435394287109375), SC_(0.34320650665410472759846385302850183249926507900385), SC_(3.7989374337321363992500232228194835442428384328783) }, + { SC_(2.539736270904541015625), SC_(0.188381969928741455078125), SC_(0.60664513287123148932168996645967883616307670876159), SC_(3.3882631942049236108254497545125740345098947059879) }, + { SC_(2.539736270904541015625), SC_(0.22103404998779296875), SC_(0.75418046769318689024626280510389922651894870065918), SC_(3.1760120980068421525868462530802458163753626136545) }, + { SC_(2.539736270904541015625), SC_(0.278498232364654541015625), SC_(0.99696859994338450962602588461832487680146529935052), SC_(2.8498938646809092776273924660633540719728165148386) }, + { SC_(2.539736270904541015625), SC_(0.308167040348052978515625), SC_(1.1167608396721874815750956320715232750965151590926), SC_(2.6983883556775732472454879863323713890715761900718) }, + { SC_(2.539736270904541015625), SC_(0.546881496906280517578125), SC_(2.0537693974299930086549193836285319461125762366927), SC_(1.6814731149266990335114284112670893662702121211471) }, + { SC_(2.539736270904541015625), SC_(0.54722058773040771484375), SC_(2.0551559213142958322292400766992089704970302823545), SC_(1.6801546387421272943620405150189924628945228748043) }, + { SC_(2.539736270904541015625), SC_(0.6323592662811279296875), SC_(2.4185680589969635905362050596047384508137625647103), SC_(1.3504767355410436804577399289250217569583832425216) }, + { SC_(2.539736270904541015625), SC_(0.814723670482635498046875), SC_(3.40979576665416423967927240319519423210777415296), SC_(0.59212166896088946445227134508857054551303877571874) }, + { SC_(2.539736270904541015625), SC_(0.835008561611175537109375), SC_(3.5574160837993229129674267171557135101187245049205), SC_(0.49470169893324134464824483154133205955707995130386) }, + { SC_(2.539736270904541015625), SC_(0.905791938304901123046875), SC_(4.2182796507799899023660816530562600398630233374474), SC_(0.1024923825265003493532574009232399351614327762936) }, + { SC_(2.539736270904541015625), SC_(0.9133758544921875), SC_(4.3110965218524397562863677756993861813110388506657), SC_(0.052913915876507499928705143631874173069654329996695) }, + { SC_(2.539736270904541015625), SC_(0.957506835460662841796875), SC_(5.0507514608729399450273591019931952059180132941558), SC_(0) }, + { SC_(2.539736270904541015625), SC_(0.964888513088226318359375), SC_(5.2367753540079862352449652338334329394906217447274), SC_(0) }, + { SC_(2.539736270904541015625), SC_(0.967694938182830810546875), SC_(5.3166351261369765913994722232174376733439995923696), SC_(0) }, + { SC_(2.539736270904541015625), SC_(0.968867778778076171875), SC_(5.3518355852884420289078324978224092379400640232473), SC_(0) }, + { SC_(2.539736270904541015625), SC_(0.992881298065185546875), SC_(6.6543700270027625401829056826390754932290120871122), SC_(0) }, + { SC_(2.539736270904541015625), SC_(0.996461331844329833984375), SC_(7.2179591549437624231339827362963118968410984819945), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.097540400922298431396484375), SC_(0.22508778216127669860840111850046509049124774877757), SC_(4.4182474958341612963906186215326419021373601918705) }, + { SC_(2.7095401287078857421875), SC_(0.12698681652545928955078125), SC_(0.40677008040551596304058142704984892724698474450013), SC_(4.1064853787719233590964661701263902837818258099719) }, + { SC_(2.7095401287078857421875), SC_(0.135477006435394287109375), SC_(0.45520478103550794436443361181077958307722151443163), SC_(4.0273711840535234079944645222273276521046853333475) }, + { SC_(2.7095401287078857421875), SC_(0.188381969928741455078125), SC_(0.73026636637142655390107377497566857551034902573012), SC_(3.6051688857008976555752955024438980784366442932919) }, + { SC_(2.7095401287078857421875), SC_(0.22103404998779296875), SC_(0.88392938865124774766856751848756090706730082877896), SC_(3.3868160972875027862660476930182731527926765710517) }, + { SC_(2.7095401287078857421875), SC_(0.278498232364654541015625), SC_(1.1363266606638255225600870950818167357150551989263), SC_(3.0511081900439111988365299857382083529598127860583) }, + { SC_(2.7095401287078857421875), SC_(0.308167040348052978515625), SC_(1.2606742963136106218231304529313368491205193889367), SC_(2.8950505886875929318857627817878016223780947076715) }, + { SC_(2.7095401287078857421875), SC_(0.546881496906280517578125), SC_(2.2302498177659995331854128657702572247393894330154), SC_(1.8455667973744983745400534580442630696797091911164) }, + { SC_(2.7095401287078857421875), SC_(0.54722058773040771484375), SC_(2.2316813572100748575831704135004677863833315667374), SC_(1.8442033129697692145260467544769069766977784230154) }, + { SC_(2.7095401287078857421875), SC_(0.6323592662811279296875), SC_(2.6066425574726804797128859441380116059728489734554), SC_(1.5029787332697810770934731568961824651116215294506) }, + { SC_(2.7095401287078857421875), SC_(0.814723670482635498046875), SC_(3.6273147425851163791281437708804229061207906824314), SC_(0.71512622824161600897697849329545315475907647131057) }, + { SC_(2.7095401287078857421875), SC_(0.835008561611175537109375), SC_(3.7791124414856923080281181724476436460102273386686), SC_(0.6135013062126620064849645584217989596131282386535) }, + { SC_(2.7095401287078857421875), SC_(0.905791938304901123046875), SC_(4.458149225470369081990154475805206149000574987061), SC_(0.20287311758458663949375602873316329115587280439004) }, + { SC_(2.7095401287078857421875), SC_(0.9133758544921875), SC_(4.5534558318934753192152596811095152683421397095551), SC_(0.15075014180575346658654817591045191201000083591655) }, + { SC_(2.7095401287078857421875), SC_(0.957506835460662841796875), SC_(5.3124762617564144681154664570520183781853730044827), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.964888513088226318359375), SC_(5.5032495514669336956769948581208817037416052591602), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.967694938182830810546875), SC_(5.5851345768544450002123999306065943091761910601596), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.968867778778076171875), SC_(5.6212251836473331425223280696624241368486034858104), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.992881298065185546875), SC_(6.9557169597554463521475489005690610777985653849856), SC_(0) }, + { SC_(2.7095401287078857421875), SC_(0.996461331844329833984375), SC_(7.5326167149449785832226654644205494408676203885886), SC_(0) }, + { SC_(4.420680999755859375), SC_(0.097540400922298431396484375), SC_(1.3396974430145122585454495048468424419264320352618), SC_(6.7283438763148574872831791875332553675646602300523) }, + { SC_(4.420680999755859375), SC_(0.12698681652545928955078125), SC_(1.5943593540803988177742830666052464453414946104313), SC_(6.3448231261522649376887252175402416326077522703472) }, + { SC_(4.420680999755859375), SC_(0.135477006435394287109375), SC_(1.6614817168246804493298951704417179835901855738303), SC_(6.2472654728602887897097583831549812995898408080858) }, + { SC_(4.420680999755859375), SC_(0.188381969928741455078125), SC_(2.0376543751759201428527313894139151409067294327461), SC_(5.7248971087658352374460042773698359549632379123834) }, + { SC_(4.420680999755859375), SC_(0.22103404998779296875), SC_(2.2446654580224159831483586795367113208776510965331), SC_(5.4534959083816989610894577161849640688969105560161) }, + { SC_(4.420680999755859375), SC_(0.278498232364654541015625), SC_(2.5807621352225720613600353790736865733555987348493), SC_(5.0343733155738426061839689866661543927206541851375) }, + { SC_(4.420680999755859375), SC_(0.308167040348052978515625), SC_(2.7448014090272585998043938931292051157965186772885), SC_(4.8387030783940425933647257151827080974800390869268) }, + { SC_(4.420680999755859375), SC_(0.546881496906280517578125), SC_(3.9981671065238395569228616860453130593511301850379), SC_(3.505557886377919314863622414565557140946176046497) }, + { SC_(4.420680999755859375), SC_(0.54722058773040771484375), SC_(3.9999907952112765051305384945750961898928212599337), SC_(3.5038021853580386246245849220792764933302027406719) }, + { SC_(4.420680999755859375), SC_(0.6323592662811279296875), SC_(4.4755370662322077623602403442109487298546581878422), SC_(3.0619503042413349619855086785373188956985791271542) }, + { SC_(4.420680999755859375), SC_(0.814723670482635498046875), SC_(5.7523737088686858401558160403666922549379961143882), SC_(2.0171468551181771134675292909683635311884497032409) }, + { SC_(4.420680999755859375), SC_(0.835008561611175537109375), SC_(5.9404758600780670389419577812214780175653318248418), SC_(1.8789308781841440866768183847400702725764547783367) }, + { SC_(4.420680999755859375), SC_(0.905791938304901123046875), SC_(6.7773282414671410921194089695719498332958307210814), SC_(1.3082106170807083740259305077843340602783571584871) }, + { SC_(4.420680999755859375), SC_(0.9133758544921875), SC_(6.8942396868242401403890538418965641039295675778748), SC_(1.2339961120705196611183391781202394749539268956262) }, + { SC_(4.420680999755859375), SC_(0.957506835460662841796875), SC_(7.821171397040121531326100235432854843584195583282), SC_(0.69094012860886707349500955292069725225637764995336) }, + { SC_(4.420680999755859375), SC_(0.964888513088226318359375), SC_(8.0530885584480679142446108828803234017581269318602), SC_(0.56731808338239770597650769892754973604608332843093) }, + { SC_(4.420680999755859375), SC_(0.967694938182830810546875), SC_(8.1525139180463156301698889503244013413378449293097), SC_(0.51578894370664249604079957647423257412791686824297) }, + { SC_(4.420680999755859375), SC_(0.968867778778076171875), SC_(8.1963130987087259540234241916685414899812024476517), SC_(0.49336751814322145986719340359087631907186828353415) }, + { SC_(4.420680999755859375), SC_(0.992881298065185546875), SC_(9.8072204903950038451972661576396667151044904421196), SC_(0) }, + { SC_(4.420680999755859375), SC_(0.996461331844329833984375), SC_(10.499064126433495605510366708137413205875937581042), SC_(0) }, + { SC_(6.16334056854248046875), SC_(0.097540400922298431396484375), SC_(2.5859664957940427509461874565129963589589688820455), SC_(8.9677396315283840656695133479880083513730127930784) }, + { SC_(6.16334056854248046875), SC_(0.12698681652545928955078125), SC_(2.9009052707869816176646796920701448375095847243642), 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SC_(21259.335818259699059193512358302672174062051718198) }, + { SC_(21533.0078125), SC_(0.992881298065185546875), SC_(21893.035484643765250702345204307305220150252817399), SC_(21173.649644247497441133933107438101115419091883006) }, + { SC_(21533.0078125), SC_(0.996461331844329833984375), SC_(21928.748866501301794010747676997623434830926924354), SC_(21138.35119254861052629794920186014308896076971475) }, + { SC_(36064.203125), SC_(0.097540400922298431396484375), SC_(35817.756768465760157590621734165018504764822234946), SC_(36309.87575645560720888470897187326305111307971067) }, + { SC_(36064.203125), SC_(0.12698681652545928955078125), SC_(35847.118451935014429046646043595383698848956280643), SC_(36280.388235416270565451383221510177119530752580683) }, + { SC_(36064.203125), SC_(0.135477006435394287109375), SC_(35854.677469154058812990446398308816060488884214194), SC_(36272.799417582977292393610184889120531754662042693) }, + { SC_(36064.203125), SC_(0.188381969928741455078125), SC_(35895.813963688688899347185287707174915963110925951), SC_(36231.519363785909686114717073499040831813259848196) }, + { SC_(36064.203125), SC_(0.22103404998779296875), SC_(35917.653456211419739834853063677815640317758960528), SC_(36209.61642914766359611814946788126355540985450165) }, + { SC_(36064.203125), SC_(0.278498232364654541015625), SC_(35952.060752264711660225756482245675795827380357506), SC_(36175.127140627721533908135757767552638737317829985) }, + { SC_(36064.203125), SC_(0.308167040348052978515625), SC_(35968.425594177917174114988015625463665216529807154), SC_(36158.731006107268024793529385791325197439496378759) }, + { SC_(36064.203125), SC_(0.546881496906280517578125), SC_(36085.90704613659583139939209269495550265372509376), SC_(36041.170493976707019744110284599564890573120559004) }, + { SC_(36064.203125), SC_(0.54722058773040771484375), SC_(36086.069626640932451371680215150290901608708630273), SC_(36041.007980928233195582697368438443122786769469145) }, + { SC_(36064.203125), SC_(0.6323592662811279296875), SC_(36127.764204300101112509183105856533197297089468966), SC_(35999.346817101912053061808800875199004294182125904) }, + { SC_(36064.203125), SC_(0.814723670482635498046875), SC_(36233.718875717808828467602793313871953177027240436), SC_(35893.621310482160314578578260464699451147711607827) }, + { SC_(36064.203125), SC_(0.835008561611175537109375), SC_(36248.690808962376388084529607619710721880422487915), SC_(35878.698428929463581276848946039910866441775986925) }, + { SC_(36064.203125), SC_(0.905791938304901123046875), SC_(36313.603520004301043829040859165025143758574372428), SC_(35814.04605003764537924388506916510248513184115016) }, + { SC_(36064.203125), SC_(0.9133758544921875), SC_(36322.466512379710977125706991116296873669955779604), SC_(35805.224607785412252967347192248072950856592829061) }, + { SC_(36064.203125), SC_(0.957506835460662841796875), SC_(36391.135299980195669560575736147367436281777633283), SC_(35736.926574297038969011121854824598284630491333953) }, + { SC_(36064.203125), SC_(0.964888513088226318359375), SC_(36407.900717612615585771720933010639586549398001689), SC_(35720.264801955017393133538609471435523920615869955) }, + { SC_(36064.203125), SC_(0.967694938182830810546875), SC_(36415.040536152281865027842381342733053433785349414), SC_(35713.170685191394470780844154374449975091285878881) }, + { SC_(36064.203125), SC_(0.968867778778076171875), SC_(36418.176877787056615649047742815339774220928179991), SC_(35710.054714220975799133006437370089582352859367611) }, + { SC_(36064.203125), SC_(0.992881298065185546875), SC_(36530.036726520261049674022412341545736190010493948), SC_(35599.039021162752923959319339473143116039188421182) }, + { SC_(36064.203125), SC_(0.996461331844329833984375), SC_(36576.19456101664404260142699288335175602354959484), SC_(35553.29611413405182964733910119676514324895954306) } + }}; +#undef SC_ + diff --git a/test/powm1_sqrtp1m1_test.cpp b/test/powm1_sqrtp1m1_test.cpp new file mode 100644 index 000000000..14fe5bb35 --- /dev/null +++ b/test/powm1_sqrtp1m1_test.cpp @@ -0,0 +1,1658 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the functions powm1 and sqrt1pm1. +// The accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", 5, 2); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void test_powm1_sqrtp1m1(T, const char* type_name) +{ +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 141> sqrtp1m1_data = { + SC_(-0.990433037281036376953125), SC_(-0.902189148255607021082179865003660033379), + SC_(-0.928576648235321044921875), SC_(-0.7327485233629588435419837213946140663952), + SC_(-0.804919183254241943359375), SC_(-0.5583204592175928547330219789723812512248), + SC_(-0.780276477336883544921875), SC_(-0.5312532425038905348288090678272629719499), + SC_(-0.775070965290069580078125), SC_(-0.525733160857803152349794525146692520785), + SC_(-0.74602639675140380859375), SC_(-0.4960420620244223284705423670972348730775), + SC_(-0.72904598712921142578125), SC_(-0.4794675678972648545670296583048773096015), + SC_(-0.7162272930145263671875), SC_(-0.4672967927771847619430106324762494277417), + SC_(-0.68477380275726318359375), SC_(-0.4385499156267435234127335149836228483716), + SC_(-0.62323606014251708984375), SC_(-0.3861890031471553130754715740017444288273), + SC_(-0.576151371002197265625), SC_(-0.3489634196162225368137418186622322770468), + SC_(-0.5579319000244140625), SC_(-0.3351179804088653673227401220715512851566), + SC_(-0.44300353527069091796875), SC_(-0.2536780421766293214167684432407843582845), + SC_(-0.40594112873077392578125), SC_(-0.2292478535422518369418742953348121057781), + SC_(-0.38366591930389404296875), SC_(-0.2149305249240001453212315877592832894955), + SC_(-0.277411997318267822265625), SC_(-0.1499482352928545459530860365223688354153), + SC_(-0.202522933483123779296875), SC_(-0.1069842854031759981174720343292165991554), + SC_(-0.156477451324462890625), SC_(-0.08156516362044655434273010203540784115386), + SC_(-0.029248714447021484375), SC_(-0.01473288619127324715091701268111937769847), + SC_(0.1431564604442703013402649929484277542953e-29), SC_(0.7157823022213515067013249647418825993242e-30), + SC_(0.1791466932348087634896446282571611213266e-29), SC_(0.8957334661740438174482231412854044374117e-30), + SC_(0.6013619202535540063110633226832922483532e-29), SC_(0.3006809601267770031555316613411940789777e-29), + SC_(0.115805324961653822428570241697281798758e-28), SC_(0.5790266248082691121428512084847326346288e-29), + SC_(0.1422457400834001098175711728787848259007e-28), SC_(0.711228700417000549087855864391394898182e-29), + SC_(0.4970121018327539153628705477876439795096e-28), SC_(0.2485060509163769576814352738907342268877e-28), + SC_(0.9660079415057497591758174164417478444323e-28), SC_(0.483003970752874879587908708209209280428e-28), + SC_(0.1232929313253182131376331095427391968754e-27), SC_(0.6164646566265910656881655476946945507336e-28), + SC_(0.3296523285617759312781860549364832953326e-27), SC_(0.1648261642808879656390930274546578154505e-27), + SC_(0.528364435768055252017009628713605422886e-27), SC_(0.26418221788402762600850481432190658932e-27), + SC_(0.886586057273120049620324386849842094685e-27), SC_(0.4432930286365600248101621933266666927236e-27), + SC_(0.2499669674831043259218157022821422146034e-26), SC_(0.1249834837415521629609078510629667512608e-26), + SC_(0.4131050397232622964314362671638736040881e-26), SC_(0.2065525198616311482157181333686170847381e-26), + SC_(0.7679738097881433551381658732998641759182e-26), SC_(0.3839869048940716775690829359127023723085e-26), + SC_(0.199929739820949207249437007767740538737e-25), SC_(0.9996486991047460362471850338422150855758e-26), + SC_(0.5151477415246978459754129800826163591626e-25), SC_(0.2575738707623489229877064867240932346063e-25), + SC_(0.101200734533556026342258477595279955025e-24), SC_(0.5060036726677801317112923751744139374608e-25), + SC_(0.2064292695896540981798546456623054911033e-24), SC_(0.1032146347948270490899273175045223276369e-24), + SC_(0.4063294332896333395257434433879773416284e-24), SC_(0.2031647166448166697628717010560376261299e-24), + SC_(0.8138195767936862452966745688936976428456e-24), SC_(0.406909788396843122648337201659060874841e-24), + SC_(0.9575550627132253801929510132578249716542e-24), SC_(0.478777531356612690096475392014950219861e-24), + SC_(0.2855160956298500804375620841706273850616e-23), SC_(0.1427580478149250402187809401860126128887e-23), + SC_(0.65201444297915461398563707001320281266e-23), SC_(0.3260072214895773069928180036030590895584e-23), + SC_(0.1310988374636350038320977491775043421995e-22), SC_(0.6554941873181750191604865975243736714241e-23), + SC_(0.2590288837798696209228010176465529547374e-22), SC_(0.1295144418899348104613996701237435743037e-22), + SC_(0.2937779542193655202274099291941187976629e-22), SC_(0.1468889771096827601137038857784795823862e-22), + SC_(0.7863513178004503049754083414326234074965e-22), SC_(0.3931756589002251524876964413613741224383e-22), + SC_(0.1903818607087388763706780167350761726053e-21), SC_(0.9519093035436943818533447771092722109619e-22), + SC_(0.3812242142377350870566942975497647799754e-21), SC_(0.1906121071188675435283289822871922426703e-21), + SC_(0.5493133580141330277178034419485741501887e-21), SC_(0.2746566790070665138588640028286254797086e-21), + SC_(0.9672153634284186955666772243312215295852e-21), SC_(0.4836076817142093477832216739707042687854e-21), + SC_(0.1702169477623814384559878647986894129041e-20), SC_(0.8510847388119071922795771513771277983779e-21), + SC_(0.4817114569977399785676754474621208412799e-20), SC_(0.2408557284988699892835476663213068137742e-20), + SC_(0.7538352992756463183303278501219690799218e-20), SC_(0.3769176496378231591644535904879420358976e-20), + SC_(0.2596305715949999708394617609422128090557e-19), SC_(0.1298152857974999854188882800497720744566e-19), + SC_(0.4444587480324321591032923385589104015025e-19), SC_(0.2222293740162160795491768745456732380306e-19), + SC_(0.9715574921498573937069095571295029856174e-19), SC_(0.4857787460749286968416557290578449901693e-19), + SC_(0.2036598542733453787268262970278076551267e-18), SC_(0.1018299271366726893582284814835737930738e-18), + SC_(0.4248971931658660264162106698360155121463e-18), SC_(0.2124485965829330131855381318229788476899e-18), + SC_(0.6521097487613458963613731825259556273977e-18), SC_(0.3260548743806729481275307007092796054895e-18), + SC_(0.1436126164096190058281493628911107407475e-17), SC_(0.718063082048095028882939519555349077068e-18), + SC_(0.3118908901459261162419055180006211003274e-17), SC_(0.1559454450729630579993578498052878596544e-17), + SC_(0.3593346613595175715618300349429858897565e-17), SC_(0.1796673306797587856195132689035439820049e-17), + SC_(0.9445874854124767215374919304693435151421e-17), SC_(0.4722937427062383596534390682373393588305e-17), + SC_(0.2566182432094081539023303073498993853718e-16), SC_(0.1283091216047040761280036193264127770924e-16), + SC_(0.3363765695149349330660137891158001366421e-16), SC_(0.1681882847574674651186419380749519252198e-16), + SC_(0.1073581901339262605326457800103412409953e-15), SC_(0.53679095066963128825600266401137964529e-16), + SC_(0.186668406231853462907965823802669547149e-15), SC_(0.9333420311592672709834617625880158673234e-16), + SC_(0.3727540802657755688795382376099496468669e-15), SC_(0.1863770401328877670715685744569461355954e-15), + SC_(0.6211646767866855090717281839829411183018e-15), SC_(0.3105823383933427063051696310530511222847e-15), + SC_(0.1561186859754253464932505224282976996619e-14), SC_(0.7805934298771264278032012284733416682125e-15), + SC_(0.3092010764722992466335682593125966377556e-14), SC_(0.1546005382361495038101520151206746652194e-14), + SC_(0.6192850577371690132255643845837767003104e-14), SC_(0.3096425288685840272203037716324248888074e-14), + SC_(0.1047879028014987723427253740737796761096e-13), SC_(0.5239395140074924891505551783318071383455e-14), + SC_(0.1978473638988408750405412206418986897916e-13), SC_(0.989236819494199482255280888212449451182e-14), + SC_(0.4041816252346730475863978426787070930004e-13), SC_(0.2020908126173344817583717046094650765545e-13), + SC_(0.9410302262901834580155480125540634617209e-13), SC_(0.4705151131450806597841891098742774312111e-13), + SC_(0.1334530223958893535574077304772799834609e-12), SC_(0.6672651119794245056505554066901540498883e-13), + SC_(0.266297021326439287136622624529991298914e-12), SC_(0.1331485106632107793053653966876952769505e-12), + SC_(0.5920415525016708979677559909760020673275e-12), SC_(0.29602077625079163483389193486561174812e-12), + SC_(0.155163989296047688526414276566356420517e-11), SC_(0.7758199464799374943373933162927267207972e-12), + SC_(0.326923297461201300961874949280172586441e-11), SC_(0.1634616487304670519279090616362400067267e-11), + SC_(0.3753785910581841633870681107509881258011e-11), SC_(0.1876892955289159453352533516401529801122e-11), + SC_(0.9579165585749116473834874341264367103577e-11), SC_(0.4789582792863088185252592080679746767978e-11), + SC_(0.1858167439361402273334533674642443656921e-10), SC_(0.9290837196763851538764282981715353513818e-11), + SC_(0.5449485307451595872407779097557067871094e-10), SC_(0.2724742653688676823559737485578304836506e-10), + SC_(0.6089519166696533147842274047434329986572e-10), SC_(0.3044759583301913769320592805847923970217e-10), + SC_(0.1337744776064297980155970435589551925659e-9), SC_(0.6688723880097794765058899684143827289977e-10), + SC_(0.2554458866654840676346793770790100097656e-9), SC_(0.1277229433245854586915920539477320784002e-9), + SC_(0.9285605062636648199259070679545402526855e-9), SC_(0.4642802530240543332889135775318577029962e-9), + SC_(0.1698227447555211711005540564656257629395e-8), SC_(0.8491137234171087978551371134983090509552e-9), + SC_(0.339355921141759608872234821319580078125e-8), SC_(0.1696779604269267531629088042873986404561e-8), + SC_(0.6313728651008432279922999441623687744141e-8), SC_(0.3156864320521319970871232105329124668304e-8), + SC_(0.8383264749056706932606175541877746582031e-8), SC_(0.4191632365743462521529019590756957227221e-8), + SC_(0.1962631124285962869180366396903991699219e-7), SC_(0.9813155573280803193195787399755096477409e-8), + SC_(0.5256384838503436185419559478759765625e-7), SC_(0.2628192384714742037346966666026188341227e-7), + SC_(0.116242290459922514855861663818359375e-6), SC_(0.5812114354092759417537613189484537667262e-7), + SC_(0.1776920584006802528165280818939208984375e-6), SC_(0.8884602525353202473263890111106254261705e-7), + SC_(0.246631174150024889968335628509521484375e-6), SC_(0.1233155794716463747702014149912298889881e-6), + SC_(0.7932688959044753573834896087646484375e-6), SC_(0.3966343692928262265327186740527956086072e-6), + SC_(0.1372093493046122603118419647216796875e-5), SC_(0.6860465111931535414103361330991297079332e-6), + SC_(0.214747751670074649155139923095703125e-5), SC_(0.1073738181893531617762304794194699568343e-5), + SC_(0.527022712049074470996856689453125e-5), SC_(0.2635110088342783512028120843144783486949e-5), + SC_(0.9233162927557714283466339111328125e-5), SC_(0.461657080741584719962950298693491037714e-5), + SC_(0.269396477960981428623199462890625e-4), SC_(0.1346973318119308516416495011442959614762e-4), + SC_(0.3208058114978484809398651123046875e-4), SC_(0.1604016193149502975581143111396377659995e-4), + SC_(0.00010957030463032424449920654296875), SC_(0.5478365169091582645735757270110530879622e-4), + SC_(0.000126518702018074691295623779296875), SC_(0.6325735026285620681070344590408428362373e-4), + SC_(0.00028976381872780621051788330078125), SC_(0.0001448714155003885621472683658275189243597), + SC_(0.000687857042066752910614013671875), SC_(0.0003438693979519525361574726383087724411954), + SC_(0.00145484809763729572296142578125), SC_(0.0007271596682270997885618336103424497281234), + SC_(0.0073254108428955078125), SC_(0.003656022172385267560713803761753901548099), + SC_(0.09376299381256103515625), SC_(0.04583124537975104196281645139037475336575), + SC_(0.0944411754608154296875), SC_(0.04615542605332570615786127417302399776704), + SC_(0.264718532562255859375), SC_(0.1245970534205822199491885900676517027912), + SC_(0.27952671051025390625), SC_(0.1311616641799057655376985599143742875242), + SC_(0.31148135662078857421875), SC_(0.1451992650280511659331021017070743457453), + SC_(0.3574702739715576171875), SC_(0.1651052630434546351309425398611544399279), + SC_(0.362719058990478515625), SC_(0.1673555837834839069815597770182209283592), + SC_(0.45167791843414306640625), SC_(0.2048559741455171397876451754340651729329), + SC_(0.58441460132598876953125), SC_(0.2587353182166570039327003862804562417757), + SC_(0.59585726261138916015625), SC_(0.263272441958340655449252054139087309968), + SC_(0.5962116718292236328125), SC_(0.2634127084326893190571263246997038824839), + SC_(0.6005609035491943359375), SC_(0.265132761234643838648662687260938123223), + SC_(0.62944734096527099609375), SC_(0.2764980771490691893425893413189776581002), + SC_(0.67001712322235107421875), SC_(0.2922914234886615056408835783570718269519), + SC_(0.6982586383819580078125), SC_(0.3031725282486421443925178739003813209589), + SC_(0.75686132907867431640625), SC_(0.3254664571684469135448954017993676523574), + SC_(0.81158387660980224609375), SC_(0.3459509190939327287182180832834475919551), + SC_(0.826751708984375), SC_(0.3515737896927326105907938446356122609441), + SC_(0.83147108554840087890625), SC_(0.3533185454830658156956304096558121255029), + SC_(0.8679864406585693359375), SC_(0.3667430046129994134441282712127036765511), + SC_(0.91433393955230712890625), SC_(0.3835945719582406373102286771719657103388), + SC_(0.91501367092132568359375), SC_(0.3838401898056457601350544781643172326518), + SC_(0.918984889984130859375), SC_(0.3852743013512272831269535677278325535267), + SC_(0.92977702617645263671875), SC_(0.3891641465919182859360893492188078577616), + SC_(0.93538987636566162109375), SC_(0.3911829054317989358513931994433199352401), + SC_(0.93773555755615234375), SC_(0.3920257029078710024155036194055376074007), + SC_(0.94118559360504150390625), SC_(0.3932643660142325996680401116017235829116), + SC_(0.96221935749053955078125), SC_(0.4007924034240546775728864480529865192659), + SC_(0.98576259613037109375), SC_(0.4091708896121758495875515950932191750449), + SC_(0.99292266368865966796875), SC_(0.4117091285702801135545007937655927942821), + }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1400> powm1_data = { + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1431564604442703013402649929484277542953e-29), SC_(-0.4876113153308343652049349438365788782568e-28), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1791466932348087634896446282571611213266e-29), SC_(-0.6101991796549119337733033929476086235147e-28), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6013619202535540063110633226832922483532e-29), SC_(-0.2048324441766037485142714404837079817647e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.115805324961653822428570241697281798758e-28), SC_(-0.3944494481885382670819636026287935618847e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1422457400834001098175711728787848259007e-28), SC_(-0.4845092719324132218645425927278781231393e-27), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4970121018327539153628705477876439795096e-28), SC_(-0.1692894082166527210362866376720838911671e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9660079415057497591758174164417478444323e-28), SC_(-0.3290360780895537432240111567751307160722e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1232929313253182131376331095427391968754e-27), SC_(-0.4199533030361316903341770129910084958631e-26), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3296523285617759312781860549364832953326e-27), SC_(-0.1122842832471787971733666733700489713877e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.528364435768055252017009628713605422886e-27), SC_(-0.1799684601724219422633144864504456459061e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.886586057273120049620324386849842094685e-27), SC_(-0.301983851933262353667369307173490332305e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2499669674831043259218157022821422146034e-26), SC_(-0.851423131205077246153257242575159396693e-25), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4131050397232622964314362671638736040881e-26), SC_(-0.1407094665264327834604146533729082291379e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7679738097881433551381658732998641759182e-26), SC_(-0.261582829282236901328884539002832986633e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.199929739820949207249437007767740538737e-25), SC_(-0.6809891995464351323188379271234568515929e-24), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.5151477415246978459754129800826163591626e-25), SC_(-0.1754666656712664684348626881941041760604e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.101200734533556026342258477595279955025e-24), SC_(-0.34470413088736644015953540038270992073e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2064292695896540981798546456623054911033e-24), SC_(-0.7031275246329503001381414764473488822476e-23), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4063294332896333395257434433879773416284e-24), SC_(-0.1384015983694437753942599100031834846519e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.8138195767936862452966745688936976428456e-24), SC_(-0.2771985511871706510319651593329877641942e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9575550627132253801929510132578249716542e-24), SC_(-0.3261569070527988349552625022548697814845e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2855160956298500804375620841706273850616e-23), SC_(-0.9725085302202836634236273890970136659468e-22), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.65201444297915461398563707001320281266e-23), SC_(-0.2220854156138726384270360939328805807496e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1310988374636350038320977491775043421995e-22), SC_(-0.4465413322989495361133815710858355990476e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2590288837798696209228010176465529547374e-22), SC_(-0.8822893101478278733146155237640398711321e-21), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2937779542193655202274099291941187976629e-22), SC_(-0.1000649598540977920935091326953457942208e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7863513178004503049754083414326234074965e-22), SC_(-0.2678424705352926224139081344679444879469e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1903818607087388763706780167350761726053e-21), SC_(-0.6484677619663470968358777401078973025876e-20), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3812242142377350870566942975497647799754e-21), SC_(-0.1298504028134944266344521819289603363198e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.5493133580141330277178034419485741501887e-21), SC_(-0.1871039617763820057590094637996686252126e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9672153634284186955666772243312215295852e-21), SC_(-0.3294473432116758998094638207256180094583e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1702169477623814384559878647986894129041e-20), SC_(-0.5797831933845967032001409153808505427486e-19), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4817114569977399785676754474621208412799e-20), SC_(-0.1640777904312887822627372613029956055284e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.7538352992756463183303278501219690799218e-20), SC_(-0.2567670510166787584153650238170661635392e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2596305715949999708394617609422128090557e-19), SC_(-0.8843387446340099351857154955251648734277e-18), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4444587480324321591032923385589104015025e-19), SC_(-0.1513889866135372642811595930728137815273e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9715574921498573937069095571295029856174e-19), SC_(-0.3309263341636914772803140564669851761043e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2036598542733453787268262970278076551267e-18), SC_(-0.6936945012060519151079315126759585887524e-17), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.4248971931658660264162106698360155121463e-18), SC_(-0.1447260421199383628860906433743869048343e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6521097487613458963613731825259556273977e-18), SC_(-0.2221178781221441480704197986591689852107e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1436126164096190058281493628911107407475e-17), SC_(-0.4891650475869226303643170206367209825168e-16), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3118908901459261162419055180006211003274e-17), SC_(-0.1062344840825147041561189485134686674784e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3593346613595175715618300349429858897565e-17), SC_(-0.1223945090048411651974452688903537147646e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.9445874854124767215374919304693435151421e-17), SC_(-0.3217399653341720695630524182811530497325e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.2566182432094081539023303073498993853718e-16), SC_(-0.8740783246589107366187111286071499874759e-15), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3363765695149349330660137891158001366421e-16), SC_(-0.114574655589157486912055633512820155344e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1073581901339262605326457800103412409953e-15), SC_(-0.3656773025840533327467103325113205753957e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.186668406231853462907965823802669547149e-15), SC_(-0.6358192065586750730853832514556047648021e-14), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3727540802657755688795382376099496468669e-15), SC_(-0.1269653544166010514919541179082417894329e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.6211646767866855090717281839829411183018e-15), SC_(-0.2115775453968538439661164062909478793222e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.1561186859754253464932505224282976996619e-14), SC_(-0.5317625036268033276607060773206517931366e-13), + SC_(0.161179845478123719842988847972264920827e-14), SC_(0.3092010764722992466335682593125966377556e-14), SC_(-0.1053182951942663933832876385506518241085e-12), + 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SC_(0.9285605062636648199259070679545402526855e-9), SC_(0.2108721880262844458250615511467757397605e-8), + SC_(9.68867778778076171875), SC_(0.1698227447555211711005540564656257629395e-8), SC_(0.3856603156494151031626014622236295405284e-8), + SC_(9.68867778778076171875), SC_(0.339355921141759608872234821319580078125e-8), SC_(0.7706630351863533996193594586001707313967e-8), + SC_(9.68867778778076171875), SC_(0.6313728651008432279922999441623687744141e-8), SC_(0.1433821247406628873576148497830599721119e-7), + SC_(9.68867778778076171875), SC_(0.8383264749056706932606175541877746582031e-8), SC_(0.1903804203900791197963208573178075630015e-7), + SC_(9.68867778778076171875), SC_(0.1962631124285962869180366396903991699219e-7), SC_(0.4457052884132293020120854209156627105415e-7), + SC_(9.68867778778076171875), SC_(0.5256384838503436185419559478759765625e-7), SC_(0.1193702973058086549839229184033913497394e-6), + SC_(9.68867778778076171875), SC_(0.116242290459922514855861663818359375e-6), SC_(0.2639813902774001345761419758813907104525e-6), + SC_(9.68867778778076171875), SC_(0.1776920584006802528165280818939208984375e-6), SC_(0.4035312768290402179724322739059024257911e-6), + SC_(9.68867778778076171875), SC_(0.246631174150024889968335628509521484375e-6), SC_(0.5600891862976372697203768774599773207812e-6), + SC_(9.68867778778076171875), SC_(0.7932688959044753573834896087646484375e-6), SC_(0.1801481940511445476740974446351987487537e-5), + SC_(9.68867778778076171875), SC_(0.1372093493046122603118419647216796875e-5), SC_(0.3115971501912317630070527687604350940031e-5), + SC_(9.68867778778076171875), SC_(0.214747751670074649155139923095703125e-5), SC_(0.487684306379282015062143358653997082528e-5), + SC_(9.68867778778076171875), SC_(0.527022712049074470996856689453125e-5), SC_(0.1196853588100977502339779488819564685125e-4), + SC_(9.68867778778076171875), SC_(0.9233162927557714283466339111328125e-5), SC_(0.2096834472868302910410196802704006676419e-4), + SC_(9.68867778778076171875), SC_(0.269396477960981428623199462890625e-4), SC_(0.6118067920858328215848432186906668843509e-4), + SC_(9.68867778778076171875), SC_(0.3208058114978484809398651123046875e-4), SC_(0.728563051879196577689828782571903259951e-4), + SC_(9.68867778778076171875), SC_(0.00010957030463032424449920654296875), SC_(0.0002488605167115786508690708436538612267867), + SC_(9.68867778778076171875), SC_(0.000126518702018074691295623779296875), SC_(0.0002873599340757661525877158215704810459439), + SC_(9.68867778778076171875), SC_(0.00028976381872780621051788330078125), SC_(0.0006582580089946178862548800972604646079047), + SC_(9.68867778778076171875), SC_(0.000687857042066752910614013671875), SC_(0.001563315133764469862768389959281037577749), + SC_(9.68867778778076171875), SC_(0.00145484809763729572296142578125), SC_(0.003309362765364723055307278727456538237784), + SC_(9.68867778778076171875), SC_(0.002847635187208652496337890625), SC_(0.006487815096079959006202372484557638978379), + SC_(9.68867778778076171875), SC_(0.0056468211114406585693359375), SC_(0.01290626953766988799660445158177424847916), + SC_(9.68867778778076171875), SC_(0.011621631681919097900390625), SC_(0.02674359636941844621991721651837741451166), + SC_(9.68867778778076171875), SC_(0.0257236398756504058837890625), SC_(0.06015731222731434671073651753326632370858), + SC_(9.68867778778076171875), SC_(0.0560617186129093170166015625), SC_(0.1357733748610437010694097215927829841061), + SC_(9.68867778778076171875), SC_(0.106835305690765380859375), SC_(0.2745822634838372037418174975910269667509), + SC_(9.68867778778076171875), SC_(0.2401093542575836181640625), SC_(0.7250883224868308476529921117348408126984), + SC_(9.68867778778076171875), SC_(0.438671648502349853515625), SC_(1.707985158316040663526167707818201486792), + SC_(9.68867778778076171875), SC_(0.903765499591827392578125), SC_(6.786671267376742594913276377075578069747), + }; +#undef SC_ + + + using namespace std; + + typedef T (*func_t)(const T&); + func_t f = &boost::math::sqrt1pm1; + + boost::math::tools::test_result result = boost::math::tools::test( + sqrtp1m1_data, + bind_func(f, 0), + extract_result(1)); + + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n" + "Test results for type " << type_name << std::endl << std::endl; + handle_test_result(result, sqrtp1m1_data[result.worst()], result.worst(), type_name, "boost::math::sqrt1pm1", "sqrt1pm1"); + + typedef T (*func2_t)(T const, T const); + func2_t f2 = &boost::math::powm1; + result = boost::math::tools::test( + powm1_data, + bind_func(f2, 0, 1), + extract_result(2)); + handle_test_result(result, powm1_data[result.worst()], result.worst(), type_name, "boost::math::powm1", "powm1"); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + test_powm1_sqrtp1m1(1.0F, "float"); + test_powm1_sqrtp1m1(1.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_powm1_sqrtp1m1(1.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_powm1_sqrtp1m1(boost::math::concepts::real_concept(), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + diff --git a/test/sinc_test.hpp b/test/sinc_test.hpp new file mode 100644 index 000000000..e2f32c0e6 --- /dev/null +++ b/test/sinc_test.hpp @@ -0,0 +1,92 @@ +// unit test file sinc.hpp for the special functions test suite + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include +#include +#include +#include + + +#include + + +#include + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(sinc_pi_test, T) +{ + using ::std::abs; + + using ::std::numeric_limits; + + using ::boost::math::sinc_pi; + + + BOOST_MESSAGE("Testing sinc_pi in the real domain for " + << string_type_name::_() << "."); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(sinc_pi(static_cast(0))-static_cast(1))) + (numeric_limits::epsilon())); +} + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(sinc_pi_complex_test, T) +{ + using ::std::abs; + using ::std::sinh; + + using ::std::numeric_limits; + + using ::boost::math::sinc_pi; + + + BOOST_MESSAGE("Testing sinc_pi in the complex domain for " + << string_type_name::_() << "."); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(sinc_pi(::std::complex(0, 1))- + ::std::complex(sinh(static_cast(1))))) + (numeric_limits::epsilon())); +} + + +void sinc_pi_manual_check() +{ + using ::boost::math::sinc_pi; + + + BOOST_MESSAGE(" "); + BOOST_MESSAGE("sinc_pi"); + + for (int i = 0; i <= 100; i++) + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << sinc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinc_pi(static_cast(i-50)/ + static_cast(50))); +#else + BOOST_MESSAGE( ::std::setw(15) + << sinc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinc_pi(static_cast(i-50)/ + static_cast(50))); +#endif + } + + BOOST_MESSAGE(" "); +} + + diff --git a/test/sinhc_test.hpp b/test/sinhc_test.hpp new file mode 100644 index 000000000..7b7e62ed5 --- /dev/null +++ b/test/sinhc_test.hpp @@ -0,0 +1,92 @@ +// unit test file sinhc.hpp for the special functions test suite + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include +#include +#include +#include + + +#include + + +#include + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(sinhc_pi_test, T) +{ + using ::std::abs; + + using ::std::numeric_limits; + + using ::boost::math::sinhc_pi; + + + BOOST_MESSAGE("Testing sinhc_pi in the real domain for " + << string_type_name::_() << "."); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(sinhc_pi(static_cast(0))-static_cast(1))) + (numeric_limits::epsilon())); +} + + +BOOST_TEST_CASE_TEMPLATE_FUNCTION(sinhc_pi_complex_test, T) +{ + using ::std::abs; + using ::std::sin; + + using ::std::numeric_limits; + + using ::boost::math::sinhc_pi; + + + BOOST_MESSAGE("Testing sinhc_pi in the complex domain for " + << string_type_name::_() << "."); + + BOOST_CHECK_PREDICATE(::std::less_equal(), + (abs(sinhc_pi(::std::complex(0, 1))- + ::std::complex(sin(static_cast(1))))) + (numeric_limits::epsilon())); +} + + +void sinhc_pi_manual_check() +{ + using ::boost::math::sinhc_pi; + + + BOOST_MESSAGE(" "); + BOOST_MESSAGE("sinc_pi"); + + for (int i = 0; i <= 100; i++) + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_MESSAGE( ::std::setw(15) + << sinhc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinhc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinhc_pi(static_cast(i-50)/ + static_cast(50))); +#else + BOOST_MESSAGE( ::std::setw(15) + << sinhc_pi(static_cast(i-50)/ + static_cast(50)) + << ::std::setw(15) + << sinhc_pi(static_cast(i-50)/ + static_cast(50))); +#endif + } + + BOOST_MESSAGE(" "); +} + + diff --git a/test/special_functions_test.cpp b/test/special_functions_test.cpp new file mode 100644 index 000000000..b6f08edbb --- /dev/null +++ b/test/special_functions_test.cpp @@ -0,0 +1,148 @@ +// test file for special functions. + +// (C) Copyright Hubert Holin 2003. +// Distributed under the Boost Software License, Version 1.0. (See +// accompanying file LICENSE_1_0.txt or copy at +// http://www.boost.org/LICENSE_1_0.txt) + + +#include + + +#include + +#include +#include +#include +#include + + +template +struct string_type_name; + +#define DEFINE_TYPE_NAME(Type) \ +template<> struct string_type_name \ +{ \ + static char const * _() \ + { \ + return #Type; \ + } \ +} + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +DEFINE_TYPE_NAME(float); +DEFINE_TYPE_NAME(double); +DEFINE_TYPE_NAME(long double); + +typedef boost::mpl::list test_types; +#else +DEFINE_TYPE_NAME(float); +DEFINE_TYPE_NAME(double); + +typedef boost::mpl::list test_types; +#endif + +// Apple GCC 4.0 uses the "double double" format for its long double, +// which means that epsilon is VERY small but useless for +// comparisons. So, don't do those comparisons. +#if (defined(__APPLE_CC__) && defined(__GNUC__) && __GNUC__ == 4) || defined(BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS) +typedef boost::mpl::list near_eps_test_types; +#else +typedef boost::mpl::list near_eps_test_types; +#endif + +#include "sinc_test.hpp" +#include "sinhc_test.hpp" +#include "atanh_test.hpp" +#include "asinh_test.hpp" +#include "acosh_test.hpp" + + + +boost::unit_test_framework::test_suite * init_unit_test_suite(int, char *[]) +{ + ::boost::unit_test::unit_test_log. + set_threshold_level(::boost::unit_test::log_messages); + + boost::unit_test_framework::test_suite * test = + BOOST_TEST_SUITE("special_functions_test"); + + BOOST_MESSAGE("Results of special functions test."); + BOOST_MESSAGE(" "); + BOOST_MESSAGE("(C) Copyright Hubert Holin 2003-2005."); + BOOST_MESSAGE("Distributed under the Boost Software License, Version 1.0."); + BOOST_MESSAGE("(See accompanying file LICENSE_1_0.txt or copy at"); + BOOST_MESSAGE("http://www.boost.org/LICENSE_1_0.txt)"); + BOOST_MESSAGE(" "); + +#define BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(fct) \ + test->add(BOOST_TEST_CASE_TEMPLATE(fct##_test, test_types)); + +#define BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR_NEAR_EPS(fct) \ + test->add(BOOST_TEST_CASE_TEMPLATE(fct##_test, near_eps_test_types)); + + +#define BOOST_SPECIAL_FUNCTIONS_COMMON_TEST \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(atanh) \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(asinh) \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(acosh) \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(sinc_pi) \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR(sinhc_pi) + +#define BOOST_SPECIAL_FUNCTIONS_TEMPLATE_TEMPLATE_TEST \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR_NEAR_EPS(sinc_pi_complex) \ + BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR_NEAR_EPS(sinhc_pi_complex) + + +#ifdef BOOST_NO_TEMPLATE_TEMPLATES + +#define BOOST_SPECIAL_FUNCTIONS_TEST \ + BOOST_SPECIAL_FUNCTIONS_COMMON_TEST \ + BOOST_MESSAGE("Warning: no template templates; curtailed functionality."); + +#else /* BOOST_NO_TEMPLATE_TEMPLATES */ + +#define BOOST_SPECIAL_FUNCTIONS_TEST \ + BOOST_SPECIAL_FUNCTIONS_COMMON_TEST \ + BOOST_SPECIAL_FUNCTIONS_TEMPLATE_TEMPLATE_TEST + +#endif /* BOOST_NO_TEMPLATE_TEMPLATES */ + + + BOOST_SPECIAL_FUNCTIONS_TEST + + +#undef BOOST_SPECIAL_FUNCTIONS_TEST + +#undef BOOST_SPECIAL_FUNCTIONS_TEMPLATE_TEMPLATE_TEST + +#undef BOOST_SPECIAL_FUNCTIONS_COMMON_TEST + +#undef BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR + +#undef BOOST_SPECIAL_FUNCTIONS_COMMON_GENERATOR_NEAR_EPS + +#ifdef BOOST_SPECIAL_FUNCTIONS_TEST_VERBOSE + + using ::std::numeric_limits; + + BOOST_MESSAGE("epsilon"); + + BOOST_MESSAGE( ::std::setw(15) << numeric_limits::epsilon() + << ::std::setw(15) << numeric_limits::epsilon() + << ::std::setw(15) << numeric_limits::epsilon()); + + BOOST_MESSAGE(" "); + + test->add(BOOST_TEST_CASE(atanh_manual_check)); + test->add(BOOST_TEST_CASE(asinh_manual_check)); + test->add(BOOST_TEST_CASE(acosh_manual_check)); + test->add(BOOST_TEST_CASE(sinc_pi_manual_check)); + test->add(BOOST_TEST_CASE(sinhc_pi_manual_check)); + +#endif /* BOOST_SPECIAL_FUNCTIONS_TEST_VERBOSE */ + + return test; +} + +#undef DEFINE_TYPE_NAME diff --git a/test/sph_bessel_data.ipp b/test/sph_bessel_data.ipp new file mode 100644 index 000000000..aa38b5617 --- /dev/null +++ b/test/sph_bessel_data.ipp @@ -0,0 +1,494 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 483> sph_bessel_data = { + SC_(0), SC_(0.177219114266335964202880859375e-2), SC_(0.999999476556507842202459130864323898236e0), + SC_(0), SC_(0.22177286446094512939453125e-2), SC_(0.9999991802801447287222232346958222796612e0), + SC_(0), SC_(0.7444499991834163665771484375e-2), SC_(0.9999907632622405689302860688925139982587e0), + SC_(0), SC_(0.1433600485324859619140625e-1), SC_(0.9999657468461303487880990241993035937654e0), + SC_(0), SC_(0.1760916970670223236083984375e-1), SC_(0.9999483203249623334100130061926184665364e0), + SC_(0), SC_(0.6152711808681488037109375e-1), SC_(0.9993691883680863109159389449304869676033e0), + SC_(0), SC_(0.11958599090576171875e0), SC_(0.9976182354925345599905760522457355689246e0), + SC_(0), SC_(0.15262925624847412109375e0), SC_(0.9961219049249727586697484380600284691174e0), + SC_(0), SC_(0.408089816570281982421875e0), SC_(0.9724739915431544258938594874310588064205e0), + SC_(0), SC_(0.6540834903717041015625e0), SC_(0.9302056396192119311058689588352213355057e0), + SC_(0), SC_(0.1097540378570556640625e1), SC_(0.8109851883728203095784514632631378051565e0), + SC_(0), SC_(0.30944411754608154296875e1), SC_(0.1523183208785146731314680065502594134681e-1), + SC_(0), SC_(0.51139926910400390625e1), SC_(-0.1799836939747169709868781821542134553205e0), + SC_(0), SC_(0.95070552825927734375e1), SC_(-0.8644582430311092690468530625290874536115e-2), + SC_(0), SC_(0.24750102996826171875e2), SC_(-0.1508556180252377401284155114952468133194e-1), + SC_(0), SC_(0.637722015380859375e2), SC_(0.1266638994943852616886089579836946490336e-1), + SC_(0), SC_(0.1252804412841796875e3), SC_(-0.2984906844146395453998586259283750185621e-2), + SC_(0), SC_(0.25554705810546875e3), SC_(-0.3447656740458789191073137012008656930551e-2), + SC_(0), SC_(0.503011474609375e3), SC_(0.6940935913226775542129988435446092279729e-3), + SC_(0), SC_(0.10074598388671875e4), SC_(0.8305987986864569002626675437293057248821e-3), + SC_(0), SC_(0.1185395751953125e4), SC_(-0.7167649812470622363222627438778746846582e-3), + SC_(0.1e1), SC_(0.177219114266335964202880859375e-2), SC_(0.5907301953593941259509561856854291087832e-3), + SC_(0.1e1), SC_(0.22177286446094512939453125e-2), SC_(0.7392425179532175006179700888774678679099e-3), + SC_(0.1e1), SC_(0.7444499991834163665771484375e-2), SC_(0.2481486244688331388548761462155911660424e-2), + SC_(0.1e1), SC_(0.1433600485324859619140625e-1), SC_(0.477857007345182708698871081176423742643e-2), + SC_(0.1e1), SC_(0.1760916970670223236083984375e-1), SC_(0.5869541227527534973977936514325779485181e-2), + SC_(0.1e1), SC_(0.6152711808681488037109375e-1), SC_(0.205012765381082643967011963466059108825e-1), + SC_(0.1e1), SC_(0.11958599090576171875e0), SC_(0.3980502019486214219412102921577104486039e-1), + SC_(0.1e1), SC_(0.15262925624847412109375e0), SC_(0.507579971871436936245493977138951657167e-1), + SC_(0.1e1), SC_(0.408089816570281982421875e0), SC_(0.1337779656209597925613024398512918494342e0), + SC_(0.1e1), SC_(0.6540834903717041015625e0), SC_(0.2088414497092880298218980583359870711751e0), + SC_(0.1e1), SC_(0.1097540378570556640625e1), SC_(0.3236312899642309880098636972650663742421e0), + SC_(0.1e1), SC_(0.30944411754608154296875e1), SC_(0.3277232784787348060184709473722850920387e0), + SC_(0.1e1), SC_(0.51139926910400390625e1), SC_(-0.1116307129673207101011381466711802872966e0), + SC_(0.1e1), SC_(0.95070552825927734375e1), SC_(0.1039199329743298950197804325967556415006e0), + SC_(0.1e1), SC_(0.24750102996826171875e2), SC_(-0.3809149370198498667373716310901593032325e-1), + SC_(0.1e1), SC_(0.637722015380859375e2), SC_(-0.9045323488777102113973090754530543316476e-2), + SC_(0.1e1), SC_(0.1252804412841796875e3), SC_(-0.7426806470289446901421322001368677195156e-2), + SC_(0.1e1), SC_(0.25554705810546875e3), SC_(0.183761395344989507935202292272917754448e-2), + SC_(0.1e1), SC_(0.503011474609375e3), SC_(-0.1861543185509194453611459536122112443011e-2), + SC_(0.1e1), SC_(0.10074598388671875e4), SC_(0.5442867382073472255455809862764039075858e-3), + SC_(0.1e1), SC_(0.1185395751953125e4), SC_(0.4442651225543710411266851455520061266312e-3), + SC_(0.2e1), SC_(0.177219114266335964202880859375e-2), SC_(0.2093773827720430565251557110538425770714e-6), + SC_(0.2e1), SC_(0.22177286446094512939453125e-2), SC_(0.3278879075515531514779472751721891924383e-6), + SC_(0.2e1), SC_(0.7444499991834163665771484375e-2), SC_(0.3694690716009003179822952040973660348991e-5), + SC_(0.2e1), SC_(0.1433600485324859619140625e-1), SC_(0.1370120120703995134662099191103188366059e-4), + SC_(0.2e1), SC_(0.1760916970670223236083984375e-1), SC_(0.2067173265753174063228459655801741280461e-4), + SC_(0.2e1), SC_(0.6152711808681488037109375e-1), SC_(0.2523041832594696207720873096523404469743e-3), + SC_(0.2e1), SC_(0.11958599090576171875e0), SC_(0.9524137960598563739357464134535327666968e-3), + SC_(0.2e1), SC_(0.15262925624847412109375e0), SC_(0.1550463428413672091078292641795407438935e-2), + SC_(0.2e1), SC_(0.408089816570281982421875e0), SC_(0.1097102611452317567405386767797158993089e-1), + SC_(0.2e1), SC_(0.6540834903717041015625e0), SC_(0.2766037955185958056378446380712964902424e-1), + SC_(0.2e1), SC_(0.1097540378570556640625e1), SC_(0.7362360493389258718677168933368717603075e-1), + SC_(0.2e1), SC_(0.30944411754608154296875e1), SC_(0.3024894557597979701575888951210976888787e0), + SC_(0.2e1), SC_(0.51139926910400390625e1), SC_(0.1144982388451644297330327843104009488548e0), + SC_(0.2e1), SC_(0.95070552825927734375e1), SC_(0.4143704967238294556855640424303560695077e-1), + SC_(0.2e1), SC_(0.24750102996826171875e2), SC_(0.1046843026490516523565262690999881523111e-1), + SC_(0.2e1), SC_(0.637722015380859375e2), SC_(-0.1309190404197185674877392619814616382613e-1), + SC_(0.2e1), SC_(0.1252804412841796875e3), SC_(0.2807062488056296863373885100415798671326e-2), + SC_(0.2e1), SC_(0.25554705810546875e3), SC_(0.3469229447658863624235064543945418606161e-2), + SC_(0.2e1), SC_(0.503011474609375e3), SC_(-0.7051959813046644551033771629828891764763e-3), + SC_(0.2e1), SC_(0.10074598388671875e4), SC_(-0.8289780291514071759326532983957910172243e-3), + SC_(0.2e1), SC_(0.1185395751953125e4), SC_(0.7178893275807378245723639629312389935114e-3), + SC_(0.4e1), SC_(0.177219114266335964202880859375e-2), SC_(0.1043783376837269415656629099992565667092e-13), + SC_(0.4e1), SC_(0.22177286446094512939453125e-2), SC_(0.2559774557671848742422562459374402479171e-13), + SC_(0.4e1), SC_(0.7444499991834163665771484375e-2), SC_(0.3250193613184550377260115454105020657549e-11), + SC_(0.4e1), SC_(0.1433600485324859619140625e-1), SC_(0.4469682677416623924037229681870123528417e-10), + SC_(0.4e1), SC_(0.1760916970670223236083984375e-1), SC_(0.101746056573412727336966799805478893789e-9), + SC_(0.4e1), SC_(0.6152711808681488037109375e-1), SC_(0.1516211386725807703466991840076574150499e-7), + SC_(0.4e1), SC_(0.11958599090576171875e0), SC_(0.216275386096930433390507967308045332362e-6), + SC_(0.4e1), SC_(0.15262925624847412109375e0), SC_(0.5736664261625388112327264738146401787856e-6), + SC_(0.4e1), SC_(0.408089816570281982421875e0), SC_(0.2912740194772400778071406390564947979695e-4), + SC_(0.4e1), SC_(0.6540834903717041015625e0), SC_(0.1899514932067494545272197267372701237686e-3), + SC_(0.4e1), SC_(0.1097540378570556640625e1), SC_(0.1453347874709174940209018570873349993662e-2), + SC_(0.4e1), SC_(0.30944411754608154296875e1), SC_(0.6180107635015316992043827444759368254146e-1), + SC_(0.4e1), SC_(0.51139926910400390625e1), SC_(0.1915321596858163372986487584900250664944e0), + SC_(0.4e1), SC_(0.95070552825927734375e1), SC_(-0.1019068788444761880653772745191206413045e0), + SC_(0.4e1), SC_(0.24750102996826171875e2), SC_(0.9030066589287021559976120569661632677975e-3), + SC_(0.4e1), 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SC_(0.2028596251539445836931564425730981539088e-27), + SC_(0.46e2), SC_(0.24750102996826171875e2), SC_(0.1506307082149153247277810883442524835901e-9), + SC_(0.46e2), SC_(0.637722015380859375e2), SC_(0.9981390073397570470311862738222580946682e-3), + SC_(0.46e2), SC_(0.1252804412841796875e3), SC_(-0.7291170921230448995307745765159879879917e-2), + SC_(0.46e2), SC_(0.25554705810546875e3), SC_(-0.3240042995538310471851852251143813655283e-2), + SC_(0.46e2), SC_(0.503011474609375e3), SC_(-0.1180752496915180759015257303074531713751e-2), + SC_(0.46e2), SC_(0.10074598388671875e4), SC_(0.8112995649167475106523147339693999640721e-4), + SC_(0.46e2), SC_(0.1185395751953125e4), SC_(0.7908406619706603256943101136482675770749e-3), + SC_(0.49e2), SC_(0.177219114266335964202880859375e-2), SC_(0.5515546045969304722008600804322812862354e-213), + SC_(0.49e2), SC_(0.22177286446094512939453125e-2), SC_(0.326650164794020092365784924346435136517e-208), + SC_(0.49e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1925360407412391019857608682831977562776e-182), + SC_(0.49e2), SC_(0.1433600485324859619140625e-1), SC_(0.1696482510953376927277246087752612819038e-168), + SC_(0.49e2), SC_(0.1760916970670223236083984375e-1), SC_(0.4034367000095565704299813933456029116899e-164), + SC_(0.49e2), SC_(0.6152711808681488037109375e-1), SC_(0.1693655026076133877660970241165014704451e-137), + SC_(0.49e2), SC_(0.11958599090576171875e0), SC_(0.2348958117281878334841541282897547765415e-123), + SC_(0.49e2), SC_(0.15262925624847412109375e0), SC_(0.3654033502135831724309937917523935502668e-118), + SC_(0.49e2), SC_(0.408089816570281982421875e0), SC_(0.3099167637766167128925774129143043527408e-97), + SC_(0.49e2), SC_(0.6540834903717041015625e0), SC_(0.3385925104782587944340438218046569787901e-87), + SC_(0.49e2), SC_(0.1097540378570556640625e1), SC_(0.3488075239147121457577900092276265954556e-76), + SC_(0.49e2), SC_(0.30944411754608154296875e1), SC_(0.3823874600343831420393015688949984870929e-54), + SC_(0.49e2), SC_(0.51139926910400390625e1), SC_(0.1727893893654003330466257673489103445204e-43), + SC_(0.49e2), SC_(0.95070552825927734375e1), SC_(0.1966295052177707644956458594445543297349e-30), + SC_(0.49e2), SC_(0.24750102996826171875e2), SC_(0.3095382191507684987784237056495861811507e-11), + SC_(0.49e2), SC_(0.637722015380859375e2), SC_(0.158295373743556601533454490387052548217e-1), + SC_(0.49e2), SC_(0.1252804412841796875e3), SC_(0.82836746548166677824971194819851133652e-2), + SC_(0.49e2), SC_(0.25554705810546875e3), SC_(0.3644444755072188782636164173218495212912e-2), + SC_(0.49e2), SC_(0.503011474609375e3), SC_(0.1873650276195313552512971867889734167198e-2), + SC_(0.49e2), SC_(0.10074598388671875e4), SC_(0.96821173121543674370030596847704670817e-3), + SC_(0.49e2), SC_(0.1185395751953125e4), SC_(-0.3882928591259993124112306803057698639751e-3), + SC_(0.52e2), SC_(0.177219114266335964202880859375e-2), SC_(0.2810426956251312594726679498943253449939e-227), + SC_(0.52e2), SC_(0.22177286446094512939453125e-2), SC_(0.3261821586222305590688396082340930337197e-222), + SC_(0.52e2), SC_(0.7444499991834163665771484375e-2), SC_(0.7272282602335146322076473310722428705234e-195), + SC_(0.52e2), SC_(0.1433600485324859619140625e-1), SC_(0.4575999118312716040572653394509627667421e-180), + SC_(0.52e2), SC_(0.1760916970670223236083984375e-1), SC_(0.2016713378994708224486485028173988784384e-175), + SC_(0.52e2), SC_(0.6152711808681488037109375e-1), SC_(0.3611413640730227943802279701896849784546e-147), + SC_(0.52e2), SC_(0.11958599090576171875e0), SC_(0.3677646815864905481023738513580609303132e-132), + SC_(0.52e2), SC_(0.15262925624847412109375e0), SC_(0.1189435235193540319708552300693475643205e-126), + SC_(0.52e2), SC_(0.408089816570281982421875e0), SC_(0.1928344047619157510429385225938387716745e-104), + SC_(0.52e2), SC_(0.6540834903717041015625e0), SC_(0.8675221916812243553880316898424564367176e-94), + SC_(0.52e2), SC_(0.1097540378570556640625e1), SC_(0.4223229826495542762298450140014968626056e-82), + SC_(0.52e2), SC_(0.30944411754608154296875e1), SC_(0.104005992095441008419804771706791846708e-58), + SC_(0.52e2), SC_(0.51139926910400390625e1), SC_(0.2131148367399257759519412699653606504304e-47), + SC_(0.52e2), SC_(0.95070552825927734375e1), SC_(0.1586614197158476471413308754970129553623e-33), + SC_(0.52e2), SC_(0.24750102996826171875e2), SC_(0.5175370600854880940253656288662351415917e-13), + SC_(0.52e2), SC_(0.637722015380859375e2), SC_(-0.17532409288934115094263569459614392385e-1), + SC_(0.52e2), SC_(0.1252804412841796875e3), SC_(-0.81931089129677844005131776262320688731e-2), + SC_(0.52e2), SC_(0.25554705810546875e3), SC_(-0.8105943479828618683212783966808076515887e-3), + SC_(0.52e2), SC_(0.503011474609375e3), SC_(0.8559984069467296788733442052899950392641e-4), + SC_(0.52e2), SC_(0.10074598388671875e4), SC_(-0.3653807379369436466708861221542025660679e-3), + SC_(0.52e2), SC_(0.1185395751953125e4), SC_(-0.6931477169897557084753887842368702759482e-3), + SC_(0.55e2), SC_(0.177219114266335964202880859375e-2), SC_(0.1208288753536924400989698078437356511703e-241), + SC_(0.55e2), SC_(0.22177286446094512939453125e-2), SC_(0.2748224243571374121394628637078216213795e-236), + SC_(0.55e2), SC_(0.7444499991834163665771484375e-2), SC_(0.2317629980513787136420907827650557993837e-207), + SC_(0.55e2), SC_(0.1433600485324859619140625e-1), SC_(0.104144686711366844962623177915396851023e-191), + SC_(0.55e2), SC_(0.1760916970670223236083984375e-1), SC_(0.8506043420003708373323912746376820821917e-187), + SC_(0.55e2), SC_(0.6152711808681488037109375e-1), SC_(0.6497467376347885988588920192401434832693e-157), + SC_(0.55e2), SC_(0.11958599090576171875e0), SC_(0.4858242908178906496481848927231899579715e-141), + SC_(0.55e2), SC_(0.15262925624847412109375e0), SC_(0.3266806562437370063677150237483657480669e-135), + SC_(0.55e2), SC_(0.408089816570281982421875e0), SC_(0.1012363516617469262510002411527525089154e-111), + SC_(0.55e2), SC_(0.6540834903717041015625e0), SC_(0.1875396080076928265201291332242100047049e-100), + SC_(0.55e2), SC_(0.1097540378570556640625e1), SC_(0.4314225131885992800618800784424280652408e-88), + SC_(0.55e2), SC_(0.30944411754608154296875e1), SC_(0.2386188946009811741115402149427915301942e-63), + SC_(0.55e2), SC_(0.51139926910400390625e1), SC_(0.2216092396198493700432387401826650688793e-51), + SC_(0.55e2), SC_(0.95070552825927734375e1), SC_(0.1077268775045601594715005289873066535698e-36), + SC_(0.55e2), SC_(0.24750102996826171875e2), SC_(0.7145924372118157260136350067139292902786e-15), + SC_(0.55e2), SC_(0.637722015380859375e2), SC_(-0.9639150724102627821461487026096968045471e-2), + SC_(0.55e2), SC_(0.1252804412841796875e3), SC_(0.7613679890890913165510416348925798112055e-2), + SC_(0.55e2), SC_(0.25554705810546875e3), SC_(-0.2628460166452094563254227164509714721701e-2), + SC_(0.55e2), SC_(0.503011474609375e3), SC_(-0.1916611872581451277353556345649407783691e-2), + SC_(0.55e2), SC_(0.10074598388671875e4), SC_(-0.8532381652732649330426131075577885254381e-3), + SC_(0.55e2), SC_(0.1185395751953125e4), SC_(0.5715720651795509548926590280953054477066e-3), + SC_(0.58e2), SC_(0.177219114266335964202880859375e-2), SC_(0.442323764221339251671141303318431108478e-256), + SC_(0.58e2), SC_(0.22177286446094512939453125e-2), SC_(0.1971583791626620727195469420538567193962e-250), + SC_(0.58e2), SC_(0.7444499991834163665771484375e-2), SC_(0.6289100883677515415757195488104887078879e-220), + SC_(0.58e2), SC_(0.1433600485324859619140625e-1), SC_(0.2018177733122065474623739428744794685423e-203), + SC_(0.58e2), SC_(0.1760916970670223236083984375e-1), SC_(0.3054795320059194548696602856000938647052e-198), + SC_(0.58e2), SC_(0.6152711808681488037109375e-1), SC_(0.9953642739298005551795428591931462815948e-167), + SC_(0.58e2), SC_(0.11958599090576171875e0), SC_(0.546461267246679212639001052047785879468e-150), + SC_(0.58e2), SC_(0.15262925624847412109375e0), SC_(0.7639710377234199236097520431614621507318e-144), + SC_(0.58e2), SC_(0.408089816570281982421875e0), SC_(0.4525408091084346067559391415753234146623e-119), + SC_(0.58e2), SC_(0.6540834903717041015625e0), SC_(0.3452008641978455477403610949257595865265e-107), + SC_(0.58e2), SC_(0.1097540378570556640625e1), SC_(0.3752484512573063001603693690152615718616e-94), + SC_(0.58e2), SC_(0.30944411754608154296875e1), SC_(0.4660345641388563178171531149615009184069e-68), + SC_(0.58e2), SC_(0.51139926910400390625e1), SC_(0.1960863006295924933138482025108297330693e-55), + SC_(0.58e2), SC_(0.95070552825927734375e1), SC_(0.6213624017861198159027984879634566021036e-40), + SC_(0.58e2), SC_(0.24750102996826171875e2), SC_(0.8252480997909935579863933179819188239925e-17), + SC_(0.58e2), SC_(0.637722015380859375e2), SC_(0.2016586308654754622690091357752537719812e-1), + SC_(0.58e2), SC_(0.1252804412841796875e3), SC_(-0.7017468118833063218312557452977246591573e-2), + SC_(0.58e2), SC_(0.25554705810546875e3), SC_(0.3961059662958721103739812991599937252217e-2), + SC_(0.58e2), SC_(0.503011474609375e3), SC_(0.1159722024061608331934095179228494399393e-2), + SC_(0.58e2), SC_(0.10074598388671875e4), SC_(0.6455961901963375666872388539662639143768e-3), + SC_(0.58e2), SC_(0.1185395751953125e4), SC_(0.5324640861533894098235525421246130031191e-3), + SC_(0.61e2), SC_(0.177219114266335964202880859375e-2), SC_(0.1390063099953273001760976643829493412235e-270), + SC_(0.61e2), SC_(0.22177286446094512939453125e-2), SC_(0.1214235746891325324123447717603508906882e-264), + SC_(0.61e2), SC_(0.7444499991834163665771484375e-2), SC_(0.1465067803501878545612089003565516614999e-232), + SC_(0.61e2), SC_(0.1433600485324859619140625e-1), SC_(0.3357425642907951097269617118707813745614e-215), + SC_(0.61e2), SC_(0.1760916970670223236083984375e-1), SC_(0.9418057496533460392723945550704596626117e-210), + SC_(0.61e2), SC_(0.6152711808681488037109375e-1), SC_(0.1309015102638367959763911559061398598334e-176), + SC_(0.61e2), SC_(0.11958599090576171875e0), SC_(0.5276720779633476596305457037624920158493e-159), + SC_(0.61e2), SC_(0.15262925624847412109375e0), SC_(0.1533750989168073201676611877485582641816e-152), + SC_(0.61e2), SC_(0.408089816570281982421875e0), SC_(0.1736609472559701112844257684577329098627e-126), + SC_(0.61e2), SC_(0.6540834903717041015625e0), SC_(0.5454707501690087691332936974401497393552e-114), + SC_(0.61e2), SC_(0.1097540378570556640625e1), SC_(0.2801873396633198777831566955045782029762e-100), + SC_(0.61e2), SC_(0.30944411754608154296875e1), SC_(0.7812083655684529175645698522139037522772e-73), + SC_(0.61e2), SC_(0.51139926910400390625e1), SC_(0.1488629979362386514563024050166990786373e-59), + SC_(0.61e2), SC_(0.95070552825927734375e1), SC_(0.3070674775920232104372515582146972644154e-43), + SC_(0.61e2), SC_(0.24750102996826171875e2), SC_(0.8059317390853095912429339684297296887014e-19), + SC_(0.61e2), SC_(0.637722015380859375e2), SC_(0.2553438361299694660955712647272386491838e-1), + SC_(0.61e2), SC_(0.1252804412841796875e3), SC_(0.6702527215895999645298014803289940554868e-2), + SC_(0.61e2), SC_(0.25554705810546875e3), SC_(-0.2435818794268515565710215283462154963534e-2), + SC_(0.61e2), SC_(0.503011474609375e3), SC_(0.1113025090458558853547625070047439913018e-2), + SC_(0.61e2), SC_(0.10074598388671875e4), SC_(0.6282813469515595099026690750450965536512e-3), + SC_(0.61e2), SC_(0.1185395751953125e4), SC_(-0.7280704607550679547528319738669934801603e-3) + }; +#undef SC_ + + diff --git a/test/sph_neumann_data.ipp b/test/sph_neumann_data.ipp new file mode 100644 index 000000000..da3467ef6 --- /dev/null +++ b/test/sph_neumann_data.ipp @@ -0,0 +1,295 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 284> sph_neumann_data = { + SC_(0), SC_(0.177219114266335964202880859375e-2), SC_(-0.5642723324792311990959765396871018960216e3), + SC_(0), SC_(0.22177286446094512939453125e-2), SC_(-0.4509106843488238999473616173421364998283e3), + SC_(0), SC_(0.7444499991834163665771484375e-2), SC_(-0.1343236336805396008478682587573788632696e3), + SC_(0), SC_(0.1433600485324859619140625e-1), SC_(-0.6974727279167912880150678975872617495088e2), + SC_(0), SC_(0.1760916970670223236083984375e-1), SC_(-0.5677979025875563956840873378239183466216e2), + SC_(0), SC_(0.6152711808681488037109375e-1), SC_(-0.1622224207701819661207370690041340570512e2), + SC_(0), SC_(0.11958599090576171875e0), SC_(-0.8302461728079151325461513362583139756373e1), + SC_(0), SC_(0.15262925624847412109375e0), SC_(-0.6475657248546134771634572625845233177042e1), + SC_(0), SC_(0.408089816570281982421875e0), SC_(-0.2249212131304610409189209411089291558038e1), + SC_(0), SC_(0.6540834903717041015625e0), SC_(-0.1213309779166084571756446746977955970241e1), + SC_(0), SC_(0.1097540378570556640625e1), SC_(-0.4152801926629981090971418551680239835571e0), + SC_(0), SC_(0.30944411754608154296875e1), SC_(0.3228009577028088632835185600650872623482e0), + SC_(0), SC_(0.51139926910400390625e1), SC_(-0.7643635410754817824148236191571837220126e-1), + SC_(0), SC_(0.95070552825927734375e1), SC_(0.1048292137319730265210439388887370801248e0), + SC_(0), SC_(0.24750102996826171875e2), SC_(-0.3748197858988794731118470575529801883541e-1), + SC_(0), SC_(0.637722015380859375e2), SC_(-0.9243942787530241818116046978138604629927e-2), + SC_(0), SC_(0.1252804412841796875e3), SC_(-0.7402980669441766401839846974565600278467e-2), + SC_(0), SC_(0.25554705810546875e3), SC_(0.1851105232770496300381256908203514858216e-2), + SC_(0), SC_(0.503011474609375e3), SC_(-0.1862923061767209624572519967199174532136e-2), + SC_(0), SC_(0.10074598388671875e4), SC_(0.5434622896619636174822759159195129760936e-3), + SC_(0), SC_(0.1185395751953125e4), SC_(0.4448697855794123235553917677921400676563e-3), + SC_(0.1e1), SC_(0.177219114266335964202880859375e-2), SC_(-0.318404765200112567469157464785027433823e6), + SC_(0.1e1), SC_(0.22177286446094512939453125e-2), SC_(-0.2033219452576704738500460434959917636461e6), + SC_(0.1e1), SC_(0.7444499991834163665771484375e-2), SC_(-0.1804433853974285124747494278925454917131e5), + SC_(0.1e1), SC_(0.1433600485324859619140625e-1), SC_(-0.486618196768193316259977336624918918276e4), + SC_(0.1e1), SC_(0.1760916970670223236083984375e-1), SC_(-0.322544443971191302525452896173067683304e4), + SC_(0.1e1), SC_(0.6152711808681488037109375e-1), SC_(-0.2646594036812657100698306327292119166165e3), + SC_(0.1e1), SC_(0.11958599090576171875e0), SC_(-0.704243267085761836707775439192578042573e2), + SC_(0.1e1), SC_(0.15262925624847412109375e0), SC_(-0.434234874553672313920027034930149194407e2), + SC_(0.1e1), SC_(0.408089816570281982421875e0), SC_(-0.6484035515444221775976367630376209298567e1), + SC_(0.1e1), SC_(0.6540834903717041015625e0), SC_(-0.2785182560801830440325676905703325489184e1), + SC_(0.1e1), SC_(0.1097540378570556640625e1), SC_(-0.1189358686761883310370162855037394165611e1), + SC_(0.1e1), SC_(0.30944411754608154296875e1), SC_(0.8908456605946127242501038405190438284591e-1), + SC_(0.1e1), SC_(0.51139926910400390625e1), SC_(0.1650371817824980920597277760628969935339e0), + SC_(0.1e1), SC_(0.95070552825927734375e1), SC_(0.1967104757813801271824802511806652238075e-1), + SC_(0.1e1), SC_(0.24750102996826171875e2), SC_(0.1357114472738293735380837736920673513642e-1), + SC_(0.1e1), SC_(0.637722015380859375e2), SC_(-0.1281134249246794610073876109037108992935e-1), + SC_(0.1e1), SC_(0.1252804412841796875e3), SC_(0.2925815571849156117920315547136950241325e-2), + SC_(0.1e1), SC_(0.25554705810546875e3), SC_(0.3454900436576811073075917245701360162869e-2), + SC_(0.1e1), SC_(0.503011474609375e3), SC_(-0.6977971312135198156036217479682334535885e-3), + SC_(0.1e1), SC_(0.10074598388671875e4), SC_(-0.830059360518605915833351761559017133065e-3), + SC_(0.1e1), SC_(0.1185395751953125e4), SC_(0.7171402734520886334015615434326112966186e-3), + SC_(0.2e1), SC_(0.177219114266335964202880859375e-2), SC_(-0.5390012807345113883197792587292120542031e9), + SC_(0.2e1), SC_(0.22177286446094512939453125e-2), SC_(-0.2750403378962024584351226505492891996551e9), + SC_(0.2e1), SC_(0.7444499991834163665771484375e-2), SC_(-0.7271410532113085385774156481477356217156e7), + SC_(0.2e1), SC_(0.1433600485324859619140625e-1), SC_(-0.1018243656809079180564791631983824027866e7), + SC_(0.2e1), SC_(0.1760916970670223236083984375e-1), SC_(-0.5494485904403902454288040447187085252735e6), + SC_(0.2e1), SC_(0.6152711808681488037109375e-1), SC_(-0.1288830239246692985269728537863836376468e5), + SC_(0.2e1), SC_(0.11958599090576171875e0), SC_(-0.1758400966704601107905829221340068593333e4), + SC_(0.2e1), SC_(0.15262925624847412109375e0), SC_(-0.8470334639256196191828910705235499636086e3), + SC_(0.2e1), SC_(0.408089816570281982421875e0), SC_(-0.4541702641837159203058389758895634766256e2), + SC_(0.2e1), SC_(0.6540834903717041015625e0), SC_(-0.1156112621471167110574129561700037138981e2), + SC_(0.2e1), SC_(0.1097540378570556640625e1), SC_(-0.2835694559566832562081631532476610393593e1), + SC_(0.2e1), SC_(0.30944411754608154296875e1), SC_(-0.2364352189394648962564115002206693193264e0), + SC_(0.2e1), SC_(0.51139926910400390625e1), SC_(0.1732514211714791641487261827982664865146e0), + SC_(0.2e1), SC_(0.95070552825927734375e1), SC_(-0.9862191389198304425904745146753435952401e-1), + SC_(0.2e1), SC_(0.24750102996826171875e2), SC_(0.3912695898400467874867999209317768037635e-1), + SC_(0.2e1), SC_(0.637722015380859375e2), SC_(0.8641265969881876225898539254984860529714e-2), + SC_(0.2e1), SC_(0.1252804412841796875e3), SC_(0.7473043056080692435099953164437164248053e-2), + SC_(0.2e1), SC_(0.25554705810546875e3), SC_(-0.1810546357287559849120539213331049290221e-2), + SC_(0.2e1), SC_(0.503011474609375e3), SC_(0.1858761344789945377984398470433340989203e-2), + SC_(0.2e1), SC_(0.10074598388671875e4), SC_(-0.5459340289665852636696979017491079500449e-3), + SC_(0.2e1), SC_(0.1185395751953125e4), SC_(-0.4430548467146395559544067187824537160368e-3), + SC_(0.4e1), SC_(0.177219114266335964202880859375e-2), SC_(-0.6006708939031862782547423129663205631395e16), + SC_(0.4e1), SC_(0.22177286446094512939453125e-2), SC_(-0.1957255048386226906846149698887287471033e16), + SC_(0.4e1), SC_(0.7444499991834163665771484375e-2), SC_(-0.4592121279331724272683855783141568753333e13), + SC_(0.4e1), SC_(0.1433600485324859619140625e-1), SC_(-0.173402349594767269025145292017473455282e12), + SC_(0.4e1), SC_(0.1760916970670223236083984375e-1), SC_(-0.6201611030027830575236750989101389849364e11), + SC_(0.4e1), SC_(0.6152711808681488037109375e-1), SC_(-0.1191170341565551054056618133292811190739e9), + SC_(0.4e1), SC_(0.11958599090576171875e0), SC_(-0.4297654348662493392476039106116454381053e7), + SC_(0.4e1), SC_(0.15262925624847412109375e0), SC_(-0.1269764721301016237085922161946420176713e7), + SC_(0.4e1), SC_(0.408089816570281982421875e0), SC_(-0.938834620739078580225698168692023203448e4), + SC_(0.4e1), SC_(0.6540834903717041015625e0), SC_(-0.9044373064697326637009564699789610374151e3), + SC_(0.4e1), SC_(0.1097540378570556640625e1), SC_(-0.7197096999109657262465719753479013842004e2), + SC_(0.4e1), SC_(0.30944411754608154296875e1), SC_(-0.8292878001802161021682270451154977977337e0), + SC_(0.4e1), SC_(0.51139926910400390625e1), SC_(-0.1672938782320394591130634026396133319056e0), + SC_(0.4e1), SC_(0.95070552825927734375e1), SC_(0.4594823416227780746480571313224025888457e-1), + SC_(0.4e1), SC_(0.24750102996826171875e2), SC_(-0.4072966707158819427744567215857101892826e-1), + SC_(0.4e1), SC_(0.637722015380859375e2), SC_(-0.7160652662228105175773913698625518194443e-2), + SC_(0.4e1), SC_(0.1252804412841796875e3), SC_(-0.7619857201446145509948734118435483624086e-2), + SC_(0.4e1), SC_(0.25554705810546875e3), SC_(0.1714938614376801147928666041967663272554e-2), + SC_(0.4e1), SC_(0.503011474609375e3), SC_(-0.1848793551823628970369126596953626884517e-2), + SC_(0.4e1), SC_(0.10074598388671875e4), SC_(0.5516825949084002964790973918927589969002e-3), + SC_(0.4e1), SC_(0.1185395751953125e4), SC_(0.4388089536910184022488035482968780382212e-3), + SC_(0.7e1), SC_(0.177219114266335964202880859375e-2), SC_(-0.1388939744137340433522522155157803129493e28), + SC_(0.7e1), SC_(0.22177286446094512939453125e-2), SC_(-0.2309408077963500230051756342440986301973e27), + SC_(0.7e1), SC_(0.7444499991834163665771484375e-2), SC_(-0.1432466589853457118157199163927226239105e23), + SC_(0.7e1), SC_(0.1433600485324859619140625e-1), SC_(-0.7574363861874272531180922496759086862498e20), + SC_(0.7e1), SC_(0.1760916970670223236083984375e-1), SC_(-0.1461712611084380044913310721560950112306e20), + SC_(0.7e1), SC_(0.6152711808681488037109375e-1), SC_(-0.6581100049139522866420337381107024877831e15), + SC_(0.7e1), SC_(0.11958599090576171875e0), SC_(-0.3232691649492879334595332596749659857755e13), + SC_(0.7e1), SC_(0.15262925624847412109375e0), SC_(-0.4592558497626386579457472880231329928345e12), + SC_(0.7e1), SC_(0.408089816570281982421875e0), SC_(-0.1768093332349639873943590983950796618094e9), + SC_(0.7e1), SC_(0.6540834903717041015625e0), SC_(-0.410071253550436848907977153943779570177e7), + SC_(0.7e1), SC_(0.1097540378570556640625e1), SC_(-0.6723738931189786892469511427815554534622e5), + SC_(0.7e1), SC_(0.30944411754608154296875e1), SC_(-0.2355914723945007502048313939035727585535e2), + SC_(0.7e1), SC_(0.51139926910400390625e1), SC_(-0.9095841466009625950437779235402980858269e0), + SC_(0.7e1), SC_(0.95070552825927734375e1), SC_(0.4137038128161563987674845255425727782835e-2), + SC_(0.7e1), SC_(0.24750102996826171875e2), SC_(0.2840265140460042677367989989206316229572e-1), + SC_(0.7e1), SC_(0.637722015380859375e2), SC_(0.1544929171899375325706484638266664284005e-1), + SC_(0.7e1), SC_(0.1252804412841796875e3), SC_(-0.1270517672039039606712892170401020621663e-2), + SC_(0.7e1), SC_(0.25554705810546875e3), SC_(-0.3630188911697934815681636175552688093357e-2), + SC_(0.7e1), SC_(0.503011474609375e3), SC_(0.7967098511734288574306139155656730858314e-3), + SC_(0.7e1), SC_(0.10074598388671875e4), SC_(0.815186884113040002201698678223896056766e-3), + SC_(0.7e1), SC_(0.1185395751953125e4), SC_(-0.7270795125271877654141198243699517038649e-3), + SC_(0.1e2), SC_(0.1433600485324859619140625e-1), SC_(-0.1245530719806351448480124937794738961635e30), + SC_(0.1e2), SC_(0.1760916970670223236083984375e-1), SC_(-0.1296992613685647077229845683943181048821e29), + SC_(0.1e2), SC_(0.6152711808681488037109375e-1), SC_(-0.1368902717654920136735378996175836140785e23), + SC_(0.1e2), SC_(0.11958599090576171875e0), SC_(-0.9156757241161448651951539042135995904994e19), + SC_(0.1e2), SC_(0.15262925624847412109375e0), SC_(-0.6256222869032686140322813275452916410343e18), + SC_(0.1e2), SC_(0.408089816570281982421875e0), SC_(-0.1257917349261965601916962527396933303977e14), + SC_(0.1e2), SC_(0.6540834903717041015625e0), SC_(-0.7063002197882333005631618484528652399524e11), + SC_(0.1e2), SC_(0.1097540378570556640625e1), SC_(-0.2427889658115064857278886600528596240123e9), + SC_(0.1e2), SC_(0.30944411754608154296875e1), SC_(-0.3394649246350136450439882104151313759251e4), + SC_(0.1e2), SC_(0.51139926910400390625e1), SC_(-0.2150737348962304792352416315737126587418e2), + SC_(0.1e2), SC_(0.95070552825927734375e1), SC_(-0.2157691001768484850020828031987882855639e0), + SC_(0.1e2), SC_(0.24750102996826171875e2), SC_(-0.1260876938605482558721300295302574251533e-1), + SC_(0.1e2), SC_(0.637722015380859375e2), SC_(-0.3658561196157287230123135947541224109443e-2), + SC_(0.1e2), SC_(0.1252804412841796875e3), SC_(0.7983606267849351237703135831852105823276e-2), + SC_(0.1e2), SC_(0.25554705810546875e3), SC_(-0.1072436715227848668749084256488184819454e-2), + SC_(0.1e2), SC_(0.503011474609375e3), SC_(0.1776245695123382255186007833872594145979e-2), + SC_(0.1e2), SC_(0.10074598388671875e4), SC_(-0.5879910859563235660929418753761246514854e-3), + SC_(0.1e2), SC_(0.1185395751953125e4), SC_(-0.411154319810368570839418603704906353283e-3), + SC_(0.13e2), SC_(0.6152711808681488037109375e-1), SC_(-0.7096588472433714153042022789489333121863e30), + SC_(0.13e2), SC_(0.11958599090576171875e0), SC_(-0.6464704713743915427301824299819781954859e26), + SC_(0.13e2), SC_(0.15262925624847412109375e0), SC_(-0.2124330495066614710929484795599742959654e25), + SC_(0.13e2), SC_(0.408089816570281982421875e0), SC_(-0.2232620432299327594678167214732483417736e19), + SC_(0.13e2), SC_(0.6540834903717041015625e0), SC_(-0.3039495964028968670551250826703930725596e16), + SC_(0.13e2), SC_(0.1097540378570556640625e1), SC_(-0.2200572297449479637006380190194549845276e13), + SC_(0.13e2), SC_(0.30944411754608154296875e1), SC_(-0.1299211572249285551909779637874602972508e7), + SC_(0.13e2), SC_(0.51139926910400390625e1), SC_(-0.1614426118247726564069541604511986952086e4), + SC_(0.13e2), SC_(0.95070552825927734375e1), SC_(-0.1192876773324610978227972311096425254541e1), + SC_(0.13e2), SC_(0.24750102996826171875e2), SC_(0.1083090330001395226011081713405820116194e-1), + SC_(0.13e2), SC_(0.637722015380859375e2), SC_(-0.1103112564518912696072370568742481339126e-1), + SC_(0.13e2), SC_(0.1252804412841796875e3), SC_(-0.2698254042376059943851702235540482168386e-2), + SC_(0.13e2), SC_(0.25554705810546875e3), SC_(0.3879446011377107269956022967187011973557e-2), + SC_(0.13e2), SC_(0.503011474609375e3), SC_(-0.1018154003350369531009009098099277689641e-2), + SC_(0.13e2), SC_(0.10074598388671875e4), SC_(-0.7782246846650067854491204003191128582844e-3), + SC_(0.13e2), SC_(0.1185395751953125e4), SC_(0.7487965903672491559847491642545850030768e-3), + SC_(0.16e2), SC_(0.408089816570281982421875e0), SC_(-0.7968764716391730011103329636034855328962e24), + SC_(0.16e2), SC_(0.6540834903717041015625e0), SC_(-0.2632119245040565674510100654677927787303e21), + SC_(0.16e2), SC_(0.1097540378570556640625e1), SC_(-0.4021306302655656019280730356166023954594e17), + SC_(0.16e2), SC_(0.30944411754608154296875e1), SC_(-0.1024734904321332361176888712960645982827e10), + SC_(0.16e2), SC_(0.51139926910400390625e1), SC_(-0.263104548681619029365595821516581062838e6), + SC_(0.16e2), SC_(0.95070552825927734375e1), SC_(-0.212434709466762829773238681303207117582e2), + SC_(0.16e2), SC_(0.24750102996826171875e2), SC_(-0.2743563165647313690782258876808431412109e-1), + SC_(0.16e2), SC_(0.637722015380859375e2), SC_(0.1592868538765449864728958003322337214849e-1), + SC_(0.16e2), SC_(0.1252804412841796875e3), SC_(-0.611189995982529806498215807361518117186e-2), + SC_(0.16e2), SC_(0.25554705810546875e3), SC_(-0.1551975946286214383853261368195650834803e-3), + SC_(0.16e2), SC_(0.503011474609375e3), SC_(-0.1610266445943186662771155287434094951638e-2), + SC_(0.16e2), SC_(0.10074598388671875e4), SC_(0.650348574795961322341087383056013722587e-3), + SC_(0.16e2), SC_(0.1185395751953125e4), SC_(0.3599072645317034685633234775339363892989e-3), + SC_(0.19e2), SC_(0.408089816570281982421875e0), SC_(-0.5008621620829854006954623536723536260922e30), + SC_(0.19e2), SC_(0.6540834903717041015625e0), SC_(-0.4015165392251303629512256297488777013034e26), + SC_(0.19e2), SC_(0.1097540378570556640625e1), SC_(-0.1295742383468043411565809880265553984875e22), + SC_(0.19e2), SC_(0.30944411754608154296875e1), SC_(-0.1440850060936814966263451261627483470922e13), + SC_(0.19e2), SC_(0.51139926910400390625e1), SC_(-0.7829560726823223330394765378802240654337e8), + SC_(0.19e2), SC_(0.95070552825927734375e1), SC_(-0.8017774502788714016096257024597920080947e3), + SC_(0.19e2), SC_(0.24750102996826171875e2), SC_(0.5104725068940265807623171340563537419839e-1), + SC_(0.19e2), SC_(0.637722015380859375e2), SC_(-0.1154540926504216331530538721227045039394e-1), + SC_(0.19e2), SC_(0.1252804412841796875e3), SC_(0.7285128790438104181486799657416599622562e-2), + SC_(0.19e2), SC_(0.25554705810546875e3), SC_(-0.3795974512769646959066511811760180003581e-2), + SC_(0.19e2), SC_(0.503011474609375e3), SC_(0.1332792253280798715550694041485478947804e-2), + SC_(0.19e2), SC_(0.10074598388671875e4), SC_(0.7140475084390964023422450272059931346758e-3), + SC_(0.19e2), SC_(0.1185395751953125e4), SC_(-0.778631877582065133128398725986021189756e-3), + SC_(0.22e2), SC_(0.1097540378570556640625e1), SC_(-0.6723360495833044373602236204563863321495e26), + SC_(0.22e2), SC_(0.30944411754608154296875e1), SC_(-0.3282992248392472051808450625423987455518e16), + SC_(0.22e2), SC_(0.51139926910400390625e1), SC_(-0.3826147075920552238475280662776521115203e11), + SC_(0.22e2), SC_(0.95070552825927734375e1), SC_(-0.5311187095528196500675385994412924026797e5), + SC_(0.22e2), SC_(0.24750102996826171875e2), SC_(-0.1306396947235207229685397581368310264004e-1), + SC_(0.22e2), SC_(0.637722015380859375e2), SC_(0.3811910734895598722091331392361204163648e-2), + SC_(0.22e2), SC_(0.1252804412841796875e3), SC_(-0.5694609931721205895231828943722573661955e-3), + SC_(0.22e2), SC_(0.25554705810546875e3), SC_(0.1872575932276900334277205173009240761345e-2), + SC_(0.22e2), SC_(0.503011474609375e3), SC_(0.1298158890481958119897857861962977581478e-2), + SC_(0.22e2), SC_(0.10074598388671875e4), SC_(-0.7329138049351166151988684874280432719372e-3), + SC_(0.22e2), SC_(0.1185395751953125e4), SC_(-0.2829751774429469196812768758194950064622e-3), + SC_(0.25e2), SC_(0.30944411754608154296875e1), SC_(-0.1132569536096268337562080171402201939034e20), + SC_(0.25e2), SC_(0.51139926910400390625e1), SC_(-0.2854197487832370443625499582893796948668e14), + SC_(0.25e2), SC_(0.95070552825927734375e1), SC_(-0.557640638109575874026675246942198134901e7), + SC_(0.25e2), SC_(0.24750102996826171875e2), SC_(-0.8289220371234890898107740191499805665166e-1), + SC_(0.25e2), SC_(0.637722015380859375e2), SC_(0.2867466802758145719509918727955672959562e-2), + SC_(0.25e2), SC_(0.1252804412841796875e3), SC_(-0.6426359266478851874122389488976604772913e-2), + SC_(0.25e2), SC_(0.25554705810546875e3), SC_(0.2788655268268470964688490819519378700901e-2), + SC_(0.25e2), SC_(0.503011474609375e3), SC_(-0.1677047836678118060016484817598253328862e-2), + SC_(0.25e2), SC_(0.10074598388671875e4), SC_(-0.6155464006245052959192750335825837759874e-3), + SC_(0.25e2), SC_(0.1185395751953125e4), SC_(0.8105378192860076835295337545544333320285e-3), + SC_(0.28e2), SC_(0.30944411754608154296875e1), SC_(-0.5621747105301521557947179246637760022417e23), + SC_(0.28e2), SC_(0.51139926910400390625e1), SC_(-0.3080131695608097771147596640961172595909e17), + SC_(0.28e2), SC_(0.95070552825927734375e1), SC_(-0.8673370399593050580914898499273415127644e9), + SC_(0.28e2), SC_(0.24750102996826171875e2), SC_(-0.2256097627150219672124131497323287189991e0), + SC_(0.28e2), SC_(0.637722015380859375e2), SC_(-0.6991956238153252791738685651136169064703e-2), + SC_(0.28e2), SC_(0.1252804412841796875e3), SC_(0.7795645183045974969575002412572072101911e-2), + SC_(0.28e2), SC_(0.25554705810546875e3), SC_(-0.3494172391974151981967644744599232334284e-2), + SC_(0.28e2), SC_(0.503011474609375e3), SC_(-0.7872272246020384880492778519333779147176e-3), + SC_(0.28e2), SC_(0.10074598388671875e4), SC_(0.825845271938107790489284932094733926616e-3), + SC_(0.28e2), SC_(0.1185395751953125e4), SC_(0.1783213113986488862036590535806820370025e-3), + SC_(0.31e2), SC_(0.30944411754608154296875e1), SC_(-0.3858743092234085837491563707437405089502e27), + SC_(0.31e2), SC_(0.51139926910400390625e1), SC_(-0.4613835194622917257572168439667276505091e20), + SC_(0.31e2), SC_(0.95070552825927734375e1), SC_(-0.1902866311295072781213762448066992585084e12), + SC_(0.31e2), SC_(0.24750102996826171875e2), SC_(-0.1242902083875520340476461862360617369865e1), + SC_(0.31e2), SC_(0.637722015380859375e2), SC_(0.8602616475850209207115672961665438799578e-2), + SC_(0.31e2), SC_(0.1252804412841796875e3), SC_(-0.3568328873757026723718279708146103263332e-2), + SC_(0.31e2), SC_(0.25554705810546875e3), SC_(-0.4708447548154973984774979548889890332232e-3), + SC_(0.31e2), SC_(0.503011474609375e3), SC_(0.1938585486734996435913653188875075463759e-2), + SC_(0.31e2), SC_(0.10074598388671875e4), SC_(0.4751515606683130102889789705916848210342e-3), + SC_(0.31e2), SC_(0.1185395751953125e4), SC_(-0.8358404490289695691960973315323391126047e-3), + SC_(0.34e2), SC_(0.51139926910400390625e1), SC_(-0.9282996603826470817559175093201479899353e23), + SC_(0.34e2), SC_(0.95070552825927734375e1), SC_(-0.5671896870965902667567112773375443299376e14), + SC_(0.34e2), SC_(0.24750102996826171875e2), SC_(-0.1196836896972653347610997098886066849879e2), + SC_(0.34e2), SC_(0.637722015380859375e2), SC_(-0.7840611740006562731867360753764337981193e-2), + SC_(0.34e2), SC_(0.1252804412841796875e3), SC_(-0.2528119461363815663298862242098595394995e-2), + SC_(0.34e2), SC_(0.25554705810546875e3), SC_(0.3790521728300347929686504759069995348496e-2), + SC_(0.34e2), SC_(0.503011474609375e3), SC_(0.6131119580874945273384687636612133212937e-4), + SC_(0.34e2), SC_(0.10074598388671875e4), SC_(-0.9142183498980430984586827864611642159497e-3), + SC_(0.34e2), SC_(0.1185395751953125e4), SC_(-0.4511848776391923835124583337569697441159e-4), + SC_(0.37e2), SC_(0.51139926910400390625e1), SC_(-0.2442375728698405242925453667580696210245e27), + SC_(0.37e2), SC_(0.95070552825927734375e1), SC_(-0.2229542216375343293897440256273780674204e17), + SC_(0.37e2), SC_(0.24750102996826171875e2), SC_(-0.1737444169853745325867053393033933403419e3), + SC_(0.37e2), SC_(0.637722015380859375e2), SC_(0.4279824575041896675727883899347662880186e-2), + SC_(0.37e2), SC_(0.1252804412841796875e3), SC_(0.6929401282911254411240566948180148883389e-2), + SC_(0.37e2), SC_(0.25554705810546875e3), SC_(-0.2511648418349492118440212332844329555998e-2), + SC_(0.37e2), SC_(0.503011474609375e3), SC_(-0.1957162644941089839867703502493036932186e-2), + SC_(0.37e2), SC_(0.10074598388671875e4), SC_(-0.2872884073535241094708945526011230038645e-3), + SC_(0.37e2), SC_(0.1185395751953125e4), SC_(0.8432140708713275871141861113932655356805e-3), + SC_(0.4e2), SC_(0.95070552825927734375e1), SC_(-0.1128026063781401159493204975300213163583e20), + SC_(0.4e2), SC_(0.24750102996826171875e2), SC_(-0.3536320450284364302528532664762113675546e4), + SC_(0.4e2), SC_(0.637722015380859375e2), SC_(0.2749332472515275038091640131810993326854e-2), + SC_(0.4e2), SC_(0.1252804412841796875e3), SC_(-0.8189335561734438098491340276810366230728e-2), + SC_(0.4e2), SC_(0.25554705810546875e3), SC_(-0.1601472101873495352874904527747422350107e-2), + SC_(0.4e2), SC_(0.503011474609375e3), SC_(0.8063429370382054903944011477925178824483e-3), + SC_(0.4e2), SC_(0.10074598388671875e4), SC_(0.9774185681131416958979641163273033546007e-3), + SC_(0.4e2), SC_(0.1185395751953125e4), SC_(-0.1146975806174666671253112473359257996345e-3), + SC_(0.43e2), SC_(0.95070552825927734375e1), SC_(-0.7198591295447037635860529334856057232135e22), + SC_(0.43e2), SC_(0.24750102996826171875e2), SC_(-0.9641200720117521887788006798691601480704e5), + SC_(0.43e2), SC_(0.637722015380859375e2), SC_(-0.1236200080963258936021132338688942409822e-1), + SC_(0.43e2), SC_(0.1252804412841796875e3), SC_(0.6766741286829961971926652018652618379149e-2), + SC_(0.43e2), SC_(0.25554705810546875e3), SC_(0.3930507468976728535272404195834679836265e-2), + SC_(0.43e2), SC_(0.503011474609375e3), SC_(0.1564028674659955458649814316707562782221e-2), + SC_(0.43e2), SC_(0.10074598388671875e4), SC_(0.5187580633057582060435802616437238457919e-4), + SC_(0.43e2), SC_(0.1185395751953125e4), SC_(-0.8191603697467604740041636746614045025268e-3), + SC_(0.46e2), SC_(0.95070552825927734375e1), SC_(-0.5695769033255529210361047038599776733494e25), + SC_(0.46e2), SC_(0.24750102996826171875e2), SC_(-0.3407542572880364861061570256982516465991e7), + SC_(0.46e2), SC_(0.637722015380859375e2), SC_(0.1892031070374081055519173538522022424373e-1), + SC_(0.46e2), SC_(0.1252804412841796875e3), SC_(-0.3930999501654252080115547096167591365109e-2), + SC_(0.46e2), SC_(0.25554705810546875e3), SC_(-0.2252774787151350219685297262688267474083e-2), + SC_(0.46e2), SC_(0.503011474609375e3), SC_(-0.160470127490624290303361714195055152753e-2), + SC_(0.46e2), SC_(0.10074598388671875e4), SC_(-0.9898053187382996942921915807261270222834e-3), + SC_(0.46e2), SC_(0.1185395751953125e4), SC_(0.2945850112272429804450537869110995190276e-3), + SC_(0.49e2), SC_(0.95070552825927734375e1), SC_(-0.5505988900115077682063953616131381316684e28), + SC_(0.49e2), SC_(0.24750102996826171875e2), SC_(-0.1522647860770080599776352342021496844231e9), + SC_(0.49e2), SC_(0.637722015380859375e2), SC_(-0.117811227613905121473227334683521728565e-1), + SC_(0.49e2), SC_(0.1252804412841796875e3), SC_(0.8581798088552142472627658285170409572199e-3), + SC_(0.49e2), SC_(0.25554705810546875e3), SC_(-0.1525297082577234374630319883497973286237e-2), + SC_(0.49e2), SC_(0.503011474609375e3), SC_(-0.6789386510763052793814186176994423811279e-3), + SC_(0.49e2), SC_(0.10074598388671875e4), SC_(0.2213660813860061497235236488103940641175e-3), + SC_(0.49e2), SC_(0.1185395751953125e4), SC_(0.7493404081650790642056171825780752281226e-3), + SC_(0.52e2), SC_(0.24750102996826171875e2), SC_(-0.8431613692546440958427835805924554720791e10), + SC_(0.52e2), SC_(0.637722015380859375e2), SC_(-0.1115225300535010626455229302801432083119e-1), + SC_(0.52e2), SC_(0.1252804412841796875e3), SC_(0.1744660661299516008597246745970354205394e-2), + SC_(0.52e2), SC_(0.25554705810546875e3), SC_(0.3871637644673301315119914349891630279192e-2), + SC_(0.52e2), SC_(0.503011474609375e3), SC_(0.1991638261007561367287975226510189975247e-2), + SC_(0.52e2), SC_(0.10074598388671875e4), SC_(0.9236248731028976631695212457160479065712e-3), + SC_(0.52e2), SC_(0.1185395751953125e4), SC_(-0.4815665463345610915674607850479789186718e-3), + SC_(0.55e2), SC_(0.24750102996826171875e2), SC_(-0.5691393603639469399255323205962188964084e12), + SC_(0.55e2), SC_(0.637722015380859375e2), SC_(0.2006392327938634300862385080463293572511e-1), + SC_(0.55e2), SC_(0.1252804412841796875e3), SC_(-0.3619108023378490361858076559653708839139e-2), + SC_(0.55e2), SC_(0.25554705810546875e3), SC_(-0.2962856754548850012378937612844037689574e-2), + SC_(0.55e2), SC_(0.503011474609375e3), SC_(-0.5505670779964820186689463080699492719812e-3), + SC_(0.55e2), SC_(0.10074598388671875e4), SC_(-0.5086536723538768197623992809520331865266e-3), + SC_(0.55e2), SC_(0.1185395751953125e4), SC_(-0.6210859629328079732529943111160023713285e-3), + SC_(0.58e2), SC_(0.24750102996826171875e2), SC_(-0.4618503933439028301329509946039169091138e14), + SC_(0.58e2), SC_(0.637722015380859375e2), SC_(0.1402558269735743898212790386772249256841e-1), + SC_(0.58e2), SC_(0.1252804412841796875e3), SC_(0.4775384619486356255055098430929380737203e-2), + SC_(0.58e2), SC_(0.25554705810546875e3), SC_(-0.2015373395162996137089683702058573806025e-3), + SC_(0.58e2), SC_(0.503011474609375e3), SC_(-0.1623050637742511470123447378958753891598e-2), + SC_(0.58e2), SC_(0.10074598388671875e4), SC_(-0.7550604452502297124697636692423906683648e-3), + SC_(0.58e2), SC_(0.1185395751953125e4), SC_(0.6549895436356671365157678660370603606525e-3), + SC_(0.61e2), SC_(0.24750102996826171875e2), SC_(-0.4452493733367375303990916467937068229415e16), + SC_(0.61e2), SC_(0.637722015380859375e2), SC_(-0.1321805471789013172474419554540756891808e-1), + SC_(0.61e2), SC_(0.1252804412841796875e3), SC_(-0.531088356762363908667551449646716451263e-2), + SC_(0.61e2), SC_(0.25554705810546875e3), SC_(0.3137411891941427755699863352585675704239e-2), + SC_(0.61e2), SC_(0.503011474609375e3), SC_(0.1656289965758384211758434873072034259055e-2), + SC_(0.61e2), SC_(0.10074598388671875e4), SC_(0.7696420775050426274332928912448241809519e-3), + SC_(0.61e2), SC_(0.1185395751953125e4), SC_(0.4272402417161322625448726249065820441606e-3) + }; +#undef SC_ + + diff --git a/test/spherical_harmonic.ipp b/test/spherical_harmonic.ipp new file mode 100644 index 000000000..0115a0edc --- /dev/null +++ b/test/spherical_harmonic.ipp @@ -0,0 +1,1010 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 1000> spherical_harmonic = { + SC_(0.2e1), SC_(0), SC_(-0.6223074436187744140625e1), SC_(-0.983176708221435546875e0), SC_(0.62736841735769885881246893757736785347239567286304e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(-0.5057456493377685546875e1), SC_(0.59339153766632080078125e0), SC_(-0.20713028443163886820218719974386053923059852163073e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(-0.4687422275543212890625e1), SC_(0.5891966342926025390625e1), SC_(-0.31480190270523966739513025513763623833894947272512e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(-0.28032686710357666015625e1), SC_(0.46800785064697265625e1), SC_(0.52655067270403872306721376357891919506022788695237e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(-0.1743031024932861328125e1), SC_(0.4387288570404052734375e1), SC_(-0.28759993950019252230305986773209798109581312882458e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(0.30552928447723388671875e1), SC_(-0.317191779613494873046875e0), SC_(0.62375382255600227118437993629928966318732328906381e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(0.3735729694366455078125e1), SC_(-0.5883162021636962890625e1), SC_(0.33428112636668248892477814957401134243912590455598e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(0.37734358310699462890625e1), SC_(-0.25506031513214111328125e1), SC_(0.30071249849279422035163175145138750883542632972971e0), SC_(0), + SC_(0.2e1), SC_(0), SC_(0.54537200927734375e1), SC_(0.2279031276702880859375e1), SC_(0.1160546117367807407152783138997786541874144507988e0), SC_(0), + SC_(0.2e1), SC_(0.1e1), SC_(-0.458073139190673828125e1), SC_(0.509933185577392578125e1), SC_(0.37939894994349071850195546472417494785116097463318e-1), SC_(-0.93107300900343613424961589916279630747721170081866e-1), + SC_(0.2e1), SC_(0.1e1), SC_(-0.97907507419586181640625e0), SC_(0.195379292964935302734375e1), SC_(-0.13365658387823648349917293511332555081805906708217e0), SC_(0.33174333022099396014687314581478847312444020861066e0), + SC_(0.2e1), SC_(0.1e1), SC_(0.475549983978271484375e1), SC_(0.5774152278900146484375e1), SC_(0.29046640079014036022889946316442406419301954589267e-1), SC_(-0.16210635922809696460530259867846941769692426758954e-1), + SC_(0.2e1), SC_(0.1e1), SC_(0.52242870330810546875e1), SC_(0.175631821155548095703125e1), SC_(-0.60855473088463839496314678179027675166851871018551e-1), SC_(0.32425114600308867975046215949339819893698567878093e0), + SC_(0.2e1), SC_(0.1e1), SC_(0.5877228260040283203125e1), SC_(-0.4302561283111572265625e1), SC_(-0.11167901750699566664933060087163004380052822920546e0), SC_(0.25707237315223816631488907449414263457063550537321e0), + SC_(0.2e1), SC_(0.2e1), SC_(-0.4902621746063232421875e1), SC_(-0.18377502262592315673828125e0), SC_(0.34758668332269754124667464441534904092678544008633e0), SC_(-0.13383731780379523410231053175504807605617305066684e0), + SC_(0.2e1), SC_(0.2e1), SC_(-0.4131989955902099609375e1), SC_(-0.24892330169677734375e1), SC_(0.71028469303627438443505218504789723476775735703371e-1), SC_(0.26061737572960702576362821401262553454136075924086e0), + SC_(0.2e1), SC_(0.2e1), SC_(-0.39159076213836669921875e1), SC_(0.589130580425262451171875e0), SC_(0.72243575677708396654934991200786489958595832424973e-1), SC_(0.17449231348731311339771539907208960289022209309613e0), + SC_(0.2e1), SC_(0.2e1), SC_(-0.35055897235870361328125e1), SC_(0.16632754802703857421875e1), SC_(-0.48123182288009319551515814304893416646752842717278e-1), SC_(-0.90036880226735999457627735549531053450913137683366e-2), + SC_(0.2e1), SC_(0.2e1), SC_(-0.12724893093109130859375e1), SC_(0.32388579845428466796875e1), SC_(0.34625198570942398650986171441955620010816501680557e0), SC_(0.6821932943579386165556674909693473588513398324536e-1), + SC_(0.2e1), SC_(0.2e1), SC_(0.19570953845977783203125e1), SC_(0.37438814640045166015625e1), SC_(0.11868540133295449383018733211573720238248966454746e0), SC_(0.30946390521414868955137848944282844227519588015905e0), + SC_(0.2e1), SC_(0.2e1), SC_(0.5749200344085693359375e1), SC_(0.623871707916259765625e1), SC_(0.9966808034179761849929083987599231751430505405574e-1), SC_(-0.88875708639419834563244802526935291667296729862762e-2), + SC_(0.2e1), SC_(0.2e1), SC_(0.5913643360137939453125e1), SC_(0.6045803070068359375e1), SC_(0.44818825447699183791615026234271299189304152367279e-1), SC_(-0.23035725896903097839072699428746139571750062847173e-1), + SC_(0.3e1), SC_(0), SC_(-0.440936374664306640625e1), SC_(-0.5062591552734375e1), SC_(0.2844959862964824854388237165468357597848603400639e0), SC_(0), + SC_(0.3e1), SC_(0), SC_(-0.3934875011444091796875e1), SC_(-0.11469180583953857421875e1), SC_(0.14121291094740439000611232468259189623852053498783e0), SC_(0), + SC_(0.3e1), SC_(0), SC_(-0.552507817745208740234375e0), SC_(-0.2055795304477214813232421875e-1), SC_(0.19783391750498753000939983604310666040001705067569e0), SC_(0), + SC_(0.3e1), SC_(0), SC_(0.2055927753448486328125e1), SC_(-0.76976108551025390625e0), SC_(0.33285173555020716750381297236585266229434503862998e0), SC_(0), + SC_(0.3e1), SC_(0.1e1), SC_(-0.6247767925262451171875e1), SC_(0.194901907444000244140625e1), SC_(0.16876892664773021474708993031012643521641026445293e-1), SC_(-0.42473255665367397599231902837920937763489895056333e-1), + SC_(0.3e1), SC_(0.1e1), SC_(-0.47100925445556640625e1), SC_(-0.22983958721160888671875e1), SC_(-0.21493410580929797924752259751422347054230070438068e0), SC_(-0.24133524984657105208106822870162658732975114112697e0), + SC_(0.3e1), SC_(0.1e1), SC_(-0.423974609375e1), SC_(0.18090667724609375e1), SC_(0.2460020537420461905141929611988448892491267501235e-2), SC_(-0.10128361841653616652755617311405848132460463121178e-1), + SC_(0.3e1), SC_(0.1e1), SC_(-0.148838031291961669921875e1), SC_(-0.3641620159149169921875e1), SC_(0.27307271430136733126056860410690628785894268952483e0), SC_(-0.14919005653183612118737489713533290688614488819696e0), + SC_(0.3e1), SC_(0.1e1), SC_(0.528886890411376953125e1), SC_(0.32004873752593994140625e1), SC_(-0.13132892216208786526729131339765405257419451394884e0), SC_(-0.77435354342006197456232571693201218623653117535312e-2), + SC_(0.3e1), SC_(0.2e1), SC_(-0.563958072662353515625e1), SC_(0.3709591388702392578125e1), SC_(0.12400538306523070140083527916246828402942458169912e0), SC_(0.26699858682144904684388476504336635512136556796993e0), + SC_(0.3e1), SC_(0.2e1), SC_(-0.28145520687103271484375e1), SC_(0.25858356952667236328125e1), SC_(-0.4427518193860363815453664754812317897095435941741e-1), SC_(0.89525230913027456917186909094786496675085768579997e-1), + SC_(0.3e1), SC_(0.2e1), SC_(0.5657657146453857421875e1), SC_(-0.11826162040233612060546875e0), SC_(0.27612706743908622809999597264410624935441554778235e0), SC_(-0.66556246601320034007511476245916059803884974109287e-1), + SC_(0.3e1), SC_(0.2e1), SC_(0.5777313232421875e1), SC_(0.92682850360870361328125e0), SC_(-0.58588098669530777703405917826904778693250781231732e-1), SC_(0.20157326960446402040769133558058970326764280176092e0), + SC_(0.3e1), SC_(0.3e1), SC_(-0.527923882007598876953125e0), SC_(-0.683784008026123046875e0), SC_(-0.24654062825223548093087660191000851588872270901477e-1), SC_(-0.47291850857448633744932273700908288294545006619689e-1), + SC_(0.3e1), SC_(0.3e1), SC_(0.1838623523712158203125e1), SC_(0.36941986083984375e1), SC_(-0.3251989759113359602179775532073865855506456807785e-1), SC_(0.37275562661891880526110138370423344373839148188933e0), + SC_(0.3e1), SC_(0.3e1), SC_(0.4064691066741943359375e1), SC_(0.4045156002044677734375e1), SC_(0.19225943415968710973448026931813683101145483967527e0), SC_(-0.88384340622034426527979275575844466979738642942222e-1), + SC_(0.4e1), SC_(0), SC_(-0.440710163116455078125e1), SC_(-0.33909590244293212890625e1), SC_(0.60872192197795412389189648659338955746589054925866e-1), SC_(0), + SC_(0.4e1), SC_(0), SC_(-0.429697513580322265625e1), SC_(-0.4541179180145263671875e1), SC_(-0.10130272828064344799880549551171672634167040868427e0), SC_(0), + SC_(0.4e1), SC_(0), SC_(-0.18641016483306884765625e1), SC_(-0.543375682830810546875e1), SC_(0.77950672871982581371063052627417982915168633930729e-1), SC_(0), + SC_(0.4e1), SC_(0), SC_(0.7485844194889068603515625e-1), SC_(0.5530133724212646484375e1), SC_(0.82273248411762862558841087329680483291796232837127e0), SC_(0), + SC_(0.4e1), SC_(0), SC_(0.35047321319580078125e1), SC_(-0.335662305355072021484375e0), SC_(0.37133004542041317082392079068126598136623939939308e0), SC_(0), + SC_(0.4e1), SC_(0.1e1), SC_(-0.30877811908721923828125e1), SC_(-0.21055171489715576171875e1), SC_(0.51530379745010133318309937545431717391992159587507e-1), SC_(0.87003979532462364316044078325872969424339205057574e-1), + SC_(0.4e1), SC_(0.1e1), SC_(-0.2515388965606689453125e1), SC_(-0.322296047210693359375e1), SC_(0.35728040642895721804296646233781584207546649366613e0), SC_(-0.29135454916493737137065666660072086321415216443954e-1), + SC_(0.4e1), SC_(0.1e1), SC_(0.38922393321990966796875e1), SC_(0.4281579494476318359375e1), SC_(0.73224312308144597581776422776064913903291741167661e-1), SC_(0.15932143480326534106072259313508337288462063133883e0), + SC_(0.4e1), SC_(0.1e1), SC_(0.539428615570068359375e1), SC_(-0.355329418182373046875e1), SC_(0.46533898855307088507984825351331498092468310453766e-1), SC_(-0.20319290220199811762971093203847032309702077858032e-1), + SC_(0.4e1), SC_(0.1e1), SC_(0.577162647247314453125e1), SC_(0.1017331600189208984375e1), SC_(0.24653136285647999037266860579180968024317536393039e0), SC_(0.39899392923702488934009644431361230633591531405502e0), + SC_(0.4e1), SC_(0.2e1), SC_(-0.109746491909027099609375e1), SC_(0.4912235260009765625e1), SC_(-0.11098204562790300318279208564336530955042958737283e0), SC_(-0.46882241953600592630431606043540984795775718139569e-1), + SC_(0.4e1), SC_(0.3e1), SC_(-0.3470681667327880859375e1), SC_(-0.6059832096099853515625e1), SC_(0.31337958445241324767620450698180315330245557945883e-1), SC_(0.24830670306818716181091950369779498234247337466209e-1), + SC_(0.4e1), SC_(0.4e1), SC_(-0.312797260284423828125e1), SC_(0.5717919826507568359375e1), SC_(-0.96954964101799260839348807502986770292227228575254e-8), SC_(-0.11740977624668344825904309478314389528362476219229e-7), + SC_(0.4e1), SC_(0.4e1), SC_(0.4036896228790283203125e1), SC_(-0.3077565670013427734375e1), SC_(0.15878540892930306515344411249534442262277798236826e0), SC_(0.41579282626431184123743601642889070990303304799171e-1), + SC_(0.4e1), SC_(0.4e1), SC_(0.473447322845458984375e1), SC_(0.107150614261627197265625e1), SC_(-0.18283689147350112716914859883900530125561571604351e0), SC_(-0.40252223561364165551747117833207254896015413001567e0), + SC_(0.4e1), SC_(0.4e1), SC_(0.5119096279144287109375e1), SC_(-0.38126966953277587890625e1), SC_(-0.28253048985912201601336039243135828403941707239604e0), SC_(-0.13898722210008351680968086768165980849714173551268e0), + SC_(0.5e1), SC_(0), SC_(-0.6133619785308837890625e1), SC_(0.28519971370697021484375e1), SC_(0.78495683565043724315160369014308547334238000668841e0), SC_(0), + SC_(0.5e1), SC_(0), SC_(-0.132062244415283203125e1), SC_(-0.3846485912799835205078125e0), SC_(0.31693347535795755694751876346186174674215495248026e0), SC_(0), + SC_(0.5e1), SC_(0), SC_(0.193621504306793212890625e1), SC_(0.3750054836273193359375e0), SC_(-0.29624781700952612310404787297827585402918731693093e0), SC_(0), + SC_(0.5e1), SC_(0), SC_(0.32320735454559326171875e1), SC_(-0.135314095020294189453125e1), SC_(-0.8790123246629224792127618910569050934884656058578e0), SC_(0), + SC_(0.5e1), SC_(0), SC_(0.455754852294921875e1), SC_(0.3585579693317413330078125e0), SC_(-0.24115920970937134101170155815017359735279175844133e0), SC_(0), + SC_(0.5e1), SC_(0.1e1), SC_(0.387013494968414306640625e0), SC_(0.11664593219757080078125e1), SC_(-0.21103277179368909749392113640951296749695200975509e0), SC_(-0.49316527556044503500327911477758500378219060039796e0), + SC_(0.5e1), SC_(0.1e1), SC_(0.369808864593505859375e1), SC_(-0.8027703762054443359375e0), SC_(0.21419837825943510697987352828754457359028432984086e0), SC_(-0.22177292654383587982333487011338866121140808680977e0), + SC_(0.5e1), SC_(0.2e1), SC_(0.6248452663421630859375e0), SC_(0.292543506622314453125e1), SC_(0.41573590536157272547449027947361423003543786240211e0), SC_(-0.19183141402020165103358558903368617865081892865933e0), + SC_(0.5e1), SC_(0.3e1), SC_(-0.14002730846405029296875e1), SC_(-0.4245142459869384765625e1), SC_(-0.24179587072105227507534954893340732889531316289458e0), SC_(0.41271158120487694865505940795712844055500009419391e-1), + SC_(0.5e1), SC_(0.3e1), SC_(0.773553669452667236328125e0), SC_(-0.150236189365386962890625e1), SC_(0.86752460384204523734238237483181189266881715764871e-1), 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SC_(0.147180497646331787109375e1), SC_(0.10235270561549462717073795224019921669009190057754e0), SC_(-0.25411603142079242733272237982972076450193559196436e0), + SC_(0.48e2), SC_(0.13e2), SC_(-0.28512251377105712890625e1), SC_(0.21358762681484222412109375e0), SC_(0.65082793160341590914926316444150324051891586307335e0), SC_(-0.24866103078135174972315060854694095694134464474465e0), + SC_(0.48e2), SC_(0.15e2), SC_(0.21509382724761962890625e1), SC_(0.505238056182861328125e1), SC_(0.33153682677233458889683871883764794291900552041288e0), SC_(0.13530604504841517586204066208483785803300406806222e0), + SC_(0.48e2), SC_(0.17e2), SC_(-0.410233306884765625e1), SC_(-0.5055477142333984375e1), SC_(0.120658610881024129421142111496549042350283006718e0), SC_(-0.24934508488272542971829338925578466799768762724609e0), + SC_(0.48e2), SC_(0.18e2), SC_(0.419954395294189453125e1), SC_(-0.31286038458347320556640625e-1), SC_(0.18229465107186350461724966401750299694404393980909e0), SC_(-0.11509073643647539446369783158526832981463008146529e0), + SC_(0.48e2), SC_(0.19e2), SC_(0.33512248992919921875e1), SC_(0.5698537349700927734375e1), SC_(-0.11430916047691646574027070138178816779368218597795e-4), SC_(-0.10096469757450613482659522651534357791031593710281e-3), + SC_(0.48e2), SC_(0.2e2), SC_(-0.4913626194000244140625e1), SC_(0.4570525646209716796875e1), SC_(0.27544694591170425084361640327912372332575356920479e0), SC_(0.86513078636045023383894712247218808351168016037179e-1), + SC_(0.48e2), SC_(0.22e2), SC_(-0.35437946319580078125e1), SC_(-0.14283583164215087890625e1), SC_(0.92521800679569304410458632278427028786030152854651e-1), SC_(-0.73615833685271087989994719034708442687869544183798e-3), + SC_(0.48e2), SC_(0.24e2), SC_(0.2529537379741668701171875e0), SC_(-0.13951661586761474609375e1), SC_(-0.14461558442027612635993431382924805648986580565293e-5), SC_(-0.26644654773805667856532999455349036511597661939098e-5), + SC_(0.48e2), SC_(0.25e2), SC_(-0.5302140712738037109375e1), SC_(0.534227657318115234375e1), SC_(0.14567724220048959327288429307006198049897616049705e-1), SC_(-0.37118356905088388538181275498694280409563892893147e0), + SC_(0.48e2), SC_(0.26e2), SC_(0.4788922786712646484375e1), SC_(0.114234256744384765625e1), SC_(-0.49889909138094330244956751684356082959364552066594e-1), SC_(-0.34351914682484557095802003482605269948574737582585e0), + SC_(0.48e2), SC_(0.29e2), SC_(-0.155845987796783447265625e1), SC_(0.265560741536319255828857421875e-2), SC_(-0.16358032708366624805995699759082827770270765931603e0), SC_(-0.12622713495579813089501286854682899269376306838204e-1), + SC_(0.48e2), SC_(0.31e2), SC_(-0.814608156681060791015625e0), SC_(-0.5973989009857177734375e1), SC_(-0.21849236027340450931517916914197181334381665924407e0), SC_(-0.35329064201860941281957627439495576583081752554927e-1), + SC_(0.48e2), SC_(0.34e2), SC_(0.5690162181854248046875e1), SC_(-0.4747711181640625e1), SC_(-0.14208000946756588475345650324158728024525891044283e-2), SC_(0.36648708355068107608507564157213823609107549413423e-2), + SC_(0.48e2), SC_(0.36e2), SC_(-0.606925201416015625e1), SC_(-0.6036619663238525390625e1), SC_(-0.33060032760649252071566767946067882261145163749527e-17), SC_(0.20197241068338690816933400483260997588417796049136e-17), + SC_(0.48e2), SC_(0.36e2), SC_(-0.557515811920166015625e1), SC_(0.549843120574951171875e1), SC_(-0.29793042772484214996215596790395794084288963000676e-1), SC_(-0.69091218291545995969013380580434842461096487036142e-3), + SC_(0.48e2), SC_(0.37e2), SC_(-0.4484233856201171875e1), SC_(-0.586126422882080078125e1), SC_(0.28783210566723256763209668589242474045102053584988e0), SC_(-0.27973722891666817142600363847230369142196912070032e-1), + SC_(0.49e2), SC_(0.4e1), SC_(0.7450397014617919921875e0), SC_(-0.486802196502685546875e1), SC_(0.43523446367954232471457024383074543728337571811041e-1), SC_(-0.31238445205075929681258187766528850038918026320874e-1), + SC_(0.49e2), SC_(0.5e1), SC_(-0.705654084682464599609375e0), SC_(0.36159880161285400390625e1), SC_(0.33812266273696200694321547558772118159278050274115e-1), SC_(-0.32761490529250158683739840580938438700363222937706e-1), + SC_(0.49e2), SC_(0.6e1), SC_(0.4099590480327606201171875e0), SC_(-0.148899996280670166015625e1), SC_(0.33872549952379279074817297974186583810635403131808e-1), SC_(0.18101089014763158990806574103071386037200021990488e-1), + SC_(0.49e2), SC_(0.6e1), SC_(0.288681316375732421875e1), SC_(0.555331707000732421875e1), SC_(-0.17299631476757984574148560709032937173460259126009e0), SC_(0.49987182677495761333169759620280021625021164232635e0), + SC_(0.49e2), SC_(0.8e1), SC_(-0.28113791942596435546875e1), SC_(0.4268764019012451171875e1), SC_(0.11234284818582161154493446715155148494092582868555e0), SC_(-0.4848176775276798667653033304055062110686751478506e-1), + SC_(0.49e2), SC_(0.8e1), SC_(-0.175365102291107177734375e1), SC_(0.3699061870574951171875e0), SC_(0.150510223757467945212888241106383078890968767236e0), SC_(-0.27752777621993241253038961455135547546359775538701e-1), + SC_(0.49e2), SC_(0.1e2), SC_(-0.623242966830730438232421875e-1), SC_(0.99178183078765869140625e0), SC_(-0.39129153600073467169496931061202903620397035718226e-4), SC_(-0.21024087848986552498673205313374207046750995885514e-4), + SC_(0.49e2), SC_(0.11e2), SC_(0.4610438823699951171875e1), SC_(0.516170978546142578125e1), SC_(-0.64931321841066713098308595750847102912382779925862e-1), SC_(-0.15212818549984179656962126137889372220056261241257e-1), + SC_(0.49e2), SC_(0.12e2), SC_(-0.492767429351806640625e1), SC_(0.5237930774688720703125e1), SC_(-0.25815646423352702571250699074506009315207777917256e0), SC_(-0.60203250835447592212607712693653576640800332594405e-2), + SC_(0.49e2), SC_(0.12e2), SC_(-0.285912036895751953125e1), SC_(0.203067779541015625e1), SC_(-0.58309335353629437699137207747030784181611219363777e0), SC_(0.55933823022969032857356665544019317581369057604275e0), + SC_(0.49e2), SC_(0.21e2), SC_(0.213364541530609130859375e0), SC_(0.3487752437591552734375e1), SC_(0.68979677728313700400977850242007578301598557836144e-5), SC_(0.10423031489346059064213401632074682208378658096143e-4), + SC_(0.49e2), SC_(0.24e2), SC_(-0.5363933563232421875e1), SC_(-0.13800899982452392578125e1), SC_(-0.39624366668844441499649851882221871259293339848909e-1), SC_(-0.29077551093957529339300484162110207541478366743715e0), + SC_(0.49e2), SC_(0.24e2), SC_(0.113704526424407958984375e1), SC_(-0.524048984050750732421875e0), SC_(-0.95716020864868532424772271176706645369949424860348e-1), SC_(0.10342521283811512447413822494381438186657088095508e-2), + SC_(0.49e2), SC_(0.26e2), SC_(0.195927083492279052734375e1), SC_(0.2748439788818359375e1), SC_(0.11971762382480258608379673285431990043583083168738e0), SC_(-0.12257642890929780276867696344794516000179864725662e0), + SC_(0.49e2), SC_(0.27e2), SC_(0.49123775959014892578125e0), SC_(0.2262770175933837890625e1), SC_(0.10204568382238142188564702604070162264838835105016e-1), SC_(0.60809845254720905762068213640063158711493179195797e-1), + SC_(0.49e2), SC_(0.28e2), SC_(-0.26242389678955078125e1), SC_(0.551108074188232421875e1), SC_(0.62995535686340908516656601005584507927420828344839e-1), SC_(0.24594729609877561046325340633244478848156201312828e-1), + SC_(0.49e2), SC_(0.29e2), SC_(-0.30029771327972412109375e1), SC_(-0.2872400760650634765625e1), SC_(-0.55339515369306716058956107221780747493058274670244e-17), SC_(-0.11662133768710831442193525927180620131415498665886e-15), + SC_(0.49e2), SC_(0.3e2), SC_(0.611942386627197265625e1), SC_(-0.63407027721405029296875e0), SC_(0.1394293153272931999882487586325865494180618327615e-14), SC_(-0.24300519585643234499387785103683410462334687962488e-15), + SC_(0.49e2), SC_(0.35e2), SC_(-0.166894090175628662109375e1), SC_(0.4225218296051025390625e1), SC_(-0.35508021992434406559798366674327649256336649541254e0), SC_(-0.82310724679645695220583934477252032989621192496738e-1), + SC_(0.49e2), SC_(0.36e2), SC_(-0.32497041225433349609375e1), SC_(0.107579600811004638671875e1), SC_(-0.17013596752796213880005913256594710378533147240406e-27), SC_(-0.28302568795764666512713804293802636201022080165826e-27), + SC_(0.49e2), SC_(0.37e2), SC_(-0.5017117023468017578125e1), SC_(0.383155155181884765625e1), SC_(-0.35232624098397971030888657233932783396857095781041e0), SC_(-0.14719428675619986819270542639579914303323853912089e0), + SC_(0.49e2), SC_(0.37e2), SC_(-0.23628532886505126953125e1), SC_(-0.123920929431915283203125e1), SC_(-0.40342907812326051474768377766202705403179945762591e-1), SC_(-0.13151134923462644400467105161114547636592907143886e0), + SC_(0.49e2), SC_(0.38e2), SC_(0.33035366535186767578125e1), SC_(0.2936229705810546875e1), SC_(-0.14617119042429598121920579635742118107020065282388e-24), SC_(0.29099331615315387488356539500057397404411535737234e-23), + SC_(0.49e2), SC_(0.4e2), SC_(0.4189170360565185546875e1), SC_(-0.5219272136688232421875e1), SC_(-0.71881448396381481201576224141351106542999167239121e-1), SC_(0.49216549050921898631651418491099543778321238775524e0) + }; +#undef SC_ + diff --git a/test/std_real_concept_check.cpp b/test/std_real_concept_check.cpp new file mode 100644 index 000000000..1554734ee --- /dev/null +++ b/test/std_real_concept_check.cpp @@ -0,0 +1,199 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false + +#include +#include + +#include "compile_test/instantiate.hpp" + +// +// The purpose of this test is to verify that our code compiles +// cleanly with a type whose std lib functions are in namespace +// std and can *not* be found by ADL. This verifies that we're +// not finding std lib functions that are in the global namespace +// for example calling ::pow(double) rather than std::pow(long double). +// This is a silent error that does the wrong thing at runtime, and +// of course we can't call std::pow() directly because we want +// the functions to be found by ADL when that's appropriate. +// +// Furthermore our code does different things internally depending +// on numeric_limits<>::digits, so there are some macros that can +// be defined that cause our concept-archetype to emulate various +// floating point types: +// +// EMULATE32: 32-bit float +// EMULATE64: 64-bit double +// EMULATE80: 80-bit long double +// EMULATE128: 128-bit long double +// +// In order to ensure total code coverage this file must be +// compiled with each of the above macros in turn, and then +// without any of the above as well! +// + +#define NULL_MACRO /**/ +#ifdef EMULATE32 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 24; + static const int digits10 = 6; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -125; + static const int min_exponent10 = -37; + static const int max_exponent = 128; + static const int max_exponent10 = 38; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE64 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 53; + static const int digits10 = 15; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -1021; + static const int min_exponent10 = -307; + static const int max_exponent = 1024; + static const int max_exponent10 = 308; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE80 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 64; + static const int digits10 = 18; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -16381; + static const int min_exponent10 = -4931; + static const int max_exponent = 16384; + static const int max_exponent10 = 4932; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif +#ifdef EMULATE128 +namespace std{ +template<> +struct numeric_limits +{ + static const bool is_specialized = true; + static boost::math::concepts::std_real_concept min NULL_MACRO() throw(); + static boost::math::concepts::std_real_concept max NULL_MACRO() throw(); + static const int digits = 113; + static const int digits10 = 33; + static const bool is_signed = true; + static const bool is_integer = false; + static const bool is_exact = false; + static const int radix = 2; + static boost::math::concepts::std_real_concept epsilon() throw(); + static boost::math::concepts::std_real_concept round_error() throw(); + static const int min_exponent = -16381; + static const int min_exponent10 = -4931; + static const int max_exponent = 16384; + static const int max_exponent10 = 4932; + static const bool has_infinity = true; + static const bool has_quiet_NaN = true; + static const bool has_signaling_NaN = true; + static const float_denorm_style has_denorm = denorm_absent; + static const bool has_denorm_loss = false; + static boost::math::concepts::std_real_concept infinity() throw(); + static boost::math::concepts::std_real_concept quiet_NaN() throw(); + static boost::math::concepts::std_real_concept signaling_NaN() throw(); + static boost::math::concepts::std_real_concept denorm_min() throw(); + static const bool is_iec559 = true; + static const bool is_bounded = false; + static const bool is_modulo = false; + static const bool traps = false; + static const bool tinyness_before = false; + static const float_round_style round_style = round_toward_zero; +}; +} +#endif + + + +int main() +{ + instantiate(boost::math::concepts::std_real_concept(0)); +} + + diff --git a/test/test_bernoulli.cpp b/test/test_bernoulli.cpp new file mode 100644 index 000000000..e86389b27 --- /dev/null +++ b/test/test_bernoulli.cpp @@ -0,0 +1,268 @@ +// test_bernoulli.cpp + +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity test for Bernoulli Cumulative Distribution Function. + +// Default domain error policy is +// #define BOOST_MATH_DOMAIN_ERROR_POLICY throw_on_error + +#include // for bernoulli_distribution +using boost::math::bernoulli_distribution; + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE_FRACTION, BOOST_CHECK_EQUAL... + +#include +using std::cout; +using std::endl; +using std::fixed; +using std::right; +using std::left; +using std::showpoint; +using std::showpos; +using std::setw; +using std::setprecision; + +#include +using std::numeric_limits; + +template // Any floating-point type RealType. +void test_spots(RealType) +{ // Parameter only provides the type, float, double... value ignored. + + // Basic sanity checks, test data may be to double precision only + // so set tolerance to 100 eps expressed as a fraction, + // or 100 eps of type double expressed as a fraction, + // whichever is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); + tolerance *= 100; + + cout << "Tolerance for type " << typeid(RealType).name() << " is " + << setprecision(3) << tolerance << " (or " << tolerance * 100 << "%)." << endl; + + // Sources of spot test values - calculator, + // or Steve Moshier's command interpreter V1.3 100 decimal digit calculator, + // Wolfram function evaluator. + + using boost::math::bernoulli_distribution; // of type RealType. + using ::boost::math::cdf; + using ::boost::math::pdf; + + BOOST_CHECK_EQUAL(bernoulli_distribution(static_cast(0.5)).success_fraction(), static_cast(0.5)); + BOOST_CHECK_EQUAL(bernoulli_distribution(static_cast(0.1L)).success_fraction(), static_cast(0.1L)); + BOOST_CHECK_EQUAL(bernoulli_distribution(static_cast(0.9L)).success_fraction(), static_cast(0.9L)); + + BOOST_CHECK_THROW( // Constructor success_fraction outside 0 to 1. + bernoulli_distribution(static_cast(2)), std::domain_error); + BOOST_CHECK_THROW( + bernoulli_distribution(static_cast(-2)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( // pdf k neither 0 nor 1. + bernoulli_distribution(static_cast(0.25L)), static_cast(-1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( // pdf k neither 0 nor 1. + bernoulli_distribution(static_cast(0.25L)), static_cast(2)), std::domain_error); + + BOOST_CHECK_EQUAL( + pdf( // OK k (or n) + bernoulli_distribution(static_cast(0.5L)), static_cast(0)), + static_cast(0.5)); // Expect 1 - p. + + BOOST_CHECK_CLOSE_FRACTION( + pdf( // OK k (or n) + bernoulli_distribution(static_cast(0.6L)), static_cast(0)), + static_cast(0.4L), tolerance); // Expect 1 - p. + + BOOST_CHECK_CLOSE_FRACTION( + pdf( // OK k (or n) + bernoulli_distribution(static_cast(0.6L)), static_cast(0)), + static_cast(0.4L), tolerance); // Expect 1- p. + + BOOST_CHECK_CLOSE_FRACTION( + pdf( // OK k (or n) + bernoulli_distribution(static_cast(0.4L)), static_cast(0)), + static_cast(0.6L), tolerance); // Expect 1- p. + + BOOST_CHECK_EQUAL( + mean(bernoulli_distribution(static_cast(0.5L))), static_cast(0.5L)); + + BOOST_CHECK_EQUAL( + mean(bernoulli_distribution(static_cast(0.1L))), + static_cast(0.1L)); + + BOOST_CHECK_CLOSE_FRACTION( + variance(bernoulli_distribution(static_cast(0.1L))), + static_cast(0.09L), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + skewness(bernoulli_distribution(static_cast(0.1L))), + static_cast(2.666666666666666666666666666666666666666666L), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + kurtosis(bernoulli_distribution(static_cast(0.1L))), + static_cast(8.11111111111111111111111111111111111111111111L), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(bernoulli_distribution(static_cast(0.1L))), + static_cast(5.11111111111111111111111111111111111111111111L), + tolerance); + + BOOST_CHECK_THROW( + quantile( + bernoulli_distribution(static_cast(2)), // prob >1 + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + bernoulli_distribution(static_cast(-1)), // prob < 0 + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + bernoulli_distribution(static_cast(0.5L)), // k >1 + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + bernoulli_distribution(static_cast(0.5L)), // k < 0 + static_cast(2)), std::domain_error + ); + + BOOST_CHECK_CLOSE_FRACTION( + cdf( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0)), + static_cast(0.4L), // 1 - p + tolerance + ); + + BOOST_CHECK_CLOSE_FRACTION( + cdf( + bernoulli_distribution(static_cast(0.6L)), + static_cast(1)), + static_cast(1), // p + tolerance + ); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(complement( + bernoulli_distribution(static_cast(0.6L)), + static_cast(1))), + static_cast(0), + tolerance + ); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(complement( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0))), + static_cast(0.6L), + tolerance + ); + + BOOST_CHECK_EQUAL( + quantile( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0.1L)), // < p + static_cast(0) + ); + + BOOST_CHECK_EQUAL( + quantile( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0.9L)), // > p + static_cast(1) + ); + + BOOST_CHECK_EQUAL( + quantile(complement( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0.1L))), // < p + static_cast(1) + ); + + BOOST_CHECK_EQUAL( + quantile(complement( + bernoulli_distribution(static_cast(0.6L)), + static_cast(0.9L))), // > p + static_cast(0) + ); +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + // Check that can generate bernoulli distribution using both convenience methods: + bernoulli_distribution bn1(0.5); // Using default RealType double. + boost::math::bernoulli bn2(0.5); // Using typedef. + + BOOST_CHECK_EQUAL(bn1.success_fraction(), 0.5); + BOOST_CHECK_EQUAL(bn2.success_fraction(), 0.5); + + BOOST_CHECK_THROW(bernoulli_distribution(-1), std::domain_error); // p outside 0 to 1. + BOOST_CHECK_THROW(bernoulli_distribution(+2), std::domain_error); // p outside 0 to 1. + BOOST_CHECK_THROW(bernoulli_distribution bn3(std::numeric_limits::quiet_NaN() ), std::domain_error); // p outside 0 to 1. + BOOST_CHECK_THROW(bernoulli_distribution bn4(std::numeric_limits::infinity() ), std::domain_error); // p outside 0 to 1. + + BOOST_CHECK_EQUAL(kurtosis(bn2) -3, kurtosis_excess(bn2)); + BOOST_CHECK_EQUAL(kurtosis_excess(bn2), -2); + + //using namespace boost::math; or + using boost::math::bernoulli; + + double tol5eps = std::numeric_limits::epsilon() * 5; // 5 eps as a fraction. + // Default bernoulli is type double, so these test values should also be type double. + BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(bernoulli(0.1)), 5.11111111111111111111111111111111111111111111111111, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(bernoulli(0.9)), 5.11111111111111111111111111111111111111111111111111, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(kurtosis(bernoulli(0.6)), 1./0.4 + 1./0.6 -3., tol5eps); + BOOST_CHECK_EQUAL(kurtosis(bernoulli(0)), +std::numeric_limits::infinity()); + BOOST_CHECK_EQUAL(kurtosis(bernoulli(1)), +std::numeric_limits::infinity()); + // + + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Running 1 test case... +Tolerance for type float is 1.19e-005 (or 0.00119%). +Tolerance for type double is 2.22e-014 (or 2.22e-012%). +Tolerance for type long double is 2.22e-014 (or 2.22e-012%). +Tolerance for type class boost::math::concepts::real_concept is 2.22e-014 (or 2.22e-012%). +*** No errors detected + +No warnings MSVC level 4 31 Jul 2007 + +*/ + + diff --git a/test/test_bessel_hooks.hpp b/test/test_bessel_hooks.hpp new file mode 100644 index 000000000..c089e4acf --- /dev/null +++ b/test/test_bessel_hooks.hpp @@ -0,0 +1,97 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double jv(double, double); + float jvf(float, float); + long double jvl(long double, long double); + double yv(double, double); + float yvf(float, float); + long double yvl(long double, long double); +} +inline float cyl_bessel_j(float a, float x) +{ return jvf(a, x); } +inline double cyl_bessel_j(double a, double x) +{ return jv(a, x); } +inline long double cyl_bessel_j(long double a, long double x) +{ +#ifdef BOOST_MSVC + return jv((double)a, x); +#else + return jvl(a, x); +#endif +} +inline float cyl_neumann(float a, float x) +{ return yvf(a, x); } +inline double cyl_neumann(double a, double x) +{ return yv(a, x); } +inline long double cyl_neumann(long double a, long double x) +{ +#ifdef BOOST_MSVC + return yv((double)a, x); +#else + return yvl(a, x); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#include +#include +#include + +namespace other{ +inline float cyl_bessel_j(float a, float x) +{ return (float)gsl_sf_bessel_Jnu(a, x); } +inline double cyl_bessel_j(double a, double x) +{ return gsl_sf_bessel_Jnu(a, x); } +inline long double cyl_bessel_j(long double a, long double x) +{ return gsl_sf_bessel_Jnu(a, x); } + +inline float cyl_bessel_i(float a, float x) +{ return (float)gsl_sf_bessel_Inu(a, x); } +inline double cyl_bessel_i(double a, double x) +{ return gsl_sf_bessel_Inu(a, x); } +inline long double cyl_bessel_i(long double a, long double x) +{ return gsl_sf_bessel_Inu(a, x); } + +inline float cyl_bessel_k(float a, float x) +{ return (float)gsl_sf_bessel_Knu(a, x); } +inline double cyl_bessel_k(double a, double x) +{ return gsl_sf_bessel_Knu(a, x); } +inline long double cyl_bessel_k(long double a, long double x) +{ return gsl_sf_bessel_Knu(a, x); } + +inline float cyl_neumann(float a, float x) +{ return (float)gsl_sf_bessel_Ynu(a, x); } +inline double cyl_neumann(double a, double x) +{ return gsl_sf_bessel_Ynu(a, x); } +inline long double cyl_neumann(long double a, long double x) +{ return gsl_sf_bessel_Ynu(a, x); } +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept cyl_bessel_j(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept cyl_bessel_i(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept cyl_bessel_k(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept cyl_neumann(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + + diff --git a/test/test_bessel_i.cpp b/test/test_bessel_i.cpp new file mode 100644 index 000000000..0a49128e0 --- /dev/null +++ b/test/test_bessel_i.cpp @@ -0,0 +1,285 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_bessel_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the bessel I function. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // Mac OS has higher error rates, why? + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 100, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 100, 50); // test function + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 15, 10); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 15, 10); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +T cyl_bessel_i_int_wrapper(T v, T x) +{ + return static_cast( + boost::math::cyl_bessel_i( + boost::math::tools::real_cast(v), x)); +} + +template +void do_test_cyl_bessel_i(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::cyl_bessel_i; +#else + pg funcp = boost::math::cyl_bessel_i; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_i against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_i", test_name); + std::cout << std::endl; + +#ifdef TEST_OTHER + if(boost::is_floating_point::value) + { + funcp = other::cyl_bessel_i; + + // + // test other::cyl_bessel_i against data: + // + result = boost::math::tools::test( + data, + boost::lambda::bind(funcp, + boost::lambda::ret(boost::lambda::_1[0]), + boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[2])); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::cyl_bessel_i"); + std::cout << std::endl; + } +#endif +} + +template +void do_test_cyl_bessel_i_int(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = cyl_bessel_i_int_wrapper; +#else + pg funcp = cyl_bessel_i_int_wrapper; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_i against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_i", test_name); + std::cout << std::endl; +} + +template +void test_bessel(T, const char* name) +{ + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 10> i0_data = {{ + SC_(0), SC_(0), SC_(1), + SC_(0), SC_(1), SC_(1.26606587775200833559824462521471753760767031135496220680814), + SC_(0), SC_(-2), SC_(2.27958530233606726743720444081153335328584110278545905407084), + SC_(0), SC_(4), SC_(11.3019219521363304963562701832171024974126165944353377060065), + SC_(0), SC_(-7), SC_(168.593908510289698857326627187500840376522679234531714193194), + SC_(0), SC_(1) / 1024, SC_(1.00000023841859331241759166109699567801556273303717896447683), + SC_(0), SC_(1) / (1024*1024), SC_(1.00000000000022737367544324498417583090700894607432256476338), + SC_(0), SC_(-1), SC_(1.26606587775200833559824462521471753760767031135496220680814), + SC_(0), SC_(100), SC_(1.07375170713107382351972085760349466128840319332527279540154e42), + SC_(0), SC_(200), SC_(2.03968717340972461954167312677945962233267573614834337894328e85), + }}; + static const boost::array, 10> i1_data = { + SC_(1), SC_(0), SC_(0), + SC_(1), SC_(1), SC_(0.565159103992485027207696027609863307328899621621092009480294), + SC_(1), SC_(-2), SC_(-1.59063685463732906338225442499966624795447815949553664713229), + SC_(1), SC_(4), SC_(9.75946515370444990947519256731268090005597033325296730692753), + SC_(1), SC_(-8), SC_(-399.873136782560098219083086145822754889628443904067647306574), + SC_(1), SC_(1)/1024, SC_(0.000488281308207663226432087816784315537514225208473395063575150), + SC_(1), SC_(1)/(1024*1024), SC_(4.76837158203179210108624277276025646653133998635956784292029E-7), + SC_(1), SC_(-10), SC_(-2670.98830370125465434103196677215254914574515378753771310849), + SC_(1), SC_(100), SC_(1.06836939033816248120614576322429526544612284405623226965918e42), + SC_(1), SC_(200), SC_(2.03458154933206270342742797713906950389661161681122964159220e85), + }; + static const boost::array, 10> in_data = { + SC_(-2), SC_(0), SC_(0), + SC_(2), SC_(1)/(1024*1024), SC_(1.13686837721624646204093977095674566928522671779753217215467e-13), + SC_(5), SC_(10), SC_(777.188286403259959907293484802339632852674154572666041953297), + SC_(-5), SC_(100), SC_(9.47009387303558124618275555002161742321578485033007130107740e41), + SC_(-5), SC_(-1), SC_(-0.000271463155956971875181073905153777342383564426758143634974124), + SC_(10), SC_(20), SC_(3.54020020901952109905289138244985607057267103782948493874391e6), + SC_(10), SC_(-5), SC_(0.00458004441917605126118647027872016953192323139337073320016447), + SC_(1e+02), SC_(9), SC_(2.74306601746058997093587654668959071522869282506446891736820e-93), + SC_(1e+02), SC_(80), SC_(4.65194832850610205318128191404145885093970505338730540776711e8), + SC_(-100), SC_(-200), SC_(4.35275044972702191438729017441198257508190719030765213981307e74), + }; + static const boost::array, 10> iv_data = { + SC_(2.25), SC_(1)/(1024*1024), SC_(2.34379212133481347189068464680335815256364262507955635911656e-15), + SC_(5.5), SC_(3.125), SC_(0.0583514045989371500460946536220735787163510569634133670181210), + SC_(-5) + T(1)/1024, SC_(2.125), SC_(0.0267920938009571023702933210070984416052633027166975342895062), + SC_(-5.5), SC_(10), SC_(597.577606961369169607937419869926705730305175364662688426534), + SC_(-5.5), SC_(100), SC_(9.22362906144706871737354069133813819358704200689067071415379e41), + SC_(-10486074)/(1024*1024), SC_(1)/1024, SC_(1.41474005665181350367684623930576333542989766867888186478185e35), + SC_(-10486074)/(1024*1024), SC_(50), SC_(1.07153277202900671531087024688681954238311679648319534644743e20), + SC_(144794)/1024, SC_(100), SC_(2066.27694757392660413922181531984160871678224178890247540320), + SC_(144794)/1024, SC_(200), SC_(2.23699739472246928794922868978337381373643889659337595319774e64), + SC_(-144794)/1024, SC_(100), SC_(2066.27694672763190927440969155740243346136463461655104698748), + }; + #undef SC_ + + do_test_cyl_bessel_i(i0_data, name, "Bessel I0: Mathworld Data"); + do_test_cyl_bessel_i(i1_data, name, "Bessel I1: Mathworld Data"); + do_test_cyl_bessel_i(in_data, name, "Bessel In: Mathworld Data"); + + do_test_cyl_bessel_i_int(i0_data, name, "Bessel I0: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_i_int(i1_data, name, "Bessel I1: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_i_int(in_data, name, "Bessel In: Mathworld Data (Integer Version)"); + + do_test_cyl_bessel_i(iv_data, name, "Bessel Iv: Mathworld Data"); + +#include "bessel_i_int_data.ipp" + do_test_cyl_bessel_i(bessel_i_int_data, name, "Bessel In: Random Data"); +#include "bessel_i_data.ipp" + do_test_cyl_bessel_i(bessel_i_data, name, "Bessel Iv: Random Data"); +} + +int test_main(int, char* []) +{ +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_bessel(0.1F, "float"); +#endif + test_bessel(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_bessel(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_bessel(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + diff --git a/test/test_bessel_j.cpp b/test/test_bessel_j.cpp new file mode 100644 index 000000000..3955e18cc --- /dev/null +++ b/test/test_bessel_j.cpp @@ -0,0 +1,544 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_bessel_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the bessel functions. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // HP-UX specific rates: + // + // Error rate for double precision are limited by the accuracy of + // the approximations use, which bracket rather than preserve the root. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + ".*J0.*Tricky.*", // test data group + ".*", 80000000000LL, 80000000000LL); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + ".*J1.*Tricky.*", // test data group + ".*", 3000000, 2000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + "double", // test type(s) + ".*Tricky.*", // test data group + ".*", 100000, 100000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + ".*J.*Tricky.*", // test data group + ".*", 3000, 500); // test function + // + // HP Tru64: + // + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*Tricky.*", // test data group + ".*", 100000, 100000); // test function + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Tricky large.*", // test data group + ".*", 3000, 1000); // test function + // + // Solaris specific rates: + // + // Error rate for double precision are limited by the accuracy of + // the approximations use, which bracket rather than preserve the root. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Sun Solaris", // platform + largest_type, // test type(s) + "Bessel J: Random Data.*Tricky.*", // test data group + ".*", 3000, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Sun Solaris", // platform + "double", // test type(s) + ".*Tricky.*", // test data group + ".*", 200000, 100000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Sun Solaris", // platform + largest_type, // test type(s) + ".*J.*tricky.*", // test data group + ".*", 400000000, 200000000); // test function + // + // Mac OS X: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + ".*J0.*Tricky.*", // test data group + ".*", 400000000, 400000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + ".*J1.*Tricky.*", // test data group + ".*", 3000000, 2000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "Bessel JN.*", // test data group + ".*", 40000, 20000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "Bessel J:.*", // test data group + ".*", 50000, 20000); // test function + + + + // + // Linux specific results: + // + // sin and cos appear to have only double precision for large + // arguments on some linux distros: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + ".*J:.*", // test data group + ".*", 40000, 30000); // test function + + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if((std::numeric_limits::digits != std::numeric_limits::digits) + && (std::numeric_limits::digits < 90)) + { + // some errors spill over into type double as well: + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*J0.*Tricky.*", // test data group + ".*", 400000, 400000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*J1.*Tricky.*", // test data group + ".*", 5000, 5000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*(JN|j).*|.*Tricky.*", // test data group + ".*", 50, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 30, 30); // test function + // + // and we have a few cases with higher limits as well: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*J0.*Tricky.*", // test data group + ".*", 400000000, 400000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*J1.*Tricky.*", // test data group + ".*", 5000000, 5000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*(JN|j).*|.*Tricky.*", // test data group + ".*", 33000, 20000); // test function + } +#endif + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*J0.*Tricky.*", // test data group + ".*", 400000000, 400000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*J1.*Tricky.*", // test data group + ".*", 5000000, 5000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*JN.*Integer.*", // test data group + ".*", 30000, 10000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*(JN|j).*|.*Tricky.*", // test data group + ".*", 1500, 700); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 40, 20); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_cyl_bessel_j(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::cyl_bessel_j; +#else + pg funcp = boost::math::cyl_bessel_j; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_j against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_j", test_name); + std::cout << std::endl; + +#ifdef TEST_OTHER + if(boost::is_floating_point::value) + { + funcp = other::cyl_bessel_j; + + // + // test other::cyl_bessel_j against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "other::cyl_bessel_j", test_name); + std::cout << std::endl; + } +#endif +} + +template +T cyl_bessel_j_int_wrapper(T v, T x) +{ + return static_cast(boost::math::cyl_bessel_j(boost::math::tools::real_cast(v), x)); +} + + +template +void do_test_cyl_bessel_j_int(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = cyl_bessel_j_int_wrapper; +#else + pg funcp = cyl_bessel_j_int_wrapper; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_j against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_j", test_name); + std::cout << std::endl; +} + +template +void do_test_sph_bessel_j(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::sph_bessel; +#else + pg funcp = boost::math::sph_bessel; +#endif + + typedef int (*cast_t)(value_type); + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test sph_bessel against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::sph_bessel", test_name); + std::cout << std::endl; +} + +template +void test_bessel(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 8> j0_data = {{ + { SC_(0), SC_(0), SC_(1) }, + { SC_(0), SC_(1), SC_(0.7651976865579665514497175261026632209093) }, + { SC_(0), SC_(-2), SC_(0.2238907791412356680518274546499486258252) }, + { SC_(0), SC_(4), SC_(-0.3971498098638473722865907684516980419756) }, + { SC_(0), SC_(-8), SC_(0.1716508071375539060908694078519720010684) }, + { SC_(0), SC_(1e-05), SC_(0.999999999975000000000156249999999565972) }, + { SC_(0), SC_(1e-10), SC_(0.999999999999999999997500000000000000000) }, + { SC_(0), SC_(-1e+01), SC_(-0.2459357644513483351977608624853287538296) }, + }}; + static const boost::array, 6> j0_tricky = {{ + // Big numbers make the accuracy of std::sin the limiting factor: + { SC_(0), SC_(1e+03), SC_(0.02478668615242017456133073111569370878617) }, + { SC_(0), SC_(1e+05), SC_(-0.001719201116235972192570601477073201747532) }, + // test at the roots: + { SC_(0), SC_(2521642)/(1024 * 1024), SC_(1.80208819970046790002973759410972422387259992955354630042138e-7) }, + { SC_(0), SC_(5788221)/(1024 * 1024), SC_(-1.37774249380686777043369399806210229535671843632174587432454e-7) }, + { SC_(0), SC_(9074091)/(1024 * 1024), SC_(1.03553057441100845081018471279571355857520645127532785991335e-7) }, + { SC_(0), SC_(12364320)/(1024 * 1024), SC_(-3.53017140778223781420794006033810387155048392363051866610931e-9) } + }}; + + static const boost::array, 8> j1_data = { + SC_(1), SC_(0), SC_(0), + SC_(1), SC_(1), SC_(0.4400505857449335159596822037189149131274), + SC_(1), SC_(-2), SC_(-0.5767248077568733872024482422691370869203), + SC_(1), SC_(4), SC_(-6.604332802354913614318542080327502872742e-02), + SC_(1), SC_(-8), SC_(-0.2346363468539146243812766515904546115488), + SC_(1), SC_(1e-05), SC_(4.999999999937500000000260416666666124132e-06), + SC_(1), SC_(1e-10), SC_(4.999999999999999999993750000000000000000e-11), + SC_(1), SC_(-1e+01), SC_(-4.347274616886143666974876802585928830627e-02), + }; + static const boost::array, 5> j1_tricky = { + // Big numbers make the accuracy of std::sin the limiting factor: + SC_(1), SC_(1e+03), SC_(4.728311907089523917576071901216916285418e-03), + SC_(1), SC_(1e+05), SC_(1.846757562882567716362123967114215743694e-03), + // test zeros: + SC_(1), SC_(4017834)/(1024*1024), SC_(3.53149033321258645807835062770856949751958513973522222203044e-7), + SC_(1), SC_(7356375)/(1024*1024), SC_(-2.31227973111067286051984021150135526024117175836722748404342e-7), + SC_(1), SC_(10667654)/(1024*1024), SC_(1.24591331097191900488116495350277530373473085499043086981229e-7), + }; + + static const boost::array, 14> jn_data = { + SC_(2), SC_(0), SC_(0), + SC_(2), SC_(1e-02), SC_(1.249989583365885362413250958437642113452e-05), + SC_(5), SC_(10), SC_(-0.2340615281867936404436949416457777864635), + SC_(5), SC_(-10), SC_(0.2340615281867936404436949416457777864635), + SC_(-5), SC_(1e+06), SC_(7.259643842453285052375779970433848914846e-04), + SC_(5), SC_(1e+06), SC_(-0.000725964384245328505237577997043384891484649290328285235308619), + SC_(-5), SC_(-1), SC_(2.497577302112344313750655409880451981584e-04), + SC_(10), SC_(10), SC_(0.2074861066333588576972787235187534280327), + SC_(10), SC_(-10), SC_(0.2074861066333588576972787235187534280327), + SC_(10), SC_(-5), SC_(1.467802647310474131107532232606627020895e-03), + SC_(-10), SC_(1e+06), SC_(-3.310793117604488741264958559035744460210e-04), + SC_(10), SC_(1e+06), SC_(-0.000331079311760448874126495855903574446020957243277028930713243), + SC_(1e+02), SC_(8e+01), SC_(4.606553064823477354141298259169874909670e-06), + SC_(1e+03), SC_(1e+05), SC_(1.283178112502480365195139312635384057363e-03), + }; + do_test_cyl_bessel_j(j0_data, name, "Bessel J0: Mathworld Data"); + do_test_cyl_bessel_j(j0_tricky, name, "Bessel J0: Mathworld Data (Tricky cases)"); + do_test_cyl_bessel_j(j1_data, name, "Bessel J1: Mathworld Data"); + do_test_cyl_bessel_j(j1_tricky, name, "Bessel J1: Mathworld Data (tricky cases)"); + do_test_cyl_bessel_j(jn_data, name, "Bessel JN: Mathworld Data"); + + do_test_cyl_bessel_j_int(j0_data, name, "Bessel J0: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_j_int(j0_tricky, name, "Bessel J0: Mathworld Data (Tricky cases) (Integer Version)"); + do_test_cyl_bessel_j_int(j1_data, name, "Bessel J1: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_j_int(j1_tricky, name, "Bessel J1: Mathworld Data (tricky cases) (Integer Version)"); + do_test_cyl_bessel_j_int(jn_data, name, "Bessel JN: Mathworld Data (Integer Version)"); + + static const boost::array, 17> jv_data = { + //SC_(-2.4), SC_(0), std::numeric_limits::infinity(), + SC_(2457)/1024, SC_(1)/1024, SC_(3.80739920118603335646474073457326714709615200130620574875292e-9), + SC_(5.5), SC_(3217)/1024, SC_(0.0281933076257506091621579544064767140470089107926550720453038), + SC_(-5.5), SC_(3217)/1024, SC_(-2.55820064470647911823175836997490971806135336759164272675969), + SC_(-5.5), SC_(1e+04), SC_(2.449843111985605522111159013846599118397e-03), + SC_(5.5), SC_(1e+04), SC_(0.00759343502722670361395585198154817047185480147294665270646578), + SC_(5.5), SC_(1e+06), SC_(-0.000747424248595630177396350688505919533097973148718960064663632), + SC_(5.125), SC_(1e+06), SC_(-0.000776600124835704280633640911329691642748783663198207360238214), + SC_(5.875), SC_(1e+06), SC_(-0.000466322721115193071631008581529503095819705088484386434589780), + SC_(0.5), SC_(101), SC_(0.0358874487875643822020496677692429287863419555699447066226409), + SC_(-5.5), SC_(1e+04), SC_(0.00244984311198560552211115901384659911839737686676766460822577), + SC_(-5.5), SC_(1e+06), SC_(0.000279243200433579511095229508894156656558211060453622750659554), + SC_(-0.5), SC_(101), SC_(0.0708184798097594268482290389188138201440114881159344944791454), + SC_(-10486074) / (1024*1024), SC_(1)/1024, SC_(1.41474013160494695750009004222225969090304185981836460288562e35), + SC_(-10486074) / (1024*1024), SC_(15), SC_(-0.0902239288885423309568944543848111461724911781719692852541489), + SC_(10486074) / (1024*1024), SC_(1e+02), SC_(-0.0547064914615137807616774867984047583596945624129838091326863), + SC_(10486074) / (1024*1024), SC_(2e+04), SC_(-0.00556783614400875611650958980796060611309029233226596737701688), + SC_(-10486074) / (1024*1024), SC_(1e+02), SC_(-0.0547613660316806551338637153942604550779513947674222863858713), + }; + do_test_cyl_bessel_j(jv_data, name, "Bessel J: Mathworld Data"); + + #undef SC_ + +#include "bessel_j_int_data.ipp" + do_test_cyl_bessel_j(bessel_j_int_data, name, "Bessel JN: Random Data"); + +#include "bessel_j_data.ipp" + do_test_cyl_bessel_j(bessel_j_data, name, "Bessel J: Random Data"); + +#include "bessel_j_large_data.ipp" + do_test_cyl_bessel_j(bessel_j_large_data, name, "Bessel J: Random Data (Tricky large values)"); + +#include "sph_bessel_data.ipp" + do_test_sph_bessel_j(sph_bessel_data, name, "Bessel j: Random Data"); +} + +int test_main(int, char* []) +{ +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + expected_results(); + BOOST_MATH_CONTROL_FP; + + test_bessel(0.1F, "float"); + test_bessel(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_bessel(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_bessel(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + diff --git a/test/test_bessel_k.cpp b/test/test_bessel_k.cpp new file mode 100644 index 000000000..3f7c286cb --- /dev/null +++ b/test/test_bessel_k.cpp @@ -0,0 +1,278 @@ +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4756) // overflow in constant arithmetic +// Constants are too big for float case, but this doesn't matter for test. +#endif + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_bessel_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the bessel K function. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double|real_concept"; +#endif + // + // On MacOS X cyl_bessel_k has much higher error levels than + // expected: given that the implementation is basically + // just a continued fraction evaluation combined with + // exponentiation, we conclude that exp and pow are less + // accurate on this platform, especially when the result + // is outside the range of a double. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 4000, 1300); // test function + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 35, 15); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +T cyl_bessel_k_int_wrapper(T v, T x) +{ + return static_cast( + boost::math::cyl_bessel_k( + boost::math::tools::real_cast(v), x)); +} + +template +void do_test_cyl_bessel_k(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::cyl_bessel_k; +#else + pg funcp = boost::math::cyl_bessel_k; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_k against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_k", test_name); + std::cout << std::endl; + +#ifdef TEST_OTHER + if(boost::is_floating_point::value) + { + funcp = other::cyl_bessel_k; + + // + // test other::cyl_bessel_k against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::cyl_bessel_k"); + std::cout << std::endl; + } +#endif +} + +template +void do_test_cyl_bessel_k_int(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = cyl_bessel_k_int_wrapper; +#else + pg funcp = cyl_bessel_k_int_wrapper; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_bessel_k against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_bessel_k", test_name); + std::cout << std::endl; +} + +template +void test_bessel(T, const char* name) +{ + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 9> k0_data = {{ + SC_(0), SC_(1), SC_(0.421024438240708333335627379212609036136219748226660472298970), + SC_(0), SC_(2), SC_(0.113893872749533435652719574932481832998326624388808882892530), + SC_(0), SC_(4), SC_(0.0111596760858530242697451959798334892250090238884743405382553), + SC_(0), SC_(8), SC_(0.000146470705222815387096584408698677921967305368833759024089154), + SC_(0), T(std::ldexp(1.0, -15)), SC_(10.5131392267382037062459525561594822400447325776672021972753), + SC_(0), T(std::ldexp(1.0, -30)), SC_(20.9103469324567717360787328239372191382743831365906131108531), + SC_(0), T(std::ldexp(1.0, -60)), SC_(41.7047623492551310138446473188663682295952219631968830346918), + SC_(0), SC_(50), SC_(3.41016774978949551392067551235295223184502537762334808993276e-23), + SC_(0), SC_(100), SC_(4.65662822917590201893900528948388635580753948544211387402671e-45), + }}; + static const boost::array, 9> k1_data = { + SC_(1), SC_(1), SC_(0.601907230197234574737540001535617339261586889968106456017768), + SC_(1), SC_(2), SC_(0.139865881816522427284598807035411023887234584841515530384442), + SC_(1), SC_(4), SC_(0.0124834988872684314703841799808060684838415849886258457917076), + SC_(1), SC_(8), SC_(0.000155369211805001133916862450622474621117065122872616157079566), + SC_(1), T(std::ldexp(1.0, -15)), SC_(32767.9998319528316432647441316539139725104728341577594326513), + SC_(1), T(std::ldexp(1.0, -30)), SC_(1.07374182399999999003003028572687332810353799544215073362305e9), + SC_(1), T(std::ldexp(1.0, -60)), SC_(1.15292150460684697599999999999999998169660198868126604634036e18), + SC_(1), SC_(50), SC_(3.44410222671755561259185303591267155099677251348256880221927e-23), + SC_(1), SC_(100), SC_(4.67985373563690928656254424202433530797494354694335352937465e-45), + }; + static const boost::array, 9> kn_data = { + SC_(2), T(std::ldexp(1.0, -30)), SC_(2.30584300921369395150000000000000000234841952009593636868109e18), + SC_(5), SC_(10), SC_(0.0000575418499853122792763740236992723196597629124356739596921536), + SC_(-5), SC_(100), SC_(5.27325611329294989461777188449044716451716555009882448801072e-45), + SC_(10), SC_(10), SC_(0.00161425530039067002345725193091329085443750382929208307802221), + SC_(10), T(std::ldexp(1.0, -30)), SC_(3.78470202927236255215249281534478864916684072926050665209083e98), + SC_(-10), SC_(1), SC_(1.80713289901029454691597861302340015908245782948536080022119e8), + SC_(100), SC_(5), SC_(7.03986019306167654653386616796116726248616158936088056952477e115), + SC_(100), SC_(80), SC_(8.39287107246490782848985384895907681748152272748337807033319e-12), + SC_(-1000), SC_(700), SC_(6.51561979144735818903553852606383312984409361984128221539405e-31), + }; + static const boost::array, 11> kv_data = { + SC_(0.5), SC_(0.875), SC_(0.558532231646608646115729767013630967055657943463362504577189), + SC_(0.5), SC_(1.125), SC_(0.383621010650189547146769320487006220295290256657827220786527), + SC_(2.25), T(std::ldexp(1.0, -30)), SC_(5.62397392719283271332307799146649700147907612095185712015604e20), + SC_(5.5), SC_(3217)/1024, SC_(1.30623288775012596319554857587765179889689223531159532808379), + SC_(-5.5), SC_(10), SC_(0.0000733045300798502164644836879577484533096239574909573072142667), + SC_(-5.5), SC_(100), SC_(5.41274555306792267322084448693957747924412508020839543293369e-45), + SC_(10240)/1024, SC_(1)/1024, SC_(2.35522579263922076203415803966825431039900000000993410734978e38), + SC_(10240)/1024, SC_(10), SC_(0.00161425530039067002345725193091329085443750382929208307802221), + SC_(144793)/1024, SC_(100), SC_(1.39565245860302528069481472855619216759142225046370312329416e-6), + SC_(144793)/1024, SC_(200), SC_(9.11950412043225432171915100042647230802198254567007382956336e-68), + SC_(-144793)/1024, SC_(50), SC_(1.30185229717525025165362673848737761549946548375142378172956e42), + }; + #undef SC_ + + do_test_cyl_bessel_k(k0_data, name, "Bessel K0: Mathworld Data"); + do_test_cyl_bessel_k(k1_data, name, "Bessel K1: Mathworld Data"); + do_test_cyl_bessel_k(kn_data, name, "Bessel Kn: Mathworld Data"); + + do_test_cyl_bessel_k_int(k0_data, name, "Bessel K0: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_k_int(k1_data, name, "Bessel K1: Mathworld Data (Integer Version)"); + do_test_cyl_bessel_k_int(kn_data, name, "Bessel Kn: Mathworld Data (Integer Version)"); + + do_test_cyl_bessel_k(kv_data, name, "Bessel Kv: Mathworld Data"); + +#include "bessel_k_int_data.ipp" + do_test_cyl_bessel_k(bessel_k_int_data, name, "Bessel Kn: Random Data"); +#include "bessel_k_data.ipp" + do_test_cyl_bessel_k(bessel_k_data, name, "Bessel Kv: Random Data"); +} + +int test_main(int, char* []) +{ +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_bessel(0.1F, "float"); +#endif + test_bessel(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_bessel(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_bessel(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + diff --git a/test/test_bessel_y.cpp b/test/test_bessel_y.cpp new file mode 100644 index 000000000..138d1daa2 --- /dev/null +++ b/test/test_bessel_y.cpp @@ -0,0 +1,463 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_bessel_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the bessel Y function. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // HP-UX and Solaris rates are very slightly higher: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX|Sun Solaris", // platform + largest_type, // test type(s) + ".*(Y[nv]|y).*Random.*", // test data group + ".*", 30000, 30000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX|Sun Solaris", // platform + largest_type, // test type(s) + ".*Y[01Nv].*", // test data group + ".*", 1300, 500); // test function + // + // Tru64: + // + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*(Y[nv]|y).*Random.*", // test data group + ".*", 30000, 30000); // test function + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Y[01Nv].*", // test data group + ".*", 400, 200); // test function + + // + // Mac OS X rates are very slightly higher: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + ".*(Y[nv1]).*", // test data group + ".*", 600000, 100000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + "long double|real_concept", // test type(s) + ".*Y[0].*", // test data group + ".*", 1200, 1000); // test function + + // + // Linux: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + ".*Yv.*Random.*", // test data group + ".*", 200000, 200000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + ".*Y[01v].*", // test data group + ".*", 2000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + ".*Yn.*", // test data group + ".*", 30000, 30000); // test function + // + // MinGW: + // + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Yv.*Random.*", // test data group + ".*", 200000, 200000); // test function + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Y[01v].*", // test data group + ".*", 2000, 1000); // test function + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Yn.*", // test data group + ".*", 30000, 30000); // test function + // + // Solaris version of long double has it's own error rates, + // again just a touch higher than msvc's 64-bit double: + // + add_expected_result( + "GNU.*", // compiler + ".*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "Y[0N].*Mathworld.*", // test data group + ".*", 2000, 2000); // test function + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if((std::numeric_limits::digits != std::numeric_limits::digits) + && (std::numeric_limits::digits < 90)) + { + // some errors spill over into type double as well: + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*Y[Nn].*", // test data group + ".*", 20, 20); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*Yv.*", // test data group + ".*", 80, 70); // test function + } +#endif + // + // defaults are based on MSVC-8 on Win32: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Y0.*Random.*", // test data group + ".*", 600, 400); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*(Y[nv]|y).*Random.*", // test data group + ".*", 2000, 2000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*(Y[nv]|y).*Random.*", // test data group + ".*", 1500, 1000); // test function + // + // Fallback for sun has to go after the general cases above: + // + add_expected_result( + "GNU.*", // compiler + ".*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "Y[0N].*", // test data group + ".*", 200, 200); // test function + // + // General fallback: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 60, 40); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_cyl_neumann_y(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::cyl_neumann; +#else + pg funcp = boost::math::cyl_neumann; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_neumann against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_neumann", test_name); + std::cout << std::endl; + +#ifdef TEST_OTHER + if(boost::is_floating_point::value) + { + funcp = other::cyl_neumann; + + // + // test other::cyl_neumann against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "other::cyl_neumann", test_name); + std::cout << std::endl; + } +#endif +} + +template +T cyl_neumann_int_wrapper(T v, T x) +{ + return static_cast(boost::math::cyl_neumann(boost::math::tools::real_cast(v), x)); +} + +template +void do_test_cyl_neumann_y_int(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = cyl_neumann_int_wrapper; +#else + pg funcp = cyl_neumann_int_wrapper; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cyl_neumann against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_neumann", test_name); + std::cout << std::endl; +} + +template +void do_test_sph_neumann_y(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::sph_neumann; +#else + pg funcp = boost::math::sph_neumann; +#endif + + typedef int (*cast_t)(value_type); + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test sph_neumann against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cyl_neumann", test_name); + std::cout << std::endl; +} + +template +void test_bessel(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 9> y0_data = {{ + SC_(0), SC_(1), SC_(0.0882569642156769579829267660235151628278175230906755467110438), + SC_(0), SC_(2), SC_(0.510375672649745119596606592727157873268139227085846135571839), + SC_(0), SC_(4), SC_(-0.0169407393250649919036351344471532182404925898980149027169321), + SC_(0), SC_(8), SC_(0.223521489387566220527323400498620359274814930781423577578334), + SC_(0), SC_(1e-05), SC_(-7.40316028370197013259676050746759072070960287586102867247159), + SC_(0), SC_(1e-10), SC_(-14.7325162726972420426916696426209144888762342592762415255386), + SC_(0), SC_(1e-20), SC_(-29.3912282502857968601858410375186700783698345615477536431464), + SC_(0), SC_(1e+03), SC_(0.00471591797762281339977326146566525500985900489680197718528000), + SC_(0), SC_(1e+05), SC_(0.00184676615886506410434074102431546125884886798090392516843524) + }}; + static const boost::array, 9> y1_data = { + SC_(1), SC_(1), SC_(-0.781212821300288716547150000047964820549906390716444607843833), + SC_(1), SC_(2), SC_(-0.107032431540937546888370772277476636687480898235053860525795), + SC_(1), SC_(4), SC_(0.397925710557100005253979972450791852271189181622908340876586), + SC_(1), SC_(8), SC_(-0.158060461731247494255555266187483550327344049526705737651263), + SC_(1), SC_(1e-10), SC_(-6.36619772367581343150789184284462611709080831190542841855708e9), + SC_(1), SC_(1e-20), SC_(-6.36619772367581343075535053490057448139324059868649274367256e19), + SC_(1), SC_(1e+01), SC_(0.249015424206953883923283474663222803260416543069658461246944), + SC_(1), SC_(1e+03), SC_(-0.0247843312923517789148623560971412909386318548648705287583490), + SC_(1), SC_(1e+05), SC_(0.00171921035008825630099494523539897102954509504993494957572726) + }; + static const boost::array, 9> yn_data = { + SC_(2), SC_(1e-20), SC_(-1.27323954473516268615107010698011489627570899691226996904849e40), + SC_(5), SC_(10), SC_(0.135403047689362303197029014762241709088405766746419538495983), + SC_(-5), SC_(1e+06), SC_(0.000331052088322609048503535570014688967096938338061796192422114), + SC_(10), SC_(10), SC_(-0.359814152183402722051986577343560609358382147846904467526222), + SC_(10), SC_(1e-10), SC_(-1.18280490494334933900960937719565669877576135140014365217993e108), + SC_(-10), SC_(1e+06), SC_(0.000725951969295187086245251366365393653610914686201194434805730), + SC_(1e+02), SC_(5), SC_(-5.08486391602022287993091563093082035595081274976837280338134e115), + SC_(1e+03), SC_(1e+05), SC_(0.00217254919137684037092834146629212647764581965821326561261181), + SC_(-1e+03), SC_(7e+02), SC_(-1.88753109980945889960843803284345261796244752396992106755091e77) + }; + static const boost::array, 9> yv_data = { + //SC_(2.25), SC_(1) / 1024, SC_(-1.01759203636941035147948317764932151601257765988969544340275e7), + SC_(0.5), SC_(1) / (1024*1024), SC_(-817.033790261762580469303126467917092806755460418223776544122), + SC_(5.5), SC_(3.125), SC_(-2.61489440328417468776474188539366752698192046890955453259866), + SC_(-5.5), SC_(3.125), SC_(-0.0274994493896489729948109971802244976377957234563871795364056), + SC_(-5.5), SC_(1e+04), SC_(-0.00759343502722670361395585198154817047185480147294665270646578), + SC_(-10486074) / (1024*1024), SC_(1)/1024, SC_(-1.50382374389531766117868938966858995093408410498915220070230e38), + SC_(-10486074) / (1024*1024), SC_(1e+02), SC_(0.0583041891319026009955779707640455341990844522293730214223545), + SC_(141.75), SC_(1e+02), SC_(-5.38829231428696507293191118661269920130838607482708483122068e9), + SC_(141.75), SC_(2e+04), SC_(-0.00376577888677186194728129112270988602876597726657372330194186), + SC_(-141.75), SC_(1e+02), SC_(-3.81009803444766877495905954105669819951653361036342457919021e9), + }; + + do_test_cyl_neumann_y(y0_data, name, "Y0: Mathworld Data"); + do_test_cyl_neumann_y(y1_data, name, "Y1: Mathworld Data"); + do_test_cyl_neumann_y(yn_data, name, "Yn: Mathworld Data"); + do_test_cyl_neumann_y_int(y0_data, name, "Y0: Mathworld Data (Integer Version)"); + do_test_cyl_neumann_y_int(y1_data, name, "Y1: Mathworld Data (Integer Version)"); + do_test_cyl_neumann_y_int(yn_data, name, "Yn: Mathworld Data (Integer Version)"); + do_test_cyl_neumann_y(yv_data, name, "Yv: Mathworld Data"); + +#include "bessel_y01_data.ipp" + do_test_cyl_neumann_y(bessel_y01_data, name, "Y0 and Y1: Random Data"); +#include "bessel_yn_data.ipp" + do_test_cyl_neumann_y(bessel_yn_data, name, "Yn: Random Data"); +#include "bessel_yv_data.ipp" + do_test_cyl_neumann_y(bessel_yv_data, name, "Yv: Random Data"); + +#include "sph_neumann_data.ipp" + do_test_sph_neumann_y(sph_neumann_data, name, "y: Random Data"); +} + +int test_main(int, char* []) +{ +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_bessel(0.1F, "float"); +#endif + test_bessel(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_bessel(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_bessel(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + diff --git a/test/test_beta.cpp b/test/test_beta.cpp new file mode 100644 index 000000000..aa9ce1482 --- /dev/null +++ b/test/test_beta.cpp @@ -0,0 +1,228 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_beta_hooks.hpp" +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the function beta. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // +#if LDBL_MANT_DIG == 106 + // Darwin: + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS.*", // platform + "(long\\s+)?double", // test type(s) + "Beta Function: Medium.*", // test data group + "boost::math::beta", 200, 35); // test function +#endif + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "(long\\s+)?double", // test type(s) + "Beta Function: Small.*", // test data group + "boost::math::beta", 8, 5); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "(long\\s+)?double", // test type(s) + "Beta Function: Medium.*", // test data group + "boost::math::beta", 130, 35); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "(long\\s+)?double", // test type(s) + "Beta Function: Divergent.*", // test data group + "boost::math::beta", 20, 6); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Beta Function: Small.*", // test data group + "boost::math::beta", 15, 15); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Beta Function: Medium.*", // test data group + "boost::math::beta", 130, 35); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Beta Function: Divergent.*", // test data group + "boost::math::beta", 25, 8); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_beta(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::beta; +#else + pg funcp = boost::math::beta; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test beta against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::beta", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::beta; + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::beta"); + } +#endif + std::cout << std::endl; +} +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and beta(a, b): + // +# include "beta_small_data.ipp" + + do_test_beta(beta_small_data, name, "Beta Function: Small Values"); + +# include "beta_med_data.ipp" + + do_test_beta(beta_med_data, name, "Beta Function: Medium Values"); + +# include "beta_exp_data.ipp" + + do_test_beta(beta_exp_data, name, "Beta Function: Divergent Values"); +} + +#undef small // VC++ #defines small char !!!!!! +template +void test_spots(T) +{ + // + // Basic sanity checks, tolerance is 20 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 20 * 100; + T small = boost::math::tools::epsilon() / 1024; + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(1), static_cast(1)), static_cast(1), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(1), static_cast(4)), static_cast(0.25), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), static_cast(1)), static_cast(0.25), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(small, static_cast(4)), 1/small, tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), small), 1/small, tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(4), static_cast(20)), static_cast(0.00002823263692828910220214568040654997176736L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::beta(static_cast(0.0125L), static_cast(0.000023L)), static_cast(43558.24045647538375006349016083320744662L), tolerance); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + test_spots(0.0F); + test_spots(0.0); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.1)); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + test_beta(0.1F, "float"); + test_beta(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_beta(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_beta_dist.cpp b/test/test_beta_dist.cpp new file mode 100644 index 000000000..5256e4a31 --- /dev/null +++ b/test/test_beta_dist.cpp @@ -0,0 +1,579 @@ +// test_beta_dist.cpp + +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity tests for the beta Distribution. + +// http://members.aol.com/iandjmsmith/BETAEX.HTM beta distribution calculator +// Appreas to be a 64-bit calculator showing 17 decimal digit (last is noisy). +// Similar to mathCAD? + +// http://www.nuhertz.com/statmat/distributions.html#Beta +// Pretty graphs and explanations for most distributions. + +// http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp +// provided 40 decimal digits accuracy incomplete beta aka beta regularized == cdf + +// http://www.ausvet.com.au/pprev/content.php?page=PPscript +// mode 0.75 5/95% 0.9 alpha 7.39 beta 3.13 +// http://www.epi.ucdavis.edu/diagnostictests/betabuster.html +// Beta Buster also calculates alpha and beta from mode & percentile estimates. +// This is NOT (yet) implemented. + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +#endif + +#include // for beta_distribution +using boost::math::beta_distribution; +using boost::math::beta; + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE_FRACTION + +#include +using std::cout; +using std::endl; +#include +using std::numeric_limits; + +template +void test_spot( + RealType a, // alpha a + RealType b, // beta b + RealType x, // Probability + RealType P, // CDF of beta(a, b) + RealType Q, // Complement of CDF + RealType tol) // Test tolerance. +{ + boost::math::beta_distribution abeta(a, b); + BOOST_CHECK_CLOSE_FRACTION(cdf(abeta, x), P, tol); + if((P < 0.99) && (Q < 0.99)) + { // We can only check this if P is not too close to 1, + // so that we can guarantee that Q is free of error, + // (and similarly for Q) + BOOST_CHECK_CLOSE_FRACTION( + cdf(complement(abeta, x)), Q, tol); + if(x != 0) + { + BOOST_CHECK_CLOSE_FRACTION( + quantile(abeta, P), x, tol); + } + else + { + // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) + && (boost::is_floating_point::value)) + { + // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(abeta, P) < boost::math::tools::epsilon() * 10); + } + } // if k + if(x != 0) + { + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(abeta, Q)), x, tol); + } + else + { // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) && (boost::is_floating_point::value)) + { // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(complement(abeta, Q)) < boost::math::tools::epsilon() * 10); + } + } // if x + // Estimate alpha & beta from mean and variance: + + BOOST_CHECK_CLOSE_FRACTION( + beta_distribution::find_alpha(mean(abeta), variance(abeta)), + abeta.alpha(), tol); + BOOST_CHECK_CLOSE_FRACTION( + beta_distribution::find_beta(mean(abeta), variance(abeta)), + abeta.beta(), tol); + + // Estimate sample alpha and beta from others: + BOOST_CHECK_CLOSE_FRACTION( + beta_distribution::find_alpha(abeta.beta(), x, P), + abeta.alpha(), tol); + BOOST_CHECK_CLOSE_FRACTION( + beta_distribution::find_beta(abeta.alpha(), x, P), + abeta.beta(), tol); + } // if((P < 0.99) && (Q < 0.99) + +} // template void test_spot + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks with 'known good' values. + // MathCAD test data is to double precision only, + // so set tolerance to 100 eps expressed as a fraction, or + // 100 eps of type double expressed as a fraction, + // whichever is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); // 0 if real_concept. + + cout << "Boost::math::tools::epsilon = " << boost::math::tools::epsilon() <() * 10; + + // Sources of spot test values: + + // MathCAD defines dbeta(x, s1, s2) pdf, s1 == alpha, s2 = beta, x = x in Wolfram + // pbeta(x, s1, s2) cdf and qbeta(x, s1, s2) inverse of cdf + // returns pr(X ,= x) when random variable X + // has the beta distribution with parameters s1)alpha) and s2(beta). + // s1 > 0 and s2 >0 and 0 < x < 1 (but allows x == 0! and x == 1!) + // dbeta(0,1,1) = 0 + // dbeta(0.5,1,1) = 1 + + using boost::math::beta_distribution; + using ::boost::math::cdf; + using ::boost::math::pdf; + + // Tests that should throw: + BOOST_CHECK_THROW(mode(beta_distribution(static_cast(1), static_cast(1))), std::domain_error); + // mode is undefined, and throws domain_error! + + // BOOST_CHECK_THROW(median(beta_distribution(static_cast(1), static_cast(1))), std::domain_error); + // median is undefined, and throws domain_error! + // But now median IS provided via derived accessor as quantile(half). + + + BOOST_CHECK_THROW( // For various bad arguments. + pdf( + beta_distribution(static_cast(-1), static_cast(1)), // bad alpha < 0. + static_cast(1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( + beta_distribution(static_cast(0), static_cast(1)), // bad alpha == 0. + static_cast(1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( + beta_distribution(static_cast(1), static_cast(0)), // bad beta == 0. + static_cast(1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( + beta_distribution(static_cast(1), static_cast(-1)), // bad beta < 0. + static_cast(1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( + beta_distribution(static_cast(1), static_cast(1)), // bad x < 0. + static_cast(-1)), std::domain_error); + + BOOST_CHECK_THROW( + pdf( + beta_distribution(static_cast(1), static_cast(1)), // bad x > 1. + static_cast(999)), std::domain_error); + + // Some exact pdf values. + + BOOST_CHECK_EQUAL( // a = b = 1 is uniform distribution. + pdf(beta_distribution(static_cast(1), static_cast(1)), + static_cast(1)), // x + static_cast(1)); + BOOST_CHECK_EQUAL( + pdf(beta_distribution(static_cast(1), static_cast(1)), + static_cast(0)), // x + static_cast(1)); + BOOST_CHECK_CLOSE_FRACTION( + pdf(beta_distribution(static_cast(1), static_cast(1)), + static_cast(0.5)), // x + static_cast(1), + tolerance); + + BOOST_CHECK_EQUAL( + beta_distribution(static_cast(1), static_cast(1)).alpha(), + static_cast(1) ); // + + BOOST_CHECK_EQUAL( + mean(beta_distribution(static_cast(1), static_cast(1))), + static_cast(0.5) ); // Exact one half. + + BOOST_CHECK_CLOSE_FRACTION( + pdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.5)), // x + static_cast(1.5), // Exactly 3/2 + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + pdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.5)), // x + static_cast(1.5), // Exactly 3/2 + tolerance); + + // CDF + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.1)), // x + static_cast(0.02800000000000000000000000000000000000000L), // Seems exact. + // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.1&a=2&b=2&digits=40 + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.0001)), // x + static_cast(2.999800000000000000000000000000000000000e-8L), + // http://members.aol.com/iandjmsmith/BETAEX.HTM 2.9998000000004 + // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.0001&a=2&b=2&digits=40 + tolerance); + + + BOOST_CHECK_CLOSE_FRACTION( + pdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.0001)), // x + static_cast(0.0005999400000000004L), // http://members.aol.com/iandjmsmith/BETAEX.HTM + // Slightly higher tolerance for real concept: + (std::numeric_limits::is_specialized ? 1 : 10) * tolerance); + + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.9999)), // x + static_cast(0.999999970002L), // http://members.aol.com/iandjmsmith/BETAEX.HTM + // Wolfram 0.9999999700020000000000000000000000000000 + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(0.5), static_cast(2)), + static_cast(0.9)), // x + static_cast(0.9961174629530394895796514664963063381217L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.1)), // x + static_cast(0.2048327646991334516491978475505189480977L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.9)), // x + static_cast(0.7951672353008665483508021524494810519023L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + quantile(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.7951672353008665483508021524494810519023L)), // x + static_cast(0.9), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.6)), // x + static_cast(0.5640942168489749316118742861695149357858L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + quantile(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.5640942168489749316118742861695149357858L)), // x + static_cast(0.6), + // Wolfram + tolerance); + + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(2), static_cast(0.5)), + static_cast(0.6)), // x + static_cast(0.1778078083562213736802876784474931812329L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + quantile(beta_distribution(static_cast(2), static_cast(0.5)), + static_cast(0.1778078083562213736802876784474931812329L)), // x + static_cast(0.6), + // Wolfram + tolerance); // gives + + BOOST_CHECK_CLOSE_FRACTION( + cdf(beta_distribution(static_cast(1), static_cast(1)), + static_cast(0.1)), // x + static_cast(0.1), // 0.1000000000000000000000000000000000000000 + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + quantile(beta_distribution(static_cast(1), static_cast(1)), + static_cast(0.1)), // x + static_cast(0.1), // 0.1000000000000000000000000000000000000000 + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(complement(beta_distribution(static_cast(0.5), static_cast(0.5)), + static_cast(0.1))), // complement of x + static_cast(0.7951672353008665483508021524494810519023L), + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + quantile(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.0280000000000000000000000000000000000L)), // x + static_cast(0.1), + // Wolfram + tolerance); + + + BOOST_CHECK_CLOSE_FRACTION( + cdf(complement(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.1))), // x + static_cast(0.9720000000000000000000000000000000000000L), // Exact. + // Wolfram + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + pdf(beta_distribution(static_cast(2), static_cast(2)), + static_cast(0.9999)), // x + static_cast(0.0005999399999999344L), // http://members.aol.com/iandjmsmith/BETAEX.HTM + tolerance*10); // Note loss of precision calculating 1-p test value. + + //void test_spot( + // RealType a, // alpha a + // RealType b, // beta b + // RealType x, // Probability + // RealType P, // CDF of beta(a, b) + // RealType Q, // Complement of CDF + // RealType tol) // Test tolerance. + + // These test quantiles and complements, and parameter estimates as well. + // Spot values using, for example: + // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.1&a=0.5&b=3&digits=40 + + test_spot( + static_cast(1), // alpha a + static_cast(1), // beta b + static_cast(0.1), // Probability p + static_cast(0.1), // Probability of result (CDF of beta), P + static_cast(0.9), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.1), // Probability p + static_cast(0.0280000000000000000000000000000000000L), // Probability of result (CDF of beta), P + static_cast(1 - 0.0280000000000000000000000000000000000L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.5), // Probability p + static_cast(0.5), // Probability of result (CDF of beta), P + static_cast(0.5), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.9), // Probability p + static_cast(0.972000000000000), // Probability of result (CDF of beta), P + static_cast(1-0.972000000000000), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.01), // Probability p + static_cast(0.0002980000000000000000000000000000000000000L), // Probability of result (CDF of beta), P + static_cast(1-0.0002980000000000000000000000000000000000000L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.001), // Probability p + static_cast(2.998000000000000000000000000000000000000E-6L), // Probability of result (CDF of beta), P + static_cast(1-2.998000000000000000000000000000000000000E-6L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.0001), // Probability p + static_cast(2.999800000000000000000000000000000000000E-8L), // Probability of result (CDF of beta), P + static_cast(1-2.999800000000000000000000000000000000000E-8L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(2), // alpha a + static_cast(2), // beta b + static_cast(0.99), // Probability p + static_cast(0.9997020000000000000000000000000000000000L), // Probability of result (CDF of beta), P + static_cast(1-0.9997020000000000000000000000000000000000L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(0.5), // alpha a + static_cast(2), // beta b + static_cast(0.5), // Probability p + static_cast(0.8838834764831844055010554526310612991060L), // Probability of result (CDF of beta), P + static_cast(1-0.8838834764831844055010554526310612991060L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(0.5), // alpha a + static_cast(3.), // beta b + static_cast(0.7), // Probability p + static_cast(0.9903963064097119299191611355232156905687L), // Probability of result (CDF of beta), P + static_cast(1-0.9903963064097119299191611355232156905687L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + + test_spot( + static_cast(0.5), // alpha a + static_cast(3.), // beta b + static_cast(0.1), // Probability p + static_cast(0.5545844446520295253493059553548880128511L), // Probability of result (CDF of beta), P + static_cast(1-0.5545844446520295253493059553548880128511L), // Complement of CDF Q = 1 - P + tolerance); // Test tolerance. + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + // Check that can generate beta distribution using one convenience methods: + beta_distribution<> mybeta11(1., 1.); // Using default RealType double. + // but that + // boost::math::beta mybeta1(1., 1.); // Using typedef fails. + // error C2039: 'beta' : is not a member of 'boost::math' + + // Basic sanity-check spot values. + + // Some simple checks using double only. + BOOST_CHECK_EQUAL(mybeta11.alpha(), 1); // + BOOST_CHECK_EQUAL(mybeta11.beta(), 1); + BOOST_CHECK_EQUAL(mean(mybeta11), 0.5); // 1 / (1 + 1) = 1/2 exactly + BOOST_CHECK_THROW(mode(mybeta11), std::domain_error); + beta_distribution<> mybeta22(2., 2.); // pdf is dome shape. + BOOST_CHECK_EQUAL(mode(mybeta22), 0.5); // 2-1 / (2+2-2) = 1/2 exactly. + beta_distribution<> mybetaH2(0.5, 2.); // + beta_distribution<> mybetaH3(0.5, 3.); // + + // Check a few values using double. + BOOST_CHECK_EQUAL(pdf(mybeta11, 1), 1); // is uniform unity over 0 to 1, + BOOST_CHECK_EQUAL(pdf(mybeta11, 0), 1); // including zero and unity. + // Although these next three have an exact result, internally they're + // *not* treated as special cases, and may be out by a couple of eps: + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.5), 1.0, 5*std::numeric_limits::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.0001), 1.0, 5*std::numeric_limits::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.9999), 1.0, 5*std::numeric_limits::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.1), 0.1, 2 * std::numeric_limits::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.5), 0.5, 2 * std::numeric_limits::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.9), 0.9, 2 * std::numeric_limits::epsilon()); + BOOST_CHECK_EQUAL(cdf(mybeta11, 1), 1.); // Exact unity expected. + + double tol = std::numeric_limits::epsilon() * 10; + BOOST_CHECK_EQUAL(pdf(mybeta22, 1), 0); // is dome shape. + BOOST_CHECK_EQUAL(pdf(mybeta22, 0), 0); + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.5), 1.5, tol); // top of dome, expect exactly 3/2. + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.0001), 5.9994000000000E-4, tol); + BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.9999), 5.9994000000000E-4, tol*50); + + BOOST_CHECK_EQUAL(cdf(mybeta22, 0.), 0); // cdf is a curved line from 0 to 1. + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.1), 0.028000000000000, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.5), 0.5, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.9), 0.972000000000000, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.0001), 2.999800000000000000000000000000000000000E-8, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.001), 2.998000000000000000000000000000000000000E-6, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.01), 0.0002980000000000000000000000000000000000000, tol); + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.1), 0.02800000000000000000000000000000000000000, tol); // exact + BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.99), 0.9997020000000000000000000000000000000000, tol); + + BOOST_CHECK_EQUAL(cdf(mybeta22, 1), 1.); // Exact unity expected. + + // Complement + + BOOST_CHECK_CLOSE_FRACTION(cdf(complement(mybeta22, 0.9)), 0.028000000000000, tol); + + // quantile. + BOOST_CHECK_CLOSE_FRACTION(quantile(mybeta22, 0.028), 0.1, tol); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(mybeta22, 1 - 0.028)), 0.1, tol); + BOOST_CHECK_EQUAL(kurtosis(mybeta11), 3+ kurtosis_excess(mybeta11)); // Check kurtosis_excess = kurtosis - 3; + BOOST_CHECK_CLOSE_FRACTION(variance(mybeta22), 0.05, tol); + BOOST_CHECK_CLOSE_FRACTION(mean(mybeta22), 0.5, tol); + BOOST_CHECK_CLOSE_FRACTION(mode(mybeta22), 0.5, tol); + BOOST_CHECK_CLOSE_FRACTION(median(mybeta22), 0.5, tol); + + BOOST_CHECK_CLOSE_FRACTION(skewness(mybeta22), 0.0, tol); + BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(mybeta22), -144.0 / 168, tol); + BOOST_CHECK_CLOSE_FRACTION(skewness(beta_distribution<>(3, 5)), 0.30983866769659335081434123198259, tol); + + BOOST_CHECK_CLOSE_FRACTION(beta_distribution::find_alpha(mean(mybeta22), variance(mybeta22)), mybeta22.alpha(), tol); // mean, variance, probability. + BOOST_CHECK_CLOSE_FRACTION(beta_distribution::find_beta(mean(mybeta22), variance(mybeta22)), mybeta22.beta(), tol);// mean, variance, probability. + + BOOST_CHECK_CLOSE_FRACTION(mybeta22.find_alpha(mybeta22.beta(), 0.8, cdf(mybeta22, 0.8)), mybeta22.alpha(), tol); + BOOST_CHECK_CLOSE_FRACTION(mybeta22.find_beta(mybeta22.alpha(), 0.8, cdf(mybeta22, 0.8)), mybeta22.beta(), tol); + + + beta_distribution rcbeta22(2, 2); // Using RealType real_concept. + cout << "numeric_limits::is_specialized " << numeric_limits::is_specialized << endl; + cout << "numeric_limits::digits " << numeric_limits::digits << endl; + cout << "numeric_limits::digits10 " << numeric_limits::digits10 << endl; + cout << "numeric_limits::epsilon " << numeric_limits::epsilon() << endl; + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#endif + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +-Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_beta_dist.exe" +Running 1 test case... +numeric_limits::is_specialized 0 +numeric_limits::digits 0 +numeric_limits::digits10 0 +numeric_limits::epsilon 0 +Boost::math::tools::epsilon = 1.19209e-007 +std::numeric_limits::epsilon = 1.19209e-007 +epsilon = 1.19209e-007, Tolerance = 0.0119209%. +Boost::math::tools::epsilon = 2.22045e-016 +std::numeric_limits::epsilon = 2.22045e-016 +epsilon = 2.22045e-016, Tolerance = 2.22045e-011%. +Boost::math::tools::epsilon = 2.22045e-016 +std::numeric_limits::epsilon = 2.22045e-016 +epsilon = 2.22045e-016, Tolerance = 2.22045e-011%. +Boost::math::tools::epsilon = 2.22045e-016 +std::numeric_limits::epsilon = 0 +epsilon = 2.22045e-016, Tolerance = 2.22045e-011%. +*** No errors detected +*/ + + + diff --git a/test/test_beta_hooks.hpp b/test/test_beta_hooks.hpp new file mode 100644 index 000000000..ff2909eb3 --- /dev/null +++ b/test/test_beta_hooks.hpp @@ -0,0 +1,81 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_BETA_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_BETA_OTHER_HOOKS_HPP + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double beta(double, double); + float betaf(float, float); + long double betal(long double, long double); + + double incbet(double, double, double); + float incbetf(float, float, float); + long double incbetl(long double, long double, long double); +} +inline float beta(float a, float b) +{ return betaf(a, b); } +inline long double beta(long double a, long double b) +{ +#ifdef BOOST_MSVC + return beta((double)a, (double)b); +#else + return betal(a, b); +#endif +} +inline float ibeta(float a, float b, float x) +{ return incbetf(a, b, x); } +inline double ibeta(double a, double b, double x) +{ return incbet(a, b, x); } +inline long double ibeta(long double a, long double b, long double x) +{ +#ifdef BOOST_MSVC + return incbet((double)a, (double)b, (double)x); +#else + return incbetl(a, b); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#include +#include +#include + +namespace other{ +inline float beta(float a, float b) +{ return (float)gsl_sf_beta(a, b); } +inline double beta(double a, double b) +{ return gsl_sf_beta(a, b); } +inline long double beta(long double a, long double b) +{ return gsl_sf_beta(a, b); } + +inline float ibeta(float a, float b, float x) +{ return (float)gsl_sf_beta_inc(a, b, x); } +inline double ibeta(double a, double b, double x) +{ return gsl_sf_beta_inc(a, b, x); } +inline long double ibeta(long double a, long double b, long double x) +{ + return gsl_sf_beta_inc((double)a, (double)b, (double)x); +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept beta(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept ibeta(boost::math::concepts::real_concept, boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + diff --git a/test/test_binomial.cpp b/test/test_binomial.cpp new file mode 100644 index 000000000..cfe9e5f1b --- /dev/null +++ b/test/test_binomial.cpp @@ -0,0 +1,742 @@ +// test_binomial.cpp + +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity test for Binomial Cumulative Distribution Function. + +#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +#endif + +#include // for binomial_distribution +using boost::math::binomial_distribution; + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE + +#include +using std::cout; +using std::endl; +#include +using std::numeric_limits; + +template +void test_spot( + RealType N, // Number of trials + RealType k, // Number of successes + RealType p, // Probability of success + RealType P, // CDF + RealType Q, // Complement of CDF + RealType tol) // Test tolerance +{ + boost::math::binomial_distribution bn(N, p); + BOOST_CHECK_CLOSE( + cdf(bn, k), P, tol); + if((P < 0.99) && (Q < 0.99)) + { + // + // We can only check this if P is not too close to 1, + // so that we can guarentee Q is free of error: + // + BOOST_CHECK_CLOSE( + cdf(complement(bn, k)), Q, tol); + if(k != 0) + { + BOOST_CHECK_CLOSE( + quantile(bn, P), k, tol); + } + else + { + // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) && (boost::is_floating_point::value)) + { + // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(bn, P) < boost::math::tools::epsilon() * 10); + } + } + if(k != 0) + { + BOOST_CHECK_CLOSE( + quantile(complement(bn, Q)), k, tol); + } + else + { + // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) && (boost::is_floating_point::value)) + { + // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(complement(bn, Q)) < boost::math::tools::epsilon() * 10); + } + } + if(k > 0) + { + // estimate success ratio: + // Note lower bound uses a different formual internally + // from upper bound, have to adjust things to prevent + // fencepost errors: + BOOST_CHECK_CLOSE( + binomial_distribution::find_lower_bound_on_p( + N, k+1, Q), + p, tol); + BOOST_CHECK_CLOSE( + binomial_distribution::find_upper_bound_on_p( + N, k, P), + p, tol); + + if(Q < P) + { + // Default method (Clopper Pearson) + BOOST_CHECK( + binomial_distribution::find_lower_bound_on_p( + N, k, Q) + <= + binomial_distribution::find_upper_bound_on_p( + N, k, Q) + ); + BOOST_CHECK(( + binomial_distribution::find_lower_bound_on_p( + N, k, Q) + <= k/N) && (k/N <= + binomial_distribution::find_upper_bound_on_p( + N, k, Q)) + ); + // Bayes Method (Jeffreys Prior) + BOOST_CHECK( + binomial_distribution::find_lower_bound_on_p( + N, k, Q, binomial_distribution::jeffreys_prior_interval) + <= + binomial_distribution::find_upper_bound_on_p( + N, k, Q, binomial_distribution::jeffreys_prior_interval) + ); + BOOST_CHECK(( + binomial_distribution::find_lower_bound_on_p( + N, k, Q, binomial_distribution::jeffreys_prior_interval) + <= k/N) && (k/N <= + binomial_distribution::find_upper_bound_on_p( + N, k, Q, binomial_distribution::jeffreys_prior_interval)) + ); + } + else + { + // Default method (Clopper Pearson) + BOOST_CHECK( + binomial_distribution::find_lower_bound_on_p( + N, k, P) + <= + binomial_distribution::find_upper_bound_on_p( + N, k, P) + ); + BOOST_CHECK( + (binomial_distribution::find_lower_bound_on_p( + N, k, P) + <= k / N) && (k/N <= + binomial_distribution::find_upper_bound_on_p( + N, k, P)) + ); + // Bayes Method (Jeffreys Prior) + BOOST_CHECK( + binomial_distribution::find_lower_bound_on_p( + N, k, P, binomial_distribution::jeffreys_prior_interval) + <= + binomial_distribution::find_upper_bound_on_p( + N, k, P, binomial_distribution::jeffreys_prior_interval) + ); + BOOST_CHECK( + (binomial_distribution::find_lower_bound_on_p( + N, k, P, binomial_distribution::jeffreys_prior_interval) + <= k / N) && (k/N <= + binomial_distribution::find_upper_bound_on_p( + N, k, P, binomial_distribution::jeffreys_prior_interval)) + ); + } + } + // + // estimate sample size: + // + BOOST_CHECK_CLOSE( + binomial_distribution::find_minimum_number_of_trials( + k, p, P), + N, tol); + BOOST_CHECK_CLOSE( + binomial_distribution::find_maximum_number_of_trials( + k, p, Q), + N, tol); + } + + // Double check consistency of CDF and PDF by computing + // the finite sum: + RealType sum = 0; + for(unsigned i = 0; i <= k; ++i) + sum += pdf(bn, RealType(i)); + BOOST_CHECK_CLOSE( + sum, P, tol); + // And complement as well: + sum = 0; + for(RealType i = N; i > k; i -= 1) + sum += pdf(bn, i); + if(P < 0.99) + { + BOOST_CHECK_CLOSE( + sum, Q, tol); + } + else + { + // Not enough information content in P for Q to be meaningful + RealType tol = (std::max)(2 * Q, boost::math::tools::epsilon()); + BOOST_CHECK(sum < tol); + } +} + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks, test data is to double precision only + // so set tolerance to 100eps expressed as a persent, or + // 100eps of type double expressed as a persent, whichever + // is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); + tolerance *= 100 * 1000; + + cout << "Tolerance = " << tolerance << "%." << endl; + + // Sources of spot test values: + + // MathCAD defines pbinom(k, n, p) + // returns pr(X ,=k) when random variable X has the binomial distribution with parameters n and p. + // 0 <= k ,= n + // 0 <= p <= 1 + // P = pbinom(30, 500, 0.05) = 0.869147702104609 + + using boost::math::binomial_distribution; + using ::boost::math::cdf; + using ::boost::math::pdf; + + // Test binomial using cdf spot values from MathCAD. + // These test quantiles and complements as well. + test_spot( + static_cast(500), // Sample size, N + static_cast(30), // Number of successes, k + static_cast(0.05), // Probability of success, p + static_cast(0.869147702104609), // Probability of result (CDF), P + static_cast(1 - 0.869147702104609), // Q = 1 - P + tolerance); + + test_spot( + static_cast(500), // Sample size, N + static_cast(250), // Number of successes, k + static_cast(0.05), // Probability of success, p + static_cast(1), // Probability of result (CDF), P + static_cast(0), // Q = 1 - P + tolerance); + + test_spot( + static_cast(500), // Sample size, N + static_cast(470), // Number of successes, k + static_cast(0.95), // Probability of success, p + static_cast(0.176470742656766), // Probability of result (CDF), P + static_cast(1 - 0.176470742656766), // Q = 1 - P + tolerance * 10); // Note higher tolerance on this test! + + test_spot( + static_cast(500), // Sample size, N + static_cast(400), // Number of successes, k + static_cast(0.05), // Probability of success, p + static_cast(1), // Probability of result (CDF), P + static_cast(0), // Q = 1 - P + tolerance); + + test_spot( + static_cast(500), // Sample size, N + static_cast(400), // Number of successes, k + static_cast(0.9), // Probability of success, p + static_cast(1.80180425681923E-11), // Probability of result (CDF), P + static_cast(1 - 1.80180425681923E-11), // Q = 1 - P + tolerance); + + test_spot( + static_cast(500), // Sample size, N + static_cast(5), // Number of successes, k + static_cast(0.05), // Probability of success, p + static_cast(9.181808267643E-7), // Probability of result (CDF), P + static_cast(1 - 9.181808267643E-7), // Q = 1 - P + tolerance); + + test_spot( + static_cast(2), // Sample size, N + static_cast(1), // Number of successes, k + static_cast(0.5), // Probability of success, p + static_cast(0.75), // Probability of result (CDF), P + static_cast(0.25), // Q = 1 - P + tolerance); + + test_spot( + static_cast(8), // Sample size, N + static_cast(3), // Number of successes, k + static_cast(0.25), // Probability of success, p + static_cast(0.8861846923828125), // Probability of result (CDF), P + static_cast(1 - 0.8861846923828125), // Q = 1 - P + tolerance); + + test_spot( + static_cast(8), // Sample size, N + static_cast(0), // Number of successes, k + static_cast(0.25), // Probability of success, p + static_cast(0.1001129150390625), // Probability of result (CDF), P + static_cast(1 - 0.1001129150390625), // Q = 1 - P + tolerance); + + test_spot( + static_cast(8), // Sample size, N + static_cast(1), // Number of successes, k + static_cast(0.25), // Probability of success, p + static_cast(0.36708068847656244), // Probability of result (CDF), P + static_cast(1 - 0.36708068847656244), // Q = 1 - P + tolerance); + + test_spot( + static_cast(8), // Sample size, N + static_cast(4), // Number of successes, k + static_cast(0.25), // Probability of success, p + static_cast(0.9727020263671875), // Probability of result (CDF), P + static_cast(1 - 0.9727020263671875), // Q = 1 - P + tolerance); + + test_spot( + static_cast(8), // Sample size, N + static_cast(7), // Number of successes, k + static_cast(0.25), // Probability of success, p + static_cast(0.9999847412109375), // Probability of result (CDF), P + static_cast(1 - 0.9999847412109375), // Q = 1 - P + tolerance); + + // Tests on PDF follow: + BOOST_CHECK_CLOSE( + pdf(binomial_distribution(static_cast(20), static_cast(0.75)), + static_cast(10)), // k. + static_cast(0.00992227527967770583927631378173), // 0.00992227527967770583927631378173 + tolerance); + + BOOST_CHECK_CLOSE( + pdf(binomial_distribution(static_cast(20), static_cast(0.5)), + static_cast(10)), // k. + static_cast(0.17619705200195312500000000000000000000), // get k=10 0.049611376398388612 p = 0.25 + tolerance); + + // Binomial pdf Test values from + // http://www.adsciengineering.com/bpdcalc/index.php for example + // http://www.adsciengineering.com/bpdcalc/index.php?n=20&p=0.25&start=0&stop=20&Submit=Generate + // Appears to use at least 80-bit long double for 32 decimal digits accuracy, + // but loses accuracy of display if leading zeros? + // (if trailings zero then are exact values?) + // so useful for testing 64-bit double accuracy. + // P = 0.25, n = 20, k = 0 to 20 + + //0 C(20,0) * 0.25^0 * 0.75^20 0.00317121193893399322405457496643 + //1 C(20,1) * 0.25^1 * 0.75^19 0.02114141292622662149369716644287 + //2 C(20,2) * 0.25^2 * 0.75^18 0.06694780759971763473004102706909 + //3 C(20,3) * 0.25^3 * 0.75^17 0.13389561519943526946008205413818 + //4 C(20,4) * 0.25^4 * 0.75^16 0.18968545486586663173511624336242 + //5 C(20,5) * 0.25^5 * 0.75^15 0.20233115185692440718412399291992 + //6 C(20,6) * 0.25^6 * 0.75^14 0.16860929321410367265343666076660 + //7 C(20,7) * 0.25^7 * 0.75^13 0.11240619547606911510229110717773 + //8 C(20,8) * 0.25^8 * 0.75^12 0.06088668921620410401374101638793 + //9 C(20,9) * 0.25^9 * 0.75^11 0.02706075076275737956166267395019 + //10 C(20,10) * 0.25^10 * 0.75^10 0.00992227527967770583927631378173 + //11 C(20,11) * 0.25^11 * 0.75^9 0.00300675008475081995129585266113 + //12 C(20,12) * 0.25^12 * 0.75^8 0.00075168752118770498782396316528 + //13 C(20,13) * 0.25^13 * 0.75^7 0.00015419231203850358724594116210 + //14 C(20,14) * 0.25^14 * 0.75^6 0.00002569871867308393120765686035 + //15 C(20,15) * 0.25^15 * 0.75^5 0.00000342649582307785749435424804 + //16 C(20,16) * 0.25^16 * 0.75^4 0.00000035692664823727682232856750 + //17 C(20,17) * 0.25^17 * 0.75^3 0.00000002799424692057073116302490 + //18 C(20,18) * 0.25^18 * 0.75^2 0.00000000155523594003170728683471 + //19 C(20,19) * 0.25^19 * 0.75^1 0.00000000005456968210637569427490 + //20 C(20,20) * 0.25^20 * 0.75^0 0.00000000000090949470177292823791 + + + BOOST_CHECK_CLOSE( + pdf(binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(10)), // k. + static_cast(0.00992227527967770583927631378173), // k=10 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 0 use different formula - only exp so more accurate. + pdf(binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(0)), // k. + static_cast(0.00317121193893399322405457496643), // k=0 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 20 use different formula - only exp so more accurate. + pdf(binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(20)), // k == n. + static_cast(0.00000000000090949470177292823791), // k=20 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 1. + pdf(binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(1)), // k. + static_cast(0.02114141292622662149369716644287), // k=1 p = 0.25 + tolerance); + + // Some exact (probably) values. + BOOST_CHECK_CLOSE( + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)), // k. + static_cast(0.10011291503906250000000000000000), // k=0 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 1. + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1)), // k. + static_cast(0.26696777343750000000000000000000), // k=1 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 2. + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(2)), // k. + static_cast(0.31146240234375000000000000000000), // k=2 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 3. + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(3)), // k. + static_cast(0.20764160156250000000000000000000), // k=3 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 7. + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(7)), // k. + static_cast(0.00036621093750000000000000000000), // k=7 p = 0.25 + tolerance); + + BOOST_CHECK_CLOSE( // k = 8. + pdf(binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(8)), // k = n. + static_cast(0.00001525878906250000000000000000), // k=8 p = 0.25 + tolerance); + + RealType tol2 = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a persent + binomial_distribution dist(static_cast(8), static_cast(0.25)); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(8 * 0.25), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(8 * 0.25 * 0.75), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , static_cast(sqrt(8 * 0.25L * 0.75L)), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol2); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(std::floor(9 * 0.25)), tol2); + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , static_cast(0.40824829046386301636621401245098L), (std::max)(tol2, static_cast(5e-29))); // test data has 32 digits only. + // kurtosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , static_cast(2.916666666666666666666666666666666666L), tol2); + // kurtosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , static_cast(-0.08333333333333333333333333333333333333L), tol2); + // Check kurtosis_excess == kurtosis -3; + BOOST_CHECK_EQUAL(kurtosis(dist), static_cast(3) + kurtosis_excess(dist)); + + // special cases for PDF: + BOOST_CHECK_EQUAL( + pdf( + binomial_distribution(static_cast(8), static_cast(0)), + static_cast(0)), static_cast(1) + ); + BOOST_CHECK_EQUAL( + pdf( + binomial_distribution(static_cast(8), static_cast(0)), + static_cast(0.0001)), static_cast(0) + ); + BOOST_CHECK_EQUAL( + pdf( + binomial_distribution(static_cast(8), static_cast(1)), + static_cast(0.001)), static_cast(0) + ); + BOOST_CHECK_EQUAL( + pdf( + binomial_distribution(static_cast(8), static_cast(1)), + static_cast(8)), static_cast(1) + ); + BOOST_CHECK_EQUAL( + pdf( + binomial_distribution(static_cast(0), static_cast(0.25)), + static_cast(0)), static_cast(1) + ); + BOOST_CHECK_THROW( + pdf( + binomial_distribution(static_cast(-1), static_cast(0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(9)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(9)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), std::domain_error + ); + + BOOST_CHECK_EQUAL( + quantile( + binomial_distribution(static_cast(16), static_cast(0.25)), + static_cast(0.01)), // Less than cdf == pdf(binomial_distribution(16, 0.25), 0) + static_cast(0) // so expect zero as best approximation. + ); + + BOOST_CHECK_EQUAL( + cdf( + binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(8)), static_cast(1) + ); + BOOST_CHECK_EQUAL( + cdf( + binomial_distribution(static_cast(8), static_cast(0)), + static_cast(7)), static_cast(1) + ); + BOOST_CHECK_EQUAL( + cdf( + binomial_distribution(static_cast(8), static_cast(1)), + static_cast(7)), static_cast(0) + ); + + { + // This is a visual sanity check that everything is OK: + binomial_distribution my8dist(8., 0.25); // Note: double values (matching the distribution definition) avoid the need for any casting. + //cout << "mean(my8dist) = " << boost::math::mean(my8dist) << endl; // mean(my8dist) = 2 + //cout << "my8dist.trials() = " << my8dist.trials() << endl; // my8dist.trials() = 8 + //cout << "my8dist.success_fraction() = " << my8dist.success_fraction() << endl; // my8dist.success_fraction() = 0.25 + BOOST_CHECK_CLOSE(my8dist.trials(), static_cast(8), tol2); + BOOST_CHECK_CLOSE(my8dist.success_fraction(), static_cast(0.25), tol2); + + //{ + // int n = static_cast(boost::math::tools::real_cast(my8dist.trials())); + // RealType sumcdf = 0.; + // for (int k = 0; k <= n; k++) + // { + // cout << k << ' ' << pdf(my8dist, static_cast(k)); + // sumcdf += pdf(my8dist, static_cast(k)); + // cout << ' ' << sumcdf; + // cout << ' ' << cdf(my8dist, static_cast(k)); + // cout << ' ' << sumcdf - cdf(my8dist, static_cast(k)) << endl; + // } // for k + // } + // n = 8, p =0.25 + //k pdf cdf + //0 0.1001129150390625 0.1001129150390625 + //1 0.26696777343749994 0.36708068847656244 + //2 0.31146240234375017 0.67854309082031261 + //3 0.20764160156249989 0.8861846923828125 + //4 0.086517333984375 0.9727020263671875 + //5 0.023071289062499997 0.9957733154296875 + //6 0.0038452148437500009 0.9996185302734375 + //7 0.00036621093749999984 0.9999847412109375 + //8 1.52587890625e-005 1 1 0 + } +#define T RealType +#include "binomial_quantile.ipp" + + for(unsigned i = 0; i < binomial_quantile_data.size(); ++i) + { + using namespace boost::math::policies; + typedef policy > P1; + typedef policy > P2; + typedef policy > P3; + typedef policy > P4; + typedef policy > P5; + typedef policy > P6; + RealType tol = boost::math::tools::epsilon() * 500; + if(!boost::is_floating_point::value) + tol *= 10; // no lanczos approximation implies less accuracy + // + // Check full real value first: + // + binomial_distribution p1(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + RealType x = quantile(p1, binomial_quantile_data[i][2]); + BOOST_CHECK_CLOSE_FRACTION(x, binomial_quantile_data[i][3], tol); + x = quantile(complement(p1, binomial_quantile_data[i][2])); + BOOST_CHECK_CLOSE_FRACTION(x, binomial_quantile_data[i][4], tol); + // + // Now with round down to integer: + // + binomial_distribution p2(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + x = quantile(p2, binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, floor(binomial_quantile_data[i][3])); + x = quantile(complement(p2, binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, floor(binomial_quantile_data[i][4])); + // + // Now with round up to integer: + // + binomial_distribution p3(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + x = quantile(p3, binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, ceil(binomial_quantile_data[i][3])); + x = quantile(complement(p3, binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, ceil(binomial_quantile_data[i][4])); + // + // Now with round to integer "outside": + // + binomial_distribution p4(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + x = quantile(p4, binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, binomial_quantile_data[i][2] < 0.5f ? floor(binomial_quantile_data[i][3]) : ceil(binomial_quantile_data[i][3])); + x = quantile(complement(p4, binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, binomial_quantile_data[i][2] < 0.5f ? ceil(binomial_quantile_data[i][4]) : floor(binomial_quantile_data[i][4])); + // + // Now with round to integer "inside": + // + binomial_distribution p5(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + x = quantile(p5, binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, binomial_quantile_data[i][2] < 0.5f ? ceil(binomial_quantile_data[i][3]) : floor(binomial_quantile_data[i][3])); + x = quantile(complement(p5, binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, binomial_quantile_data[i][2] < 0.5f ? floor(binomial_quantile_data[i][4]) : ceil(binomial_quantile_data[i][4])); + // + // Now with round to nearest integer: + // + binomial_distribution p6(binomial_quantile_data[i][0], binomial_quantile_data[i][1]); + x = quantile(p6, binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, floor(binomial_quantile_data[i][3] + 0.5f)); + x = quantile(complement(p6, binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, floor(binomial_quantile_data[i][4] + 0.5f)); + } + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + // Check that can generate binomial distribution using one convenience methods: + binomial_distribution<> mybn2(1., 0.5); // Using default RealType double. + // but that + // boost::math::binomial mybn1(1., 0.5); // Using typedef fails + // error C2039: 'binomial' : is not a member of 'boost::math' + + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating point type). +#ifdef TEST_FLOAT + test_spots(0.0F); // Test float. +#endif +#ifdef TEST_DOUBLE + test_spots(0.0); // Test double. +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_spots(0.0L); // Test long double. +#endif +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_binomial.exe" +Running 1 test case... +Tolerance = 0.0119209%. +Tolerance = 2.22045e-011%. +Tolerance = 2.22045e-011%. +Tolerance = 2.22045e-011%. +*** No errors detected + +*/ diff --git a/test/test_binomial_coeff.cpp b/test/test_binomial_coeff.cpp new file mode 100644 index 000000000..f8557a4f8 --- /dev/null +++ b/test/test_binomial_coeff.cpp @@ -0,0 +1,197 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include "functor.hpp" +#include + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the function binomial_coefficient. +// The accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these function. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*large.*", // test data group + ".*", 100, 20); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*large.*", // test data group + ".*", 200, 100); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 100, 30); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", 2, 1); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +T binomial_wrapper(T n, T k) +{ + return boost::math::binomial_coefficient( + boost::math::tools::real_cast(n), + boost::math::tools::real_cast(k)); +} + +template +void test_binomial(T, const char* type_name) +{ + using namespace std; + + typedef T (*func_t)(T, T); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + func_t f = &binomial_wrapper; +#else + func_t f = &binomial_wrapper; +#endif + +#include "binomial_data.ipp" + + boost::math::tools::test_result result = boost::math::tools::test( + binomial_data, + bind_func(f, 0, 1), + extract_result(2)); + + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n" + "Test results for small arguments and type " << type_name << std::endl << std::endl; + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + handle_test_result(result, binomial_data[result.worst()], result.worst(), type_name, "binomial_coefficient", "Binomials: small arguments"); + std::cout << std::endl; + +#include "binomial_large_data.ipp" + + result = boost::math::tools::test( + binomial_large_data, + bind_func(f, 0, 1), + extract_result(2)); + + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n" + "Test results for large arguments and type " << type_name << std::endl << std::endl; + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + handle_test_result(result, binomial_large_data[result.worst()], result.worst(), type_name, "binomial_coefficient", "Binomials: large arguments"); + std::cout << std::endl; +} + +template +void test_spots(T, const char* name) +{ + T tolerance = boost::math::tools::epsilon() * 50 * 100; // 50 eps as a percentage + if(!std::numeric_limits::is_specialized) + tolerance *= 10; // beta function not so accurate without lanczos support + + std::cout << "Testing spot checks for type " << name << " with tolerance " << tolerance << "%\n"; + + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 0), static_cast(1)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 1), static_cast(20)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 2), static_cast(190)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 3), static_cast(1140)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 20), static_cast(1)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 19), static_cast(20)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 18), static_cast(190)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 17), static_cast(1140)); + BOOST_CHECK_EQUAL(boost::math::binomial_coefficient(20, 10), static_cast(184756L)); + + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(100, 5), static_cast(7.528752e7L), tolerance); + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(100, 81), static_cast(1.323415729392122674e20L), tolerance); + + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(300, 3), static_cast(4.45510e6L), tolerance); + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(300, 7), static_cast(4.043855956140000e13L), tolerance); + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(300, 290), static_cast(1.3983202332417017700000000e18L), tolerance); + BOOST_CHECK_CLOSE(boost::math::binomial_coefficient(300, 275), static_cast(1.953265141442868389822364184842211512000000e36L), tolerance); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + + test_spots(1.0F, "float"); + test_spots(1.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(1.0L, "long double"); + test_spots(boost::math::concepts::real_concept(), "real_concept"); +#endif + + test_binomial(1.0F, "float"); + test_binomial(1.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_binomial(1.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_binomial(boost::math::concepts::real_concept(), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_carlson.cpp b/test/test_carlson.cpp new file mode 100644 index 000000000..f29ca77d1 --- /dev/null +++ b/test/test_carlson.cpp @@ -0,0 +1,411 @@ +// Copyright 2006 John Maddock +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Carlson Elliptic Integrals. +// There are two sets of tests, spot +// tests which compare our results with the published test values, +// in Numerical Computation of Real or Complex Elliptic Integrals, +// B. C. Carlson: http://arxiv.org/abs/math.CA/9409227 +// However, the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // real long doubles: + // + if(boost::math::policies::digits >() > 53) + { + add_expected_result( + ".*", // compiler + ".*", // stdlib + BOOST_PLATFORM, // platform + largest_type, // test type(s) + ".*RJ.*", // test data group + ".*", 1000, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + BOOST_PLATFORM, // platform + "real_concept", // test type(s) + ".*RJ.*", // test data group + ".*", 1000, 50); // test function + } + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*RJ.*", // test data group + ".*", 180, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*RJ.*", // test data group + ".*", 180, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 15, 8); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 15, 8); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_ellint_rf(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp)(value_type, value_type, value_type) = boost::math::ellint_rf; +#else + value_type (*fp)(value_type, value_type, value_type) = boost::math::ellint_rf; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_rf", test); + + std::cout << std::endl; + +} + +template +void do_test_ellint_rc(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp)(value_type, value_type) = boost::math::ellint_rc; +#else + value_type (*fp)(value_type, value_type) = boost::math::ellint_rc; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_rc", test); + + std::cout << std::endl; + +} + +template +void do_test_ellint_rj(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp)(value_type, value_type, value_type, value_type) = boost::math::ellint_rj; +#else + value_type (*fp)(value_type, value_type, value_type, value_type) = boost::math::ellint_rj; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp, 0, 1, 2, 3), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_rf", test); + + std::cout << std::endl; + +} + +template +void do_test_ellint_rd(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp)(value_type, value_type, value_type) = boost::math::ellint_rd; +#else + value_type (*fp)(value_type, value_type, value_type) = boost::math::ellint_rd; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_rd", test); + + std::cout << std::endl; + +} + +template +void test_spots(T, const char* type_name) +{ + using namespace boost::math; + using namespace std; + // Spot values from Numerical Computation of Real or Complex + // Elliptic Integrals, B. C. Carlson: http://arxiv.org/abs/math.CA/9409227 + // RF: + T tolerance = (std::max)(T(1e-13f), tools::epsilon() * 5) * 100; // Note 5eps expressed as a persentage!!! + T eps2 = 2 * tools::epsilon(); + BOOST_CHECK_CLOSE(ellint_rf(T(1), T(2), T(0)), T(1.3110287771461), tolerance); + BOOST_CHECK_CLOSE(ellint_rf(T(0.5), T(1), T(0)), T(1.8540746773014), tolerance); + BOOST_CHECK_CLOSE(ellint_rf(T(2), T(3), T(4)), T(0.58408284167715), tolerance); + // RC: + BOOST_CHECK_CLOSE_FRACTION(ellint_rc(T(0), T(1)/4), boost::math::constants::pi(), eps2); + BOOST_CHECK_CLOSE_FRACTION(ellint_rc(T(9)/4, T(2)), log(T(2)), eps2); + BOOST_CHECK_CLOSE_FRACTION(ellint_rc(T(1)/4, T(-2)), log(T(2))/3, eps2); + // RJ: + BOOST_CHECK_CLOSE(ellint_rj(T(0), T(1), T(2), T(3)), T(0.77688623778582), tolerance); + BOOST_CHECK_CLOSE(ellint_rj(T(2), T(3), T(4), T(5)), T(0.14297579667157), tolerance); + BOOST_CHECK_CLOSE(ellint_rj(T(2), T(3), T(4), T(-0.5)), T(0.24723819703052), tolerance); + BOOST_CHECK_CLOSE(ellint_rj(T(2), T(3), T(4), T(-5)), T(-0.12711230042964), tolerance); + // RD: + BOOST_CHECK_CLOSE(ellint_rd(T(0), T(2), T(1)), T(1.7972103521034), tolerance); + BOOST_CHECK_CLOSE(ellint_rd(T(2), T(3), T(4)), T(0.16510527294261), tolerance); + + // Sanity/consistency checks from Numerical Computation of Real or Complex + // Elliptic Integrals, B. C. Carlson: http://arxiv.org/abs/math.CA/9409227 + std::tr1::mt19937 ran; + std::tr1::uniform_real ur(0, 1000); + T eps40 = 40 * tools::epsilon(); + + for(unsigned i = 0; i < 1000; ++i) + { + T x = ur(ran); + T y = ur(ran); + T z = ur(ran); + T lambda = ur(ran); + T mu = x * y / lambda; + // RF, eq 49: + T s1 = ellint_rf(x+lambda, y+lambda, lambda) + + ellint_rf(x + mu, y + mu, mu); + T s2 = ellint_rf(x, y, T(0)); + BOOST_CHECK_CLOSE_FRACTION(s1, s2, eps40); + // RC is degenerate case of RF: + s1 = ellint_rc(x, y); + s2 = ellint_rf(x, y, y); + BOOST_CHECK_CLOSE_FRACTION(s1, s2, eps40); + // RC, eq 50 (Note have to assume y = x): + T mu2 = x * x / lambda; + s1 = ellint_rc(lambda, x+lambda) + + ellint_rc(mu2, x + mu2); + s2 = ellint_rc(T(0), x); + BOOST_CHECK_CLOSE_FRACTION(s1, s2, eps40); + /* + T p = ????; // no closed form for a, b and p??? + s1 = ellint_rj(x+lambda, y+lambda, lambda, p+lambda) + + ellint_rj(x+mu, y+mu, mu, p+mu); + s2 = ellint_rj(x, y, T(0), p) + - 3 * ellint_rc(a, b); + */ + // RD, eq 53: + s1 = ellint_rd(lambda, x+lambda, y+lambda) + + ellint_rd(mu, x+mu, y+mu); + s2 = ellint_rd(T(0), x, y) + - 3 / (y * sqrt(x+y+lambda+mu)); + BOOST_CHECK_CLOSE_FRACTION(s1, s2, eps40); + // RD is degenerate case of RJ: + s1 = ellint_rd(x, y, z); + s2 = ellint_rj(x, y, z, z); + BOOST_CHECK_CLOSE_FRACTION(s1, s2, eps40); + } + + // + // Now random spot values: + // +#include "ellint_rf_data.ipp" + + do_test_ellint_rf(ellint_rf_data, type_name, "RF: Random data"); + +#include "ellint_rc_data.ipp" + + do_test_ellint_rc(ellint_rc_data, type_name, "RC: Random data"); + +#include "ellint_rj_data.ipp" + + do_test_ellint_rj(ellint_rj_data, type_name, "RJ: Random data"); + +#include "ellint_rd_data.ipp" + + do_test_ellint_rd(ellint_rd_data, type_name, "RD: Random data"); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + + boost::math::ellint_rj(1.778e-31, 1.407e+18, 10.05, -4.83e-10); + + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} + +/* + +test_carlson.cpp +Linking... +Embedding manifest... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_carlson.exe" +Running 1 test case... +Tests run with Microsoft Visual C++ version 8.0, Dinkumware standard library version 405, Win32 +Testing: RF: Random data +boost::math::ellint_rf Max = 0 RMS Mean=0 +Testing: RC: Random data +boost::math::ellint_rc Max = 0 RMS Mean=0 +Testing: RJ: Random data +boost::math::ellint_rf Max = 0 RMS Mean=0 +Testing: RD: Random data +boost::math::ellint_rd Max = 0 RMS Mean=0 +Testing: RF: Random data +boost::math::ellint_rf Max = 2.949 RMS Mean=0.7498 + worst case at row: 377 + { 3.418e+025, 2.594e-005, 3.264e-012, 6.169e-012 } +Testing: RC: Random data +boost::math::ellint_rc Max = 2.396 RMS Mean=0.6283 + worst case at row: 10 + { 1.97e-029, 3.224e-025, 2.753e+012 } +Testing: RJ: Random data +boost::math::ellint_rf Max = 152.9 RMS Mean=11.15 + worst case at row: 633 + { 1.876e+016, 0.000278, 3.796e-006, -4.412e-005, -1.656e-005 } +Testing: RD: Random data +boost::math::ellint_rd Max = 2.586 RMS Mean=0.8614 + worst case at row: 45 + { 2.111e-020, 8.757e-026, 1.923e-023, 1.004e+033 } +Testing: RF: Random data +boost::math::ellint_rf Max = 2.949 RMS Mean=0.7498 + worst case at row: 377 + { 3.418e+025, 2.594e-005, 3.264e-012, 6.169e-012 } +Testing: RC: Random data +boost::math::ellint_rc Max = 2.396 RMS Mean=0.6283 + worst case at row: 10 + { 1.97e-029, 3.224e-025, 2.753e+012 } +Testing: RJ: Random data +boost::math::ellint_rf Max = 152.9 RMS Mean=11.15 + worst case at row: 633 + { 1.876e+016, 0.000278, 3.796e-006, -4.412e-005, -1.656e-005 } +Testing: RD: Random data +boost::math::ellint_rd Max = 2.586 RMS Mean=0.8614 + worst case at row: 45 + { 2.111e-020, 8.757e-026, 1.923e-023, 1.004e+033 } +Testing: RF: Random data +boost::math::ellint_rf Max = 2.949 RMS Mean=0.7498 + worst case at row: 377 + { 3.418e+025, 2.594e-005, 3.264e-012, 6.169e-012 } +Testing: RC: Random data +boost::math::ellint_rc Max = 2.396 RMS Mean=0.6283 + worst case at row: 10 + { 1.97e-029, 3.224e-025, 2.753e+012 } +Testing: RJ: Random data +boost::math::ellint_rf Max = 152.9 RMS Mean=11.15 + worst case at row: 633 + { 1.876e+016, 0.000278, 3.796e-006, -4.412e-005, -1.656e-005 } +Testing: RD: Random data +boost::math::ellint_rd Max = 2.586 RMS Mean=0.8614 + worst case at row: 45 + { 2.111e-020, 8.757e-026, 1.923e-023, 1.004e+033 } +*** No errors detected + +*/ diff --git a/test/test_cauchy.cpp b/test/test_cauchy.cpp new file mode 100644 index 000000000..14e6cea71 --- /dev/null +++ b/test/test_cauchy.cpp @@ -0,0 +1,756 @@ +// Copyright John Maddock 2006, 2007. +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_cauchy.cpp Test Cauchy distribution + +#ifdef _MSC_VER +# pragma warning(disable: 4100) // unreferenced formal parameter. +// Seems an entirely spurious warning - formal parameter T IS used - get error if /* T */ +//# pragma warning(disable: 4535) // calling _set_se_translator() requires /EHa (in Boost.test) +// Enable C++ Exceptions Yes With SEH Exceptions (/EHa) prevents warning 4535. +# pragma warning(disable: 4127) // conditional expression is constant +#endif + +// #define BOOST_MATH_ASSERT_UNDEFINED_POLICY false +// To compile even if Cauchy mean is used. + +#include // for real_concept +#include + using boost::math::cauchy_distribution; + +#include // Boost.Test +#include + +#include + using std::cout; + using std::endl; + +template +void test_spots(RealType T) +{ + // Check some bad parameters to the distribution, + BOOST_CHECK_THROW(boost::math::cauchy_distribution nbad1(0, 0), std::domain_error); // zero sd + BOOST_CHECK_THROW(boost::math::cauchy_distribution nbad1(0, -1), std::domain_error); // negative scale (shape) + cauchy_distribution C01; + + BOOST_CHECK_EQUAL(C01.location(), 0); // Check standard values. + BOOST_CHECK_EQUAL(C01.scale(), 1); + + // Tests on extreme values of random variate x, if has numeric_limit infinity etc. + if(std::numeric_limits::has_infinity) + { + BOOST_CHECK_EQUAL(pdf(C01, +std::numeric_limits::infinity()), 0); // x = + infinity, pdf = 0 + BOOST_CHECK_EQUAL(pdf(C01, -std::numeric_limits::infinity()), 0); // x = - infinity, pdf = 0 + BOOST_CHECK_EQUAL(cdf(C01, +std::numeric_limits::infinity()), 1); // x = + infinity, cdf = 1 + BOOST_CHECK_EQUAL(cdf(C01, -std::numeric_limits::infinity()), 0); // x = - infinity, cdf = 0 + BOOST_CHECK_EQUAL(cdf(complement(C01, +std::numeric_limits::infinity())), 0); // x = + infinity, cdf = 0 + BOOST_CHECK_EQUAL(cdf(complement(C01, -std::numeric_limits::infinity())), 1); // x = - infinity, cdf = 1 + BOOST_CHECK_THROW(boost::math::cauchy_distribution nbad1(std::numeric_limits::infinity(), static_cast(1)), std::domain_error); // +infinite mean + BOOST_CHECK_THROW(boost::math::cauchy_distribution nbad1(-std::numeric_limits::infinity(), static_cast(1)), std::domain_error); // -infinite mean + BOOST_CHECK_THROW(boost::math::cauchy_distribution nbad1(static_cast(0), std::numeric_limits::infinity()), std::domain_error); // infinite sd + } + + if (std::numeric_limits::has_quiet_NaN) + { // No longer allow x to be NaN, so these tests should throw. + BOOST_CHECK_THROW(pdf(C01, +std::numeric_limits::quiet_NaN()), std::domain_error); // x = NaN + BOOST_CHECK_THROW(cdf(C01, +std::numeric_limits::quiet_NaN()), std::domain_error); // x = NaN + BOOST_CHECK_THROW(cdf(complement(C01, +std::numeric_limits::quiet_NaN())), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(quantile(C01, +std::numeric_limits::quiet_NaN()), std::domain_error); // p = + infinity + BOOST_CHECK_THROW(quantile(complement(C01, +std::numeric_limits::quiet_NaN())), std::domain_error); // p = + infinity + } + + // Basic sanity checks. + // 50eps as a percentage, up to a maximum of double precision + // (that's the limit of our test data). + RealType tolerance = (std::max)( + static_cast(boost::math::tools::epsilon()), + boost::math::tools::epsilon()); + tolerance *= 50 * 100; + + cout << "Tolerance for type " << typeid(T).name() << " is " << tolerance << " %" << endl; + + // These first sets of test values were calculated by punching numbers + // into a calculator, and using the formulas on the Mathworld website: + // http://mathworld.wolfram.com/CauchyDistribution.html + // and values from MathCAD 200 Professional, + // CDF: + // + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(0.125)), // x + static_cast(0.53958342416056554201085167134004L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(-0.125)), // x + static_cast(0.46041657583943445798914832865996L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(0.5)), // x + static_cast(0.64758361765043327417540107622474L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(-0.5)), // x + static_cast(0.35241638234956672582459892377526L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(1.0)), // x + static_cast(0.75), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(-1.0)), // x + static_cast(0.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(2.0)), // x + static_cast(0.85241638234956672582459892377526L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(-2.0)), // x + static_cast(0.14758361765043327417540107622474L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(10.0)), // x + static_cast(0.9682744825694464304850228813987L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(), + static_cast(-10.0)), // x + static_cast(0.031725517430553569514977118601302L), // probability. + tolerance); // % + + // + // Complements: + // + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(0.125))), // x + static_cast(0.46041657583943445798914832865996L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(-0.125))), // x + static_cast(0.53958342416056554201085167134004L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(0.5))), // x + static_cast(0.35241638234956672582459892377526L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(-0.5))), // x + static_cast(0.64758361765043327417540107622474L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(1.0))), // x + static_cast(0.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(-1.0))), // x + static_cast(0.75), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(2.0))), // x + static_cast(0.14758361765043327417540107622474L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(-2.0))), // x + static_cast(0.85241638234956672582459892377526L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(10.0))), // x + static_cast(0.031725517430553569514977118601302L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(), + static_cast(-10.0))), // x + static_cast(0.9682744825694464304850228813987L), // probability. + tolerance); // % + + // + // Quantiles: + // + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.53958342416056554201085167134004L)), + static_cast(0.125), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.46041657583943445798914832865996L)), + static_cast(-0.125), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.64758361765043327417540107622474L)), + static_cast(0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.35241638234956672582459892377526)), + static_cast(-0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.75)), + static_cast(1.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.25)), + static_cast(-1.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.85241638234956672582459892377526L)), + static_cast(2.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.14758361765043327417540107622474L)), + static_cast(-2.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.9682744825694464304850228813987L)), + static_cast(10.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(), + static_cast(0.031725517430553569514977118601302L)), + static_cast(-10.0), + tolerance); // % + + // + // Quantile from complement: + // + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.46041657583943445798914832865996L))), + static_cast(0.125), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.53958342416056554201085167134004L))), + static_cast(-0.125), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.35241638234956672582459892377526L))), + static_cast(0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.64758361765043327417540107622474L))), + static_cast(-0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.25))), + static_cast(1.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.75))), + static_cast(-1.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.14758361765043327417540107622474L))), + static_cast(2.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.85241638234956672582459892377526L))), + static_cast(-2.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.031725517430553569514977118601302L))), + static_cast(10.0), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(), + static_cast(0.9682744825694464304850228813987L))), + static_cast(-10.0), + tolerance); // % + + // + // PDF + // + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(0.125)), // x + static_cast(0.31341281101173235351410956479511L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(-0.125)), // x + static_cast(0.31341281101173235351410956479511L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(0.5)), // x + static_cast(0.25464790894703253723021402139602L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(-0.5)), // x + static_cast(0.25464790894703253723021402139602L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(1.0)), // x + static_cast(0.15915494309189533576888376337251L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(-1.0)), // x + static_cast(0.15915494309189533576888376337251L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(2.0)), // x + static_cast(0.063661977236758134307553505349006L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(-2.0)), // x + static_cast(0.063661977236758134307553505349006L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(10.0)), // x + static_cast(0.0031515830315226799162155200667825L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(), + static_cast(-10.0)), // x + static_cast(0.0031515830315226799162155200667825L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(2, 5), + static_cast(1)), // x + static_cast(0.061213439650728975295724524374044L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + cauchy_distribution(-2, 0.25), + static_cast(1)), // x + static_cast(0.0087809623774838805941453110826215L), // probability. + tolerance); // % + + // + // The following test values were calculated using MathCad, + // precision seems to be about 10^-13. + // + tolerance = (std::max)(tolerance, static_cast(1e-11)); + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(1, 1), + static_cast(0.125)), // x + static_cast(0.271189304634946L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(1, 1), + static_cast(0.125))), // x + static_cast(1 - 0.271189304634946L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(1, 1), + static_cast(0.271189304634946L)), // x + static_cast(0.125), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(1, 1), + static_cast(1 - 0.271189304634946L))), // x + static_cast(0.125), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(0.125)), // x + static_cast(0.539583424160566L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(0.5)), // x + static_cast(0.647583617650433L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(1)), // x + static_cast(0.750000000000000), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(2)), // x + static_cast(0.852416382349567), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(10)), // x + static_cast(0.968274482569447), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(100)), // x + static_cast(0.996817007235092), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-0.125)), // x + static_cast(0.460416575839434), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-0.5)), // x + static_cast(0.352416382349567), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-1)), // x + static_cast(0.2500000000000000), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-2)), // x + static_cast(0.147583617650433), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-10)), // x + static_cast(0.031725517430554), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(0, 1), + static_cast(-100)), // x + static_cast(3.18299276490824E-3), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(1, 5), + static_cast(1.25)), // x + static_cast(0.515902251256176), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(2, 2), + static_cast(1.25)), // x + static_cast(0.385799748780092), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(4, 0.125), + static_cast(3)), // x + static_cast(0.039583424160566), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(-2, static_cast(0.0001)), + static_cast(-3)), // x + static_cast(3.1830988512275777e-5), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(4, 50), + static_cast(-3)), // x + static_cast(0.455724386698215), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(-4, 50), + static_cast(-3)), // x + static_cast(0.506365349100973), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(1, 5), + static_cast(1.25))), // x + static_cast(1-0.515902251256176), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(2, 2), + static_cast(1.25))), // x + static_cast(1-0.385799748780092), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(4, 0.125), + static_cast(3))), // x + static_cast(1-0.039583424160566), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + cauchy_distribution(-2, static_cast(0.001)), + static_cast(-3)), // x + static_cast(0.000318309780080539), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(4, 50), + static_cast(-3))), // x + static_cast(1-0.455724386698215), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(cauchy_distribution(-4, 50), + static_cast(-3))), // x + static_cast(1-0.506365349100973), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(1, 5), + static_cast(0.515902251256176)), // x + static_cast(1.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(2, 2), + static_cast(0.385799748780092)), // x + static_cast(1.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(4, 0.125), + static_cast(0.039583424160566)), // x + static_cast(3), // probability. + tolerance); // % + /* + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(-2, 0.0001), + static_cast(-3)), // x + static_cast(0.000015915494296), // probability. + tolerance); // % + */ + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(4, 50), + static_cast(0.455724386698215)), // x + static_cast(-3), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(-4, 50), + static_cast(0.506365349100973)), // x + static_cast(-3), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(1, 5), + static_cast(1-0.515902251256176))), // x + static_cast(1.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(2, 2), + static_cast(1-0.385799748780092))), // x + static_cast(1.25), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(4, 0.125), + static_cast(1-0.039583424160566))), // x + static_cast(3), // probability. + tolerance); // % + /* + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + cauchy_distribution(-2, 0.0001), + static_cast(-3)), // x + static_cast(0.000015915494296), // probability. + tolerance); // % + */ + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(4, 50), + static_cast(1-0.455724386698215))), // x + static_cast(-3), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(cauchy_distribution(-4, 50), + static_cast(1-0.506365349100973))), // x + static_cast(-3), // probability. + tolerance); // % + + cauchy_distribution dist; // default (0, 1) + BOOST_CHECK_EQUAL( + mode(dist), + static_cast(0)); + BOOST_CHECK_EQUAL( + median(dist), + static_cast(0)); + // + // Things that now don't compile (BOOST-STATIC_ASSERT_FAILURE) by default. + // #define BOOST_MATH_ASSERT_UNDEFINED_POLICY false + // To compile even if Cauchy mean is used. + // See policy reference, mathematically undefined function policies + // + //BOOST_CHECK_THROW( + // mean(dist), + // std::domain_error); + //BOOST_CHECK_THROW( + // variance(dist), + // std::domain_error); + //BOOST_CHECK_THROW( + // standard_deviation(dist), + // std::domain_error); + //BOOST_CHECK_THROW( + // kurtosis(dist), + // std::domain_error); + //BOOST_CHECK_THROW( + // kurtosis_excess(dist), + // std::domain_error); + //BOOST_CHECK_THROW( + // skewness(dist), + // std::domain_error); + + BOOST_CHECK_THROW( + quantile(dist, RealType(0.0)), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(1.0)), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, RealType(0.0))), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, RealType(1.0))), + std::overflow_error); + + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + // Check that can generate cauchy distribution using the two convenience methods: + boost::math::cauchy mycd1(1.); // Using typedef + cauchy_distribution<> mycd2(1.); // Using default RealType double. + cauchy_distribution<> C01; // Using default RealType double for Standard Cauchy. + BOOST_CHECK_EQUAL(C01.location(), 0); // Check standard values. + BOOST_CHECK_EQUAL(C01.scale(), 1); + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + + +/* +Output: + +Running 1 test case... +Tolerance for type float is 0.000596046 % +Tolerance for type double is 1.11022e-012 % +Tolerance for type long double is 1.11022e-012 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-012 % +*** No errors detected + + +*/ diff --git a/test/test_cbrt.cpp b/test/test_cbrt.cpp new file mode 100644 index 000000000..adc30da5c --- /dev/null +++ b/test/test_cbrt.cpp @@ -0,0 +1,135 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the function cbrt. The accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; + + add_expected_result( + "Borland.*", // compiler + ".*", // stdlib + ".*", // platform + "long double", // test type(s) + ".*", // test data group + ".*", 10, 6); // test function +} + +struct negative_cbrt +{ + negative_cbrt(){} + + template + typename S::value_type operator()(const S& row) + { + return boost::math::cbrt(-row[1]); + } +}; + + +template +void do_test_cbrt(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::cbrt; +#else + pg funcp = boost::math::cbrt; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test cbrt against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 1), + extract_result(0)); + result += boost::math::tools::test( + data, + negative_cbrt(), + negate(extract_result(0))); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cbrt", test_name); + std::cout << std::endl; +} +template +void test_cbrt(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "cbrt_data.ipp" + + do_test_cbrt(cbrt_data, name, "cbrt Function"); + +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + test_cbrt(0.1F, "float"); + test_cbrt(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_cbrt(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_cbrt(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif + return 0; +} + + diff --git a/test/test_chi_squared.cpp b/test/test_chi_squared.cpp new file mode 100644 index 000000000..49a788e16 --- /dev/null +++ b/test/test_chi_squared.cpp @@ -0,0 +1,560 @@ +// test_chi_squared.cpp + +// Copyright Paul A. Bristow 2006. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for chi_squared_distribution +using boost::math::chi_squared_distribution; +using boost::math::chi_squared; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE + +#include +using std::cout; +using std::endl; +#include +using std::numeric_limits; + +template +RealType naive_pdf(RealType df, RealType x) +{ + using namespace std; // For ADL of std functions. + RealType e = log(x) * ((df / 2) - 1) - (x / 2) - boost::math::lgamma(df/2); + e -= log(static_cast(2)) * df / 2; + return exp(e); +} + +template +void test_spot( + RealType df, // Degrees of freedom + RealType cs, // Chi Square statistic + RealType P, // CDF + RealType Q, // Complement of CDF + RealType tol) // Test tolerance +{ + boost::math::chi_squared_distribution dist(df); + BOOST_CHECK_CLOSE( + cdf(dist, cs), P, tol); + BOOST_CHECK_CLOSE( + pdf(dist, cs), naive_pdf(dist.degrees_of_freedom(), cs), tol); + if((P < 0.99) && (Q < 0.99)) + { + // + // We can only check this if P is not too close to 1, + // so that we can guarentee Q is free of error: + // + BOOST_CHECK_CLOSE( + cdf(complement(dist, cs)), Q, tol); + BOOST_CHECK_CLOSE( + quantile(dist, P), cs, tol); + BOOST_CHECK_CLOSE( + quantile(complement(dist, Q)), cs, tol); + } +} + +// +// This test data is taken from the tables of upper and lower +// critical values of the Chi Squared distribution available +// at http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm +// +double q[] = { 0.10, 0.05, 0.025, 0.01, 0.001 }; +double upper_critical_values[][6] = { + { 1, 2.706, 3.841, 5.024, 6.635, 10.828 }, + { 2, 4.605, 5.991, 7.378, 9.210, 13.816 }, + { 3, 6.251, 7.815, 9.348, 11.345, 16.266 }, + { 4, 7.779, 9.488, 11.143, 13.277, 18.467 }, + { 5, 9.236, 11.070, 12.833, 15.086, 20.515 }, + { 6, 10.645, 12.592, 14.449, 16.812, 22.458 }, + { 7, 12.017, 14.067, 16.013, 18.475, 24.322 }, + { 8, 13.362, 15.507, 17.535, 20.090, 26.125 }, + { 9, 14.684, 16.919, 19.023, 21.666, 27.877 }, + { 10, 15.987, 18.307, 20.483, 23.209, 29.588 }, + { 11, 17.275, 19.675, 21.920, 24.725, 31.264 }, + { 12, 18.549, 21.026, 23.337, 26.217, 32.910 }, + { 13, 19.812, 22.362, 24.736, 27.688, 34.528 }, + { 14, 21.064, 23.685, 26.119, 29.141, 36.123 }, + { 15, 22.307, 24.996, 27.488, 30.578, 37.697 }, + { 16, 23.542, 26.296, 28.845, 32.000, 39.252 }, + { 17, 24.769, 27.587, 30.191, 33.409, 40.790 }, + { 18, 25.989, 28.869, 31.526, 34.805, 42.312 }, + { 19, 27.204, 30.144, 32.852, 36.191, 43.820 }, + { 20, 28.412, 31.410, 34.170, 37.566, 45.315 }, + { 21, 29.615, 32.671, 35.479, 38.932, 46.797 }, + { 22, 30.813, 33.924, 36.781, 40.289, 48.268 }, + { 23, 32.007, 35.172, 38.076, 41.638, 49.728 }, + { 24, 33.196, 36.415, 39.364, 42.980, 51.179 }, + { 25, 34.382, 37.652, 40.646, 44.314, 52.620 }, + { 26, 35.563, 38.885, 41.923, 45.642, 54.052 }, + { 27, 36.741, 40.113, 43.195, 46.963, 55.476 }, + { 28, 37.916, 41.337, 44.461, 48.278, 56.892 }, + { 29, 39.087, 42.557, 45.722, 49.588, 58.301 }, + { 30, 40.256, 43.773, 46.979, 50.892, 59.703 }, + { 31, 41.422, 44.985, 48.232, 52.191, 61.098 }, + { 32, 42.585, 46.194, 49.480, 53.486, 62.487 }, + { 33, 43.745, 47.400, 50.725, 54.776, 63.870 }, + { 34, 44.903, 48.602, 51.966, 56.061, 65.247 }, + { 35, 46.059, 49.802, 53.203, 57.342, 66.619 }, + { 36, 47.212, 50.998, 54.437, 58.619, 67.985 }, + { 37, 48.363, 52.192, 55.668, 59.893, 69.347 }, + { 38, 49.513, 53.384, 56.896, 61.162, 70.703 }, + { 39, 50.660, 54.572, 58.120, 62.428, 72.055 }, + { 40, 51.805, 55.758, 59.342, 63.691, 73.402 }, + { 41, 52.949, 56.942, 60.561, 64.950, 74.745 }, + { 42, 54.090, 58.124, 61.777, 66.206, 76.084 }, + { 43, 55.230, 59.304, 62.990, 67.459, 77.419 }, + { 44, 56.369, 60.481, 64.201, 68.710, 78.750 }, + { 45, 57.505, 61.656, 65.410, 69.957, 80.077 }, + { 46, 58.641, 62.830, 66.617, 71.201, 81.400 }, + { 47, 59.774, 64.001, 67.821, 72.443, 82.720 }, + { 48, 60.907, 65.171, 69.023, 73.683, 84.037 }, + { 49, 62.038, 66.339, 70.222, 74.919, 85.351 }, + { 50, 63.167, 67.505, 71.420, 76.154, 86.661 }, + { 51, 64.295, 68.669, 72.616, 77.386, 87.968 }, + { 52, 65.422, 69.832, 73.810, 78.616, 89.272 }, + { 53, 66.548, 70.993, 75.002, 79.843, 90.573 }, + { 54, 67.673, 72.153, 76.192, 81.069, 91.872 }, + { 55, 68.796, 73.311, 77.380, 82.292, 93.168 }, + { 56, 69.919, 74.468, 78.567, 83.513, 94.461 }, + { 57, 71.040, 75.624, 79.752, 84.733, 95.751 }, + { 58, 72.160, 76.778, 80.936, 85.950, 97.039 }, + { 59, 73.279, 77.931, 82.117, 87.166, 98.324 }, + { 60, 74.397, 79.082, 83.298, 88.379, 99.607 }, + { 61, 75.514, 80.232, 84.476, 89.591, 100.888 }, + { 62, 76.630, 81.381, 85.654, 90.802, 102.166 }, + { 63, 77.745, 82.529, 86.830, 92.010, 103.442 }, + { 64, 78.860, 83.675, 88.004, 93.217, 104.716 }, + { 65, 79.973, 84.821, 89.177, 94.422, 105.988 }, + { 66, 81.085, 85.965, 90.349, 95.626, 107.258 }, + { 67, 82.197, 87.108, 91.519, 96.828, 108.526 }, + { 68, 83.308, 88.250, 92.689, 98.028, 109.791 }, + { 69, 84.418, 89.391, 93.856, 99.228, 111.055 }, + { 70, 85.527, 90.531, 95.023, 100.425, 112.317 }, + { 71, 86.635, 91.670, 96.189, 101.621, 113.577 }, + { 72, 87.743, 92.808, 97.353, 102.816, 114.835 }, + { 73, 88.850, 93.945, 98.516, 104.010, 116.092 }, + { 74, 89.956, 95.081, 99.678, 105.202, 117.346 }, + { 75, 91.061, 96.217, 100.839, 106.393, 118.599 }, + { 76, 92.166, 97.351, 101.999, 107.583, 119.850 }, + { 77, 93.270, 98.484, 103.158, 108.771, 121.100 }, + { 78, 94.374, 99.617, 104.316, 109.958, 122.348 }, + { 79, 95.476, 100.749, 105.473, 111.144, 123.594 }, + { 80, 96.578, 101.879, 106.629, 112.329, 124.839 }, + { 81, 97.680, 103.010, 107.783, 113.512, 126.083 }, + { 82, 98.780, 104.139, 108.937, 114.695, 127.324 }, + { 83, 99.880, 105.267, 110.090, 115.876, 128.565 }, + { 84, 100.980, 106.395, 111.242, 117.057, 129.804 }, + { 85, 102.079, 107.522, 112.393, 118.236, 131.041 }, + { 86, 103.177, 108.648, 113.544, 119.414, 132.277 }, + { 87, 104.275, 109.773, 114.693, 120.591, 133.512 }, + { 88, 105.372, 110.898, 115.841, 121.767, 134.746 }, + { 89, 106.469, 112.022, 116.989, 122.942, 135.978 }, + { 90, 107.565, 113.145, 118.136, 124.116, 137.208 }, + { 91, 108.661, 114.268, 119.282, 125.289, 138.438 }, + { 92, 109.756, 115.390, 120.427, 126.462, 139.666 }, + { 93, 110.850, 116.511, 121.571, 127.633, 140.893 }, + { 94, 111.944, 117.632, 122.715, 128.803, 142.119 }, + { 95, 113.038, 118.752, 123.858, 129.973, 143.344 }, + { 96, 114.131, 119.871, 125.000, 131.141, 144.567 }, + { 97, 115.223, 120.990, 126.141, 132.309, 145.789 }, + { 98, 116.315, 122.108, 127.282, 133.476, 147.010 }, + { 99, 117.407, 123.225, 128.422, 134.642, 148.230 }, + { 100, 118.498, 124.342, 129.561, 135.807, 149.449 }, + {100, 118.498, 124.342, 129.561, 135.807, 149.449 } +}; + +double lower_critical_values[][6] = { + /* + These have fewer than 4 significant digits, leave them out + of the tests for now: + 1., .016, .004, .001, .000, .000, + 2., .211, .103, .051, .020, .002, + 3., .584, .352, .216, .115, .024, + 4., 1.064, .711, .484, .297, .091, + 5., 1.610, 1.145, .831, .554, .210, + 6., 2.204, 1.635, 1.237, .872, .381, + 7., 2.833, 2.167, 1.690, 1.239, .598, + 8., 3.490, 2.733, 2.180, 1.646, .857, + */ + { 9., 4.168, 3.325, 2.700, 2.088, 1.152 }, + { 10., 4.865, 3.940, 3.247, 2.558, 1.479 }, + { 11., 5.578, 4.575, 3.816, 3.053, 1.834 }, + { 12., 6.304, 5.226, 4.404, 3.571, 2.214 }, + { 13., 7.042, 5.892, 5.009, 4.107, 2.617 }, + { 14., 7.790, 6.571, 5.629, 4.660, 3.041 }, + { 15., 8.547, 7.261, 6.262, 5.229, 3.483 }, + { 16., 9.312, 7.962, 6.908, 5.812, 3.942 }, + { 17., 10.085, 8.672, 7.564, 6.408, 4.416 }, + { 18., 10.865, 9.390, 8.231, 7.015, 4.905 }, + { 19., 11.651, 10.117, 8.907, 7.633, 5.407 }, + { 20., 12.443, 10.851, 9.591, 8.260, 5.921 }, + { 21., 13.240, 11.591, 10.283, 8.897, 6.447 }, + { 22., 14.041, 12.338, 10.982, 9.542, 6.983 }, + { 23., 14.848, 13.091, 11.689, 10.196, 7.529 }, + { 24., 15.659, 13.848, 12.401, 10.856, 8.085 }, + { 25., 16.473, 14.611, 13.120, 11.524, 8.649 }, + { 26., 17.292, 15.379, 13.844, 12.198, 9.222 }, + { 27., 18.114, 16.151, 14.573, 12.879, 9.803 }, + { 28., 18.939, 16.928, 15.308, 13.565, 10.391 }, + { 29., 19.768, 17.708, 16.047, 14.256, 10.986 }, + { 30., 20.599, 18.493, 16.791, 14.953, 11.588 }, + { 31., 21.434, 19.281, 17.539, 15.655, 12.196 }, + { 32., 22.271, 20.072, 18.291, 16.362, 12.811 }, + { 33., 23.110, 20.867, 19.047, 17.074, 13.431 }, + { 34., 23.952, 21.664, 19.806, 17.789, 14.057 }, + { 35., 24.797, 22.465, 20.569, 18.509, 14.688 }, + { 36., 25.643, 23.269, 21.336, 19.233, 15.324 }, + { 37., 26.492, 24.075, 22.106, 19.960, 15.965 }, + { 38., 27.343, 24.884, 22.878, 20.691, 16.611 }, + { 39., 28.196, 25.695, 23.654, 21.426, 17.262 }, + { 40., 29.051, 26.509, 24.433, 22.164, 17.916 }, + { 41., 29.907, 27.326, 25.215, 22.906, 18.575 }, + { 42., 30.765, 28.144, 25.999, 23.650, 19.239 }, + { 43., 31.625, 28.965, 26.785, 24.398, 19.906 }, + { 44., 32.487, 29.787, 27.575, 25.148, 20.576 }, + { 45., 33.350, 30.612, 28.366, 25.901, 21.251 }, + { 46., 34.215, 31.439, 29.160, 26.657, 21.929 }, + { 47., 35.081, 32.268, 29.956, 27.416, 22.610 }, + { 48., 35.949, 33.098, 30.755, 28.177, 23.295 }, + { 49., 36.818, 33.930, 31.555, 28.941, 23.983 }, + { 50., 37.689, 34.764, 32.357, 29.707, 24.674 }, + { 51., 38.560, 35.600, 33.162, 30.475, 25.368 }, + { 52., 39.433, 36.437, 33.968, 31.246, 26.065 }, + { 53., 40.308, 37.276, 34.776, 32.018, 26.765 }, + { 54., 41.183, 38.116, 35.586, 32.793, 27.468 }, + { 55., 42.060, 38.958, 36.398, 33.570, 28.173 }, + { 56., 42.937, 39.801, 37.212, 34.350, 28.881 }, + { 57., 43.816, 40.646, 38.027, 35.131, 29.592 }, + { 58., 44.696, 41.492, 38.844, 35.913, 30.305 }, + { 59., 45.577, 42.339, 39.662, 36.698, 31.020 }, + { 60., 46.459, 43.188, 40.482, 37.485, 31.738 }, + { 61., 47.342, 44.038, 41.303, 38.273, 32.459 }, + { 62., 48.226, 44.889, 42.126, 39.063, 33.181 }, + { 63., 49.111, 45.741, 42.950, 39.855, 33.906 }, + { 64., 49.996, 46.595, 43.776, 40.649, 34.633 }, + { 65., 50.883, 47.450, 44.603, 41.444, 35.362 }, + { 66., 51.770, 48.305, 45.431, 42.240, 36.093 }, + { 67., 52.659, 49.162, 46.261, 43.038, 36.826 }, + { 68., 53.548, 50.020, 47.092, 43.838, 37.561 }, + { 69., 54.438, 50.879, 47.924, 44.639, 38.298 }, + { 70., 55.329, 51.739, 48.758, 45.442, 39.036 }, + { 71., 56.221, 52.600, 49.592, 46.246, 39.777 }, + { 72., 57.113, 53.462, 50.428, 47.051, 40.519 }, + { 73., 58.006, 54.325, 51.265, 47.858, 41.264 }, + { 74., 58.900, 55.189, 52.103, 48.666, 42.010 }, + { 75., 59.795, 56.054, 52.942, 49.475, 42.757 }, + { 76., 60.690, 56.920, 53.782, 50.286, 43.507 }, + { 77., 61.586, 57.786, 54.623, 51.097, 44.258 }, + { 78., 62.483, 58.654, 55.466, 51.910, 45.010 }, + { 79., 63.380, 59.522, 56.309, 52.725, 45.764 }, + { 80., 64.278, 60.391, 57.153, 53.540, 46.520 }, + { 81., 65.176, 61.261, 57.998, 54.357, 47.277 }, + { 82., 66.076, 62.132, 58.845, 55.174, 48.036 }, + { 83., 66.976, 63.004, 59.692, 55.993, 48.796 }, + { 84., 67.876, 63.876, 60.540, 56.813, 49.557 }, + { 85., 68.777, 64.749, 61.389, 57.634, 50.320 }, + { 86., 69.679, 65.623, 62.239, 58.456, 51.085 }, + { 87., 70.581, 66.498, 63.089, 59.279, 51.850 }, + { 88., 71.484, 67.373, 63.941, 60.103, 52.617 }, + { 89., 72.387, 68.249, 64.793, 60.928, 53.386 }, + { 90., 73.291, 69.126, 65.647, 61.754, 54.155 }, + { 91., 74.196, 70.003, 66.501, 62.581, 54.926 }, + { 92., 75.100, 70.882, 67.356, 63.409, 55.698 }, + { 93., 76.006, 71.760, 68.211, 64.238, 56.472 }, + { 94., 76.912, 72.640, 69.068, 65.068, 57.246 }, + { 95., 77.818, 73.520, 69.925, 65.898, 58.022 }, + { 96., 78.725, 74.401, 70.783, 66.730, 58.799 }, + { 97., 79.633, 75.282, 71.642, 67.562, 59.577 }, + { 98., 80.541, 76.164, 72.501, 68.396, 60.356 }, + { 99., 81.449, 77.046, 73.361, 69.230, 61.137 }, + {100., 82.358, 77.929, 74.222, 70.065, 61.918 } +}; + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks, test data is to three decimal places only + // so set tolerance to 0.001 expressed as a persentage. + + RealType tolerance = 0.001f * 100; + + cout << "Tolerance = " << tolerance << "%." << endl; + + using boost::math::chi_squared_distribution; + using ::boost::math::chi_squared; + using ::boost::math::cdf; + using ::boost::math::pdf; + + for(unsigned i = 0; i < sizeof(upper_critical_values) / sizeof(upper_critical_values[0]); ++i) + { + test_spot( + static_cast(upper_critical_values[i][0]), // degrees of freedom + static_cast(upper_critical_values[i][1]), // Chi Squared statistic + static_cast(1 - q[0]), // Probability of result (CDF), P + static_cast(q[0]), // Q = 1 - P + tolerance); + test_spot( + static_cast(upper_critical_values[i][0]), // degrees of freedom + static_cast(upper_critical_values[i][2]), // Chi Squared statistic + static_cast(1 - q[1]), // Probability of result (CDF), P + static_cast(q[1]), // Q = 1 - P + tolerance); + test_spot( + static_cast(upper_critical_values[i][0]), // degrees of freedom + static_cast(upper_critical_values[i][3]), // Chi Squared statistic + static_cast(1 - q[2]), // Probability of result (CDF), P + static_cast(q[2]), // Q = 1 - P + tolerance); + test_spot( + static_cast(upper_critical_values[i][0]), // degrees of freedom + static_cast(upper_critical_values[i][4]), // Chi Squared statistic + static_cast(1 - q[3]), // Probability of result (CDF), P + static_cast(q[3]), // Q = 1 - P + tolerance); + test_spot( + static_cast(upper_critical_values[i][0]), // degrees of freedom + static_cast(upper_critical_values[i][5]), // Chi Squared statistic + static_cast(1 - q[4]), // Probability of result (CDF), P + static_cast(q[4]), // Q = 1 - P + tolerance); + } + + for(unsigned i = 0; i < sizeof(lower_critical_values) / sizeof(lower_critical_values[0]); ++i) + { + test_spot( + static_cast(lower_critical_values[i][0]), // degrees of freedom + static_cast(lower_critical_values[i][1]), // Chi Squared statistic + static_cast(q[0]), // Probability of result (CDF), P + static_cast(1 - q[0]), // Q = 1 - P + tolerance); + test_spot( + static_cast(lower_critical_values[i][0]), // degrees of freedom + static_cast(lower_critical_values[i][2]), // Chi Squared statistic + static_cast(q[1]), // Probability of result (CDF), P + static_cast(1 - q[1]), // Q = 1 - P + tolerance); + test_spot( + static_cast(lower_critical_values[i][0]), // degrees of freedom + static_cast(lower_critical_values[i][3]), // Chi Squared statistic + static_cast(q[2]), // Probability of result (CDF), P + static_cast(1 - q[2]), // Q = 1 - P + tolerance); + test_spot( + static_cast(lower_critical_values[i][0]), // degrees of freedom + static_cast(lower_critical_values[i][4]), // Chi Squared statistic + static_cast(q[3]), // Probability of result (CDF), P + static_cast(1 - q[3]), // Q = 1 - P + tolerance); + test_spot( + static_cast(lower_critical_values[i][0]), // degrees of freedom + static_cast(lower_critical_values[i][5]), // Chi Squared statistic + static_cast(q[4]), // Probability of result (CDF), P + static_cast(1 - q[4]), // Q = 1 - P + tolerance); + } + + RealType tol2 = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a percentage + chi_squared_distribution dist(static_cast(8)); + RealType x = 7; + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(8), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(16), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , static_cast(4), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol2); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(6), tol2); + + BOOST_CHECK_CLOSE( + median(dist), + quantile( + chi_squared_distribution(static_cast(8)), + static_cast(0.5)), static_cast(tol2)); + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , static_cast(1), tol2); + // kurtosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , static_cast(4.5), tol2); + // kurtosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , static_cast(1.5), tol2); + // special cases: + BOOST_CHECK_THROW( + pdf( + chi_squared_distribution(static_cast(1)), + static_cast(0)), std::overflow_error + ); + BOOST_CHECK_EQUAL( + pdf(chi_squared_distribution(2), static_cast(0)) + , static_cast(0.5f)); + BOOST_CHECK_EQUAL( + pdf(chi_squared_distribution(3), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(chi_squared_distribution(1), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(chi_squared_distribution(2), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(chi_squared_distribution(3), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(complement(chi_squared_distribution(1), static_cast(0))) + , static_cast(1)); + BOOST_CHECK_EQUAL( + cdf(complement(chi_squared_distribution(2), static_cast(0))) + , static_cast(1)); + BOOST_CHECK_EQUAL( + cdf(complement(chi_squared_distribution(3), static_cast(0))) + , static_cast(1)); + + BOOST_CHECK_THROW( + pdf( + chi_squared_distribution(static_cast(-1)), + static_cast(1)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + chi_squared_distribution(static_cast(8)), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + chi_squared_distribution(static_cast(-1)), + static_cast(1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + chi_squared_distribution(static_cast(8)), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf(complement( + chi_squared_distribution(static_cast(-1)), + static_cast(1))), std::domain_error + ); + BOOST_CHECK_THROW( + cdf(complement( + chi_squared_distribution(static_cast(8)), + static_cast(-1))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + chi_squared_distribution(static_cast(-1)), + static_cast(0.5)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + chi_squared_distribution(static_cast(8)), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + chi_squared_distribution(static_cast(8)), + static_cast(1.1)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + chi_squared_distribution(static_cast(-1)), + static_cast(0.5))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + chi_squared_distribution(static_cast(8)), + static_cast(-1))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + chi_squared_distribution(static_cast(8)), + static_cast(1.1))), std::domain_error + ); + + // This first test value is taken from an example here: + // http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm + // Subsequent tests just test our empirically generated values, they + // catch regressions, but otherwise aren't worth much. + BOOST_CHECK_EQUAL( + ceil(chi_squared_distribution::find_degrees_of_freedom( + 55, 0.05f, 0.01f, 100)), static_cast(170)); + BOOST_CHECK_EQUAL( + ceil(chi_squared_distribution::find_degrees_of_freedom( + 10, 0.05f, 0.01f, 100)), static_cast(3493)); + BOOST_CHECK_EQUAL( + ceil(chi_squared_distribution::find_degrees_of_freedom( + -55, 0.05f, 0.01f, 100)), static_cast(49)); + BOOST_CHECK_EQUAL( + ceil(chi_squared_distribution::find_degrees_of_freedom( + -10, 0.05f, 0.01f, 100)), static_cast(2826)); +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + // Check that can generate chi_squared distribution using the two convenience methods: + chi_squared_distribution<> mychisqr(8); + chi_squared mychisqr2(8); + + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#endif + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_chi_squared.exe" +Running 1 test case... +Tolerance = 0.1%. +Tolerance = 0.1%. +Tolerance = 0.1%. +Tolerance = 0.1%. +*** No errors detected + +*/ + + + diff --git a/test/test_classify.cpp b/test/test_classify.cpp new file mode 100644 index 000000000..ab457b597 --- /dev/null +++ b/test/test_classify.cpp @@ -0,0 +1,166 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include + +#ifdef _MSC_VER +#pragma warning(disable: 4127) // conditional expression is constant +#endif + +template +void test_classify(T t, const char* type) +{ + std::cout << "Testing type " << type << std::endl; + t = 2; + T u = 2; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + if(std::numeric_limits::is_specialized) + { + t = (std::numeric_limits::max)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + t = (std::numeric_limits::min)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NORMAL); + } + if(std::numeric_limits::has_denorm) + { + t /= 2; + if(t != 0) + { + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_SUBNORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_SUBNORMAL); + } + t = std::numeric_limits::denorm_min(); + if((t != 0) && (t < (std::numeric_limits::min)())) + { + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_SUBNORMAL); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_SUBNORMAL); + } + } + else + { + std::cout << "Denormalised forms not tested" << std::endl; + } + t = 0; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_ZERO); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_ZERO); + t /= -u; // create minus zero if it exists + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_ZERO); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_ZERO); + // inifinity: + if(std::numeric_limits::has_infinity) + { + // At least one std::numeric_limits::infinity)() returns zero + // (Compaq true64 cxx), hence the check. + t = (std::numeric_limits::infinity)(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); +#if !defined(__BORLANDC__) && !(defined(__DECCXX) && !defined(_IEEE_FP)) + // divide by zero on Borland triggers a C++ exception :-( + // divide by zero on Compaq CXX triggers a C style signal :-( + t = 2; + u = 0; + t /= u; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); + t = -2; + t /= u; + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_INFINITE); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_INFINITE); +#else + std::cout << "Infinities from divide by zero not tested" << std::endl; +#endif + } + else + { + std::cout << "Infinity not tested" << std::endl; + } +#ifndef __BORLANDC__ + // NaN's: + // Note that Borland throws an exception if we even try to obtain a Nan + // by calling std::numeric_limits::quiet_NaN() !!!!!!! + if(std::numeric_limits::has_quiet_NaN) + { + t = std::numeric_limits::quiet_NaN(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NAN); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NAN); + } + else + { + std::cout << "Quiet NaN's not tested" << std::endl; + } + if(std::numeric_limits::has_signaling_NaN) + { + t = std::numeric_limits::signaling_NaN(); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(t), (int)FP_NAN); + BOOST_CHECK_EQUAL((::boost::math::fpclassify)(-t), (int)FP_NAN); + } + else + { + std::cout << "Signaling NaN's not tested" << std::endl; + } +#endif +} + +int test_main(int, char* [] ) +{ + BOOST_MATH_CONTROL_FP; + // start by printing some information: +#ifdef isnan + std::cout << "Platform has isnan macro." << std::endl; +#endif +#ifdef fpclassify + std::cout << "Platform has fpclassify macro." << std::endl; +#endif +#ifdef BOOST_HAS_FPCLASSIFY + std::cout << "Platform has FP_NORMAL macro." << std::endl; +#endif + std::cout << "FP_ZERO: " << (int)FP_ZERO << std::endl; + std::cout << "FP_NORMAL: " << (int)FP_NORMAL << std::endl; + std::cout << "FP_INFINITE: " << (int)FP_INFINITE << std::endl; + std::cout << "FP_NAN: " << (int)FP_NAN << std::endl; + std::cout << "FP_SUBNORMAL: " << (int)FP_SUBNORMAL << std::endl; + + // then run the tests: + test_classify(float(0), "float"); + test_classify(double(0), "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_classify((long double)(0), "long double"); + test_classify((boost::math::concepts::real_concept)(0), "real_concept"); +#endif + return 0; +} + +/* +Autorun "i:\Boost-sandbox\math_toolkit\libs\math\test\MSVC80\debug\test_classify.exe" +Running 1 test case... +FP_ZERO: 0 +FP_NORMAL: 1 +FP_INFINITE: 2 +FP_NAN: 3 +FP_SUBNORMAL: 4 +Testing type float +Testing type double +Testing type long double +Testing type real_concept +Denormalised forms not tested +Infinity not tested +Quiet NaN's not tested +Signaling NaN's not tested +Test suite "Test Program" passed with: + 79 assertions out of 79 passed + 1 test case out of 1 passed + Test case "test_main_caller( argc, argv )" passed with: + 79 assertions out of 79 passed + +*/ diff --git a/test/test_constants.cpp b/test/test_constants.cpp new file mode 100644 index 000000000..06c250d11 --- /dev/null +++ b/test/test_constants.cpp @@ -0,0 +1,88 @@ +// Copyright Paul Bristow 2007. +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_constants.cpp + +#include // for real_concept +#include // Boost.Test +#include + +#include +#include + +template +void test_spots(RealType) +{ + // Basic sanity checks for constants. + + RealType tolerance = static_cast(2e-15); // double + //cout << "Tolerance for type " << typeid(T).name() << " is " << tolerance << "." << endl; + + using namespace boost::math::constants; + using namespace std; // Help ADL of std exp, log... + using std::exp; + + BOOST_CHECK_CLOSE_FRACTION(static_cast(3.14159265358979323846264338327950288419716939937510L), pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(sqrt(3.14159265358979323846264338327950288419716939937510L)), root_pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(sqrt(3.14159265358979323846264338327950288419716939937510L/2)), root_half_pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(sqrt(3.14159265358979323846264338327950288419716939937510L * 2)), root_two_pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(sqrt(log(4.0L))), root_ln_four(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(2.71828182845904523536028747135266249775724709369995L), e(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(0.5), half(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(0.57721566490153286060651209008240243104259335L), euler(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(sqrt(2.0L)), root_two(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(log(2.0L)), ln_two(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(log(log(2.0L))), ln_ln_two(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(1)/3, third(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(2)/3, twothirds(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(0.14159265358979323846264338327950288419716939937510L), pi_minus_three(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(4. - 3.14159265358979323846264338327950288419716939937510L), four_minus_pi(), tolerance); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_CHECK_CLOSE_FRACTION(static_cast(pow((4 - 3.14159265358979323846264338327950288419716939937510L), 1.5L)), pow23_four_minus_pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(exp(-0.5L)), exp_minus_half(), tolerance); +#else + BOOST_CHECK_CLOSE_FRACTION(static_cast(pow((4 - 3.14159265358979323846264338327950288419716939937510), 1.5)), pow23_four_minus_pi(), tolerance); + BOOST_CHECK_CLOSE_FRACTION(static_cast(exp(-0.5)), exp_minus_half(), tolerance); +#endif + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_constants.exe" +Running 1 test case... +*** No errors detected + +*/ + + + diff --git a/test/test_digamma.cpp b/test/test_digamma.cpp new file mode 100644 index 000000000..035b94496 --- /dev/null +++ b/test/test_digamma.cpp @@ -0,0 +1,182 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the digamma function. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*Negative.*", // test data group + ".*", 300, 40); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", 3, 3); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_digamma(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::digamma; +#else + pg funcp = boost::math::digamma; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test digamma against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::digamma", test_name); + std::cout << std::endl; +} + +template +void test_digamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "digamma_data.ipp" + + do_test_digamma(digamma_data, name, "Digamma Function: Large Values"); + +# include "digamma_root_data.ipp" + + do_test_digamma(digamma_root_data, name, "Digamma Function: Near the Positive Root"); + +# include "digamma_small_data.ipp" + + do_test_digamma(digamma_small_data, name, "Digamma Function: Near Zero"); + +# include "digamma_neg_data.ipp" + + do_test_digamma(digamma_neg_data, name, "Digamma Function: Negative Values"); + +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // Basic sanity checks, tolerance is 3 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 300; + // + // Special tolerance (200eps) for when we're very near the root, + // and T has more than 64-bits in it's mantissa: + // + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(0.125)), static_cast(-8.3884926632958548678027429230863430000514460424495L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(0.5)), static_cast(-1.9635100260214234794409763329987555671931596046604L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1)), static_cast(-0.57721566490153286060651209008240243104215933593992L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5)), static_cast(0.036489973978576520559023667001244432806840395339566L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5) - static_cast(1)/32), static_cast(0.00686541147073577672813890866512415766586241385896200579891429L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(2)), static_cast(0.42278433509846713939348790991759756895784066406008L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(8)), static_cast(2.0156414779556099965363450527747404261006978069172L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(12)), static_cast(2.4426616799758120167383652547949424463027180089374L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(22)), static_cast(3.0681430398611966699248760264450329818421699570581L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(50)), static_cast(3.9019896734278921969539597028823666609284424880275L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(500)), static_cast(6.2136077650889917423827750552855712637776544784569L), tolerance); + // + // negative values: + // + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(-0.125)), static_cast(7.1959829284523046176757814502538535827603450463013L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(-10.125)), static_cast(9.9480538258660761287008034071425343357982429855241L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(-10.875)), static_cast(-5.1527360383841562620205965901515879492020193154231L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(-1.5)), static_cast(0.70315664064524318722569033366791109947350706200623L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif + + expected_results(); + + test_digamma(0.1F, "float"); + test_digamma(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_digamma(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_digamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + diff --git a/test/test_dist_overloads.cpp b/test/test_dist_overloads.cpp new file mode 100644 index 000000000..ef1008df0 --- /dev/null +++ b/test/test_dist_overloads.cpp @@ -0,0 +1,101 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_dist_overloads.cpp + +#include // for real_concept +#include + using boost::math::normal_distribution; + +#include // Boost.Test +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; + +template +void test_spots(RealType) +{ + // Basic sanity checks, + // 2 eps as a percentage: + RealType tolerance = boost::math::tools::epsilon() * 2 * 100; + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + for(int i = -4; i <= 4; ++i) + { + BOOST_CHECK_CLOSE( + ::boost::math::cdf(normal_distribution(), i), + ::boost::math::cdf(normal_distribution(), static_cast(i)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::pdf(normal_distribution(), i), + ::boost::math::pdf(normal_distribution(), static_cast(i)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::cdf(complement(normal_distribution(), i)), + ::boost::math::cdf(complement(normal_distribution(), static_cast(i))), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::hazard(normal_distribution(), i), + ::boost::math::hazard(normal_distribution(), static_cast(i)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::chf(normal_distribution(), i), + ::boost::math::chf(normal_distribution(), static_cast(i)), + tolerance); + } + for(float f = 0.01f; f < 1; f += 0.01f) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile(normal_distribution(), f), + ::boost::math::quantile(normal_distribution(), static_cast(f)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::quantile(complement(normal_distribution(), f)), + ::boost::math::quantile(complement(normal_distribution(), static_cast(f))), + tolerance); + } +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Running 1 test case... +Tolerance for type float is 2.38419e-005 % +Tolerance for type double is 4.44089e-014 % +Tolerance for type long double is 4.44089e-014 % +Tolerance for type class boost::math::concepts::real_concept is 4.44089e-014 % +*** No errors detected + +*/ + diff --git a/test/test_ellint_1.cpp b/test/test_ellint_1.cpp new file mode 100644 index 000000000..e99a18902 --- /dev/null +++ b/test/test_ellint_1.cpp @@ -0,0 +1,220 @@ +// Copyright Xiaogang Zhang 2006 +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4756) // overflow in constant arithmetic +// Constants are too big for float case, but this doesn't matter for test. +#endif + +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Elliptic Integrals of the first kind. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 5, 3); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 5, 3); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + + +template +void do_test_ellint_f(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp2)(value_type, value_type) = boost::math::ellint_1; +#else + value_type (*fp2)(value_type, value_type) = boost::math::ellint_1; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp2, 1, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_1", test); + + std::cout << std::endl; + +} + +template +void do_test_ellint_k(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + boost::math::tools::test_result result; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp1)(value_type) = boost::math::ellint_1; +#else + value_type (*fp1)(value_type) = boost::math::ellint_1; +#endif + result = boost::math::tools::test( + data, + bind_func(fp1, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_1", test); + + std::cout << std::endl; +} + +template +void test_spots(T, const char* type_name) +{ + // Function values calculated on http://functions.wolfram.com/ + // Note that Mathematica's EllipticF accepts k^2 as the second parameter. + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 19> data1 = { + SC_(0), SC_(0), SC_(0), + SC_(-10), SC_(0), SC_(-10), + SC_(-1), SC_(-1), SC_(-1.2261911708835170708130609674719067527242483502207), + SC_(-4), SC_(0.875), SC_(-5.3190556182262405182189463092940736859067548232647), + SC_(8), SC_(-0.625), SC_(9.0419973860310100524448893214394562615252527557062), + SC_(1e-05), SC_(0.875), SC_(0.000010000000000127604166668510945638036143355898993088), + SC_(1e+05), SC_(10)/1024, SC_(100002.38431454899771096037307519328741455615271038), + SC_(1e-20), SC_(1), SC_(1.0000000000000000000000000000000000000000166666667e-20), + SC_(1e-20), SC_(1e-20), SC_(1.000000000000000e-20), + SC_(1e+20), SC_(400)/1024, SC_(1.0418143796499216839719289963154558027005142709763e20), + SC_(1e+50), SC_(0.875), SC_(1.3913251718238765549409892714295358043696028445944e50), + SC_(2), SC_(0.5), SC_(2.1765877052210673672479877957388515321497888026770), + SC_(4), SC_(0.5), SC_(4.2543274975235836861894752787874633017836785640477), + SC_(6), SC_(0.5), SC_(6.4588766202317746302999080620490579800463614807916), + SC_(10), SC_(0.5), SC_(10.697409951222544858346795279378531495869386960090), + SC_(-2), SC_(0.5), SC_(-2.1765877052210673672479877957388515321497888026770), + SC_(-4), SC_(0.5), SC_(-4.2543274975235836861894752787874633017836785640477), + SC_(-6), SC_(0.5), SC_(-6.4588766202317746302999080620490579800463614807916), + SC_(-10), SC_(0.5), SC_(-10.697409951222544858346795279378531495869386960090), + }; + #undef SC_ + + do_test_ellint_f(data1, type_name, "Elliptic Integral F: Mathworld Data"); + +#include "ellint_f_data.ipp" + + do_test_ellint_f(ellint_f_data, type_name, "Elliptic Integral F: Random Data"); + + // Function values calculated on http://functions.wolfram.com/ + // Note that Mathematica's EllipticK accepts k^2 as the second parameter. + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 9> data2 = { + SC_(0), SC_(1.5707963267948966192313216916397514420985846996876), + SC_(0.125), SC_(1.5769867712158131421244030532288080803822271060839), + SC_(0.25), SC_(1.5962422221317835101489690714979498795055744578951), + SC_(300)/1024, SC_(1.6062331054696636704261124078746600894998873503208), + SC_(400)/1024, SC_(1.6364782007562008756208066125715722889067992997614), + SC_(-0.5), SC_(1.6857503548125960428712036577990769895008008941411), + SC_(-0.75), SC_(1.9109897807518291965531482187613425592531451316788), + 1-SC_(1)/8, SC_(2.185488469278223686913080323730158689730428415766), + 1-SC_(1)/1024, SC_(4.5074135978990422666372495313621124487894807327687), + }; + #undef SC_ + + do_test_ellint_k(data2, type_name, "Elliptic Integral K: Mathworld Data"); + +#include "ellint_k_data.ipp" + + do_test_ellint_k(ellint_k_data, type_name, "Elliptic Integral K: Random Data"); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} diff --git a/test/test_ellint_2.cpp b/test/test_ellint_2.cpp new file mode 100644 index 000000000..55a93b8d8 --- /dev/null +++ b/test/test_ellint_2.cpp @@ -0,0 +1,212 @@ +// Copyright Xiaogang Zhang 2006 +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4756) // overflow in constant arithmetic +// Constants are too big for float case, but this doesn't matter for test. +#endif + +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Elliptic Integrals of the second kind. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 15, 6); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 15, 6); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + + +template +void do_test_ellint_e2(const T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp2)(value_type, value_type) = boost::math::ellint_2; +#else + value_type (*fp2)(value_type, value_type) = boost::math::ellint_2; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp2, 1, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_2", test); + + std::cout << std::endl; +} + +template +void do_test_ellint_e1(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + boost::math::tools::test_result result; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp1)(value_type) = boost::math::ellint_2; +#else + value_type (*fp1)(value_type) = boost::math::ellint_2; +#endif + result = boost::math::tools::test( + data, + bind_func(fp1, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_2", test); + + std::cout << std::endl; +} + +template +void test_spots(T, const char* type_name) +{ + // Function values calculated on http://functions.wolfram.com/ + // Note that Mathematica's EllipticE accepts k^2 as the second parameter. + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 10> data1 = { + SC_(0), SC_(0), SC_(0), + SC_(-10), SC_(0), SC_(-10), + SC_(-1), SC_(-1), SC_(-0.84147098480789650665250232163029899962256306079837), + SC_(-4), SC_(900) / 1024, SC_(-3.1756145986492562317862928524528520686391383168377), + SC_(8), SC_(-600) / 1024, SC_(7.2473147180505693037677015377802777959345489333465), + SC_(1e-05), SC_(800) / 1024, SC_(9.999999999898274739584436515967055859383969942432E-6), + SC_(1e+05), SC_(100) / 1024, SC_(99761.153306972066658135668386691227343323331995888), + SC_(1e+10), SC_(-0.5), SC_(9.3421545766487137036576748555295222252286528414669e9), + ldexp(SC_(1), 66), SC_(400) / 1024, SC_(7.0886102721911705466476846969992069994308167515242e19), + ldexp(SC_(1), 166), SC_(900) / 1024, SC_(7.1259011068364515942912094521783688927118026465790e49), + }; + #undef SC_ + + do_test_ellint_e2(data1, type_name, "Elliptic Integral E: Mathworld Data"); + +#include "ellint_e2_data.ipp" + + do_test_ellint_e2(ellint_e2_data, type_name, "Elliptic Integral E: Random Data"); + + // Function values calculated on http://functions.wolfram.com/ + // Note that Mathematica's EllipticE accepts k^2 as the second parameter. + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 10> data2 = { + SC_(-1), SC_(1), + SC_(0), SC_(1.5707963267948966192313216916397514420985846996876), + SC_(100) / 1024, SC_(1.5670445330545086723323795143598956428788609133377), + SC_(200) / 1024, SC_(1.5557071588766556854463404816624361127847775545087), + SC_(300) / 1024, SC_(1.5365278991162754883035625322482669608948678755743), + SC_(400) / 1024, SC_(1.5090417763083482272165682786143770446401437564021), + SC_(-0.5), SC_(1.4674622093394271554597952669909161360253617523272), + SC_(-600) / 1024, SC_(1.4257538571071297192428217218834579920545946473778), + SC_(-800) / 1024, SC_(1.2927868476159125056958680222998765985004489572909), + SC_(-900) / 1024, SC_(1.1966864890248739524112920627353824133420353430982), + }; + #undef SC_ + + do_test_ellint_e1(data2, type_name, "Elliptic Integral E: Mathworld Data"); + +#include "ellint_e_data.ipp" + + do_test_ellint_e1(ellint_e_data, type_name, "Elliptic Integral E: Random Data"); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_spots(0.0F, "float"); +#endif + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} diff --git a/test/test_ellint_3.cpp b/test/test_ellint_3.cpp new file mode 100644 index 000000000..006f41ead --- /dev/null +++ b/test/test_ellint_3.cpp @@ -0,0 +1,247 @@ +// Copyright Xiaogang Zhang 2006 +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4756) // overflow in constant arithmetic +// Constants are too big for float case, but this doesn't matter for test. +#endif + +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Elliptic Integrals of the third kind. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Large.*", // test data group + ".*", 50, 20); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*Large.*", // test data group + ".*", 50, 20); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 15, 8); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 15, 8); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_ellint_pi3(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp2)(value_type, value_type, value_type) = boost::math::ellint_3; +#else + value_type (*fp2)(value_type, value_type, value_type) = boost::math::ellint_3; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp2, 2, 0, 1), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_3", test); + + std::cout << std::endl; + +} + +template +void do_test_ellint_pi2(T& data, const char* type_name, const char* test) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << "Testing: " << test << std::endl; + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + value_type (*fp2)(value_type, value_type) = boost::math::ellint_3; +#else + value_type (*fp2)(value_type, value_type) = boost::math::ellint_3; +#endif + boost::math::tools::test_result result; + + result = boost::math::tools::test( + data, + bind_func(fp2, 1, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), + type_name, "boost::math::ellint_3", test); + + std::cout << std::endl; + +} + +template +void test_spots(T, const char* type_name) +{ + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 21> data1 = { + SC_(1), SC_(-1), SC_(0), SC_(-1.557407724654902230506974807458360173087), + SC_(0), SC_(-4), SC_(0.4), SC_(-4.153623371196831087495427530365430979011), + SC_(0), SC_(8), SC_(-0.6), SC_(8.935930619078575123490612395578518914416), + SC_(0), SC_(0.5), SC_(0.25), SC_(0.501246705365439492445236118603525029757890291780157969500480), + SC_(0), SC_(0.5), SC_(0), SC_(0.5), + SC_(-2), SC_(0.5), SC_(0), SC_(0.437501067017546278595664813509803743009132067629603474488486), + SC_(0.25), SC_(0.5), SC_(0), SC_(0.510269830229213412212501938035914557628394166585442994564135), + SC_(0.75), SC_(0.5), SC_(0), SC_(0.533293253875952645421201146925578536430596894471541312806165), + SC_(0.75), SC_(0.75), SC_(0), SC_(0.871827580412760575085768367421866079353646112288567703061975), + SC_(1), SC_(0.25), SC_(0), SC_(0.255341921221036266504482236490473678204201638800822621740476), + SC_(2), SC_(0.25), SC_(0), SC_(0.261119051639220165094943572468224137699644963125853641716219), + SC_(1023)/1024, SC_(1.5), SC_(0), SC_(13.2821612239764190363647953338544569682942329604483733197131), + SC_(0.5), SC_(-1), SC_(0.5), SC_(-1.228014414316220642611298946293865487807), + SC_(0.5), SC_(1e+10), SC_(0.5), SC_(1.536591003599172091573590441336982730551e+10), + SC_(-1e+05), SC_(10), SC_(0.75), SC_(0.0347926099493147087821620459290460547131012904008557007934290), + SC_(-1e+10), SC_(10), SC_(0.875), SC_(0.000109956202759561502329123384755016959364346382187364656768212), + SC_(-1e+10), SC_(1e+20), SC_(0.875), SC_(1.00000626665567332602765201107198822183913978895904937646809e15), + SC_(-1e+10), SC_(1608)/1024, SC_(0.875), SC_(0.0000157080616044072676127333183571107873332593142625043567690379), + 1-SC_(1) / 1024, SC_(1e+20), SC_(0.875), SC_(6.43274293944380717581167058274600202023334985100499739678963e21), + SC_(50), SC_(0.1), SC_(0.25), SC_(0.124573770342749525407523258569507331686458866564082916835900), + SC_(1.125), SC_(1), SC_(0.25), SC_(1.77299767784815770192352979665283069318388205110727241629752), + //SC_(1.125), SC_(10), SC_(0.25), SC_(0.662467818678976949597336360256848770217429434745967677192487), + //SC_(1.125), SC_(3), SC_(0.25), SC_(-0.142697285116693775525461312178015106079842313950476205580178), + }; + #undef SC_ + + do_test_ellint_pi3(data1, type_name, "Elliptic Integral PI: Mathworld Data"); + +#include "ellint_pi3_data.ipp" + + do_test_ellint_pi3(ellint_pi3_data, type_name, "Elliptic Integral PI: Random Data"); + +#include "ellint_pi3_large_data.ipp" + + do_test_ellint_pi3(ellint_pi3_large_data, type_name, "Elliptic Integral PI: Large Random Data"); + + // function values calculated on http://functions.wolfram.com/ + #define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 9> data2 = { + SC_(0), SC_(0.2), SC_(1.586867847454166237308008033828114192951), + SC_(0), SC_(0.4), SC_(1.639999865864511206865258329748601457626), + SC_(0), SC_(0), SC_(1.57079632679489661923132169163975144209858469968755291048747), + SC_(0.5), SC_(0), SC_(2.221441469079183123507940495030346849307), + SC_(-4), SC_(0.3), SC_(0.712708870925620061597924858162260293305195624270730660081949), + SC_(-1e+05), SC_(-0.5), SC_(0.00496944596485066055800109163256108604615568144080386919012831), + SC_(-1e+10), SC_(-0.75), SC_(0.0000157080225184890546939710019277357161497407143903832703317801), + SC_(1) / 1024, SC_(-0.875), SC_(2.18674503176462374414944618968850352696579451638002110619287), + SC_(1023)/1024, SC_(-0.875), SC_(101.045289804941384100960063898569538919135722087486350366997), + }; + #undef SC_ + + do_test_ellint_pi2(data2, type_name, "Complete Elliptic Integral PI: Mathworld Data"); + +#include "ellint_pi2_data.ipp" + + do_test_ellint_pi2(ellint_pi2_data, type_name, "Complete Elliptic Integral PI: Random Data"); + + // Special cases, exceptions etc: + BOOST_CHECK_THROW(boost::math::ellint_3(T(1.0001), T(-1), T(0)), std::domain_error); + BOOST_CHECK_THROW(boost::math::ellint_3(T(0.5), T(20), T(1.5)), std::domain_error); + BOOST_CHECK_THROW(boost::math::ellint_3(T(1.0001), T(-1)), std::domain_error); + BOOST_CHECK_THROW(boost::math::ellint_3(T(0.5), T(1)), std::domain_error); + BOOST_CHECK_THROW(boost::math::ellint_3(T(0.5), T(2)), std::domain_error); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} diff --git a/test/test_erf.cpp b/test/test_erf.cpp new file mode 100644 index 000000000..70880439c --- /dev/null +++ b/test/test_erf.cpp @@ -0,0 +1,509 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_erf_hooks.hpp" +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the functions erf, erfc, and the inverses +// erf_inv and erfc_inv. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // On MacOS X erfc has much higher error levels than + // expected: given that the implementation is basically + // just a rational function evaluation combined with + // exponentiation, we conclude that exp and pow are less + // accurate on this platform, especially when the result + // is outside the range of a double. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "Erf Function:.*Large.*", // test data group + "boost::math::erfc", 4300, 1300); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "Erf Function:.*", // test data group + "boost::math::erfc", 40, 10); // test function + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Erf Function:.*", // test data group + "boost::math::erfc?", 20, 6); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Inverse Erfc.*", // test data group + "boost::math::erfc_inv", 80, 10); // test function + + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + "Erf Function:.*", // test data group + "boost::math::erfc?", 2, 2); // test function + add_expected_result( + ".*aCC.*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + "Inverse Erfc.*", // test data group + "boost::math::erfc_inv", 80, 10); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + "Inverse Erf.*", // test data group + "boost::math::erfc?_inv", 18, 4); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_erf(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::erf; +#else + pg funcp = boost::math::erf; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test erf against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::erf", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::erf; + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::erf"); + } +#endif + // + // test erfc against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::erfc; +#else + funcp = boost::math::erfc; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::erfc", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::erfc; + result = boost::math::tools::test( + data, + bind(funcp, 0), + extract_result(2)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::erfc"); + } +#endif + std::cout << std::endl; +} + +template +void do_test_erf_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); + + boost::math::tools::test_result result; + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + // + // test erf_inv against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::erf_inv; +#else + pg funcp = boost::math::erf_inv; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::erf_inv", test_name); + std::cout << std::endl; +} + +template +void do_test_erfc_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::erf; +#else + pg funcp = boost::math::erf; +#endif + + boost::math::tools::test_result result; + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + // + // test erfc_inv against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::erfc_inv; +#else + funcp = boost::math::erfc_inv; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::erfc_inv", test_name); + std::cout << std::endl; +} + +template +void test_erf(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "erf_small_data.ipp" + + do_test_erf(erf_small_data, name, "Erf Function: Small Values"); + +# include "erf_data.ipp" + + do_test_erf(erf_data, name, "Erf Function: Medium Values"); + +# include "erf_large_data.ipp" + + do_test_erf(erf_large_data, name, "Erf Function: Large Values"); + +# include "erf_inv_data.ipp" + + do_test_erf_inv(erf_inv_data, name, "Inverse Erf Function"); + +# include "erfc_inv_data.ipp" + + do_test_erfc_inv(erfc_inv_data, name, "Inverse Erfc Function"); + +# include "erfc_inv_big_data.ipp" + + if(std::numeric_limits::min_exponent <= -4500) + { + do_test_erfc_inv(erfc_inv_big_data, name, "Inverse Erfc Function: extreme values"); + } +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 10 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 1000; + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0.125)), static_cast(0.85968379519866618260697055347837660181302041685015L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0.5)), static_cast(0.47950012218695346231725334610803547126354842424204L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(1)), static_cast(0.15729920705028513065877936491739074070393300203370L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(5)), static_cast(1.5374597944280348501883434853833788901180503147234e-12L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(-0.125)), static_cast(1.1403162048013338173930294465216233981869795831498L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(-0.5)), static_cast(1.5204998778130465376827466538919645287364515757580L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erfc(static_cast(0)), static_cast(1), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0.125)), static_cast(0.14031620480133381739302944652162339818697958314985L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0.5)), static_cast(0.52049987781304653768274665389196452873645157575796L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(1)), static_cast(0.84270079294971486934122063508260925929606699796630L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(5)), static_cast(0.9999999999984625402055719651498116565146166211099L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(-0.125)), static_cast(-0.14031620480133381739302944652162339818697958314985L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(-0.5)), static_cast(-0.52049987781304653768274665389196452873645157575796L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::erf(static_cast(0)), static_cast(0), tolerance); + + tolerance = boost::math::tools::epsilon() * 100 * 200; // 200 eps %. +#if defined(__CYGWIN__) + // some platforms long double is only reliably accurate to double precision: + if(sizeof(T) == sizeof(long double)) + tolerance = boost::math::tools::epsilon() * 100 * 200; // 200 eps %. +#endif + + for(T i = -0.95f; i < 1; i += 0.125f) + { + T inv = boost::math::erf_inv(i); + T b = boost::math::erf(inv); + BOOST_CHECK_CLOSE(b, i, tolerance); + } + for(T j = 0.125f; j < 2; j += 0.125f) + { + T inv = boost::math::erfc_inv(j); + T b = boost::math::erfc(inv); + BOOST_CHECK_CLOSE(b, j, tolerance); + } +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif + + expected_results(); + + test_erf(0.1F, "float"); + test_erf(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_erf(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_erf(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + +/* + +Output: + +test_erf.cpp +Compiling manifest to resources... +Linking... +Embedding manifest... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_erf.exe" +Running 1 test case... +Testing basic sanity checks for type float +Testing basic sanity checks for type double +Testing basic sanity checks for type long double +Testing basic sanity checks for type real_concept +Tests run with Microsoft Visual C++ version 8.0, Dinkumware standard library version 405, Win32 +Testing Erf Function: Small Values with type float +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 0 RMS Mean=0 +Testing Erf Function: Medium Values with type float +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 0 RMS Mean=0 +Testing Erf Function: Large Values with type float +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 0 RMS Mean=0 +Testing Inverse Erf Function with type float +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf_inv Max = 0 RMS Mean=0 +Testing Inverse Erfc Function with type float +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erfc_inv Max = 0 RMS Mean=0 +Testing Erf Function: Small Values with type double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 0.7857 RMS Mean=0.06415 + worst case at row: 149 + { 0.3343, 0.3636, 0.6364 } +Testing Erf Function: Medium Values with type double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0.9219 RMS Mean=0.1016 + worst case at row: 273 + { 0.5252, 0.5424, 0.4576 } +boost::math::erfc Max = 1.08 RMS Mean=0.3224 + worst case at row: 287 + { 0.8461, 0.7685, 0.2315 } +Testing Erf Function: Large Values with type double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 1.048 RMS Mean=0.2032 + worst case at row: 50 + { 20.96, 1, 4.182e-193 } +Testing Inverse Erf Function with type double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf_inv Max = 1.124 RMS Mean=0.5082 + worst case at row: 98 + { 0.9881, 1.779 } +Testing Inverse Erfc Function with type double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erfc_inv Max = 1.124 RMS Mean=0.5006 + worst case at row: 98 + { 1.988, -1.779 } +Testing Erf Function: Small Values with type long double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 0.7857 RMS Mean=0.06415 + worst case at row: 149 + { 0.3343, 0.3636, 0.6364 } +Testing Erf Function: Medium Values with type long double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0.9219 RMS Mean=0.1016 + worst case at row: 273 + { 0.5252, 0.5424, 0.4576 } +boost::math::erfc Max = 1.08 RMS Mean=0.3224 + worst case at row: 287 + { 0.8461, 0.7685, 0.2315 } +Testing Erf Function: Large Values with type long double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 1.048 RMS Mean=0.2032 + worst case at row: 50 + { 20.96, 1, 4.182e-193 } +Testing Inverse Erf Function with type long double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf_inv Max = 1.124 RMS Mean=0.5082 + worst case at row: 98 + { 0.9881, 1.779 } +Testing Inverse Erfc Function with type long double +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erfc_inv Max = 1.124 RMS Mean=0.5006 + worst case at row: 98 + { 1.988, -1.779 } +Testing Erf Function: Small Values with type real_concept +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 1.271 RMS Mean=0.5381 + worst case at row: 144 + { 0.0109, 0.0123, 0.9877 } +boost::math::erfc Max = 0.7857 RMS Mean=0.07777 + worst case at row: 149 + { 0.3343, 0.3636, 0.6364 } +Testing Erf Function: Medium Values with type real_concept +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 22.5 RMS Mean=4.224 + worst case at row: 233 + { -0.7852, -0.7332, 1.733 } +Peak error greater than expected value of 20 +i:/boost-sandbox/math_toolkit/libs/math/test/handle_test_result.hpp(146): error in "test_main_caller( argc, argv )": check bounds.first >= max_error_found failed +boost::math::erfc Max = 97.77 RMS Mean=8.373 + worst case at row: 289 + { 0.9849, 0.8363, 0.1637 } +Peak error greater than expected value of 20 +i:/boost-sandbox/math_toolkit/libs/math/test/handle_test_result.hpp(146): error in "test_main_caller( argc, argv )": check bounds.first >= max_error_found failed +Mean error greater than expected value of 6 +i:/boost-sandbox/math_toolkit/libs/math/test/handle_test_result.hpp(151): error in "test_main_caller( argc, argv )": check bounds.second >= mean_error_found failed +Testing Erf Function: Large Values with type real_concept +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf Max = 0 RMS Mean=0 +boost::math::erfc Max = 1.395 RMS Mean=0.2908 + worst case at row: 11 + { 10.99, 1, 1.87e-054 } +Testing Inverse Erf Function with type real_concept +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erf_inv Max = 1.124 RMS Mean=0.5082 + worst case at row: 98 + { 0.9881, 1.779 } +Testing Inverse Erfc Function with type real_concept +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +boost::math::erfc_inv Max = 1.124 RMS Mean=0.5006 + worst case at row: 98 + { 1.988, -1.779 } +Test suite "Test Program" failed with: + 181 assertions out of 184 passed + 3 assertions out of 184 failed + 1 test case out of 1 failed + Test case "test_main_caller( argc, argv )" failed with: + 181 assertions out of 184 passed + 3 assertions out of 184 failed +Build Time 0:15 + +*/ diff --git a/test/test_erf_hooks.hpp b/test/test_erf_hooks.hpp new file mode 100644 index 000000000..e1c42af50 --- /dev/null +++ b/test/test_erf_hooks.hpp @@ -0,0 +1,105 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_ERF_OTHER_HOOKS_HPP + +#ifdef TEST_NATIVE +namespace other{ +inline float erf(float a) +{ + return ::erff(a); +} +inline float erfc(float a) +{ + return ::erfcf(a); +} +inline double erf(double a) +{ + return ::erf(a); +} +inline double erfc(double a) +{ + return ::erfc(a); +} +inline long double erf(long double a) +{ + return ::erfl(a); +} +inline long double erfc(long double a) +{ + return ::erfcl(a); +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double erf(double); + float erff(float); + long double erfl(long double); +} +inline float erf(float a) +{ return erff(a); } +inline long double erf(long double a) +{ +#ifdef BOOST_MSVC + return erf((double)a); +#else + return erfl(a); +#endif +} +extern "C" { + double erfc(double); + float erfcf(float); + long double erfcl(long double); +} +inline float erfc(float a) +{ return erfcf(a); } +inline long double erfc(long double a) +{ +#ifdef BOOST_MSVC + return erfc((double)a); +#else + return erfcl(a); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#include + +namespace other{ +inline float erf(float a) +{ return (float)gsl_sf_erf(a); } +inline double erf(double a) +{ return gsl_sf_erf(a); } +inline long double erf(long double a) +{ return gsl_sf_erf(a); } +inline float erfc(float a) +{ return (float)gsl_sf_erfc(a); } +inline double erfc(double a) +{ return gsl_sf_erfc(a); } +inline long double erfc(long double a) +{ return gsl_sf_erfc(a); } +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept erf(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept erfc(boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + diff --git a/test/test_error_handling.cpp b/test/test_error_handling.cpp new file mode 100644 index 000000000..04d31a314 --- /dev/null +++ b/test/test_error_handling.cpp @@ -0,0 +1,193 @@ +// Copyright Paul A. Bristow 2006-7. +// Copyright John Maddock 2006-7. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Test error handling mechanism produces the expected error messages. +// for example Error in function boost::math::test_function(float, float, float): Domain Error evaluating function at 0 + +// Define some custom dummy error handlers that do nothing but throw, +// in order to check that they are otherwise undefined. +// The user MUST define them before they can be used. +// +struct user_defined_error{}; + +namespace boost{ namespace math{ namespace policies{ + +template +T user_domain_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +template +T user_pole_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +template +T user_overflow_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +template +T user_underflow_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +template +T user_denorm_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +template +T user_evaluation_error(const char* , const char* , const T& ) +{ + throw user_defined_error(); +} + +}}} // namespaces + +#include +#include +#include +#include // for test_main +// +// Define some policies: +// +using namespace boost::math::policies; +policy< + domain_error, + pole_error, + overflow_error, + underflow_error, + denorm_error, + evaluation_error > throw_policy; +policy< + domain_error, + pole_error, + overflow_error, + underflow_error, + denorm_error, + evaluation_error > errno_policy; +policy< + domain_error, + pole_error, + overflow_error, + underflow_error, + denorm_error, + evaluation_error > user_policy; +policy<> default_policy; + +#define TEST_EXCEPTION(expression, exception)\ + BOOST_CHECK_THROW(expression, exception);\ + try{ expression; }catch(const exception& e){ std::cout << e.what() << std::endl; } + +template +void test_error(T) +{ + const char* func = "boost::math::test_function<%1%>(%1%, %1%, %1%)"; + const char* msg1 = "Error while handling value %1%"; + const char* msg2 = "Error message goes here..."; + + // Check that exception is thrown, catch and show the message, for example: + // Error in function boost::math::test_function(float, float, float): Error while handling value 0 + + TEST_EXCEPTION(boost::math::policies::raise_domain_error(func, msg1, T(0.0), throw_policy), std::domain_error); + TEST_EXCEPTION(boost::math::policies::raise_domain_error(func, 0, T(0.0), throw_policy), std::domain_error); + TEST_EXCEPTION(boost::math::policies::raise_pole_error(func, msg1, T(0.0), throw_policy), std::domain_error); + TEST_EXCEPTION(boost::math::policies::raise_pole_error(func, 0, T(0.0), throw_policy), std::domain_error); + TEST_EXCEPTION(boost::math::policies::raise_overflow_error(func, msg2, throw_policy), std::overflow_error); + TEST_EXCEPTION(boost::math::policies::raise_overflow_error(func, 0, throw_policy), std::overflow_error); + TEST_EXCEPTION(boost::math::policies::raise_underflow_error(func, msg2, throw_policy), std::underflow_error); + TEST_EXCEPTION(boost::math::policies::raise_underflow_error(func, 0, throw_policy), std::underflow_error); + TEST_EXCEPTION(boost::math::policies::raise_denorm_error(func, msg2, T(0), throw_policy), std::underflow_error); + TEST_EXCEPTION(boost::math::policies::raise_denorm_error(func, 0, T(0), throw_policy), std::underflow_error); + TEST_EXCEPTION(boost::math::policies::raise_evaluation_error(func, msg1, T(1.25), throw_policy), boost::math::evaluation_error); + TEST_EXCEPTION(boost::math::policies::raise_evaluation_error(func, 0, T(1.25), throw_policy), boost::math::evaluation_error); + // + // Now try user error handlers: these should all throw user_error(): + // - because by design these are undefined and must be defined by the user ;-) + BOOST_CHECK_THROW(boost::math::policies::raise_domain_error(func, msg1, T(0.0), user_policy), user_defined_error); + BOOST_CHECK_THROW(boost::math::policies::raise_pole_error(func, msg1, T(0.0), user_policy), user_defined_error); + BOOST_CHECK_THROW(boost::math::policies::raise_overflow_error(func, msg2, user_policy), user_defined_error); + BOOST_CHECK_THROW(boost::math::policies::raise_underflow_error(func, msg2, user_policy), user_defined_error); + BOOST_CHECK_THROW(boost::math::policies::raise_denorm_error(func, msg2, T(0), user_policy), user_defined_error); + BOOST_CHECK_THROW(boost::math::policies::raise_evaluation_error(func, msg1, T(0.0), user_policy), user_defined_error); + +} + +int test_main(int, char* []) +{ + // Test error handling. + // (Parameter value, arbitrarily zero, only communicates the floating point type FPT). + test_error(0.0F); // Test float. + test_error(0.0); // Test double. + test_error(0.0L); // Test long double. + test_error(boost::math::concepts::real_concept(0.0L)); // Test concepts. + return 0; +} // int test_main(int, char* []) + +/* + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_error_handling.exe" +Running 1 test case... +Error in function boost::math::test_function(float, float, float): Error while handling value 0 +Error in function boost::math::test_function(float, float, float): Domain Error evaluating function at 0 +Error in function boost::math::test_function(float, float, float): Error while handling value 0 +Error in function boost::math::test_function(float, float, float): Evaluation of function at pole 0 +Error in function boost::math::test_function(float, float, float): Error message goes here... +Error in function boost::math::test_function(float, float, float): Overflow Error +Error in function boost::math::test_function(float, float, float): Error message goes here... +Error in function boost::math::test_function(float, float, float): Underflow Error +Error in function boost::math::test_function(float, float, float): Error message goes here... +Error in function boost::math::test_function(float, float, float): Denorm Error +Error in function boost::math::test_function(float, float, float): Error while handling value 1.25 +Error in function boost::math::test_function(float, float, float): Internal Evaluation Error, best value so far was 1.25 +Error in function boost::math::test_function(double, double, double): Error while handling value 0 +Error in function boost::math::test_function(double, double, double): Domain Error evaluating function at 0 +Error in function boost::math::test_function(double, double, double): Error while handling value 0 +Error in function boost::math::test_function(double, double, double): Evaluation of function at pole 0 +Error in function boost::math::test_function(double, double, double): Error message goes here... +Error in function boost::math::test_function(double, double, double): Overflow Error +Error in function boost::math::test_function(double, double, double): Error message goes here... +Error in function boost::math::test_function(double, double, double): Underflow Error +Error in function boost::math::test_function(double, double, double): Error message goes here... +Error in function boost::math::test_function(double, double, double): Denorm Error +Error in function boost::math::test_function(double, double, double): Error while handling value 1.25 +Error in function boost::math::test_function(double, double, double): Internal Evaluation Error, best value so far was 1.25 +Error in function boost::math::test_function(long double, long double, long double): Error while handling value 0 +Error in function boost::math::test_function(long double, long double, long double): Domain Error evaluating function at 0 +Error in function boost::math::test_function(long double, long double, long double): Error while handling value 0 +Error in function boost::math::test_function(long double, long double, long double): Evaluation of function at pole 0 +Error in function boost::math::test_function(long double, long double, long double): Error message goes here... +Error in function boost::math::test_function(long double, long double, long double): Overflow Error +Error in function boost::math::test_function(long double, long double, long double): Error message goes here... +Error in function boost::math::test_function(long double, long double, long double): Underflow Error +Error in function boost::math::test_function(long double, long double, long double): Error message goes here... +Error in function boost::math::test_function(long double, long double, long double): Denorm Error +Error in function boost::math::test_function(long double, long double, long double): Error while handling value 1.25 +Error in function boost::math::test_function(long double, long double, long double): Internal Evaluation Error, best value so far was 1.25 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error while handling value 0 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Domain Error evaluating function at 0 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error while handling value 0 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Evaluation of function at pole 0 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error message goes here... +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Overflow Error +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error message goes here... +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Underflow Error +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error message goes here... +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Denorm Error +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Error while handling value 1.25 +Error in function boost::math::test_function(class boost::math::concepts::real_concept, class boost::math::concepts::real_concept, class boost::math::concepts::real_concept): Internal Evaluation Error, best value so far was 1.25 +*** No errors detected + +*/ + diff --git a/test/test_exponential_dist.cpp b/test/test_exponential_dist.cpp new file mode 100644 index 000000000..3a2a7b5fa --- /dev/null +++ b/test/test_exponential_dist.cpp @@ -0,0 +1,291 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_exponential_dist.cpp + +#include // for real_concept +#include + using boost::math::exponential_distribution; + +#include // Boost.Test +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; + +template +void test_spot(RealType l, RealType x, RealType p, RealType q, RealType tolerance) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + exponential_distribution(l), + x), + p, + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(exponential_distribution(l), + x)), + q, + tolerance); // % + if(p < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + exponential_distribution(l), + p), + x, + tolerance); // % + } + if(q < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(exponential_distribution(l), + q)), + x, + tolerance); // % + } +} + +template +void test_spots(RealType T) +{ + // Basic sanity checks. + // 50 eps as a percentage, up to a maximum of double precision + // (that's the limit of our test data: obtained by punching + // numbers into a calculator). + RealType tolerance = (std::max)( + static_cast(boost::math::tools::epsilon()), + boost::math::tools::epsilon()); + tolerance *= 50 * 100; + // # pragma warning(disable: 4100) // unreferenced formal parameter. + // prevent his spurious warning. + if (T != 0) + { + cout << "Expect parameter T == 0!" << endl; + } + cout << "Tolerance for type " << typeid(T).name() << " is " << tolerance << " %" << endl; + + test_spot( + static_cast(0.5), // lambda + static_cast(0.125), // x + static_cast(0.060586937186524213880289175377695L), // p + static_cast(0.93941306281347578611971082462231L), //q + tolerance); + test_spot( + static_cast(0.5), // lambda + static_cast(5), // x + static_cast(0.91791500137610120483047132553284L), // p + static_cast(0.08208499862389879516952867446716L), //q + tolerance); + test_spot( + static_cast(2), // lambda + static_cast(0.125), // x + static_cast(0.22119921692859513175482973302168L), // p + static_cast(0.77880078307140486824517026697832L), //q + tolerance); + test_spot( + static_cast(2), // lambda + static_cast(5), // x + static_cast(0.99995460007023751514846440848444L), // p + static_cast(4.5399929762484851535591515560551e-5L), //q + tolerance); + + // + // Some spot tests generated by MathCAD pexp(x,r): + // + test_spot( + static_cast(1), // lambda + static_cast(1), // x + static_cast(6.321205588285580E-001L), // p + static_cast(1-6.321205588285580E-001L), //q + tolerance); + test_spot( + static_cast(2), // lambda + static_cast(1), // x + static_cast(8.646647167633870E-001L), // p + static_cast(1-8.646647167633870E-001L), //q + tolerance); + test_spot( + static_cast(1), // lambda + static_cast(0.5), // x + static_cast(3.934693402873670E-001L), // p + static_cast(1-3.934693402873670E-001L), //q + tolerance); + test_spot( + static_cast(0.1), // lambda + static_cast(1), // x + static_cast(9.516258196404040E-002L), // p + static_cast(1-9.516258196404040E-002L), //q + tolerance); + test_spot( + static_cast(10), // lambda + static_cast(1), // x + static_cast(9.999546000702380E-001L), // p + static_cast(1-9.999546000702380E-001L), //q + tolerance*10000); // we loose four digits to cancellation + test_spot( + static_cast(0.1), // lambda + static_cast(10), // x + static_cast(6.321205588285580E-001L), // p + static_cast(1-6.321205588285580E-001L), //q + tolerance); + test_spot( + static_cast(1), // lambda + static_cast(0.01), // x + static_cast(9.950166250831950E-003L), // p + static_cast(1-9.950166250831950E-003L), //q + tolerance); + test_spot( + static_cast(1), // lambda + static_cast(0.0001), // x + static_cast(9.999500016666250E-005L), // p + static_cast(1-9.999500016666250E-005L), //q + tolerance); + /* + // This test data appears to be erroneous, MathCad appears + // to suffer from cancellation error as x -> 0 + test_spot( + static_cast(1), // lambda + static_cast(0.0000001), // x + static_cast(9.999999499998730E-008L), // p + static_cast(1-9.999999499998730E-008L), //q + tolerance); + */ + + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + exponential_distribution(0.5), + static_cast(0.125)), // x + static_cast(0.46970653140673789305985541231115L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + exponential_distribution(0.5), + static_cast(5)), // x + static_cast(0.04104249931194939758476433723358L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + exponential_distribution(2), + static_cast(0.125)), // x + static_cast(1.5576015661428097364903405339566L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + exponential_distribution(2), + static_cast(5)), // x + static_cast(9.0799859524969703071183031121101e-5L), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::mean( + exponential_distribution(2)), + static_cast(0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::standard_deviation( + exponential_distribution(2)), + static_cast(0.5), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::mode( + exponential_distribution(2)), + static_cast(0), + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::median( + exponential_distribution(4)), + static_cast(0.693147180559945309417232121458176568075500134360255254) / 4, + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::skewness( + exponential_distribution(2)), + static_cast(2), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis( + exponential_distribution(2)), + static_cast(9), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis_excess( + exponential_distribution(2)), + static_cast(6), + tolerance); // % + + // + // Things that are errors: + // + exponential_distribution dist(0.5); + BOOST_CHECK_THROW( + quantile(dist, RealType(1.0)), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, RealType(0.0))), + std::overflow_error); + BOOST_CHECK_THROW( + pdf(dist, RealType(-1)), + std::domain_error); + BOOST_CHECK_THROW( + cdf(dist, RealType(-1)), + std::domain_error); + BOOST_CHECK_THROW( + cdf(exponential_distribution(-1), RealType(1)), + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(-1)), + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(2)), + std::domain_error); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can generate exponential distribution using the two convenience methods: + boost::math::exponential mycexp1(1.); // Using typedef + exponential_distribution<> myexp2(1.); // Using default RealType double. + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Running 1 test case... +Tolerance for type float is 0.000596046 % +Tolerance for type double is 1.11022e-012 % +Tolerance for type long double is 1.11022e-012 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-012 % +*** No errors detected + +*/ diff --git a/test/test_extreme_value.cpp b/test/test_extreme_value.cpp new file mode 100644 index 000000000..a1575cdcf --- /dev/null +++ b/test/test_extreme_value.cpp @@ -0,0 +1,222 @@ +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_extreme_value.cpp + +#include // for real_concept +#include + using boost::math::extreme_value_distribution; + +#include // Boost.Test +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; + +template +void test_spot(RealType a, RealType b, RealType x, RealType p, RealType q, RealType tolerance) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + extreme_value_distribution(a, b), + x), + p, + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(extreme_value_distribution(a, b), + x)), + q, + tolerance); // % + if((p < 0.999) && (p > 0)) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + extreme_value_distribution(a, b), + p), + x, + tolerance); // % + } + if((q < 0.999) && (q > 0)) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(extreme_value_distribution(a, b), + q)), + x, + tolerance); // % + } +} + +template +void test_spots(RealType) +{ + // Basic sanity checks. + // 50eps as a percentage, up to a maximum of double precision + // (that's the limit of our test data). + RealType tolerance = (std::max)( + static_cast(boost::math::tools::epsilon()), + boost::math::tools::epsilon()); + tolerance *= 50 * 100; + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + // Results calculated by punching numbers into a calculator, + // and using the formula at http://mathworld.wolfram.com/ExtremeValueDistribution.html + test_spot( + static_cast(0.5), // a + static_cast(1.5), // b + static_cast(0.125), // x + static_cast(0.27692033409990891617007608217222L), // p + static_cast(0.72307966590009108382992391782778L), //q + tolerance); + test_spot( + static_cast(0.5), // a + static_cast(2), // b + static_cast(-5), // x + static_cast(1.6087601139887776413169427645933e-7L), // p + static_cast(0.99999983912398860112223586830572L), //q + tolerance); + test_spot( + static_cast(0.5), // a + static_cast(0.25), // b + static_cast(0.75), // x + static_cast(0.69220062755534635386542199718279L), // p + static_cast(0.30779937244465364613457800281721), //q + tolerance); + test_spot( + static_cast(0.5), // a + static_cast(0.25), // b + static_cast(5), // x + static_cast(0.99999998477002037126351248727041L), // p + static_cast(1.5229979628736487512729586276294e-8L), //q + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + extreme_value_distribution(0.5, 2), + static_cast(0.125)), // x + static_cast(0.18052654830890205978204427757846L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + extreme_value_distribution(1, 3), + static_cast(5)), // x + static_cast(0.0675057324099851209129017326286L), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + extreme_value_distribution(1, 3), + static_cast(0)), // x + static_cast(0.11522236828583456431277265757312L), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::mean( + extreme_value_distribution(2, 3)), + static_cast(3.731646994704598581819536270246L), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::standard_deviation( + extreme_value_distribution(-1, 0.5)), + static_cast(0.6412749150809320477720181798355L), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::mode( + extreme_value_distribution(2, 3)), + static_cast(2), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::median( + extreme_value_distribution(0, 1)), + static_cast(+0.36651292058166432701243915823266946945426344783710526305367771367056), + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::skewness( + extreme_value_distribution(2, 3)), + static_cast(1.1395470994046486574927930193898461120875997958366L), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis( + extreme_value_distribution(2, 3)), + static_cast(5.4), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis_excess( + extreme_value_distribution(2, 3)), + static_cast(2.4), + tolerance); // % + + // + // Things that are errors: + // + extreme_value_distribution dist(0.5, 2); + BOOST_CHECK_THROW( + quantile(dist, RealType(1.0)), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, RealType(0.0))), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(0.0)), + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, RealType(1.0))), + std::overflow_error); + BOOST_CHECK_THROW( + cdf(extreme_value_distribution(0, -1), RealType(1)), + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(-1)), + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(2)), + std::domain_error); +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + + // Check that can generate extreme_value distribution using the two convenience methods: + boost::math::extreme_value mycev1(1.); // Using typedef + extreme_value_distribution<> myev2(1.); // Using default RealType double. + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +-Running 1 test case... +Tolerance for type float is 0.000596046 % +Tolerance for type double is 1.11022e-012 % +Tolerance for type long double is 1.11022e-012 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-012 % +*** No errors detected +*/ + + diff --git a/test/test_factorials.cpp b/test/test_factorials.cpp new file mode 100644 index 000000000..190abce44 --- /dev/null +++ b/test/test_factorials.cpp @@ -0,0 +1,341 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +# pragma warning(disable: 4245) // int/unsigned int conversion +#endif + +// Return infinities not exceptions: +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include + +#include + using std::cout; + using std::endl; + +template +T naive_falling_factorial(T x, unsigned n) +{ + if(n == 0) + return 1; + T result = x; + while(--n) + { + x -= 1; + result *= x; + } + return result; +} + +template +void test_spots(T) +{ + // + // Basic sanity checks. + // + T tolerance = boost::math::tools::epsilon() * 100 * 2; // 2 eps as a percent. + BOOST_CHECK_CLOSE( + ::boost::math::factorial(0), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::factorial(1), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::factorial(10), + static_cast(3628800L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::unchecked_factorial(0), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::unchecked_factorial(1), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::unchecked_factorial(10), + static_cast(3628800L), tolerance); + + // + // Try some double factorials: + // + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(0), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(1), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(2), + static_cast(2), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(5), + static_cast(15), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(10), + static_cast(3840), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(19), + static_cast(6.547290750e8L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(24), + static_cast(1.961990553600000e12L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(33), + static_cast(6.33265987076285062500000e18L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(42), + static_cast(1.0714547155728479551488000000e26L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::double_factorial(47), + static_cast(1.19256819277443412353990764062500000e30L), tolerance); + + if((std::numeric_limits::has_infinity) && (std::numeric_limits::max_exponent <= 1024)) + { + BOOST_CHECK_EQUAL( + ::boost::math::double_factorial(320), + std::numeric_limits::infinity()); + BOOST_CHECK_EQUAL( + ::boost::math::double_factorial(301), + std::numeric_limits::infinity()); + } + // + // Rising factorials: + // + tolerance = boost::math::tools::epsilon() * 100 * 20; // 20 eps as a percent. + if(std::numeric_limits::is_specialized == 0) + tolerance *= 5; // higher error rates without Lanczos support + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(3), 4), + static_cast(360), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(7), -4), + static_cast(0.00277777777777777777777777777777777777777777777777777777777778L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(120.5f), 8), + static_cast(5.58187566784927180664062500e16L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(120.5f), -4), + static_cast(5.15881498170104646868208445266116850161120996179812063177241e-9L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(5000.25f), 8), + static_cast(3.92974581976666067544013393509103775024414062500000e29L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(5000.25f), -7), + static_cast(1.28674092710208810281923019294164707555099052561945725535047e-26L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(30.25), 21), + static_cast(3.93286957998925490693364184100209193343633629069699964020401e33L), tolerance * 2); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(30.25), -21), + static_cast(3.35010902064291983728782493133164809108646650368560147505884e-27L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), 21), + static_cast(-9.76168312768123676601980433377916854311706629232503473758698e26L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), -21), + static_cast(-1.50079704000923674318934280259377728203516775215430875839823e-34L), 2 * tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), 5), + static_cast(-1.78799177197265625000000e7L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), -5), + static_cast(-2.47177487004482195012362027432181137141899692171397467859150e-8L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), 6), + static_cast(4.5146792242309570312500000e8L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-30.25), -6), + static_cast(6.81868929667537089689274558433603136943171564610751635473516e-10L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-3), 6), + static_cast(0), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-3.25), 6), + static_cast(2.99926757812500L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-5.25), 6), + static_cast(50.987548828125000000000000L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-5.25), 13), + static_cast(127230.91046623885631561279296875000L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-3.25), -6), + static_cast(0.0000129609865918182348202632178291407500332449622510474437452125L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-5.25), -6), + static_cast(2.50789821857946332294524052303699065683926911849535903362649e-6L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::rising_factorial(static_cast(-5.25), -13), + static_cast(-1.38984989447269128946284683518361786049649013886981662962096e-14L), tolerance); + + // + // Falling factorials: + // + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.25), 0), + static_cast(naive_falling_factorial(30.25L, 0)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.25), 1), + static_cast(naive_falling_factorial(30.25L, 1)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.25), 2), + static_cast(naive_falling_factorial(30.25L, 2)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.25), 5), + static_cast(naive_falling_factorial(30.25L, 5)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.25), 22), + static_cast(naive_falling_factorial(30.25L, 22)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(100.5), 6), + static_cast(naive_falling_factorial(100.5L, 6)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(30.75), 30), + static_cast(naive_falling_factorial(30.75L, 30)), + tolerance * 3); + if(boost::math::policies::digits >() > 50) + { + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-30.75L), 30), + static_cast(naive_falling_factorial(-30.75L, 30)), + tolerance * 3); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-30.75L), 27), + static_cast(naive_falling_factorial(-30.75L, 27)), + tolerance * 3); + } + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-12.0), 6), + static_cast(naive_falling_factorial(-12.0L, 6)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-12), 5), + static_cast(naive_falling_factorial(-12.0L, 5)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-3.0), 6), + static_cast(naive_falling_factorial(-3.0L, 6)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(-3), 5), + static_cast(naive_falling_factorial(-3.0L, 5)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3.0), 6), + static_cast(naive_falling_factorial(3.0L, 6)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3), 5), + static_cast(naive_falling_factorial(3.0L, 5)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3.25), 4), + static_cast(naive_falling_factorial(3.25L, 4)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3.25), 5), + static_cast(naive_falling_factorial(3.25L, 5)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3.25), 6), + static_cast(naive_falling_factorial(3.25L, 6)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(3.25), 7), + static_cast(naive_falling_factorial(3.25L, 7)), + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::falling_factorial(static_cast(8.25), 12), + static_cast(naive_falling_factorial(8.25L, 12)), + tolerance); + + + tolerance = boost::math::tools::epsilon() * 100 * 20; // 20 eps as a percent. + unsigned i = boost::math::max_factorial::value; + if((boost::is_floating_point::value) && (sizeof(T) <= sizeof(double))) + { + // Without Lanczos support, tgamma isn't accurate enough for this test: + BOOST_CHECK_CLOSE( + ::boost::math::unchecked_factorial(i), + boost::math::tgamma(static_cast(i+1)), tolerance); + } + + i += 10; + while(boost::math::lgamma(static_cast(i+1)) < boost::math::tools::log_max_value()) + { + BOOST_CHECK_CLOSE( + ::boost::math::factorial(i), + boost::math::tgamma(static_cast(i+1)), tolerance); + i += 10; + } +} // template void test_spots(T) + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + test_spots(0.0F); + test_spots(0.0); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0.)); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + if (std::numeric_limits::digits == std::numeric_limits::digits) + { + cout << "Types double and long double have the same number of floating-point significand bits (" + << std::numeric_limits::digits << ") on this platform." << endl; + } + if (std::numeric_limits::digits == std::numeric_limits::digits) + { + cout << "Types float and double have the same number of floating-point significand bits (" + << std::numeric_limits::digits << ") on this platform." << endl; + } + + using boost::math::max_factorial; + cout << "max factorial for float " << max_factorial::value << endl; + cout << "max factorial for double " << max_factorial::value << endl; + cout << "max factorial for long double " << max_factorial::value << endl; + cout << "max factorial for real_concept " << max_factorial::value << endl; + + + + return 0; +} + +/* + +Output is: + +Running 1 test case... +Types double and long double have the same number of floating-point significand bits (53) on this platform. +max factorial for float 34 +max factorial for double 170 +max factorial for long double 170 +max factorial for real_concept 100 +*** No errors detected + +*/ + + + + diff --git a/test/test_find_location.cpp b/test/test_find_location.cpp new file mode 100644 index 000000000..7f1a8b61f --- /dev/null +++ b/test/test_find_location.cpp @@ -0,0 +1,172 @@ +// test_find_location.cpp + +// Copyright John Maddock 2007. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity test for find_location function. + +// Default domain error policy is +// #define BOOST_MATH_DOMAIN_ERROR_POLICY throw_on_error + +#include // for normal_distribution + using boost::math::normal; // Default type double. + using boost::math::normal_distribution; // All floating-point types. +#include // for cauchy_distribution + using boost::math::cauchy; +#include // for cauchy_distribution + using boost::math::pareto; +#include + using boost::math::find_location; + using boost::math::complement;// will be needed by users who want complement, +#include + using boost::math::policies::policy; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE_FRACTION, BOOST_CHECK_EQUAL... +#include // for real_concept + +#include + using std::cout; using std::endl; using std::fixed; + using std::right; using std::left; using std::showpoint; + using std::showpos; using std::setw; using std::setprecision; + +#include + using std::numeric_limits; + +template // Any floating-point type RealType. +void test_spots(RealType) +{ // Parameter only provides the type, float, double... value ignored. + + // Basic sanity checks, test data may be to double precision only + // so set tolerance to 100 eps expressed as a fraction, + // or 100 eps of type double expressed as a fraction, + // whichever is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); + tolerance *= 100; // 100 eps as a fraction. + + cout << "Tolerance for type " << typeid(RealType).name() << " is " + << setprecision(3) << tolerance << " (or " << tolerance * 100 << "%)." << endl; + + BOOST_CHECK_THROW( // Probability outside 0 to 1. + find_location >( + static_cast(0.), static_cast(-1.), static_cast(0.) ), + std::domain_error); + + normal_distribution n; // standard N(0,1) + BOOST_CHECK_EQUAL(n.location(), 0); // aka mean. + BOOST_CHECK_EQUAL(n.scale(), 1); // aka standard_deviation. + + // Check for 'bad' arguments. + BOOST_CHECK_THROW(find_location(0., -1., 0.), std::domain_error); // p below 0 to 1. + BOOST_CHECK_THROW(find_location(0., 2., 0.), std::domain_error); // p above 0 to 1. + BOOST_CHECK_THROW(find_location(numeric_limits::infinity(), 0.5, 0.), + std::domain_error); // z not finite. + BOOST_CHECK_THROW(find_location(numeric_limits::quiet_NaN(), -1., 0.), + std::domain_error); // z not finite + BOOST_CHECK_THROW(find_location(0., -1., numeric_limits::quiet_NaN()), + std::domain_error); // scale not finite + + BOOST_CHECK_THROW(find_location(complement(0., -1., 0.)), std::domain_error); // p below 0 to 1. + BOOST_CHECK_THROW(find_location(complement(0., 2., 0.)), std::domain_error); // p above 0 to 1. + BOOST_CHECK_THROW(find_location(complement(numeric_limits::infinity(), 0.5, 0.)), + std::domain_error); // z not finite. + BOOST_CHECK_THROW(find_location(complement(numeric_limits::quiet_NaN(), -1., 0.)), + std::domain_error); // z not finite + BOOST_CHECK_THROW(find_location(complement(0., -1., numeric_limits::quiet_NaN())), + std::domain_error); // scale not finite + + //// Check for ab-use with unsuitable distribution(s) when concept check implemented. + // BOOST_CHECK_THROW(find_location(0., 0.5, 0.), std::domain_error); // pareto can't be used with find_location. + + // Check doesn't throw when an ignore_error for domain_error policy is used. + using boost::math::policies::policy; + using boost::math::policies::domain_error; + using boost::math::policies::ignore_error; + + // Define a (bad?) policy to ignore domain errors ('bad' arguments): + typedef policy > ignore_domain_policy; + // Using a typedef is convenient, especially if it is re-used. + + BOOST_CHECK_NO_THROW(find_location(0, -1, 1, + ignore_domain_policy())); // probability outside [0, 1] + BOOST_CHECK_NO_THROW(find_location(numeric_limits::infinity(), -1, 1, + ignore_domain_policy())); // z not finite. + + BOOST_CHECK_NO_THROW(find_location(complement(0, -1, 1, + ignore_domain_policy()))); // probability outside [0, 1] + BOOST_CHECK_NO_THROW(find_location(complement(numeric_limits::infinity(), -1, 1, + ignore_domain_policy()))); // z not finite. + + // Find location to give a probability p (0.05) of z (-2) + RealType sd = static_cast(1); // normal default standard deviation = 1. + RealType z = static_cast(-2); // z to give prob p + RealType p = static_cast(0.05); // only 5% will be below z + RealType l = find_location >(z, p, sd); + // cout << z << " " << p << " " << sd << " " << l << endl; + + normal_distribution np05pc(l, sd); // Same standard_deviation (scale) but with mean(location) shifted. + // cout << "Normal distribution with mean = " << l << " has " << "fraction <= " << z << " = " << cdf(np05pc, z) << endl; + // Check cdf such that only fraction p really is below offset mean l. + BOOST_CHECK_CLOSE_FRACTION(p, cdf(np05pc, z), tolerance); + + // Check that some policies can be applied (though not used here). + l = find_location >(z, p, sd, policy<>()); // Default policy, needs using boost::math::policies::policy; + l = find_location >(z, p, sd, boost::math::policies::policy<>()); // Default policy, fully specified. + l = find_location >(z, p, sd, ignore_domain_policy()); // find_location with new policy, using typedef. + l = find_location >(z, p, sd, policy >()); // New policy, without typedef. + + // Check that can use the complement version. + RealType q = 1 - p; // complement. + // cout << "find_location >(complement(z, q, sd)) = " << endl; + l = find_location >(complement(z, q, sd)); + + normal_distribution np95pc(l, sd); // Same standard_deviation (scale) but with mean(location) shifted + // cout << "Normal distribution with mean = " << l << " has " << "fraction <= " << z << " = " << cdf(np95pc, z) << endl; + BOOST_CHECK_CLOSE_FRACTION(q, cdf(np95pc, z), tolerance); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating-point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. + #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. + #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. + #endif + #else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; + #endif + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_find_location.exe" +Running 1 test case... +Tolerance for type float is 1.19e-005 (or 0.00119%). +Tolerance for type double is 2.22e-014 (or 2.22e-012%). +Tolerance for type long double is 2.22e-014 (or 2.22e-012%). +Tolerance for type class boost::math::concepts::real_concept is 2.22e-014 (or 2.22e-012%). +*** No errors detected + +*/ + + diff --git a/test/test_find_scale.cpp b/test/test_find_scale.cpp new file mode 100644 index 000000000..4312253a4 --- /dev/null +++ b/test/test_find_scale.cpp @@ -0,0 +1,207 @@ +// test_find_scale.cpp + +// Copyright John Maddock 2007. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity test for find_scale function. + +// Default distribution domain error policy is +// #define BOOST_MATH_DOMAIN_ERROR_POLICY throw_on_error + +#include // for normal_distribution + using boost::math::normal; // Default type double. + using boost::math::normal_distribution; // All floating-point types. +#include // for cauchy_distribution + using boost::math::cauchy; +#include // for cauchy_distribution + using boost::math::pareto; +#include + using boost::math::find_scale; + using boost::math::complement;// will be needed by users who want complement, +#include + using boost::math::policies::policy; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE_FRACTION, BOOST_CHECK_EQUAL... +#include // for real_concept + +#include + using std::cout; using std::endl; using std::fixed; + using std::right; using std::left; using std::showpoint; + using std::showpos; using std::setw; using std::setprecision; + +#include + using std::numeric_limits; + +template // Any floating-point type RealType. +void test_spots(RealType) +{ // Parameter only provides the type, float, double... value ignored. + + // Basic sanity checks, test data may be to double precision only + // so set tolerance to 100 eps expressed as a fraction, + // or 100 eps of type double expressed as a fraction, + // whichever is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); + tolerance *= 100; // 100 eps as a fraction. + + cout << "Tolerance for type " << typeid(RealType).name() << " is " + << setprecision(3) << tolerance << " (or " << tolerance * 100 << "%)." << endl; + + BOOST_CHECK_THROW( // Probability outside 0 to 1. + find_scale >( + static_cast(0.), static_cast(-1.), static_cast(0.) ), + std::domain_error); + + normal_distribution n; // standard N(0,1) + BOOST_CHECK_EQUAL(n.location(), 0); // aka mean. + BOOST_CHECK_EQUAL(n.scale(), 1); // aka standard_deviation. + + // Check for 'bad' arguments. + BOOST_CHECK_THROW(find_scale(0., -1., 0.), std::domain_error); // p below 0 to 1. + BOOST_CHECK_THROW(find_scale(0., 2., 0.), std::domain_error); // p above 0 to 1. + BOOST_CHECK_THROW(find_scale(numeric_limits::infinity(), 0.5, 0.), + std::domain_error); // z not finite. + BOOST_CHECK_THROW(find_scale(numeric_limits::quiet_NaN(), -1., 0.), + std::domain_error); // z not finite + BOOST_CHECK_THROW(find_scale(0., -1., numeric_limits::quiet_NaN()), + std::domain_error); // scale not finite + + + BOOST_CHECK_THROW(find_scale(complement(0., -1., 0.)), std::domain_error); // p below 0 to 1. + BOOST_CHECK_THROW(find_scale(complement(0., 2., 0.)), std::domain_error); // p above 0 to 1. + BOOST_CHECK_THROW(find_scale(complement(numeric_limits::infinity(), 0.5, 0.)), + std::domain_error); // z not finite. + BOOST_CHECK_THROW(find_scale(complement(numeric_limits::quiet_NaN(), -1., 0.)), + std::domain_error); // z not finite + BOOST_CHECK_THROW(find_scale(complement(0., -1., numeric_limits::quiet_NaN())), + std::domain_error); // scale not finite + + BOOST_CHECK_THROW(find_scale(complement(0., -1., 0.)), std::domain_error); // p below 0 to 1. + + + // Check for ab-use with unsuitable distribution(s), for example, + // pareto distribution (and most others) can't be used with find_scale (or find_location) + // because they lack the scale and location attributes. + // BOOST_CHECK_THROW(find_scale(0., 0.5, 0.), std::domain_error); + // correctly fails to compile in find_scale() at + // BOOST_STATIC_ASSERT(::boost::math::tools::is_scaled_distribution::value); + + // Check doesn't throw when an ignore_error for domain_error policy is used. + using boost::math::policies::policy; + using boost::math::policies::domain_error; + using boost::math::policies::ignore_error; + + // Define a (bad?) policy to ignore domain errors ('bad' arguments): + typedef policy > ignore_domain_policy; + // Using a typedef is convenient, especially if it is re-used. + + BOOST_CHECK_NO_THROW(find_scale(0, -1, 1, + ignore_domain_policy())); // probability outside [0, 1] + BOOST_CHECK_NO_THROW(find_scale(numeric_limits::infinity(), -1, 1, + ignore_domain_policy())); // z not finite. + BOOST_CHECK_NO_THROW(find_scale(complement(0, -1, 1, ignore_domain_policy()))); // probability outside [0, 1] + BOOST_CHECK_NO_THROW(find_scale(complement(numeric_limits::infinity(), -1, 1, + ignore_domain_policy()))); // z not finite. + + RealType l = 0.; // standard normal distribution. + RealType sd = static_cast(1); // normal default standard deviation = 1. + normal_distribution n01(l, sd); // mean(location) = 0, standard_deviation (scale) = 1. + RealType z = static_cast(-2); // z to give prob p + //cout << "Standard normal distribution with standard deviation = " << sd + // << " has " << "fraction <= " << z << " = " << cdf(n01, z) << endl; + // Standard normal distribution with standard deviation = 1 has fraction <= -2 = 0.0227501 + + //normal_distribution np001pc(l, sd); // Same mean(location) but with standard_deviation (scale) changed. + //cout << "Normal distribution with standard deviation = " << s + // << " has " << "fraction <= " << z << " = " << cdf(np001pc, z) << endl; + + // Find scale to give a probability p (0.001) of z (-2) + RealType p = static_cast(0.001); // only 0.1% to be below z (-2). + // location (mean) remains at zero. + RealType s = find_scale >(z, p, l); + //cout << "Mean " << l << ", z " << z << ", p " << p + // << ", sd " << sd << ", find_scale " << s + // << ", difference in sd " << s - sd << endl; + // Mean 0, z -2, p 0.001, sd 1, find_scale 0.64720053440907599, difference in sd -0.352799 + + cout.precision(17); + BOOST_CHECK_CLOSE_FRACTION(s, static_cast(0.64720053440907599L), tolerance); + + normal_distribution np001pc(l, s); // Same mean(location) but with standard_deviation (scale) changed. + //cout << "Normal distribution with standard deviation = " << s + // << " has " << "fraction <= " << z << " = " << cdf(np001pc, z) << endl; + // Normal distribution with standard deviation = 0.647201 has fraction <= -2 = 0.001 + + // Check cdf such that only fraction p really is below changed standard deviation s. + BOOST_CHECK_CLOSE_FRACTION(p, cdf(np001pc, z), tolerance); + + // Check that some policies can be applied (though results not used here). + s = find_scale >(z, p, l, policy<>()); // Default policy, needs using boost::math::policies::policy; + s = find_scale >(z, p, l, boost::math::policies::policy<>()); // Default policy, fully specified. + s = find_scale >(z, p, l, ignore_domain_policy()); // find_scale with new policy, using typedef. + s = find_scale >(z, p, l, policy >()); // New policy, without typedef. + + // Check that can use the complement version too. + RealType q = 1 - p; // complement. + s = find_scale >(complement(z, q, l)); // Implicit default policy. + BOOST_CHECK_CLOSE_FRACTION(s, static_cast(0.64720053440907599L), tolerance); + s = find_scale >(complement(z, q, l, policy<>())); // Explicit default policy. + BOOST_CHECK_CLOSE_FRACTION(s, static_cast(0.64720053440907599L), tolerance); + + normal_distribution np95pc(l, s); // Same mean(location) but with new standard_deviation (scale). + + //cout << "Mean " << l << ", z " << z << ", q " << q + //<< ", sd " << sd << ", find_scale " << s + //<< ", difference in sd " << s - sd << endl; + + //cout << "Normal distribution with standard deviation = " << s + // << " has " << "fraction <= " << z << " = " << cdf(np001pc, z) << endl; + BOOST_CHECK_CLOSE_FRACTION(q, cdf(complement(np95pc, z)), tolerance); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating-point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. + #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. + #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. + #endif + #else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; + #endif + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_find_scale.exe" +Running 1 test case... +Tolerance for type float is 1.19e-005 (or 0.00119%). +Tolerance for type double is 2.22e-014 (or 2.22e-012%). +Tolerance for type long double is 2.22e-014 (or 2.22e-012%). +Tolerance for type class boost::math::concepts::real_concept is 2.22e-014 (or 2.22e-012%). +*** No errors detected + + +*/ + + diff --git a/test/test_fisher_f.cpp b/test/test_fisher_f.cpp new file mode 100644 index 000000000..978b08efd --- /dev/null +++ b/test/test_fisher_f.cpp @@ -0,0 +1,549 @@ +// test_fisher_squared.cpp + +// Copyright Paul A. Bristow 2006. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for fisher_f_distribution +using boost::math::fisher_f_distribution; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE + +#include +using std::cout; +using std::endl; +#include +using std::numeric_limits; + +template +RealType naive_pdf(RealType df1, RealType df2, RealType x) +{ + // + // Calculate the PDF naively using direct evaluation + // of equation 2 from http://mathworld.wolfram.com/F-Distribution.html + // + // Our actual PDF implementation uses a completely different method, + // so this is a good sanity check that our math is correct. + // + using namespace std; // For ADL of std functions. + RealType e = boost::math::lgamma((df1 + df2) / 2); + e += log(df1) * df1 / 2; + e += log(df2) * df2 / 2; + e += log(x) * ((df1 / 2) - 1); + e -= boost::math::lgamma(df1 / 2); + e -= boost::math::lgamma(df2 / 2); + e -= log(df2 + x * df1) * (df1 + df2) / 2; + return exp(e); +} + +template +void test_spot( + RealType df1, // Degrees of freedom 1 + RealType df2, // Degrees of freedom 2 + RealType cs, // Chi Square statistic + RealType P, // CDF + RealType Q, // Complement of CDF + RealType tol) // Test tolerance +{ + boost::math::fisher_f_distribution dist(df1, df2); + BOOST_CHECK_CLOSE( + cdf(dist, cs), P, tol); + BOOST_CHECK_CLOSE( + pdf(dist, cs), naive_pdf(dist.degrees_of_freedom1(), dist.degrees_of_freedom2(), cs), tol); + if((P < 0.999) && (Q < 0.999)) + { + // + // We can only check this if P is not too close to 1, + // so that we can guarentee Q is free of error: + // + BOOST_CHECK_CLOSE( + cdf(complement(dist, cs)), Q, tol); + BOOST_CHECK_CLOSE( + quantile(dist, P), cs, tol); + BOOST_CHECK_CLOSE( + quantile(complement(dist, Q)), cs, tol); + } +} + +// +// This test data is taken from the tables of upper +// critical values of the F distribution available +// at http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm +// +double q[] = { 0.10, 0.05, 0.025, 0.01, 0.001 }; +double upper_critical_values[][10] = { + { 161.448,199.500,215.707,224.583,230.162,233.986,236.768,238.882,240.543,241.882 }, + { 18.513, 19.000, 19.164, 19.247, 19.296, 19.330, 19.353, 19.371, 19.385, 19.396 }, + { 10.128, 9.552, 9.277, 9.117, 9.013, 8.941, 8.887, 8.845, 8.812, 8.786 }, + { 7.709, 6.944, 6.591, 6.388, 6.256, 6.163, 6.094, 6.041, 5.999, 5.964 }, + { 6.608, 5.786, 5.409, 5.192, 5.050, 4.950, 4.876, 4.818, 4.772, 4.735 }, + { 5.987, 5.143, 4.757, 4.534, 4.387, 4.284, 4.207, 4.147, 4.099, 4.060 }, + { 5.591, 4.737, 4.347, 4.120, 3.972, 3.866, 3.787, 3.726, 3.677, 3.637 }, + { 5.318, 4.459, 4.066, 3.838, 3.687, 3.581, 3.500, 3.438, 3.388, 3.347 }, + { 5.117, 4.256, 3.863, 3.633, 3.482, 3.374, 3.293, 3.230, 3.179, 3.137 }, + { 4.965, 4.103, 3.708, 3.478, 3.326, 3.217, 3.135, 3.072, 3.020, 2.978 }, + { 4.844, 3.982, 3.587, 3.357, 3.204, 3.095, 3.012, 2.948, 2.896, 2.854 }, + { 4.747, 3.885, 3.490, 3.259, 3.106, 2.996, 2.913, 2.849, 2.796, 2.753 }, + { 4.667, 3.806, 3.411, 3.179, 3.025, 2.915, 2.832, 2.767, 2.714, 2.671 }, + { 4.600, 3.739, 3.344, 3.112, 2.958, 2.848, 2.764, 2.699, 2.646, 2.602 }, + { 4.543, 3.682, 3.287, 3.056, 2.901, 2.790, 2.707, 2.641, 2.588, 2.544 }, + { 4.494, 3.634, 3.239, 3.007, 2.852, 2.741, 2.657, 2.591, 2.538, 2.494 }, + { 4.451, 3.592, 3.197, 2.965, 2.810, 2.699, 2.614, 2.548, 2.494, 2.450 }, + { 4.414, 3.555, 3.160, 2.928, 2.773, 2.661, 2.577, 2.510, 2.456, 2.412 }, + { 4.381, 3.522, 3.127, 2.895, 2.740, 2.628, 2.544, 2.477, 2.423, 2.378 }, + { 4.351, 3.493, 3.098, 2.866, 2.711, 2.599, 2.514, 2.447, 2.393, 2.348 }, + { 4.325, 3.467, 3.072, 2.840, 2.685, 2.573, 2.488, 2.420, 2.366, 2.321 }, + { 4.301, 3.443, 3.049, 2.817, 2.661, 2.549, 2.464, 2.397, 2.342, 2.297 }, + { 4.279, 3.422, 3.028, 2.796, 2.640, 2.528, 2.442, 2.375, 2.320, 2.275 }, + { 4.260, 3.403, 3.009, 2.776, 2.621, 2.508, 2.423, 2.355, 2.300, 2.255 }, + { 4.242, 3.385, 2.991, 2.759, 2.603, 2.490, 2.405, 2.337, 2.282, 2.236 }, + { 4.225, 3.369, 2.975, 2.743, 2.587, 2.474, 2.388, 2.321, 2.265, 2.220 }, + { 4.210, 3.354, 2.960, 2.728, 2.572, 2.459, 2.373, 2.305, 2.250, 2.204 }, + { 4.196, 3.340, 2.947, 2.714, 2.558, 2.445, 2.359, 2.291, 2.236, 2.190 }, + { 4.183, 3.328, 2.934, 2.701, 2.545, 2.432, 2.346, 2.278, 2.223, 2.177 }, + { 4.171, 3.316, 2.922, 2.690, 2.534, 2.421, 2.334, 2.266, 2.211, 2.165 }, + { 4.160, 3.305, 2.911, 2.679, 2.523, 2.409, 2.323, 2.255, 2.199, 2.153 }, + { 4.149, 3.295, 2.901, 2.668, 2.512, 2.399, 2.313, 2.244, 2.189, 2.142 }, + { 4.139, 3.285, 2.892, 2.659, 2.503, 2.389, 2.303, 2.235, 2.179, 2.133 }, + { 4.130, 3.276, 2.883, 2.650, 2.494, 2.380, 2.294, 2.225, 2.170, 2.123 }, + { 4.121, 3.267, 2.874, 2.641, 2.485, 2.372, 2.285, 2.217, 2.161, 2.114 }, + { 4.113, 3.259, 2.866, 2.634, 2.477, 2.364, 2.277, 2.209, 2.153, 2.106 }, + { 4.105, 3.252, 2.859, 2.626, 2.470, 2.356, 2.270, 2.201, 2.145, 2.098 }, + { 4.098, 3.245, 2.852, 2.619, 2.463, 2.349, 2.262, 2.194, 2.138, 2.091 }, + { 4.091, 3.238, 2.845, 2.612, 2.456, 2.342, 2.255, 2.187, 2.131, 2.084 }, + { 4.085, 3.232, 2.839, 2.606, 2.449, 2.336, 2.249, 2.180, 2.124, 2.077 }, + { 4.079, 3.226, 2.833, 2.600, 2.443, 2.330, 2.243, 2.174, 2.118, 2.071 }, + { 4.073, 3.220, 2.827, 2.594, 2.438, 2.324, 2.237, 2.168, 2.112, 2.065 }, + { 4.067, 3.214, 2.822, 2.589, 2.432, 2.318, 2.232, 2.163, 2.106, 2.059 }, + { 4.062, 3.209, 2.816, 2.584, 2.427, 2.313, 2.226, 2.157, 2.101, 2.054 }, + { 4.057, 3.204, 2.812, 2.579, 2.422, 2.308, 2.221, 2.152, 2.096, 2.049 }, + { 4.052, 3.200, 2.807, 2.574, 2.417, 2.304, 2.216, 2.147, 2.091, 2.044 }, + { 4.047, 3.195, 2.802, 2.570, 2.413, 2.299, 2.212, 2.143, 2.086, 2.039 }, + { 4.043, 3.191, 2.798, 2.565, 2.409, 2.295, 2.207, 2.138, 2.082, 2.035 }, + { 4.038, 3.187, 2.794, 2.561, 2.404, 2.290, 2.203, 2.134, 2.077, 2.030 }, + { 4.034, 3.183, 2.790, 2.557, 2.400, 2.286, 2.199, 2.130, 2.073, 2.026 }, + { 4.030, 3.179, 2.786, 2.553, 2.397, 2.283, 2.195, 2.126, 2.069, 2.022 }, + { 4.027, 3.175, 2.783, 2.550, 2.393, 2.279, 2.192, 2.122, 2.066, 2.018 }, + { 4.023, 3.172, 2.779, 2.546, 2.389, 2.275, 2.188, 2.119, 2.062, 2.015 }, + { 4.020, 3.168, 2.776, 2.543, 2.386, 2.272, 2.185, 2.115, 2.059, 2.011 }, + { 4.016, 3.165, 2.773, 2.540, 2.383, 2.269, 2.181, 2.112, 2.055, 2.008 }, + { 4.013, 3.162, 2.769, 2.537, 2.380, 2.266, 2.178, 2.109, 2.052, 2.005 }, + { 4.010, 3.159, 2.766, 2.534, 2.377, 2.263, 2.175, 2.106, 2.049, 2.001 }, + { 4.007, 3.156, 2.764, 2.531, 2.374, 2.260, 2.172, 2.103, 2.046, 1.998 }, + { 4.004, 3.153, 2.761, 2.528, 2.371, 2.257, 2.169, 2.100, 2.043, 1.995 }, + { 4.001, 3.150, 2.758, 2.525, 2.368, 2.254, 2.167, 2.097, 2.040, 1.993 }, + { 3.998, 3.148, 2.755, 2.523, 2.366, 2.251, 2.164, 2.094, 2.037, 1.990 }, + { 3.996, 3.145, 2.753, 2.520, 2.363, 2.249, 2.161, 2.092, 2.035, 1.987 }, + { 3.993, 3.143, 2.751, 2.518, 2.361, 2.246, 2.159, 2.089, 2.032, 1.985 }, + { 3.991, 3.140, 2.748, 2.515, 2.358, 2.244, 2.156, 2.087, 2.030, 1.982 }, + { 3.989, 3.138, 2.746, 2.513, 2.356, 2.242, 2.154, 2.084, 2.027, 1.980 }, + { 3.986, 3.136, 2.744, 2.511, 2.354, 2.239, 2.152, 2.082, 2.025, 1.977 }, + { 3.984, 3.134, 2.742, 2.509, 2.352, 2.237, 2.150, 2.080, 2.023, 1.975 }, + { 3.982, 3.132, 2.740, 2.507, 2.350, 2.235, 2.148, 2.078, 2.021, 1.973 }, + { 3.980, 3.130, 2.737, 2.505, 2.348, 2.233, 2.145, 2.076, 2.019, 1.971 }, + { 3.978, 3.128, 2.736, 2.503, 2.346, 2.231, 2.143, 2.074, 2.017, 1.969 }, + { 3.976, 3.126, 2.734, 2.501, 2.344, 2.229, 2.142, 2.072, 2.015, 1.967 }, + { 3.974, 3.124, 2.732, 2.499, 2.342, 2.227, 2.140, 2.070, 2.013, 1.965 }, + { 3.972, 3.122, 2.730, 2.497, 2.340, 2.226, 2.138, 2.068, 2.011, 1.963 }, + { 3.970, 3.120, 2.728, 2.495, 2.338, 2.224, 2.136, 2.066, 2.009, 1.961 }, + { 3.968, 3.119, 2.727, 2.494, 2.337, 2.222, 2.134, 2.064, 2.007, 1.959 }, + { 3.967, 3.117, 2.725, 2.492, 2.335, 2.220, 2.133, 2.063, 2.006, 1.958 }, + { 3.965, 3.115, 2.723, 2.490, 2.333, 2.219, 2.131, 2.061, 2.004, 1.956 }, + { 3.963, 3.114, 2.722, 2.489, 2.332, 2.217, 2.129, 2.059, 2.002, 1.954 }, + { 3.962, 3.112, 2.720, 2.487, 2.330, 2.216, 2.128, 2.058, 2.001, 1.953 }, + { 3.960, 3.111, 2.719, 2.486, 2.329, 2.214, 2.126, 2.056, 1.999, 1.951 }, + { 3.959, 3.109, 2.717, 2.484, 2.327, 2.213, 2.125, 2.055, 1.998, 1.950 }, + { 3.957, 3.108, 2.716, 2.483, 2.326, 2.211, 2.123, 2.053, 1.996, 1.948 }, + { 3.956, 3.107, 2.715, 2.482, 2.324, 2.210, 2.122, 2.052, 1.995, 1.947 }, + { 3.955, 3.105, 2.713, 2.480, 2.323, 2.209, 2.121, 2.051, 1.993, 1.945 }, + { 3.953, 3.104, 2.712, 2.479, 2.322, 2.207, 2.119, 2.049, 1.992, 1.944 }, + { 3.952, 3.103, 2.711, 2.478, 2.321, 2.206, 2.118, 2.048, 1.991, 1.943 }, + { 3.951, 3.101, 2.709, 2.476, 2.319, 2.205, 2.117, 2.047, 1.989, 1.941 }, + { 3.949, 3.100, 2.708, 2.475, 2.318, 2.203, 2.115, 2.045, 1.988, 1.940 }, + { 3.948, 3.099, 2.707, 2.474, 2.317, 2.202, 2.114, 2.044, 1.987, 1.939 }, + { 3.947, 3.098, 2.706, 2.473, 2.316, 2.201, 2.113, 2.043, 1.986, 1.938 }, + { 3.946, 3.097, 2.705, 2.472, 2.315, 2.200, 2.112, 2.042, 1.984, 1.936 }, + { 3.945, 3.095, 2.704, 2.471, 2.313, 2.199, 2.111, 2.041, 1.983, 1.935 }, + { 3.943, 3.094, 2.703, 2.470, 2.312, 2.198, 2.110, 2.040, 1.982, 1.934 }, + { 3.942, 3.093, 2.701, 2.469, 2.311, 2.197, 2.109, 2.038, 1.981, 1.933 }, + { 3.941, 3.092, 2.700, 2.467, 2.310, 2.196, 2.108, 2.037, 1.980, 1.932 }, + { 3.940, 3.091, 2.699, 2.466, 2.309, 2.195, 2.106, 2.036, 1.979, 1.931 }, + { 3.939, 3.090, 2.698, 2.465, 2.308, 2.194, 2.105, 2.035, 1.978, 1.930 }, + { 3.938, 3.089, 2.697, 2.465, 2.307, 2.193, 2.104, 2.034, 1.977, 1.929 }, + { 3.937, 3.088, 2.696, 2.464, 2.306, 2.192, 2.103, 2.033, 1.976, 1.928 }, + { 3.936, 3.087, 2.696, 2.463, 2.305, 2.191, 2.103, 2.032, 1.975, 1.927 } +}; + + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks, test data is to three decimal places only + // so set tolerance to 0.002 expressed as a persentage. Note that + // we can't even get full 3 digit accuracy since the data we're + // using as input has *already been rounded*, leading to even + // greater differences in output. As an accuracy test this is + // pretty useless, but it is an excellent sanity check. + + RealType tolerance = 0.002f * 100; + cout << "Tolerance = " << tolerance << "%." << endl; + + using boost::math::fisher_f_distribution; + using ::boost::math::fisher_f; + using ::boost::math::cdf; + using ::boost::math::pdf; + + for(unsigned i = 0; i < sizeof(upper_critical_values) / sizeof(upper_critical_values[0]); ++i) + { + for(unsigned j = 0; j < sizeof(upper_critical_values[0])/sizeof(upper_critical_values[0][0]); ++j) + { + test_spot( + static_cast(j+1), // degrees of freedom 1 + static_cast(i+1), // degrees of freedom 2 + static_cast(upper_critical_values[i][j]), // test statistic F + static_cast(0.95), // Probability of result (CDF), P + static_cast(0.05), // Q = 1 - P + tolerance); + } + } + + // http://www.vias.org/simulations/simusoft_distcalc.html + // Distcalc version 1.2 Copyright 2002 H Lohninger, TU Wein + // H.Lohninger: Teach/Me Data Analysis, Springer-Verlag, Berlin-New York-Tokyo, 1999. ISBN 3-540-14743-8 + // The Windows calculator is available zipped distcalc.exe for download at: + // http://www.vias.org/simulations/simu_stat.html + + // This interactive Windows program was used to find some combination for which the + // result appears to be exact. No doubt this can be done analytically too, + // by mathematicians! + + // Some combinations for which the result is 'exact', or at least is to 40 decimal digits. + // 40 decimal digits includes 128-bit significand User Defined Floating-Point types. + // These all pass tests at near epsilon accuracy for the floating-point type. + tolerance = boost::math::tools::epsilon() * 5 * 100; + cout << "Tolerance = " << tolerance << "%." << endl; + BOOST_CHECK_CLOSE( + cdf(fisher_f_distribution( + static_cast(1.), // df1 + static_cast(2.)), // df2 + static_cast(2.)/static_cast(3.) ), // F + static_cast(0.5), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(1.), // df1 + static_cast(2.)), // df2 + static_cast(1.6L))), // F + static_cast(0.333333333333333333333333333333333333L), // probability. + tolerance * 100); // needs higher tolerance at 128-bit precision - value not exact? + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(1.), // df1 + static_cast(2.)), // df2 + static_cast(6.5333333333333333333333333333333333L))), // F + static_cast(0.125L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(1.))), // F + static_cast(0.5L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(3.))), // F + static_cast(0.25L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(3.))), // F + static_cast(0.25L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(7.))), // F + static_cast(0.125L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(9.))), // F + static_cast(0.1L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(19.))), // F + static_cast(0.05L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(29.))), // F + static_cast(0.03333333333333333333333333333333333333333L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(99.))), // F + static_cast(0.01L), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(4.), // df1 + static_cast(4.)), // df2 + static_cast(9.))), // F + static_cast(0.028L), // probability. + tolerance*10); // not quite exact??? + + BOOST_CHECK_CLOSE( + cdf(complement(fisher_f_distribution( + static_cast(8.), // df1 + static_cast(8.)), // df2 + static_cast(1.))), // F + static_cast(0.5L), // probability. + tolerance); + +// Inverse tests + + BOOST_CHECK_CLOSE( + quantile(complement(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(0.03333333333333333333333333333333333333333L))), // probability + static_cast(29.), // F expected. + tolerance*10); + + BOOST_CHECK_CLOSE( + quantile(fisher_f_distribution( + static_cast(2.), // df1 + static_cast(2.)), // df2 + static_cast(1.0L - 0.03333333333333333333333333333333333333333L)), // probability + static_cast(29.), // F expected. + tolerance*10); + + +// Also note limit cases for F(1, infinity) == normal distribution +// F(1, n2) == Student's t distribution +// F(n1, infinity) == Chisq distribution + +// These might allow some further cross checks? + + RealType tol2 = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a percent + cout << "Tolerance = " << tol2 << "%." << endl; + fisher_f_distribution dist(static_cast(8), static_cast(6)); + RealType x = 7; + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(6)/static_cast(4), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(2 * 6 * 6 * (8 + 6 - 2)) / static_cast(8 * 16 * 2), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , sqrt(static_cast(2 * 6 * 6 * (8 + 6 - 2)) / static_cast(8 * 16 * 2)), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol2); + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(6*6)/static_cast(8*8), tol2); + + fisher_f_distribution dist2(static_cast(8), static_cast(12)); + BOOST_CHECK_CLOSE( + skewness(dist2) + , static_cast(26 * sqrt(64.0L)) / (12*6), tol2); + BOOST_CHECK_CLOSE( + kurtosis_excess(dist2) + , static_cast(6272) * 12 / 3456, tol2); + BOOST_CHECK_CLOSE( + kurtosis(dist2) + , static_cast(6272) * 12 / 3456 + 3, tol2); + // special cases: + BOOST_CHECK_THROW( + pdf( + fisher_f_distribution(static_cast(1), static_cast(1)), + static_cast(0)), std::overflow_error + ); + BOOST_CHECK_EQUAL( + pdf(fisher_f_distribution(2, 2), static_cast(0)) + , static_cast(1.0f)); + BOOST_CHECK_EQUAL( + pdf(fisher_f_distribution(3, 3), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(fisher_f_distribution(1, 1), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(fisher_f_distribution(2, 2), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(fisher_f_distribution(3, 3), static_cast(0)) + , static_cast(0.0f)); + BOOST_CHECK_EQUAL( + cdf(complement(fisher_f_distribution(1, 1), static_cast(0))) + , static_cast(1)); + BOOST_CHECK_EQUAL( + cdf(complement(fisher_f_distribution(2, 2), static_cast(0))) + , static_cast(1)); + BOOST_CHECK_EQUAL( + cdf(complement(fisher_f_distribution(3, 3), static_cast(0))) + , static_cast(1)); + + BOOST_CHECK_THROW( + pdf( + fisher_f_distribution(-1, 2), + static_cast(1)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + fisher_f_distribution(1, -1), + static_cast(1)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( + fisher_f_distribution(8, 2), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + fisher_f_distribution(-1, 1), + static_cast(1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( + fisher_f_distribution(8, 4), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf(complement( + fisher_f_distribution(-1, 2), + static_cast(1))), std::domain_error + ); + BOOST_CHECK_THROW( + cdf(complement( + fisher_f_distribution(8, 4), + static_cast(-1))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + fisher_f_distribution(-1, 2), + static_cast(0.5)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + fisher_f_distribution(8, 8), + static_cast(-1)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( + fisher_f_distribution(8, 8), + static_cast(1.1)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + fisher_f_distribution(2, -1), + static_cast(0.5))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + fisher_f_distribution(8, 8), + static_cast(-1))), std::domain_error + ); + BOOST_CHECK_THROW( + quantile(complement( + fisher_f_distribution(8, 8), + static_cast(1.1))), std::domain_error + ); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + + // Check that can generate fisher distribution using the two convenience methods: + boost::math::fisher_f myf1(1., 2); // Using typedef + fisher_f_distribution<> myf2(1., 2); // Using default RealType double. + + + // Basic sanity-check spot values. + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. + test_spots(0.0); // Test double. +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#endif + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_fisher.exe" +Running 1 test case... +Tolerance = 0.2%. +Tolerance = 5.96046e-005%. +Tolerance = 5.96046e-005%. +Tolerance = 0.2%. +Tolerance = 1.11022e-013%. +Tolerance = 1.11022e-013%. +Tolerance = 0.2%. +Tolerance = 1.11022e-013%. +Tolerance = 1.11022e-013%. +Tolerance = 0.2%. +Tolerance = 1.11022e-013%. +Tolerance = 1.11022e-013%. +*** No errors detected + +*/ + + + diff --git a/test/test_gamma.cpp b/test/test_gamma.cpp new file mode 100644 index 000000000..f5801084d --- /dev/null +++ b/test/test_gamma.cpp @@ -0,0 +1,497 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_gamma_hooks.hpp" +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the functions tgamma and lgamma, and the +// function tgamma1pm1. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // G++ on Darwin: results are just slightly worse than we might hope for + // but still pretty good: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::tgamma", 100, 15); // test function + + // + // G++ on Linux, result vary a bit by processor type, + // on Itanium results are *much* better than listed here, + // but x86 appears to have much less accurate std::pow + // that throws off the results: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::tgamma", 400, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::lgamma", 30, 10); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + "near (1|2|-10)", // test data group + "boost::math::tgamma", 10, 5); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + "near (1|2|-10)", // test data group + "boost::math::lgamma", 50, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + "tgamma1pm1.*", // test data group + "boost::math::tgamma1pm1", 50, 15); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::tgamma", 220, 70); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + "real_concept", // test type(s) + "near (0|-55)", // test data group + "boost::math::(t|l)gamma", 130, 80); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + "real_concept", // test type(s) + "tgamma1pm1.*", // test data group + "boost::math::tgamma1pm1", 40, 10); // test function + // + // HP-UX results: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::tgamma", 5, 4); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + "near (0|-55)", // test data group + "boost::math::tgamma", 10, 5); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + "near (1|2|-10)", // test data group + "boost::math::lgamma", 250, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::lgamma", 50, 20); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX", // platform + "real_concept", // test type(s) + "tgamma1pm1.*", // test data group + "boost::math::tgamma1pm1", 200, 80); // test function + // + // Tru64: + // + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::lgamma", 50, 20); // test function + // + // Sun OS: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::tgamma", 300, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "Sun.*", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::tgamma", 300, 50); // test function + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::tgamma", 4, 1); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "factorials", // test data group + "boost::math::lgamma", 9, 1); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "near (0|-55)", // test data group + "boost::math::(t|l)gamma", 200, 100); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "near (1|2|-10)", // test data group + "boost::math::tgamma", 10, 5); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "near (1|2|-10)", // test data group + "boost::math::lgamma", 14, 7); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "tgamma1pm1.*", // test data group + "boost::math::tgamma1pm1", 30, 9); // test function + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::tgamma", 70, 25); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "factorials", // test data group + "boost::math::lgamma", 40, 4); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "near.*", // test data group + "boost::math::tgamma", 80, 60); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "near.*", // test data group + "boost::math::lgamma", 10000000, 10000000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "tgamma1pm1.*", // test data group + "boost::math::tgamma1pm1", 20, 5); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_gamma(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::tgamma; +#else + pg funcp = boost::math::tgamma; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::tgamma; + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::tgamma"); + } +#endif + // + // test lgamma against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::lgamma; +#else + funcp = boost::math::lgamma; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::lgamma", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::lgamma; + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(2)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::lgamma"); + } +#endif + + std::cout << std::endl; +} + +template +void do_test_gammap1m1(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::tgamma1pm1; +#else + pg funcp = boost::math::tgamma1pm1; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma1pm1 against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma1pm1", test_name); + std::cout << std::endl; +} + +template +void test_gamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value, gamma and lgamma: + // + // gamma and lgamma at integer and half integer values: + // boost::array, N> factorials; + // + // gamma and lgamma for z near 0: + // boost::array, N> near_0; + // + // gamma and lgamma for z near 1: + // boost::array, N> near_1; + // + // gamma and lgamma for z near 2: + // boost::array, N> near_2; + // + // gamma and lgamma for z near -10: + // boost::array, N> near_m10; + // + // gamma and lgamma for z near -55: + // boost::array, N> near_m55; + // + // The last two cases are chosen more or less at random, + // except that one is even and the other odd, and both are + // at negative poles. The data near zero also tests near + // a pole, the data near 1 and 2 are to probe lgamma as + // the result -> 0. + // +# include "test_gamma_data.ipp" + + do_test_gamma(factorials, name, "factorials"); + do_test_gamma(near_0, name, "near 0"); + do_test_gamma(near_1, name, "near 1"); + do_test_gamma(near_2, name, "near 2"); + do_test_gamma(near_m10, name, "near -10"); + do_test_gamma(near_m55, name, "near -55"); + + // + // And now tgamma1pm1 which computes gamma(1+dz)-1: + // + do_test_gammap1m1(gammap1m1_data, name, "tgamma1pm1(dz)"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 50 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 5000; + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(3.5)), static_cast(3.3233509704478425511840640312646472177454052302295L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.125)), static_cast(7.5339415987976119046992298412151336246104195881491L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-0.125)), static_cast(-8.7172188593831756100190140408231437691829605421405L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-3.125)), static_cast(1.1668538708507675587790157356605097019141636072094L), tolerance); + // Lower tolerance on this one, is only really needed on Linux x86 systems, result is mostly down to std lib accuracy: + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(-53249.0/1024)), static_cast(-1.2646559519067605488251406578743995122462767733517e-65L), tolerance * 3); + + int sign = 1; + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(3.5), &sign), static_cast(1.2009736023470742248160218814507129957702389154682L), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(0.125), &sign), static_cast(2.0194183575537963453202905211670995899482809521344L), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-0.125), &sign), static_cast(2.1653002489051702517540619481440174064962195287626L), tolerance); + BOOST_CHECK(sign == -1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-3.125), &sign), static_cast(0.1543111276840418242676072830970532952413339012367L), tolerance); + BOOST_CHECK(sign == 1); + BOOST_CHECK_CLOSE(::boost::math::lgamma(static_cast(-53249.0/1024), &sign), static_cast(-149.43323093420259741100038126078721302600128285894L), tolerance); + BOOST_CHECK(sign == -1); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_spots(0.0F); +#endif + test_spots(0.0); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); + test_spots(boost::math::concepts::real_concept(0.1)); +#endif + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_gamma(0.1F, "float"); +#endif + test_gamma(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_gamma(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_gamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_gamma_data.ipp b/test/test_gamma_data.ipp new file mode 100644 index 000000000..185b41252 --- /dev/null +++ b/test/test_gamma_data.ipp @@ -0,0 +1,582 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 198> factorials = { { + { SC_(1), SC_(1), SC_(0) }, + { SC_(2), SC_(1), SC_(0) }, + { SC_(3), SC_(2), SC_(0.6931471805599453094172321214581765680755) }, + { SC_(4), SC_(6), SC_(1.791759469228055000812477358380702272723) }, + { SC_(5), SC_(24), SC_(3.178053830347945619646941601297055408874) }, + { SC_(6), SC_(120), SC_(4.7874917427820459942477009345232430484) }, + { SC_(7), SC_(720), SC_(6.579251212010100995060178292903945321123) }, + { SC_(8), SC_(5040), SC_(8.52516136106541430016553103634712505076) }, + { SC_(9), SC_(40320), SC_(10.60460290274525022841722740072165475499) }, + { SC_(10), SC_(362880), SC_(12.80182748008146961120771787456670616428) }, + { SC_(11), SC_(3628800), SC_(15.10441257307551529522570932925107037188) }, + { SC_(12), SC_(39916800), SC_(17.5023078458738858392876529072161996717) }, + { SC_(13), SC_(479001600), SC_(19.9872144956618861495173623870550785125) }, + { SC_(14), SC_(6227020800.0), SC_(22.55216385312342288557084982862039711731) }, + { SC_(15), SC_(87178291200.0), SC_(25.19122118273868150009343469352175341502) }, + { SC_(16), SC_(1307674368000.0), SC_(27.89927138384089156608943926367046675919) }, + { SC_(17), SC_(20922789888000.0), SC_(30.6718601060806728037583677495031730315) }, + { SC_(18), SC_(355687428096000.0), SC_(33.50507345013688888400790236737629956708) }, + { SC_(19), SC_(6402373705728000.0), SC_(36.39544520803305357621562496267952754445) }, + { SC_(20), SC_(121645100408832000.0), SC_(39.33988418719949403622465239456738108169) }, + { SC_(21), SC_(0.243290200817664e19), SC_(42.33561646075348502965987597070992185737) }, + { SC_(22), SC_(0.5109094217170944e20), SC_(45.38013889847690802616047395107562729165) }, + { SC_(23), SC_(0.112400072777760768e22), SC_(48.47118135183522387963964965049893315955) }, + { SC_(24), SC_(0.2585201673888497664e23), SC_(51.60667556776437357044640248230912927799) }, + { SC_(25), SC_(0.62044840173323943936e24), SC_(54.78472939811231919009334408360618468687) }, + { SC_(26), SC_(0.15511210043330985984e26), SC_(58.00360522298051993929486275005855996592) }, + { SC_(27), SC_(0.403291461126605635584e27), SC_(61.2617017610020019847655823130820551388) }, + { SC_(28), SC_(0.10888869450418352160768e29), SC_(64.55753862700633105895131802384963225274) }, + { SC_(29), SC_(0.304888344611713860501504e30), SC_(67.88974313718153498289113501020916511853) }, + { SC_(30), SC_(0.8841761993739701954543616e31), SC_(71.25703896716800901007440704257107672402) }, + { SC_(31), SC_(0.26525285981219105863630848e33), SC_(74.65823634883016438548764373417796663627) }, + { SC_(32), SC_(0.822283865417792281772556288e34), SC_(78.09222355331531063141680805872032384672) }, + { SC_(33), SC_(0.26313083693369353016721801216e36), SC_(81.5579594561150371785029686660112066871) }, + { SC_(34), SC_(0.868331761881188649551819440128e37), SC_(85.05446701758151741396015748089886169157) }, + { SC_(35), SC_(0.29523279903960414084761860964352e39), SC_(88.58082754219767880362692422023016479523) }, + { SC_(36), SC_(0.103331479663861449296666513375232e41), SC_(92.13617560368709248333303629689953216439) }, + { SC_(37), SC_(0.3719933267899012174679994481508352e42), SC_(95.71969454214320248495799101366093670984) }, + { SC_(38), SC_(0.137637530912263450463159795815809024e44), SC_(99.33061245478742692932608668469238387374) }, + { SC_(39), SC_(0.5230226174666011117600072241000742912e45), SC_(102.9681986145138126987523462380384139791) }, + { SC_(40), SC_(0.203978820811974433586402817399028973568e47), SC_(106.6317602606434591262010789165262582885) }, + { SC_(41), SC_(0.815915283247897734345611269596115894272e48), SC_(110.3206397147573954290535346141269756323) }, + { SC_(42), SC_(0.3345252661316380710817006205344075166515e50), SC_(114.0342117814617032329202979871643832206) }, + { SC_(43), SC_(0.1405006117752879898543142606244511569936e52), SC_(117.771881399745071538838128088988265223) }, + { SC_(44), SC_(0.6041526306337383563735513206851399750726e53), SC_(121.5330815154386339623109706023341122586) }, + { SC_(45), SC_(0.265827157478844876804362581101461589032e55), SC_(125.3172711493568951252073784232155946945) }, + { SC_(46), SC_(0.1196222208654801945619631614956577150644e57), SC_(129.1239336391272148825986282302868337433) }, + { SC_(47), SC_(0.5502622159812088949850305428800254892962e58), SC_(132.9525750356163098828226131835552064299) }, + { SC_(48), SC_(0.2586232415111681806429643551536119799692e60), SC_(136.8027226373263684696435638533273801388) }, + { SC_(49), SC_(0.1241391559253607267086228904737337503852e62), SC_(140.6739236482342593987077375760826121157) }, + { SC_(50), SC_(0.6082818640342675608722521633212953768876e63), SC_(144.565743946344886008918443062968971575) }, + { SC_(51), SC_(0.3041409320171337804361260816606476884438e65), SC_(148.4777669517730320675371938508795234221) }, + { SC_(52), SC_(0.1551118753287382280224243016469303211063e67), SC_(152.4095925844973578391819737056751756623) }, + { SC_(53), SC_(0.8065817517094387857166063685640376697529e68), SC_(156.3608363030787851940699253901568474033) }, + { SC_(54), SC_(0.427488328406002556429801375338939964969e70), SC_(160.3311282166309070282143945291859051737) }, + { SC_(55), SC_(0.2308436973392413804720927426830275810833e72), SC_(164.3201122631951814118173623614116588557) }, + { SC_(56), SC_(0.1269640335365827592596510084756651695958e74), SC_(168.327445448427652330480065272602975795) }, + { SC_(57), SC_(0.7109985878048634518540456474637249497365e75), SC_(172.3527971391628015638371143804206852289) }, + { SC_(58), SC_(0.4052691950487721675568060190543232213498e77), SC_(176.3958484069973517152413870492310644708) }, + { SC_(59), SC_(0.2350561331282878571829474910515074683829e79), SC_(180.4562914175437710518418912030511526443) }, + { SC_(60), SC_(0.1386831185456898357379390197203894063459e81), SC_(184.5338288614494905024579415767708502684) }, + { SC_(61), SC_(0.8320987112741390144276341183223364380754e82), SC_(188.6281734236715911872884103898359167487) }, + { SC_(62), SC_(0.507580213877224798800856812176625227226e84), SC_(192.7390472878449024360397994932615314951) }, + { SC_(63), SC_(0.3146997326038793752565312235495076408801e86), SC_(196.8661816728899939913861959392620652736) }, + { SC_(64), SC_(0.1982608315404440064116146708361898137545e88), SC_(201.0093163992815266792820391565502964125) }, + { SC_(65), SC_(0.1268869321858841641034333893351614808029e90), SC_(205.168199482641198535785431885299355821) }, + { SC_(66), SC_(0.8247650592082470666723170306785496252186e91), SC_(209.3425867525368356464396786600908620653) }, + { SC_(67), SC_(0.5443449390774430640037292402478427526443e93), SC_(213.5322414945632611913140995964366936378) }, + { SC_(68), SC_(0.3647111091818868528824985909660546442717e95), SC_(217.7369341139542272509841715928004163884) }, + { SC_(69), SC_(0.2480035542436830599600990418569171581047e97), SC_(221.9564418191303339500681704535898960601) }, + { SC_(70), SC_(0.1711224524281413113724683388812728390923e99), SC_(226.1905483237275933322701685223226178832) }, + { SC_(71), SC_(0.1197857166996989179607278372168909873646e101), SC_(230.4390435657769523213935127204501618205) }, + { SC_(72), SC_(0.8504785885678623175211676442399260102886e102), SC_(234.7017234428182677427229672529631959172) }, + { SC_(73), SC_(0.6123445837688608686152407038527467274078e104), SC_(238.9783895618343230537651540911827770308) }, + { SC_(74), SC_(0.4470115461512684340891257138125051110077e106), SC_(243.2688490029827141828572629486213196017) }, + { SC_(75), SC_(0.3307885441519386412259530282212537821457e108), SC_(247.5729140961868839366425907411109433336) }, + { SC_(76), SC_(0.2480914081139539809194647711659403366093e110), SC_(251.8904022097231943772393546444858443173) }, + { SC_(77), SC_(0.188549470166605025498793226086114655823e112), SC_(256.2211355500095254560828463192900509907) }, + { SC_(78), SC_(0.1451830920282858696340707840863082849837e114), SC_(260.5649409718632093052501426406983600202) }, + { SC_(79), SC_(0.1132428117820629783145752115873204622873e116), SC_(264.9216497985528010421161074406443808977) }, + { SC_(80), SC_(0.8946182130782975286851441715398316520698e117), SC_(269.2910976510198225362890529821257918199) }, + { SC_(81), SC_(0.7156945704626380229481153372318653216558e119), SC_(273.6731242856937041485587408011846857317) }, + { SC_(82), SC_(0.5797126020747367985879734231578109105412e121), SC_(278.0675734403661429141397217488747885503) }, + { SC_(83), SC_(0.4753643337012841748421382069894049466438e123), SC_(282.4742926876303960274237172433703727067) }, + { SC_(84), SC_(0.3945523969720658651189747118012061057144e125), SC_(286.8931332954269939508991894666617431598) }, + { SC_(85), SC_(0.3314240134565353266999387579130131288001e127), SC_(291.3239500942703075662342516899438017302) }, + { SC_(86), SC_(0.2817104114380550276949479442260611594801e129), SC_(295.7666013507606240210845456410431159053) }, + { SC_(87), SC_(0.2422709538367273238176552320344125971528e131), SC_(300.220948647014131753974620275847139509) }, + { SC_(88), SC_(0.210775729837952771721360051869938959523e133), SC_(304.6868567656687154725531375451315768191) }, + { SC_(89), SC_(0.1854826422573984391147968456455462843802e135), SC_(309.1641935801469219448667774874712358232) }, + { SC_(90), SC_(0.1650795516090846108121691926245361930984e137), SC_(313.6528299498790617831845930281410850426) }, + { SC_(91), SC_(0.1485715964481761497309522733620825737886e139), SC_(318.1526396202093268499930749566705006595) }, + { SC_(92), SC_(0.1352001527678402962551665687594951421476e141), SC_(322.6634991267261768911519151416789989939) }, + { SC_(93), SC_(0.1243841405464130725547532432587355307758e143), SC_(327.1852877037752172007931322164055482485) }, + { SC_(94), SC_(0.1156772507081641574759205162306240436215e145), SC_(331.7178871969284731381175417778704311636) }, + { SC_(95), SC_(0.1087366156656743080273652852567866010042e147), SC_(336.2611819791984770343557245691007814406) }, + { SC_(96), SC_(0.103299784882390592625997020993947270954e149), SC_(340.8150588707990178689655113342148226173) }, + { SC_(97), SC_(0.9916779348709496892095714015418938011582e150), SC_(345.3794070622668541074469171784282311623) }, + { SC_(98), SC_(0.9619275968248211985332842594956369871234e152), SC_(349.9541180407702369295636388001321928762) }, + { SC_(99), SC_(0.942689044888324774562618574305724247381e154), SC_(354.5390855194408088491915764084767289035) }, + { SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519) }, + { SC_(2.5), SC_(1.329340388179137020473625612505858887098), SC_(0.2846828704729191596324946696827019243201) }, + { SC_(3.5), SC_(3.323350970447842551184064031264647217745), SC_(1.20097360234707422481602188145071299577) }, + { SC_(4.5), SC_(11.63172839656744892914422410942626526211), SC_(2.453736570842442220504142503435716157332) }, + { SC_(5.5), SC_(52.34277778455352018114900849241819367949), SC_(3.957813967618716293877400855822590998551) }, + { SC_(6.5), SC_(287.8852778150443609963195467083000652372), SC_(5.662562059857141528522112312329543730298) }, + { SC_(7.5), SC_(1871.254305797788346476077053603950424042), SC_(7.534364236758732955158367632436685767027) }, + { SC_(8.5), SC_(14034.40729348341259857057790202962818031), SC_(9.549267257300997711737140081127222543125) }, + { SC_(9.5), SC_(119292.4619946090070878499121672518395327), SC_(11.68933342079726848256944257754217251064) }, + { SC_(10.5), SC_(1133278.38894878556733457416558889247556), SC_(13.9406252194037636331612378879718494798) }, + { SC_(11.5), SC_(11899423.08396224845701302873868337099338), SC_(16.29200047656724132024460374687937834601) }, + { SC_(12.5), SC_(136843365.4655658572556498304948587664239), SC_(18.73434751193644570163412445723139789638) }, + { SC_(13.5), SC_(1710542068.319573215695622881185734580299), SC_(21.26007615624470114141841100222559660735) }, + { SC_(14.5), SC_(23092317922.31423841189090889600741683403), SC_(23.86276584168908490618691459153499715322) }, + { SC_(15.5), SC_(334838609873.5564569724181789921075440935), SC_(26.53691449111561362395295450243873219064) }, + { SC_(16.5), SC_(5189998453040.125083072481774377666933449), SC_(29.27775451504081456046488670552291283301) }, + { SC_(17.5), SC_(85634974475162.06387069594927723150440191), SC_(32.08111489594734948650484339895239126941) }, + { SC_(18.5), SC_(1498612053315336.117737179112351551327033), SC_(34.94331577687681785679372335416358207049) }, + { SC_(19.5), SC_(27724322986333718.17813781357850369955012), SC_(37.86108650896109699174458690373685266632) }, + { SC_(20.5), SC_(540624298233507504.4736873647808221412273), SC_(40.83150097453079810977608746076652040769) }, + { SC_(21.5), SC_(11082798113786903841.71059097800685389516), SC_(43.851925860675160604225618712345751428) }, + { SC_(22.5), SC_(238280159446418432596.7777060271473587459), SC_(46.91997879580877771828122910423342189548) }, + { SC_(23.5), SC_(5361303587544414733427.498385610815571784), SC_(50.03349410501915216625524678984648437622) }, + { SC_(24.5), SC_(125990634307293746235546.2120618541659369), SC_(53.19049452616926544365896533816048151704) }, + { SC_(25.5), SC_(3086770540528696782770882.195515427065454), SC_(56.38916764371994674445243870358866440824) }, + { SC_(26.5), SC_(78712648783481767960657495.98564339016909), SC_(59.6278460958843272066799864369261400804) }, + { SC_(27.5), SC_(2085885192762266850957423643.619549839481), SC_(62.90499082887650373140722345449702128269) }, + { SC_(28.5), SC_(57361842800962338401329150199.53762058572), SC_(66.21917683354902934065269424423016165396) }, + { SC_(29.5), SC_(1634812519827426644437880780686.822186693), SC_(69.56908092082363418263973479158236432777) }, + { SC_(30.5), SC_(48226969334909086010917483030261.25450745), SC_(72.95347118416940832383855304384388538376) }, + { SC_(31.5), SC_(1470922564714727123332983232422968.262477), SC_(76.371197867782774263172710025811323562) }, + { SC_(32.5), SC_(46334060788513904384988971821323500.26803), SC_(79.82118541361436164165132112164137813285) }, + { SC_(33.5), SC_(1505856975626701892512141584193013758.711), SC_(83.30242550295005344288833577497470780911) }, + { SC_(34.5), SC_(50446208683494513399156743070465960916.82), SC_(86.8139709417810741931411756498802539916) }, + { SC_(35.5), SC_(1740394199580560712270907635931075651630.0), SC_(90.35493026581838826592594159715479924661) }, + { SC_(36.5), SC_(61783994085109905285617221075553185632870.0), SC_(93.9244629622997583778381640082096567753) }, + { SC_(37.5), SC_(0.22551157841065115429250285692576912756e43), SC_(97.52177522288820419751304074419002277813) }, + { SC_(38.5), SC_(0.8456684190399418285968857134716342283499e44), SC_(101.1461161558645693286925725261067471938) }, + { SC_(39.5), SC_(0.3255823413303776040098009996865791779147e46), SC_(104.7967743971583078684426367260568796551) }, + { SC_(40.5), SC_(0.1286050248254991535838713948761987752763e48), SC_(108.4730750690653840531983501460801140092) }, + { SC_(41.5), SC_(0.5208503505432715720146791492486050398691e49), SC_(112.1743770431778775093620989723120402597) }, + { SC_(42.5), SC_(0.2161528954754577023860918469381710915457e51), SC_(115.9000704704145301234203390741452341447) }, + { SC_(43.5), SC_(0.9186498057706952351408903494872271390691e52), SC_(119.6495745463449012688534009037863717517) }, + { SC_(44.5), SC_(0.3996126655102524272862873020269438054951e54), SC_(123.4223354844395396780146860516126324938) }, + { SC_(45.5), SC_(0.1778276361520623301423978494019899934453e56), SC_(127.2178246736117342069152694708243051451) }, + { SC_(46.5), SC_(0.8091157444918836021479102147790544701761e57), SC_(131.0355369995686389386568775343746269115) }, + { SC_(47.5), SC_(0.3762388211887258749987782498722603286319e59), SC_(134.8749893121619495665640549743813332585) }, + { SC_(48.5), SC_(0.1787134400646447906244196686893236561001e61), SC_(138.7357190232025450917566096180371978672) }, + { SC_(49.5), SC_(0.8667601843135272345284353931432197320857e62), SC_(142.6172828211459826044560991182829830129) }, + { SC_(50.5), SC_(0.4290462912351959810915755196058937673824e64), SC_(146.519255490720627221891301048634987154) }, + { SC_(51.5), SC_(0.2166683770737739704512456374009763525281e66), SC_(150.4412288270019413633582671940897997428) }, + { SC_(52.5), SC_(0.111584214192993594782391503261502821552e68), SC_(154.3828106346716318247096373876850639954) }, + { SC_(53.5), SC_(0.5858171245132163726075553921228898131479e69), SC_(158.3436238042692098863937625798187805011) }, + { SC_(54.5), SC_(0.3134121616145707593450421347857460500341e71), SC_(162.3233054581711707502809292753838809346) }, + { SC_(55.5), SC_(0.1708096280799410638430479634582315972686e73), SC_(166.3215061598403691412410136061349060176) }, + { SC_(56.5), SC_(0.9479934358436729043289161971931853648408e74), SC_(170.337889180592757967587122392630702318) }, + { SC_(57.5), SC_(0.535616291251675190945837651414149731135e76), SC_(174.372129818745153226752021764788547422) }, + { SC_(58.5), SC_(0.3079793674697132347938566495631360954026e78), SC_(178.4239147665484579827423018083667546119) }, + { SC_(59.5), SC_(0.1801679299697822423544061399944346158105e80), SC_(182.4929415207862687921690476023189480579) }, + { SC_(60.5), SC_(0.1071999183320204342008716532966885964073e82), SC_(186.5789178333378528681067028421770777551) }, + { SC_(61.5), SC_(0.648559505908723626915273502444966008264e83), SC_(190.6815611983746486468133578766491597866) }, + { SC_(62.5), SC_(0.3988640961338650305528932040036540950824e85), SC_(194.8005983731871208326581343651509165116) }, + { SC_(63.5), SC_(0.2492900600836656440955582525022838094265e87), SC_(198.9357649299294766470431802433713028621) }, + { SC_(64.5), SC_(0.1582991881531276840006794903389502189858e89), SC_(203.0868048358281226106733889296294186889) }, + { SC_(65.5), SC_(0.1021029763587673561804382712686228912459e91), SC_(207.253470059629849416124244558439614861) }, + { SC_(66.5), SC_(0.6687744951499261829818706768094799376603e92), SC_(211.435520202271055650856436448150964186) }, + { SC_(67.5), SC_(0.4447350392747009116829440000783041585441e94), SC_(215.6327221499328641065535845020238208848) }, + { SC_(68.5), SC_(0.3001961515104231153859872000528553070173e96), SC_(219.8448497478113482459228474245594090702) }, + { SC_(69.5), SC_(0.2056343637846398340394012320362058853068e98), SC_(224.0716834930795278518208054310810978927) }, + { SC_(70.5), SC_(0.1429158828303246846573838562651630902882e100), SC_(228.3130102456502742995923582284984651071) }, + { SC_(71.5), SC_(0.1007556973953789026834556186669399786532e102), SC_(232.5686229554684972683913220137349879525) }, + { SC_(72.5), SC_(0.7204032363769591541867076734686208473705e103), SC_(236.8383204051684592390895209118072592891) }, + { SC_(73.5), SC_(0.5222923463732953867853630632647501143436e105), SC_(241.121906967029088331456320155937181966) }, + { SC_(74.5), SC_(0.3838848745843721092872418514995913340426e107), SC_(245.4191923732478793236450387582878905619) }, + { SC_(75.5), SC_(0.2859942315653572214189951793671955438617e109), SC_(249.7299914986333931552202349119338344817) }, + { SC_(76.5), SC_(0.2159256448318447021713413604222326356156e111), SC_(254.0541241548883721745992390899601342039) }, + { SC_(77.5), SC_(0.1651831182963611971610761407230079662459e113), SC_(258.3914148957208623282220320602201355807) }, + { SC_(78.5), SC_(0.1280169166796799277998340090603311738406e115), SC_(262.7416928320801636393347235965305038626) }, + { SC_(79.5), SC_(0.1004932795935487433228696971123599714649e117), SC_(267.1047914568685263873419367114758025432) }, + { SC_(80.5), SC_(0.7989215727687125094168140920432617731457e118), SC_(271.4805484785288126034644189659692094502) }, + { SC_(81.5), SC_(0.6431318660788135700805353440948257273823e120), SC_(275.8688056629533302899592924197644087302) }, + { SC_(82.5), SC_(0.5241524708542330596156363054372829678165e122), SC_(280.2694086832001473146069926644269820387) }, + { SC_(83.5), SC_(0.4324257884547422741828999519857584484486e124), SC_(284.6822069765407826152477086910826481146) }, + { SC_(84.5), SC_(0.3610755333597097989427214599081083044546e126), SC_(289.1070536083975924130902273463692509124) }, + { SC_(85.5), SC_(0.3051088256889547801065996336223515172642e128), SC_(293.5438051427607205757799701080417115539) }, + { SC_(86.5), SC_(0.2608680459640563369911426867471105472609e130), SC_(297.9923215187034351091622558923164399324) }, + { SC_(87.5), SC_(0.2256508597589087314973384240362506233806e132), SC_(302.4524659326412687466785241592992548335) }, + { SC_(88.5), SC_(0.1974445022890451400601711210317192954581e134), SC_(306.9241047260048374915681634477366332741) }, + { SC_(89.5), SC_(0.1747383845258049489532514421130715764804e136), SC_(311.4071072780187213241622269369206800347) }, + { SC_(90.5), SC_(0.1563908541505954293131600406911990609499e138), SC_(315.9013459032995310109227992454839056422) }, + { SC_(91.5), SC_(0.1415337230062888635284098368255351501597e140), SC_(320.4066957540054114483446541626587380014) }, + { SC_(92.5), SC_(0.1295033565507543101284950006953646623961e142), SC_(324.9230347262868870790740563815487018843) }, + { SC_(93.5), SC_(0.1197906048094477368688578756432123127164e144), SC_(329.4502433708052665886256792643481601196) }, + { SC_(94.5), SC_(0.1120042154968336339723821137264035123898e146), SC_(333.988204807099907903519925338728239387) }, + { SC_(95.5), SC_(0.1058439836445077841039010974714513192084e148), SC_(338.5368046415996049733937816714808196625) }, + { SC_(96.5), SC_(0.101081004380504933819225548085236009844e150), SC_(343.095930889086289536826499502225010241) }, + { SC_(97.5), SC_(0.9754316922718726113555265390225274949948e151), SC_(347.6654738974312297792641984341498924371) }, + { SC_(98.5), SC_(0.95104589996507579607163837554696430762e153), SC_(352.245326275435031271896458324405747818) }, + { SC_(99.5), SC_(0.9367802114655996591305637999137598430057e155), SC_(356.835382823613074469259023532110402225) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 41> near_1 = { { + { SC_(0.5), SC_(1.772453850905516027298167483341145182798), SC_(0.5723649429247000870717136756765293558236) }, + { SC_(0.625), SC_(1.434518848090556775636019739456423136632), SC_(0.3608294954889401811849576858227794878574) }, + { SC_(0.75), SC_(1.225416702465177645129098303362890526851), SC_(0.2032809514312953714814329718624296997597) }, + { SC_(0.875), SC_(1.089652357422896951252376755102892971148), SC_(0.08585870722533432350236558376948770226972) }, + { SC_(0.875), SC_(1.089652357422896951252376755102892971148), SC_(0.08585870722533432350236558376948770226972) }, + { SC_(0.9375), SC_(1.040177011186767171459762817361921128614), SC_(0.03939090173458230065822754634095384450336) }, + { SC_(0.96875), SC_(1.019032525056673950564730047928136024882), SC_(0.01885367233441289053559206570480080890174) }, + { SC_(0.984375), SC_(1.009263984715686303151364227987264939567), SC_(0.009221337197578781045045446027854805411837) }, + { SC_(0.9921875), SC_(1.004570300975031369541819081985749775618), SC_(0.004559888861804558865096599455042458443847) }, + { SC_(0.99609375), SC_(1.002269894807266338070518646683408822875), SC_(0.002267322487909119869224853055660925945503) }, + { SC_(0.998046875), SC_(1.001131154070271719475148653701632936175), SC_(0.001130514797538261731446855551493110644591) }, + { SC_(0.9990234375), SC_(1.000564631256105134179583334797007222469), SC_(0.0005644719118551233842574575277837954032273) }, + { SC_(0.99951171875), SC_(1.000282079501403060312266009234616105409), SC_(0.0002820397244605020216499093258679218227703) }, + { SC_(0.999755859375), SC_(1.000140980758729162527664293184201592009), SC_(0.0001409708218759223705137152436852900410971) }, + { SC_(0.9998779296875), SC_(1.000070475636328154358952919083132410835), SC_(0.7047315303717008615499045661742960649129e-4) }, + { SC_(0.99993896484375), SC_(1.000035234133024267207862236519063784064), SC_(0.3523351231678223923789189421658142521323e-4) }, + { SC_(0.999969482421875), SC_(1.00001761614530457531538182039987833925), SC_(0.1761599014210985977615986870219441253152e-4) }, + { SC_(0.9999847412109375), SC_(1.000008807842360071243230645907272632701), SC_(0.8807803571255486980463673710461571970668e-5) }, + { SC_(0.99999237060546875), SC_(1.000004403863608190592851185888812529877), SC_(0.4403853911211722516241836921789602523957e-5) }, + { SC_(0.999996185302734375), SC_(1.000002201917411285169225005738085227502), SC_(0.2201914987068584780432891571768087250967e-5) }, + { SC_(1), SC_(1), SC_(0) }, + { SC_(1.000003814697265625), SC_(0.9999977981113740328314320876032760067576), SC_(-0.2201891050127487637714181831108975522931e-5) }, + { SC_(1.00000762939453125), SC_(0.9999955962515330814147665236857616584616), SC_(-0.4403758163447332570221840098229107074209e-5) }, + { SC_(1.0000152587890625), SC_(0.9999911926182050168670695717547199129391), SC_(-0.8807420580197905194061157631901679899078e-5) }, + { SC_(1.000030517578125), SC_(0.99998238569695577840308912119540189507), SC_(-0.1761445817787918059338915122874759423839e-4) }, + { SC_(1.00006103515625), SC_(0.9999647732360171681023427372223702590454), SC_(-0.352273844598538899122262467277208094941e-4) }, + { SC_(1.0001220703125), SC_(0.9999295538398379138630109146622273453912), SC_(-0.7044864160936656733821449407577769386531e-4) }, + { SC_(1.000244140625), SC_(0.9998591371459403420587898072239427065107), SC_(-0.0001408727761632663509703058417841195184664) }, + { SC_(1.00048828125), SC_(0.9997183921173586652300583206642029951561), SC_(-0.0002816475415868068318239205463917072406589) }, + { SC_(1.0009765625), SC_(0.999437255220281084345213592031181905165), SC_(-0.0005629031799912046317021499216851450277288) }, + { SC_(1.001953125), SC_(0.998876391856702293840181665944397250951), SC_(-0.001124239864176365593088235002674045499535) }, + { SC_(1.00390625), SC_(0.9977602892435009045255150050890440579513), SC_(-0.002242222659961150144765481909230529929808) }, + { SC_(1.0078125), SC_(0.995550440714294209465143243929765941358), SC_(-0.004459488037952299086670078922352109198758) }, + { SC_(1.015625), SC_(0.9912190698420517341764818662191450397148), SC_(-0.008819709705733069204889229698627066380001) }, + { SC_(1.03125), SC_(0.98290109928362691478263486825456935047), SC_(-0.01724677500176806740289126202224623179737) }, + { SC_(1.0625), SC_(0.9675800675995248847599762987154317516646), SC_(-0.03295710029357781908319883575047418315706) }, + { SC_(1.125), SC_(0.9417426998497014880874037301518917030763), SC_(-0.06002318412603958293140584320743011427822) }, + { SC_(1.125), SC_(0.9417426998497014880874037301518917030763), SC_(-0.06002318412603958293140584320743011427822) }, + { SC_(1.25), SC_(0.9064024770554770779826712889669180007488), SC_(-0.09827183642181316146385380269663584022562) }, + { SC_(1.375), SC_(0.8889135691562253407424275640662446912078), SC_(-0.1177552707410787744513620333179885042465) }, + { SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 41> near_2 = { { + { SC_(1.5), SC_(0.8862269254527580136490837416705725913988), SC_(-0.1207822376352452223455184457816472122519) }, + { SC_(1.625), SC_(0.8965742800565979847725123371602644603951), SC_(-0.1091741337567953724659793453255625768435) }, + { SC_(1.75), SC_(0.9190625268488832338468237275221678951384), SC_(-0.08440112102048555595778603413139773174384) }, + { SC_(1.875), SC_(0.9534458127450348323458296607150313497544), SC_(-0.0476726853991882996439780371618622723197) }, + { SC_(1.875), SC_(0.9534458127450348323458296607150313497544), SC_(-0.0476726853991882996439780371618622723197) }, + { SC_(1.9375), SC_(0.9751659479875942232435276412768010580753), SC_(-0.02514761940298887101469636934303908362555) }, + { SC_(1.96875), SC_(0.9871877586486528896095822339303817741048), SC_(-0.01289502598016741062140421704372482102582) }, + { SC_(1.984375), SC_(0.9934942349545037046646241619249639248858), SC_(-0.006527019770560387562504065432973464109098) }, + { SC_(1.9921875), SC_(0.9967220954986639369672736204077361054958), SC_(-0.003283288599221334008087443035901123210745) }, + { SC_(1.99609375), SC_(0.9983547780306754539374306832198017571606), SC_(-0.001646576833227209223092930587911738897203) }, + { SC_(1.998046875), SC_(0.9991758197849782200230487539873719343464), SC_(-0.0008245200382650888261806366903737014930725) }, + { SC_(1.9990234375), SC_(0.9995875173583940940094860854466195201034), SC_(-0.0004125677359714894017106176239696704326377) }, + { SC_(1.99951171875), SC_(0.9997936605172715158492229105972945155141), SC_(-0.0002063607736483724433350542340289891606905) }, + { SC_(1.999755859375), SC_(0.9998968057145986134157190626438734177924), SC_(-0.0001031996102979920861817501746961436019694) }, + { SC_(1.9998779296875), SC_(0.9999483967208452041447977734631271456482), SC_(-0.5160461064981215542788033369454594603258e-4) }, + { SC_(1.99993896484375), SC_(0.9999741968262534527384281628962293685498), SC_(-0.2580350665416168648325053559023655125278e-4) }, + { SC_(1.999969482421875), SC_(0.9999870980295774847216527132886600812439), SC_(-0.1290205365365156795229453393309637521044e-4) }, + { SC_(1.9999847412109375), SC_(0.9999935489189005625751513729787156979985), SC_(-0.6451101907750591399976874019296260011728e-5) }, + { SC_(1.99999237060546875), SC_(0.9999967744354781276641509840822795646932), SC_(-0.3225569724016764801116581064181763189594e-5) }, + { SC_(1.999996185302734375), SC_(0.9999983872117460118412633921783404380531), SC_(-0.1612789554532533173009818453505550062943e-5) }, + { SC_(2), SC_(1), SC_(0) }, + { SC_(2.000003814697265625), SC_(1.00000161280024011931074434129623713986), SC_(0.161279893955840184299760314572495677042e-5) }, + { SC_(2.00000762939453125), SC_(1.000003225612466396944257297966490462185), SC_(0.322560726412023958566345675090317173141e-5) }, + { SC_(2.0000152587890625), SC_(1.000006451272877535864519325623917830418), SC_(0.6451252068164492211696167502987139983584e-5) }, + { SC_(2.000030517578125), SC_(1.000012902737534909133631207655399313341), SC_(0.1290265429530719797568036294024708700466e-4) }, + { SC_(2.00006103515625), SC_(1.000025806242196124228325546227327679105), SC_(0.2580590922078463500093254292124318294867e-4) }, + { SC_(2.0001220703125), SC_(1.000051615552953128452105520486771074315), SC_(0.5161422091631080428484087308642550680092e-4) }, + { SC_(2.000244140625), SC_(1.000103243380595112650112753954221989398), SC_(0.0001032380513640963581901732440393341809948) }, + { SC_(2.00048828125), SC_(1.000206535863509719265815185078589813025), SC_(0.0002065145379145443567131291462537026096587) }, + { SC_(2.0009765625), SC_(1.000413268164832140091644464679649856244), SC_(0.000413182793064254264258674986332041644883) }, + { SC_(2.001953125), SC_(1.000827322309547415507838270760694901832), SC_(0.000826980267085383846585814529167493000815) }, + { SC_(2.00390625), SC_(1.001657790373358329933817798077673136303), SC_(0.001656417755696172869171861186612377080916) }, + { SC_(2.0078125), SC_(1.003328178532374632976589675522967237775), SC_(0.003322652404102649860792821138784654479368) }, + { SC_(2.015625), SC_(1.00670686780833379252298939537881918096), SC_(0.006684476830232184945964816343819769497868) }, + { SC_(2.03125), SC_(1.013616758636240255869592207887524642672), SC_(0.01352488366498562096813694557452593229433) }, + { SC_(2.0625), SC_(1.028053821824495190057474817385146236144), SC_(0.02766752152285702349740729628994608012915) }, + { SC_(2.125), SC_(1.059460537330914174098329196420878165961), SC_(0.05775985153034387160738826626309159079026) }, + { SC_(2.125), SC_(1.059460537330914174098329196420878165961), SC_(0.05775985153034387160738826626309159079026) }, + { SC_(2.25), SC_(1.133003096319346347478339111208647500936), SC_(0.124871714892396594302441287613198663149) }, + { SC_(2.375), SC_(1.222256157589809843520837900591086450411), SC_(0.2006984603774558413588851802726110913487) }, + { SC_(2.5), SC_(1.329340388179137020473625612505858887098), SC_(0.2846828704729191596324946696827019243201) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_0 = { { + { SC_(-0.5), SC_(-3.544907701811032054596334966682290365595), SC_(1.265512123484645396488945797134705923899) }, + { SC_(-0.375), SC_(-3.825383594908151401696052638550461697686), SC_(1.341658748500666418041408813274783487436) }, + { SC_(-0.25), SC_(-4.901666809860710580516393213451562107405), SC_(1.589575312551185990315897214778782835911) }, + { SC_(-0.125), SC_(-8.717218859383175610019014040823143769183), SC_(2.165300248905170251754061948144017406496) }, + { SC_(-0.125), SC_(-8.717218859383175610019014040823143769183), SC_(2.165300248905170251754061948144017406496) }, + { SC_(-0.0625), SC_(-16.64283217898827474335620507779073805782), SC_(2.811979623974363538327156032173660116805) }, + { SC_(-0.03125), SC_(-32.60904080181356641807136153370035279623), SC_(3.484589575134139437621752672995683649279) }, + { SC_(-0.015625), SC_(-64.59289502180392340168731059118495613226), SC_(4.168104420557250637548438174776914213865) }, + { SC_(-0.0078125), SC_(-128.5849985248040153013528424941759712791), SC_(4.856590152781421724785721449662278434972) }, + { SC_(-0.00390625), SC_(-256.581093070660182546052773550952658656), SC_(5.54744476696747159520708182472107347055) }, + { SC_(-0.001953125), SC_(-512.5791508839791203712761106952360633215), SC_(6.239455139837046046486535948675082223324) }, + { SC_(-0.0009765625), SC_(-1024.578182406251657399893334832135395808), SC_(6.932036277511308217556578672109549476158) }, + { SC_(-0.00048828125), SC_(-2048.577698818873467519520786912493783878), SC_(7.624901025883858905611203245365810170653) }, + { SC_(-0.000244140625), SC_(-4096.57745718775464971331294488248972087), SC_(8.317907137541219635377299172741804106947) }, + { SC_(-0.0001220703125), SC_(-8192.577336412800240508542313129020709561), SC_(9.010983820432326192510172569412912814588) }, + { SC_(-0.6103515625e-4), SC_(-16384.5772760354695939336148831283410381), SC_(9.704095761351551114080487592308688534482) }, + { SC_(-0.30517578125e-4), SC_(-32768.57724584934032393443149086321342054), SC_(10.39722532438932175111825798174135071555) }, + { SC_(-0.152587890625e-4), SC_(-65536.57723075690962899636361017901925666), SC_(11.09036369676269620616269440700453555078) }, + { SC_(-0.762939453125e-5), SC_(-131072.577223210852757386190636818435916), SC_(11.78350647337298147181546230662592344689) }, + { SC_(-0.3814697265625e-5), SC_(-262144.5772194378639394013199042046138782), SC_(12.47665145199400263809495861913874999345) }, + { SC_(0.3814697265625e-5), SC_(262143.4227881080344625629331726731855155), SC_(12.47664704818796544202254047206534711638) }, + { SC_(0.762939453125e-5), SC_(131071.4227918809440471962777925401520979), SC_(11.78349766576090681276037584294890342818) }, + { SC_(0.152587890625e-4), SC_(65535.42279942668398540027145451732421438), SC_(11.09034608153854475277051988217319318753) }, + { SC_(0.30517578125e-4), SC_(32767.42281451784694671242432333092929765), SC_(10.39719009394100176207788843272141977354) }, + { SC_(0.6103515625e-4), SC_(16383.4228446989052821887834066513143242), SC_(9.704025300454774477951337474167744232248) }, + { SC_(0.0001220703125), SC_(8191.422905055952190365785412912966413445), SC_(9.010842898637679655856679364462219607288) }, + { SC_(0.000244140625), SC_(4095.423025749771641072803050389269325868), SC_(8.317625293943180446655815151656334697388) }, + { SC_(0.00048828125), SC_(2047.42326705635054639115944072028773408), SC_(7.62433733861781159675772941549355054159) }, + { SC_(0.0009765625), SC_(1023.423749345567830369498718239930270889), SC_(6.930908902419461889540619064660080535727) }, + { SC_(0.001953125), SC_(511.4247126306315744461730129635313924869), SC_(6.23720038517533141916200085812091506718) }, + { SC_(0.00390625), SC_(255.4266340463362315585318413027952788355), SC_(5.542935221819601325193091489756182014674) }, + { SC_(0.0078125), SC_(127.4304564114296588115383352230100404938), SC_(4.84757077588166486683395477128488386733) }, + { SC_(0.015625), SC_(63.43802046989131098729483943802528254174), SC_(4.150063373653938787298503499050432342073) }, + { SC_(0.03125), SC_(31.45283517707606127304431578414621921504), SC_(3.44848912779795847968326934526863660858) }, + { SC_(0.0625), SC_(15.48128108159239815615962077944690802663), SC_(2.739631621946203418585729650082232089145) }, + { SC_(0.125), SC_(7.53394159879761190469922984121513362461), SC_(2.019418357553796345320290521167099589948) }, + { SC_(0.125), SC_(7.53394159879761190469922984121513362461), SC_(2.019418357553796345320290521167099589948) }, + { SC_(0.25), SC_(3.625609908221908311930685155867672002995), SC_(1.288022524698077457370610440219717295925) }, + { SC_(0.375), SC_(2.370436184416600908646473504176652509887), SC_(0.8630739822706474624050890941340154953325) }, + { SC_(0.5), SC_(1.772453850905516027298167483341145182798), SC_(0.5723649429247000870717136756765293558236) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_m10 = { { + { SC_(-10.5), SC_(-0.2640121820547716316246385325311240439682e-6), SC_(-15.14727059071784114610117639552631963436) }, + { SC_(-10.375), SC_(-0.3853824777091100016167565620752110328498e-6), SC_(-14.76902954720701012688042720516103787762) }, + { SC_(-10.25), SC_(-0.67808180432946731304891004492754985848e-6), SC_(-14.20399790093109065161116876070387206737) }, + { SC_(-10.125), SC_(-0.1684831262052517456216882327889895477394e-5), SC_(-13.293845140389538484236785089823191703) }, + { SC_(-10.125), SC_(-0.1684831262052517456216882327889895477394e-5), SC_(-13.293845140389538484236785089823191703) }, + { SC_(-10.0625), SC_(-0.3830328107020316635889325503807551568979e-5), SC_(-12.47256009081165579039929355805785844157) }, + { SC_(-10.03125), SC_(-0.820629952953019778657392524711140975814e-5), SC_(-11.71060846332720387792243958913415617938) }, + { SC_(-10.015625), SC_(-0.1700699874643038383461557487368349913225e-4), SC_(-10.98188560766303889037266091827081352261) }, + { SC_(-10.0078125), SC_(-0.3463458256516313677243695342138690766685e-4), SC_(-10.27065787896126424410583159193311350064) }, + { SC_(-10.00390625), SC_(-0.6990332874758209033426446699035926999779e-4), SC_(-9.568397288290682386853008683681967849367) }, + { SC_(-10.001953125), SC_(-0.0001404477363900451708521851118608466504012), SC_(-8.870675121382310094805966387146254501189) }, + { SC_(-10.0009765625), SC_(-0.000281540041401883411812666338203440274148), SC_(-8.175235877509399161097817609681419866606) }, + { SC_(-10.00048828125), SC_(-0.000563726404385624063397951573975953521758), SC_(-7.480941522771659111292091779019988346536) }, + { SC_(-10.000244140625), SC_(-0.001128100008866690679009902807702968873353), SC_(-6.787220469493554911614571257238006425123) }, + { SC_(-10.0001220703125), SC_(-0.002256847657595184957233851276605048793085), SC_(-6.093786281167310238545736320487058532071) }, + { SC_(-10.00006103515625), SC_(-0.004514343175062893955072428860155326550553), SC_(-5.400495578872419834979145899538225218653) }, + { SC_(-10.000030517578125), SC_(-0.009029334320035575647839997474459714791226), SC_(-4.707276632982054173483611347195057396393) }, + { SC_(-10.0000152587890625), SC_(-0.01805931666500754906434806474181437991942), SC_(-4.01409356864116185854622778461065066607) }, + { SC_(-10.00000762939453125), SC_(-0.03611928138246679574769960801636554264536), SC_(-3.320928445911808853642423587159533484381) }, + { SC_(-10.000003814697265625), SC_(-0.07223921083114343778444263107651750106133), SC_(-2.627772294197426150094657771000510514181) }, + { SC_(-9.999996185302734375), SC_(0.07224050699108014022628046448675851453374), SC_(-2.627754351749084271591553302449947161499) }, + { SC_(-9.99999237060546875), SC_(0.03612057754240364069840471083444586728855), SC_(-3.320892561015125097640948165422843317137) }, + { SC_(-9.9999847412109375), SC_(0.01806061282494496405052242015364798732676), SC_(-4.014021798847794354581145064057511795502) }, + { SC_(-9.999969482421875), SC_(0.009030630479975270775894162659645956519741), SC_(-4.707133093395319229856390889602684453056) }, + { SC_(-9.99993896484375), SC_(0.004515639335011709650690623512369371764253), SC_(-5.400208499698950462148264857327856480671) }, + { SC_(-9.9998779296875), SC_(0.002258143817580482923824809786660261248415), SC_(-6.093212122820375608272453375481778776963) }, + { SC_(-9.999755859375), SC_(0.00112939616899791774095825334343857186485), SC_(-6.786072152799718574175843462405206935614) }, + { SC_(-9.99951171875), SC_(0.0005650225651005676902373042742306456894427), SC_(-7.478644889384249821277500306107773873963) }, + { SC_(-9.9990234375), SC_(0.000282836204451696233602080064105049188933), SC_(-8.170642610736687659976646968506313698216) }, + { SC_(-9.998046875), SC_(0.0001417439087793817388175538505551912206884), SC_(-8.861488587853743723990903404527520549863) }, + { SC_(-9.99609375), SC_(0.7119953849576511774290270246050426493188e-4), SC_(-9.550024221368402701862865053477154735263) }, + { SC_(-9.9921875), SC_(0.3593094176075641243360143967272818899195e-4), SC_(-10.23391174619552949434163600960412224542) }, + { SC_(-9.984375), SC_(0.1830395592410737084371847753712605958898e-4), SC_(-10.90839335076217169965796414369624068766) }, + { SC_(-9.96875), SC_(0.9505651717966543211040666528035462899479e-5), SC_(-11.56362401857045907442600014919001566007) }, + { SC_(-9.9375), SC_(0.5139309871979484321498376816774659227047e-5), SC_(-12.17859175366355833461560769108992933228) }, + { SC_(-9.875), SC_(0.3033137834984411836791652786842133042328e-5), SC_(-12.70591288519170921633064985576606088605) }, + { SC_(-9.875), SC_(0.3033137834984411836791652786842133042328e-5), SC_(-12.70591288519170921633064985576606088605) }, + { SC_(-9.75), SC_(0.2197547155462853887868709703817446704567e-5), SC_(-13.02816874893114553892881831326149507953) }, + { SC_(-9.625), SC_(0.2248214426276410648574460028763198846825e-5), SC_(-13.00537424512744790544056216033820032846) }, + { SC_(-9.5), SC_(0.2772127911575102132058704591576802461667e-5), SC_(-12.79589533355436345901781053661879076815) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 40> near_m55 = { { + { SC_(-55.5), SC_(0.3313939247684676728377268347296671102738e-73), SC_(-169.1931592947433577934436950412776436064) }, + { SC_(-55.375), SC_(0.5931885913251829148011431845907438457576e-73), SC_(-168.6109546898941580979033836431457031234) }, + { SC_(-55.25), SC_(0.128134265213561204650151776242601279892e-72), SC_(-167.8408033134133338381597995376176917559) }, + { SC_(-55.125), SC_(0.3913191103476296924523858848738104144085e-72), SC_(-166.7243586084319013982262850893209212539) }, + { SC_(-55.125), SC_(0.3913191103476296924523858848738104144085e-72), SC_(-166.7243586084319013982262850893209212539) }, + { SC_(-55.0625), SC_(0.986732607487198018754292852511389106856e-72), SC_(-165.7994828862238820223463651561184383051) }, + { SC_(-55.03125), SC_(0.2226660343675615375290464186909601183619e-71), SC_(-164.9856238363615162450940197678605306484) }, + { SC_(-55.015625), SC_(0.4736069454740001055763353712089973133951e-71), SC_(-164.2309191328226253115354307486462442346) }, + { SC_(-55.0078125), SC_(0.977114118293834733521912343859491871413e-71), SC_(-163.5066934315334134525852432438180103697) }, + { SC_(-55.00390625), SC_(0.1984981336813056540008261636773673389822e-70), SC_(-162.7979320905915813518737104001603528406) }, + { SC_(-55.001953125), SC_(0.4001152702583918680236220298606474660766e-70), SC_(-162.0969591073259992663458061954608696748) }, + { SC_(-55.0009765625), SC_(0.8033716579133006533597549451686567408424e-70), SC_(-161.399894344940666691934457343256688548) }, + { SC_(-55.00048828125), SC_(0.1609895558223411179719291838965309144853e-69), SC_(-160.7047872033597077904481004857770657217) }, + { SC_(-55.000244140625), SC_(0.3222948938373906907742658942729222148632e-69), SC_(-160.0106597497627978468127607946271352327) }, + { SC_(-55.0001220703125), SC_(0.6449058492709206027088103024110200975353e-69), SC_(-159.3170223595527956784727838139907674684) }, + { SC_(-55.00006103515625), SC_(0.1290127899946570906748117205135768082314e-68), SC_(-158.6236300558849309281660984198839415596) }, + { SC_(-55.000030517578125), SC_(0.2580572071228903781598785159467838581806e-68), SC_(-157.9303603092003033255947568117785799832) }, + { SC_(-55.0000152587890625), SC_(0.5161460448765772204325064962419091742529e-68), SC_(-157.2371518444353362397006108889220356827) }, + { SC_(-55.00000762939453125), SC_(0.1032323722132728230665906965174285853426e-67), SC_(-156.5439740214872098856520824742320518588) }, + { SC_(-55.000003814697265625), SC_(0.2064679077519460715554797358664728667652e-67), SC_(-155.8508115196617565149201605014267824236) }, + { SC_(-54.999996185302734375), SC_(-0.2064742345776389872444192676691131506575e-67), SC_(-155.8507808769879053144034254273344043224) }, + { SC_(-54.99999237060546875), SC_(-0.1032386990389669331883069223703195417382e-67), SC_(-156.543912736139507484654652677755511734) }, + { SC_(-54.9999847412109375), SC_(-0.5162093131335660989607568848213731056733e-68), SC_(-157.2370292737399314379940741096346901828) }, + { SC_(-54.999969482421875), SC_(-0.258120475380070365932807313377307528986e-68), SC_(-157.9301151678094937244882657625299630878) }, + { SC_(-54.99993896484375), SC_(-0.129076058252601515431668371755594450098e-68), SC_(-158.6231397731033117444057763960015760334) }, + { SC_(-54.9998779296875), SC_(-0.6455385318809423304687659289501264852653e-69), SC_(-159.3160417939895574585734203633457962732) }, + { SC_(-54.999755859375), SC_(-0.3229275765697223526481784197301709161285e-69), SC_(-160.0086986186363225879842786767213113941) }, + { SC_(-54.99951171875), SC_(-0.1616222390439127298761439614470246404862e-69), SC_(-160.70086494110676672055309472267163668) }, + { SC_(-54.9990234375), SC_(-0.8096985096986489455731873769112931053315e-70), SC_(-161.3920498204348601342401201789602276343) }, + { SC_(-54.998046875), SC_(-0.4064422003228156070372890110113661523823e-70), SC_(-162.0812700583149908077227373306519861132) }, + { SC_(-54.99609375), SC_(-0.2048253768707794232109022149495560465895e-70), SC_(-162.7665539925744016887591545718340828031) }, + { SC_(-54.9921875), SC_(-0.1040399076590405899054222579682253865197e-70), SC_(-163.4439372355377521596244089121137210432) }, + { SC_(-54.984375), SC_(-0.5369420317824801296090020359957584504561e-71), SC_(-164.1054067411408869986598886088213583004) }, + { SC_(-54.96875), SC_(-0.2862019916619447982849309567142574291595e-71), SC_(-164.7345990554747140245090340482576614581) }, + { SC_(-54.9375), SC_(-0.1630184748946170270451320476784206632433e-71), SC_(-165.2974333442636798880161449262335724888) }, + { SC_(-54.875), SC_(-0.1068084665867092473960298264876009169786e-71), SC_(-165.7202596830189416811721153572453840111) }, + { SC_(-54.875), SC_(-0.1068084665867092473960298264876009169786e-71), SC_(-165.7202596830189416811721153572453840111) }, + { SC_(-54.75), SC_(-0.9545836185625177195542164628822374598687e-72), SC_(-165.8326067306542059861208673933569437798) }, + { SC_(-54.625), SC_(-0.1206186475649396145903288691554707530168e-71), SC_(-165.5986629859610336595561650158366255984) }, + { SC_(-54.5), SC_(-0.183923628246499558424938393274965246202e-71), SC_(-165.1767762739909689670975862547818473059) } + } }; +#undef SC_ + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 141> gammap1m1_data = { { + { SC_(-0.4952165186405181884765625), SC_(0.7559827693907095754807809442951050489732) }, + { SC_(-0.4642883241176605224609375), SC_(0.6574328869566978138139138799311066062094) }, + { SC_(-0.4024595916271209716796875), SC_(0.4948624198600628575485791492257182331098) }, + { SC_(-0.3901382386684417724609375), SC_(0.466994695902624837582482771934607569504) }, + { SC_(-0.3875354826450347900390625), SC_(0.4612760821854033810460931227828015343156) }, + { SC_(-0.373013198375701904296875), SC_(0.4303940312894863907637421435297095229576) }, + { SC_(-0.364522993564605712890625), SC_(0.4131121158463471061555369932726425070959) }, + { SC_(-0.35811364650726318359375), SC_(0.4004260060282522370607617707987066040664) }, + { SC_(-0.342386901378631591796875), SC_(0.3705491389143732767395411311340356537958) }, + { SC_(-0.311618030071258544921875), SC_(0.3168394199636867612664810682360364987851) }, + { SC_(-0.2880756855010986328125), SC_(0.2795634828731819295069040561405863199118) }, + { SC_(-0.27896595001220703125), SC_(0.2659494215075372040950689557832739194382) }, + { SC_(-0.221501767635345458984375), SC_(0.1892800230490205183297725744929501015544) }, + { SC_(-0.202970564365386962890625), SC_(0.1675837744862475402389911769714649852773) }, + { SC_(-0.191832959651947021484375), SC_(0.1551782986717939911912971517092154248689) }, + { SC_(-0.1387059986591339111328125), SC_(0.1019358076187795495751950704491293377337) }, + { SC_(-0.1012614667415618896484375), SC_(0.06964849559574637322009665674759450635903) }, + { SC_(-0.0782387256622314453125), SC_(0.05168944861403749320872745190411594799105) }, + { SC_(-0.0146243572235107421875), SC_(0.008655823217365420051945171708362679166147) }, + { SC_(0.1431564604442703013402649929484277542953e-29), SC_(-0.8263215150028947028222714725368343067077e-30) }, + { SC_(0.1791466932348087634896446282571611213266e-29), SC_(-0.1034062776504410791508686126702507453693e-29) }, + { SC_(0.6013619202535540063110633226832922483532e-29), SC_(-0.3471155206456177563387123499681270341265e-29) }, + { SC_(0.115805324961653822428570241697281798758e-28), SC_(-0.6684464764687909153739175517973613571261e-29) }, + { SC_(0.1422457400834001098175711728787848259007e-28), SC_(-0.8210646944165041873250036406993552111264e-29) }, + { SC_(0.4970121018327539153628705477876439795096e-28), SC_(-0.2868831708235014101261168518349749234889e-28) }, + { SC_(0.9660079415057497591758174164417478444323e-28), SC_(-0.5575949162564024099347963251548294827165e-28) }, + { SC_(0.1232929313253182131376331095427391968754e-27), SC_(-0.7116661133260258147780879126862696392479e-28) }, + { SC_(0.3296523285617759312781860549364832953326e-27), SC_(-0.1902804880171240659658521971091414147824e-27) }, + { SC_(0.528364435768055252017009628713605422886e-27), SC_(-0.3049802291021812635024061181080113889506e-27) }, + { SC_(0.886586057273120049620324386849842094685e-27), SC_(-0.5117513605413324831805933701656815521236e-27) }, + { SC_(0.2499669674831043259218157022821422146034e-26), SC_(-0.1442848493391799075204112945868205575921e-26) }, + { SC_(0.4131050397232622964314362671638736040881e-26), SC_(-0.2384507001780369908735157814349257650306e-26) }, + { SC_(0.7679738097881433551381658732998641759182e-26), SC_(-0.4432865132438264916724504940164877771849e-26) }, + { SC_(0.199929739820949207249437007767740538737e-25), SC_(-0.1154025777043396680338534232061371984153e-25) }, + { SC_(0.5151477415246978459754129800826163591626e-25), SC_(-0.2973513461467014566158126478567428007026e-25) }, + { SC_(0.101200734533556026342258477595279955025e-24), SC_(-0.5841464927231005972586520769367701801272e-25) }, + { SC_(0.2064292695896540981798546456623054911033e-24), SC_(-0.1191542081013299677372796818685763621433e-24) }, + { SC_(0.4063294332896333395257434433879773416284e-24), SC_(-0.2345397140053387487331153502526122099635e-24) }, + { SC_(0.8138195767936862452966745688936976428456e-24), SC_(-0.4697494081288516881709792363579127898501e-24) }, + { SC_(0.9575550627132253801929510132578249716542e-24), SC_(-0.5527157822038433842858279196451987644551e-24) }, + { SC_(0.2855160956298500804375620841706273850616e-23), SC_(-0.1648043629790735548426012109835882111878e-23) }, + { SC_(0.65201444297915461398563707001320281266e-23), SC_(-0.3763529502296153146061378052316774680657e-23) }, + { SC_(0.1310988374636350038320977491775043421995e-22), SC_(-0.7567230263439006455136418784154172627278e-23) }, + { SC_(0.2590288837798696209228010176465529547374e-22), SC_(-0.1495155293796993236481538654450094828034e-22) }, + { SC_(0.2937779542193655202274099291941187976629e-22), SC_(-0.1695732371781431498672748630538267989951e-22) }, + { SC_(0.7863513178004503049754083414326234074965e-22), SC_(-0.4538942987503834952636621253829968594981e-22) }, + { SC_(0.1903818607087388763706780167350761726053e-21), SC_(-0.109891392314185724619218859435147914583e-21) }, + { SC_(0.3812242142377350870566942975497647799754e-21), SC_(-0.2200485882977986685255989911406099731464e-21) }, + { SC_(0.5493133580141330277178034419485741501887e-21), SC_(-0.3170722751854215599980143372792630827825e-21) }, + { SC_(0.9672153634284186955666772243312215295852e-21), SC_(-0.5582918591043124472447977064584587087382e-21) }, + { SC_(0.1702169477623814384559878647986894129041e-20), SC_(-0.9825188868017248805929057833465618943933e-21) }, + { SC_(0.4817114569977399785676754474621208412799e-20), SC_(-0.2780513989416366360400948471972900830387e-20) }, + { SC_(0.7538352992756463183303278501219690799218e-20), SC_(-0.4351255434976382024407146526549699565796e-20) }, + { SC_(0.2596305715949999708394617609422128090557e-19), SC_(-0.1498628330119729391523576917959646915902e-19) }, + { SC_(0.4444587480324321591032923385589104015025e-19), SC_(-0.2565485517668431887674899070788880246089e-19) }, + { SC_(0.9715574921498573937069095571295029856174e-19), SC_(-0.5607982038213457280620337061143959968827e-19) }, + { SC_(0.2036598542733453787268262970278076551267e-18), SC_(-0.1175556581981383412558675525031327289634e-18) }, + { SC_(0.4248971931658660264162106698360155121463e-18), SC_(-0.2452573158680304024128136998709195441154e-18) }, + { SC_(0.6521097487613458963613731825259556273977e-18), SC_(-0.3764079622200518159115220576707372289309e-18) }, + { SC_(0.1436126164096190058281493628911107407475e-17), SC_(-0.8289545186912702312307596583951073368993e-18) }, + { SC_(0.3118908901459261162419055180006211003274e-17), SC_(-0.1800283075323116855097674712567357424907e-17) }, + { SC_(0.3593346613595175715618300349429858897565e-17), SC_(-0.2074135954788010816821644850649799519675e-17) }, + { SC_(0.9445874854124767215374919304693435151421e-17), SC_(-0.5452306934500297136990208691880314661067e-17) }, + { SC_(0.2566182432094081539023303073498993853718e-16), SC_(-0.1481240698799817909729203530750507585117e-16) }, + { SC_(0.3363765695149349330660137891158001366421e-16), SC_(-0.1941618252298598450712101459398214835952e-16) }, + { SC_(0.1073581901339262605326457800103412409953e-15), SC_(-0.6196882910077942026172170471644452240087e-16) }, + { SC_(0.186668406231853462907965823802669547149e-15), SC_(-0.1077479282192287023344331697438668504475e-15) }, + { SC_(0.3727540802657755688795382376099496468669e-15), SC_(-0.2151594942853688563255514533064804448857e-15) }, + { SC_(0.6211646767866855090717281839829411183018e-15), SC_(-0.3585459819247720488263755672803206966864e-15) }, + { SC_(0.1561186859754253464932505224282976996619e-14), SC_(-0.9011415112885851355703262394946230181441e-15) }, + { SC_(0.3092010764722992466335682593125966377556e-14), SC_(-0.1784757049442269726369719145429686196432e-14) }, + { SC_(0.6192850577371690132255643845837767003104e-14), SC_(-0.3574610363653403859138387348420733335946e-14) }, + { SC_(0.1047879028014987723427253740737796761096e-13), SC_(-0.6048521898920322580919537460083385958307e-14) }, + { SC_(0.1978473638988408750405412206418986897916e-13), SC_(-0.1142005977018810929368580238123776497134e-13) }, + { SC_(0.4041816252346730475863978426787070930004e-13), SC_(-0.2332999655507978182935820166341591465938e-13) }, + { SC_(0.9410302262901834580155480125540634617209e-13), SC_(-0.5431773877604405885692410233977439758357e-13) }, + { SC_(0.1334530223958893535574077304772799834609e-12), SC_(-0.7703117505534481432167486795079037095016e-13) }, + { SC_(0.266297021326439287136622624529991298914e-12), SC_(-0.1537108122261681912683838906901126667239e-12) }, + { SC_(0.5920415525016708979677559909760020673275e-12), SC_(-0.3417356583762410657228951913532598329392e-12) }, + { SC_(0.155163989296047688526414276566356420517e-11), SC_(-0.8956308525005437046951559037146364454014e-12) }, + { SC_(0.326923297461201300961874949280172586441e-11), SC_(-0.1887052485148118271327652592003779278951e-11) }, + { SC_(0.3753785910581841633870681107509881258011e-11), SC_(-0.2166744030260566997416579153839836535144e-11) }, + { SC_(0.9579165585749116473834874341264367103577e-11), SC_(-0.5529244432689301568675461523208969657637e-11) }, + { SC_(0.1858167439361402273334533674642443656921e-10), SC_(-0.1072563353975220567027440177997243696045e-10) }, + { SC_(0.5449485307451595872407779097557067871094e-10), SC_(-0.3145528284818088265306644806429650505592e-10) }, + { SC_(0.6089519166696533147842274047434329986572e-10), SC_(-0.3514965854368603547185748339618264962e-10) }, + { SC_(0.1337744776064297980155970435589551925659e-9), SC_(-0.7721672402075083279577276691763777798079e-10) }, + { SC_(0.2554458866654840676346793770790100097656e-9), SC_(-0.1474473672534405166880467009401481006846e-9) }, + { SC_(0.9285605062636648199259070679545402526855e-9), SC_(-0.5359796691714968347006887450221011388011e-9) }, + { SC_(0.1698227447555211711005540564656257629395e-8), SC_(-0.980243482442200345890191122898173278899e-9) }, + { SC_(0.339355921141759608872234821319580078125e-8), SC_(-0.1958815525210918994679361737180445367869e-8) }, + { SC_(0.6313728651008432279922999441623687744141e-8), SC_(-0.3644383041872783822741274272616254061398e-8) }, + { SC_(0.8383264749056706932606175541877746582031e-8), SC_(-0.4838951666662356901644608730186546413827e-8) }, + { SC_(0.1962631124285962869180366396903991699219e-7), SC_(-0.1132861391263510827635198807950784606364e-7) }, + { SC_(0.5256384838503436185419559478759765625e-7), SC_(-0.3034067396263077516870264512967026960836e-7) }, + { SC_(0.116242290459922514855861663818359375e-6), SC_(-0.6709685761311096545819618213723265238395e-7) }, + { SC_(0.1776920584006802528165280818939208984375e-6), SC_(-0.1025666084085592538025029643343934584008e-6) }, + { SC_(0.246631174150024889968335628509521484375e-6), SC_(-0.142359317011220184093241753035251343271e-6) }, + { SC_(0.7932688959044753573834896087646484375e-6), SC_(-0.4578866108069128537168607745624614794722e-6) }, + { SC_(0.1372093493046122603118419647216796875e-5), SC_(-0.791991995861101927407880386227298092424e-6) }, + { SC_(0.214747751670074649155139923095703125e-5), SC_(-0.1239553101482841564081262476002391564028e-5) }, + { SC_(0.527022712049074470996856689453125e-5), SC_(-0.3042030180348038757030022358659056016131e-5) }, + { SC_(0.9233162927557714283466339111328125e-5), SC_(-0.5329441960781667799420300524501852822265e-5) }, + { SC_(0.269396477960981428623199462890625e-4), SC_(-0.1554926893050895772896453957379474045684e-4) }, + { SC_(0.3208058114978484809398651123046875e-4), SC_(-0.1851639610824637542072201121307897455015e-4) }, + { SC_(0.00010957030463032424449920654296875), SC_(-0.6323382317252073303439803607481865268353e-4) }, + { SC_(0.000126518702018074691295623779296875), SC_(-0.7301274674392621832822527575061291232133e-4) }, + { SC_(0.00028976381872780621051788330078125), SC_(-0.0001671731931845284705043154051351730706298) }, + { SC_(0.000687857042066752910614013671875), SC_(-0.0003965741858362509385654340724912705823236) }, + { SC_(0.00145484809763729572296142578125), SC_(-0.0008376704829300962209037156359622989708524) }, + { SC_(0.00366270542144775390625), SC_(-0.002100946766981816464155333982357752904999) }, + { SC_(0.046881496906280517578125), SC_(-0.02497588947336944943591732868959193811385) }, + { SC_(0.04722058773040771484375), SC_(-0.02514197077286474061941968460870171431946) }, + { SC_(0.1323592662811279296875), SC_(-0.06091072639354085529687250076608036167983) }, + { SC_(0.139763355255126953125), SC_(-0.06350237679012056526225278765134891026831) }, + { SC_(0.155740678310394287109375), SC_(-0.06883441875617310404657299803607175885806) }, + { SC_(0.17873513698577880859375), SC_(-0.07590347542832235360535014668688262042636) }, + { SC_(0.1813595294952392578125), SC_(-0.07666619009216679212771233211297326572625) }, + { SC_(0.225838959217071533203125), SC_(-0.08828121583826801836069461556215827733624) }, + { SC_(0.292207300662994384765625), SC_(-0.1013142147509511845410398110728244632327) }, + { SC_(0.297928631305694580078125), SC_(-0.1022125377703106514269725748156778633457) }, + { SC_(0.29810583591461181640625), SC_(-0.1022398124529602122123098193784073635017) }, + { SC_(0.30028045177459716796875), SC_(-0.1025718475377491669113360177547935208456) }, + { SC_(0.314723670482635498046875), SC_(-0.1046527148786850527567817015912636515928) }, + { SC_(0.335008561611175537109375), SC_(-0.1072166100764849221291653907100739024018) }, + { SC_(0.34912931919097900390625), SC_(-0.1087597918806720999316678995289547437333) }, + { SC_(0.378430664539337158203125), SC_(-0.1113472284261879798379642546535625292991) }, + { SC_(0.405791938304901123046875), SC_(-0.1130363541503776146472634379575969350382) }, + { SC_(0.4133758544921875), SC_(-0.1133834162443148976453781329005327033269) }, + { SC_(0.415735542774200439453125), SC_(-0.1134808289122708542397946052703635909288) }, + { SC_(0.43399322032928466796875), SC_(-0.1140666251072475325460987089801937766512) }, + { SC_(0.457166969776153564453125), SC_(-0.1143882508090213200510081514983929178479) }, + { SC_(0.457506835460662841796875), SC_(-0.1143895043011533114503086722896019260013) }, + { SC_(0.4594924449920654296875), SC_(-0.1143948425572877686750521180317509748309) }, + { SC_(0.464888513088226318359375), SC_(-0.1143922664386800836288584840918904877633) }, + { SC_(0.467694938182830810546875), SC_(-0.1143810844126184461958335384166685988825) }, + { SC_(0.468867778778076171875), SC_(-0.114374421494214380845931849419009841586) }, + { SC_(0.470592796802520751953125), SC_(-0.1143624939837521390307936225841266176506) }, + { SC_(0.481109678745269775390625), SC_(-0.1142351916017016455044428590981136786372) }, + { SC_(0.492881298065185546875), SC_(-0.1139822218283491394475161618135268104636) }, + { SC_(0.496461331844329833984375), SC_(-0.1138823102618022293565213085801050842487) } + } }; +#undef SC_ + + diff --git a/test/test_gamma_dist.cpp b/test/test_gamma_dist.cpp new file mode 100644 index 000000000..0023de948 --- /dev/null +++ b/test/test_gamma_dist.cpp @@ -0,0 +1,256 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_gamma_dist.cpp + +// http://en.wikipedia.org/wiki/Gamma_distribution +// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm +// Also: +// Weisstein, Eric W. "Gamma Distribution." +// From MathWorld--A Wolfram Web Resource. +// http://mathworld.wolfram.com/GammaDistribution.html + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::gamma_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + +template +RealType NaivePDF(RealType shape, RealType scale, RealType x) +{ + // Deliberately naive PDF calculator again which + // we'll compare our pdf function. However some + // published values to compare against would be better.... + using namespace std; + RealType result = log(x) * (shape - 1) - x / scale - boost::math::lgamma(shape) - log(scale) * shape; + return exp(result); +} + +template +void check_gamma(RealType shape, RealType scale, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + gamma_distribution(shape, scale), // distribution. + x), // random variable. + p, // probability. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement( + gamma_distribution(shape, scale), // distribution. + x)), // random variable. + q, // probability complement. + tol); // %tolerance. + if(p < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + gamma_distribution(shape, scale), // distribution. + p), // probability. + x, // random variable. + tol); // %tolerance. + } + if(q < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement( + gamma_distribution(shape, scale), // distribution. + q)), // probability complement. + x, // random variable. + tol); // %tolerance. + } + // PDF: + BOOST_CHECK_CLOSE( + boost::math::pdf( + gamma_distribution(shape, scale), // distribution. + x), // random variable. + NaivePDF(shape, scale, x), // PDF + tol); // %tolerance. +} + +template +void test_spots(RealType) +{ + // Basic sanity checks + // + // 15 decimal places expressed as a persentage. + // The first tests use values generated by MathCAD, + // and should be accurate to around double precision. + // + RealType tolerance = (std::max)(5e-14f, boost::math::tools::real_cast(std::numeric_limits::epsilon() * 20)) * 100; + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + check_gamma( + static_cast(0.5), + static_cast(1), + static_cast(0.5), + static_cast(0.682689492137085), + static_cast(1-0.682689492137085), + tolerance); + check_gamma( + static_cast(2), + static_cast(1), + static_cast(0.5), + static_cast(0.090204010431050), + static_cast(1-0.090204010431050), + tolerance); + check_gamma( + static_cast(40), + static_cast(1), + static_cast(10), + static_cast(7.34163631456064E-13), + static_cast(1-7.34163631456064E-13), + tolerance); + + // + // Some more test data generated by the online + // calculator at http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm + // This has the advantage of supporting the scale parameter as well + // as shape, but has only a few digits accuracy, and produces + // some deeply suspect values if the shape parameter is < 1 + // (it doesn't agree with MathCAD or this implementation). + // To be fair the incomplete gamma is tricky to get right in this area... + // + tolerance = 1e-5f * 100; // 5 decimal places as a persentage + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + check_gamma( + static_cast(2), + static_cast(1)/5, + static_cast(0.1), + static_cast(0.090204), + static_cast(1-0.090204), + tolerance); + check_gamma( + static_cast(2), + static_cast(1)/5, + static_cast(0.5), + static_cast(1-0.287298), + static_cast(0.287298), + tolerance); + check_gamma( + static_cast(3), + static_cast(2), + static_cast(1), + static_cast(0.014388), + static_cast(1-0.014388), + tolerance * 10); // one less decimal place in the test value + check_gamma( + static_cast(3), + static_cast(2), + static_cast(5), + static_cast(0.456187), + static_cast(1-0.456187), + tolerance); + + + RealType tol2 = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a persentage + gamma_distribution dist(8, 3); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(8*3), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(8*3*3), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , sqrt(static_cast(8*3*3)), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol2); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(7 * 3), tol2); + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , 2 / sqrt(static_cast(8)), tol2); + // kertosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , 3 + 6 / static_cast(8), tol2); + // kertosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , 6 / static_cast(8), tol2); + + BOOST_CHECK_CLOSE( + median(dist), static_cast(23.007748327502412), // double precision test value + (std::max)(tol2, static_cast(std::numeric_limits::epsilon() * 2 * 100))); // 2 eps as persent + // Rely on default definition in derived accessors. + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_gamma_dist.exe" +Running 1 test case... +Tolerance for type float is 0.000238419 % +Tolerance for type float is 0.001 % +Tolerance for type double is 5e-012 % +Tolerance for type double is 0.001 % +Tolerance for type long double is 5e-012 % +Tolerance for type long double is 0.001 % +Tolerance for type class boost::math::concepts::real_concept is 5e-012 % +Tolerance for type class boost::math::concepts::real_concept is 0.001 % +*** No errors detected + +*/ + + diff --git a/test/test_gamma_hooks.hpp b/test/test_gamma_hooks.hpp new file mode 100644 index 000000000..f33fd9a2f --- /dev/null +++ b/test/test_gamma_hooks.hpp @@ -0,0 +1,176 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_GAMMA_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_GAMMA_OTHER_HOOKS_HPP + +#ifdef TEST_CEPHES +namespace other{ +extern "C" { + double gamma(double); + float gammaf(float); + long double gammal(long double); + double lgam(double); + float lgamf(float); + long double lgaml(long double); + float igamf(float, float); + double igam(double, double); + long double igaml(long double, long double); + float igamcf(float, float); + double igamc(double, double); + long double igamcl(long double, long double); +} +inline float tgamma(float x) +{ return gammaf(x); } +inline double tgamma(double x) +{ return gamma(x); } +inline long double tgamma(long double x) +{ +#ifdef BOOST_MSVC + return gamma((double)x); +#else + return gammal(x); +#endif +} +inline float lgamma(float x) +{ return lgamf(x); } +inline double lgamma(double x) +{ return lgam(x); } +inline long double lgamma(long double x) +{ +#ifdef BOOST_MSVC + return lgam((double)x); +#else + return lgaml(x); +#endif +} +inline float gamma_q(float x, float y) +{ return igamcf(x, y); } +inline double gamma_q(double x, double y) +{ return igamc(x, y); } +inline long double gamma_q(long double x, long double y) +{ +#ifdef BOOST_MSVC + return igamc((double)x, (double)y); +#else + return igamcl(x, y); +#endif +} +inline float gamma_p(float x, float y) +{ return igamf(x, y); } +inline double gamma_p(double x, double y) +{ return igam(x, y); } +inline long double gamma_p(long double x, long double y) +{ +#ifdef BOOST_MSVC + return igam((double)x, (double)y); +#else + return igaml(x, y); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_NATIVE +#include +namespace other{ +#if defined(__FreeBSD__) +// no float version: +inline float tgamma(float x) +{ return ::tgamma(x); } +#else +inline float tgamma(float x) +{ return ::tgammaf(x); } +#endif +inline double tgamma(double x) +{ return ::tgamma(x); } +inline long double tgamma(long double x) +{ +#if defined(__CYGWIN__) || defined(__FreeBSD__) + // no long double versions: + return ::tgamma(x); +#else + return ::tgammal(x); +#endif +} +#if defined(__FreeBSD__) +inline float lgamma(float x) +{ return ::lgamma(x); } +#else +inline float lgamma(float x) +{ return ::lgammaf(x); } +#endif +inline double lgamma(double x) +{ return ::lgamma(x); } +inline long double lgamma(long double x) +{ +#if defined(__CYGWIN__) || defined(__FreeBSD__) + // no long double versions: + return ::lgamma(x); +#else + return ::lgammal(x); +#endif +} +} +#define TEST_OTHER +#endif + +#ifdef TEST_GSL +#define TEST_OTHER +#include + +namespace other{ +float tgamma(float z) +{ + return (float)gsl_sf_gamma(z); +} +double tgamma(double z) +{ + return gsl_sf_gamma(z); +} +long double tgamma(long double z) +{ + return gsl_sf_gamma(z); +} +float lgamma(float z) +{ + return (float)gsl_sf_lngamma(z); +} +double lgamma(double z) +{ + return gsl_sf_lngamma(z); +} +long double lgamma(long double z) +{ + return gsl_sf_lngamma(z); +} +inline float gamma_q(float x, float y) +{ return (float)gsl_sf_gamma_inc_Q(x, y); } +inline double gamma_q(double x, double y) +{ return gsl_sf_gamma_inc_Q(x, y); } +inline long double gamma_q(long double x, long double y) +{ return gsl_sf_gamma_inc_Q(x, y); } +inline float gamma_p(float x, float y) +{ return (float)gsl_sf_gamma_inc_P(x, y); } +inline double gamma_p(double x, double y) +{ return gsl_sf_gamma_inc_P(x, y); } +inline long double gamma_p(long double x, long double y) +{ return gsl_sf_gamma_inc_P(x, y); } +} +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept tgamma(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept lgamma(boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept gamma_q(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept gamma_p(boost::math::concepts::real_concept, boost::math::concepts::real_concept){ return 0; } +} +#endif + +#endif + + diff --git a/test/test_hermite.cpp b/test/test_hermite.cpp new file mode 100644 index 000000000..6e207376b --- /dev/null +++ b/test/test_hermite.cpp @@ -0,0 +1,207 @@ +// Copyright John Maddock 2006, 2007 +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4127) // conditional expression is constant +# pragma warning(disable : 4512) // assignment operator could not be generated +# pragma warning(disable : 4756) // overflow in constant arithmetic +// Constants are too big for float case, but this doesn't matter for test. +#endif + +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_legendre_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Hermite polynomials. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + "boost::math::hermite", 10, 5); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + "boost::math::hermite", 10, 5); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_hermite(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::hermite; +#else + pg funcp = boost::math::hermite; +#endif + + typedef unsigned (*cast_t)(value_type); + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test hermite against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::hermite", test_name); + + std::cout << std::endl; +} + +template +void test_hermite(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "hermite.ipp" + + do_test_hermite(hermite, name, "Hermite Polynomials"); +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 100 epsilon: + // These spots were generated by MathCAD, precision is + // 14-16 digits. + // + T tolerance = (std::max)(boost::math::tools::epsilon() * 100, static_cast(1e-14)); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(0, static_cast(1)), static_cast(1.L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(1)), static_cast(2.L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(2)), static_cast(4.L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(10)), static_cast(20), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(100)), static_cast(200), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(1e6)), static_cast(2e6), tolerance); + if(std::numeric_limits::max_exponent >= std::numeric_limits::max_exponent) + { + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(1, static_cast(1e307)), static_cast(2e307), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(99, static_cast(100)), static_cast(4.967223743011310E+227L), tolerance); + } + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(10, static_cast(30)), static_cast(5.896624628001300E+17L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(10, static_cast(1000)), static_cast(1.023976960161280E+33L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(10, static_cast(10)), static_cast(8.093278209760000E+12L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(10, static_cast(-10)), static_cast(8.093278209760000E+12L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(3, static_cast(-10)), static_cast(-7.880000000000000E+3L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(3, static_cast(-1000)), static_cast(-7.999988000000000E+9L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::hermite(3, static_cast(-1000000)), static_cast(-7.999999999988000E+18L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + + boost::math::hermite(51, 915.0); + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_spots(0.0F, "float"); +#endif + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif + + expected_results(); + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_hermite(0.1F, "float"); +#endif + test_hermite(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_hermite(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_hermite(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_ibeta.cpp b/test/test_ibeta.cpp new file mode 100644 index 000000000..40a4611f7 --- /dev/null +++ b/test/test_ibeta.cpp @@ -0,0 +1,582 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_beta_hooks.hpp" +#include "handle_test_result.hpp" + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete beta functions beta, +// betac, ibeta and ibetac. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Darwin: just one special case for real_concept: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + "real_concept", // test type(s) + "(?i).*large.*", // test data group + ".*", 400000, 50000); // test function + + // + // Linux - results depend quite a bit on the + // processor type, and how good the std::pow + // function is for that processor. + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + largest_type, // test type(s) + "(?i).*small.*", // test data group + ".*", 350, 100); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + largest_type, // test type(s) + "(?i).*medium.*", // test data group + ".*", 300, 80); // test function + // + // Deficiencies in pow function really kick in here for + // large arguments. Note also that the tests here get + // *very* extreme due to the increased exponent range + // of 80-bit long doubles. Also effect Mac OS. + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux|Mac OS", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 200000, 10000); // test function +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux|Mac OS|Sun.*", // platform + "double", // test type(s) + "(?i).*large.*", // test data group + ".*", 40, 20); // test function +#endif + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux|Mac OS", // platform + "real_concept", // test type(s) + "(?i).*medium.*", // test data group + ".*", 350, 100); // test function + + // + // HP-UX: + // + // Large value tests include some with *very* extreme + // results, thanks to the large exponent range of + // 128-bit long doubles. + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 200000, 10000); // test function + // + // Tru64: + // + add_expected_result( + ".*Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 130000, 10000); // test function + // + // Sun OS: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 130000, 10000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "(?i).*small.*", // test data group + ".*", 130, 30); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "(?i).*medium.*", // test data group + ".*", 200, 40); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + "real_concept", // test type(s) + "(?i).*medium.*", // test data group + ".*", 200, 40); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + "real_concept", // test type(s) + "(?i).*small.*", // test data group + ".*", 130, 30); // test function + // + // MinGW: + // + add_expected_result( + "[^|]*mingw[^|]*", // compiler + "[^|]*", // stdlib + ".*", // platform + "double", // test type(s) + "(?i).*large.*", // test data group + ".*", 20, 10); // test function + add_expected_result( + "[^|]*mingw[^|]*", // compiler + "[^|]*", // stdlib + ".*", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 200000, 10000); // test function + +#ifdef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + // + // No long doubles: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + BOOST_PLATFORM, // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 13000, 500); // test function +#endif + // + // Catch all cases come last: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "(?i).*small.*", // test data group + ".*", 60, 10); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "(?i).*medium.*", // test data group + ".*", 150, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "(?i).*large.*", // test data group + ".*", 5000, 500); // test function + + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "(?i).*small.*", // test data group + ".*", 60, 15); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "(?i).*medium.*", // test data group + ".*", 200, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "(?i).*large.*", // test data group + ".*", 200000, 50000); // test function + + // catch all default is 2eps for all types: + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "[^|]*", // test type(s) + "[^|]*", // test data group + ".*", 2, 2); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_beta(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::beta; +#else + pg funcp = boost::math::beta; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test beta against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::beta", test_name); + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::betac; +#else + funcp = boost::math::betac; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::betac", test_name); + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibeta; +#else + funcp = boost::math::ibeta; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(5)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibeta", test_name); + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibetac; +#else + funcp = boost::math::ibetac; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(6)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibetac", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::ibeta; + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(5)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::ibeta"); + } +#endif + std::cout << std::endl; +} + +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // +# include "ibeta_small_data.ipp" + + do_test_beta(ibeta_small_data, name, "Incomplete Beta Function: Small Values"); + +# include "ibeta_data.ipp" + + do_test_beta(ibeta_data, name, "Incomplete Beta Function: Medium Values"); + +# include "ibeta_large_data.ipp" + + do_test_beta(ibeta_large_data, name, "Incomplete Beta Function: Large and Diverse Values"); + +# include "ibeta_int_data.ipp" + + do_test_beta(ibeta_int_data, name, "Incomplete Beta Function: Small Integer Values"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 30 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 3000; + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(159) / 10000, //(0.015964560210704803L), + static_cast(1184) / 1000000000L,//(1.1846856068586931e-005L), + static_cast(6917) / 10000),//(0.69176378846168518L)), + static_cast(0.000075393541456247525676062058821484095548666733251733L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4243) / 100,//(42.434902191162109L), + static_cast(3001) / 10000, //(0.30012050271034241L), + static_cast(9157) / 10000), //(0.91574394702911377L)), + static_cast(0.0028387319012616013434124297160711532419664289474798L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(9713) / 1000, //(9.7131776809692383L), + static_cast(9940) / 100, //(99.406852722167969L), + static_cast(8391) / 100000), //(0.083912998437881470L)), + static_cast(0.46116895440368248909937863372410093344466819447476L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(72.5), + static_cast(1.125), + static_cast(0.75)), + static_cast(1.3423066982487051710597194786268004978931316494920e-9L), tolerance*3); // extra tolerance needed on linux X86EM64 + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4985)/1000, //(4.9854421615600586L), + static_cast(1066)/1000, //(1.0665277242660522L), + static_cast(7599)/10000), //(0.75997146964073181L)), + static_cast(0.27533431334486812211032939156910472371928659321347L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(6813)/1000, //(6.8127136230468750L), + static_cast(1056)/1000, //(1.0562920570373535L), + static_cast(1741)/10000), //(0.17416560649871826L)), + static_cast(7.6736128722762245852815040810349072461658078840945e-6L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4898)/10000, //(0.48983201384544373L), + static_cast(2251)/10000, //(0.22512593865394592L), + static_cast(2003)/10000), //(0.20032680034637451L)), + static_cast(0.17089223868046209692215231702890838878342349377008L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4049)/1000, //(4.0498137474060059L), + static_cast(1540)/10000, //(0.15403440594673157L), + static_cast(6537)/10000), //(0.65370121598243713L)), + static_cast(0.017273988301528087878279199511703371301647583919670L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(7269)/1000, //(7.2695474624633789L), + static_cast(1190)/10000, //(0.11902070045471191L), + static_cast(8003)/10000), //(0.80036874115467072L)), + static_cast(0.013334694467796052900138431733772122625376753696347L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(2726)/1000, //(2.7266697883605957L), + static_cast(1151)/100000, //(0.011510574258863926L), + static_cast(8665)/100000), //(0.086654007434844971L)), + static_cast(5.8218877068298586420691288375690562915515260230173e-6L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(3431)/10000, //(0.34317314624786377L), + static_cast(4634)/100000, //0.046342257410287857L), + static_cast(7582)/10000), //(0.75823287665843964L)), + static_cast(0.15132819929418661038699397753916091907278005695387L), tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.34317314624786377L), + static_cast(0.046342257410287857L), + static_cast(0)), + static_cast(0), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibetac( + static_cast(0.34317314624786377L), + static_cast(0.046342257410287857L), + static_cast(0)), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(0.34317314624786377L), + static_cast(0.046342257410287857L), + static_cast(1)), + static_cast(1), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibetac( + static_cast(0.34317314624786377L), + static_cast(0.046342257410287857L), + static_cast(1)), + static_cast(0), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(1), + static_cast(4634)/100000, //(0.046342257410287857L), + static_cast(32)/100), + static_cast(0.017712849440718489999419956301675684844663359595318L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(4634)/100000, //(0.046342257410287857L), + static_cast(1), + static_cast(32)/100), + static_cast(0.94856839398626914764591440181367780660208493234722L), tolerance); + + // try with some integer arguments: + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(3), + static_cast(8), + static_cast(0.25)), + static_cast(0.474407196044921875000000000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(6), + static_cast(8), + static_cast(0.25)), + static_cast(0.0802125930786132812500000000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(12), + static_cast(1), + static_cast(0.25)), + static_cast(5.96046447753906250000000000000000000000000000000000000000000e-8L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta( + static_cast(1), + static_cast(8), + static_cast(0.25)), + static_cast(0.899887084960937500000000000000000000000000000000000000000000L), tolerance); + + // very naive check on derivative: + using namespace std; // For ADL of std functions + tolerance = boost::math::tools::epsilon() * 10000; // 100 eps + BOOST_CHECK_CLOSE( + ::boost::math::ibeta_derivative( + static_cast(2), + static_cast(3), + static_cast(0.5)), + pow(static_cast(0.5), static_cast(2)) * pow(static_cast(0.5), static_cast(1)) / boost::math::beta(static_cast(2), static_cast(3)), tolerance); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif +#ifdef TEST_FLOAT + test_spots(0.0F); +#endif +#ifdef TEST_DOUBLE + test_spots(0.0); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_spots(0.0L); +#endif +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.1)); +#endif +#endif +#endif + +#ifdef TEST_FLOAT + test_beta(0.1F, "float"); +#endif +#ifdef TEST_DOUBLE + test_beta(0.1, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_beta(0.1L, "long double"); +#endif +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#ifdef TEST_REAL_CONCEPT + test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + + diff --git a/test/test_ibeta_inv.cpp b/test/test_ibeta_inv.cpp new file mode 100644 index 000000000..fb6a3705c --- /dev/null +++ b/test/test_ibeta_inv.cpp @@ -0,0 +1,379 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_beta_hooks.hpp" +#include "handle_test_result.hpp" + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete beta function inverses +// ibeta_inv and ibetac_inv. There are three sets of tests: +// 1) Spot tests which compare our results with selected values +// computed using the online special function calculator at +// functions.wolfram.com, +// 2) TODO!!!! Accuracy tests use values generated with NTL::RR at +// 1000-bit precision and our generic versions of these functions. +// 3) Round trip sanity checks, use the test data for the forward +// functions, and verify that we can get (approximately) back +// where we started. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + // Note that permitted max errors are really pretty high + // at around 10000eps. The reason for this is that even + // if the forward function is off by 1eps, it's enough to + // throw out the inverse by ~7000eps. In other words the + // forward function may flatline, so that many x-values + // all map to about the same p. Trying to invert in this + // region is almost futile. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + // + // Linux etc, + // Extended exponent range of long double + // causes more extreme test cases to be executed: + // + if(std::numeric_limits::digits == 64) + { + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 20, 10); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "long double", // test type(s) + ".*", // test data group + ".*", 200000, 100000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 5000000L, 500000); // test function + } +#endif + // + // MinGW, + // Extended exponent range of long double + // causes more extreme test cases to be executed: + // + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 10, 10); // test function + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 300000, 20000); // test function + + // + // HP-UX and Solaris: + // Extended exponent range of long double + // causes more extreme test cases to be executed: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "HP-UX|Sun Solaris", // platform + "long double", // test type(s) + ".*", // test data group + ".*", 200000, 100000); // test function + + // + // HP Tru64: + // Extended exponent range of long double + // causes more extreme test cases to be executed: + // + add_expected_result( + "HP Tru64.*", // compiler + ".*", // stdlib + ".*", // platform + "long double", // test type(s) + ".*", // test data group + ".*", 200000, 100000); // test function + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 10000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 500000, 500000); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void test_inverses(const T& data) +{ + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::policies::digits >()/2)) * 100; + if(boost::math::policies::digits >() < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete beta is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + BOOST_CHECK_EQUAL(boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]), value_type(0)); + else if((1 - data[i][5] > 0.001) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + } + else if(1 == data[i][5]) + BOOST_CHECK_EQUAL(boost::math::ibeta_inv(data[i][0], data[i][1], data[i][5]), value_type(1)); + + if(data[i][6] == 0) + BOOST_CHECK_EQUAL(boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]), value_type(1)); + else if((1 - data[i][6] > 0.001) + && (fabs(data[i][6]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][6]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]); + BOOST_CHECK_CLOSE(data[i][2], inv, precision); + } + else if(data[i][6] == 1) + BOOST_CHECK_EQUAL(boost::math::ibetac_inv(data[i][0], data[i][1], data[i][6]), value_type(0)); + } +} + +template +void test_inverses2(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::ibeta_inv; +#else + pg funcp = boost::math::ibeta_inv; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test ibeta_inv(T, T, T) against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibeta_inv", test_name); + // + // test ibetac_inv(T, T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibetac_inv; +#else + funcp = boost::math::ibetac_inv; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibetac_inv", test_name); +} + + +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // +# include "ibeta_small_data.ipp" + + test_inverses(ibeta_small_data); + +# include "ibeta_data.ipp" + + test_inverses(ibeta_data); + +# include "ibeta_large_data.ipp" + + test_inverses(ibeta_large_data); + +# include "ibeta_inv_data.ipp" + + test_inverses2(ibeta_inv_data, name, "Inverse incomplete beta"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 100 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 10000; + BOOST_CHECK_CLOSE( + ::boost::math::ibeta_inv( + static_cast(1), + static_cast(2), + static_cast(0.5)), + static_cast(0.29289321881345247559915563789515096071516406231153L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta_inv( + static_cast(3), + static_cast(0.5), + static_cast(0.5)), + static_cast(0.92096723292382700385142816696980724853063433975470L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta_inv( + static_cast(20.125), + static_cast(0.5), + static_cast(0.5)), + static_cast(0.98862133312917003480022776106012775747685870929920L), tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::ibeta_inv( + static_cast(40), + static_cast(80), + static_cast(0.5)), + static_cast(0.33240456430025026300937492802591128972548660643778L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + expected_results(); +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif +#ifdef TEST_FLOAT + test_spots(0.0F); +#endif +#ifdef TEST_DOUBLE + test_spots(0.0); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_spots(0.0L); +#endif +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.1)); +#endif +#endif + +#ifdef TEST_FLOAT + test_beta(0.1F, "float"); +#endif +#ifdef TEST_DOUBLE + test_beta(0.1, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_beta(0.1L, "long double"); +#endif +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#ifdef TEST_REAL_CONCEPT + test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + + diff --git a/test/test_ibeta_inv_ab.cpp b/test/test_ibeta_inv_ab.cpp new file mode 100644 index 000000000..3110e41bd --- /dev/null +++ b/test/test_ibeta_inv_ab.cpp @@ -0,0 +1,315 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#ifdef TEST_GSL +#include +#include +#endif + +#include "handle_test_result.hpp" + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete beta function inverses +// ibeta_inva and ibetac_inva. There are three sets of tests: +// 1) TODO!!!! Accuracy tests use values generated with NTL::RR at +// 1000-bit precision and our generic versions of these functions. +// 2) Round trip sanity checks, use the test data for the forward +// functions, and verify that we can get (approximately) back +// where we started. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Linux: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 3000, 500); // test function + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 500, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "float|double", // test type(s) + ".*", // test data group + ".*", 5, 3); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 1000000, 500000); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void test_inverses(const T& data) +{ + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::policies::digits >()/2)) * 100; + if(boost::math::policies::digits >() < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete beta is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + { + BOOST_CHECK_EQUAL(boost::math::ibeta_inva(data[i][1], data[i][2], data[i][5]), boost::math::tools::max_value()); + BOOST_CHECK_EQUAL(boost::math::ibeta_invb(data[i][0], data[i][2], data[i][5]), boost::math::tools::min_value()); + } + else if((1 - data[i][5] > 0.001) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::ibeta_inva(data[i][1], data[i][2], data[i][5]); + BOOST_CHECK_CLOSE(data[i][0], inv, precision); + inv = boost::math::ibeta_invb(data[i][0], data[i][2], data[i][5]); + BOOST_CHECK_CLOSE(data[i][1], inv, precision); + } + else if(1 == data[i][5]) + { + BOOST_CHECK_EQUAL(boost::math::ibeta_inva(data[i][1], data[i][2], data[i][5]), boost::math::tools::min_value()); + BOOST_CHECK_EQUAL(boost::math::ibeta_invb(data[i][0], data[i][2], data[i][5]), boost::math::tools::max_value()); + } + + if(data[i][6] == 0) + { + BOOST_CHECK_EQUAL(boost::math::ibetac_inva(data[i][1], data[i][2], data[i][6]), boost::math::tools::min_value()); + BOOST_CHECK_EQUAL(boost::math::ibetac_invb(data[i][0], data[i][2], data[i][6]), boost::math::tools::max_value()); + } + else if((1 - data[i][6] > 0.001) + && (fabs(data[i][6]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][6]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::ibetac_inva(data[i][1], data[i][2], data[i][6]); + BOOST_CHECK_CLOSE(data[i][0], inv, precision); + inv = boost::math::ibetac_invb(data[i][0], data[i][2], data[i][6]); + BOOST_CHECK_CLOSE(data[i][1], inv, precision); + } + else if(data[i][6] == 1) + { + BOOST_CHECK_EQUAL(boost::math::ibetac_inva(data[i][1], data[i][2], data[i][6]), boost::math::tools::max_value()); + BOOST_CHECK_EQUAL(boost::math::ibetac_invb(data[i][0], data[i][2], data[i][6]), boost::math::tools::min_value()); + } + } +} + +template +void test_inverses2(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::ibeta_inva; +#else + pg funcp = boost::math::ibeta_inva; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test ibeta_inva(T, T, T) against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibeta_inva", test_name); + // + // test ibetac_inva(T, T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibetac_inva; +#else + funcp = boost::math::ibetac_inva; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibetac_inva", test_name); + // + // test ibeta_invb(T, T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibeta_invb; +#else + funcp = boost::math::ibeta_invb; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(5)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibeta_invb", test_name); + // + // test ibetac_invb(T, T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::ibetac_invb; +#else + funcp = boost::math::ibetac_invb; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1, 2), + extract_result(6)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ibetac_invb", test_name); +} + +template +void test_beta(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // + std::cout << "Running sanity checks for type " << name << std::endl; + +# include "ibeta_small_data.ipp" + + test_inverses(ibeta_small_data); + +# include "ibeta_data.ipp" + + test_inverses(ibeta_data); + +# include "ibeta_large_data.ipp" + + test_inverses(ibeta_large_data); +#ifndef FULL_TEST + if(boost::is_floating_point::value){ +#endif + // + // This accuracy test is normally only enabled for "real" + // floating point types and not for class real_concept. + // The reason is that these tests are exceptionally slow + // to complete when T doesn't have Lanczos support defined for it. + // +# include "ibeta_inva_data.ipp" + + test_inverses2(ibeta_inva_data, name, "Inverse incomplete beta"); +#ifndef FULL_TEST + } +#endif +} + +int test_main(int, char* []) +{ + expected_results(); +#ifdef TEST_GSL + gsl_set_error_handler_off(); +#endif + +#ifdef TEST_FLOAT + test_beta(0.1F, "float"); +#endif +#ifdef TEST_DOUBLE + test_beta(0.1, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_beta(0.1L, "long double"); +#endif +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#ifdef TEST_REAL_CONCEPT + test_beta(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + diff --git a/test/test_igamma.cpp b/test/test_igamma.cpp new file mode 100644 index 000000000..2940ca09e --- /dev/null +++ b/test/test_igamma.cpp @@ -0,0 +1,518 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_gamma_hooks.hpp" +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete gamma functions tgamma, +// tgamma_lower, gamma_p and gamma_q. There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Linux: + // + // These should not really be needed, but on *some* Linux + // versions these error rates are quite large and appear to + // be related to the accuracy of powl and expl. On Itanium + // or Xeon machines the error rates are much lower than this. + // Worst cases appear to be AMD64 machines. + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 1000, 200); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + largest_type, // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 1000, 200); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 600, 200); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + "real_concept", // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 600, 200); // test function + + // + // Mac OS X: + // It's not clear why these should be required, but see notes above + // about Linux. + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 5000, 1000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 40, 15); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 2000, 300); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 5000, 1000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + "real_concept", // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 40, 15); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + "real_concept", // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 2000, 300); // test function + // + // HP-UX: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "HP-UX", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 500, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "HP-UX", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 500, 100); // test function + // + // Sun OS: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 500, 100); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + largest_type, // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 100, 30); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 500, 100); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Sun.*", // platform + "real_concept", // test type(s) + "[^|]*integer[^|]*", // test data group + "[^|]*", 100, 30); // test function + + // + // Mac OS X: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Mac OS", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 100, 50); // test function + + // + // Large exponent range causes more extreme test cases to be evaluated: + // + if(std::numeric_limits::max_exponent > std::numeric_limits::max_exponent) + { + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + ".*", 40000, 3000); // test function + } + + + // + // Catch all cases come last: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 50, 20); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 20, 10); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + "boost::math::gamma_q", 500, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "Cygwin", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + "boost::math::gamma_p", 700, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + "boost::math::gamma_p", 350, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*integer[^|]*", // test data group + ".*", 20, 10); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 200, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*small[^|]*", // test data group + ".*", 20, 10); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*large[^|]*", // test data group + ".*", 1000000, 100000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*integer[^|]*", // test data group + ".*", 40, 10); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_gamma_2(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::tgamma; +#else + pg funcp = boost::math::tgamma; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma(T, T) against data: + // + if(data[0][2] > 0) + { + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma", test_name); + // + // test tgamma_lower(T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::tgamma_lower; +#else + funcp = boost::math::tgamma_lower; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma_lower", test_name); + } + // + // test gamma_q(T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::gamma_q; +#else + funcp = boost::math::gamma_q; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_q", test_name); +#if defined(TEST_CEPHES) || defined(TEST_GSL) + // + // test other gamma_q(T, T) against data: + // + if(boost::is_floating_point::value) + { + funcp = other::gamma_q; + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(3)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::gamma_q"); + } +#endif + // + // test gamma_p(T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::gamma_p; +#else + funcp = boost::math::gamma_p; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(5)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_p", test_name); +#if defined(TEST_CEPHES) || defined(TEST_GSL) + // + // test other gamma_p(T, T) against data: + // + if(boost::is_floating_point::value) + { + funcp = other::gamma_p; + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(5)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::gamma_p"); + } +#endif + std::cout << std::endl; +} + +template +void test_gamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // First the data for the incomplete gamma function, each + // row has the following 6 entries: + // Parameter a, parameter z, + // Expected tgamma(a, z), Expected gamma_q(a, z) + // Expected tgamma_lower(a, z), Expected gamma_p(a, z) + // +# include "igamma_med_data.ipp" + + do_test_gamma_2(igamma_med_data, name, "tgamma(a, z) medium values"); + +# include "igamma_small_data.ipp" + + do_test_gamma_2(igamma_small_data, name, "tgamma(a, z) small values"); + +# include "igamma_big_data.ipp" + + do_test_gamma_2(igamma_big_data, name, "tgamma(a, z) large values"); + +# include "igamma_int_data.ipp" + + do_test_gamma_2(igamma_int_data, name, "tgamma(a, z) integer and half integer values"); +} + +template +void test_spots(T) +{ + // + // basic sanity checks, tolerance is 10 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 1000; +#if (defined(macintosh) || defined(__APPLE__) || defined(__APPLE_CC__)) + tolerance *= 10; +#endif + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(1)), static_cast(23.912163676143750903709045060494956383977723517065L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(5)), static_cast(10.571838841565097874621959975919877646444998907920L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(10)), static_cast(0.70206451384706574414638719662835463671916532623256L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(5), static_cast(100)), static_cast(3.8734332808745531496973774140085644548465762343719e-36L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(0.5)), static_cast(0.56241823159440712427949495730204306902676756479651L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(9)/10), static_cast(0.31853210360412109873859360390443790076576777747449L), tolerance*10); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(5)), static_cast(0.0027746032604128093194908357272603294120210079791437L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma(static_cast(0.5), static_cast(100)), static_cast(3.7017478604082789202535664481339075721362102520338e-45L), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(1)), static_cast(0.087836323856249096290954939505043616022276482935091L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(5)), static_cast(13.428161158434902125378040024080122353555001092080L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(10)), static_cast(23.297935486152934255853612803371645363280834673767L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::tgamma_lower(static_cast(5), static_cast(100)), static_cast(23.999999999999999999999999999999999996126566719125L), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(5), static_cast(1)), static_cast(0.99634015317265628765454354418728984933240514654437L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(5), static_cast(5)), static_cast(0.44049328506521241144258166566332823526854162116334L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(5), static_cast(10)), static_cast(0.029252688076961072672766133192848109863298555259690L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(5), static_cast(100)), static_cast(1.6139305336977304790405739225035685228527400976549e-37L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(1.5), static_cast(2)), static_cast(0.26146412994911062220282207597592120190281060919079L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q(static_cast(20.5), static_cast(22)), static_cast(0.34575332043467326814971590879658406632570278929072L), tolerance); + + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(5), static_cast(1)), static_cast(0.0036598468273437123454564558127101506675948534556288L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(5), static_cast(5)), static_cast(0.55950671493478758855741833433667176473145837883666L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(5), static_cast(10)), static_cast(0.97074731192303892732723386680715189013670144474031L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(5), static_cast(100)), static_cast(0.9999999999999999999999999999999999998386069466302L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(1.5), static_cast(2)), static_cast(0.73853587005088937779717792402407879809718939080921L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_p(static_cast(20.5), static_cast(22)), static_cast(0.65424667956532673185028409120341593367429721070928L), tolerance); + + // naive check on derivative function: + using namespace std; // For ADL of std functions + tolerance = boost::math::tools::epsilon() * 5000; // 50 eps + BOOST_CHECK_CLOSE(::boost::math::gamma_p_derivative(static_cast(20.5), static_cast(22)), + exp(static_cast(-22)) * pow(static_cast(22), static_cast(19.5)) / boost::math::tgamma(static_cast(20.5)), tolerance); + +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_spots(0.0F); +#endif + test_spots(0.0); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.1)); +#endif +#endif + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_gamma(0.1F, "float"); +#endif + test_gamma(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_gamma(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_gamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_igamma_inv.cpp b/test/test_igamma_inv.cpp new file mode 100644 index 000000000..a9530ddee --- /dev/null +++ b/test/test_igamma_inv.cpp @@ -0,0 +1,440 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "test_gamma_hooks.hpp" +#include "handle_test_result.hpp" + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete gamma function inverses +// gamma_p_inv and gamma_q_inv. There are three sets of tests: +// 1) Spot tests which compare our results with selected values +// computed using the online special function calculator at +// functions.wolfram.com, +// 2) Accuracy tests use values generated with NTL::RR at +// 1000-bit precision and our generic versions of these functions. +// 3) Round trip sanity checks, use the test data for the forward +// functions, and verify that we can get (approximately) back +// where we started. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Large exponent range causes more extreme test cases to be evaluated: + // + if(std::numeric_limits::max_exponent > std::numeric_limits::max_exponent) + { + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 200000, 10000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 70000, 8000); // test function + } + // + // These high error rates are seen on on some Linux + // architectures: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux.*", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 350, 5); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux.*", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + "[^|]*", 150, 5); // test function + + + // + // Catch all cases come last: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 20, 5); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*large[^|]*", // test data group + "[^|]*", 5, 2); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 2100, 500); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "float|double", // test type(s) + "[^|]*small[^|]*", // test data group + "boost::math::gamma_p_inv", 500, 60); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "float|double", // test type(s) + "[^|]*", // test data group + "boost::math::gamma_q_inv", 350, 60); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "float|double", // test type(s) + "[^|]*", // test data group + "[^|]*", 4, 2); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*medium[^|]*", // test data group + "[^|]*", 20, 5); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*large[^|]*", // test data group + "[^|]*", 1000, 500); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*small[^|]*", // test data group + "[^|]*", 3700, 500); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +#define BOOST_CHECK_CLOSE_EX(a, b, prec, i) \ + {\ + unsigned int failures = boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed;\ + BOOST_CHECK_CLOSE(a, b, prec); \ + if(failures != boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed)\ + {\ + std::cerr << "Failure was at row " << i << std::endl;\ + std::cerr << std::setprecision(35); \ + std::cerr << "{ " << data[i][0] << " , " << data[i][1] << " , " << data[i][2];\ + std::cerr << " , " << data[i][3] << " , " << data[i][4] << " , " << data[i][5] << " } " << std::endl;\ + }\ + } + +template +void do_test_gamma_2(const T& data, const char* type_name, const char* test_name) +{ + // + // test gamma_p_inv(T, T) against data: + // + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << test_name << " with type " << type_name << std::endl; + + // + // These sanity checks test for a round trip accuracy of one half + // of the bits in T, unless T is type float, in which case we check + // for just one decimal digit. The problem here is the sensitivity + // of the functions, not their accuracy. This test data was generated + // for the forward functions, which means that when it is used as + // the input to the inverses then it is necessarily inexact. This rounding + // of the input is what makes the data unsuitable for use as an accuracy check, + // and also demonstrates that you can't in general round-trip these functions. + // It is however a useful sanity check. + // + value_type precision = static_cast(ldexp(1.0, 1-boost::math::policies::digits >()/2)) * 100; + if(boost::math::policies::digits >() < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated to float + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete gamma is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_p_inv(data[i][0], data[i][5]), value_type(0)); + else if((1 - data[i][5] > 0.001) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::gamma_p_inv(data[i][0], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][1], inv, precision, i); + } + else if(1 == data[i][5]) + BOOST_CHECK_EQUAL(boost::math::gamma_p_inv(data[i][0], data[i][5]), boost::math::tools::max_value()); + else + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_p_inv(data[i][0], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][1], inv, 100000, i); + } + + if(data[i][3] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_q_inv(data[i][0], data[i][3]), boost::math::tools::max_value()); + else if((1 - data[i][3] > 0.001) && (fabs(data[i][3]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::gamma_q_inv(data[i][0], data[i][3]); + BOOST_CHECK_CLOSE_EX(data[i][1], inv, precision, i); + } + else if(1 == data[i][3]) + BOOST_CHECK_EQUAL(boost::math::gamma_q_inv(data[i][0], data[i][3]), value_type(0)); + else if(fabs(data[i][3]) > 2 * boost::math::tools::min_value()) + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_q_inv(data[i][0], data[i][3]); + BOOST_CHECK_CLOSE_EX(data[i][1], inv, 100, i); + } + } + std::cout << std::endl; +} + +template +void do_test_gamma_inv(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::gamma_p_inv; +#else + pg funcp = boost::math::gamma_p_inv; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test gamma_p_inv(T, T) against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_p_inv", test_name); + // + // test gamma_q_inv(T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::gamma_q_inv; +#else + funcp = boost::math::gamma_q_inv; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_q_inv", test_name); +} + +template +void test_gamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // First the data for the incomplete gamma function, each + // row has the following 6 entries: + // Parameter a, parameter z, + // Expected tgamma(a, z), Expected gamma_q(a, z) + // Expected tgamma_lower(a, z), Expected gamma_p(a, z) + // +# include "igamma_med_data.ipp" + + do_test_gamma_2(igamma_med_data, name, "Running round trip sanity checks on incomplete gamma medium sized values"); + +# include "igamma_small_data.ipp" + + do_test_gamma_2(igamma_small_data, name, "Running round trip sanity checks on incomplete gamma small values"); + +# include "igamma_big_data.ipp" + + do_test_gamma_2(igamma_big_data, name, "Running round trip sanity checks on incomplete gamma large values"); + +# include "gamma_inv_data.ipp" + + do_test_gamma_inv(gamma_inv_data, name, "incomplete gamma inverse(a, z) medium values"); + +# include "gamma_inv_big_data.ipp" + + do_test_gamma_inv(gamma_inv_big_data, name, "incomplete gamma inverse(a, z) large values"); + +# include "gamma_inv_small_data.ipp" + + do_test_gamma_inv(gamma_inv_small_data, name, "incomplete gamma inverse(a, z) small values"); +} + +template +void test_spots(T, const char* type_name) +{ + std::cout << "Running spot checks for type " << type_name << std::endl; + // + // basic sanity checks, tolerance is 100 epsilon expressed as a percentage: + // + T tolerance = boost::math::tools::epsilon() * 10000; + if(tolerance < 1e-25f) + tolerance = 1e-25f; // limit of test data? + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(1)/100, static_cast(1.0/128)), static_cast(0.35767144525455121503672919307647515332256996883787L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(1)/100, static_cast(0.5)), static_cast(4.4655350189103486773248562646452806745879516124613e-31L), tolerance*10); + // + // We can't test in this region against Mathworld's data as the results produced + // by functions.wolfram.com appear to be in error, and do *not* round trip with + // their own version of gamma_q. Using our output from the inverse as input to + // their version of gamma_q *does* round trip however. It should be pointed out + // that the functions in this area are very sensitive with nearly infinite + // first derivatives, it's also questionable how useful these functions are + // in this part of the domain. + // + //BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(1e-2), static_cast(1.0-1.0/128)), static_cast(3.8106736649978161389878528903698068142257930575497e-181L), tolerance); + // + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(0.5), static_cast(1.0/128)), static_cast(3.5379794687984498627918583429482809311448951189097L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(0.5), static_cast(1.0/2)), static_cast(0.22746821155978637597125832348982469815821055329511L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(0.5), static_cast(1.0-1.0/128)), static_cast(0.000047938431649305382237483273209405461203600840052182L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10), static_cast(1.0/128)), static_cast(19.221865946801723949866005318845155649972164294057L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10), static_cast(1.0/2)), static_cast(9.6687146147141311517500637401166726067778162022664L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10), static_cast(1.0-1.0/128)), static_cast(3.9754602513640844712089002210120603689809432130520L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10000), static_cast(1.0/128)), static_cast(10243.369973939134157953734588122880006091919872879L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10000), static_cast(1.0/2)), static_cast(9999.6666686420474237369661574633153551436435884101L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::gamma_q_inv(static_cast(10000), static_cast(1.0-1.0/128)), static_cast(9759.8597223369324083191194574874497413261589080204L), tolerance); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS +#ifdef TEST_FLOAT + test_spots(0.0F, "float"); +#endif +#endif +#ifdef TEST_DOUBLE + test_spots(0.0, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_spots(0.0L, "long double"); +#endif +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#endif + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS +#ifdef TEST_FLOAT + test_gamma(0.1F, "float"); +#endif +#endif +#ifdef TEST_DOUBLE + test_gamma(0.1, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_gamma(0.1L, "long double"); +#endif +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_gamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_igamma_inva.cpp b/test/test_igamma_inva.cpp new file mode 100644 index 000000000..e16f9e24a --- /dev/null +++ b/test/test_igamma_inva.cpp @@ -0,0 +1,307 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the incomplete gamma function inverses +// gamma_p_inva and gamma_q_inva. There are two sets of tests: +// 2) TODO: Accuracy tests use values generated with NTL::RR at +// 1000-bit precision and our generic versions of these functions. +// 3) Round trip sanity checks, use the test data for the forward +// functions, and verify that we can get (approximately) back +// where we started. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Linux: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux", // platform + largest_type, // test type(s) + "[^|]*", // test data group + "[^|]*", 800, 200); // test function + + // + // Catch all cases come last: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*", // test data group + "[^|]*", 3000, 1000); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*", // test data group + "[^|]*", 300, 100); // test function + // this one has to come last in case double *is* the widest + // float type: + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "float|double", // test type(s) + "[^|]*", // test data group + "[^|]*", 10, 5); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +#define BOOST_CHECK_CLOSE_EX(a, b, prec, i) \ + {\ + unsigned int failures = boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed;\ + BOOST_CHECK_CLOSE(a, b, prec); \ + if(failures != boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed)\ + {\ + std::cerr << "Failure was at row " << i << std::endl;\ + std::cerr << std::setprecision(35); \ + std::cerr << "{ " << data[i][0] << " , " << data[i][1] << " , " << data[i][2];\ + std::cerr << " , " << data[i][3] << " , " << data[i][4] << " , " << data[i][5] << " } " << std::endl;\ + }\ + } + +template +void do_test_gamma_2(const T& data, const char* type_name, const char* test_name) +{ + // + // test gamma_p_inva(T, T) against data: + // + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + std::cout << test_name << " with type " << type_name << std::endl; + + // + // These sanity checks test for a round trip accuracy of one half + // of the bits in T, unless T is type float, in which case we check + // for just one decimal digit. The problem here is the sensitivity + // of the functions, not their accuracy. This test data was generated + // for the forward functions, which means that when it is used as + // the input to the inverses then it is necessarily inexact. This rounding + // of the input is what makes the data unsuitable for use as an accuracy check, + // and also demonstrates that you can't in general round-trip these functions. + // It is however a useful sanity check. + // + value_type precision = static_cast(ldexp(1.0, 1-boost::math::policies::digits >()/2)) * 100; + if(boost::math::policies::digits >() < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated to float + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete gamma is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_p_inva(data[i][1], data[i][5]), boost::math::tools::max_value()); + else if((1 - data[i][5] > 0.001) && (fabs(data[i][5]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::gamma_p_inva(data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][0], inv, precision, i); + } + else if(1 == data[i][5]) + BOOST_CHECK_EQUAL(boost::math::gamma_p_inva(data[i][1], data[i][5]), boost::math::tools::min_value()); + else if(data[i][5] > 2 * boost::math::tools::min_value()) + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_p_inva(data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][0], inv, 100, i); + } + + if(data[i][3] == 0) + BOOST_CHECK_EQUAL(boost::math::gamma_q_inva(data[i][1], data[i][3]), boost::math::tools::min_value()); + else if((1 - data[i][3] > 0.001) + && (fabs(data[i][3]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][3]) > 2 * boost::math::tools::min_value())) + { + value_type inv = boost::math::gamma_q_inva(data[i][1], data[i][3]); + BOOST_CHECK_CLOSE_EX(data[i][0], inv, precision, i); + } + else if(1 == data[i][3]) + BOOST_CHECK_EQUAL(boost::math::gamma_q_inva(data[i][1], data[i][3]), boost::math::tools::max_value()); + else if(data[i][3] > 2 * boost::math::tools::min_value()) + { + // not enough bits in our input to get back to x, but we should be in + // the same ball park: + value_type inv = boost::math::gamma_q_inva(data[i][1], data[i][3]); + BOOST_CHECK_CLOSE_EX(data[i][0], inv, 100, i); + } + } + std::cout << std::endl; +} + +template +void do_test_gamma_inva(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::gamma_p_inva; +#else + pg funcp = boost::math::gamma_p_inva; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test gamma_p_inva(T, T) against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_p_inva", test_name); + // + // test gamma_q_inva(T, T) against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::gamma_q_inva; +#else + funcp = boost::math::gamma_q_inva; +#endif + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::gamma_q_inva", test_name); +} + +template +void test_gamma(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // First the data for the incomplete gamma function, each + // row has the following 6 entries: + // Parameter a, parameter z, + // Expected tgamma(a, z), Expected gamma_q(a, z) + // Expected tgamma_lower(a, z), Expected gamma_p(a, z) + // +# include "igamma_med_data.ipp" + + do_test_gamma_2(igamma_med_data, name, "Running round trip sanity checks on incomplete gamma medium sized values"); + +# include "igamma_small_data.ipp" + + do_test_gamma_2(igamma_small_data, name, "Running round trip sanity checks on incomplete gamma small values"); + +# include "igamma_big_data.ipp" + + do_test_gamma_2(igamma_big_data, name, "Running round trip sanity checks on incomplete gamma large values"); + +# include "igamma_inva_data.ipp" + + do_test_gamma_inva(igamma_inva_data, name, "Incomplete gamma inverses."); +} + +int test_main(int, char* []) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS +#ifdef TEST_FLOAT + test_gamma(0.1F, "float"); +#endif +#endif +#ifdef TEST_DOUBLE + test_gamma(0.1, "double"); +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_gamma(0.1L, "long double"); +#endif +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_gamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_instantiate1.cpp b/test/test_instantiate1.cpp new file mode 100644 index 000000000..ab45fa1a6 --- /dev/null +++ b/test/test_instantiate1.cpp @@ -0,0 +1,20 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "compile_test/instantiate.hpp" + +extern void other_test(); + +int main(int argc, char* []) +{ + if(argc > 10000) + { + instantiate(double(0)); + instantiate_mixed(double(0)); + other_test(); + } +} + + diff --git a/test/test_instantiate2.cpp b/test/test_instantiate2.cpp new file mode 100644 index 000000000..6996636a6 --- /dev/null +++ b/test/test_instantiate2.cpp @@ -0,0 +1,14 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "compile_test/instantiate.hpp" + +void other_test() +{ + instantiate(double(0)); + instantiate_mixed(double(0)); +} + + diff --git a/test/test_laguerre.cpp b/test/test_laguerre.cpp new file mode 100644 index 000000000..50e45271e --- /dev/null +++ b/test/test_laguerre.cpp @@ -0,0 +1,292 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_legendre_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Laguerre polynomials. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Linux special cases, error rates seem to be much higer here + // even though the implementation contains nothing but basic + // arithmetic? + // + if((std::numeric_limits::digits <= 64) + && (std::numeric_limits::digits != std::numeric_limits::digits)) + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 10, 5); // test function +#endif + } + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux.*|Mac OS|Sun.*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 40000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "linux.*|Mac OS|Sun.*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 40000, 1000); // test function + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 40000, 1000); // test function + add_expected_result( + ".*mingw.*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 40000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "IBM Aix", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 5000, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + "IBM Aix", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 5000, 500); // test function + + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 4000, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 4000, 500); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_laguerre2(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::laguerre; +#else + pg funcp = boost::math::laguerre; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test laguerre against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::laguerre(n, x)", test_name); + + std::cout << std::endl; +} + +template +void do_test_laguerre3(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::laguerre; +#else + pg funcp = boost::math::laguerre; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test laguerre against data: + // + result = boost::math::tools::test( + data, + bind_func_int2(funcp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::laguerre(n, m, x)", test_name); + std::cout << std::endl; +} + +template +void test_laguerre(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "laguerre2.ipp" + + do_test_laguerre2(laguerre2, name, "Laguerre Polynomials"); + +# include "laguerre3.ipp" + + do_test_laguerre3(laguerre3, name, "Associated Laguerre Polynomials"); +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 100 epsilon: + // + T tolerance = boost::math::tools::epsilon() * 100; + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(1, static_cast(0.5L)), static_cast(0.5L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(4, static_cast(0.5L)), static_cast(-0.3307291666666666666666666666666666666667L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(7, static_cast(0.5L)), static_cast(-0.5183392237103174603174603174603174603175L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(20, static_cast(0.5L)), static_cast(0.3120174870800154148915399248893113634676L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(50, static_cast(0.5L)), static_cast(-0.3181388060269979064951118308575628226834L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(1, static_cast(-0.5L)), static_cast(1.5L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(4, static_cast(-0.5L)), static_cast(3.835937500000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(7, static_cast(-0.5L)), static_cast(7.950934709821428571428571428571428571429L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(20, static_cast(-0.5L)), static_cast(76.12915699869631476833699787070874048223L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(50, static_cast(-0.5L)), static_cast(2307.428631277506570629232863491518399720L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(1, static_cast(4.5L)), static_cast(-3.500000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(4, static_cast(4.5L)), static_cast(0.08593750000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(7, static_cast(4.5L)), static_cast(-1.036928013392857142857142857142857142857L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(20, static_cast(4.5L)), static_cast(1.437239150257817378525582974722170737587L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(50, static_cast(4.5L)), static_cast(-0.7795068145562651416494321484050019245248L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(4, 5, static_cast(0.5L)), static_cast(88.31510416666666666666666666666666666667L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(10, 0, static_cast(2.5L)), static_cast(-0.8802526766660982969576719576719576719577L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(10, 1, static_cast(4.5L)), static_cast(1.564311458042689732142857142857142857143L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(10, 6, static_cast(8.5L)), static_cast(20.51596541066649098875661375661375661376L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(10, 12, static_cast(12.5L)), static_cast(-199.5560968456234671241181657848324514991L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::laguerre(50, 40, static_cast(12.5L)), static_cast(-4.996769495006119488583146995907246595400e16L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_spots(0.0F, "float"); +#endif + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif + + expected_results(); + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_laguerre(0.1F, "float"); +#endif + test_laguerre(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_laguerre(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_laguerre(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_legendre.cpp b/test/test_legendre.cpp new file mode 100644 index 000000000..c742bb20b --- /dev/null +++ b/test/test_legendre.cpp @@ -0,0 +1,385 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "test_legendre_hooks.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the legendre polynomials. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // Linux: + // + if((std::numeric_limits::digits <= 64) + && (std::numeric_limits::digits != std::numeric_limits::digits)) + { +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 10, 5); // test function +#endif + } + if(std::numeric_limits::digits == 64) + { + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_p", 1000, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_q", 7000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_p", 1000, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_q", 7000, 1000); // test function + } + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_p", 500, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_q", 5400, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*", // test data group + "boost::math::legendre_p", 300, 80); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Legendre Polynomials.*", // test data group + "boost::math::legendre_q", 100, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + "Associated Legendre Polynomials.*", // test data group + ".*", 200, 20); // test function + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_p", 500, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*Large.*", // test data group + "boost::math::legendre_q", 5400, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*", // test data group + "boost::math::legendre_p", 300, 80); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Legendre Polynomials.*", // test data group + "boost::math::legendre_q", 100, 50); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + "Associated Legendre Polynomials.*", // test data group + ".*", 200, 20); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_legendre_p(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(int, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::legendre_p; +#else + pg funcp = boost::math::legendre_p; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test legendre_p against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::legendre_p", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::legendre_p; + result = boost::math::tools::test( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::legendre_p"); + } +#endif + + typedef value_type (*pg2)(unsigned, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg2 funcp2 = boost::math::legendre_q; +#else + pg2 funcp2 = boost::math::legendre_q; +#endif + + // + // test legendre_q against data: + // + result = boost::math::tools::test( + data, + bind_func_int1(funcp2, 0, 1), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::legendre_q", test_name); +#ifdef TEST_OTHER + if(::boost::is_floating_point::value){ + funcp = other::legendre_q; + result = boost::math::tools::test( + data, + bind_func_int1(funcp2, 0, 1), + extract_result(3)); + print_test_result(result, data[result.worst()], result.worst(), type_name, "other::legendre_q"); + } +#endif + + + std::cout << std::endl; +} + +template +void do_test_assoc_legendre_p(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(int, int, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::legendre_p; +#else + pg funcp = boost::math::legendre_p; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test legendre_p against data: + // + result = boost::math::tools::test( + data, + bind_func_int2(funcp, 0, 1, 2), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::legendre_p", test_name); + std::cout << std::endl; +} + +template +void test_legendre_p(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "legendre_p.ipp" + + do_test_legendre_p(legendre_p, name, "Legendre Polynomials: Small Values"); + +# include "legendre_p_large.ipp" + + do_test_legendre_p(legendre_p_large, name, "Legendre Polynomials: Large Values"); + +# include "assoc_legendre_p.ipp" + + do_test_assoc_legendre_p(assoc_legendre_p, name, "Associated Legendre Polynomials: Small Values"); + +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 100 epsilon: + // + T tolerance = boost::math::tools::epsilon() * 100; + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(1, static_cast(0.5L)), static_cast(0.5L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-1, static_cast(0.5L)), static_cast(1L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(4, static_cast(0.5L)), static_cast(-0.2890625000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-4, static_cast(0.5L)), static_cast(-0.4375000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(7, static_cast(0.5L)), static_cast(0.2231445312500000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-7, static_cast(0.5L)), static_cast(0.3232421875000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(40, static_cast(0.5L)), static_cast(-0.09542943523261546936538467572384923220258L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-40, static_cast(0.5L)), static_cast(-0.1316993126940266257030910566308990611306L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(4, 2, static_cast(0.5L)), static_cast(4.218750000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-4, 2, static_cast(0.5L)), static_cast(5.625000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(7, 5, static_cast(0.5L)), static_cast(-5696.789530152175143607977274672800795328L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-7, 4, static_cast(0.5L)), static_cast(465.1171875000000000000000000000000000000L), tolerance); + if(std::numeric_limits::max_exponent > std::numeric_limits::max_exponent) + { + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(40, 30, static_cast(0.5L)), static_cast(-7.855722083232252643913331343916012143461e45L), tolerance); + } + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-40, 20, static_cast(0.5L)), static_cast(4.966634149702370788037088925152355134665e30L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(4, 2, static_cast(-0.5L)), static_cast(4.218750000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-4, 2, static_cast(-0.5L)), static_cast(-5.625000000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(7, 5, static_cast(-0.5L)), static_cast(-5696.789530152175143607977274672800795328L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-7, 4, static_cast(-0.5L)), static_cast(465.1171875000000000000000000000000000000L), tolerance); + if(std::numeric_limits::max_exponent > std::numeric_limits::max_exponent) + { + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(40, 30, static_cast(-0.5L)), static_cast(-7.855722083232252643913331343916012143461e45L), tolerance); + } + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-40, 20, static_cast(-0.5L)), static_cast(-4.966634149702370788037088925152355134665e30L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(4, -2, static_cast(0.5L)), static_cast(0.01171875000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-4, -2, static_cast(0.5L)), static_cast(0.04687500000000000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(7, -5, static_cast(0.5L)), static_cast(0.00002378609812640364935569308025139290054701L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-7, -4, static_cast(0.5L)), static_cast(0.0002563476562500000000000000000000000000000L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(40, -30, static_cast(0.5L)), static_cast(-2.379819988646847616996471299410611801239e-48L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_p(-40, -20, static_cast(0.5L)), static_cast(4.356454600748202401657099008867502679122e-33L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_q(1, static_cast(0.5L)), static_cast(-0.7253469278329725771511886907693685738381L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_q(4, static_cast(0.5L)), static_cast(0.4401745259867706044988642951843745400835L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_q(7, static_cast(0.5L)), static_cast(-0.3439152932669753451878700644212067616780L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::legendre_q(40, static_cast(0.5L)), static_cast(0.1493671665503550095010454949479907886011L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif + + expected_results(); + + test_legendre_p(0.1F, "float"); + test_legendre_p(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_legendre_p(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_legendre_p(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_legendre_hooks.hpp b/test/test_legendre_hooks.hpp new file mode 100644 index 000000000..3c962615f --- /dev/null +++ b/test/test_legendre_hooks.hpp @@ -0,0 +1,41 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_TEST_LEGENDRE_OTHER_HOOKS_HPP +#define BOOST_MATH_TEST_LEGENDRE_OTHER_HOOKS_HPP + +#ifdef TEST_GSL +#include + +namespace other{ +inline float legendre_p(int l, float a) +{ return (float)gsl_sf_legendre_Pl (l, a); } +inline double legendre_p(int l, double a) +{ return gsl_sf_legendre_Pl (l, a); } +inline long double legendre_p(int l, long double a) +{ return gsl_sf_legendre_Pl (l, a); } + +inline float legendre_q(int l, float a) +{ return (float)gsl_sf_legendre_Ql (l, a); } +inline double legendre_q(int l, double a) +{ return gsl_sf_legendre_Ql (l, a); } +inline long double legendre_q(int l, long double a) +{ return gsl_sf_legendre_Ql (l, a); } + +} +#define TEST_OTHER +#endif + +#ifdef TEST_OTHER +namespace other{ + boost::math::concepts::real_concept legendre_p(int, boost::math::concepts::real_concept){ return 0; } + boost::math::concepts::real_concept legendre_q(int, boost::math::concepts::real_concept){ return 0; } +} +#endif + + +#endif + + diff --git a/test/test_lognormal.cpp b/test/test_lognormal.cpp new file mode 100644 index 000000000..b964b9c0c --- /dev/null +++ b/test/test_lognormal.cpp @@ -0,0 +1,308 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007 + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_lognormal.cpp + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::lognormal_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; +#include + +template +void check_lognormal(RealType loc, RealType scale, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + lognormal_distribution(loc, scale), // distribution. + x), // random variable. + p, // probability. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement( + lognormal_distribution(loc, scale), // distribution. + x)), // random variable. + q, // probability complement. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + lognormal_distribution(loc, scale), // distribution. + p), // probability. + x, // random variable. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement( + lognormal_distribution(loc, scale), // distribution. + q)), // probability complement. + x, // random variable. + tol); // %tolerance. +} + +template +void test_spots(RealType) +{ + + // Basic sanity checks. + RealType tolerance = 5e-3 * 100; + // Some tests only pass at 1e-4 because values generated by + // http://faculty.vassar.edu/lowry/VassarStats.html + // give only 5 or 6 *fixed* places, so small values have fewer digits. + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + using std::exp; + + // + // These test values were generated for the normal distribution + // using the online calculator at http://faculty.vassar.edu/lowry/VassarStats.html + // and then exponentiating the random variate x. + // + + check_lognormal( + static_cast(0), // location + static_cast(5), // scale + static_cast(1), // x + static_cast(0.5), // p + static_cast(0.5), // q + tolerance); + check_lognormal( + static_cast(2), // location + static_cast(2), // scale + static_cast(exp(1.8)), // x + static_cast(0.46017), // p + static_cast(1-0.46017), // q + tolerance); + check_lognormal( + static_cast(2), // location + static_cast(2), // scale + static_cast(exp(2.2)), // x + static_cast(1-0.46017), // p + static_cast(0.46017), // q + tolerance); + check_lognormal( + static_cast(2), // location + static_cast(2), // scale + static_cast(exp(-1.4)), // x + static_cast(0.04457), // p + static_cast(1-0.04457), // q + tolerance); + check_lognormal( + static_cast(2), // location + static_cast(2), // scale + static_cast(exp(5.4)), // x + static_cast(1-0.04457), // p + static_cast(0.04457), // q + tolerance); + + check_lognormal( + static_cast(-3), // location + static_cast(5), // scale + static_cast(exp(-5.0)), // x + static_cast(0.34458), // p + static_cast(1-0.34458), // q + tolerance); + check_lognormal( + static_cast(-3), // location + static_cast(5), // scale + static_cast(exp(-1.0)), // x + static_cast(1-0.34458), // p + static_cast(0.34458), // q + tolerance); + check_lognormal( + static_cast(-3), // location + static_cast(5), // scale + static_cast(exp(-9.0)), // x + static_cast(0.11507), // p + static_cast(1-0.11507), // q + tolerance); + check_lognormal( + static_cast(-3), // location + static_cast(5), // scale + static_cast(exp(3.0)), // x + static_cast(1-0.11507), // p + static_cast(0.11507), // q + tolerance); + + // + // Tests for PDF + // + tolerance = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a percentage + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + BOOST_CHECK_CLOSE( + pdf(lognormal_distribution(), static_cast(1)), + static_cast(0.3989422804014326779399460599343818684759L), // 1/sqrt(2*pi) + tolerance); + BOOST_CHECK_CLOSE( + pdf(lognormal_distribution(3), exp(static_cast(3))), + static_cast(0.3989422804014326779399460599343818684759L) / exp(static_cast(3)), + tolerance); + BOOST_CHECK_CLOSE( + pdf(lognormal_distribution(3, 5), exp(static_cast(3))), + static_cast(0.3989422804014326779399460599343818684759L / (5 * exp(static_cast(3)))), + tolerance); + // + // Spot checks for location = -5, scale = 6, + // use relation to normal to test: + // + for(RealType x = -15; x < 5; x += 0.125) + { + BOOST_CHECK_CLOSE( + pdf(lognormal_distribution(-5, 6), exp(x)), + pdf(boost::math::normal_distribution(-5, 6), x) / exp(x), + tolerance); + } + + // + // These test values were obtained by punching numbers into + // a calculator, using the formulas at http://mathworld.wolfram.com/LogNormalDistribution.html + // + tolerance = (std::max)( + boost::math::tools::epsilon(), + static_cast(boost::math::tools::epsilon())) * 5 * 100; // 5 eps as a percentage + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + lognormal_distribution dist(8, 3); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(268337.28652087445695647967378715L), tolerance); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(583389737628117.49553037857325892L), tolerance); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , static_cast(24153462.228594009489719473727471L), tolerance); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tolerance); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tolerance); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tolerance); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(0.36787944117144232159552377016146L), tolerance); + + BOOST_CHECK_CLOSE( + median(dist) + , static_cast(exp(dist.location())), tolerance); + + BOOST_CHECK_CLOSE( + median(dist), + quantile(dist, static_cast(0.5)), tolerance); + + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , static_cast(729551.38304660255658441529235697L), tolerance); + // kertosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , static_cast(4312295840576303.2363383232038251L), tolerance); + // kertosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , static_cast(4312295840576300.2363383232038251L), tolerance); + + BOOST_CHECK_CLOSE( + range(dist).first + , static_cast(0), tolerance); + + // + // Special cases: + // + BOOST_CHECK(pdf(dist, 0) == 0); + BOOST_CHECK(cdf(dist, 0) == 0); + BOOST_CHECK(cdf(complement(dist, 0)) == 1); + BOOST_CHECK(quantile(dist, 0) == 0); + BOOST_CHECK(quantile(complement(dist, 1)) == 0); + + // + // Error checks: + // + BOOST_CHECK_THROW(lognormal_distribution(0, -1), std::domain_error); + BOOST_CHECK_THROW(pdf(dist, -1), std::domain_error); + BOOST_CHECK_THROW(cdf(dist, -1), std::domain_error); + BOOST_CHECK_THROW(cdf(complement(dist, -1)), std::domain_error); + BOOST_CHECK_THROW(quantile(dist, 1), std::overflow_error); + BOOST_CHECK_THROW(quantile(complement(dist, 0)), std::overflow_error); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + + // Check that can generate lognormal distribution using the two convenience methods: + boost::math::lognormal myf1(1., 2); // Using typedef + lognormal_distribution<> myf2(1., 2); // Using default RealType double. + + // Test range and support using double only, + // because it supports numeric_limits max for a pseudo-infinity. + BOOST_CHECK_EQUAL(range(myf2).first, 0); // range 0 to +infinity + BOOST_CHECK_EQUAL(range(myf2).second, (std::numeric_limits::max)()); + BOOST_CHECK_EQUAL(support(myf2).first, 0); // support 0 to + infinity. + BOOST_CHECK_EQUAL(support(myf2).second, (std::numeric_limits::max)()); + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* +Running 1 test case... +Tolerance for type float is 0.5 % +Tolerance for type float is 5.96046e-005 % +Tolerance for type float is 5.96046e-005 % +Tolerance for type double is 0.5 % +Tolerance for type double is 1.11022e-013 % +Tolerance for type double is 1.11022e-013 % +Tolerance for type long double is 0.5 % +Tolerance for type long double is 1.11022e-013 % +Tolerance for type long double is 1.11022e-013 % +Tolerance for type class boost::math::concepts::real_concept is 0.5 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-013 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-013 % +*** No errors detected +*/ + + diff --git a/test/test_minima.cpp b/test/test_minima.cpp new file mode 100644 index 000000000..8d2b4a50b --- /dev/null +++ b/test/test_minima.cpp @@ -0,0 +1,59 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include + +template +struct poly_test +{ + // minima is at (3,4): + T operator()(const T& v) + { + T a = v - 3; + return 3 * a * a + 4; + } +}; + +template +void test_minima(T, const char* /* name */) +{ + std::pair m = boost::math::tools::brent_find_minima(poly_test(), T(-10), T(10), 50); + BOOST_CHECK_CLOSE(m.first, T(3), T(0.001)); + BOOST_CHECK_CLOSE(m.second, T(4), T(0.001)); + + T (*fp)(T); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + fp = boost::math::lgamma; +#else + fp = boost::math::lgamma; +#endif + + m = boost::math::tools::brent_find_minima(fp, T(0.5), T(10), 50); + BOOST_CHECK_CLOSE(m.first, T(1.461632), T(0.1)); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + fp = boost::math::tgamma; +#else + fp = boost::math::tgamma; +#endif + m = boost::math::tools::brent_find_minima(fp, T(0.5), T(10), 50); + BOOST_CHECK_CLOSE(m.first, T(1.461632), T(0.1)); +} + +int test_main(int, char* []) +{ + test_minima(0.1f, "float"); + test_minima(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_minima(0.1L, "long double"); +#endif + return 0; +} + + diff --git a/test/test_negative_binomial.cpp b/test/test_negative_binomial.cpp new file mode 100644 index 000000000..64417d28c --- /dev/null +++ b/test/test_negative_binomial.cpp @@ -0,0 +1,850 @@ +// test_negative_binomial.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Tests for Negative Binomial Distribution. + +// Note that these defines must be placed BEFORE #includes. +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error +// because several tests overflow & underflow by design. +#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +#endif + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +#include // for negative_binomial_distribution +using boost::math::negative_binomial_distribution; + +#include + using boost::math::lgamma; // log gamma + +#include // for real_concept +using ::boost::math::concepts::real_concept; + +#include // for test_main +#include // for BOOST_CHECK_CLOSE + +#include +using std::cout; +using std::endl; +using std::setprecision; +using std::showpoint; +#include +using std::numeric_limits; + +template +void test_spot( // Test a single spot value against 'known good' values. + RealType N, // Number of successes. + RealType k, // Number of failures. + RealType p, // Probability of success_fraction. + RealType P, // CDF probability. + RealType Q, // Complement of CDF. + RealType tol) // Test tolerance. +{ + boost::math::negative_binomial_distribution bn(N, p); + BOOST_CHECK_EQUAL(N, bn.successes()); + BOOST_CHECK_EQUAL(p, bn.success_fraction()); + BOOST_CHECK_CLOSE( + cdf(bn, k), P, tol); + + if((P < 0.99) && (Q < 0.99)) + { + // We can only check this if P is not too close to 1, + // so that we can guarantee that Q is free of error: + // + BOOST_CHECK_CLOSE( + cdf(complement(bn, k)), Q, tol); + if(k != 0) + { + BOOST_CHECK_CLOSE( + quantile(bn, P), k, tol); + } + else + { + // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) + && (boost::is_floating_point::value)) + { + // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(bn, P) < boost::math::tools::epsilon() * 10); + } + } + if(k != 0) + { + BOOST_CHECK_CLOSE( + quantile(complement(bn, Q)), k, tol); + } + else + { + // Just check quantile is very small: + if((std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) + && (boost::is_floating_point::value)) + { + // Limit where this is checked: if exponent range is very large we may + // run out of iterations in our root finding algorithm. + BOOST_CHECK(quantile(complement(bn, Q)) < boost::math::tools::epsilon() * 10); + } + } + // estimate success ratio: + BOOST_CHECK_CLOSE( + negative_binomial_distribution::find_lower_bound_on_p( + N+k, N, P), + p, tol); + // Note we bump up the sample size here, purely for the sake of the test, + // internally the function has to adjust the sample size so that we get + // the right upper bound, our test undoes this, so we can verify the result. + BOOST_CHECK_CLOSE( + negative_binomial_distribution::find_upper_bound_on_p( + N+k+1, N, Q), + p, tol); + + if(Q < P) + { + // + // We check two things here, that the upper and lower bounds + // are the right way around, and that they do actually bracket + // the naive estimate of p = successes / (sample size) + // + BOOST_CHECK( + negative_binomial_distribution::find_lower_bound_on_p( + N+k, N, Q) + <= + negative_binomial_distribution::find_upper_bound_on_p( + N+k, N, Q) + ); + BOOST_CHECK( + negative_binomial_distribution::find_lower_bound_on_p( + N+k, N, Q) + <= + N / (N+k) + ); + BOOST_CHECK( + N / (N+k) + <= + negative_binomial_distribution::find_upper_bound_on_p( + N+k, N, Q) + ); + } + else + { + // As above but when P is small. + BOOST_CHECK( + negative_binomial_distribution::find_lower_bound_on_p( + N+k, N, P) + <= + negative_binomial_distribution::find_upper_bound_on_p( + N+k, N, P) + ); + BOOST_CHECK( + negative_binomial_distribution::find_lower_bound_on_p( + N+k, N, P) + <= + N / (N+k) + ); + BOOST_CHECK( + N / (N+k) + <= + negative_binomial_distribution::find_upper_bound_on_p( + N+k, N, P) + ); + } + + // Estimate sample size: + BOOST_CHECK_CLOSE( + negative_binomial_distribution::find_minimum_number_of_trials( + k, p, P), + N+k, tol); + BOOST_CHECK_CLOSE( + negative_binomial_distribution::find_maximum_number_of_trials( + k, p, Q), + N+k, tol); + + // Double check consistency of CDF and PDF by computing the finite sum: + RealType sum = 0; + for(unsigned i = 0; i <= k; ++i) + { + sum += pdf(bn, RealType(i)); + } + BOOST_CHECK_CLOSE(sum, P, tol); + + // Complement is not possible since sum is to infinity. + } // +} // test_spot + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks, test data is to double precision only + // so set tolerance to 1000 eps expressed as a percent, or + // 1000 eps of type double expressed as a percent, whichever + // is the larger. + + RealType tolerance = (std::max) + (boost::math::tools::epsilon(), + static_cast(std::numeric_limits::epsilon())); + tolerance *= 100 * 1000; + + cout << "Tolerance = " << tolerance << "%." << endl; + + RealType tol1eps = boost::math::tools::epsilon() * 2; // Very tight, suit exact values. + //RealType tol2eps = boost::math::tools::epsilon() * 2; // Tight, suit exact values. + RealType tol5eps = boost::math::tools::epsilon() * 5; // Wider 5 epsilon. + cout << "Tolerance 5 eps = " << tol5eps << "%." << endl; + + // Sources of spot test values: + + // MathCAD defines pbinom(k, r, p) (at about 64-bit double precision, about 16 decimal digits) + // returns pr(X , k) when random variable X has the binomial distribution with parameters r and p. + // 0 <= k + // r > 0 + // 0 <= p <= 1 + // P = pbinom(30, 500, 0.05) = 0.869147702104609 + + // And functions.wolfram.com + + using boost::math::negative_binomial_distribution; + using ::boost::math::negative_binomial; + using ::boost::math::cdf; + using ::boost::math::pdf; + + // Test negative binomial using cdf spot values from MathCAD cdf = pnbinom(k, r, p). + // These test quantiles and complements as well. + + test_spot( // pnbinom(1,2,0.5) = 0.5 + static_cast(2), // successes r + static_cast(1), // Number of failures, k + static_cast(0.5), // Probability of success as fraction, p + static_cast(0.5), // Probability of result (CDF), P + static_cast(0.5), // complement CCDF Q = 1 - P + tolerance); + + test_spot( // pbinom(0, 2, 0.25) + static_cast(2), // successes r + static_cast(0), // Number of failures, k + static_cast(0.25), + static_cast(0.0625), // Probability of result (CDF), P + static_cast(0.9375), // Q = 1 - P + tolerance); + + test_spot( // pbinom(48,8,0.25) + static_cast(8), // successes r + static_cast(48), // Number of failures, k + static_cast(0.25), // Probability of success, p + static_cast(9.826582228110670E-1), // Probability of result (CDF), P + static_cast(1 - 9.826582228110670E-1), // Q = 1 - P + tolerance); + + test_spot( // pbinom(2,5,0.4) + static_cast(5), // successes r + static_cast(2), // Number of failures, k + static_cast(0.4), // Probability of success, p + static_cast(9.625600000000020E-2), // Probability of result (CDF), P + static_cast(1 - 9.625600000000020E-2), // Q = 1 - P + tolerance); + + test_spot( // pbinom(10,100,0.9) + static_cast(100), // successes r + static_cast(10), // Number of failures, k + static_cast(0.9), // Probability of success, p + static_cast(4.535522887695670E-1), // Probability of result (CDF), P + static_cast(1 - 4.535522887695670E-1), // Q = 1 - P + tolerance); + + test_spot( // pbinom(1,100,0.991) + static_cast(100), // successes r + static_cast(1), // Number of failures, k + static_cast(0.991), // Probability of success, p + static_cast(7.693413044217000E-1), // Probability of result (CDF), P + static_cast(1 - 7.693413044217000E-1), // Q = 1 - P + tolerance); + + test_spot( // pbinom(10,100,0.991) + static_cast(100), // successes r + static_cast(10), // Number of failures, k + static_cast(0.991), // Probability of success, p + static_cast(9.999999940939000E-1), // Probability of result (CDF), P + static_cast(1 - 9.999999940939000E-1), // Q = 1 - P + tolerance); + +if(std::numeric_limits::is_specialized) +{ // An extreme value test that takes 3 minutes using the real concept type + // for which numeric_limits::is_specialized == false, deliberately + // and for which there is no Lanczos approximation defined (also deliberately) + // giving a very slow computation, but with acceptable accuracy. + // A possible enhancement might be to use a normal approximation for + // extreme values, but this is not implemented. + test_spot( // pbinom(100000,100,0.001) + static_cast(100), // successes r + static_cast(100000), // Number of failures, k + static_cast(0.001), // Probability of success, p + static_cast(5.173047534260320E-1), // Probability of result (CDF), P + static_cast(1 - 5.173047534260320E-1), // Q = 1 - P + tolerance*1000); // *1000 is OK 0.51730475350664229 versus + + // functions.wolfram.com + // for I[0.001](100, 100000+1) gives: + // Wolfram 0.517304753506834882009032744488738352004003696396461766326713 + // JM nonLanczos 0.51730475350664229 differs at the 13th decimal digit. + // MathCAD 0.51730475342603199 differs at 10th decimal digit. +} + // End of single spot tests using RealType + + + // Tests on PDF: + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(2), static_cast(0.5)), + static_cast(0) ), // k = 0. + static_cast(0.25), // 0 + tolerance); + + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(4), static_cast(0.5)), + static_cast(0)), // k = 0. + static_cast(0.0625), // exact 1/16 + tolerance); + + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(0)), // k = 0 + static_cast(9.094947017729270E-13), // pbinom(0,20,0.25) = 9.094947017729270E-13 + tolerance); + + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(20), static_cast(0.2)), + static_cast(0)), // k = 0 + static_cast(1.0485760000000003e-014), // MathCAD 1.048576000000000E-14 + tolerance); + + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(10), static_cast(0.1)), + static_cast(0)), // k = 0. + static_cast(1e-10), // MathCAD says zero, but suffers cancellation error? + tolerance); + + BOOST_CHECK_CLOSE( + pdf(negative_binomial_distribution(static_cast(20), static_cast(0.1)), + static_cast(0)), // k = 0. + static_cast(1e-20), // MathCAD says zero, but suffers cancellation error? + tolerance); + + + BOOST_CHECK_CLOSE( // . + pdf(negative_binomial_distribution(static_cast(20), static_cast(0.9)), + static_cast(0)), // k. + static_cast(1.215766545905690E-1), // k=20 p = 0.9 + tolerance); + + // Tests on cdf: + // MathCAD pbinom k, r, p) == failures, successes, probability. + + BOOST_CHECK_CLOSE(cdf( + negative_binomial_distribution(static_cast(2), static_cast(0.5)), // successes = 2,prob 0.25 + static_cast(0) ), // k = 0 + static_cast(0.25), // probability 1/4 + tolerance); + + BOOST_CHECK_CLOSE(cdf(complement( + negative_binomial_distribution(static_cast(2), static_cast(0.5)), // successes = 2,prob 0.25 + static_cast(0) )), // k = 0 + static_cast(0.75), // probability 3/4 + tolerance); + BOOST_CHECK_CLOSE( // k = 1. + cdf(negative_binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(1)), // k =1. + static_cast(1.455191522836700E-11), + tolerance); + + BOOST_CHECK_SMALL( // Check within an epsilon with CHECK_SMALL + cdf(negative_binomial_distribution(static_cast(20), static_cast(0.25)), + static_cast(1)) - + static_cast(1.455191522836700E-11), + tolerance ); + + // Some exact (probably - judging by trailing zeros) values. + BOOST_CHECK_CLOSE( + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)), // k. + static_cast(1.525878906250000E-5), + tolerance); + + BOOST_CHECK_CLOSE( + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)), // k. + static_cast(1.525878906250000E-5), + tolerance); + + BOOST_CHECK_SMALL( + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)) - + static_cast(1.525878906250000E-5), + tolerance ); + + BOOST_CHECK_CLOSE( // k = 1. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1)), // k. + static_cast(1.068115234375010E-4), + tolerance); + + BOOST_CHECK_CLOSE( // k = 2. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(2)), // k. + static_cast(4.158020019531300E-4), + tolerance); + + BOOST_CHECK_CLOSE( // k = 3. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(3)), // k.bristow + static_cast(1.188278198242200E-3), + tolerance); + + BOOST_CHECK_CLOSE( // k = 4. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(4)), // k. + static_cast(2.781510353088410E-3), + tolerance); + + BOOST_CHECK_CLOSE( // k = 5. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(5)), // k. + static_cast(5.649328231811500E-3), + tolerance); + + BOOST_CHECK_CLOSE( // k = 6. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(6)), // k. + static_cast(1.030953228473680E-2), + tolerance); + + BOOST_CHECK_CLOSE( // k = 7. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(7)), // k. + static_cast(1.729983836412430E-2), + tolerance); + + BOOST_CHECK_CLOSE( // k = 8. + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(8)), // k = n. + static_cast(2.712995628826370E-2), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(48)), // k + static_cast(9.826582228110670E-1), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(64)), // k + static_cast(9.990295004935590E-1), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(5), static_cast(0.4)), + static_cast(26)), // k + static_cast(9.989686246611190E-1), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(5), static_cast(0.4)), + static_cast(2)), // k failures + static_cast(9.625600000000020E-2), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(50), static_cast(0.9)), + static_cast(20)), // k + static_cast(9.999970854144170E-1), + tolerance); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(500), static_cast(0.7)), + static_cast(200)), // k + static_cast(2.172846379930550E-1), + tolerance* 2); + + BOOST_CHECK_CLOSE( // + cdf(negative_binomial_distribution(static_cast(50), static_cast(0.7)), + static_cast(20)), // k + static_cast(4.550203671301790E-1), + tolerance); + + // Tests of other functions, mean and other moments ... + + negative_binomial_distribution dist(static_cast(8), static_cast(0.25)); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist), static_cast(8 * (1 - 0.25) /0.25), tol5eps); + BOOST_CHECK_CLOSE( + mode(dist), static_cast(21), tol1eps); + // variance: + BOOST_CHECK_CLOSE( + variance(dist), static_cast(8 * (1 - 0.25) / (0.25 * 0.25)), tol5eps); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist), // 9.79795897113271239270 + static_cast(9.797958971132712392789136298823565567864L), // using functions.wolfram.com + // 9.79795897113271152534 == sqrt(8 * (1 - 0.25) / (0.25 * 0.25))) + tol5eps * 100); + BOOST_CHECK_CLOSE( + skewness(dist), // + static_cast(0.71443450831176036), + // using http://mathworld.wolfram.com/skewness.html + tolerance); + BOOST_CHECK_CLOSE( + kurtosis_excess(dist), // + static_cast(0.7604166666666666666666666666666666666666L), // using Wikipedia Kurtosis(excess) formula + tol5eps * 100); + BOOST_CHECK_CLOSE( + kurtosis(dist), // true + static_cast(3.76041666666666666666666666666666666666666L), // + tol5eps * 100); + // hazard: + RealType x = static_cast(0.125); + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol5eps); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x), -log(cdf(complement(dist, x))), tol5eps); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol5eps); + + // Special cases for PDF: + BOOST_CHECK_EQUAL( + pdf( + negative_binomial_distribution(static_cast(8), static_cast(0)), // + static_cast(0)), + static_cast(0) ); + + BOOST_CHECK_EQUAL( + pdf( + negative_binomial_distribution(static_cast(8), static_cast(0)), + static_cast(0.0001)), + static_cast(0) ); + + BOOST_CHECK_EQUAL( + pdf( + negative_binomial_distribution(static_cast(8), static_cast(1)), + static_cast(0.001)), + static_cast(0) ); + + BOOST_CHECK_EQUAL( + pdf( + negative_binomial_distribution(static_cast(8), static_cast(1)), + static_cast(8)), + static_cast(0) ); + + BOOST_CHECK_SMALL( + pdf( + negative_binomial_distribution(static_cast(2), static_cast(0.25)), + static_cast(0))- + static_cast(0.0625), + 2 * boost::math::tools::epsilon() ); // Expect exact, but not quite. + // numeric_limits::epsilon()); // Not suitable for real concept! + + // Quantile boundary cases checks: + BOOST_CHECK_EQUAL( + quantile( // zero P < cdf(0) so should be exactly zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)), + static_cast(0)); + + BOOST_CHECK_EQUAL( + quantile( // min P < cdf(0) so should be exactly zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(boost::math::tools::min_value())), + static_cast(0)); + + BOOST_CHECK_CLOSE_FRACTION( + quantile( // Small P < cdf(0) so should be near zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(boost::math::tools::epsilon())), // + static_cast(0), + tol5eps); + + BOOST_CHECK_CLOSE( + quantile( // Small P < cdf(0) so should be exactly zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0.0001)), + static_cast(0.95854156929288470), + tolerance); + + //BOOST_CHECK( // Fails with overflow for real_concept + //quantile( // Small P near 1 so k failures should be big. + //negative_binomial_distribution(static_cast(8), static_cast(0.25)), + //static_cast(1 - boost::math::tools::epsilon())) <= + //static_cast(189.56999032670058) // 106.462769 for float + //); + + if(std::numeric_limits::has_infinity) + { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() + // Note that infinity is not implemented for real_concept, so these tests + // are only done for types, like built-in float, double.. that have infinity. + // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. + // #define BOOST_MATH_THROW_ON_OVERFLOW_POLICY == throw_on_error would throw here. + // #define BOOST_MAT_DOMAIN_ERROR_POLICY IS defined throw_on_error, + // so the throw path of error handling is tested below with BOOST_CHECK_THROW tests. + + BOOST_CHECK( + quantile( // At P == 1 so k failures should be infinite. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1)) == + //static_cast(boost::math::tools::infinity()) + static_cast(std::numeric_limits::infinity()) ); + + BOOST_CHECK_EQUAL( + quantile( // At 1 == P so should be infinite. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1)), // + std::numeric_limits::infinity() ); + + BOOST_CHECK_EQUAL( + quantile(complement( // Q zero 1 so P == 1 < cdf(0) so should be exactly infinity. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0))), + std::numeric_limits::infinity() ); + } // test for infinity using std::numeric_limits<>::infinity() + else + { // real_concept case, so check it throws rather than returning infinity. + BOOST_CHECK_EQUAL( + quantile( // At P == 1 so k failures should be infinite. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1)), + boost::math::tools::max_value() ); + + BOOST_CHECK_EQUAL( + quantile(complement( // Q zero 1 so P == 1 < cdf(0) so should be exactly infinity. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0))), + boost::math::tools::max_value()); + } + BOOST_CHECK( // Should work for built-in and real_concept. + quantile(complement( // Q very near to 1 so P nearly 1 < so should be large > 384. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(boost::math::tools::min_value()))) + >= static_cast(384) ); + + BOOST_CHECK_EQUAL( + quantile( // P == 0 < cdf(0) so should be zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(0)), + static_cast(0)); + + // Quantile Complement boundary cases: + + BOOST_CHECK_EQUAL( + quantile(complement( // Q = 1 so P = 0 < cdf(0) so should be exactly zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1))), + static_cast(0) + ); + + BOOST_CHECK_EQUAL( + quantile(complement( // Q very near 1 so P == epsilon < cdf(0) so should be exactly zero. + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(1 - boost::math::tools::epsilon()))), + static_cast(0) + ); + + // Check that duff arguments throw domain_error: + BOOST_CHECK_THROW( + pdf( // Negative successes! + negative_binomial_distribution(static_cast(-1), static_cast(0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( // Negative success_fraction! + negative_binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + pdf( // Success_fraction > 1! + negative_binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), + std::domain_error + ); + BOOST_CHECK_THROW( + pdf( // Negative k argument ! + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(-1)), + std::domain_error + ); + //BOOST_CHECK_THROW( + //pdf( // Unlike binomial there is NO limit on k (failures) + //negative_binomial_distribution(static_cast(8), static_cast(0.25)), + //static_cast(9)), std::domain_error + //); + BOOST_CHECK_THROW( + cdf( // Negative k argument ! + negative_binomial_distribution(static_cast(8), static_cast(0.25)), + static_cast(-1)), + std::domain_error + ); + BOOST_CHECK_THROW( + cdf( // Negative success_fraction! + negative_binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + cdf( // Success_fraction > 1! + negative_binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( // Negative success_fraction! + negative_binomial_distribution(static_cast(8), static_cast(-0.25)), + static_cast(0)), std::domain_error + ); + BOOST_CHECK_THROW( + quantile( // Success_fraction > 1! + negative_binomial_distribution(static_cast(8), static_cast(1.25)), + static_cast(0)), std::domain_error + ); + // End of check throwing 'duff' out-of-domain values. + +#define T RealType +#include "negative_binomial_quantile.ipp" + + for(unsigned i = 0; i < negative_binomial_quantile_data.size(); ++i) + { + using namespace boost::math::policies; + typedef policy > P1; + typedef policy > P2; + typedef policy > P3; + typedef policy > P4; + typedef policy > P5; + typedef policy > P6; + RealType tol = boost::math::tools::epsilon() * 700; + if(!boost::is_floating_point::value) + tol *= 10; // no lanczos approximation implies less accuracy + // + // Check full real value first: + // + negative_binomial_distribution p1(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + RealType x = quantile(p1, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_CLOSE_FRACTION(x, negative_binomial_quantile_data[i][3], tol); + x = quantile(complement(p1, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_CLOSE_FRACTION(x, negative_binomial_quantile_data[i][4], tol); + // + // Now with round down to integer: + // + negative_binomial_distribution p2(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + x = quantile(p2, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, floor(negative_binomial_quantile_data[i][3])); + x = quantile(complement(p2, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, floor(negative_binomial_quantile_data[i][4])); + // + // Now with round up to integer: + // + negative_binomial_distribution p3(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + x = quantile(p3, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, ceil(negative_binomial_quantile_data[i][3])); + x = quantile(complement(p3, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, ceil(negative_binomial_quantile_data[i][4])); + // + // Now with round to integer "outside": + // + negative_binomial_distribution p4(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + x = quantile(p4, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, negative_binomial_quantile_data[i][2] < 0.5f ? floor(negative_binomial_quantile_data[i][3]) : ceil(negative_binomial_quantile_data[i][3])); + x = quantile(complement(p4, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, negative_binomial_quantile_data[i][2] < 0.5f ? ceil(negative_binomial_quantile_data[i][4]) : floor(negative_binomial_quantile_data[i][4])); + // + // Now with round to integer "inside": + // + negative_binomial_distribution p5(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + x = quantile(p5, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, negative_binomial_quantile_data[i][2] < 0.5f ? ceil(negative_binomial_quantile_data[i][3]) : floor(negative_binomial_quantile_data[i][3])); + x = quantile(complement(p5, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, negative_binomial_quantile_data[i][2] < 0.5f ? floor(negative_binomial_quantile_data[i][4]) : ceil(negative_binomial_quantile_data[i][4])); + // + // Now with round to nearest integer: + // + negative_binomial_distribution p6(negative_binomial_quantile_data[i][0], negative_binomial_quantile_data[i][1]); + x = quantile(p6, negative_binomial_quantile_data[i][2]); + BOOST_CHECK_EQUAL(x, floor(negative_binomial_quantile_data[i][3] + 0.5f)); + x = quantile(complement(p6, negative_binomial_quantile_data[i][2])); + BOOST_CHECK_EQUAL(x, floor(negative_binomial_quantile_data[i][4] + 0.5f)); + } + + return; +} // template void test_spots(RealType) // Any floating-point type RealType. + +int test_main(int, char* []) +{ + // Check that can generate negative_binomial distribution using the two convenience methods: + using namespace boost::math; + negative_binomial mynb1(2., 0.5); // Using typedef - default type is double. + negative_binomial_distribution<> myf2(2., 0.5); // Using default RealType double. + + // Basic sanity-check spot values. + + // Test some simple double only examples. + negative_binomial_distribution my8dist(8., 0.25); + // 8 successes (r), 0.25 success fraction = 35% or 1 in 4 successes. + // Note: double values (matching the distribution definition) avoid the need for any casting. + + // Check accessor functions return exact values for double at least. + BOOST_CHECK_EQUAL(my8dist.successes(), static_cast(8)); + BOOST_CHECK_EQUAL(my8dist.success_fraction(), static_cast(1./4.)); + + // (Parameter value, arbitrarily zero, only communicates the floating point type). +#ifdef TEST_FLOAT + test_spots(0.0F); // Test float. +#endif +#ifdef TEST_DOUBLE + test_spots(0.0); // Test double. +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS +#ifdef TEST_LDOUBLE + test_spots(0.0L); // Test long double. +#endif + #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif + #endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_negative_binomial.exe" +Running 1 test case... +Tolerance = 0.0119209%. +Tolerance 5 eps = 5.96046e-007%. +Tolerance = 2.22045e-011%. +Tolerance 5 eps = 1.11022e-015%. +Tolerance = 2.22045e-011%. +Tolerance 5 eps = 1.11022e-015%. +Tolerance = 2.22045e-011%. +Tolerance 5 eps = 1.11022e-015%. +*** No errors detected + +*/ diff --git a/test/test_normal.cpp b/test/test_normal.cpp new file mode 100644 index 000000000..a4012e071 --- /dev/null +++ b/test/test_normal.cpp @@ -0,0 +1,329 @@ +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_normal.cpp + +// http://en.wikipedia.org/wiki/Normal_distribution +// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm +// Also: +// Weisstein, Eric W. "Normal Distribution." +// From MathWorld--A Wolfram Web Resource. +// http://mathworld.wolfram.com/NormalDistribution.html + +#ifdef _MSC_VER +#pragma warning (disable: 4127) // conditional expression is constant +// caused by using if(std::numeric_limits::has_infinity) +// and if (std::numeric_limits::has_quiet_NaN) +#endif + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::normal_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + +template +RealType NaivePDF(RealType mean, RealType sd, RealType x) +{ + // Deliberately naive PDF calculator again which + // we'll compare our pdf function. However some + // published values to compare against would be better.... + using namespace std; + return exp(-(x-mean)*(x-mean)/(2*sd*sd))/(sd * sqrt(2*boost::math::constants::pi())); +} + +template +void check_normal(RealType mean, RealType sd, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + normal_distribution(mean, sd), // distribution. + x), // random variable. + p, // probability. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement( + normal_distribution(mean, sd), // distribution. + x)), // random variable. + q, // probability complement. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + normal_distribution(mean, sd), // distribution. + p), // probability. + x, // random variable. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement( + normal_distribution(mean, sd), // distribution. + q)), // probability complement. + x, // random variable. + tol); // %tolerance. +} + +template +void test_spots(RealType) +{ + // Basic sanity checks + RealType tolerance = 1e-2f; // 1e-4 (as %) + // Some tests only pass at 1e-4 because values generated by + // http://faculty.vassar.edu/lowry/VassarStats.html + // give only 5 or 6 *fixed* places, so small values have fewer digits. + + // Check some bad parameters to the distribution, + BOOST_CHECK_THROW(boost::math::normal_distribution nbad1(0, 0), std::domain_error); // zero sd + BOOST_CHECK_THROW(boost::math::normal_distribution nbad1(0, -1), std::domain_error); // negative sd + + // Tests on extreme values of random variate x, if has numeric_limit infinity etc. + normal_distribution N01; + if(std::numeric_limits::has_infinity) + { + BOOST_CHECK_EQUAL(pdf(N01, +std::numeric_limits::infinity()), 0); // x = + infinity, pdf = 0 + BOOST_CHECK_EQUAL(pdf(N01, -std::numeric_limits::infinity()), 0); // x = - infinity, pdf = 0 + BOOST_CHECK_EQUAL(cdf(N01, +std::numeric_limits::infinity()), 1); // x = + infinity, cdf = 1 + BOOST_CHECK_EQUAL(cdf(N01, -std::numeric_limits::infinity()), 0); // x = - infinity, cdf = 0 + BOOST_CHECK_EQUAL(cdf(complement(N01, +std::numeric_limits::infinity())), 0); // x = + infinity, c cdf = 0 + BOOST_CHECK_EQUAL(cdf(complement(N01, -std::numeric_limits::infinity())), 1); // x = - infinity, c cdf = 1 + BOOST_CHECK_THROW(boost::math::normal_distribution nbad1(std::numeric_limits::infinity(), static_cast(1)), std::domain_error); // +infinite mean + BOOST_CHECK_THROW(boost::math::normal_distribution nbad1(-std::numeric_limits::infinity(), static_cast(1)), std::domain_error); // -infinite mean + BOOST_CHECK_THROW(boost::math::normal_distribution nbad1(static_cast(0), std::numeric_limits::infinity()), std::domain_error); // infinite sd + } + + if (std::numeric_limits::has_quiet_NaN) + { + // No longer allow x to be NaN, then these tests should throw. + BOOST_CHECK_THROW(pdf(N01, +std::numeric_limits::quiet_NaN()), std::domain_error); // x = NaN + BOOST_CHECK_THROW(cdf(N01, +std::numeric_limits::quiet_NaN()), std::domain_error); // x = NaN + BOOST_CHECK_THROW(cdf(complement(N01, +std::numeric_limits::quiet_NaN())), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(quantile(N01, +std::numeric_limits::quiet_NaN()), std::domain_error); // p = + infinity + BOOST_CHECK_THROW(quantile(complement(N01, +std::numeric_limits::quiet_NaN())), std::domain_error); // p = + infinity + } + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + check_normal( + static_cast(5), + static_cast(2), + static_cast(4.8), + static_cast(0.46017), + static_cast(1 - 0.46017), + tolerance); + + check_normal( + static_cast(5), + static_cast(2), + static_cast(5.2), + static_cast(1 - 0.46017), + static_cast(0.46017), + tolerance); + + check_normal( + static_cast(5), + static_cast(2), + static_cast(2.2), + static_cast(0.08076), + static_cast(1 - 0.08076), + tolerance); + + check_normal( + static_cast(5), + static_cast(2), + static_cast(7.8), + static_cast(1 - 0.08076), + static_cast(0.08076), + tolerance); + + check_normal( + static_cast(-3), + static_cast(5), + static_cast(-4.5), + static_cast(0.38209), + static_cast(1 - 0.38209), + tolerance); + + check_normal( + static_cast(-3), + static_cast(5), + static_cast(-1.5), + static_cast(1 - 0.38209), + static_cast(0.38209), + tolerance); + + check_normal( + static_cast(-3), + static_cast(5), + static_cast(-8.5), + static_cast(0.13567), + static_cast(1 - 0.13567), + tolerance); + + check_normal( + static_cast(-3), + static_cast(5), + static_cast(2.5), + static_cast(1 - 0.13567), + static_cast(0.13567), + tolerance); + + // + // Tests for PDF: we know that the peak value is at 1/sqrt(2*pi) + // + tolerance = boost::math::tools::epsilon() * 5 * 100; // 5 eps as a percentage + BOOST_CHECK_CLOSE( + pdf(normal_distribution(), static_cast(0)), + static_cast(0.3989422804014326779399460599343818684759L), // 1/sqrt(2*pi) + tolerance); + BOOST_CHECK_CLOSE( + pdf(normal_distribution(3), static_cast(3)), + static_cast(0.3989422804014326779399460599343818684759L), + tolerance); + BOOST_CHECK_CLOSE( + pdf(normal_distribution(3, 5), static_cast(3)), + static_cast(0.3989422804014326779399460599343818684759L / 5), + tolerance); + + // + // Spot checks for mean = -5, sd = 6: + // + for(RealType x = -15; x < 5; x += 0.125) + { + BOOST_CHECK_CLOSE( + pdf(normal_distribution(-5, 6), x), + NaivePDF(RealType(-5), RealType(6), x), + tolerance); + } + + RealType tol2 = boost::math::tools::epsilon() * 5; + normal_distribution dist(8, 3); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(8), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(9), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , static_cast(3), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tol2); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , static_cast(8), tol2); + + BOOST_CHECK_CLOSE( + median(dist) + , static_cast(8), tol2); + + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , static_cast(0), tol2); + // kertosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , static_cast(3), tol2); + // kertosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , static_cast(0), tol2); + + normal_distribution norm01(0, 1); // Test default (0, 1) + BOOST_CHECK_CLOSE( + mean(norm01), + static_cast(0), 0); // Mean == zero + + normal_distribution defsd_norm01(0); // Test default (0, sd = 1) + BOOST_CHECK_CLOSE( + mean(defsd_norm01), + static_cast(0), 0); // Mean == zero + + normal_distribution def_norm01; // Test default (0, sd = 1) + BOOST_CHECK_CLOSE( + mean(def_norm01), + static_cast(0), 0); // Mean == zero + + BOOST_CHECK_CLOSE( + standard_deviation(def_norm01), + static_cast(1), 0); // Mean == zero + + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can generate normal distribution using the two convenience methods: + boost::math::normal myf1(1., 2); // Using typedef + normal_distribution<> myf2(1., 2); // Using default RealType double. + boost::math::normal myn01; // Use default values. + // Note NOT myn01() as the compiler will interpret as a function! + + // Check the synonyms, provided to allow generic use of find_location and find_scale. + BOOST_CHECK_EQUAL(myn01.mean(), myn01.location()); + BOOST_CHECK_EQUAL(myn01.standard_deviation(), myn01.scale()); + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_normal.exe" +Running 1 test case... +Tolerance for type float is 0.01 % +Tolerance for type double is 0.01 % +Tolerance for type long double is 0.01 % +Tolerance for type class boost::math::concepts::real_concept is 0.01 % +*** No errors detected + +*/ + + diff --git a/test/test_pareto.cpp b/test/test_pareto.cpp new file mode 100644 index 000000000..7c80d6cef --- /dev/null +++ b/test/test_pareto.cpp @@ -0,0 +1,333 @@ +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_pareto.cpp + +// http://en.wikipedia.org/wiki/pareto_distribution +// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm +// Also: +// Weisstein, Eric W. "pareto Distribution." +// From MathWorld--A Wolfram Web Resource. +// http://mathworld.wolfram.com/paretoDistribution.html + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +# pragma warning(disable: 4100) // unreferenced formal parameter. +#endif + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::pareto_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + + template + void check_pareto(RealType location, RealType shape, RealType x, RealType p, RealType q, RealType tol) + { + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + pareto_distribution(location, shape), // distribution. + x), // random variable. + p, // probability. + tol); // tolerance eps. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + complement( + pareto_distribution(location, shape), // distribution. + x)), // random variable. + q, // probability complement. + tol); // tolerance eps. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + pareto_distribution(location, shape), // distribution. + p), // probability. + x, // random variable. + tol); // tolerance eps. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + complement( + pareto_distribution(location, shape), // distribution. + q)), // probability complement. + x, // random variable. + tol); // tolerance eps. + } // check_pareto + +template +void test_spots(RealType) +{ + // Basic sanity checks. + // + // Tolerance are based on units of epsilon, but capped at + // double precision, since that's the limit of out test data: + // + RealType tol = (std::max)((RealType)boost::math::tools::epsilon(), boost::math::tools::epsilon()); + RealType tol5eps = tol * 5; + RealType tol10eps = tol * 10; + RealType tol100eps = tol * 100; + RealType tol1000eps = tol * 1000; + + check_pareto( + static_cast(1.1L), // + static_cast(5.5L), + static_cast(2.2L), + static_cast(0.97790291308792L), + static_cast(0.0220970869120796L), + tol10eps * 4); + + check_pareto( + static_cast(0.5L), + static_cast(10.1L), + static_cast(1.5L), + static_cast(0.99998482686481L), + static_cast(1.51731351900608e-005L), + tol100eps * 1000); // Much less accurate as p close to unity. + + check_pareto( + static_cast(0.1L), + static_cast(2.3L), + static_cast(1.5L), + static_cast(0.99802762220697L), + static_cast(0.00197237779302972L), + tol1000eps); + + // Example from 23.3 page 259 + check_pareto( + static_cast(2.30444301457005L), + static_cast(4), + static_cast(2.4L), + static_cast(0.15L), + static_cast(0.85L), + tol100eps); + + check_pareto( + static_cast(2), + static_cast(3), + static_cast(3.4L), + static_cast(0.796458375737838L), + static_cast(0.203541624262162L), + tol10eps); + + check_pareto( // Probability near 0.5 + static_cast(2), + static_cast(2), + static_cast(3), + static_cast(0.5555555555555555555555555555555555555556L), + static_cast(0.4444444444444444444444444444444444444444L), + tol5eps); // accurate. + + + // Tests for: + + // pdf for shapes 1, 2 & 3 (exact) + BOOST_CHECK_CLOSE_FRACTION( + pdf(pareto_distribution(1, 1), 1), + static_cast(1), // + tol5eps); + + BOOST_CHECK_CLOSE_FRACTION( pdf(pareto_distribution(1, 2), 1), + static_cast(2), // + tol5eps); + + BOOST_CHECK_CLOSE_FRACTION( pdf(pareto_distribution(1, 3), 1), + static_cast(3), // + tol5eps); + + // cdf + BOOST_CHECK_EQUAL( // x = location + cdf(pareto_distribution(1, 1), 1), + static_cast(0) ); + + // Compare with values from StatCalc K. Krishnamoorthy, ISBN 1-58488-635-8 eq 23.1.3 + BOOST_CHECK_CLOSE_FRACTION( // small x + cdf(pareto_distribution(2, 5), static_cast(3.4)), + static_cast(0.929570372227626L), tol5eps); + + BOOST_CHECK_CLOSE_FRACTION( // small x + cdf(pareto_distribution(2, 5), static_cast(3.4)), + static_cast(1 - 0.0704296277723743L), tol5eps); + + BOOST_CHECK_CLOSE_FRACTION( // small x + cdf(complement(pareto_distribution(2, 5), static_cast(3.4))), + static_cast(0.0704296277723743L), tol5eps); + + // quantile + BOOST_CHECK_EQUAL( // x = location + quantile(pareto_distribution(1, 1), 0), + static_cast(1) ); + + BOOST_CHECK_EQUAL( // x = location + quantile(complement(pareto_distribution(1, 1), 1)), + static_cast(1) ); + + BOOST_CHECK_CLOSE_FRACTION( // small x + cdf(complement(pareto_distribution(2, 5), static_cast(3.4))), + static_cast(0.0704296277723743L), tol5eps); + + using namespace std; // ADL of std names. + + pareto_distribution pareto15(1, 5); + // Note: shape must be big enough (5) that all moments up to kurtosis are defined + // to allow all functions to be tested. + + // mean: + BOOST_CHECK_CLOSE_FRACTION( + mean(pareto15), static_cast(1.25), tol5eps); // 1.25 == 5/4 + BOOST_CHECK_EQUAL( + mean(pareto15), static_cast(1.25)); // 1.25 == 5/4 (expect exact so check equal) + + pareto_distribution p12(1, 2); // + BOOST_CHECK_EQUAL( + mean(p12), static_cast(2)); // Exactly two. + + // variance: + BOOST_CHECK_CLOSE_FRACTION( + variance(pareto15), static_cast(0.10416666666666667L), tol5eps); + // std deviation: + BOOST_CHECK_CLOSE_FRACTION( + standard_deviation(pareto15), static_cast(0.32274861218395140L), tol5eps); + // hazard: No independent test values found yet. + //BOOST_CHECK_CLOSE_FRACTION( + // hazard(pareto15, x), pdf(pareto15, x) / cdf(complement(pareto15, x)), tol5eps); + //// cumulative hazard: + //BOOST_CHECK_CLOSE_FRACTION( + // chf(pareto15, x), -log(cdf(complement(pareto15, x))), tol5eps); + //// coefficient_of_variation: + BOOST_CHECK_CLOSE_FRACTION( + coefficient_of_variation(pareto15), static_cast(0.25819888974716110L), tol5eps); + // mode: + BOOST_CHECK_CLOSE_FRACTION( + mode(pareto15), static_cast(1), tol5eps); + + BOOST_CHECK_CLOSE_FRACTION( + median(pareto15), static_cast(1.1486983549970351L), tol5eps); + + // skewness: + BOOST_CHECK_CLOSE_FRACTION( + skewness(pareto15), static_cast(4.6475800154489004L), tol5eps); + // kertosis: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis(pareto15), static_cast(73.8L), tol5eps); + // kertosis excess: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(pareto15), static_cast(70.8L), tol5eps); + // Check difference between kurtosis and excess: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(pareto15), kurtosis(pareto15) - static_cast(3L), tol5eps); + // Check kurtosis excess = kurtosis - 3; + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can generate pareto distribution using the two convenience methods: + boost::math::pareto myp1(1., 1); // Using typedef + pareto_distribution<> myp2(1., 1); // Using default RealType double. + boost::math::pareto pareto11; // Use default values (location = 1, shape = 1). + // Note NOT pareto11() as the compiler will interpret as a function! + // Basic sanity-check spot values. + + BOOST_CHECK_EQUAL(pareto11.location(), 1); // Check defaults again. + BOOST_CHECK_EQUAL(pareto11.shape(), 1); + BOOST_CHECK_EQUAL(myp1.location(), 1); + BOOST_CHECK_EQUAL(myp1.shape(), 1); + BOOST_CHECK_EQUAL(myp2.location(), 1); + BOOST_CHECK_EQUAL(myp2.shape(), 1); + + // Test range and support using double only, + // because it supports numeric_limits max for pseudo-infinity. + BOOST_CHECK_EQUAL(range(myp2).first, 0); // range 0 to +infinity + BOOST_CHECK_EQUAL(range(myp2).second, (numeric_limits::max)()); + BOOST_CHECK_EQUAL(support(myp2).first, myp2.location()); // support location to + infinity. + BOOST_CHECK_EQUAL(support(myp2).second, (numeric_limits::max)()); + + // Check some bad parameters to the distribution. + BOOST_CHECK_THROW(boost::math::pareto mypm1(-1, 1), std::domain_error); // Using typedef + BOOST_CHECK_THROW(boost::math::pareto myp0(0, 1), std::domain_error); // Using typedef + BOOST_CHECK_THROW(boost::math::pareto myp1m1(1, -1), std::domain_error); // Using typedef + BOOST_CHECK_THROW(boost::math::pareto myp10(1, 0), std::domain_error); // Using typedef + + // Check some moments that should fail because shape not big enough. + BOOST_CHECK_THROW(variance(myp2), std::domain_error); + BOOST_CHECK_THROW(standard_deviation(myp2), std::domain_error); + BOOST_CHECK_THROW(skewness(myp2), std::domain_error); + BOOST_CHECK_THROW(kurtosis(myp2), std::domain_error); + BOOST_CHECK_THROW(kurtosis_excess(myp2), std::domain_error); + + // Test on extreme values of distribution parameters, + // using just double because it has numeric_limit infinity etc. + BOOST_CHECK_THROW(boost::math::pareto mypinf1(+std::numeric_limits::infinity(), 1), std::domain_error); // Using typedef + BOOST_CHECK_THROW(boost::math::pareto myp1inf(1, +std::numeric_limits::infinity()), std::domain_error); // Using typedef + BOOST_CHECK_THROW(boost::math::pareto mypinf1(+std::numeric_limits::infinity(), +std::numeric_limits::infinity()), std::domain_error); // Using typedef + + // Test on extreme values of random variate x, using just double because it has numeric_limit infinity etc.. + // No longer allow x to be + or - infinity, then these tests should throw. + BOOST_CHECK_THROW(pdf(pareto11, +std::numeric_limits::infinity()), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(pdf(pareto11, -std::numeric_limits::infinity()), std::domain_error); // x = - infinity + BOOST_CHECK_THROW(cdf(pareto11, +std::numeric_limits::infinity()), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(cdf(pareto11, -std::numeric_limits::infinity()), std::domain_error); // x = - infinity + + BOOST_CHECK_EQUAL(pdf(pareto11, 0.5), 0); // x < location but > 0 + BOOST_CHECK_EQUAL(pdf(pareto11, (std::numeric_limits::min)()), 0); // x almost zero but > 0 + BOOST_CHECK_EQUAL(pdf(pareto11, 1), 1); // x == location, result == shape == 1 + BOOST_CHECK_EQUAL(pdf(pareto11, +(std::numeric_limits::max)()), 0); // x = +max, pdf has fallen to zero. + + BOOST_CHECK_THROW(pdf(pareto11, 0), std::domain_error); // x == 0 + BOOST_CHECK_THROW(pdf(pareto11, -1), std::domain_error); // x = -1 + BOOST_CHECK_THROW(pdf(pareto11, -(std::numeric_limits::max)()), std::domain_error); // x = - max + BOOST_CHECK_THROW(pdf(pareto11, -(std::numeric_limits::min)()), std::domain_error); // x = - min + + BOOST_CHECK_EQUAL(cdf(pareto11, 1), 0); // x == location, cdf = zero. + BOOST_CHECK_EQUAL(cdf(pareto11, +(std::numeric_limits::max)()), 1); // x = + max, cdf = unity. + + BOOST_CHECK_THROW(cdf(pareto11, 0), std::domain_error); // x == 0 + BOOST_CHECK_THROW(cdf(pareto11, -(std::numeric_limits::min)()), std::domain_error); // x = - min, + BOOST_CHECK_THROW(cdf(pareto11, -(std::numeric_limits::max)()), std::domain_error); // x = - max, + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tol5eps = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tol5eps = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Compiling... +test_pareto.cpp +Linking... +Embedding manifest... +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_pareto.exe" +Running 1 test case... +*** No errors detected + + + +*/ + + diff --git a/test/test_poisson.cpp b/test/test_poisson.cpp new file mode 100644 index 000000000..361637cd7 --- /dev/null +++ b/test/test_poisson.cpp @@ -0,0 +1,623 @@ +// test_poisson.cpp + +// Copyright Paul A. Bristow 2007. +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// Basic sanity test for Poisson Cumulative Distribution Function. + +#define BOOST_MATH_DISCRETE_QUANTILE_POLICY real + +#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT) +# define TEST_FLOAT +# define TEST_DOUBLE +# define TEST_LDOUBLE +# define TEST_REAL_CONCEPT +#endif + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +#endif + +#include // Boost.Test +#include + +#include + using boost::math::poisson_distribution; +#include // for real_concept +#include // for real_concept + +#include // for (incomplete) gamma. +// using boost::math::qamma_Q; + +#include + using std::cout; + using std::endl; + using std::setprecision; + using std::showpoint; + using std::ios; +#include + using std::numeric_limits; + +template // Any floating-point type RealType. +void test_spots(RealType) +{ + // Basic sanity checks, tolerance is about numeric_limits::digits10 decimal places, + // guaranteed for type RealType, eg 6 for float, 15 for double, + // expressed as a percentage (so -2) for BOOST_CHECK_CLOSE, + + int decdigits = numeric_limits::digits10; + // May eb >15 for 80 and 128-bit FP typtes. + if (decdigits <= 0) + { // decdigits is not defined, for example real concept, + // so assume precision of most test data is double (for example, MathCAD). + decdigits = numeric_limits::digits10; // == 15 for 64-bit + } + if (decdigits > 15 ) // numeric_limits::digits10) + { // 15 is the accuracy of the MathCAD test data. + decdigits = 15; // numeric_limits::digits10; + } + + decdigits -= 1; // Perhaps allow some decimal digit(s) margin of numerical error. + RealType tolerance = static_cast(std::pow(10., static_cast(2-decdigits))); // 1e-6 (-2 so as %) + tolerance *= 2; // Allow some bit(s) small margin (2 means + or - 1 bit) of numerical error. + // Typically 2e-13% = 2e-15 as fraction for double. + + // Sources of spot test values: + + // Many be some combinations for which the result is 'exact', + // or at least is good to 40 decimal digits. + // 40 decimal digits includes 128-bit significand User Defined Floating-Point types, + + // Best source of accurate values is: + // Mathworld online calculator (40 decimal digits precision, suitable for up to 128-bit significands) + // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=GammaRegularized + // GammaRegularized is same as gamma incomplete, gamma or gamma_q(a, x) or Q(a, z). + + // http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/PoissonDistribution.html + + // MathCAD defines ppois(k, lambda== mean) as k integer, k >=0. + // ppois(0, 5) = 6.73794699908547e-3 + // ppois(1, 5) = 0.040427681994513; + // ppois(10, 10) = 5.830397501929850E-001 + // ppois(10, 1) = 9.999999899522340E-001 + // ppois(5,5) = 0.615960654833065 + + // qpois returns inverse Poission distribution, that is the smallest (floor) k so that ppois(k, lambda) >= p + // p is real number, real mean lambda > 0 + // k is approximately the integer for which probability(X <= k) = p + // when random variable X has the Poisson distribution with parameters lambda. + // Uses discrete bisection. + // qpois(6.73794699908547e-3, 5) = 1 + // qpois(0.040427681994513, 5) = + + // Test Poisson with spot values from MathCAD 'known good'. + + using boost::math::poisson_distribution; + using ::boost::math::poisson; + using ::boost::math::cdf; + using ::boost::math::pdf; + + // Check that bad arguments throw. + BOOST_CHECK_THROW( + cdf(poisson_distribution(static_cast(0)), // mean zero is bad. + static_cast(0)), // even for a good k. + std::domain_error); // Expected error to be thrown. + + BOOST_CHECK_THROW( + cdf(poisson_distribution(static_cast(-1)), // mean negative is bad. + static_cast(0)), + std::domain_error); + + BOOST_CHECK_THROW( + cdf(poisson_distribution(static_cast(1)), // mean unit OK, + static_cast(-1)), // but negative events is bad. + std::domain_error); + + BOOST_CHECK_THROW( + cdf(poisson_distribution(static_cast(0)), // mean zero is bad. + static_cast(99999)), // for any k events. + std::domain_error); + + BOOST_CHECK_THROW( + cdf(poisson_distribution(static_cast(0)), // mean zero is bad. + static_cast(99999)), // for any k events. + std::domain_error); + + BOOST_CHECK_THROW( + quantile(poisson_distribution(static_cast(0)), // mean zero. + static_cast(0.5)), // probability OK. + std::domain_error); + + BOOST_CHECK_THROW( + quantile(poisson_distribution(static_cast(-1)), + static_cast(-1)), // bad probability. + std::domain_error); + + BOOST_CHECK_THROW( + quantile(poisson_distribution(static_cast(1)), + static_cast(-1)), // bad probability. + std::domain_error); + + // Check some test values. + + BOOST_CHECK_CLOSE( // mode + mode(poisson_distribution(static_cast(4))), // mode = mean = 4. + static_cast(4), // mode. + tolerance); + + //BOOST_CHECK_CLOSE( // mode + // median(poisson_distribution(static_cast(4))), // mode = mean = 4. + // static_cast(4), // mode. + // tolerance); + poisson_distribution dist4(static_cast(40)); + + BOOST_CHECK_CLOSE( // median + median(dist4), // mode = mean = 4. median = 40.328333333333333 + quantile(dist4, static_cast(0.5)), // 39.332839138842637 + tolerance); + + // PDF + BOOST_CHECK_CLOSE( + pdf(poisson_distribution(static_cast(4)), // mean 4. + static_cast(0)), + static_cast(1.831563888873410E-002), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + pdf(poisson_distribution(static_cast(4)), // mean 4. + static_cast(2)), + static_cast(1.465251111098740E-001), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + pdf(poisson_distribution(static_cast(20)), // mean big. + static_cast(1)), // k small + static_cast(4.122307244877130E-008), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + pdf(poisson_distribution(static_cast(4)), // mean 4. + static_cast(20)), // K>> mean + static_cast(8.277463646553730E-009), // probability. + tolerance); + + // CDF + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(0)), // zero k events. + static_cast(3.678794411714420E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(1)), // one k event. + static_cast(7.357588823428830E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(2)), // two k events. + static_cast(9.196986029286060E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(10)), // two k events. + static_cast(9.999999899522340E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(15)), // two k events. + static_cast(9.999999999999810E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(16)), // two k events. + static_cast(9.999999999999990E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(17)), // two k events. + static_cast(1.), // probability unity for double. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(33)), // k events at limit for float unchecked_factorial table. + static_cast(1.), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(100)), // mean 100. + static_cast(33)), // k events at limit for float unchecked_factorial table. + static_cast(6.328271240363390E-15), // probability is tiny. + tolerance * static_cast(2e11)); // 6.3495253382825722e-015 MathCAD + // Note that there two tiny probability are much more different. + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(100)), // mean 100. + static_cast(34)), // k events at limit for float unchecked_factorial table. + static_cast(1.898481372109020E-14), // probability is tiny. + tolerance*static_cast(2e11)); // 1.8984813721090199e-014 MathCAD + + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(33)), // mean = k + static_cast(33)), // k events above limit for float unchecked_factorial table. + static_cast(5.461191812386560E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(33)), // mean = k-1 + static_cast(34)), // k events above limit for float unchecked_factorial table. + static_cast(6.133535681502950E-1), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1)), // mean unity. + static_cast(34)), // k events above limit for float unchecked_factorial table. + static_cast(1.), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(5.)), // mean + static_cast(5)), // k events. + static_cast(0.615960654833065), // probability. + tolerance); + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(5.)), // mean + static_cast(1)), // k events. + static_cast(0.040427681994512805), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(5.)), // mean + static_cast(0)), // k events (uses special case formula, not gamma). + static_cast(0.006737946999085467), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(1.)), // mean + static_cast(0)), // k events (uses special case formula, not gamma). + static_cast(0.36787944117144233), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(10.)), // mean + static_cast(10)), // k events. + static_cast(0.5830397501929856), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(4.)), // mean + static_cast(5)), // k events. + static_cast(0.785130387030406), // probability. + tolerance); + + // complement CDF + BOOST_CHECK_CLOSE( // Complement CDF + cdf(complement(poisson_distribution(static_cast(4.)), // mean + static_cast(5))), // k events. + static_cast(1 - 0.785130387030406), // probability. + tolerance); + + BOOST_CHECK_CLOSE( // Complement CDF + cdf(complement(poisson_distribution(static_cast(4.)), // mean + static_cast(0))), // Zero k events (uses special case formula, not gamma). + static_cast(0.98168436111126578), // probability. + tolerance); + BOOST_CHECK_CLOSE( // Complement CDF + cdf(complement(poisson_distribution(static_cast(1.)), // mean + static_cast(0))), // Zero k events (uses special case formula, not gamma). + static_cast(0.63212055882855767), // probability. + tolerance); + + // Example where k is bigger than max_factorial (>34 for float) + // (therefore using log gamma so perhaps less accurate). + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(40.)), // mean + static_cast(40)), // k events. + static_cast(0.5419181783625430), // probability. + tolerance); + + // Quantile & complement. + BOOST_CHECK_CLOSE( + boost::math::quantile( + poisson_distribution(5), // mean. + static_cast(0.615960654833065)), // probability. + static_cast(5.), // Expect k = 5 + tolerance/5); // + + // EQUAL is too optimistic - fails [5.0000000000000124 != 5] + // BOOST_CHECK_EQUAL(boost::math::quantile( // + // poisson_distribution(5.), // mean. + // static_cast(0.615960654833065)), // probability. + // static_cast(5.)); // Expect k = 5 events. + + BOOST_CHECK_CLOSE(boost::math::quantile( + poisson_distribution(4), // mean. + static_cast(0.785130387030406)), // probability. + static_cast(5.), // Expect k = 5 events. + tolerance/5); + + // Check on quantile of other examples of inverse of cdf. + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(10.)), // mean + static_cast(10)), // k events. + static_cast(0.5830397501929856), // probability. + tolerance); + + BOOST_CHECK_CLOSE(boost::math::quantile( // inverse of cdf above. + poisson_distribution(10.), // mean. + static_cast(0.5830397501929856)), // probability. + static_cast(10.), // Expect k = 10 events. + tolerance/5); + + + BOOST_CHECK_CLOSE( + cdf(poisson_distribution(static_cast(4.)), // mean + static_cast(5)), // k events. + static_cast(0.785130387030406), // probability. + tolerance); + + BOOST_CHECK_CLOSE(boost::math::quantile( // inverse of cdf above. + poisson_distribution(4.), // mean. + static_cast(0.785130387030406)), // probability. + static_cast(5.), // Expect k = 10 events. + tolerance/5); + + + + //BOOST_CHECK_CLOSE(boost::math::quantile( + // poisson_distribution(5), // mean. + // static_cast(0.785130387030406)), // probability. + // // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob + // static_cast(6.), // Expect k = 6 events. + // tolerance/5); + + //BOOST_CHECK_CLOSE(boost::math::quantile( + // poisson_distribution(5), // mean. + // static_cast(0.77)), // probability. + // // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob + // static_cast(7.), // Expect k = 6 events. + // tolerance/5); + + //BOOST_CHECK_CLOSE(boost::math::quantile( + // poisson_distribution(5), // mean. + // static_cast(0.75)), // probability. + // // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob + // static_cast(6.), // Expect k = 6 events. + // tolerance/5); + + BOOST_CHECK_CLOSE( + boost::math::quantile( + complement( + poisson_distribution(4), + static_cast(1 - 0.785130387030406))), // complement. + static_cast(5), // Expect k = 5 events. + tolerance/5); + + BOOST_CHECK_EQUAL(boost::math::quantile( // Check case when probability < cdf(0) (== pdf(0)) + poisson_distribution(1), // mean is small, so cdf and pdf(0) are about 0.35. + static_cast(0.0001)), // probability < cdf(0). + static_cast(0)); // Expect k = 0 events exactly. + + BOOST_CHECK_EQUAL( + boost::math::quantile( + complement( + poisson_distribution(1), + static_cast(0.9999))), // complement, so 1-probability < cdf(0) + static_cast(0)); // Expect k = 0 events exactly. + + // + // Test quantile policies against test data: + // +#define T RealType +#include "poisson_quantile.ipp" + + for(unsigned i = 0; i < poisson_quantile_data.size(); ++i) + { + using namespace boost::math::policies; + typedef policy > P1; + typedef policy > P2; + typedef policy > P3; + typedef policy > P4; + typedef policy > P5; + typedef policy > P6; + RealType tol = boost::math::tools::epsilon() * 20; + if(!boost::is_floating_point::value) + tol *= 7; + // + // Check full real value first: + // + poisson_distribution p1(poisson_quantile_data[i][0]); + RealType x = quantile(p1, poisson_quantile_data[i][1]); + BOOST_CHECK_CLOSE_FRACTION(x, poisson_quantile_data[i][2], tol); + x = quantile(complement(p1, poisson_quantile_data[i][1])); + BOOST_CHECK_CLOSE_FRACTION(x, poisson_quantile_data[i][3], tol); + // + // Now with round down to integer: + // + poisson_distribution p2(poisson_quantile_data[i][0]); + x = quantile(p2, poisson_quantile_data[i][1]); + BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][2])); + x = quantile(complement(p2, poisson_quantile_data[i][1])); + BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][3])); + // + // Now with round up to integer: + // + poisson_distribution p3(poisson_quantile_data[i][0]); + x = quantile(p3, poisson_quantile_data[i][1]); + BOOST_CHECK_EQUAL(x, ceil(poisson_quantile_data[i][2])); + x = quantile(complement(p3, poisson_quantile_data[i][1])); + BOOST_CHECK_EQUAL(x, ceil(poisson_quantile_data[i][3])); + // + // Now with round to integer "outside": + // + poisson_distribution p4(poisson_quantile_data[i][0]); + x = quantile(p4, poisson_quantile_data[i][1]); + BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? floor(poisson_quantile_data[i][2]) : ceil(poisson_quantile_data[i][2])); + x = quantile(complement(p4, poisson_quantile_data[i][1])); + BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? ceil(poisson_quantile_data[i][3]) : floor(poisson_quantile_data[i][3])); + // + // Now with round to integer "inside": + // + poisson_distribution p5(poisson_quantile_data[i][0]); + x = quantile(p5, poisson_quantile_data[i][1]); + BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? ceil(poisson_quantile_data[i][2]) : floor(poisson_quantile_data[i][2])); + x = quantile(complement(p5, poisson_quantile_data[i][1])); + BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? floor(poisson_quantile_data[i][3]) : ceil(poisson_quantile_data[i][3])); + // + // Now with round to nearest integer: + // + poisson_distribution p6(poisson_quantile_data[i][0]); + x = quantile(p6, poisson_quantile_data[i][1]); + BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][2] + 0.5f)); + x = quantile(complement(p6, poisson_quantile_data[i][1])); + BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][3] + 0.5f)); + } + +} // template void test_spots(RealType) + +// + +int test_main(int, char* []) +{ + // Check that can construct normal distribution using the two convenience methods: + using namespace boost::math; + poisson myp1(2); // Using typedef + poisson_distribution<> myp2(2); // Using default RealType double. + + // Basic sanity-check spot values. + + // Some plain double examples & tests: + cout.precision(17); // double max_digits10 + cout.setf(ios::showpoint); + + poisson mypoisson(4.); // // mean = 4, default FP type is double. + cout << "mean(mypoisson, 4.) == " << mean(mypoisson) << endl; + cout << "mean(mypoisson, 0.) == " << mean(mypoisson) << endl; + cout << "cdf(mypoisson, 2.) == " << cdf(mypoisson, 2.) << endl; + cout << "pdf(mypoisson, 2.) == " << pdf(mypoisson, 2.) << endl; + + // poisson mydudpoisson(0.); + // throws (if BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error). + + + BOOST_CHECK_THROW(poisson mydudpoisson(-1), std::domain_error);// Mean must be > 0. + BOOST_CHECK_THROW(poisson mydudpoisson(-1), std::logic_error);// Mean must be > 0. + // Passes the check because logic_error is a parent???? + // BOOST_CHECK_THROW(poisson mydudpoisson(-1), std::overflow_error); // fails the check + // because overflow_error is unrelated - except from std::exception + BOOST_CHECK_THROW(cdf(mypoisson, -1), std::domain_error); // k must be >= 0 + + BOOST_CHECK_EQUAL(mean(mypoisson), 4.); + BOOST_CHECK_CLOSE( + pdf(mypoisson, 2.), // k events = 2. + 1.465251111098740E-001, // probability. + 5e-13); + + BOOST_CHECK_CLOSE( + cdf(mypoisson, 2.), // k events = 2. + 0.238103305553545, // probability. + 5e-13); + + +#if 0 + // Compare cdf from finite sum of pdf and gamma_q. + using boost::math::cdf; + using boost::math::pdf; + + double mean = 4.; + cout.precision(17); // double max_digits10 + cout.setf(ios::showpoint); + cout << showpoint << endl; // Ensure trailing zeros are shown. + // This also helps show the expected precision max_digits10 + //cout.unsetf(ios::showpoint); // No trailing zeros are shown. + + cout << "k pdf sum cdf diff" << endl; + double sum = 0.; + for (int i = 0; i <= 50; i++) + { + cout << i << ' ' ; + double p = pdf(poisson_distribution(mean), static_cast(i)); + sum += p; + + cout << p << ' ' << sum << ' ' + << cdf(poisson_distribution(mean), static_cast(i)) << ' '; + { + cout << boost::math::gamma_q(i+1, mean); // cdf + double diff = boost::math::gamma_q(i+1, mean) - sum; // cdf -sum + cout << setprecision (2) << ' ' << diff; // 0 0 to 4, 1 eps 5 to 9, 10 to 20 2 eps, 21 upwards 3 eps + + } + BOOST_CHECK_CLOSE( + cdf(mypoisson, static_cast(i)), + sum, // of pdfs. + 4e-14); // Fails at 2e-14 + // This call puts the precision etc back to default 6 !!! + cout << setprecision(17) << showpoint; + + + cout << endl; + } + + cout << cdf(poisson_distribution(5), static_cast(0)) << ' ' << endl; // 0.006737946999085467 + cout << cdf(poisson_distribution(5), static_cast(1)) << ' ' << endl; // 0.040427681994512805 + cout << cdf(poisson_distribution(2), static_cast(3)) << ' ' << endl; // 0.85712346049854715 + + { // Compare approximate formula in Wikipedia with quantile(half) + for (int i = 1; i < 100; i++) + { + poisson_distribution distn(static_cast(i)); + cout << i << ' ' << median(distn) << ' ' << quantile(distn, 0.5) << ' ' + << median(distn) - quantile(distn, 0.5) << endl; // formula appears to be out-by-one?? + } // so quantile(half) used via derived accressors. + } +#endif + + // (Parameter value, arbitrarily zero, only communicates the floating-point type). +#ifdef TEST_POISSON + test_spots(0.0F); // Test float. +#endif +#ifdef TEST_DOUBLE + test_spots(0.0); // Test double. +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if (numeric_limits::digits10 > numeric_limits::digits10) + { // long double is better than double (so not MSVC where they are same). +#ifdef TEST_LDOUBLE + test_spots(0.0L); // Test long double. +#endif + } + + #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) +#ifdef TEST_REAL_CONCEPT + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif + #endif +#endif + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_poisson.exe" +Running 1 test case... +mean(mypoisson, 4.) == 4.0000000000000000 +mean(mypoisson, 0.) == 4.0000000000000000 +cdf(mypoisson, 2.) == 0.23810330555354431 +pdf(mypoisson, 2.) == 0.14652511110987343 +*** No errors detected + +*/ diff --git a/test/test_policy.cpp b/test/test_policy.cpp new file mode 100644 index 000000000..e0410ea1a --- /dev/null +++ b/test/test_policy.cpp @@ -0,0 +1,153 @@ + +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include // for test_main + +template +bool check_same(const P1&, const P2&) +{ + if(!boost::is_same::value) + { + std::cout << "P1 = " << typeid(P1).name() << std::endl; + std::cout << "P2 = " << typeid(P2).name() << std::endl; + } + return boost::is_same::value; +} + + +int test_main(int, char* []) +{ + using namespace boost::math::policies; + using namespace boost; + BOOST_CHECK(is_domain_error >::value); + BOOST_CHECK(0 == is_domain_error >::value); + BOOST_CHECK(is_pole_error >::value); + BOOST_CHECK(0 == is_pole_error >::value); + BOOST_CHECK(is_digits10 >::value); + BOOST_CHECK(0 == is_digits10 >::value); + + BOOST_CHECK((is_same::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same::evaluation_error_type, evaluation_error >::value)); + + BOOST_CHECK((is_same >::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same >::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, policy<>::denorm_error_type >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, policy<>::denorm_error_type >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, policy<>::evaluation_error_type >::value)); + BOOST_CHECK((is_same >::precision_type, digits2<20> >::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, policy<>::denorm_error_type >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, policy<>::evaluation_error_type >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type >::value)); + BOOST_CHECK((is_same >::promote_float_type, promote_float >::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, policy<>::denorm_error_type >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, policy<>::evaluation_error_type >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type >::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, promote_double >::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same >::domain_error_type, policy<>::domain_error_type >::value)); + BOOST_CHECK((is_same >::pole_error_type, policy<>::pole_error_type >::value)); + BOOST_CHECK((is_same >::overflow_error_type, policy<>::overflow_error_type >::value)); + BOOST_CHECK((is_same >::underflow_error_type, policy<>::underflow_error_type >::value)); + BOOST_CHECK((is_same >::denorm_error_type, policy<>::denorm_error_type >::value)); + BOOST_CHECK((is_same >::evaluation_error_type, policy<>::evaluation_error_type >::value)); + BOOST_CHECK((is_same >::precision_type, policy<>::precision_type >::value)); + BOOST_CHECK((is_same >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same >::discrete_quantile_type, discrete_quantile >::value)); + + return 0; +} // int test_main(int, char* []) + + + diff --git a/test/test_policy_2.cpp b/test/test_policy_2.cpp new file mode 100644 index 000000000..b51a2d4e8 --- /dev/null +++ b/test/test_policy_2.cpp @@ -0,0 +1,109 @@ + +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include // for test_main + +template +bool check_same(const P1&, const P2&) +{ + if(!boost::is_same::value) + { + std::cout << "P1 = " << typeid(P1).name() << std::endl; + std::cout << "P2 = " << typeid(P2).name() << std::endl; + } + return boost::is_same::value; +} + + +int test_main(int, char* []) +{ + using namespace boost::math::policies; + using namespace boost; + + BOOST_CHECK((is_same, overflow_error >::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same, overflow_error >::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same, overflow_error >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same, overflow_error >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same, overflow_error >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same, overflow_error >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same, overflow_error >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same, overflow_error >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same, overflow_error >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same, overflow_error >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same, domain_error >::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same, domain_error >::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same, domain_error >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same, domain_error >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same, domain_error >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same, domain_error >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same, domain_error >::precision_type, policy<>::precision_type>::value)); + BOOST_CHECK((is_same, domain_error >::promote_float_type, policy<>::promote_float_type>::value)); + BOOST_CHECK((is_same, domain_error >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same, domain_error >::discrete_quantile_type, policy<>::discrete_quantile_type>::value)); + + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::precision_type, digits2<20> >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::promote_float_type, promote_float >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error >::discrete_quantile_type, discrete_quantile >::value)); + + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::precision_type, digits2<20> >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::promote_float_type, promote_float >::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same, promote_float, discrete_quantile, denorm_error, domain_error > >::type::discrete_quantile_type, discrete_quantile >::value)); + + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::domain_error_type, domain_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::pole_error_type, pole_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::overflow_error_type, overflow_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::underflow_error_type, underflow_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::denorm_error_type, denorm_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::evaluation_error_type, evaluation_error >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::precision_type, digits2<20> >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::promote_float_type, promote_float >::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::promote_double_type, policy<>::promote_double_type>::value)); + BOOST_CHECK((is_same, digits2<20>, promote_float, discrete_quantile, denorm_error, domain_error >::type::discrete_quantile_type, discrete_quantile >::value)); + + BOOST_CHECK(check_same(make_policy(), policy<>())); + BOOST_CHECK(check_same(make_policy(denorm_error()), normalise > >::type())); + BOOST_CHECK(check_same(make_policy(digits2<20>()), normalise > >::type())); + BOOST_CHECK(check_same(make_policy(promote_float()), normalise > >::type())); + BOOST_CHECK(check_same(make_policy(domain_error()), normalise > >::type())); + BOOST_CHECK(check_same(make_policy(pole_error()), normalise > >::type())); + + BOOST_CHECK(check_same(make_policy(domain_error()), policy >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error()), policy, pole_error >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error()), policy, pole_error, overflow_error >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error()), policy, pole_error, overflow_error, underflow_error >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error()), policy, pole_error, overflow_error, underflow_error, denorm_error >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error(), digits2<10>()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error, digits2<10> >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error(), digits10<5>()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error, digits2<19> >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error(), digits2<10>(), promote_float()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error, digits2<10>, promote_float >())); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error(), digits2<10>(), promote_float(), promote_double()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error, digits2<10>, promote_float, promote_double >())); + BOOST_CHECK(check_same(make_policy(domain_error(), pole_error(), overflow_error(), underflow_error(), denorm_error(), evaluation_error(), digits2<10>(), promote_float(), promote_double(), discrete_quantile()), policy, pole_error, overflow_error, underflow_error, denorm_error, evaluation_error, digits2<10>, promote_float, promote_double, discrete_quantile >())); +#endif + return 0; +} // int test_main(int, char* []) + + + diff --git a/test/test_policy_sf.cpp b/test/test_policy_sf.cpp new file mode 100644 index 000000000..7c19749d5 --- /dev/null +++ b/test/test_policy_sf.cpp @@ -0,0 +1,138 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file provides very basic sanity checks for the special functions +// declared with BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS, basically we +// just want to make sure that the inline forwarding functions do +// actually forward to the right function!! +// + +namespace test{ + + typedef boost::math::policies::policy<> policy; + + BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(policy); + +} + +#define TEST_POLICY_SF(call)\ + BOOST_CHECK_EQUAL(boost::math::call , test::call) + +// +// Prevent some macro conflicts just in case: +// +#undef fpclassify +#undef isnormal +#undef isinf +#undef isfinite +#undef isnan + +int test_main(int, char* []) +{ + int i; + TEST_POLICY_SF(tgamma(3.0)); + TEST_POLICY_SF(tgamma1pm1(0.25)); + TEST_POLICY_SF(lgamma(50.0)); + TEST_POLICY_SF(lgamma(50.0, &i)); + TEST_POLICY_SF(digamma(12.0)); + TEST_POLICY_SF(tgamma_ratio(12.0, 13.5)); + TEST_POLICY_SF(tgamma_delta_ratio(100.0, 0.25)); + TEST_POLICY_SF(factorial(8)); + TEST_POLICY_SF(unchecked_factorial(3)); + TEST_POLICY_SF(double_factorial(5)); + TEST_POLICY_SF(rising_factorial(20.5, 5)); + TEST_POLICY_SF(falling_factorial(10.2, 7)); + TEST_POLICY_SF(tgamma(12.0, 13.0)); + TEST_POLICY_SF(tgamma_lower(12.0, 13.0)); + TEST_POLICY_SF(gamma_p(12.0, 13.0)); + TEST_POLICY_SF(gamma_q(12.0, 15.0)); + TEST_POLICY_SF(gamma_p_inv(12.0, 0.25)); + TEST_POLICY_SF(gamma_q_inv(15.0, 0.25)); + TEST_POLICY_SF(gamma_p_inva(12.0, 0.25)); + TEST_POLICY_SF(gamma_q_inva(12.0, 0.25)); + TEST_POLICY_SF(erf(2.5)); + TEST_POLICY_SF(erfc(2.5)); + TEST_POLICY_SF(erf_inv(0.25)); + TEST_POLICY_SF(erfc_inv(0.25)); + TEST_POLICY_SF(beta(12.0, 15.0)); + TEST_POLICY_SF(beta(12.0, 15.0, 0.25)); + TEST_POLICY_SF(betac(12.0, 15.0, 0.25)); + TEST_POLICY_SF(ibeta(12.0, 15.0, 0.25)); + TEST_POLICY_SF(ibetac(12.0, 15.0, 0.25)); + TEST_POLICY_SF(ibeta_inv(12.0, 15.0, 0.25)); + TEST_POLICY_SF(ibetac_inv(12.0, 15.0, 0.25)); + TEST_POLICY_SF(ibeta_inva(12.0, 0.75, 0.25)); + TEST_POLICY_SF(ibetac_inva(12.0, 0.75, 0.25)); + TEST_POLICY_SF(ibeta_invb(12.0, 0.75, 0.25)); + TEST_POLICY_SF(ibetac_invb(12.0, 0.75, 0.25)); + TEST_POLICY_SF(gamma_p_derivative(12.0, 15.0)); + TEST_POLICY_SF(ibeta_derivative(12.0, 15.75, 0.25)); + TEST_POLICY_SF(fpclassify(12.0)); + TEST_POLICY_SF(isfinite(12.0)); + TEST_POLICY_SF(isnormal(12.0)); + TEST_POLICY_SF(isnan(12.0)); + TEST_POLICY_SF(isinf(12.0)); + TEST_POLICY_SF(log1p(0.0025)); + TEST_POLICY_SF(expm1(0.0025)); + TEST_POLICY_SF(cbrt(30.0)); + TEST_POLICY_SF(sqrt1pm1(0.0025)); + TEST_POLICY_SF(powm1(1.0025, 12.0)); + TEST_POLICY_SF(legendre_p(5, 0.75)); + TEST_POLICY_SF(legendre_p(7, 3, 0.75)); + TEST_POLICY_SF(legendre_q(5, 0.75)); + TEST_POLICY_SF(legendre_next(2, 0.25, 12.0, 5.0)); + TEST_POLICY_SF(legendre_next(2, 2, 0.25, 12.0, 5.0)); + TEST_POLICY_SF(laguerre(5, 12.2)); + TEST_POLICY_SF(laguerre(7, 3, 5.0)); + TEST_POLICY_SF(laguerre_next(2, 5.0, 12.0, 5.0)); + TEST_POLICY_SF(laguerre_next(5, 3, 5.0, 20.0, 10.0)); + TEST_POLICY_SF(hermite(1, 2.0)); + TEST_POLICY_SF(hermite_next(2, 2.0, 3.0, 2.0)); + TEST_POLICY_SF(spherical_harmonic_r(5, 4, 0.75, 0.25)); + TEST_POLICY_SF(spherical_harmonic_i(5, 4, 0.75, 0.25)); + TEST_POLICY_SF(ellint_1(0.25)); + TEST_POLICY_SF(ellint_1(0.25, 0.75)); + TEST_POLICY_SF(ellint_2(0.25)); + TEST_POLICY_SF(ellint_2(0.25, 0.75)); + TEST_POLICY_SF(ellint_3(0.25, 0.75)); + TEST_POLICY_SF(ellint_3(0.25, 0.125, 0.75)); + TEST_POLICY_SF(ellint_rc(3.0, 5.0)); + TEST_POLICY_SF(ellint_rd(2.0, 3.0, 4.0)); + TEST_POLICY_SF(ellint_rf(2.0, 3.0, 4.0)); + TEST_POLICY_SF(ellint_rj(2.0, 3.0, 5.0, 0.25)); + TEST_POLICY_SF(hypot(5.0, 3.0)); + TEST_POLICY_SF(sinc_pi(3.0)); + TEST_POLICY_SF(sinhc_pi(2.0)); + TEST_POLICY_SF(asinh(12.0)); + TEST_POLICY_SF(acosh(5.0)); + TEST_POLICY_SF(atanh(0.75)); + TEST_POLICY_SF(sin_pi(5.0)); + TEST_POLICY_SF(cos_pi(6.0)); + TEST_POLICY_SF(cyl_neumann(2.0, 5.0)); + TEST_POLICY_SF(cyl_neumann(2, 5.0)); + TEST_POLICY_SF(cyl_bessel_j(2.0, 5.0)); + TEST_POLICY_SF(cyl_bessel_j(2, 5.0)); + TEST_POLICY_SF(cyl_bessel_i(3.0, 5.0)); + TEST_POLICY_SF(cyl_bessel_i(3, 5.0)); + TEST_POLICY_SF(cyl_bessel_k(3.0, 5.0)); + TEST_POLICY_SF(cyl_bessel_k(3, 5.0)); + TEST_POLICY_SF(sph_bessel(3, 5.0)); + TEST_POLICY_SF(sph_bessel(3, 5)); + TEST_POLICY_SF(sph_neumann(3, 5.0)); + TEST_POLICY_SF(sph_neumann(3, 5)); + return 0; +} + + + + diff --git a/test/test_rational_instances/test_rational.hpp b/test/test_rational_instances/test_rational.hpp new file mode 100644 index 000000000..30820e726 --- /dev/null +++ b/test/test_rational_instances/test_rational.hpp @@ -0,0 +1,5480 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#ifndef BOOST_MATH_TEST_RATIONAL_HPP +#define BOOST_MATH_TEST_RATIONAL_HPP + +#include +#include +#include +#include +#include + +template +void do_test_spots1(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Everything past this point is generated by the program + // ../tools/generate_rational_test.cpp + // + + // + // Polynomials of order 0 + // + static const U n1c[1] = { 2 }; + static const boost::array n1a = { 2 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.125), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.25), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.75), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(1.0f - 1.0f/64.0f), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(6.5), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(10247.25), 1), + static_cast(0.2e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.125)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.25)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(0.75)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(6.5)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1c, static_cast(10247.25)), + static_cast(0.2e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(0.125)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(0.25)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(0.75)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(6.5)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n1a, static_cast(10247.25)), + static_cast(0.2e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.125), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.25), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.75), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(1.0f - 1.0f/64.0f), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(6.5f), 1), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(10247.25f), 1), + static_cast(0.2e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.125)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.25)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(0.75)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(6.5f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1c, static_cast(10247.25f)), + static_cast(0.2e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(0.125)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(0.25)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(0.75)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(6.5f)), + static_cast(0.2e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n1a, static_cast(10247.25f)), + static_cast(0.2e1L), + tolerance); + + // + // Rational functions of order 0 + // + static const U d1c[1] = { 3 }; + static const boost::array d1a = { 3 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.125), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.25), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.75), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(1.0f - 1.0f/64.0f), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(6.5f), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(10247.25f), 1), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.125)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.25)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(0.75)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(6.5f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1c, d1c, static_cast(10247.25f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(0.125)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(0.25)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(0.75)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(6.5f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n1a, d1a, static_cast(10247.25f)), + static_cast(0.6666666666666666666666666666666666666667e0L), + tolerance); + + // + // Polynomials of order 1 + // + static const U n2c[2] = { 3, 1 }; + static const boost::array n2a = { 3, 1 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.125), 2), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.25), 2), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.75), 2), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f), 2), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(6.5), 2), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(10247.25), 2), + static_cast(0.1025025e5L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.125)), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.25)), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(0.75)), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(6.5)), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2c, static_cast(10247.25)), + static_cast(0.1025025e5L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(0.125)), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(0.25)), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(0.75)), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(6.5)), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n2a, static_cast(10247.25)), + static_cast(0.1025025e5L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.125), 2), + static_cast(0.3015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.25), 2), + static_cast(0.30625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.75), 2), + static_cast(0.35625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f), 2), + static_cast(0.3968994140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(6.5f), 2), + static_cast(0.4525e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(10247.25f), 2), + static_cast(0.1050061355625e9L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.125)), + static_cast(0.3015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.25)), + static_cast(0.30625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(0.75)), + static_cast(0.35625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3968994140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(6.5f)), + static_cast(0.4525e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2c, static_cast(10247.25f)), + static_cast(0.1050061355625e9L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(0.125)), + static_cast(0.3015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(0.25)), + static_cast(0.30625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(0.75)), + static_cast(0.35625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3968994140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(6.5f)), + static_cast(0.4525e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n2a, static_cast(10247.25f)), + static_cast(0.1050061355625e9L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.125), 2), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.25), 2), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.75), 2), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f), 2), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(6.5f), 2), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(10247.25f), 2), + static_cast(0.1025025e5L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.125)), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.25)), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(0.75)), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(6.5f)), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2c, static_cast(10247.25f)), + static_cast(0.1025025e5L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(0.125)), + static_cast(0.3125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(0.25)), + static_cast(0.325e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(0.75)), + static_cast(0.375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(6.5f)), + static_cast(0.95e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n2a, static_cast(10247.25f)), + static_cast(0.1025025e5L), + tolerance); + + // + // Rational functions of order 1 + // + static const U d2c[2] = { 5, 9 }; + static const boost::array d2a = { 5, 9 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.125), 2), + static_cast(0.5102040816326530612244897959183673469388e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.25), 2), + static_cast(0.4482758620689655172413793103448275862069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.75), 2), + static_cast(0.3191489361702127659574468085106382978723e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(1.0f - 1.0f/64.0f), 2), + static_cast(0.2874859075535512965050732807215332581736e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(6.5f), 2), + static_cast(0.1496062992125984251968503937007874015748e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(10247.25f), 2), + static_cast(0.1111376148281068304596377002122405609873e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.125)), + static_cast(0.5102040816326530612244897959183673469388e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.25)), + static_cast(0.4482758620689655172413793103448275862069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(0.75)), + static_cast(0.3191489361702127659574468085106382978723e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2874859075535512965050732807215332581736e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(6.5f)), + static_cast(0.1496062992125984251968503937007874015748e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2c, d2c, static_cast(10247.25f)), + static_cast(0.1111376148281068304596377002122405609873e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(0.125)), + static_cast(0.5102040816326530612244897959183673469388e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(0.25)), + static_cast(0.4482758620689655172413793103448275862069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(0.75)), + static_cast(0.3191489361702127659574468085106382978723e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.2874859075535512965050732807215332581736e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(6.5f)), + static_cast(0.1496062992125984251968503937007874015748e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n2a, d2a, static_cast(10247.25f)), + static_cast(0.1111376148281068304596377002122405609873e0L), + tolerance); +} + +template +void do_test_spots2(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 2 + // + static const U n3c[3] = { 10, 6, 11 }; + static const boost::array n3a = { 10, 6, 11 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.125), 3), + static_cast(0.10921875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.25), 3), + static_cast(0.121875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.75), 3), + static_cast(0.206875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f), 3), + static_cast(0.26565185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(6.5), 3), + static_cast(0.51375e3L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(10247.25), 3), + static_cast(0.11551289516875e10L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.125)), + static_cast(0.10921875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.25)), + static_cast(0.121875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(0.75)), + static_cast(0.206875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26565185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(6.5)), + static_cast(0.51375e3L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3c, static_cast(10247.25)), + static_cast(0.11551289516875e10L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(0.125)), + static_cast(0.10921875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(0.25)), + static_cast(0.121875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(0.75)), + static_cast(0.206875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26565185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(6.5)), + static_cast(0.51375e3L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n3a, static_cast(10247.25)), + static_cast(0.11551289516875e10L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.125), 3), + static_cast(0.10096435546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.25), 3), + static_cast(0.1041796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.75), 3), + static_cast(0.1685546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f), 3), + static_cast(0.26142410933971405029296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(6.5f), 3), + static_cast(0.198991875e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(10247.25f), 3), + static_cast(0.12128916726310335635546875e18L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.125)), + static_cast(0.10096435546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.25)), + static_cast(0.1041796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(0.75)), + static_cast(0.1685546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26142410933971405029296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(6.5f)), + static_cast(0.198991875e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3c, static_cast(10247.25f)), + static_cast(0.12128916726310335635546875e18L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(0.125)), + static_cast(0.10096435546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(0.25)), + static_cast(0.1041796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(0.75)), + static_cast(0.1685546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26142410933971405029296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(6.5f)), + static_cast(0.198991875e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n3a, static_cast(10247.25f)), + static_cast(0.12128916726310335635546875e18L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.125), 3), + static_cast(0.10771484375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.25), 3), + static_cast(0.11671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.75), 3), + static_cast(0.19140625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f), 3), + static_cast(0.26398639678955078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(6.5f), 3), + static_cast(0.3069875e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(10247.25f), 3), + static_cast(0.11836265072405359375e14L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.125)), + static_cast(0.10771484375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.25)), + static_cast(0.11671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(0.75)), + static_cast(0.19140625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26398639678955078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(6.5f)), + static_cast(0.3069875e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3c, static_cast(10247.25f)), + static_cast(0.11836265072405359375e14L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(0.125)), + static_cast(0.10771484375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(0.25)), + static_cast(0.11671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(0.75)), + static_cast(0.19140625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.26398639678955078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(6.5f)), + static_cast(0.3069875e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n3a, static_cast(10247.25f)), + static_cast(0.11836265072405359375e14L), + tolerance); + + // + // Rational functions of order 2 + // + static const U d3c[3] = { 3, 4, 10 }; + static const boost::array d3a = { 3, 4, 10 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.125), 3), + static_cast(0.2987179487179487179487179487179487179487e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.25), 3), + static_cast(0.2635135135135135135135135135135135135135e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.75), 3), + static_cast(0.1779569892473118279569892473118279569892e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(1.0f - 1.0f/64.0f), 3), + static_cast(0.1597671277126831703520981998649164537633e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(6.5f), 3), + static_cast(0.1137873754152823920265780730897009966777e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(10247.25f), 3), + static_cast(0.1100015619716026431429617996316152069115e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.125)), + static_cast(0.2987179487179487179487179487179487179487e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.25)), + static_cast(0.2635135135135135135135135135135135135135e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(0.75)), + static_cast(0.1779569892473118279569892473118279569892e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1597671277126831703520981998649164537633e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(6.5f)), + static_cast(0.1137873754152823920265780730897009966777e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3c, d3c, static_cast(10247.25f)), + static_cast(0.1100015619716026431429617996316152069115e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(0.125)), + static_cast(0.2987179487179487179487179487179487179487e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(0.25)), + static_cast(0.2635135135135135135135135135135135135135e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(0.75)), + static_cast(0.1779569892473118279569892473118279569892e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1597671277126831703520981998649164537633e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(6.5f)), + static_cast(0.1137873754152823920265780730897009966777e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n3a, d3a, static_cast(10247.25f)), + static_cast(0.1100015619716026431429617996316152069115e1L), + tolerance); +} + +template +void do_test_spots3(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 3 + // + static const U n4c[4] = { 1, 4, 9, 11 }; + static const boost::array n4a = { 1, 4, 9, 11 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.125), 4), + static_cast(0.1662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.25), 4), + static_cast(0.2734375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.75), 4), + static_cast(0.13703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f), 4), + static_cast(0.24150836944580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(6.5), 4), + static_cast(0.3428125e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(10247.25), 4), + static_cast(0.11837210107094921875e14L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.125)), + static_cast(0.1662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.25)), + static_cast(0.2734375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(0.75)), + static_cast(0.13703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.24150836944580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(6.5)), + static_cast(0.3428125e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4c, static_cast(10247.25)), + static_cast(0.11837210107094921875e14L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(0.125)), + static_cast(0.1662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(0.25)), + static_cast(0.2734375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(0.75)), + static_cast(0.13703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.24150836944580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(6.5)), + static_cast(0.3428125e4L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n4a, static_cast(10247.25)), + static_cast(0.11837210107094921875e14L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.125), 4), + static_cast(0.1064739227294921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.25), 4), + static_cast(0.1287841796875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.75), 4), + static_cast(0.8055419921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f), 4), + static_cast(0.23334727106775972060859203338623046875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(6.5f), 4), + static_cast(0.845843359375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(10247.25f), 4), + static_cast(0.12736106409103529349764202508544921875e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.125)), + static_cast(0.1064739227294921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.25)), + static_cast(0.1287841796875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(0.75)), + static_cast(0.8055419921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.23334727106775972060859203338623046875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(6.5f)), + static_cast(0.845843359375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4c, static_cast(10247.25f)), + static_cast(0.12736106409103529349764202508544921875e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(0.125)), + static_cast(0.1064739227294921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(0.25)), + static_cast(0.1287841796875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(0.75)), + static_cast(0.8055419921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.23334727106775972060859203338623046875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(6.5f)), + static_cast(0.845843359375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n4a, static_cast(10247.25f)), + static_cast(0.12736106409103529349764202508544921875e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.125), 4), + static_cast(0.1517913818359375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.25), 4), + static_cast(0.21513671875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.75), 4), + static_cast(0.104072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f), 4), + static_cast(0.23689246584661304950714111328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(6.5f), 4), + static_cast(0.13013059375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(10247.25f), 4), + static_cast(0.12428804224649080826343115234375e22L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.125)), + static_cast(0.1517913818359375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.25)), + static_cast(0.21513671875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(0.75)), + static_cast(0.104072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.23689246584661304950714111328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(6.5f)), + static_cast(0.13013059375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4c, static_cast(10247.25f)), + static_cast(0.12428804224649080826343115234375e22L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(0.125)), + static_cast(0.1517913818359375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(0.25)), + static_cast(0.21513671875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(0.75)), + static_cast(0.104072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.23689246584661304950714111328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(6.5f)), + static_cast(0.13013059375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n4a, static_cast(10247.25f)), + static_cast(0.12428804224649080826343115234375e22L), + tolerance); + + // + // Rational functions of order 3 + // + static const U d4c[4] = { 10, 2, 5, 4 }; + static const boost::array d4a = { 10, 2, 5, 4 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.125), 4), + static_cast(0.1608087679516250944822373393801965230537e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.25), 4), + static_cast(0.2514367816091954022988505747126436781609e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.75), 4), + static_cast(0.8564453125e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(1.0f - 1.0f/64.0f), 4), + static_cast(0.1170714951947222939292918160495461743806e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(6.5f), 4), + static_cast(0.2572219095854436315888201087975989495404e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(10247.25f), 4), + static_cast(0.2749884125808399380227005558292823797886e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.125)), + static_cast(0.1608087679516250944822373393801965230537e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.25)), + static_cast(0.2514367816091954022988505747126436781609e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(0.75)), + static_cast(0.8564453125e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1170714951947222939292918160495461743806e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(6.5f)), + static_cast(0.2572219095854436315888201087975989495404e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4c, d4c, static_cast(10247.25f)), + static_cast(0.2749884125808399380227005558292823797886e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(0.125)), + static_cast(0.1608087679516250944822373393801965230537e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(0.25)), + static_cast(0.2514367816091954022988505747126436781609e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(0.75)), + static_cast(0.8564453125e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1170714951947222939292918160495461743806e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(6.5f)), + static_cast(0.2572219095854436315888201087975989495404e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n4a, d4a, static_cast(10247.25f)), + static_cast(0.2749884125808399380227005558292823797886e1L), + tolerance); +} + +template +void do_test_spots4(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 4 + // + static const U n5c[5] = { 10, 10, 4, 11, 9 }; + static const boost::array n5a = { 10, 10, 4, 11, 9 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.125), 5), + static_cast(0.11336181640625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.25), 5), + static_cast(0.1295703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.75), 5), + static_cast(0.2723828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f), 5), + static_cast(0.42662663042545318603515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(6.5), 5), + static_cast(0.193304375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(10247.25), 5), + static_cast(0.9924842756673782995703125e17L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.125)), + static_cast(0.11336181640625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.25)), + static_cast(0.1295703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(0.75)), + static_cast(0.2723828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.42662663042545318603515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(6.5)), + static_cast(0.193304375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5c, static_cast(10247.25)), + static_cast(0.9924842756673782995703125e17L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(0.125)), + static_cast(0.11336181640625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(0.25)), + static_cast(0.1295703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(0.75)), + static_cast(0.2723828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.42662663042545318603515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(6.5)), + static_cast(0.193304375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n5a, static_cast(10247.25)), + static_cast(0.9924842756673782995703125e17L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.125), 5), + static_cast(0.10157269060611724853515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.25), 5), + static_cast(0.106434478759765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.75), 5), + static_cast(0.197494049072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f), 5), + static_cast(0.4138858164296656028113829961512237787247e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(6.5f), 5), + static_cast(0.2951521370703125e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(10247.25f), 5), + static_cast(0.1094211231602999407223950000397888253311e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.125)), + static_cast(0.10157269060611724853515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.25)), + static_cast(0.106434478759765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(0.75)), + static_cast(0.197494049072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4138858164296656028113829961512237787247e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(6.5f)), + static_cast(0.2951521370703125e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5c, static_cast(10247.25f)), + static_cast(0.1094211231602999407223950000397888253311e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(0.125)), + static_cast(0.10157269060611724853515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(0.25)), + static_cast(0.106434478759765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(0.75)), + static_cast(0.197494049072265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4138858164296656028113829961512237787247e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(6.5f)), + static_cast(0.2951521370703125e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n5a, static_cast(10247.25f)), + static_cast(0.1094211231602999407223950000397888253311e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.125), 5), + static_cast(0.11258152484893798828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.25), 5), + static_cast(0.1257379150390625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.75), 5), + static_cast(0.2299920654296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f), 5), + static_cast(0.4188681309761682314274366945028305053711e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(6.5f), 5), + static_cast(0.45408105703125e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(10247.25f), 5), + static_cast(0.106780963829612765105169679718983215332e30L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.125)), + static_cast(0.11258152484893798828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.25)), + static_cast(0.1257379150390625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(0.75)), + static_cast(0.2299920654296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4188681309761682314274366945028305053711e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(6.5f)), + static_cast(0.45408105703125e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5c, static_cast(10247.25f)), + static_cast(0.106780963829612765105169679718983215332e30L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(0.125)), + static_cast(0.11258152484893798828125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(0.25)), + static_cast(0.1257379150390625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(0.75)), + static_cast(0.2299920654296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4188681309761682314274366945028305053711e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(6.5f)), + static_cast(0.45408105703125e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n5a, static_cast(10247.25f)), + static_cast(0.106780963829612765105169679718983215332e30L), + tolerance); + + // + // Rational functions of order 4 + // + static const U d5c[5] = { 6, 9, 6, 2, 5 }; + static const boost::array d5a = { 6, 9, 6, 2, 5 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.125), 5), + static_cast(0.1569265605461489066882963263374902835513e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.25), 5), + static_cast(0.1493471409275101305718144979738856371004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.75), 5), + static_cast(0.1468309117708991366603495472731101284481e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(1.0f - 1.0f/64.0f), 5), + static_cast(0.1564121691159921277310988862398683772017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(6.5f), 5), + static_cast(0.1973991741181125982091000185089449263153e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(10247.25f), 5), + static_cast(0.1800144410401676792233921448870747965702e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.125)), + static_cast(0.1569265605461489066882963263374902835513e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.25)), + static_cast(0.1493471409275101305718144979738856371004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(0.75)), + static_cast(0.1468309117708991366603495472731101284481e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1564121691159921277310988862398683772017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(6.5f)), + static_cast(0.1973991741181125982091000185089449263153e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5c, d5c, static_cast(10247.25f)), + static_cast(0.1800144410401676792233921448870747965702e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(0.125)), + static_cast(0.1569265605461489066882963263374902835513e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(0.25)), + static_cast(0.1493471409275101305718144979738856371004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(0.75)), + static_cast(0.1468309117708991366603495472731101284481e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1564121691159921277310988862398683772017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(6.5f)), + static_cast(0.1973991741181125982091000185089449263153e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n5a, d5a, static_cast(10247.25f)), + static_cast(0.1800144410401676792233921448870747965702e1L), + tolerance); +} + +template +void do_test_spots5(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 5 + // + static const U n6c[6] = { 6, 8, 12, 5, 7, 5 }; + static const boost::array n6a = { 6, 8, 12, 5, 7, 5 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.125), 6), + static_cast(0.7199127197265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.25), 6), + static_cast(0.88603515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.75), 6), + static_cast(0.242607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f), 6), + static_cast(0.41466238017193973064422607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(6.5), 6), + static_cast(0.7244809375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(10247.25), 6), + static_cast(0.5650228315695522094919501953125e21L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.125)), + static_cast(0.7199127197265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.25)), + static_cast(0.88603515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(0.75)), + static_cast(0.242607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.41466238017193973064422607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(6.5)), + static_cast(0.7244809375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6c, static_cast(10247.25)), + static_cast(0.5650228315695522094919501953125e21L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(0.125)), + static_cast(0.7199127197265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(0.25)), + static_cast(0.88603515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(0.75)), + static_cast(0.242607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.41466238017193973064422607421875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(6.5)), + static_cast(0.7244809375e5L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n6a, static_cast(10247.25)), + static_cast(0.5650228315695522094919501953125e21L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.125), 6), + static_cast(0.6127949182875454425811767578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.25), 6), + static_cast(0.654820728302001953125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.75), 6), + static_cast(0.1616912555694580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f), 6), + static_cast(0.4001137167344577683526091194110563264985e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(6.5f), 6), + static_cast(0.6958411633369140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.125)), + static_cast(0.6127949182875454425811767578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.25)), + static_cast(0.654820728302001953125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(0.75)), + static_cast(0.1616912555694580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4001137167344577683526091194110563264985e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6c, static_cast(6.5f)), + static_cast(0.6958411633369140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6a, static_cast(0.125)), + static_cast(0.6127949182875454425811767578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6a, static_cast(0.25)), + static_cast(0.654820728302001953125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6a, static_cast(0.75)), + static_cast(0.1616912555694580078125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4001137167344577683526091194110563264985e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n6a, static_cast(6.5f)), + static_cast(0.6958411633369140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.125), 6), + static_cast(0.7023593463003635406494140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.25), 6), + static_cast(0.8192829132080078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.75), 6), + static_cast(0.19558834075927734375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f), 6), + static_cast(0.4055123471588142408661425974969461094588e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(6.5f), 6), + static_cast(0.107052491744140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(10247.25f), 6), + static_cast(0.6229253367792843599034768117351560896265e37L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.125)), + static_cast(0.7023593463003635406494140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.25)), + static_cast(0.8192829132080078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(0.75)), + static_cast(0.19558834075927734375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4055123471588142408661425974969461094588e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(6.5f)), + static_cast(0.107052491744140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6c, static_cast(10247.25f)), + static_cast(0.6229253367792843599034768117351560896265e37L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(0.125)), + static_cast(0.7023593463003635406494140625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(0.25)), + static_cast(0.8192829132080078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(0.75)), + static_cast(0.19558834075927734375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4055123471588142408661425974969461094588e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(6.5f)), + static_cast(0.107052491744140625e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n6a, static_cast(10247.25f)), + static_cast(0.6229253367792843599034768117351560896265e37L), + tolerance); + + // + // Rational functions of order 5 + // + static const U d6c[6] = { 5, 11, 7, 12, 10, 5 }; + static const boost::array d6a = { 5, 11, 7, 12, 10, 5 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.125), 6), + static_cast(0.1105787665293227020667219792530925829572e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.25), 6), + static_cast(0.1052430112515949425820670455863588910799e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.75), 6), + static_cast(0.9120378868534087154447666948125848966555e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(1.0f - 1.0f/64.0f), 6), + static_cast(0.8626539746676637108178973543021882567334e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(6.5f), 6), + static_cast(0.9109197333022141385660730837432462113756e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(10247.25f), 6), + static_cast(0.9999414458034701919327379302959430043671e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.125)), + static_cast(0.1105787665293227020667219792530925829572e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.25)), + static_cast(0.1052430112515949425820670455863588910799e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(0.75)), + static_cast(0.9120378868534087154447666948125848966555e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8626539746676637108178973543021882567334e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(6.5f)), + static_cast(0.9109197333022141385660730837432462113756e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6c, d6c, static_cast(10247.25f)), + static_cast(0.9999414458034701919327379302959430043671e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(0.125)), + static_cast(0.1105787665293227020667219792530925829572e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(0.25)), + static_cast(0.1052430112515949425820670455863588910799e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(0.75)), + static_cast(0.9120378868534087154447666948125848966555e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8626539746676637108178973543021882567334e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(6.5f)), + static_cast(0.9109197333022141385660730837432462113756e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n6a, d6a, static_cast(10247.25f)), + static_cast(0.9999414458034701919327379302959430043671e0L), + tolerance); +} + +template +void do_test_spots6(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 6 + // + static const U n7c[7] = { 3, 4, 11, 5, 10, 7, 9 }; + static const boost::array n7a = { 3, 4, 11, 5, 10, 7, 9 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.125), 7), + static_cast(0.3684329986572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.25), 7), + static_cast(0.4813720703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.75), 7), + static_cast(0.20723876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f), 7), + static_cast(0.46413680258337990380823612213134765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(6.5), 7), + static_cast(0.779707859375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(10247.25), 7), + static_cast(0.10421241651331160693970241510986328125e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.125)), + static_cast(0.3684329986572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.25)), + static_cast(0.4813720703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(0.75)), + static_cast(0.20723876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.46413680258337990380823612213134765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(6.5)), + static_cast(0.779707859375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7c, static_cast(10247.25)), + static_cast(0.10421241651331160693970241510986328125e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(0.125)), + static_cast(0.3684329986572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(0.25)), + static_cast(0.4813720703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(0.75)), + static_cast(0.20723876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.46413680258337990380823612213134765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(6.5)), + static_cast(0.779707859375e6L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n7a, static_cast(10247.25)), + static_cast(0.10421241651331160693970241510986328125e26L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.125), 7), + static_cast(0.3065205223058001138269901275634765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.25), 7), + static_cast(0.3294349253177642822265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.75), 7), + static_cast(0.11300772249698638916015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f), 7), + static_cast(0.4400013192129567626077980251194602528964e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(6.5f), 7), + static_cast(0.52166734985505126953125e11L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.125)), + static_cast(0.3065205223058001138269901275634765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.25)), + static_cast(0.3294349253177642822265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(0.75)), + static_cast(0.11300772249698638916015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4400013192129567626077980251194602528964e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7c, static_cast(6.5f)), + static_cast(0.52166734985505126953125e11L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7a, static_cast(0.125)), + static_cast(0.3065205223058001138269901275634765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7a, static_cast(0.25)), + static_cast(0.3294349253177642822265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7a, static_cast(0.75)), + static_cast(0.11300772249698638916015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4400013192129567626077980251194602528964e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n7a, static_cast(6.5f)), + static_cast(0.52166734985505126953125e11L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.125), 7), + static_cast(0.3521641784464009106159210205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.25), 7), + static_cast(0.41773970127105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.75), 7), + static_cast(0.140676963329315185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f), 7), + static_cast(0.4465092766607814731253821207562770823074e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(6.5f), 7), + static_cast(0.802565153877001953125e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.125)), + static_cast(0.3521641784464009106159210205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.25)), + static_cast(0.41773970127105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(0.75)), + static_cast(0.140676963329315185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4465092766607814731253821207562770823074e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7c, static_cast(6.5f)), + static_cast(0.802565153877001953125e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7a, static_cast(0.125)), + static_cast(0.3521641784464009106159210205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7a, static_cast(0.25)), + static_cast(0.41773970127105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7a, static_cast(0.75)), + static_cast(0.140676963329315185546875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4465092766607814731253821207562770823074e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n7a, static_cast(6.5f)), + static_cast(0.802565153877001953125e10L), + tolerance); + // + // Rational functions of order 6 + // + static const U d7c[7] = { 2, 8, 10, 8, 1, 11, 1 }; + static const boost::array d7a = { 2, 8, 10, 8, 1, 11, 1 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.125), 7), + static_cast(0.1161348466465698540596242849979738853664e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.25), 7), + static_cast(0.1010247476558897371522262642824204539632e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.75), 7), + static_cast(0.103079575951134804308491906398377636644e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(1.0f - 1.0f/64.0f), 7), + static_cast(0.1183671559390403425417413542214181025989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(6.5f), 7), + static_cast(0.3757457624476840396478985178609736319955e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(10247.25f), 7), + static_cast(0.8991031618241406513349732955332873069556e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.125)), + static_cast(0.1161348466465698540596242849979738853664e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.25)), + static_cast(0.1010247476558897371522262642824204539632e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(0.75)), + static_cast(0.103079575951134804308491906398377636644e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1183671559390403425417413542214181025989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(6.5f)), + static_cast(0.3757457624476840396478985178609736319955e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7c, d7c, static_cast(10247.25f)), + static_cast(0.8991031618241406513349732955332873069556e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(0.125)), + static_cast(0.1161348466465698540596242849979738853664e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(0.25)), + static_cast(0.1010247476558897371522262642824204539632e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(0.75)), + static_cast(0.103079575951134804308491906398377636644e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1183671559390403425417413542214181025989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(6.5f)), + static_cast(0.3757457624476840396478985178609736319955e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n7a, d7a, static_cast(10247.25f)), + static_cast(0.8991031618241406513349732955332873069556e1L), + tolerance); +} + +template +void do_test_spots7(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 7 + // + static const U n8c[8] = { 9, 5, 6, 1, 12, 2, 11, 1 }; + static const boost::array n8a = { 9, 5, 6, 1, 12, 2, 11, 1 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.125), 8), + static_cast(0.9723736286163330078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.25), 8), + static_cast(0.1069219970703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.75), 8), + static_cast(0.2290960693359375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f), 8), + static_cast(0.4470947054706607559637632220983505249023e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(6.5), 8), + static_cast(0.13650267734375e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(10247.25), 8), + static_cast(0.1187728773094625678513681460864459228516e29L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.125)), + static_cast(0.9723736286163330078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.25)), + static_cast(0.1069219970703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(0.75)), + static_cast(0.2290960693359375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4470947054706607559637632220983505249023e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(6.5)), + static_cast(0.13650267734375e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8c, static_cast(10247.25)), + static_cast(0.1187728773094625678513681460864459228516e29L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(0.125)), + static_cast(0.9723736286163330078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(0.25)), + static_cast(0.1069219970703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(0.75)), + static_cast(0.2290960693359375e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4470947054706607559637632220983505249023e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(6.5)), + static_cast(0.13650267734375e7L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n8a, static_cast(10247.25)), + static_cast(0.1187728773094625678513681460864459228516e29L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.125), 8), + static_cast(0.9079594375725946520105935633182525634766e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.25), 8), + static_cast(0.93363673128187656402587890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.75), 8), + static_cast(0.155691558457911014556884765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f), 8), + static_cast(0.4258457138999176910226338119632657870077e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(6.5f), 8), + static_cast(0.30319406120433428955078125e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.125)), + static_cast(0.9079594375725946520105935633182525634766e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.25)), + static_cast(0.93363673128187656402587890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(0.75)), + static_cast(0.155691558457911014556884765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4258457138999176910226338119632657870077e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8c, static_cast(6.5f)), + static_cast(0.30319406120433428955078125e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8a, static_cast(0.125)), + static_cast(0.9079594375725946520105935633182525634766e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8a, static_cast(0.25)), + static_cast(0.93363673128187656402587890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8a, static_cast(0.75)), + static_cast(0.155691558457911014556884765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4258457138999176910226338119632657870077e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n8a, static_cast(6.5f)), + static_cast(0.30319406120433428955078125e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.125), 8), + static_cast(0.9636755005807572160847485065460205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.25), 8), + static_cast(0.1034546925127506256103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.75), 8), + static_cast(0.1775887446105480194091796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f), 8), + static_cast(0.4311765982475354321499772058039525455316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(6.5f), 8), + static_cast(0.466452401928975830078125e11L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.125)), + static_cast(0.9636755005807572160847485065460205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.25)), + static_cast(0.1034546925127506256103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(0.75)), + static_cast(0.1775887446105480194091796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4311765982475354321499772058039525455316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8c, static_cast(6.5f)), + static_cast(0.466452401928975830078125e11L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8a, static_cast(0.125)), + static_cast(0.9636755005807572160847485065460205078125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8a, static_cast(0.25)), + static_cast(0.1034546925127506256103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8a, static_cast(0.75)), + static_cast(0.1775887446105480194091796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4311765982475354321499772058039525455316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n8a, static_cast(6.5f)), + static_cast(0.466452401928975830078125e11L), + tolerance); + // + // Rational functions of order 7 + // + static const U d8c[8] = { 7, 10, 10, 11, 2, 4, 1, 7 }; + static const boost::array d8a = { 7, 10, 10, 11, 2, 4, 1, 7 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.125), 8), + static_cast(0.1153693678861771369296601206130394355814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.25), 8), + static_cast(0.103714470093009762768860970830101771981e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.75), 8), + static_cast(0.834289461108456229648481346061946410909e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(1.0f - 1.0f/64.0f), 8), + static_cast(0.8981362736388035283180082845549732926578e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(6.5f), 8), + static_cast(0.383383275200627223098571833372266642165e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(10247.25f), 8), + static_cast(0.1430085023641929377426860779365964126311e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.125)), + static_cast(0.1153693678861771369296601206130394355814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.25)), + static_cast(0.103714470093009762768860970830101771981e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(0.75)), + static_cast(0.834289461108456229648481346061946410909e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8981362736388035283180082845549732926578e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(6.5f)), + static_cast(0.383383275200627223098571833372266642165e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8c, d8c, static_cast(10247.25f)), + static_cast(0.1430085023641929377426860779365964126311e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(0.125)), + static_cast(0.1153693678861771369296601206130394355814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(0.25)), + static_cast(0.103714470093009762768860970830101771981e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(0.75)), + static_cast(0.834289461108456229648481346061946410909e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8981362736388035283180082845549732926578e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(6.5f)), + static_cast(0.383383275200627223098571833372266642165e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n8a, d8a, static_cast(10247.25f)), + static_cast(0.1430085023641929377426860779365964126311e0L), + tolerance); +} + +template +void do_test_spots8(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 8 + // + static const U n9c[9] = { 3, 9, 3, 9, 5, 6, 10, 7, 10 }; + static const boost::array n9a = { 3, 9, 3, 9, 5, 6, 10, 7, 10 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.125), 9), + static_cast(0.419089901447296142578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.25), 9), + static_cast(0.5606536865234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.75), 9), + static_cast(0.21955535888671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f), 9), + static_cast(0.577754343750660055434309470001608133316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(6.5), 9), + static_cast(0.3613143234375e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(10247.25), 9), + static_cast(0.1215873306624182859977656082228297326996e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.125)), + static_cast(0.419089901447296142578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.25)), + static_cast(0.5606536865234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(0.75)), + static_cast(0.21955535888671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.577754343750660055434309470001608133316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(6.5)), + static_cast(0.3613143234375e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9c, static_cast(10247.25)), + static_cast(0.1215873306624182859977656082228297326996e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(0.125)), + static_cast(0.419089901447296142578125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(0.25)), + static_cast(0.5606536865234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(0.75)), + static_cast(0.21955535888671875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.577754343750660055434309470001608133316e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(6.5)), + static_cast(0.3613143234375e8L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n9a, static_cast(10247.25)), + static_cast(0.1215873306624182859977656082228297326996e34L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.125), 9), + static_cast(0.3141392057908696244794555241242051124573e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.25), 9), + static_cast(0.35764986560679972171783447265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.75), 9), + static_cast(0.119936861679889261722564697265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f), 9), + static_cast(0.5392583642412261279423815221065846904997e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(6.5f), 9), + static_cast(0.103274449934495763275146484375e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.125)), + static_cast(0.3141392057908696244794555241242051124573e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.25)), + static_cast(0.35764986560679972171783447265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(0.75)), + static_cast(0.119936861679889261722564697265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5392583642412261279423815221065846904997e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9c, static_cast(6.5f)), + static_cast(0.103274449934495763275146484375e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9a, static_cast(0.125)), + static_cast(0.3141392057908696244794555241242051124573e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9a, static_cast(0.25)), + static_cast(0.35764986560679972171783447265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9a, static_cast(0.75)), + static_cast(0.119936861679889261722564697265625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5392583642412261279423815221065846904997e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n9a, static_cast(6.5f)), + static_cast(0.103274449934495763275146484375e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.125), 9), + static_cast(0.4131136463269569958356441929936408996582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.25), 9), + static_cast(0.530599462427198886871337890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.75), 9), + static_cast(0.1499158155731856822967529296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f), 9), + static_cast(0.5473418303402932093382923399178003205076e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(6.5f), 9), + static_cast(0.1588837691300188665771484375e14L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.125)), + static_cast(0.4131136463269569958356441929936408996582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.25)), + static_cast(0.530599462427198886871337890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(0.75)), + static_cast(0.1499158155731856822967529296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5473418303402932093382923399178003205076e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9c, static_cast(6.5f)), + static_cast(0.1588837691300188665771484375e14L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9a, static_cast(0.125)), + static_cast(0.4131136463269569958356441929936408996582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9a, static_cast(0.25)), + static_cast(0.530599462427198886871337890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9a, static_cast(0.75)), + static_cast(0.1499158155731856822967529296875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5473418303402932093382923399178003205076e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n9a, static_cast(6.5f)), + static_cast(0.1588837691300188665771484375e14L), + tolerance); + // + // Rational functions of order 8 + // + static const U d9c[9] = { 12, 3, 10, 4, 6, 6, 6, 10, 7 }; + static const boost::array d9a = { 12, 3, 10, 4, 6, 6, 6, 10, 7 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.125), 9), + static_cast(0.3341827920887278954826517708316980450243e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.25), 9), + static_cast(0.4162555242250224594229316089330159747956e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.75), 9), + static_cast(0.7844550246723573100342976024389406178802e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(1.0f - 1.0f/64.0f), 9), + static_cast(0.959335028097323235424759017468360386113e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(6.5f), 9), + static_cast(0.1302420407483849169746727326286868117535e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(10247.25f), 9), + static_cast(0.1428469874366314841691622991213446692856e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.125)), + static_cast(0.3341827920887278954826517708316980450243e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.25)), + static_cast(0.4162555242250224594229316089330159747956e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(0.75)), + static_cast(0.7844550246723573100342976024389406178802e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.959335028097323235424759017468360386113e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(6.5f)), + static_cast(0.1302420407483849169746727326286868117535e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9c, d9c, static_cast(10247.25f)), + static_cast(0.1428469874366314841691622991213446692856e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(0.125)), + static_cast(0.3341827920887278954826517708316980450243e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(0.25)), + static_cast(0.4162555242250224594229316089330159747956e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(0.75)), + static_cast(0.7844550246723573100342976024389406178802e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.959335028097323235424759017468360386113e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(6.5f)), + static_cast(0.1302420407483849169746727326286868117535e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n9a, d9a, static_cast(10247.25f)), + static_cast(0.1428469874366314841691622991213446692856e1L), + tolerance); +} + +template +void do_test_spots9(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 9 + // + static const U n10c[10] = { 3, 4, 2, 6, 8, 1, 2, 4, 8, 8 }; + static const boost::array n10a = { 3, 4, 2, 6, 8, 1, 2, 4, 8, 8 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.125), 10), + static_cast(0.3544962465763092041015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.25), 10), + static_cast(0.4251861572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.75), 10), + static_cast(0.14716278076171875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f), 10), + static_cast(0.4243246072286939307716124858416151255369e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(6.5), 10), + static_cast(0.193326261328125e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(10247.25), 10), + static_cast(0.9967777935240642903307419028007759631098e37L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.125)), + static_cast(0.3544962465763092041015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.25)), + static_cast(0.4251861572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(0.75)), + static_cast(0.14716278076171875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4243246072286939307716124858416151255369e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(6.5)), + static_cast(0.193326261328125e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10c, static_cast(10247.25)), + static_cast(0.9967777935240642903307419028007759631098e37L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(0.125)), + static_cast(0.3544962465763092041015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(0.25)), + static_cast(0.4251861572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(0.75)), + static_cast(0.14716278076171875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.4243246072286939307716124858416151255369e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(6.5)), + static_cast(0.193326261328125e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n10a, static_cast(10247.25)), + static_cast(0.9967777935240642903307419028007759631098e37L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.125), 10), + static_cast(0.3063011647232116718697625401546247303486e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.25), 10), + static_cast(0.3259400503826327621936798095703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.75), 10), + static_cast(0.8067807371844537556171417236328125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f), 10), + static_cast(0.3922779295817342542834377568121069117613e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(6.5f), 10), + static_cast(0.3514067090785774022613525390625e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.125)), + static_cast(0.3063011647232116718697625401546247303486e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.25)), + static_cast(0.3259400503826327621936798095703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(0.75)), + static_cast(0.8067807371844537556171417236328125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3922779295817342542834377568121069117613e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10c, static_cast(6.5f)), + static_cast(0.3514067090785774022613525390625e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10a, static_cast(0.125)), + static_cast(0.3063011647232116718697625401546247303486e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10a, static_cast(0.25)), + static_cast(0.3259400503826327621936798095703125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10a, static_cast(0.75)), + static_cast(0.8067807371844537556171417236328125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3922779295817342542834377568121069117613e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n10a, static_cast(6.5f)), + static_cast(0.3514067090785774022613525390625e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.125), 10), + static_cast(0.3504093177856933749581003212369978427887e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.25), 10), + static_cast(0.40376020153053104877471923828125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.75), 10), + static_cast(0.97570764957927167415618896484375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f), 10), + static_cast(0.3980283729084284487958732767615054341702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(6.5f), 10), + static_cast(0.54062570627473700347900390625e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.125)), + static_cast(0.3504093177856933749581003212369978427887e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.25)), + static_cast(0.40376020153053104877471923828125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(0.75)), + static_cast(0.97570764957927167415618896484375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3980283729084284487958732767615054341702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10c, static_cast(6.5f)), + static_cast(0.54062570627473700347900390625e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10a, static_cast(0.125)), + static_cast(0.3504093177856933749581003212369978427887e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10a, static_cast(0.25)), + static_cast(0.40376020153053104877471923828125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10a, static_cast(0.75)), + static_cast(0.97570764957927167415618896484375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.3980283729084284487958732767615054341702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n10a, static_cast(6.5f)), + static_cast(0.54062570627473700347900390625e15L), + tolerance); + // + // Rational functions of order 9 + // + static const U d10c[10] = { 3, 11, 1, 12, 8, 8, 7, 10, 8, 8 }; + static const boost::array d10a = { 3, 11, 1, 12, 8, 8, 7, 10, 8, 8 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.125), 10), + static_cast(0.8027011456035955638016996485812217742707e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.25), 10), + static_cast(0.7037718026559713894599659542655668311705e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.75), 10), + static_cast(0.5819711007563332500606442409694389231031e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(1.0f - 1.0f/64.0f), 10), + static_cast(0.6021345078884753739086911192013079405483e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(6.5f), 10), + static_cast(0.9827105949744728065574239430024037810213e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(10247.25f), 10), + static_cast(0.9999999928576761766405011572543053028339e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.125)), + static_cast(0.8027011456035955638016996485812217742707e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.25)), + static_cast(0.7037718026559713894599659542655668311705e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(0.75)), + static_cast(0.5819711007563332500606442409694389231031e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6021345078884753739086911192013079405483e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(6.5f)), + static_cast(0.9827105949744728065574239430024037810213e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10c, d10c, static_cast(10247.25f)), + static_cast(0.9999999928576761766405011572543053028339e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(0.125)), + static_cast(0.8027011456035955638016996485812217742707e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(0.25)), + static_cast(0.7037718026559713894599659542655668311705e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(0.75)), + static_cast(0.5819711007563332500606442409694389231031e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6021345078884753739086911192013079405483e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(6.5f)), + static_cast(0.9827105949744728065574239430024037810213e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n10a, d10a, static_cast(10247.25f)), + static_cast(0.9999999928576761766405011572543053028339e0L), + tolerance); +} + +template +void do_test_spots10(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 10 + // + static const U n11c[11] = { 2, 2, 8, 11, 3, 4, 10, 11, 5, 1, 6 }; + static const boost::array n11a = { 2, 2, 8, 11, 3, 4, 10, 11, 5, 1, 6 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.125), 11), + static_cast(0.239738257043063640594482421875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.25), 11), + static_cast(0.31906986236572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.75), 11), + static_cast(0.187007007598876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f), 11), + static_cast(0.5807897685780276847943015550157497273176e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(6.5), 11), + static_cast(0.85061053443359375e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.125)), + static_cast(0.239738257043063640594482421875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.25)), + static_cast(0.31906986236572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(0.75)), + static_cast(0.187007007598876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5807897685780276847943015550157497273176e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11c, static_cast(6.5)), + static_cast(0.85061053443359375e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11a, static_cast(0.125)), + static_cast(0.239738257043063640594482421875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11a, static_cast(0.25)), + static_cast(0.31906986236572265625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11a, static_cast(0.75)), + static_cast(0.187007007598876953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5807897685780276847943015550157497273176e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n11a, static_cast(6.5)), + static_cast(0.85061053443359375e9L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.125), 11), + static_cast(0.2033245269357184586631048794913567689946e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.25), 11), + static_cast(0.2158985776148256263695657253265380859375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.75), 11), + static_cast(0.8727145384755203849636018276214599609375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f), 11), + static_cast(0.5363972553738812062759598952966094072427e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(6.5f), 11), + static_cast(0.1092297265410211371166019439697265625e18L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.125)), + static_cast(0.2033245269357184586631048794913567689946e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.25)), + static_cast(0.2158985776148256263695657253265380859375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(0.75)), + static_cast(0.8727145384755203849636018276214599609375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5363972553738812062759598952966094072427e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11c, static_cast(6.5f)), + static_cast(0.1092297265410211371166019439697265625e18L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11a, static_cast(0.125)), + static_cast(0.2033245269357184586631048794913567689946e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11a, static_cast(0.25)), + static_cast(0.2158985776148256263695657253265380859375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11a, static_cast(0.75)), + static_cast(0.8727145384755203849636018276214599609375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.5363972553738812062759598952966094072427e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n11a, static_cast(6.5f)), + static_cast(0.1092297265410211371166019439697265625e18L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.125), 11), + static_cast(0.2265962154857476693048390359308541519567e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.25), 11), + static_cast(0.26359431045930250547826290130615234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.75), 11), + static_cast(0.109695271796736051328480243682861328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f), 11), + static_cast(0.544594037205212653994625925380682572437e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(6.5f), 11), + static_cast(0.16804573314003253556400299072265625e17L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.125)), + static_cast(0.2265962154857476693048390359308541519567e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.25)), + static_cast(0.26359431045930250547826290130615234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(0.75)), + static_cast(0.109695271796736051328480243682861328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.544594037205212653994625925380682572437e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11c, static_cast(6.5f)), + static_cast(0.16804573314003253556400299072265625e17L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11a, static_cast(0.125)), + static_cast(0.2265962154857476693048390359308541519567e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11a, static_cast(0.25)), + static_cast(0.26359431045930250547826290130615234375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11a, static_cast(0.75)), + static_cast(0.109695271796736051328480243682861328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.544594037205212653994625925380682572437e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n11a, static_cast(6.5f)), + static_cast(0.16804573314003253556400299072265625e17L), + tolerance); + // + // Rational functions of order 10 + // + static const U d11c[11] = { 4, 1, 3, 9, 11, 8, 11, 2, 6, 6, 4 }; + static const boost::array d11a = { 4, 1, 3, 9, 11, 8, 11, 2, 6, 6, 4 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.125), 11), + static_cast(0.5718365676248588095654568811483084403598e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.25), 11), + static_cast(0.6888631839546304707516922567568791812269e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.75), 11), + static_cast(0.9783539912974912482969079012097816310129e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(1.0f - 1.0f/64.0f), 11), + static_cast(0.9694017102874332007392886881642471036972e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(6.5f), 11), + static_cast(0.1243900864392932237542421996384079347041e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(10247.25f), 11), + static_cast(0.1499804844733304585200728061706913399511e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.125)), + static_cast(0.5718365676248588095654568811483084403598e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.25)), + static_cast(0.6888631839546304707516922567568791812269e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(0.75)), + static_cast(0.9783539912974912482969079012097816310129e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9694017102874332007392886881642471036972e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(6.5f)), + static_cast(0.1243900864392932237542421996384079347041e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11c, d11c, static_cast(10247.25f)), + static_cast(0.1499804844733304585200728061706913399511e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(0.125)), + static_cast(0.5718365676248588095654568811483084403598e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(0.25)), + static_cast(0.6888631839546304707516922567568791812269e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(0.75)), + static_cast(0.9783539912974912482969079012097816310129e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9694017102874332007392886881642471036972e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(6.5f)), + static_cast(0.1243900864392932237542421996384079347041e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n11a, d11a, static_cast(10247.25f)), + static_cast(0.1499804844733304585200728061706913399511e1L), + tolerance); +} + +template +void do_test_spots11(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 11 + // + static const U n12c[12] = { 10, 12, 4, 1, 12, 7, 11, 5, 12, 5, 10, 6 }; + static const boost::array n12a = { 10, 12, 4, 1, 12, 7, 11, 5, 12, 5, 10, 6 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.125), 12), + static_cast(0.1156764154392294585704803466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.25), 12), + static_cast(0.13322539806365966796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.75), 12), + static_cast(0.32148390293121337890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f), 12), + static_cast(0.8737331822870016474402916385744166660743e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(6.5), 12), + static_cast(0.67419250750654296875e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.125)), + static_cast(0.1156764154392294585704803466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.25)), + static_cast(0.13322539806365966796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(0.75)), + static_cast(0.32148390293121337890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8737331822870016474402916385744166660743e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12c, static_cast(6.5)), + static_cast(0.67419250750654296875e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12a, static_cast(0.125)), + static_cast(0.1156764154392294585704803466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12a, static_cast(0.25)), + static_cast(0.13322539806365966796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12a, static_cast(0.75)), + static_cast(0.32148390293121337890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8737331822870016474402916385744166660743e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n12a, static_cast(6.5)), + static_cast(0.67419250750654296875e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.125), 12), + static_cast(0.101884810991335118067599354446661763518e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.25), 12), + static_cast(0.1076605959896767217287560924887657165527e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.75), 12), + static_cast(0.2041755737073560794669901952147483825684e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f), 12), + static_cast(0.8060387357327405373376954971672611741429e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(6.5f), 12), + static_cast(0.4778085851102157284559772014617919921875e19L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.125)), + static_cast(0.101884810991335118067599354446661763518e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.25)), + static_cast(0.1076605959896767217287560924887657165527e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(0.75)), + static_cast(0.2041755737073560794669901952147483825684e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8060387357327405373376954971672611741429e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12c, static_cast(6.5f)), + static_cast(0.4778085851102157284559772014617919921875e19L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12a, static_cast(0.125)), + static_cast(0.101884810991335118067599354446661763518e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12a, static_cast(0.25)), + static_cast(0.1076605959896767217287560924887657165527e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12a, static_cast(0.75)), + static_cast(0.2041755737073560794669901952147483825684e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8060387357327405373376954971672611741429e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n12a, static_cast(6.5f)), + static_cast(0.4778085851102157284559772014617919921875e19L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.125), 12), + static_cast(0.1150784879306809445407948355732941081442e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.25), 12), + static_cast(0.1306423839587068869150243699550628662109e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.75), 12), + static_cast(0.2389007649431414392893202602863311767578e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f), 12), + static_cast(0.8172456997919903871367065368048367483356e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(6.5f), 12), + static_cast(0.73509013093879343685534954071044921875e18L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.125)), + static_cast(0.1150784879306809445407948355732941081442e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.25)), + static_cast(0.1306423839587068869150243699550628662109e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(0.75)), + static_cast(0.2389007649431414392893202602863311767578e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8172456997919903871367065368048367483356e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12c, static_cast(6.5f)), + static_cast(0.73509013093879343685534954071044921875e18L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12a, static_cast(0.125)), + static_cast(0.1150784879306809445407948355732941081442e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12a, static_cast(0.25)), + static_cast(0.1306423839587068869150243699550628662109e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12a, static_cast(0.75)), + static_cast(0.2389007649431414392893202602863311767578e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8172456997919903871367065368048367483356e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n12a, static_cast(6.5f)), + static_cast(0.73509013093879343685534954071044921875e18L), + tolerance); + // + // Rational functions of order 11 + // + static const U d12c[12] = { 12, 5, 2, 8, 3, 2, 6, 9, 2, 8, 9, 12 }; + static const boost::array d12a = { 12, 5, 2, 8, 3, 2, 6, 9, 2, 8, 9, 12 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.125), 12), + static_cast(0.9128003783370762743953357962892418132189e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.25), 12), + static_cast(0.9857041905689267091438933694601440838819e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.75), 12), + static_cast(0.1248112763387283893598834927961632655902e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(1.0f - 1.0f/64.0f), 12), + static_cast(0.1227813945781309965073515980672922656926e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(6.5f), 12), + static_cast(0.5670462630528956417277364302989555872917e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(10247.25f), 12), + static_cast(0.5000447249679368028702341332079904080375e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.125)), + static_cast(0.9128003783370762743953357962892418132189e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.25)), + static_cast(0.9857041905689267091438933694601440838819e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(0.75)), + static_cast(0.1248112763387283893598834927961632655902e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1227813945781309965073515980672922656926e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(6.5f)), + static_cast(0.5670462630528956417277364302989555872917e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12c, d12c, static_cast(10247.25f)), + static_cast(0.5000447249679368028702341332079904080375e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(0.125)), + static_cast(0.9128003783370762743953357962892418132189e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(0.25)), + static_cast(0.9857041905689267091438933694601440838819e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(0.75)), + static_cast(0.1248112763387283893598834927961632655902e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1227813945781309965073515980672922656926e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(6.5f)), + static_cast(0.5670462630528956417277364302989555872917e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n12a, d12a, static_cast(10247.25f)), + static_cast(0.5000447249679368028702341332079904080375e0L), + tolerance); +} + +template +void do_test_spots12(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 12 + // + static const U n13c[13] = { 4, 11, 7, 1, 1, 1, 8, 11, 10, 12, 8, 2, 1 }; + static const boost::array n13a = { 4, 11, 7, 1, 1, 1, 8, 11, 10, 12, 8, 2, 1 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.125), 13), + static_cast(0.5486639239141368307173252105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.25), 13), + static_cast(0.7210838854312896728515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.75), 13), + static_cast(0.22524036943912506103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f), 13), + static_cast(0.7013317633407797455061737623412208147977e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(6.5), 13), + static_cast(0.8801602436469970703125e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.125)), + static_cast(0.5486639239141368307173252105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.25)), + static_cast(0.7210838854312896728515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(0.75)), + static_cast(0.22524036943912506103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7013317633407797455061737623412208147977e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13c, static_cast(6.5)), + static_cast(0.8801602436469970703125e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13a, static_cast(0.125)), + static_cast(0.5486639239141368307173252105712890625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13a, static_cast(0.25)), + static_cast(0.7210838854312896728515625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13a, static_cast(0.75)), + static_cast(0.22524036943912506103515625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7013317633407797455061737623412208147977e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n13a, static_cast(6.5)), + static_cast(0.8801602436469970703125e10L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.125), 13), + static_cast(0.4173587859727185607499727379291788731397e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.25), 13), + static_cast(0.4715104623414053008900737040676176548004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.75), 13), + static_cast(0.1338397611887558369403450342360883951187e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f), 13), + static_cast(0.6407202225497595548875186059951665936749e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(6.5f), 13), + static_cast(0.3403521961549553788009664398431777954102e20L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.125)), + static_cast(0.4173587859727185607499727379291788731397e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.25)), + static_cast(0.4715104623414053008900737040676176548004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(0.75)), + static_cast(0.1338397611887558369403450342360883951187e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6407202225497595548875186059951665936749e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13c, static_cast(6.5f)), + static_cast(0.3403521961549553788009664398431777954102e20L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13a, static_cast(0.125)), + static_cast(0.4173587859727185607499727379291788731397e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13a, static_cast(0.25)), + static_cast(0.4715104623414053008900737040676176548004e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13a, static_cast(0.75)), + static_cast(0.1338397611887558369403450342360883951187e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6407202225497595548875186059951665936749e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n13a, static_cast(6.5f)), + static_cast(0.3403521961549553788009664398431777954102e20L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.125), 13), + static_cast(0.5388702877817484859997819034334309851175e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.25), 13), + static_cast(0.6860418493656212035602948162704706192017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.75), 13), + static_cast(0.165119681585007782587126712314784526825e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f), 13), + static_cast(0.6502554641775335160762093775188993967491e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(6.5f), 13), + static_cast(0.5236187633153159677245637536048889160156e19L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.125)), + static_cast(0.5388702877817484859997819034334309851175e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.25)), + static_cast(0.6860418493656212035602948162704706192017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(0.75)), + static_cast(0.165119681585007782587126712314784526825e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6502554641775335160762093775188993967491e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13c, static_cast(6.5f)), + static_cast(0.5236187633153159677245637536048889160156e19L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13a, static_cast(0.125)), + static_cast(0.5388702877817484859997819034334309851175e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13a, static_cast(0.25)), + static_cast(0.6860418493656212035602948162704706192017e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13a, static_cast(0.75)), + static_cast(0.165119681585007782587126712314784526825e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.6502554641775335160762093775188993967491e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n13a, static_cast(6.5f)), + static_cast(0.5236187633153159677245637536048889160156e19L), + tolerance); + // + // Rational functions of order 12 + // + static const U d13c[13] = { 4, 7, 1, 6, 11, 4, 9, 11, 1, 10, 1, 11, 12 }; + static const boost::array d13a = { 4, 7, 1, 6, 11, 4, 9, 11, 1, 10, 1, 11, 12 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.125), 13), + static_cast(0.1118537310443419140823936840990235560775e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.25), 13), + static_cast(0.12106743933208147312270004711488386001e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.75), 13), + static_cast(0.1042994706832119738384940706309786965245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(1.0f - 1.0f/64.0f), 13), + static_cast(0.8849937505065081554917467830175954168154e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(6.5f), 13), + static_cast(0.1125049235146435310211414108934070096386e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(10247.25f), 13), + static_cast(0.8334214878002966556152610218152238753709e-1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.125)), + static_cast(0.1118537310443419140823936840990235560775e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.25)), + static_cast(0.12106743933208147312270004711488386001e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(0.75)), + static_cast(0.1042994706832119738384940706309786965245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8849937505065081554917467830175954168154e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(6.5f)), + static_cast(0.1125049235146435310211414108934070096386e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13c, d13c, static_cast(10247.25f)), + static_cast(0.8334214878002966556152610218152238753709e-1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(0.125)), + static_cast(0.1118537310443419140823936840990235560775e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(0.25)), + static_cast(0.12106743933208147312270004711488386001e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(0.75)), + static_cast(0.1042994706832119738384940706309786965245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8849937505065081554917467830175954168154e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(6.5f)), + static_cast(0.1125049235146435310211414108934070096386e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n13a, d13a, static_cast(10247.25f)), + static_cast(0.8334214878002966556152610218152238753709e-1L), + tolerance); +} +template +void do_test_spots13(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 13 + // + static const U n14c[14] = { 5, 5, 3, 5, 12, 8, 10, 5, 5, 9, 2, 10, 3, 3 }; + static const boost::array n14a = { 5, 5, 3, 5, 12, 8, 10, 5, 5, 9, 2, 10, 3, 3 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.125), 14), + static_cast(0.5684855352437807596288621425628662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.25), 14), + static_cast(0.657317422330379486083984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.75), 14), + static_cast(0.2256699807941913604736328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f), 14), + static_cast(0.7710103154409585858207746641830426369713e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(6.5), 14), + static_cast(0.1372059025011920166015625e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.125)), + static_cast(0.5684855352437807596288621425628662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.25)), + static_cast(0.657317422330379486083984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(0.75)), + static_cast(0.2256699807941913604736328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7710103154409585858207746641830426369713e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14c, static_cast(6.5)), + static_cast(0.1372059025011920166015625e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14a, static_cast(0.125)), + static_cast(0.5684855352437807596288621425628662109375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14a, static_cast(0.25)), + static_cast(0.657317422330379486083984375e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14a, static_cast(0.75)), + static_cast(0.2256699807941913604736328125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7710103154409585858207746641830426369713e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n14a, static_cast(6.5)), + static_cast(0.1372059025011920166015625e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.125), 14), + static_cast(0.5078877218214320312311861835604176527818e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.25), 14), + static_cast(0.5325630803958699699407475236512254923582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.75), 14), + static_cast(0.1183906394403697492911931021808413788676e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f), 14), + static_cast(0.7015764398304385537424317046920004042802e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(6.5f), 14), + static_cast(0.4205557544065332561152158209905028343201e22L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.125)), + static_cast(0.5078877218214320312311861835604176527818e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.25)), + static_cast(0.5325630803958699699407475236512254923582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(0.75)), + static_cast(0.1183906394403697492911931021808413788676e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7015764398304385537424317046920004042802e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14c, static_cast(6.5f)), + static_cast(0.4205557544065332561152158209905028343201e22L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14a, static_cast(0.125)), + static_cast(0.5078877218214320312311861835604176527818e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14a, static_cast(0.25)), + static_cast(0.5325630803958699699407475236512254923582e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14a, static_cast(0.75)), + static_cast(0.1183906394403697492911931021808413788676e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7015764398304385537424317046920004042802e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n14a, static_cast(6.5f)), + static_cast(0.4205557544065332561152158209905028343201e22L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.125), 14), + static_cast(0.5631017745714562498494894684833412222547e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.25), 14), + static_cast(0.6302523215834798797629900946049019694328e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.75), 14), + static_cast(0.1411875192538263323882574695744551718235e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f), 14), + static_cast(0.7119189230023502768177083984172702519672e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(6.5f), 14), + static_cast(0.6470088529331280863353320322930812835693e21L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.125)), + static_cast(0.5631017745714562498494894684833412222547e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.25)), + static_cast(0.6302523215834798797629900946049019694328e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(0.75)), + static_cast(0.1411875192538263323882574695744551718235e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7119189230023502768177083984172702519672e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14c, static_cast(6.5f)), + static_cast(0.6470088529331280863353320322930812835693e21L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14a, static_cast(0.125)), + static_cast(0.5631017745714562498494894684833412222547e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14a, static_cast(0.25)), + static_cast(0.6302523215834798797629900946049019694328e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14a, static_cast(0.75)), + static_cast(0.1411875192538263323882574695744551718235e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7119189230023502768177083984172702519672e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n14a, static_cast(6.5f)), + static_cast(0.6470088529331280863353320322930812835693e21L), + tolerance); + // + // Rational functions of order 13 + // + static const U d14c[14] = { 1, 2, 1, 8, 5, 8, 2, 11, 3, 6, 5, 9, 7, 10 }; + static const boost::array d14a = { 1, 2, 1, 8, 5, 8, 2, 11, 3, 6, 5, 9, 7, 10 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.125), 14), + static_cast(0.4431848049037056640776200482297774574883e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.25), 14), + static_cast(0.3830343514018902583833597521450902906851e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.75), 14), + static_cast(0.1657621200711495617835320787645038062003e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(1.0f - 1.0f/64.0f), 14), + static_cast(0.1117985091800992684003226640313705018678e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(6.5f), 14), + static_cast(0.3280672161259396135368350556553925507681e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(10247.25f), 14), + static_cast(0.3000087891957249835881866421118445480875e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.125)), + static_cast(0.4431848049037056640776200482297774574883e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.25)), + static_cast(0.3830343514018902583833597521450902906851e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(0.75)), + static_cast(0.1657621200711495617835320787645038062003e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1117985091800992684003226640313705018678e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(6.5f)), + static_cast(0.3280672161259396135368350556553925507681e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14c, d14c, static_cast(10247.25f)), + static_cast(0.3000087891957249835881866421118445480875e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(0.125)), + static_cast(0.4431848049037056640776200482297774574883e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(0.25)), + static_cast(0.3830343514018902583833597521450902906851e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(0.75)), + static_cast(0.1657621200711495617835320787645038062003e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1117985091800992684003226640313705018678e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(6.5f)), + static_cast(0.3280672161259396135368350556553925507681e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n14a, d14a, static_cast(10247.25f)), + static_cast(0.3000087891957249835881866421118445480875e0L), + tolerance); +} + +template +void do_test_spots14(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 14 + // + static const U n15c[15] = { 6, 2, 10, 2, 4, 11, 4, 6, 5, 5, 10, 12, 8, 6, 2 }; + static const boost::array n15a = { 6, 2, 10, 2, 4, 11, 4, 6, 5, 5, 10, 12, 8, 6, 2 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.125), 15), + static_cast(0.6411486971785507193999364972114562988281e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.25), 15), + static_cast(0.7184068299829959869384765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.75), 15), + static_cast(0.21735079027712345123291015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f), 15), + static_cast(0.8299984286646360949452954853655163780233e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(6.5), 15), + static_cast(0.7599432765217742919921875e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.125)), + static_cast(0.6411486971785507193999364972114562988281e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.25)), + static_cast(0.7184068299829959869384765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(0.75)), + static_cast(0.21735079027712345123291015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8299984286646360949452954853655163780233e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15c, static_cast(6.5)), + static_cast(0.7599432765217742919921875e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15a, static_cast(0.125)), + static_cast(0.6411486971785507193999364972114562988281e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15a, static_cast(0.25)), + static_cast(0.7184068299829959869384765625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15a, static_cast(0.75)), + static_cast(0.21735079027712345123291015625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8299984286646360949452954853655163780233e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n15a, static_cast(6.5)), + static_cast(0.7599432765217742919921875e12L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.125), 15), + static_cast(0.603369928436724862526100235763586452224e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.25), 15), + static_cast(0.6164622568840771005271861326946236658841e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.75), 15), + static_cast(0.1204200081304656236302896843426424311474e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f), 15), + static_cast(0.7438435296991670658675730793779505088179e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(6.5f), 15), + static_cast(0.123975649339443727300884144392229616642e24L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.125)), + static_cast(0.603369928436724862526100235763586452224e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.25)), + static_cast(0.6164622568840771005271861326946236658841e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(0.75)), + static_cast(0.1204200081304656236302896843426424311474e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7438435296991670658675730793779505088179e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15c, static_cast(6.5f)), + static_cast(0.123975649339443727300884144392229616642e24L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15a, static_cast(0.125)), + static_cast(0.603369928436724862526100235763586452224e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15a, static_cast(0.25)), + static_cast(0.6164622568840771005271861326946236658841e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15a, static_cast(0.75)), + static_cast(0.1204200081304656236302896843426424311474e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7438435296991670658675730793779505088179e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n15a, static_cast(6.5f)), + static_cast(0.123975649339443727300884144392229616642e24L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.125), 15), + static_cast(0.6269594274937989002088018861086916177919e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.25), 15), + static_cast(0.6658490275363084021087445307784946635365e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.75), 15), + static_cast(0.1405600108406208315070529124568565748632e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f), 15), + static_cast(0.7546981889007411462781694774633148026087e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(6.5f), 15), + static_cast(0.1907317682145288112321802221418917179108e23L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.125)), + static_cast(0.6269594274937989002088018861086916177919e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.25)), + static_cast(0.6658490275363084021087445307784946635365e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(0.75)), + static_cast(0.1405600108406208315070529124568565748632e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7546981889007411462781694774633148026087e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15c, static_cast(6.5f)), + static_cast(0.1907317682145288112321802221418917179108e23L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15a, static_cast(0.125)), + static_cast(0.6269594274937989002088018861086916177919e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15a, static_cast(0.25)), + static_cast(0.6658490275363084021087445307784946635365e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15a, static_cast(0.75)), + static_cast(0.1405600108406208315070529124568565748632e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7546981889007411462781694774633148026087e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n15a, static_cast(6.5f)), + static_cast(0.1907317682145288112321802221418917179108e23L), + tolerance); + // + // Rational functions of order 14 + // + static const U d15c[15] = { 7, 10, 7, 1, 10, 11, 11, 10, 7, 1, 10, 9, 8, 2, 3 }; + static const boost::array d15a = { 7, 10, 7, 1, 10, 11, 11, 10, 7, 1, 10, 9, 8, 2, 3 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.125), 15), + static_cast(0.7665435387801205084357966990293734431712e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.25), 15), + static_cast(0.7179510450452743437979498898508621162489e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.75), 15), + static_cast(0.7245051156202447814210400449367214490615e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(1.0f - 1.0f/64.0f), 15), + static_cast(0.8564567127146107292325365412425895512358e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(6.5f), 15), + static_cast(0.8943937154363365123476971638815814742919e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(10247.25f), 15), + static_cast(0.6668184674989563469735607034881326982176e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.125)), + static_cast(0.7665435387801205084357966990293734431712e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.25)), + static_cast(0.7179510450452743437979498898508621162489e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(0.75)), + static_cast(0.7245051156202447814210400449367214490615e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8564567127146107292325365412425895512358e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(6.5f)), + static_cast(0.8943937154363365123476971638815814742919e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15c, d15c, static_cast(10247.25f)), + static_cast(0.6668184674989563469735607034881326982176e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(0.125)), + static_cast(0.7665435387801205084357966990293734431712e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(0.25)), + static_cast(0.7179510450452743437979498898508621162489e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(0.75)), + static_cast(0.7245051156202447814210400449367214490615e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8564567127146107292325365412425895512358e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(6.5f)), + static_cast(0.8943937154363365123476971638815814742919e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n15a, d15a, static_cast(10247.25f)), + static_cast(0.6668184674989563469735607034881326982176e0L), + tolerance); +} + +template +void do_test_spots15(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 15 + // + static const U n16c[16] = { 5, 3, 4, 6, 7, 9, 7, 6, 12, 12, 7, 12, 10, 3, 6, 2 }; + static const boost::array n16a = { 5, 3, 4, 6, 7, 9, 7, 6, 12, 12, 7, 12, 10, 3, 6, 2 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.125), 16), + static_cast(0.545123276921327715172083117067813873291e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.25), 16), + static_cast(0.613219709135591983795166015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.75), 16), + static_cast(0.2195364624075591564178466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f), 16), + static_cast(0.9836568924934260045838041177111855604603e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(6.5), 16), + static_cast(0.474557924297607818603515625e13L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.125)), + static_cast(0.545123276921327715172083117067813873291e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.25)), + static_cast(0.613219709135591983795166015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(0.75)), + static_cast(0.2195364624075591564178466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9836568924934260045838041177111855604603e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16c, static_cast(6.5)), + static_cast(0.474557924297607818603515625e13L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16a, static_cast(0.125)), + static_cast(0.545123276921327715172083117067813873291e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16a, static_cast(0.25)), + static_cast(0.613219709135591983795166015625e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16a, static_cast(0.75)), + static_cast(0.2195364624075591564178466796875e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9836568924934260045838041177111855604603e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n16a, static_cast(6.5)), + static_cast(0.474557924297607818603515625e13L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.125), 16), + static_cast(0.5047874876401281302859159688868887179229e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.25), 16), + static_cast(0.5204705680902215943553490440365294489311e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.75), 16), + static_cast(0.1080276729363941941482185615797106947866e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f), 16), + static_cast(0.8750139513999183433264859554731006794881e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(6.5f), 16), + static_cast(0.5231073382834479254225049897383445873857e25L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.125)), + static_cast(0.5047874876401281302859159688868887179229e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.25)), + static_cast(0.5204705680902215943553490440365294489311e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(0.75)), + static_cast(0.1080276729363941941482185615797106947866e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8750139513999183433264859554731006794881e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16c, static_cast(6.5f)), + static_cast(0.5231073382834479254225049897383445873857e25L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16a, static_cast(0.125)), + static_cast(0.5047874876401281302859159688868887179229e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16a, static_cast(0.25)), + static_cast(0.5204705680902215943553490440365294489311e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16a, static_cast(0.75)), + static_cast(0.1080276729363941941482185615797106947866e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8750139513999183433264859554731006794881e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n16a, static_cast(6.5f)), + static_cast(0.5231073382834479254225049897383445873857e25L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.125), 16), + static_cast(0.5382999011210250422873277510951097433836e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.25), 16), + static_cast(0.5818822723608863774213961761461177957244e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.75), 16), + static_cast(0.1273702305818589255309580821062809263822e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f), 16), + static_cast(0.8881094109459487932205571611155308490038e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(6.5f), 16), + static_cast(0.8047805204360737314192426765205301344395e24L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.125)), + static_cast(0.5382999011210250422873277510951097433836e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.25)), + static_cast(0.5818822723608863774213961761461177957244e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(0.75)), + static_cast(0.1273702305818589255309580821062809263822e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8881094109459487932205571611155308490038e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16c, static_cast(6.5f)), + static_cast(0.8047805204360737314192426765205301344395e24L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16a, static_cast(0.125)), + static_cast(0.5382999011210250422873277510951097433836e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16a, static_cast(0.25)), + static_cast(0.5818822723608863774213961761461177957244e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16a, static_cast(0.75)), + static_cast(0.1273702305818589255309580821062809263822e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8881094109459487932205571611155308490038e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n16a, static_cast(6.5f)), + static_cast(0.8047805204360737314192426765205301344395e24L), + tolerance); + // + // Rational functions of order 15 + // + static const U d16c[16] = { 9, 5, 2, 4, 4, 12, 11, 5, 1, 4, 1, 7, 9, 6, 6, 10 }; + static const boost::array d16a = { 9, 5, 2, 4, 4, 12, 11, 5, 1, 4, 1, 7, 9, 6, 6, 10 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.125), 16), + static_cast(0.5639916333371058636233240074655101514789e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.25), 16), + static_cast(0.5858114501425065583332351775671440891069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.75), 16), + static_cast(0.9254727283150680371198227978127749853934e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(1.0f - 1.0f/64.0f), 16), + static_cast(0.1155192269792597191440811772688208898465e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(6.5f), 16), + static_cast(0.2736469218296852258039959401695110274611e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(10247.25f), 16), + static_cast(0.2000468408076577399196121638600795513365e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.125)), + static_cast(0.5639916333371058636233240074655101514789e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.25)), + static_cast(0.5858114501425065583332351775671440891069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(0.75)), + static_cast(0.9254727283150680371198227978127749853934e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1155192269792597191440811772688208898465e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(6.5f)), + static_cast(0.2736469218296852258039959401695110274611e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16c, d16c, static_cast(10247.25f)), + static_cast(0.2000468408076577399196121638600795513365e0L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(0.125)), + static_cast(0.5639916333371058636233240074655101514789e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(0.25)), + static_cast(0.5858114501425065583332351775671440891069e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(0.75)), + static_cast(0.9254727283150680371198227978127749853934e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.1155192269792597191440811772688208898465e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(6.5f)), + static_cast(0.2736469218296852258039959401695110274611e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n16a, d16a, static_cast(10247.25f)), + static_cast(0.2000468408076577399196121638600795513365e0L), + tolerance); +} + +template +void do_test_spots16(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 16 + // + static const U n17c[17] = { 9, 6, 11, 2, 9, 10, 6, 3, 3, 3, 4, 9, 10, 2, 3, 2, 2 }; + static const boost::array n17a = { 9, 6, 11, 2, 9, 10, 6, 3, 3, 3, 4, 9, 10, 2, 3, 2, 2 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.125), 17), + static_cast(0.9928308216189925872185995103791356086731e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.25), 17), + static_cast(0.112653836444951593875885009765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.75), 17), + static_cast(0.288157429718412458896636962890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f), 17), + static_cast(0.8499409476622319741012411965028222548509e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(6.5), 17), + static_cast(0.24291309657542805938720703125e14L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.125)), + static_cast(0.9928308216189925872185995103791356086731e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.25)), + static_cast(0.112653836444951593875885009765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(0.75)), + static_cast(0.288157429718412458896636962890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8499409476622319741012411965028222548509e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17c, static_cast(6.5)), + static_cast(0.24291309657542805938720703125e14L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17a, static_cast(0.125)), + static_cast(0.9928308216189925872185995103791356086731e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17a, static_cast(0.25)), + static_cast(0.112653836444951593875885009765625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17a, static_cast(0.75)), + static_cast(0.288157429718412458896636962890625e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8499409476622319741012411965028222548509e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n17a, static_cast(6.5)), + static_cast(0.24291309657542805938720703125e14L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.125), 17), + static_cast(0.9096443722112564415334975111453014308977e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.25), 17), + static_cast(0.9418604266644799371155545586464796770088e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.75), 17), + static_cast(0.1800748238050061998856161277204890325265e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f), 17), + static_cast(0.7725414891276907880696219594586980069116e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(6.5f), 17), + static_cast(0.2112416072820759278619692697965107844211e27L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.125)), + static_cast(0.9096443722112564415334975111453014308977e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.25)), + static_cast(0.9418604266644799371155545586464796770088e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(0.75)), + static_cast(0.1800748238050061998856161277204890325265e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7725414891276907880696219594586980069116e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17c, static_cast(6.5f)), + static_cast(0.2112416072820759278619692697965107844211e27L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17a, static_cast(0.125)), + static_cast(0.9096443722112564415334975111453014308977e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17a, static_cast(0.25)), + static_cast(0.9418604266644799371155545586464796770088e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17a, static_cast(0.75)), + static_cast(0.1800748238050061998856161277204890325265e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7725414891276907880696219594586980069116e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n17a, static_cast(6.5f)), + static_cast(0.2112416072820759278619692697965107844211e27L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.125), 17), + static_cast(0.9771549776900515322679800891624114471814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.25), 17), + static_cast(0.1067441706657919748462218234585918708035e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.75), 17), + static_cast(0.210099765073341599847488170293985376702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f), 17), + static_cast(0.7833754810186065148643778635770900387673e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(6.5f), 17), + static_cast(0.3249870881262706582491835681484781298786e26L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.125)), + static_cast(0.9771549776900515322679800891624114471814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.25)), + static_cast(0.1067441706657919748462218234585918708035e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(0.75)), + static_cast(0.210099765073341599847488170293985376702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7833754810186065148643778635770900387673e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17c, static_cast(6.5f)), + static_cast(0.3249870881262706582491835681484781298786e26L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17a, static_cast(0.125)), + static_cast(0.9771549776900515322679800891624114471814e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17a, static_cast(0.25)), + static_cast(0.1067441706657919748462218234585918708035e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17a, static_cast(0.75)), + static_cast(0.210099765073341599847488170293985376702e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7833754810186065148643778635770900387673e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n17a, static_cast(6.5f)), + static_cast(0.3249870881262706582491835681484781298786e26L), + tolerance); + // + // Rational functions of order 16 + // + static const U d17c[17] = { 7, 12, 3, 11, 10, 2, 5, 10, 4, 11, 10, 6, 2, 12, 1, 2, 1 }; + static const boost::array d17a = { 7, 12, 3, 11, 10, 2, 5, 10, 4, 11, 10, 6, 2, 12, 1, 2, 1 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.125), 17), + static_cast(0.1158375951946763080209403673166347533318e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.25), 17), + static_cast(0.1082966720285644924561131930917359028989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.75), 17), + static_cast(0.9405394059495731884564238556014726773139e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(1.0f - 1.0f/64.0f), 17), + static_cast(0.8654376623250381987950547557864849720673e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(6.5f), 17), + static_cast(0.1737601600028386310221746538845672721834e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(10247.25f), 17), + static_cast(0.1999804873273333726353229695741031917586e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.125)), + static_cast(0.1158375951946763080209403673166347533318e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.25)), + static_cast(0.1082966720285644924561131930917359028989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(0.75)), + static_cast(0.9405394059495731884564238556014726773139e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8654376623250381987950547557864849720673e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(6.5f)), + static_cast(0.1737601600028386310221746538845672721834e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17c, d17c, static_cast(10247.25f)), + static_cast(0.1999804873273333726353229695741031917586e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(0.125)), + static_cast(0.1158375951946763080209403673166347533318e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(0.25)), + static_cast(0.1082966720285644924561131930917359028989e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(0.75)), + static_cast(0.9405394059495731884564238556014726773139e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8654376623250381987950547557864849720673e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(6.5f)), + static_cast(0.1737601600028386310221746538845672721834e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n17a, d17a, static_cast(10247.25f)), + static_cast(0.1999804873273333726353229695741031917586e1L), + tolerance); +} + +template +void do_test_spots17(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 17 + // + static const U n18c[18] = { 6, 5, 7, 8, 12, 9, 4, 1, 3, 5, 1, 5, 12, 8, 4, 6, 1, 10 }; + static const boost::array n18a = { 6, 5, 7, 8, 12, 9, 4, 1, 3, 5, 1, 5, 12, 8, 4, 6, 1, 10 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.125), 18), + static_cast(0.6753220299099845114199069939786568284035e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.25), 18), + static_cast(0.7869269511546008288860321044921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.75), 18), + static_cast(0.25590585466590709984302520751953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f), 18), + static_cast(0.9432688198514753127308115619023015607828e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(6.5), 18), + static_cast(0.680830157865510950897216796875e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.125)), + static_cast(0.6753220299099845114199069939786568284035e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.25)), + static_cast(0.7869269511546008288860321044921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(0.75)), + static_cast(0.25590585466590709984302520751953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9432688198514753127308115619023015607828e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18c, static_cast(6.5)), + static_cast(0.680830157865510950897216796875e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18a, static_cast(0.125)), + static_cast(0.6753220299099845114199069939786568284035e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18a, static_cast(0.25)), + static_cast(0.7869269511546008288860321044921875e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18a, static_cast(0.75)), + static_cast(0.25590585466590709984302520751953125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9432688198514753127308115619023015607828e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n18a, static_cast(6.5)), + static_cast(0.680830157865510950897216796875e15L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.125), 18), + static_cast(0.6079865225649211447450201991749585071453e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.25), 18), + static_cast(0.6341988806453956744898332268528529098717e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.75), 18), + static_cast(0.1439378129034890784564674906867431936064e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f), 18), + static_cast(0.8374908316692411299816073455081075151876e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(6.5f), 18), + static_cast(0.4367459075155664096657376949959960997093e29L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.125)), + static_cast(0.6079865225649211447450201991749585071453e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.25)), + static_cast(0.6341988806453956744898332268528529098717e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(0.75)), + static_cast(0.1439378129034890784564674906867431936064e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8374908316692411299816073455081075151876e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18c, static_cast(6.5f)), + static_cast(0.4367459075155664096657376949959960997093e29L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18a, static_cast(0.125)), + static_cast(0.6079865225649211447450201991749585071453e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18a, static_cast(0.25)), + static_cast(0.6341988806453956744898332268528529098717e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18a, static_cast(0.75)), + static_cast(0.1439378129034890784564674906867431936064e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8374908316692411299816073455081075151876e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n18a, static_cast(6.5f)), + static_cast(0.4367459075155664096657376949959960997093e29L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.125), 18), + static_cast(0.6638921805193691579601615933996680571626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.25), 18), + static_cast(0.7367955225815826979593329074114116394867e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.75), 18), + static_cast(0.1719170838713187712752899875823242581419e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f), 18), + static_cast(0.8498319559814513066479820652780774757461e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(6.5f), 18), + static_cast(0.6719167807931790917934426081938401533989e28L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.125)), + static_cast(0.6638921805193691579601615933996680571626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.25)), + static_cast(0.7367955225815826979593329074114116394867e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(0.75)), + static_cast(0.1719170838713187712752899875823242581419e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8498319559814513066479820652780774757461e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18c, static_cast(6.5f)), + static_cast(0.6719167807931790917934426081938401533989e28L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18a, static_cast(0.125)), + static_cast(0.6638921805193691579601615933996680571626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18a, static_cast(0.25)), + static_cast(0.7367955225815826979593329074114116394867e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18a, static_cast(0.75)), + static_cast(0.1719170838713187712752899875823242581419e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8498319559814513066479820652780774757461e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n18a, static_cast(6.5f)), + static_cast(0.6719167807931790917934426081938401533989e28L), + tolerance); + // + // Rational functions of order 17 + // + static const U d18c[18] = { 6, 2, 11, 2, 12, 4, 1, 5, 7, 12, 5, 7, 5, 7, 5, 7, 2, 9 }; + static const boost::array d18a = { 6, 2, 11, 2, 12, 4, 1, 5, 7, 12, 5, 7, 5, 7, 5, 7, 2, 9 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.125), 18), + static_cast(0.1050457095586493221315235501384188635425e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.25), 18), + static_cast(0.1082394733850278225180153358339433604182e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.75), 18), + static_cast(0.1117650157149023587583496399377426188558e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(1.0f - 1.0f/64.0f), 18), + static_cast(0.9882013696252427983431096654690843223851e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(6.5f), 18), + static_cast(0.1086348665080997883580887205673821889245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(10247.25f), 18), + static_cast(0.1111097856940394090609547290690047016487e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.125)), + static_cast(0.1050457095586493221315235501384188635425e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.25)), + static_cast(0.1082394733850278225180153358339433604182e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(0.75)), + static_cast(0.1117650157149023587583496399377426188558e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9882013696252427983431096654690843223851e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(6.5f)), + static_cast(0.1086348665080997883580887205673821889245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18c, d18c, static_cast(10247.25f)), + static_cast(0.1111097856940394090609547290690047016487e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(0.125)), + static_cast(0.1050457095586493221315235501384188635425e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(0.25)), + static_cast(0.1082394733850278225180153358339433604182e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(0.75)), + static_cast(0.1117650157149023587583496399377426188558e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.9882013696252427983431096654690843223851e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(6.5f)), + static_cast(0.1086348665080997883580887205673821889245e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n18a, d18a, static_cast(10247.25f)), + static_cast(0.1111097856940394090609547290690047016487e1L), + tolerance); +} + +template +void do_test_spots18(T, U) +{ + // + // Tolerance is 4 eps expressed as a persentage: + // + T tolerance = boost::math::tools::epsilon() * 4 * 100; + + // + // Polynomials of order 18 + // + static const U n19c[19] = { 7, 2, 4, 2, 4, 3, 9, 1, 9, 3, 7, 2, 10, 4, 2, 5, 11, 3, 9 }; + static const boost::array n19a = { 7, 2, 4, 2, 4, 3, 9, 1, 9, 3, 7, 2, 10, 4, 2, 5, 11, 3, 9 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.125), 19), + static_cast(0.7317509740045990140888676478425622917712e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.25), 19), + static_cast(0.7802219584656995721161365509033203125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.75), 19), + static_cast(0.17609299321266007609665393829345703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f), 19), + static_cast(0.8327888521934174049333133442289010778727e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(6.5), 19), + static_cast(0.4179028813935817562572479248046875e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.125)), + static_cast(0.7317509740045990140888676478425622917712e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.25)), + static_cast(0.7802219584656995721161365509033203125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(0.75)), + static_cast(0.17609299321266007609665393829345703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8327888521934174049333133442289010778727e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19c, static_cast(6.5)), + static_cast(0.4179028813935817562572479248046875e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19a, static_cast(0.125)), + static_cast(0.7317509740045990140888676478425622917712e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19a, static_cast(0.25)), + static_cast(0.7802219584656995721161365509033203125e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19a, static_cast(0.75)), + static_cast(0.17609299321266007609665393829345703125e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8327888521934174049333133442289010778727e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_polynomial(n19a, static_cast(6.5)), + static_cast(0.4179028813935817562572479248046875e16L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.125), 19), + static_cast(0.7032234433238304833197011488540991991734e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.25), 19), + static_cast(0.7141177719741940632063571816842174888595e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.75), 19), + static_cast(0.107671418972012709727788225142148040292e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f), 19), + static_cast(0.7205306905267900876445365429703520869133e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(6.5f), 19), + static_cast(0.1670453627683043936397442394984734181614e31L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.125)), + static_cast(0.7032234433238304833197011488540991991734e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.25)), + static_cast(0.7141177719741940632063571816842174888595e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(0.75)), + static_cast(0.107671418972012709727788225142148040292e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7205306905267900876445365429703520869133e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19c, static_cast(6.5f)), + static_cast(0.1670453627683043936397442394984734181614e31L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19a, static_cast(0.125)), + static_cast(0.7032234433238304833197011488540991991734e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19a, static_cast(0.25)), + static_cast(0.7141177719741940632063571816842174888595e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19a, static_cast(0.75)), + static_cast(0.107671418972012709727788225142148040292e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7205306905267900876445365429703520869133e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_even_polynomial(n19a, static_cast(6.5f)), + static_cast(0.1670453627683043936397442394984734181614e31L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.125), 19), + static_cast(0.7257875465906438665576091908327935933871e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.25), 19), + static_cast(0.7564710878967762528254287267368699554382e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.75), 19), + static_cast(0.120228558629350279637050966856197387056e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f), 19), + static_cast(0.7308565745034058033214339484143259295627e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(6.5f), 19), + static_cast(0.2569928657973913748303757530804975664022e30L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.125)), + static_cast(0.7257875465906438665576091908327935933871e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.25)), + static_cast(0.7564710878967762528254287267368699554382e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(0.75)), + static_cast(0.120228558629350279637050966856197387056e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7308565745034058033214339484143259295627e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19c, static_cast(6.5f)), + static_cast(0.2569928657973913748303757530804975664022e30L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19a, static_cast(0.125)), + static_cast(0.7257875465906438665576091908327935933871e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19a, static_cast(0.25)), + static_cast(0.7564710878967762528254287267368699554382e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19a, static_cast(0.75)), + static_cast(0.120228558629350279637050966856197387056e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.7308565745034058033214339484143259295627e2L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_odd_polynomial(n19a, static_cast(6.5f)), + static_cast(0.2569928657973913748303757530804975664022e30L), + tolerance); + // + // Rational functions of order 18 + // + static const U d19c[19] = { 3, 2, 3, 3, 10, 6, 2, 6, 9, 8, 8, 10, 5, 7, 6, 4, 6, 9, 7 }; + static const boost::array d19a = { 3, 2, 3, 3, 10, 6, 2, 6, 9, 8, 8, 10, 5, 7, 6, 4, 6, 9, 7 }; + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.125), 19), + static_cast(0.2213824709533496994632324982010106870288e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.25), 19), + static_cast(0.2063899260318829004588039751781791436716e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.75), 19), + static_cast(0.1085337280046506127330919833145641152667e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(1.0f - 1.0f/64.0f), 19), + static_cast(0.8530644127293554041631338796773621506306e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(6.5f), 19), + static_cast(0.1140014406058131768268407547610620204626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(10247.25f), 19), + static_cast(0.1285594810697167354761598106316532892359e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.125)), + static_cast(0.2213824709533496994632324982010106870288e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.25)), + static_cast(0.2063899260318829004588039751781791436716e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(0.75)), + static_cast(0.1085337280046506127330919833145641152667e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8530644127293554041631338796773621506306e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(6.5f)), + static_cast(0.1140014406058131768268407547610620204626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19c, d19c, static_cast(10247.25f)), + static_cast(0.1285594810697167354761598106316532892359e1L), + tolerance); + + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(0.125)), + static_cast(0.2213824709533496994632324982010106870288e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(0.25)), + static_cast(0.2063899260318829004588039751781791436716e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(0.75)), + static_cast(0.1085337280046506127330919833145641152667e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(1.0f - 1.0f/64.0f)), + static_cast(0.8530644127293554041631338796773621506306e0L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(6.5f)), + static_cast(0.1140014406058131768268407547610620204626e1L), + tolerance); + BOOST_CHECK_CLOSE( + boost::math::tools::evaluate_rational(n19a, d19a, static_cast(10247.25f)), + static_cast(0.1285594810697167354761598106316532892359e1L), + tolerance); +} + +template +void do_test_spots(T t, U u) +{ + do_test_spots1(t, u); + do_test_spots2(t, u); + do_test_spots3(t, u); + do_test_spots4(t, u); + do_test_spots5(t, u); + do_test_spots6(t, u); + do_test_spots7(t, u); + do_test_spots8(t, u); + do_test_spots9(t, u); + do_test_spots10(t, u); + do_test_spots11(t, u); + do_test_spots12(t, u); + do_test_spots13(t, u); + do_test_spots14(t, u); + do_test_spots15(t, u); + do_test_spots16(t, u); + do_test_spots17(t, u); + do_test_spots18(t, u); +} + +#endif diff --git a/test/test_rational_instances/test_rational_double1.cpp b/test/test_rational_instances/test_rational_double1.cpp new file mode 100644 index 000000000..4cc42ec0f --- /dev/null +++ b/test/test_rational_instances/test_rational_double1.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(double, double); + diff --git a/test/test_rational_instances/test_rational_double2.cpp b/test/test_rational_instances/test_rational_double2.cpp new file mode 100644 index 000000000..254bbcd90 --- /dev/null +++ b/test/test_rational_instances/test_rational_double2.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(double, int); + diff --git a/test/test_rational_instances/test_rational_double3.cpp b/test/test_rational_instances/test_rational_double3.cpp new file mode 100644 index 000000000..d371b988b --- /dev/null +++ b/test/test_rational_instances/test_rational_double3.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(double, unsigned); + diff --git a/test/test_rational_instances/test_rational_double4.cpp b/test/test_rational_instances/test_rational_double4.cpp new file mode 100644 index 000000000..64e31eccd --- /dev/null +++ b/test/test_rational_instances/test_rational_double4.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "test_rational.hpp" + +#ifdef BOOST_HAS_LONG_LONG +template void do_test_spots(double, unsigned long long); +#endif diff --git a/test/test_rational_instances/test_rational_double5.cpp b/test/test_rational_instances/test_rational_double5.cpp new file mode 100644 index 000000000..f5ed5c863 --- /dev/null +++ b/test/test_rational_instances/test_rational_double5.cpp @@ -0,0 +1,9 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(double, float); diff --git a/test/test_rational_instances/test_rational_float1.cpp b/test/test_rational_instances/test_rational_float1.cpp new file mode 100644 index 000000000..227c48267 --- /dev/null +++ b/test/test_rational_instances/test_rational_float1.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(float, float); + diff --git a/test/test_rational_instances/test_rational_float2.cpp b/test/test_rational_instances/test_rational_float2.cpp new file mode 100644 index 000000000..81b843d83 --- /dev/null +++ b/test/test_rational_instances/test_rational_float2.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(float, int); + diff --git a/test/test_rational_instances/test_rational_float3.cpp b/test/test_rational_instances/test_rational_float3.cpp new file mode 100644 index 000000000..aa7a1ae7d --- /dev/null +++ b/test/test_rational_instances/test_rational_float3.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(float, unsigned); + diff --git a/test/test_rational_instances/test_rational_float4.cpp b/test/test_rational_instances/test_rational_float4.cpp new file mode 100644 index 000000000..49f792016 --- /dev/null +++ b/test/test_rational_instances/test_rational_float4.cpp @@ -0,0 +1,11 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +#ifdef BOOST_HAS_LONG_LONG +template void do_test_spots(float, unsigned long long); +#endif diff --git a/test/test_rational_instances/test_rational_ldouble1.cpp b/test/test_rational_instances/test_rational_ldouble1.cpp new file mode 100644 index 000000000..5b69bbafb --- /dev/null +++ b/test/test_rational_instances/test_rational_ldouble1.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(long double, long double); + diff --git a/test/test_rational_instances/test_rational_ldouble2.cpp b/test/test_rational_instances/test_rational_ldouble2.cpp new file mode 100644 index 000000000..593c33f33 --- /dev/null +++ b/test/test_rational_instances/test_rational_ldouble2.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(long double, int); + diff --git a/test/test_rational_instances/test_rational_ldouble3.cpp b/test/test_rational_instances/test_rational_ldouble3.cpp new file mode 100644 index 000000000..82281c381 --- /dev/null +++ b/test/test_rational_instances/test_rational_ldouble3.cpp @@ -0,0 +1,10 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(long double, unsigned); + diff --git a/test/test_rational_instances/test_rational_ldouble4.cpp b/test/test_rational_instances/test_rational_ldouble4.cpp new file mode 100644 index 000000000..9b945b186 --- /dev/null +++ b/test/test_rational_instances/test_rational_ldouble4.cpp @@ -0,0 +1,11 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +#ifdef BOOST_HAS_LONG_LONG +template void do_test_spots(long double, unsigned long long); +#endif diff --git a/test/test_rational_instances/test_rational_ldouble5.cpp b/test/test_rational_instances/test_rational_ldouble5.cpp new file mode 100644 index 000000000..cccd06926 --- /dev/null +++ b/test/test_rational_instances/test_rational_ldouble5.cpp @@ -0,0 +1,9 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include "test_rational.hpp" + +template void do_test_spots(long double, float); diff --git a/test/test_rational_instances/test_rational_real_concept1.cpp b/test/test_rational_instances/test_rational_real_concept1.cpp new file mode 100644 index 000000000..144cc33df --- /dev/null +++ b/test/test_rational_instances/test_rational_real_concept1.cpp @@ -0,0 +1,15 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + +#include "test_rational.hpp" +#include + +template void do_test_spots(boost::math::concepts::real_concept, boost::math::concepts::real_concept); + +#endif diff --git a/test/test_rational_instances/test_rational_real_concept2.cpp b/test/test_rational_instances/test_rational_real_concept2.cpp new file mode 100644 index 000000000..2c04fc62e --- /dev/null +++ b/test/test_rational_instances/test_rational_real_concept2.cpp @@ -0,0 +1,15 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + +#include "test_rational.hpp" +#include + +template void do_test_spots(boost::math::concepts::real_concept, int); + +#endif diff --git a/test/test_rational_instances/test_rational_real_concept3.cpp b/test/test_rational_instances/test_rational_real_concept3.cpp new file mode 100644 index 000000000..e238dcc3e --- /dev/null +++ b/test/test_rational_instances/test_rational_real_concept3.cpp @@ -0,0 +1,15 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + +#include "test_rational.hpp" +#include + +template void do_test_spots(boost::math::concepts::real_concept, unsigned); + +#endif diff --git a/test/test_rational_instances/test_rational_real_concept4.cpp b/test/test_rational_instances/test_rational_real_concept4.cpp new file mode 100644 index 000000000..e74537f75 --- /dev/null +++ b/test/test_rational_instances/test_rational_real_concept4.cpp @@ -0,0 +1,16 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + +#include "test_rational.hpp" +#include + +#ifdef BOOST_HAS_LONG_LONG +template void do_test_spots(boost::math::concepts::real_concept, unsigned long long); +#endif +#endif diff --git a/test/test_rational_instances/test_rational_real_concept5.cpp b/test/test_rational_instances/test_rational_real_concept5.cpp new file mode 100644 index 000000000..1bd217c37 --- /dev/null +++ b/test/test_rational_instances/test_rational_real_concept5.cpp @@ -0,0 +1,15 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + +#include "test_rational.hpp" +#include + +template void do_test_spots(boost::math::concepts::real_concept, float); + +#endif diff --git a/test/test_rationals.cpp b/test/test_rationals.cpp new file mode 100644 index 000000000..7987cdfc9 --- /dev/null +++ b/test/test_rationals.cpp @@ -0,0 +1,47 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include + +template +void do_test_spots(T, U); + +template +void test_spots(T t, const char* n) +{ + std::cout << "Testing basic sanity checks for type " << n << std::endl; + do_test_spots(t, int(0)); + do_test_spots(t, unsigned(0)); +#ifdef BOOST_HAS_LONG_LONG + do_test_spots(t, (unsigned long long)(0)); +#endif + do_test_spots(t, float(0)); + do_test_spots(t, T(0)); +} + +int test_main(int, char* []) +{ + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + diff --git a/test/test_rayleigh.cpp b/test/test_rayleigh.cpp new file mode 100644 index 000000000..be876ad24 --- /dev/null +++ b/test/test_rayleigh.cpp @@ -0,0 +1,319 @@ +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_rayleigh.cpp + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +# pragma warning(disable: 4100) // unreferenced formal parameter. +#endif + +#include // for real_concept +#include + using boost::math::rayleigh_distribution; + +#include // Boost.Test +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; + +template +void test_spot(RealType s, RealType x, RealType p, RealType q, RealType tolerance) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + rayleigh_distribution(s), + x), + p, + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement(rayleigh_distribution(s), + x)), + q, + tolerance); // % + // Special extra tests for p and q near to unity. + if(p < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + rayleigh_distribution(s), + p), + x, + tolerance); // % + } + if(q < 0.999) + { + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement(rayleigh_distribution(s), + q)), + x, + tolerance); // % + } +} // void test_spot + +template +void test_spots(RealType T) +{ + using namespace std; // ADL of std names. + // Basic sanity checks. + // 50 eps as a percentage, up to a maximum of double precision + // (that's the limit of our test data: obtained by punching + // numbers into a calculator). + RealType tolerance = (std::max)( + static_cast(boost::math::tools::epsilon()), + boost::math::tools::epsilon()); + tolerance *= 10 * 100; // 10 eps as a percent + cout << "Tolerance for type " << typeid(T).name() << " is " << tolerance << " %" << endl; + + using namespace boost::math::constants; + + // Things that are errors: + rayleigh_distribution dist(0.5); + + BOOST_CHECK_THROW( + quantile(dist, + RealType(1.)), // quantile unity should overflow. + std::overflow_error); + BOOST_CHECK_THROW( + quantile(complement(dist, + RealType(0.))), // quantile complement zero should overflow. + std::overflow_error); + BOOST_CHECK_THROW( + pdf(dist, RealType(-1)), // Bad negative x. + std::domain_error); + BOOST_CHECK_THROW( + cdf(dist, RealType(-1)), // Bad negative x. + std::domain_error); + BOOST_CHECK_THROW( + cdf(rayleigh_distribution(-1), // bad sigma < 0 + RealType(1)), + std::domain_error); + BOOST_CHECK_THROW( + cdf(rayleigh_distribution(0), // bad sigma == 0 + RealType(1)), + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(-1)), // negative quantile probability. + std::domain_error); + BOOST_CHECK_THROW( + quantile(dist, RealType(2)), // > unity quantile probability. + std::domain_error); + + test_spot( + static_cast(1.L), // sigma + static_cast(1.L), // x + static_cast(1 - exp_minus_half()), // p + static_cast(exp_minus_half()), // q + tolerance); + + test_spot( + static_cast(0.5L), // sigma + static_cast(0.5L), // x + static_cast(1 - exp_minus_half()), // p + static_cast(exp_minus_half()), //q + tolerance); + + test_spot( + static_cast(3.L), // sigma + static_cast(3.L), // x + static_cast(1 - exp_minus_half()), // p + static_cast(exp_minus_half()), //q + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + rayleigh_distribution(1.L), + static_cast(1.L)), // x + static_cast(exp_minus_half()), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + rayleigh_distribution(0.5L), + static_cast(0.5L)), // x + static_cast(2 * exp_minus_half()), // probability. + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::pdf( + rayleigh_distribution(2.L), + static_cast(2.L)), // x + static_cast(exp_minus_half() /2), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::mean( + rayleigh_distribution(1.L)), + static_cast(root_half_pi()), + tolerance); // % + BOOST_CHECK_CLOSE( + ::boost::math::variance( + rayleigh_distribution(root_two())), + static_cast(four_minus_pi()), + tolerance * 100); // % + + BOOST_CHECK_CLOSE( + ::boost::math::mode( + rayleigh_distribution(1.L)), + static_cast(1.L), + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::median( + rayleigh_distribution(1.L)), + static_cast(sqrt(log(4.L))), // sigma * sqrt(log_four) + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::skewness( + rayleigh_distribution(1.L)), + static_cast(2.L * root_pi()) * (pi() - 3) / (pow((4 - pi()), static_cast(1.5L))), + tolerance * 100); // % + + BOOST_CHECK_CLOSE( + ::boost::math::skewness( + rayleigh_distribution(1.L)), + static_cast(0.63111065781893713819189935154422777984404221106391L), + tolerance * 100); // % + + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis_excess( + rayleigh_distribution(1.L)), + static_cast(0.2450893006876380628486604106197544154170667057995L), + tolerance * 1000); // % + + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis( + rayleigh_distribution(1.L)), + static_cast(3.2450893006876380628486604106197544154170667057995L), + tolerance * 100); // % + + + BOOST_CHECK_CLOSE( + ::boost::math::kurtosis_excess(rayleigh_distribution(2)), + ::boost::math::kurtosis(rayleigh_distribution(2)) -3, + tolerance* 100); // % + return; + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can generate rayleigh distribution using the two convenience methods: + boost::math::rayleigh ray1(1.); // Using typedef + rayleigh_distribution<> ray2(1.); // Using default RealType double. + + using namespace boost::math::constants; + // Basic sanity-check spot values. + + // Double only tests. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::pdf( + rayleigh_distribution(1.), + static_cast(1)), // x + static_cast(exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::pdf( + rayleigh_distribution(0.5), + static_cast(0.5)), // x + static_cast(2 * exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::pdf( + rayleigh_distribution(2.), + static_cast(2)), // x + static_cast(exp_minus_half() /2 ), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + rayleigh_distribution(1.), + static_cast(1)), // x + static_cast(1- exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + rayleigh_distribution(2.), + static_cast(2)), // x + static_cast(1- exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + rayleigh_distribution(3.), + static_cast(3)), // x + static_cast(1- exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + rayleigh_distribution(4.), + static_cast(4)), // x + static_cast(1- exp_minus_half()), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf(complement( + rayleigh_distribution(4.), + static_cast(4))), // x + static_cast(exp_minus_half()), // q = 1 - p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + rayleigh_distribution(4.), + static_cast(1- exp_minus_half())), // x + static_cast(4), // p + 1e-15); // % + + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile(complement( + rayleigh_distribution(4.), + static_cast(exp_minus_half()))), // x + static_cast(4), // p + 1e-15); // % + + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output is: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_rayleigh.exe" +Running 1 test case... +Tolerance for type float is 0.000119209 % +Tolerance for type double is 2.22045e-013 % +Tolerance for type long double is 2.22045e-013 % +Tolerance for type class boost::math::concepts::real_concept is 2.22045e-013 % +*** No errors detected + +*/ + + diff --git a/test/test_remez.cpp b/test/test_remez.cpp new file mode 100644 index 000000000..661217ccd --- /dev/null +++ b/test/test_remez.cpp @@ -0,0 +1,181 @@ +// Copyright John Maddock 2006 +// Copyright Paul A. Bristow 2007 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifdef _MSC_VER +# pragma warning(disable : 4267) // conversion from 'size_t' to 'const unsigned int', possible loss of data +# pragma warning(disable : 4180) // qualifier applied to function type has no meaning; ignored +# pragma warning(disable : 4224) // nonstandard extension used : formal parameter 'function_ptr' was previously defined as a type +#endif + +#include +#include +#include +#include + + +void test_polynomial() +{ +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + double (*f)(double) = boost::math::expm1; +#else + double (*f)(double) = boost::math::expm1; +#endif + std::cout << "Testing expm1 approximation, pinned to origin, abolute error, 6 term polynomial\n"; + boost::math::tools::remez_minimax approx1(f, 6, 0, -1, 1, true, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx1.iterate(); + std::cout << approx1.error_term() << " " << approx1.max_error() << " " << approx1.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing expm1 approximation, pinned to origin, relative error, 6 term polynomial\n"; + boost::math::tools::remez_minimax approx2(f, 6, 0, -1, 1, true, true); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx2.iterate(); + std::cout << approx2.error_term() << " " << approx2.max_error() << " " << approx2.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + + f = std::exp; + std::cout << "Testing exp approximation, not pinned to origin, abolute error, 6 term polynomial\n"; + boost::math::tools::remez_minimax approx3(f, 6, 0, -1, 1, false, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx3.iterate(); + std::cout << approx3.error_term() << " " << approx3.max_error() << " " << approx3.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing exp approximation, not pinned to origin, relative error, 6 term polynomial\n"; + boost::math::tools::remez_minimax approx4(f, 6, 0, -1, 1, false, true); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx4.iterate(); + std::cout << approx4.error_term() << " " << approx4.max_error() << " " << approx4.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + + f = std::cos; + std::cout << "Testing cos approximation, not pinned to origin, abolute error, 5 term polynomial\n"; + boost::math::tools::remez_minimax approx5(f, 5, 0, -1, 1, false, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx5.iterate(); + std::cout << approx5.error_term() << " " << approx5.max_error() << " " << approx5.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing cos approximation, not pinned to origin, relative error, 5 term polynomial\n"; + boost::math::tools::remez_minimax approx6(f, 5, 0, -1, 1, false, true); + for(unsigned i = 0; i < 7; ++i) + { + approx6.iterate(); + std::cout << approx6.error_term() << " " << approx6.max_error() << " " << approx6.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + + f = std::sin; + std::cout << "Testing sin approximation, pinned to origin, abolute error, 4 term polynomial\n"; + boost::math::tools::remez_minimax approx7(f, 4, 0, 0, 1, true, false); + for(unsigned i = 0; i < 7; ++i) + { + approx7.iterate(); + std::cout << approx7.error_term() << " " << approx7.max_error() << " " << approx7.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing sin approximation, pinned to origin, relative error, 4 term polynomial\n"; + boost::math::tools::remez_minimax approx8(f, 4, 0, 0, 1, true, true); + for(unsigned i = 0; i < 7; ++i) + { + approx8.iterate(); + std::cout << approx8.error_term() << " " << approx8.max_error() << " " << approx8.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; +} + +void test_rational() +{ +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + double (*f)(double) = boost::math::expm1; +#else + double (*f)(double) = boost::math::expm1; +#endif + std::cout << "Testing expm1 approximation, pinned to origin, abolute error, 3+3 term rational\n"; + boost::math::tools::remez_minimax approx1(f, 3, 3, -1, 1, true, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx1.iterate(); + std::cout << approx1.error_term() << " " << approx1.max_error() << " " << approx1.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; +#if 0 + // + // This one causes UBLAS to fail on some systems, so disabled for now. + // + std::cout << "Testing expm1 approximation, pinned to origin, relative error, 3+3 term rational\n"; + boost::math::tools::remez_minimax approx2(f, 3, 3, -1, 1, true, true); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx2.iterate(); + std::cout << approx2.error_term() << " " << approx2.max_error() << " " << approx2.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; +#endif + f = std::exp; + std::cout << "Testing exp approximation, not pinned to origin, abolute error, 3+3 term rational\n"; + boost::math::tools::remez_minimax approx3(f, 3, 3, -1, 1, false, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx3.iterate(); + std::cout << approx3.error_term() << " " << approx3.max_error() << " " << approx3.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing exp approximation, not pinned to origin, relative error, 3+3 term rational\n"; + boost::math::tools::remez_minimax approx4(f, 3, 3, -1, 1, false, true); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx4.iterate(); + std::cout << approx4.error_term() << " " << approx4.max_error() << " " << approx4.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + + f = std::cos; + std::cout << "Testing cos approximation, not pinned to origin, abolute error, 2+2 term rational\n"; + boost::math::tools::remez_minimax approx5(f, 2, 2, 0, 1, false, false); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx5.iterate(); + std::cout << approx5.error_term() << " " << approx5.max_error() << " " << approx5.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; + std::cout << "Testing cos approximation, not pinned to origin, relative error, 2+2 term rational\n"; + boost::math::tools::remez_minimax approx6(f, 2, 2, 0, 1, false, true); + std::cout << "Interpolation Error: " << approx1.max_error() << std::endl; + for(unsigned i = 0; i < 7; ++i) + { + approx6.iterate(); + std::cout << approx6.error_term() << " " << approx6.max_error() << " " << approx6.max_change() << std::endl; + } + std::cout << "~~~~~~~~~~~~~~~~~~~~~~~~~" << std::endl; +} + +int test_main(int, char* []) +{ + test_polynomial(); + test_rational(); + return 0; +} + + diff --git a/test/test_roots.cpp b/test/test_roots.cpp new file mode 100644 index 000000000..2ac488e5a --- /dev/null +++ b/test/test_roots.cpp @@ -0,0 +1,315 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include + +#define BOOST_CHECK_CLOSE_EX(a, b, prec, i) \ + {\ + unsigned int failures = boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed;\ + BOOST_CHECK_CLOSE(a, b, prec); \ + if(failures != boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed)\ + {\ + std::cerr << "Failure was at row " << i << std::endl;\ + std::cerr << std::setprecision(35); \ + std::cerr << "{ " << data[i][0] << " , " << data[i][1] << " , " << data[i][2];\ + std::cerr << " , " << data[i][3] << " , " << data[i][4] << " , " << data[i][5] << " } " << std::endl;\ + }\ + } + + +// +// Implement various versions of inverse of the incomplete beta +// using different root finding algorithms, and deliberately "bad" +// starting conditions: that way we get all the pathological cases +// we could ever wish for!!! +// + +template +struct ibeta_roots_1 // for first order algorithms +{ + ibeta_roots_1(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + T operator()(const T& x) + { + return boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target; + } +private: + T a, b, target; + bool invert; +}; + +template +struct ibeta_roots_2 // for second order algorithms +{ + ibeta_roots_2(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + std::tr1::tuple operator()(const T& x) + { + typedef typename boost::math::lanczos::lanczos::type L; + T f = boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target; + T f1 = invert ? + -boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true, Policy()) + : boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true, Policy()); + T y = 1 - x; + if(y == 0) + y = boost::math::tools::min_value() * 8; + f1 /= y * x; + + // make sure we don't have a zero derivative: + if(f1 == 0) + f1 = (invert ? -1 : 1) * boost::math::tools::min_value() * 64; + + return std::tr1::make_tuple(f, f1); + } +private: + T a, b, target; + bool invert; +}; + +template +struct ibeta_roots_3 // for third order algorithms +{ + ibeta_roots_3(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + std::tr1::tuple operator()(const T& x) + { + typedef typename boost::math::lanczos::lanczos::type L; + T f = boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target; + T f1 = invert ? + -boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true, Policy()) + : boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true, Policy()); + T y = 1 - x; + if(y == 0) + y = boost::math::tools::min_value() * 8; + f1 /= y * x; + T f2 = f1 * (-y * a + (b - 2) * x + 1) / (y * x); + if(invert) + f2 = -f2; + + // make sure we don't have a zero derivative: + if(f1 == 0) + f1 = (invert ? -1 : 1) * boost::math::tools::min_value() * 64; + + return std::tr1::make_tuple(f, f1, f2); + } +private: + T a, b, target; + bool invert; +}; + +double inverse_ibeta_bisect(double a, double b, double z) +{ + typedef boost::math::policies::policy<> pol; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + boost::math::tools::eps_tolerance tol(precision); + return boost::math::tools::bisect(ibeta_roots_1(a, b, z, invert), min, max, tol).first; +} + +double inverse_ibeta_newton(double a, double b, double z) +{ + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::newton_raphson_iterate(ibeta_roots_2 >(a, b, z, invert), guess, min, max, precision); +} + +double inverse_ibeta_halley(double a, double b, double z) +{ + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::halley_iterate(ibeta_roots_3 >(a, b, z, invert), guess, min, max, precision); +} + +double inverse_ibeta_schroeder(double a, double b, double z) +{ + double guess = 0.5; + bool invert = false; + int bits = std::numeric_limits::digits; + + // + // special cases, we need to have these because there may be other + // possible answers: + // + if(z == 1) return 1; + if(z == 0) return 0; + + // + // We need a good estimate of the error in the incomplete beta function + // so that we don't set the desired precision too high. Assume that 3-bits + // are lost each time the arguments increase by a factor of 10: + // + using namespace std; + int bits_lost = static_cast(ceil(log10((std::max)(a, b)) * 3)); + if(bits_lost < 0) + bits_lost = 3; + else + bits_lost += 3; + int precision = bits - bits_lost; + + double min = 0; + double max = 1; + return boost::math::tools::schroeder_iterate(ibeta_roots_3 >(a, b, z, invert), guess, min, max, precision); +} + + +template +void test_inverses(const T& data) +{ + using namespace std; + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + value_type precision = static_cast(ldexp(1.0, 1-boost::math::policies::digits >()/2)) * 100; + if(boost::math::policies::digits >() < 50) + precision = 1; // 1% or two decimal digits, all we can hope for when the input is truncated + + for(unsigned i = 0; i < data.size(); ++i) + { + // + // These inverse tests are thrown off if the output of the + // incomplete beta is too close to 1: basically there is insuffient + // information left in the value we're using as input to the inverse + // to be able to get back to the original value. + // + if(data[i][5] == 0) + { + BOOST_CHECK_EQUAL(inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]), value_type(0)); + BOOST_CHECK_EQUAL(inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]), value_type(0)); + } + else if((1 - data[i][5] > 0.001) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value()) + && (fabs(data[i][5]) > 2 * boost::math::tools::min_value())) + { + value_type inv = inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][2], inv, precision, i); + inv = inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][2], inv, precision, i); + inv = inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][2], inv, precision, i); + inv = inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]); + BOOST_CHECK_CLOSE_EX(data[i][2], inv, precision, i); + } + else if(1 == data[i][5]) + { + BOOST_CHECK_EQUAL(inverse_ibeta_halley(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_schroeder(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_newton(data[i][0], data[i][1], data[i][5]), value_type(1)); + BOOST_CHECK_EQUAL(inverse_ibeta_bisect(data[i][0], data[i][1], data[i][5]), value_type(1)); + } + + } +} + +template +void test_beta(T, const char* /* name */) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // five items, input value a, input value b, integration limits x, beta(a, b, x) and ibeta(a, b, x): + // +# include "ibeta_small_data.ipp" + + test_inverses(ibeta_small_data); + +# include "ibeta_data.ipp" + + test_inverses(ibeta_data); + +# include "ibeta_large_data.ipp" + + test_inverses(ibeta_large_data); +} + +int test_main(int, char* []) +{ + test_beta(0.1, "double"); + return 0; +} + + + diff --git a/test/test_spherical_harmonic.cpp b/test/test_spherical_harmonic.cpp new file mode 100644 index 000000000..3b68be35c --- /dev/null +++ b/test/test_spherical_harmonic.cpp @@ -0,0 +1,314 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the Spherical Harmonic Functions. +// There are two sets of tests, spot +// tests which compare our results with selected values computed +// using the online special function calculator at +// functions.wolfram.com, while the bulk of the accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if((std::numeric_limits::digits <= 64) && + (std::numeric_limits::digits != std::numeric_limits::digits)) + { + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*", // test data group + ".*", 10, 5); // test function + } +#endif + // + // Catch all cases come last: + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 30000, 1000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*", // test data group + ".*", 30000, 1000); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void do_test_spherical_harmonic(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(unsigned, int, value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::spherical_harmonic_r; +#else + pg funcp = boost::math::spherical_harmonic_r; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test Spheric Harmonic against data: + // + result = boost::math::tools::test( + data, + bind_func_int2(funcp, 0, 1, 2, 3), + extract_result(4)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::spherical_harmonic_r", test_name); + +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::spherical_harmonic_i; +#else + funcp = boost::math::spherical_harmonic_i; +#endif + // + // test Spheric Harmonic against data: + // + result = boost::math::tools::test( + data, + bind_func_int2(funcp, 0, 1, 2, 3), + extract_result(5)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::spherical_harmonic_i", test_name); + + std::cout << std::endl; +} + +template +void test_complex_spherical_harmonic(const T& data, const char* /* name */, boost::mpl::true_ const &) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + for(unsigned i = 0; i < sizeof(data) / sizeof(data[0]); ++i) + { + // + // Sanity check that the complex version does the same thing as the real + // and imaginary versions: + // + std::complex r = boost::math::spherical_harmonic( + boost::math::tools::real_cast(data[i][0]), + boost::math::tools::real_cast(data[i][1]), + data[i][2], + data[i][3]); + value_type re = boost::math::spherical_harmonic_r( + boost::math::tools::real_cast(data[i][0]), + boost::math::tools::real_cast(data[i][1]), + data[i][2], + data[i][3]); + value_type im = boost::math::spherical_harmonic_i( + boost::math::tools::real_cast(data[i][0]), + boost::math::tools::real_cast(data[i][1]), + data[i][2], + data[i][3]); + BOOST_CHECK_CLOSE_FRACTION(std::real(r), re, value_type(5)); + BOOST_CHECK_CLOSE_FRACTION(std::imag(r), im, value_type(5)); + } +} + +template +void test_complex_spherical_harmonic(const T& /* data */, const char* /* name */, boost::mpl::false_ const &) +{ + // T is not a built in type, can't use std::complex with it... +} + +template +void test_spherical_harmonic(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "spherical_harmonic.ipp" + + do_test_spherical_harmonic(spherical_harmonic, name, "Spherical Harmonics"); + + test_complex_spherical_harmonic(spherical_harmonic, name, boost::is_floating_point()); +} + +template +void test_spots(T, const char* t) +{ + std::cout << "Testing basic sanity checks for type " << t << std::endl; + // + // basic sanity checks, tolerance is 100 epsilon: + // + T tolerance = boost::math::tools::epsilon() * 100; + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(3, 2, static_cast(0.5), static_cast(0)), static_cast(0.2061460599687871330692286791802688341213L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 10, static_cast(0.75), static_cast(-0.25)), static_cast(0.06197787102219208244041677775577045124092L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 10, static_cast(0.75), static_cast(-0.25)), static_cast(0.04629885158895932341185988759669916977920L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, 15, static_cast(-0.75), static_cast(2.25)), static_cast(0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, 15, static_cast(-0.75), static_cast(2.25)), static_cast(-0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, 15, static_cast(-0.75), static_cast(-2.25)), static_cast(0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, 15, static_cast(-0.75), static_cast(-2.25)), static_cast(0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, 15, static_cast(0.75), static_cast(-2.25)), static_cast(-0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, 15, static_cast(0.75), static_cast(-2.25)), static_cast(-0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, 15, static_cast(0.75), static_cast(2.25)), static_cast(-0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, 15, static_cast(0.75), static_cast(2.25)), static_cast(0.2933918444656603582282372590387544902135L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(-0.75), static_cast(2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(-0.75), static_cast(2.25)), static_cast(0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(-0.75), static_cast(-2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(-0.75), static_cast(-2.25)), static_cast(-0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(0.75), static_cast(-2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(0.75), static_cast(-2.25)), static_cast(-0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(0.75), static_cast(2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(0.75), static_cast(2.25)), static_cast(0.0293201066685263879566422194539567289974L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(39, 15, static_cast(-0.75), static_cast(2.25)), static_cast(0.1757594233240278196989039119899901986211L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(39, 15, static_cast(-0.75), static_cast(2.25)), static_cast(-0.1837126108841860058078729532035715580790L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(39, 15, static_cast(-0.75), static_cast(-2.25)), static_cast(0.1757594233240278196989039119899901986211L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(39, 15, static_cast(-0.75), static_cast(-2.25)), static_cast(0.1837126108841860058078729532035715580790L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(39, 15, static_cast(0.75), static_cast(-2.25)), static_cast(-0.1757594233240278196989039119899901986211L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(39, 15, static_cast(0.75), static_cast(-2.25)), static_cast(-0.1837126108841860058078729532035715580790L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(39, 15, static_cast(0.75), static_cast(2.25)), static_cast(-0.1757594233240278196989039119899901986211L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(39, 15, static_cast(0.75), static_cast(2.25)), static_cast(0.1837126108841860058078729532035715580790L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(19, 14, static_cast(-0.75), static_cast(2.25)), static_cast(0.2341701030303444033808969389588343934828L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(19, 14, static_cast(-0.75), static_cast(2.25)), static_cast(0.0197340092863212879172432610952871202640L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(19, 14, static_cast(-0.75), static_cast(-2.25)), static_cast(0.2341701030303444033808969389588343934828L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(19, 14, static_cast(-0.75), static_cast(-2.25)), static_cast(-0.0197340092863212879172432610952871202640L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(19, 14, static_cast(0.75), static_cast(-2.25)), static_cast(0.2341701030303444033808969389588343934828L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(19, 14, static_cast(0.75), static_cast(-2.25)), static_cast(-0.0197340092863212879172432610952871202640L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(19, 14, static_cast(0.75), static_cast(2.25)), static_cast(0.2341701030303444033808969389588343934828L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(19, 14, static_cast(0.75), static_cast(2.25)), static_cast(0.0197340092863212879172432610952871202640L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, -15, static_cast(-0.75), static_cast(2.25)), static_cast(-0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, -15, static_cast(-0.75), static_cast(2.25)), static_cast(-0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, -15, static_cast(-0.75), static_cast(-2.25)), static_cast(-0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, -15, static_cast(-0.75), static_cast(-2.25)), static_cast(0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, -15, static_cast(0.75), static_cast(-2.25)), static_cast(0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, -15, static_cast(0.75), static_cast(-2.25)), static_cast(-0.2933918444656603582282372590387544902135L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(40, -15, static_cast(0.75), static_cast(2.25)), static_cast(0.2806904825045745687343492963236868973484L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(40, -15, static_cast(0.75), static_cast(2.25)), static_cast(0.2933918444656603582282372590387544902135L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, -14, static_cast(-0.75), static_cast(2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, -14, static_cast(-0.75), static_cast(2.25)), static_cast(-0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, -14, static_cast(-0.75), static_cast(-2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, -14, static_cast(-0.75), static_cast(-2.25)), static_cast(0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, -14, static_cast(0.75), static_cast(-2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, -14, static_cast(0.75), static_cast(-2.25)), static_cast(0.0293201066685263879566422194539567289974L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, -14, static_cast(0.75), static_cast(2.25)), static_cast(0.3479218186133435466692822481919867452442L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, -14, static_cast(0.75), static_cast(2.25)), static_cast(-0.0293201066685263879566422194539567289974L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(-4), static_cast(2.25)), static_cast(0.5253373768014719124617844890495875474590L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(-4), static_cast(2.25)), static_cast(0.0442712905622650144694916590407495495699L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(-4), static_cast(-2.25)), static_cast(0.5253373768014719124617844890495875474590L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(-4), static_cast(-2.25)), static_cast(-0.0442712905622650144694916590407495495699L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(4), static_cast(-2.25)), static_cast(0.5253373768014719124617844890495875474590L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(4), static_cast(-2.25)), static_cast(-0.0442712905622650144694916590407495495699L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 14, static_cast(4), static_cast(2.25)), static_cast(0.5253373768014719124617844890495875474590L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 14, static_cast(4), static_cast(2.25)), static_cast(0.0442712905622650144694916590407495495699L), tolerance); + + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 15, static_cast(-4), static_cast(2.25)), static_cast(-0.2991140325667575801827063718821420263438L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 15, static_cast(-4), static_cast(2.25)), static_cast(0.3126490678888350710506307405826667514065L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 15, static_cast(-4), static_cast(-2.25)), static_cast(-0.2991140325667575801827063718821420263438L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 15, static_cast(-4), static_cast(-2.25)), static_cast(-0.3126490678888350710506307405826667514065L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 15, static_cast(4), static_cast(-2.25)), static_cast(0.2991140325667575801827063718821420263438L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 15, static_cast(4), static_cast(-2.25)), static_cast(0.3126490678888350710506307405826667514065L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(20, 15, static_cast(4), static_cast(2.25)), static_cast(0.2991140325667575801827063718821420263438L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(20, 15, static_cast(4), static_cast(2.25)), static_cast(-0.3126490678888350710506307405826667514065L), tolerance); + + BOOST_CHECK_EQUAL(::boost::math::spherical_harmonic_r(10, 15, static_cast(-0.75), static_cast(2.25)), static_cast(0)); + BOOST_CHECK_EQUAL(::boost::math::spherical_harmonic_i(10, 15, static_cast(-0.75), static_cast(2.25)), static_cast(0)); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_r(53, 42, static_cast(-8.75), static_cast(-2.25)), static_cast(-0.0008147976618889536159592309471859037113647L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(::boost::math::spherical_harmonic_i(53, 42, static_cast(-8.75), static_cast(-2.25)), static_cast(0.0002099802242493057018193798824353982612756L), tolerance); +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + test_spots(0.0F, "float"); + test_spots(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L, "long double"); + test_spots(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif + + expected_results(); + + test_spherical_harmonic(0.1F, "float"); + test_spherical_harmonic(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spherical_harmonic(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS + test_spherical_harmonic(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + + diff --git a/test/test_students_t.cpp b/test/test_students_t.cpp new file mode 100644 index 000000000..153f360c7 --- /dev/null +++ b/test/test_students_t.cpp @@ -0,0 +1,416 @@ +// Copyright Paul A. Bristow 2006. +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_students_t.cpp + +// http://en.wikipedia.org/wiki/Student%27s_t_distribution +// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm + +// Basic sanity test for Student's t probability (quantile) (0. < p < 1). +// and Student's t probability Quantile (0. < p < 1). + +#include // Boost.Test +#include + +#include + using boost::math::students_t_distribution; +#include // for real_concept +#include // for real_concept + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + +template +RealType naive_pdf(RealType v, RealType t) +{ + // Calculate the pdf of the students t in a deliberately + // naive way, using equation (5) from + // http://mathworld.wolfram.com/Studentst-Distribution.html + // This is equivalent to, but a different method + // to the one in the actual implementation, so can be used as + // a very basic sanity check. However some published values + // would be nice.... + + using namespace std; // for ADL + using boost::math::beta; + + //return pow(v / (v + t*t), (1+v) / 2) / (sqrt(v) * beta(v/2, RealType(0.5f))); + RealType result = boost::math::tgamma_ratio((v+1)/2, v/2); + result /= sqrt(v * boost::math::constants::pi()); + result /= pow(1 + t*t/v, (v+1)/2); + return result; +} + +template +void test_spots(RealType) +{ + // Basic sanity checks + RealType tolerance = 1e-4; // 1e-6 (as %) + // Some tests only pass at 1e-5 because probability value is less accurate, + // a digit in 6th decimal place, although calculated using + // a t-distribution generator (claimed 6 decimal digits) at + // http://faculty.vassar.edu/lowry/VassarStats.html + // http://faculty.vassar.edu/lowry/tsamp.html + // df = 5, +/-t = 2.0, 1-tailed = 0.050970, 2-tailed = 0.101939 + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + // http://en.wikipedia.org/wiki/Student%27s_t_distribution#Table_of_selected_values + // Using tabulated value of t = 3.182 for 0.975, 3 df, one-sided. + + // http://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf refers to: + + // A lookup table of quantiles of the RealType distribution + // for 1 to 25 in steps of 0.1 is provided in CSV form at: + // www.mth.kcl.ac.uk/~shaww/web_page/papers/Tsupp/tquantiles.csv + // gives accurate t of -3.1824463052837 and 3 degrees of freedom. + // Values below are from this source, saved as tquantiles.xls. + // DF are across the columns, probabilities down the rows + // and the t- values (quantiles) are shown. + // These values are probably accurate to nearly 64-bit double + // (perhaps 14 decimal digits). + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(2), // degrees_of_freedom + static_cast(-6.96455673428326)), // t + static_cast(0.01), // probability. + tolerance); // % + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(5), // degrees_of_freedom + static_cast(-3.36492999890721)), // t + static_cast(0.01), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(1), // degrees_of_freedom + static_cast(-31830.988607907)), // t + static_cast(0.00001), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(25.), // degrees_of_freedom + static_cast(-5.2410429995425)), // t + static_cast(0.00001), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(1), // degrees_of_freedom + static_cast(-63661.97723)), // t + static_cast(0.000005), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(5.), // degrees_of_freedom + static_cast(-17.89686614)), // t + static_cast(0.000005), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(25.), // degrees_of_freedom + static_cast(-5.510848412)), // t + static_cast(0.000005), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(10.), // degrees_of_freedom + static_cast(-1.812461123)), // t + static_cast(0.05), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(10), // degrees_of_freedom + static_cast(1.812461123)), // t + static_cast(0.95), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement( + students_t_distribution(10), // degrees_of_freedom + static_cast(1.812461123))), // t + static_cast(0.05), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(10), // degrees_of_freedom + static_cast(9.751995491)), // t + static_cast(0.999999), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(10.), // degrees_of_freedom - for ALL degrees_of_freedom! + static_cast(0.)), // t + static_cast(0.5), // probability. + tolerance); + + + // Student's t Inverse function tests. + // Special cases + + BOOST_CHECK_THROW(boost::math::quantile( + students_t_distribution(1.), // degrees_of_freedom (ignored). + static_cast(0)), std::overflow_error); // t == -infinity. + + BOOST_CHECK_THROW(boost::math::quantile( + students_t_distribution(1.), // degrees_of_freedom (ignored). + static_cast(1)), std::overflow_error); // t == +infinity. + + BOOST_CHECK_EQUAL(boost::math::quantile( + students_t_distribution(1.), // degrees_of_freedom (ignored). + static_cast(0.5)), // probability == half - special case. + static_cast(0)); // t == zero. + + BOOST_CHECK_EQUAL(boost::math::quantile( + complement( + students_t_distribution(1.), // degrees_of_freedom (ignored). + static_cast(0.5))), // probability == half - special case. + static_cast(0)); // t == zero. + + BOOST_CHECK_CLOSE(boost::math::quantile( + students_t_distribution(1.), // degrees_of_freedom (ignored). + static_cast(0.5)), // probability == half - special case. + static_cast(0), // t == zero. + tolerance); + + BOOST_CHECK_CLOSE( // Tests of p middling. + ::boost::math::cdf( + students_t_distribution(5.), // degrees_of_freedom + static_cast(-0.559429644)), // t + static_cast(0.3), // probability. + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + students_t_distribution(5.), // degrees_of_freedom + static_cast(0.3)), // probability. + static_cast(-0.559429644), // t + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement( + students_t_distribution(5.), // degrees_of_freedom + static_cast(0.7))), // probability. + static_cast(-0.559429644), // t + tolerance); + + BOOST_CHECK_CLOSE( // Tests of p high. + ::boost::math::cdf( + students_t_distribution(5.), // degrees_of_freedom + static_cast(1.475884049)), // t + static_cast(0.9), // probability. + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + students_t_distribution(5.), // degrees_of_freedom + static_cast(0.9)), // probability. + static_cast(1.475884049), // t + tolerance); + + BOOST_CHECK_CLOSE( // Tests of p low. + ::boost::math::cdf( + students_t_distribution(5.), // degrees_of_freedom + static_cast(-1.475884049)), // t + static_cast(0.1), // probability. + tolerance); + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + students_t_distribution(5.), // degrees_of_freedom + static_cast(0.1)), // probability. + static_cast(-1.475884049), // t + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + students_t_distribution(2.), // degrees_of_freedom + static_cast(-6.96455673428326)), // t + static_cast(0.01), // probability. + tolerance); + + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + students_t_distribution(2.), // degrees_of_freedom + static_cast(0.01)), // probability. + static_cast(-6.96455673428326), // t + tolerance); + + + // Student's t pdf tests. + // for PDF checks, use 100 eps tolerance expressed as a percent: + tolerance = boost::math::tools::epsilon() * 10000; + + for(unsigned i = 1; i < 20; i += 3) + { + for(RealType r = -10; r < 10; r += 0.125) + { + //std::cout << "df=" << i << " t=" << r << std::endl; + BOOST_CHECK_CLOSE( + boost::math::pdf( + students_t_distribution(static_cast(i)), + r), + naive_pdf(static_cast(i), r), + tolerance); + } + } + + RealType tol2 = boost::math::tools::epsilon() * 5; + students_t_distribution dist(8); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(0), tol2); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , static_cast(8.0L / 6.0L), tol2); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , static_cast(sqrt(8.0L / 6.0L)), tol2); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tol2); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tol2); + // coefficient_of_variation: + BOOST_CHECK_THROW( + coefficient_of_variation(dist), + std::overflow_error); + // mode: + BOOST_CHECK_CLOSE( + mean(dist) + , static_cast(0), tol2); + // median: + BOOST_CHECK_CLOSE( + median(dist) + , static_cast(0), tol2); + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist) + , static_cast(0), tol2); + // kurtosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , static_cast(4.5), tol2); + // kurtosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist) + , static_cast(1.5), tol2); + + // Parameter estimation. These results are close to but + // not identical to those reported on the NIST website at + // http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm + // the NIST results appear to be calculated using a normal + // approximation, which slightly under-estimates the degrees of + // freedom required, particularly when the result is small. + // + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(0.5), + static_cast(0.005), + static_cast(0.01), + static_cast(1.0))), + 99); + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(1.5), + static_cast(0.005), + static_cast(0.01), + static_cast(1.0))), + 14); + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(0.5), + static_cast(0.025), + static_cast(0.01), + static_cast(1.0))), + 76); + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(1.5), + static_cast(0.025), + static_cast(0.01), + static_cast(1.0))), + 11); + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(0.5), + static_cast(0.05), + static_cast(0.01), + static_cast(1.0))), + 65); + BOOST_CHECK_EQUAL( + ceil(students_t_distribution::find_degrees_of_freedom( + static_cast(1.5), + static_cast(0.05), + static_cast(0.01), + static_cast(1.0))), + 9); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can construct students_t distribution using the two convenience methods: + using namespace boost::math; + students_t myst1(2); // Using typedef + students_t_distribution<> myst2(2); // Using default RealType double. + //students_t_distribution myst3(2); // Using explicit RealType double. + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_students_t.exe" +Running 1 test case... +Tolerance for type float is 0.0001 % +Tolerance for type double is 0.0001 % +Tolerance for type long double is 0.0001 % +Tolerance for type class boost::math::concepts::real_concept is 0.0001 % +*** No errors detected + +*/ + + diff --git a/test/test_tgamma_ratio.cpp b/test/test_tgamma_ratio.cpp new file mode 100644 index 000000000..4652cdd18 --- /dev/null +++ b/test/test_tgamma_ratio.cpp @@ -0,0 +1,249 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the gamma ratio functions tgamma_ratio, +// and tgamma_delta_ratio. The accuracy tests +// use values generated with NTL::RR at 1000-bit precision +// and our generic versions of these functions. +// +// Note that when this file is first run on a new platform many of +// these tests will fail: the default accuracy is 1 epsilon which +// is too tight for most platforms. In this situation you will +// need to cast a human eye over the error rates reported and make +// a judgement as to whether they are acceptable. Either way please +// report the results to the Boost mailing list. Acceptable rates of +// error are marked up below as a series of regular expressions that +// identify the compiler/stdlib/platform/data-type/test-data/test-function +// along with the maximum expected peek and RMS mean errors for that +// test. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double"; + } + else + { + largest_type = "long double"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // HP-UX + // This is a weird one, HP-UX and Mac OS X show up errors at float + // precision, that don't show up on other platforms. + // There appears to be some kind of rounding issue going on (not enough + // precision in the input to get the answer right): + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "HP-UX|Mac OS|linux|.*(bsd|BSD).*", // platform + "float", // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_ratio[^|]*", 35, 8); // test function + // + // Linux AMD x86em64 has slightly higher rates: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux.*", // platform + largest_type, // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_ratio[^|]*", 300, 100); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "linux.*", // platform + "real_concept", // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_ratio[^|]*", 300, 100); // test function + // + // Catch all cases come last: + // + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_delta_ratio[^|]*", 30, 20); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + largest_type, // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_ratio[^|]*", 100, 50); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_delta_ratio[^|]*", 40, 15); // test function + add_expected_result( + "[^|]*", // compiler + "[^|]*", // stdlib + "[^|]*", // platform + "real_concept", // test type(s) + "[^|]*", // test data group + "boost::math::tgamma_ratio[^|]*", 150, 50); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +struct negative_tgamma_ratio +{ + template + typename Row::value_type operator()(const Row& row) + { + return boost::math::tgamma_delta_ratio(row[0], -row[1]); + } +}; + +template +void do_test_tgamma_delta_ratio(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::tgamma_delta_ratio; +#else + pg funcp = boost::math::tgamma_delta_ratio; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma_delta_ratio against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma_delta_ratio(a, delta)", test_name); + result = boost::math::tools::test( + data, + negative_tgamma_ratio(), + extract_result(3)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma_delta_ratio(a -delta)", test_name); +} + +template +void do_test_tgamma_ratio(const T& data, const char* type_name, const char* test_name) +{ + typedef typename T::value_type row_type; + typedef typename row_type::value_type value_type; + + typedef value_type (*pg)(value_type, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::tgamma_ratio; +#else + pg funcp = boost::math::tgamma_ratio; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test tgamma_ratio against data: + // + result = boost::math::tools::test( + data, + bind_func(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::tgamma_ratio(a, b)", test_name); +} + +template +void test_tgamma_ratio(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // +# include "tgamma_delta_ratio_data.ipp" + + do_test_tgamma_delta_ratio(tgamma_delta_ratio_data, name, "tgamma + small delta ratios"); + +# include "tgamma_delta_ratio_int.ipp" + + do_test_tgamma_delta_ratio(tgamma_delta_ratio_int, name, "tgamma + small integer ratios"); + +# include "tgamma_delta_ratio_int2.ipp" + + do_test_tgamma_delta_ratio(tgamma_delta_ratio_int2, name, "integer tgamma ratios"); + +# include "tgamma_ratio_data.ipp" + + do_test_tgamma_ratio(tgamma_ratio_data, name, "tgamma ratios"); + +} + +int test_main(int, char* []) +{ + BOOST_MATH_CONTROL_FP; + expected_results(); + +#ifndef BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS + test_tgamma_ratio(0.1F, "float"); +#endif + test_tgamma_ratio(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_tgamma_ratio(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_tgamma_ratio(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + return 0; +} + + diff --git a/test/test_toms748_solve.cpp b/test/test_toms748_solve.cpp new file mode 100644 index 000000000..d926a6085 --- /dev/null +++ b/test/test_toms748_solve.cpp @@ -0,0 +1,270 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include + +#include +#include +#include + +// +// Test functor implements the same test cases as used by +// "Algorithm 748: Enclosing Zeros of Continuous Functions" +// by G.E. Alefeld, F.A. Potra and Yixun Shi. +// +// Plus two more: one for inverting the incomplete gamma, +// and one for inverting the incomplete beta. +// +template +struct toms748tester +{ + toms748tester(unsigned i) : k(i), ip(0), a(0), b(0){} + toms748tester(unsigned i, int ip_) : k(i), ip(ip_), a(0), b(0){} + toms748tester(unsigned i, T a_, T b_) : k(i), ip(0), a(a_), b(b_){} + + static unsigned total_calls() + { + return invocations; + } + static void reset() + { + invocations = 0; + } + + T operator()(T x) + { + using namespace std; + + ++invocations; + switch(k) + { + case 1: + return sin(x) - x / 2; + case 2: + { + T r = 0; + for(int i = 1; i <= 20; ++i) + { + T p = (2 * i - 5); + T q = x - i * i; + r += p * p / (q * q * q); + } + r *= -2; + return r; + } + case 3: + return a * x * exp(b * x); + case 4: + return pow(x, b) - a; + case 5: + return sin(x) - 0.5; + case 6: + return 2 * x * exp(-T(ip)) - 2 * exp(-ip * x) + 1; + case 7: + return (1 + (1 - ip) * (1 - ip)) * x - (1 - ip * x) * (1 - ip * x); + case 8: + return x * x - pow(1 - x, a); + case 9: + return (1 + (1 - ip) * (1 - ip) * (1 - ip) * (1 - ip)) * x - (1 - ip * x) * (1 - ip * x) * (1 - ip * x) * (1 - ip * x); + case 10: + return exp(-ip * x) * (x - 1) + pow(x, T(ip)); + case 11: + return (ip * x - 1) / ((ip - 1) * x); + case 12: + return pow(x, T(1)/ip) - pow(T(ip), T(1)/ip); + case 13: + return x == 0 ? 0 : x * exp(-1 / (x * x)); + case 14: + return x >= 0 ? (T(ip)/20) * (x / 1.5f + sin(x) - 1) : -T(ip)/20; + case 15: + { + T d = 2e-3f / (1+ip); + if(x > d) + return exp(1.0) - 1.859; + else if(x > 0) + return exp((ip+1)*x*1000 / 2) - 1.859; + return -0.859f; + } + case 16: + { + return boost::math::gamma_q(x, a) - b; + } + case 17: + return boost::math::ibeta(x, a, 0.5) - b; + } + return 0; + } +private: + int k; // index of problem. + int ip; // integer parameter + T a, b; // real parameter + + static unsigned invocations; +}; + +template +unsigned toms748tester::invocations = 0; + +boost::uintmax_t total = 0; +boost::uintmax_t invocations = 0; + +template +void run_test(T a, T b, int id) +{ + boost::uintmax_t c = 1000; + std::pair r = toms748_solve(toms748tester(id), + a, + b, + boost::math::tools::eps_tolerance(std::numeric_limits::digits), + c); + BOOST_CHECK_EQUAL(c, toms748tester::total_calls()); + total += c; + invocations += toms748tester::total_calls(); + std::cout << "Function " << id << "\nresult={" << r.first << ", " << r.second << "} total calls=" << toms748tester::total_calls() << "\n\n"; + toms748tester::reset(); +} + +template +void run_test(T a, T b, int id, int p) +{ + boost::uintmax_t c = 1000; + std::pair r = toms748_solve(toms748tester(id, p), + a, + b, + boost::math::tools::eps_tolerance(std::numeric_limits::digits), + c); + BOOST_CHECK_EQUAL(c, toms748tester::total_calls()); + total += c; + invocations += toms748tester::total_calls(); + std::cout << "Function " << id << "\nresult={" << r.first << ", " << r.second << "} total calls=" << toms748tester::total_calls() << "\n\n"; + toms748tester::reset(); +} + +template +void run_test(T a, T b, int id, T p1, T p2) +{ + boost::uintmax_t c = 1000; + std::pair r = toms748_solve(toms748tester(id, p1, p2), + a, + b, + boost::math::tools::eps_tolerance(std::numeric_limits::digits), + c); + BOOST_CHECK_EQUAL(c, toms748tester::total_calls()); + total += c; + invocations += toms748tester::total_calls(); + std::cout << "Function " << id << "\n Result={" << r.first << ", " << r.second << "} total calls=" << toms748tester::total_calls() << "\n\n"; + toms748tester::reset(); +} + +int test_main(int, char* []) +{ + std::cout << std::setprecision(18); + run_test(3.14/2, 3.14, 1); + + for(int i = 1; i <= 10; i += 1) + { + run_test(i*i + 1e-9, (i+1)*(i+1) - 1e-9, 2); + } + + run_test(-9.0, 31.0, 3, -40.0, -1.0); + run_test(-9.0, 31.0, 3, -100.0, -2.0); + run_test(-9.0, 31.0, 3, -200.0, -3.0); + + for(int n = 4; n <= 12; n += 2) + { + run_test(0.0, 5.0, 4, 0.2, double(n)); + } + for(int n = 4; n <= 12; n += 2) + { + run_test(0.0, 5.0, 4, 1.0, double(n)); + } + for(int n = 8; n <= 14; n += 2) + { + run_test(-0.95, 4.05, 4, 1.0, double(n)); + } + run_test(0.0, 1.5, 5); + for(int n = 1; n <= 5; ++n) + { + run_test(0.0, 1.0, 6, n); + } + for(int n = 20; n <= 100; n += 20) + { + run_test(0.0, 1.0, 6, n); + } + run_test(0.0, 1.0, 7, 5); + run_test(0.0, 1.0, 7, 10); + run_test(0.0, 1.0, 7, 20); + run_test(0.0, 1.0, 8, 2); + run_test(0.0, 1.0, 8, 5); + run_test(0.0, 1.0, 8, 10); + run_test(0.0, 1.0, 8, 15); + run_test(0.0, 1.0, 8, 20); + run_test(0.0, 1.0, 9, 1); + run_test(0.0, 1.0, 9, 2); + run_test(0.0, 1.0, 9, 4); + run_test(0.0, 1.0, 9, 5); + run_test(0.0, 1.0, 9, 8); + run_test(0.0, 1.0, 9, 15); + run_test(0.0, 1.0, 9, 20); + run_test(0.0, 1.0, 10, 1); + run_test(0.0, 1.0, 10, 5); + run_test(0.0, 1.0, 10, 10); + run_test(0.0, 1.0, 10, 15); + run_test(0.0, 1.0, 10, 20); + + run_test(0.01, 1.0, 11, 2); + run_test(0.01, 1.0, 11, 5); + run_test(0.01, 1.0, 11, 15); + run_test(0.01, 1.0, 11, 20); + + for(int n = 2; n <= 6; ++n) + run_test(1.0, 100.0, 12, n); + for(int n = 7; n <= 33; n+=2) + run_test(1.0, 100.0, 12, n); + + run_test(-1.0, 4.0, 13); + + for(int n = 1; n <= 40; ++n) + run_test(-1e4, 3.14/2, 14, n); + + for(int n = 20; n <= 40; ++n) + run_test(-1e4, 1e-4, 15, n); + + for(int n = 100; n <= 1000; n+=100) + run_test(-1e4, 1e-4, 15, n); + + std::cout << "Total iterations consumed = " << total << std::endl; + std::cout << "Total function invocations consumed = " << invocations << std::endl << std::endl; + + BOOST_CHECK(invocations < 3150); + + std::cout << std::setprecision(18); + + for(int n = 5; n <= 100; n += 10) + run_test(sqrt(double(n)), double(n+1), 16, (double)n, 0.4); + + for(int n = 5; n <= 100; n += 10) + run_test(double(n / 2), double(2*n), 17, (double)n, 0.4); + + + for(int n = 4; n <= 12; n += 2) + { + boost::uintmax_t c = 1000; + std::pair r = bracket_and_solve_root(toms748tester(4, 0.2, double(n)), + 2.0, + 2.0, + true, + boost::math::tools::eps_tolerance(std::numeric_limits::digits), + c); + std::cout << std::setprecision(18); + std::cout << "Function " << 4 << "\n Result={" << r.first << ", " << r.second << "} total calls=" << toms748tester::total_calls() << "\n\n"; + toms748tester::reset(); + BOOST_CHECK(c < 20); + } + + return 0; +} + diff --git a/test/test_triangular.cpp b/test/test_triangular.cpp new file mode 100644 index 000000000..789e2f8a3 --- /dev/null +++ b/test/test_triangular.cpp @@ -0,0 +1,705 @@ +// Copyright Paul Bristow 2006, 2007. +// Copyright John Maddock 2006, 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_triangular.cpp + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +# pragma warning(disable: 4305) // truncation from 'long double' to 'float' +#endif + +#include // for real_concept +#include // Boost.Test +#include + +#include +using boost::math::triangular_distribution; +#include +#include + +#include +using std::cout; +using std::endl; +using std::scientific; +using std::fixed; +using std::left; +using std::right; +using std::setw; +using std::setprecision; +using std::showpos; +#include +using std::numeric_limits; + +template +void check_triangular(RealType lower, RealType mode, RealType upper, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + triangular_distribution(lower, mode, upper), // distribution. + x), // random variable. + p, // probability. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + complement( + triangular_distribution(lower, mode, upper), // distribution. + x)), // random variable. + q, // probability complement. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + triangular_distribution(lower,mode, upper), // distribution. + p), // probability. + x, // random variable. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + complement( + triangular_distribution(lower, mode, upper), // distribution. + q)), // probability complement. + x, // random variable. + tol); // tolerance. +} // void check_triangular + +template +void test_spots(RealType) +{ + // Basic sanity checks: + // + // Some test values were generated for the triangular distribution + // using the online calculator at + // http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm + // + // Tolerance is just over 5 epsilon expressed as a fraction: + RealType tolerance = boost::math::tools::epsilon() * 5; // 5 eps as a fraction. + RealType tol5eps = boost::math::tools::epsilon() * 5; // 5 eps as a fraction. + + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; + + using namespace std; // for ADL of std::exp; + + // Tests on construction + // Default should be 0, 0, 1 + BOOST_CHECK_EQUAL(triangular_distribution().lower(), -1); + BOOST_CHECK_EQUAL(triangular_distribution().mode(), 0); + BOOST_CHECK_EQUAL(triangular_distribution().upper(), 1); + BOOST_CHECK_EQUAL(support(triangular_distribution()).first, triangular_distribution().lower()); + BOOST_CHECK_EQUAL(support(triangular_distribution()).second, triangular_distribution().upper()); + + if (std::numeric_limits::has_quiet_NaN == true) + { + BOOST_CHECK_THROW( // duff parameter lower. + triangular_distribution(static_cast(std::numeric_limits::quiet_NaN()), 0, 0), + std::domain_error); + + BOOST_CHECK_THROW( // duff parameter mode. + triangular_distribution(0, static_cast(std::numeric_limits::quiet_NaN()), 0), + std::domain_error); + + BOOST_CHECK_THROW( // duff parameter upper. + triangular_distribution(0, 0, static_cast(std::numeric_limits::quiet_NaN())), + std::domain_error); + } // quiet_NaN tests. + + BOOST_CHECK_THROW( // duff parameters upper < lower. + triangular_distribution(1, 0, -1), + std::domain_error); + + BOOST_CHECK_THROW( // duff parameters upper == lower. + triangular_distribution(0, 0, 0), + std::domain_error); + BOOST_CHECK_THROW( // duff parameters mode < lower. + triangular_distribution(0, -1, 1), + std::domain_error); + + BOOST_CHECK_THROW( // duff parameters mode > upper. + triangular_distribution(0, 2, 1), + std::domain_error); + + // Tests for PDF + // // triangular_distribution() default is (0, 0, 1), mode == lower. + BOOST_CHECK_CLOSE_FRACTION( // x == lower == mode + pdf(triangular_distribution(0, 0, 1), static_cast(0)), + static_cast(2), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + pdf(triangular_distribution(0, 0, 1), static_cast(1)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x > upper + pdf(triangular_distribution(0, 0, 1), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x < lower + pdf(triangular_distribution(0, 0, 1), static_cast(2)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x < lower + pdf(triangular_distribution(0, 0, 1), static_cast(2)), + static_cast(0), + tolerance); + + // triangular_distribution() (0, 1, 1) mode == upper + BOOST_CHECK_CLOSE_FRACTION( // x == lower + pdf(triangular_distribution(0, 1, 1), static_cast(0)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + pdf(triangular_distribution(0, 1, 1), static_cast(1)), + static_cast(2), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x > upper + pdf(triangular_distribution(0, 1, 1), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x < lower + pdf(triangular_distribution(0, 1, 1), static_cast(2)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x < middle so Wiki says special case pdf = 2 * x + pdf(triangular_distribution(0, 1, 1), static_cast(0.25)), + static_cast(0.5), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x < middle so Wiki says special case cdf = x * x + cdf(triangular_distribution(0, 1, 1), static_cast(0.25)), + static_cast(0.25 * 0.25), + tolerance); + + // triangular_distribution() (0, 0.5, 1) mode == middle. + BOOST_CHECK_CLOSE_FRACTION( // x == lower + pdf(triangular_distribution(0, 0.5, 1), static_cast(0)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + pdf(triangular_distribution(0, 0.5, 1), static_cast(1)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x > upper + pdf(triangular_distribution(0, 0.5, 1), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x < lower + pdf(triangular_distribution(0, 0.5, 1), static_cast(2)), + static_cast(0), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == mode + pdf(triangular_distribution(0, 0.5, 1), static_cast(0.5)), + static_cast(2), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == half mode + pdf(triangular_distribution(0, 0.5, 1), static_cast(0.25)), + static_cast(1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x == half mode + pdf(triangular_distribution(0, 0.5, 1), static_cast(0.75)), + static_cast(1), + tolerance); + + if(std::numeric_limits::has_infinity) + { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() + // Note that infinity is not implemented for real_concept, so these tests + // are only done for types, like built-in float, double.. that have infinity. + // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. + // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. + // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path + // of error handling is tested below with BOOST_CHECK_THROW tests. + + BOOST_CHECK_THROW( // x == infinity NOT OK. + pdf(triangular_distribution(0, 0, 1), static_cast(std::numeric_limits::infinity())), + std::domain_error); + + BOOST_CHECK_THROW( // x == minus infinity not OK too. + pdf(triangular_distribution(0, 0, 1), static_cast(-std::numeric_limits::infinity())), + std::domain_error); + } + if(std::numeric_limits::has_quiet_NaN) + { // BOOST_CHECK tests for NaN using std::numeric_limits<>::has_quiet_NaN() - should throw. + BOOST_CHECK_THROW( + pdf(triangular_distribution(0, 0, 1), static_cast(std::numeric_limits::quiet_NaN())), + std::domain_error); + BOOST_CHECK_THROW( + pdf(triangular_distribution(0, 0, 1), static_cast(-std::numeric_limits::quiet_NaN())), + std::domain_error); + } // test for x = NaN using std::numeric_limits<>::quiet_NaN() + + // cdf + BOOST_CHECK_EQUAL( // x < lower + cdf(triangular_distribution(0, 1, 1), static_cast(-1)), + static_cast(0) ); + BOOST_CHECK_CLOSE_FRACTION( // x == lower + cdf(triangular_distribution(0, 1, 1), static_cast(0)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x == upper + cdf(triangular_distribution(0, 1, 1), static_cast(1)), + static_cast(1), + tolerance); + BOOST_CHECK_EQUAL( // x > upper + cdf(triangular_distribution(0, 1, 1), static_cast(2)), + static_cast(1)); + + BOOST_CHECK_CLOSE_FRACTION( // x == mode + cdf(triangular_distribution(-1, 0, 1), static_cast(0)), + //static_cast((mode - lower) / (upper - lower)), + static_cast(0.5), // (0 --1) / (1 -- 1) = 0.5 + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(0, 1, 1), static_cast(0.9L)), + static_cast(0.81L), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(-1, 0, 1), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(-1, 0, 1), static_cast(-0.5L)), + static_cast(0.125L), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(-1, 0, 1), static_cast(0)), + static_cast(0.5), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(-1, 0, 1), static_cast(+0.5L)), + static_cast(0.875L), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(triangular_distribution(-1, 0, 1), static_cast(1)), + static_cast(1), + tolerance); + + // cdf complement + BOOST_CHECK_EQUAL( // x < lower + cdf(complement(triangular_distribution(0, 0, 1), static_cast(-1))), + static_cast(1)); + BOOST_CHECK_EQUAL( // x == lower + cdf(complement(triangular_distribution(0, 0, 1), static_cast(0))), + static_cast(1)); + + BOOST_CHECK_EQUAL( // x == mode + cdf(complement(triangular_distribution(-1, 0, 1), static_cast(0))), + static_cast(0.5)); + + BOOST_CHECK_EQUAL( // x == mode + cdf(complement(triangular_distribution(0, 0, 1), static_cast(0))), + static_cast(1)); + BOOST_CHECK_EQUAL( // x == mode + cdf(complement(triangular_distribution(0, 1, 1), static_cast(1))), + static_cast(0)); + + BOOST_CHECK_EQUAL( // x > upper + cdf(complement(triangular_distribution(0, 0, 1), static_cast(2))), + static_cast(0)); + BOOST_CHECK_EQUAL( // x == upper + cdf(complement(triangular_distribution(0, 0, 1), static_cast(1))), + static_cast(0)); + + BOOST_CHECK_CLOSE_FRACTION( // x = -0.5 + cdf(complement(triangular_distribution(-1, 0, 1), static_cast(-0.5))), + static_cast(0.875L), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x = +0.5 + cdf(complement(triangular_distribution(-1, 0, 1), static_cast(0.5))), + static_cast(0.125), + tolerance); + + triangular_distribution triang; // Using typedef == triangular_distribution tristd; + triangular_distribution tristd(0, 0.5, 1); // 'Standard' triangular distribution. + + BOOST_CHECK_CLOSE_FRACTION( // median of Standard triangular is sqrt(mode/2) if c > 1/2 else 1 - sqrt((1-c)/2) + median(tristd), + static_cast(0.5), + tolerance); + triangular_distribution tri011(0, 1, 1); // Using default RealType double. + triangular_distribution tri0q1(0, 0.25, 1); // mode is near bottom. + triangular_distribution tri0h1(0, 0.5, 1); // Equilateral triangle - mode is the middle. + triangular_distribution trim12(-1, -0.5, 2); // mode is negative. + + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.02L), static_cast(0.0016L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.5L), static_cast(0.66666666666666666666666666666666666666666666667L), tolerance); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.98L), static_cast(0.9994666666666666666666666666666666666666666666L), tolerance); + + // quantile + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast(0.0016L)), static_cast(0.02L), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast(0.66666666666666666666666666666666666666666666667L)), static_cast(0.5), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, static_cast(0.3333333333333333333333333333333333333333333333333L))), static_cast(0.5), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, static_cast(0.999466666666666666666666666666666666666666666666666L)), static_cast(98) / 100, 10 * tol5eps); + + BOOST_CHECK_CLOSE_FRACTION(pdf(trim12, 0), static_cast(0.533333333333333333333333333333333333333333333L), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(trim12, 0), static_cast(0.466666666666666666666666666666666666666666667L), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(complement(trim12, 0)), static_cast(1 - 0.466666666666666666666666666666666666666666667L), tol5eps); + + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, static_cast(1 - 0.999466666666666666666666666666666666666666666666L))), static_cast(0.98L), 10 * tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast(1))), static_cast(0), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast(0.5))), static_cast(0.5), tol5eps); // OK + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast(1 - 0.02L))), static_cast(0.1L), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, static_cast(1 - 0.98L))), static_cast(0.9L), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0)), static_cast(1), tol5eps); + + RealType xs [] = {0, 0.01L, 0.02L, 0.05L, 0.1L, 0.2L, 0.3L, 0.4L, 0.5L, 0.6L, 0.7L, 0.8L, 0.9L, 0.95L, 0.98L, 0.99L, 1}; + + const triangular_distribution& distr = triang; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(distr, 1.)), static_cast(-1), tol5eps); + const triangular_distribution* distp = &triang; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*distp, 1.)), static_cast(-1), tol5eps); + + const triangular_distribution* dists [] = {&tristd, &tri011, &tri0q1, &tri0h1, &trim12}; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*dists[1], 1.)), static_cast(0), tol5eps); + + for (int i = 0; i < 5; i++) + { + const triangular_distribution* const dist = dists[i]; + // cout << "Distribution " << i << endl; + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.5L), quantile(complement(*dist, 0.5L)), tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98L), quantile(complement(*dist, 1.L - 0.98L)),tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98L), quantile(complement(*dist, 1.L - 0.98L)),tol5eps); + } // for i + + // quantile complement + for (int i = 0; i < 5; i++) + { + const triangular_distribution* const dist = dists[i]; + //cout << "Distribution " << i << endl; + BOOST_CHECK_EQUAL(quantile(complement(*dists[i], 1.)), quantile(*dists[i], 0.)); + for (unsigned j = 0; j < sizeof(xs) /sizeof(RealType); j++) + { + RealType x = xs[j]; + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], x), quantile(complement(*dist, 1 - x)), tol5eps); + } // for j + } // for i + + + check_triangular( + static_cast(0), // lower + static_cast(0.5), // mode + static_cast(1), // upper + static_cast(0.5), // x + static_cast(0.5), // p + static_cast(1 - 0.5), // q + tolerance); + + // Some Not-standard triangular tests. + check_triangular( + static_cast(-1), // lower + static_cast(0), // mode + static_cast(1), // upper + static_cast(0), // x + static_cast(0.5), // p + static_cast(1 - 0.5), // q = 1 - p + tolerance); + + check_triangular( + static_cast(1), // lower + static_cast(1), // mode + static_cast(3), // upper + static_cast(2), // x + static_cast(0.75), // p + static_cast(1 - 0.75), // q = 1 - p + tolerance); + + check_triangular( + static_cast(-1), // lower + static_cast(1), // mode + static_cast(2), // upper + static_cast(1), // x + static_cast(0.66666666666666666666666666666666666666666667L), // p + static_cast(0.33333333333333333333333333333333333333333333L), // q = 1 - p + tolerance); + tolerance = (std::max)( + boost::math::tools::epsilon(), + static_cast(boost::math::tools::epsilon())) * 10; // 10 eps as a fraction. + cout << "Tolerance (as fraction) for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; + triangular_distribution tridef; // (-1, 0, 1) // default + RealType x = static_cast(0.5); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE_FRACTION( + mean(tridef), static_cast(0), tolerance); + // variance: + BOOST_CHECK_CLOSE_FRACTION( + variance(tridef), static_cast(0.16666666666666666666666666666666666666666667L), tolerance); + // was 0.0833333333333333333333333333333333333333333L + + // std deviation: + BOOST_CHECK_CLOSE_FRACTION( + standard_deviation(tridef), sqrt(variance(tridef)), tolerance); + // hazard: + BOOST_CHECK_CLOSE_FRACTION( + hazard(tridef, x), pdf(tridef, x) / cdf(complement(tridef, x)), tolerance); + // cumulative hazard: + BOOST_CHECK_CLOSE_FRACTION( + chf(tridef, x), -log(cdf(complement(tridef, x))), tolerance); + // coefficient_of_variation: + if (mean(tridef) != 0) + { + BOOST_CHECK_CLOSE_FRACTION( + coefficient_of_variation(tridef), standard_deviation(tridef) / mean(tridef), tolerance); + } + // mode: + BOOST_CHECK_CLOSE_FRACTION( + mode(tridef), static_cast(0), tolerance); + // skewness: + BOOST_CHECK_CLOSE_FRACTION( + median(trim12), static_cast(-0.13397459621556151), tolerance); + BOOST_CHECK_EQUAL( + skewness(tridef), static_cast(0)); + // kurtosis: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(tridef), kurtosis(tridef) - static_cast(3L), tolerance); + // kurtosis excess = kurtosis - 3; + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(tridef), static_cast(-0.6), tolerance); // for all distributions. + + if(std::numeric_limits::has_infinity) + { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() + // Note that infinity is not implemented for real_concept, so these tests + // are only done for types, like built-in float, double.. that have infinity. + // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. + // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. + // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path + // of error handling is tested below with BOOST_CHECK_THROW tests. + + using boost::math::policies::policy; + using boost::math::policies::domain_error; + using boost::math::policies::ignore_error; + + // Define a (bad?) policy to ignore domain errors ('bad' arguments): + typedef policy > inf_policy; // domain error returns infinity. + triangular_distribution tridef_inf(-1, 0., 1); + // But can't use BOOST_CHECK_EQUAL(?, quiet_NaN) + using boost::math::isnan; + BOOST_CHECK((isnan)(pdf(tridef_inf, std::numeric_limits::infinity()))); + } // test for infinity using std::numeric_limits<>::infinity() + else + { // real_concept case, does has_infinfity == false, so can't check it throws. + // cout << std::numeric_limits::infinity() << ' ' + // << boost::math::fpclassify(std::numeric_limits::infinity()) << endl; + // value of std::numeric_limits::infinity() is zero, so FPclassify is zero, + // so (boost::math::isfinite)(std::numeric_limits::infinity()) does not detect infinity. + // so these tests would never throw. + //BOOST_CHECK_THROW(pdf(tridef, std::numeric_limits::infinity()), std::domain_error); + //BOOST_CHECK_THROW(pdf(tridef, std::numeric_limits::quiet_NaN()), std::domain_error); + // BOOST_CHECK_THROW(pdf(tridef, boost::math::tools::max_value() * 2), std::domain_error); // Doesn't throw. + BOOST_CHECK_EQUAL(pdf(tridef, boost::math::tools::max_value()), 0); + } + // Special cases: + BOOST_CHECK(pdf(tridef, -1) == 0); + BOOST_CHECK(pdf(tridef, 1) == 0); + BOOST_CHECK(cdf(tridef, 0) == 0.5); + BOOST_CHECK(pdf(tridef, 1) == 0); + BOOST_CHECK(cdf(tridef, 1) == 1); + BOOST_CHECK(cdf(complement(tridef, -1)) == 1); + BOOST_CHECK(cdf(complement(tridef, 1)) == 0); + BOOST_CHECK(quantile(tridef, 1) == 1); + BOOST_CHECK(quantile(complement(tridef, 1)) == -1); + + BOOST_CHECK_EQUAL(support(trim12).first, trim12.lower()); + BOOST_CHECK_EQUAL(support(trim12).second, trim12.upper()); + + // Error checks: + if(std::numeric_limits::has_quiet_NaN) + { // BOOST_CHECK tests for quiet_NaN (not for real_concept, for example - see notes above). + BOOST_CHECK_THROW(triangular_distribution(0, std::numeric_limits::quiet_NaN()), std::domain_error); + BOOST_CHECK_THROW(triangular_distribution(0, -std::numeric_limits::quiet_NaN()), std::domain_error); + } + BOOST_CHECK_THROW(triangular_distribution(1, 0), std::domain_error); // lower > upper! + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // double toleps = std::numeric_limits::epsilon(); // 5 eps as a fraction. + double tol5eps = std::numeric_limits::epsilon() * 5; // 5 eps as a fraction. + // double tol50eps = std::numeric_limits::epsilon() * 50; // 50 eps as a fraction. + double tol500eps = std::numeric_limits::epsilon() * 500; // 500 eps as a fraction. + + // Check that can construct triangular distribution using the two convenience methods: + using namespace boost::math; + triangular triang; // Using typedef + // == triangular_distribution triang; + + BOOST_CHECK_EQUAL(triang.lower(), -1); // Check default. + BOOST_CHECK_EQUAL(triang.mode(), 0); + BOOST_CHECK_EQUAL(triang.upper(), 1); + + triangular tristd (0, 0.5, 1); // Using typedef + + BOOST_CHECK_EQUAL(tristd.lower(), 0); + BOOST_CHECK_EQUAL(tristd.mode(), 0.5); + BOOST_CHECK_EQUAL(tristd.upper(), 1); + + //cout << "X range from " << range(tristd).first << " to " << range(tristd).second << endl; + //cout << "Supported from "<< support(tristd).first << ' ' << support(tristd).second << endl; + + BOOST_CHECK_EQUAL(support(tristd).first, tristd.lower()); + BOOST_CHECK_EQUAL(support(tristd).second, tristd.upper()); + + triangular_distribution<> tri011(0, 1, 1); // Using default RealType double. + // mode is upper + BOOST_CHECK_EQUAL(tri011.lower(), 0); // Check defaults again. + BOOST_CHECK_EQUAL(tri011.mode(), 1); // Check defaults again. + BOOST_CHECK_EQUAL(tri011.upper(), 1); + BOOST_CHECK_EQUAL(mode(tri011), 1); + + BOOST_CHECK_EQUAL(pdf(tri011, 0), 0); + BOOST_CHECK_EQUAL(pdf(tri011, 0.1), 0.2); + BOOST_CHECK_EQUAL(pdf(tri011, 0.5), 1); + BOOST_CHECK_EQUAL(pdf(tri011, 0.9), 1.8); + BOOST_CHECK_EQUAL(pdf(tri011, 1), 2); + + BOOST_CHECK_EQUAL(cdf(tri011, 0), 0); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri011, 0.1), 0.01, tol5eps); + BOOST_CHECK_EQUAL(cdf(tri011, 0.5), 0.25); + BOOST_CHECK_EQUAL(cdf(tri011, 0.9), 0.81); + BOOST_CHECK_EQUAL(cdf(tri011, 1), 1); + BOOST_CHECK_EQUAL(cdf(tri011, 9), 1); + BOOST_CHECK_EQUAL(mean(tri011), 0.666666666666666666666666666666666666666666666666667); + BOOST_CHECK_EQUAL(variance(tri011), 1./18.); + + triangular tri0h1(0, 0.5, 1); // Equilateral triangle - mode is the middle. + BOOST_CHECK_EQUAL(tri0h1.lower(), 0); + BOOST_CHECK_EQUAL(tri0h1.mode(), 0.5); + BOOST_CHECK_EQUAL(tri0h1.upper(), 1); + BOOST_CHECK_EQUAL(mean(tri0h1), 0.5); + BOOST_CHECK_EQUAL(mode(tri0h1), 0.5); + BOOST_CHECK_EQUAL(pdf(tri0h1, -1), 0); + BOOST_CHECK_EQUAL(cdf(tri0h1, -1), 0); + BOOST_CHECK_EQUAL(pdf(tri0h1, 1), 0); + BOOST_CHECK_EQUAL(pdf(tri0h1, 999), 0); + BOOST_CHECK_EQUAL(cdf(tri0h1, 999), 1); + BOOST_CHECK_EQUAL(cdf(tri0h1, 1), 1); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0h1, 0.1), 0.02, tol5eps); + BOOST_CHECK_EQUAL(cdf(tri0h1, 0.5), 0.5); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0h1, 0.9), 0.98, tol5eps); + + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.), 0., tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.02), 0.1, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.5), 0.5, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 0.98), 0.9, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0h1, 1.), 1., tol5eps); + + triangular tri0q1(0, 0.25, 1); // mode is near bottom. + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.02), 0.0016, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.5), 0.66666666666666666666666666666666666666666666667, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(tri0q1, 0.98), 0.99946666666666661, tol5eps); + + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.0016), 0.02, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.66666666666666666666666666666666666666666666667), 0.5, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, 0.3333333333333333333333333333333333333333333333333)), 0.5, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(tri0q1, 0.99946666666666661), 0.98, 10 * tol5eps); + + triangular trim12(-1, -0.5, 2); // mode is negative. + BOOST_CHECK_CLOSE_FRACTION(pdf(trim12, 0), 0.533333333333333333333333333333333333333333333, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(trim12, 0), 0.466666666666666666666666666666666666666666667, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(cdf(complement(trim12, 0)), 1 - 0.466666666666666666666666666666666666666666667, tol5eps); + + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0q1, 1 - 0.99946666666666661)), 0.98, 10 * tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1.)), 0., tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0.5)), 0.5, tol5eps); // OK + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1. - 0.02)), 0.1, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 1. - 0.98)), 0.9, tol5eps); + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(tri0h1, 0)), 1., tol5eps); + + double xs [] = {0., 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.98, 0.99, 1.}; + + const triangular_distribution& distr = tristd; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(distr, 1.)), 0., tol5eps); + const triangular_distribution* distp = &tristd; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*distp, 1.)), 0., tol5eps); + + const triangular_distribution* dists [] = {&tristd, &tri011, &tri0q1, &tri0h1, &trim12}; + BOOST_CHECK_CLOSE_FRACTION(quantile(complement(*dists[1], 1.)), 0., tol5eps); + + for (int i = 0; i < 5; i++) + { + const triangular_distribution* const dist = dists[i]; + cout << "Distribution " << i << endl; + BOOST_CHECK_EQUAL(quantile(complement(*dists[i], 1.)), quantile(*dists[i], 0.)); + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.5), quantile(complement(*dist, 0.5)), tol5eps); // OK + BOOST_CHECK_CLOSE_FRACTION(quantile(*dists[i], 0.98), quantile(complement(*dist, 1. - 0.98)),tol5eps); + // cout << setprecision(17) << median(*dist) << endl; + } + + cout << showpos << setprecision(2) << endl; + + //triangular_distribution& dist = trim12; + for (unsigned i = 0; i < sizeof(xs) /sizeof(double); i++) + { + double x = xs[i] * (trim12.upper() - trim12.lower()) + trim12.lower(); + double dx = cdf(trim12, x); + double cx = cdf(complement(trim12, x)); + //cout << fixed << showpos << setprecision(3) + // << xs[i] << ", " << x << ", " << pdf(trim12, x) << ", " << dx << ", " << cx << ",, " ; + + BOOST_CHECK_CLOSE_FRACTION(cx, 1 - dx, tol500eps); // cx == 1 - dx + + // << setprecision(2) << scientific << cr - x << ", " // difference x - quan(cdf) + // << setprecision(3) << fixed + // << quantile(trim12, dx) << ", " + // << quantile(complement(trim12, 1 - dx)) << ", " + // << quantile(complement(trim12, cx)) << ", " + // << endl; + BOOST_CHECK_CLOSE_FRACTION(quantile(trim12, dx), quantile(complement(trim12, 1 - dx)), tol500eps); + } + cout << endl; + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % + #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. + #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. + #endif + #else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; + #endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_triangular.exe" +Running 1 test case... +Distribution 0 +Distribution 1 +Distribution 2 +Distribution 3 +Distribution 4 +Tolerance for type float is 5.96046e-007. +Tolerance for type double is 1.11022e-015. +Tolerance for type long double is 1.11022e-015. +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-015. +*** No errors detected + + + +*/ + + + + diff --git a/test/test_uniform.cpp b/test/test_uniform.cpp new file mode 100644 index 000000000..274f55d3d --- /dev/null +++ b/test/test_uniform.cpp @@ -0,0 +1,450 @@ +// Copyright Paul Bristow 2007. +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_uniform.cpp + +#ifdef _MSC_VER +# pragma warning(disable: 4127) // conditional expression is constant. +# pragma warning(disable: 4100) // unreferenced formal parameter. +#endif + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::uniform_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + +template +void check_uniform(RealType lower, RealType upper, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + uniform_distribution(lower, upper), // distribution. + x), // random variable. + p, // probability. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::cdf( + complement( + uniform_distribution(lower, upper), // distribution. + x)), // random variable. + q, // probability complement. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + uniform_distribution(lower, upper), // distribution. + p), // probability. + x, // random variable. + tol); // tolerance. + BOOST_CHECK_CLOSE_FRACTION( + ::boost::math::quantile( + complement( + uniform_distribution(lower, upper), // distribution. + q)), // probability complement. + x, // random variable. + tol); // tolerance. +} // void check_uniform + +template +void test_spots(RealType) +{ + // Basic sanity checks + // + // These test values were generated for the normal distribution + // using the online calculator at + // http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm + // + // Tolerance is just over 5 decimal digits expressed as a fraction: + // that's the limit of the test data. + RealType tolerance = 2e-5f; + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; + + using std::exp; + + // Tests for PDF + // + BOOST_CHECK_CLOSE_FRACTION( // x == upper + pdf(uniform_distribution(0, 1), static_cast(0)), + static_cast(1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x == lower + pdf(uniform_distribution(0, 1), static_cast(1)), + static_cast(1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x > upper + pdf(uniform_distribution(0, 1), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x < lower + pdf(uniform_distribution(0, 1), static_cast(2)), + static_cast(0), + tolerance); + + if(std::numeric_limits::has_infinity) + { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() + // Note that infinity is not implemented for real_concept, so these tests + // are only done for types, like built-in float, double.. that have infinity. + // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. + // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. + // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path + // of error handling is tested below with BOOST_CHECK_THROW tests. + + BOOST_CHECK_THROW( // x == infinity should NOT be OK. + pdf(uniform_distribution(0, 1), static_cast(std::numeric_limits::infinity())), + std::domain_error); + + BOOST_CHECK_THROW( // x == minus infinity should be OK too. + pdf(uniform_distribution(0, 1), static_cast(-std::numeric_limits::infinity())), + std::domain_error); + } + if(std::numeric_limits::has_quiet_NaN) + { // BOOST_CHECK tests for NaN using std::numeric_limits<>::has_quiet_NaN() - should throw. + BOOST_CHECK_THROW( + pdf(uniform_distribution(0, 1), static_cast(std::numeric_limits::quiet_NaN())), + std::domain_error); + BOOST_CHECK_THROW( + pdf(uniform_distribution(0, 1), static_cast(-std::numeric_limits::quiet_NaN())), + std::domain_error); + } // test for x = NaN using std::numeric_limits<>::quiet_NaN() + + // cdf + BOOST_CHECK_EQUAL( // x < lower + cdf(uniform_distribution(0, 1), static_cast(-1)), + static_cast(0) ); + BOOST_CHECK_CLOSE_FRACTION( + cdf(uniform_distribution(0, 1), static_cast(0)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(uniform_distribution(0, 1), static_cast(0.5)), + static_cast(0.5), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(uniform_distribution(0, 1), static_cast(0.1)), + static_cast(0.1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + cdf(uniform_distribution(0, 1), static_cast(0.9)), + static_cast(0.9), + tolerance); + BOOST_CHECK_EQUAL( // x > upper + cdf(uniform_distribution(0, 1), static_cast(2)), + static_cast(1)); + + // cdf complement + BOOST_CHECK_EQUAL( // x < lower + cdf(complement(uniform_distribution(0, 1), static_cast(0))), + static_cast(1)); + BOOST_CHECK_EQUAL( // x == 0 + cdf(complement(uniform_distribution(0, 1), static_cast(0))), + static_cast(1)); + BOOST_CHECK_CLOSE_FRACTION( // x = 0.1 + cdf(complement(uniform_distribution(0, 1), static_cast(0.1))), + static_cast(0.9), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x = 0.5 + cdf(complement(uniform_distribution(0, 1), static_cast(0.5))), + static_cast(0.5), + tolerance); + BOOST_CHECK_EQUAL( // x == 1 + cdf(complement(uniform_distribution(0, 1), static_cast(1))), + static_cast(0)); + BOOST_CHECK_EQUAL( // x > upper + cdf(complement(uniform_distribution(0, 1), static_cast(2))), + static_cast(1)); + + // quantile + + BOOST_CHECK_CLOSE_FRACTION( + quantile(uniform_distribution(0, 1), static_cast(0.9)), + static_cast(0.9), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(uniform_distribution(0, 1), static_cast(0.1)), + static_cast(0.1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(uniform_distribution(0, 1), static_cast(0.5)), + static_cast(0.5), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(uniform_distribution(0, 1), static_cast(0)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(uniform_distribution(0, 1), static_cast(1)), + static_cast(1), + tolerance); + + // quantile complement + + BOOST_CHECK_CLOSE_FRACTION( + quantile(complement(uniform_distribution(0, 1), static_cast(0.1))), + static_cast(0.9), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(complement(uniform_distribution(0, 1), static_cast(0.9))), + static_cast(0.1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(complement(uniform_distribution(0, 1), static_cast(0.5))), + static_cast(0.5), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(complement(uniform_distribution(0, 1), static_cast(0))), + static_cast(1), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( + quantile(complement(uniform_distribution(0, 1), static_cast(1))), + static_cast(1), + tolerance); + + // Some tests using a different location & scale, neight zero or unity. + BOOST_CHECK_CLOSE_FRACTION( // x == mid + pdf(uniform_distribution(-1, 2), static_cast(1)), + static_cast(0.3333333333333333333333333333333333333333333333333333), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + pdf(uniform_distribution(-1, 2), static_cast(+2)), + static_cast(0.3333333333333333333333333333333333333333333333333333), // 1 / (2 - -1) = 1/3 + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == lower + cdf(uniform_distribution(-1, 2), static_cast(-1)), + static_cast(0), + tolerance); + BOOST_CHECK_CLOSE_FRACTION( // x == upper + cdf(uniform_distribution(-1, 2), static_cast(0)), + static_cast(0.3333333333333333333333333333333333333333333333333333), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + cdf(uniform_distribution(-1, 2), static_cast(1)), + static_cast(0.6666666666666666666666666666666666666666666666666667), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == lower + cdf(uniform_distribution(-1, 2), static_cast(2)), + static_cast(1), + tolerance); + + BOOST_CHECK_CLOSE_FRACTION( // x == upper + quantile(uniform_distribution(-1, 2), static_cast(0.6666666666666666666666666666666666666666666666666667)), + static_cast(1), + tolerance); + + check_uniform( + static_cast(0), // lower + static_cast(1), // upper + static_cast(0.5), // x + static_cast(0.5), // p + static_cast(1 - 0.5), // q + tolerance); + + // Some Not-standard uniform tests. + check_uniform( + static_cast(-1), // lower + static_cast(1), // upper + static_cast(0), // x + static_cast(0.5), // p + static_cast(1 - 0.5), // q = 1 - p + tolerance); + + check_uniform( + static_cast(1), // lower + static_cast(3), // upper + static_cast(2), // x + static_cast(0.5), // p + static_cast(1 - 0.5), // q = 1 - p + tolerance); + + check_uniform( + static_cast(-1), // lower + static_cast(2), // upper + static_cast(1), // x + static_cast(0.66666666666666666666666666666666666666666667), // p + static_cast(0.33333333333333333333333333333333333333333333), // q = 1 - p + tolerance); + tolerance = (std::max)( + boost::math::tools::epsilon(), + static_cast(boost::math::tools::epsilon())) * 5; // 5 eps as a fraction. + cout << "Tolerance (as fraction) for type " << typeid(RealType).name() << " is " << tolerance << "." << endl; + uniform_distribution distu01(0, 1); + RealType x = static_cast(0.5); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE_FRACTION( + mean(distu01), static_cast(0.5), tolerance); + // variance: + BOOST_CHECK_CLOSE_FRACTION( + variance(distu01), static_cast(0.0833333333333333333333333333333333333333333), tolerance); + // std deviation: + BOOST_CHECK_CLOSE_FRACTION( + standard_deviation(distu01), sqrt(variance(distu01)), tolerance); + // hazard: + BOOST_CHECK_CLOSE_FRACTION( + hazard(distu01, x), pdf(distu01, x) / cdf(complement(distu01, x)), tolerance); + // cumulative hazard: + BOOST_CHECK_CLOSE_FRACTION( + chf(distu01, x), -log(cdf(complement(distu01, x))), tolerance); + // coefficient_of_variation: + BOOST_CHECK_CLOSE_FRACTION( + coefficient_of_variation(distu01), standard_deviation(distu01) / mean(distu01), tolerance); + // mode: + BOOST_CHECK_CLOSE_FRACTION( + mode(distu01), static_cast(0), tolerance); + BOOST_CHECK_CLOSE_FRACTION( + median(distu01), static_cast(0.5), tolerance); + // skewness: + BOOST_CHECK_EQUAL( + skewness(distu01), static_cast(0)); + // kertosis: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis(distu01), kurtosis_excess(distu01) + static_cast(3), tolerance); + // kertosis excess: + BOOST_CHECK_CLOSE_FRACTION( + kurtosis_excess(distu01), static_cast(-1.2), tolerance); + + if(std::numeric_limits::has_infinity) + { // BOOST_CHECK tests for infinity using std::numeric_limits<>::infinity() + // Note that infinity is not implemented for real_concept, so these tests + // are only done for types, like built-in float, double, long double, that have infinity. + // Note that these assume that BOOST_MATH_OVERFLOW_ERROR_POLICY is NOT throw_on_error. + // #define BOOST_MATH_OVERFLOW_ERROR_POLICY == throw_on_error would give a throw here. + // #define BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error IS defined, so the throw path + // of error handling is tested below with BOOST_CHECK_THROW tests. + + BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits::infinity()), std::domain_error); + BOOST_CHECK_THROW(pdf(distu01, -std::numeric_limits::infinity()), std::domain_error); + } // test for infinity using std::numeric_limits<>::infinity() + else + { // real_concept case, does has_infinfity == false, so can't check it throws. + // cout << std::numeric_limits::infinity() << ' ' + // << boost::math::fpclassify(std::numeric_limits::infinity()) << endl; + // value of std::numeric_limits::infinity() is zero, so FPclassify is zero, + // so (boost::math::isfinite)(std::numeric_limits::infinity()) does not detect infinity. + // so these tests would never throw. + //BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits::infinity()), std::domain_error); + //BOOST_CHECK_THROW(pdf(distu01, std::numeric_limits::quiet_NaN()), std::domain_error); + // BOOST_CHECK_THROW(pdf(distu01, boost::math::tools::max_value() * 2), std::domain_error); // Doesn't throw. + BOOST_CHECK_EQUAL(pdf(distu01, boost::math::tools::max_value()), 0); + } + // Special cases: + BOOST_CHECK(pdf(distu01, 0) == 1); + BOOST_CHECK(cdf(distu01, 0) == 0); + BOOST_CHECK(pdf(distu01, 1) == 1); + BOOST_CHECK(cdf(distu01, 1) == 1); + BOOST_CHECK(cdf(complement(distu01, 0)) == 1); + BOOST_CHECK(cdf(complement(distu01, 1)) == 0); + BOOST_CHECK(quantile(distu01, 0) == 0); + BOOST_CHECK(quantile(complement(distu01, 0)) == 1); + BOOST_CHECK(quantile(distu01, 1) == 1); + BOOST_CHECK(quantile(complement(distu01, 1)) == 1); + + // Error checks: + if(std::numeric_limits::has_quiet_NaN) + { // BOOST_CHECK tests for constructing with quiet_NaN (not for real_concept, for example - see notes above). + BOOST_CHECK_THROW(uniform_distribution(0, std::numeric_limits::quiet_NaN()), std::domain_error); + BOOST_CHECK_THROW(uniform_distribution(0, -std::numeric_limits::quiet_NaN()), std::domain_error); + } + BOOST_CHECK_THROW(uniform_distribution(1, 0), std::domain_error); // lower > upper! + BOOST_CHECK_THROW(uniform_distribution(1, 1), std::domain_error); // lower == upper! + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + // Check that can construct uniform distribution using the two convenience methods: + using namespace boost::math; + uniform unistd; // Using typedef + // == uniform_distribution unistd; + BOOST_CHECK_EQUAL(unistd.lower(), 0); // Check defaults. + BOOST_CHECK_EQUAL(unistd.upper(), 1); + uniform_distribution<> myu01(0, 1); // Using default RealType double. + BOOST_CHECK_EQUAL(myu01.lower(), 0); // Check defaults again. + BOOST_CHECK_EQUAL(myu01.upper(), 1); + + // Test on extreme values of random variate x, using just double because it has numeric_limit infinity etc.. + // No longer allow x to be + or - infinity, then these tests should throw. + BOOST_CHECK_THROW(pdf(unistd, +std::numeric_limits::infinity()), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(pdf(unistd, -std::numeric_limits::infinity()), std::domain_error); // x = - infinity + BOOST_CHECK_THROW(cdf(unistd, +std::numeric_limits::infinity()), std::domain_error); // x = + infinity + BOOST_CHECK_THROW(cdf(unistd, -std::numeric_limits::infinity()), std::domain_error); // x = - infinity + + BOOST_CHECK_EQUAL(pdf(unistd, +(std::numeric_limits::max)()), 0); // x = + max + BOOST_CHECK_EQUAL(pdf(unistd, -(std::numeric_limits::min)()), 0); // x = - min + BOOST_CHECK_EQUAL(cdf(unistd, +(std::numeric_limits::max)()), 1); // x = + max + BOOST_CHECK_EQUAL(cdf(unistd, -(std::numeric_limits::min)()), 0); // x = - min + + BOOST_CHECK_THROW(uniform_distribution<> zinf(0, +std::numeric_limits::infinity()), std::domain_error); // zero to infinity using default RealType double. + + uniform_distribution<> zmax(0, +(std::numeric_limits::max)()); // zero to max using default RealType double. + BOOST_CHECK_EQUAL(zmax.lower(), 0); // Check defaults again. + BOOST_CHECK_EQUAL(zmax.upper(), +(std::numeric_limits::max)()); + + BOOST_CHECK_EQUAL(pdf(zmax, -1), 0); // pdf is 1/(0 - max) = almost zero for all x + BOOST_CHECK_EQUAL(pdf(zmax, 0), (std::numeric_limits::min)()/4); // x = + BOOST_CHECK_EQUAL(pdf(zmax, 1), (std::numeric_limits::min)()/4); // x = + BOOST_CHECK_THROW(pdf(zmax, +std::numeric_limits::infinity()), std::domain_error); // pdf is 1/(0 - infinity) = zero for all x + BOOST_CHECK_THROW(pdf(zmax, -std::numeric_limits::infinity()), std::domain_error); + BOOST_CHECK_EQUAL(pdf(zmax, +(std::numeric_limits::max)()), (std::numeric_limits::min)()/4); // x = + BOOST_CHECK_EQUAL(pdf(zmax, -(std::numeric_limits::max)()), 0); // x = + + // Ensure NaN throws an exception. + BOOST_CHECK_THROW(uniform_distribution<> zNaN(0, std::numeric_limits::quiet_NaN()), std::domain_error); + BOOST_CHECK_THROW(pdf(unistd, std::numeric_limits::quiet_NaN()), std::domain_error); + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_uniform.exe" +Running 1 test case... +Tolerance for type float is 2e-005. +Tolerance (as fraction) for type float is 5.96046e-007. +Tolerance for type double is 2e-005. +Tolerance (as fraction) for type double is 1.11022e-015. +Tolerance for type long double is 2e-005. +Tolerance (as fraction) for type long double is 1.11022e-015. +Tolerance for type class boost::math::concepts::real_concept is 2e-005. +Tolerance (as fraction) for type class boost::math::concepts::real_concept is 1.11022e-015. +*** No errors detected + +*/ + + + diff --git a/test/test_weibull.cpp b/test/test_weibull.cpp new file mode 100644 index 000000000..32c049219 --- /dev/null +++ b/test/test_weibull.cpp @@ -0,0 +1,378 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +// test_weibull.cpp + +#include // for real_concept +#include // Boost.Test +#include + +#include + using boost::math::weibull_distribution; +#include + +#include + using std::cout; + using std::endl; + using std::setprecision; +#include + using std::numeric_limits; + +template +void check_weibull(RealType shape, RealType scale, RealType x, RealType p, RealType q, RealType tol) +{ + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + weibull_distribution(shape, scale), // distribution. + x), // random variable. + p, // probability. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::cdf( + complement( + weibull_distribution(shape, scale), // distribution. + x)), // random variable. + q, // probability complement. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + weibull_distribution(shape, scale), // distribution. + p), // probability. + x, // random variable. + tol); // %tolerance. + BOOST_CHECK_CLOSE( + ::boost::math::quantile( + complement( + weibull_distribution(shape, scale), // distribution. + q)), // probability complement. + x, // random variable. + tol); // %tolerance. +} + +template +void test_spots(RealType) +{ + // Basic sanity checks + // + // These test values were generated for the normal distribution + // using the online calculator at + // http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm + // + // Tolerance is just over 5 decimal digits expressed as a persentage: + // that's the limit of the test data. + RealType tolerance = 2e-5f * 100; + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + + using std::exp; + + check_weibull( + static_cast(0.25), // shape + static_cast(0.5), // scale + static_cast(0.1), // x + static_cast(0.487646), // p + static_cast(1-0.487646), // q + tolerance); + check_weibull( + static_cast(0.25), // shape + static_cast(0.5), // scale + static_cast(0.5), // x + static_cast(1-0.367879), // p + static_cast(0.367879), // q + tolerance); + check_weibull( + static_cast(0.25), // shape + static_cast(0.5), // scale + static_cast(1), // x + static_cast(1-0.304463), // p + static_cast(0.304463), // q + tolerance); + check_weibull( + static_cast(0.25), // shape + static_cast(0.5), // scale + static_cast(2), // x + static_cast(1-0.243117), // p + static_cast(0.243117), // q + tolerance); + check_weibull( + static_cast(0.25), // shape + static_cast(0.5), // scale + static_cast(5), // x + static_cast(1-0.168929), // p + static_cast(0.168929), // q + tolerance); + + check_weibull( + static_cast(0.5), // shape + static_cast(2), // scale + static_cast(0.1), // x + static_cast(0.200371), // p + static_cast(1-0.200371), // q + tolerance); + check_weibull( + static_cast(0.5), // shape + static_cast(2), // scale + static_cast(0.5), // x + static_cast(0.393469), // p + static_cast(1-0.393469), // q + tolerance); + check_weibull( + static_cast(0.5), // shape + static_cast(2), // scale + static_cast(1), // x + static_cast(1-0.493069), // p + static_cast(0.493069), // q + tolerance); + check_weibull( + static_cast(0.5), // shape + static_cast(2), // scale + static_cast(2), // x + static_cast(1-0.367879), // p + static_cast(0.367879), // q + tolerance); + check_weibull( + static_cast(0.5), // shape + static_cast(2), // scale + static_cast(5), // x + static_cast(1-0.205741), // p + static_cast(0.205741), // q + tolerance); + + check_weibull( + static_cast(2), // shape + static_cast(0.25), // scale + static_cast(0.1), // x + static_cast(0.147856), // p + static_cast(1-0.147856), // q + tolerance); + check_weibull( + static_cast(2), // shape + static_cast(0.25), // scale + static_cast(0.5), // x + static_cast(1-0.018316), // p + static_cast(0.018316), // q + tolerance); + + /* + This test value came from + http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm + but appears to be grossly incorrect: certainly it does not agree with the values + I get from pushing numbers into a calculator (0.0001249921878255106610615995196123). + Strangely other test values generated for the same shape and scale parameters do look OK. + check_weibull( + static_cast(3), // shape + static_cast(2), // scale + static_cast(0.1), // x + static_cast(1.25E-40), // p + static_cast(1-1.25E-40), // q + tolerance); + */ + check_weibull( + static_cast(3), // shape + static_cast(2), // scale + static_cast(0.5), // x + static_cast(0.015504), // p + static_cast(1-0.015504), // q + tolerance * 10); // few digits in test value + check_weibull( + static_cast(3), // shape + static_cast(2), // scale + static_cast(1), // x + static_cast(0.117503), // p + static_cast(1-0.117503), // q + tolerance); + check_weibull( + static_cast(3), // shape + static_cast(2), // scale + static_cast(2), // x + static_cast(1-0.367879), // p + static_cast(0.367879), // q + tolerance); + + // + // Tests for PDF + // + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.25, 0.5), static_cast(0.1)), + static_cast(0.856579), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.25, 0.5), static_cast(0.5)), + static_cast(0.183940), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.25, 0.5), static_cast(5)), + static_cast(0.015020), + tolerance * 10); // fewer digits in test value + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.5, 2), static_cast(0.1)), + static_cast(0.894013), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.5, 2), static_cast(0.5)), + static_cast(0.303265), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(0.5, 2), static_cast(1)), + static_cast(0.174326), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(2, 0.25), static_cast(0.1)), + static_cast(2.726860), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(2, 0.25), static_cast(0.5)), + static_cast(0.293050), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(3, 2), static_cast(1)), + static_cast(0.330936), + tolerance); + BOOST_CHECK_CLOSE( + pdf(weibull_distribution(3, 2), static_cast(2)), + static_cast(0.551819), + tolerance); + + // + // These test values were obtained using the formulas at + // http://en.wikipedia.org/wiki/Weibull_distribution + // which are subtly different to (though mathematically + // the same as) the ones on the Mathworld site + // http://mathworld.wolfram.com/WeibullDistribution.html + // which are the ones used in the implementation. + // The assumption is that if both computation methods + // agree then the implementation is probably correct... + // What's not clear is which method is more accurate. + // + tolerance = (std::max)( + boost::math::tools::epsilon(), + static_cast(boost::math::tools::epsilon())) * 5 * 100; // 5 eps as a percentage + cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl; + weibull_distribution dist(2, 3); + RealType x = static_cast(0.125); + using namespace std; // ADL of std names. + // mean: + BOOST_CHECK_CLOSE( + mean(dist) + , dist.scale() * boost::math::tgamma(1 + 1 / dist.shape()), tolerance); + // variance: + BOOST_CHECK_CLOSE( + variance(dist) + , dist.scale() * dist.scale() * boost::math::tgamma(1 + 2 / dist.shape()) - mean(dist) * mean(dist), tolerance); + // std deviation: + BOOST_CHECK_CLOSE( + standard_deviation(dist) + , sqrt(variance(dist)), tolerance); + // hazard: + BOOST_CHECK_CLOSE( + hazard(dist, x) + , pdf(dist, x) / cdf(complement(dist, x)), tolerance); + // cumulative hazard: + BOOST_CHECK_CLOSE( + chf(dist, x) + , -log(cdf(complement(dist, x))), tolerance); + // coefficient_of_variation: + BOOST_CHECK_CLOSE( + coefficient_of_variation(dist) + , standard_deviation(dist) / mean(dist), tolerance); + // mode: + BOOST_CHECK_CLOSE( + mode(dist) + , dist.scale() * pow((dist.shape() - 1) / dist.shape(), 1/dist.shape()), tolerance); + // median: + BOOST_CHECK_CLOSE( + median(dist) + , dist.scale() * pow(log(static_cast(2)), 1 / dist.shape()), tolerance); + // skewness: + BOOST_CHECK_CLOSE( + skewness(dist), + (boost::math::tgamma(1 + 3/dist.shape()) * pow(dist.scale(), RealType(3)) - 3 * mean(dist) * variance(dist) - pow(mean(dist), RealType(3))) / pow(standard_deviation(dist), RealType(3)), + tolerance * 100); + // kertosis: + BOOST_CHECK_CLOSE( + kurtosis(dist) + , kurtosis_excess(dist) + 3, tolerance); + // kertosis excess: + BOOST_CHECK_CLOSE( + kurtosis_excess(dist), + (pow(dist.scale(), RealType(4)) * boost::math::tgamma(1 + 4/dist.shape()) + - 3 * variance(dist) * variance(dist) + - 4 * skewness(dist) * variance(dist) * standard_deviation(dist) * mean(dist) + - 6 * variance(dist) * mean(dist) * mean(dist) + - pow(mean(dist), RealType(4))) / (variance(dist) * variance(dist)), + tolerance * 1000); + + // + // Special cases: + // + BOOST_CHECK(pdf(dist, 0) == 0); + BOOST_CHECK(cdf(dist, 0) == 0); + BOOST_CHECK(cdf(complement(dist, 0)) == 1); + BOOST_CHECK(quantile(dist, 0) == 0); + BOOST_CHECK(quantile(complement(dist, 1)) == 0); + + // + // Error checks: + // + BOOST_CHECK_THROW(weibull_distribution(0, -1), std::domain_error); + BOOST_CHECK_THROW(weibull_distribution(-1, 1), std::domain_error); + BOOST_CHECK_THROW(pdf(dist, -1), std::domain_error); + BOOST_CHECK_THROW(cdf(dist, -1), std::domain_error); + BOOST_CHECK_THROW(cdf(complement(dist, -1)), std::domain_error); + BOOST_CHECK_THROW(quantile(dist, 1), std::overflow_error); + BOOST_CHECK_THROW(quantile(complement(dist, 0)), std::overflow_error); + +} // template void test_spots(RealType) + +int test_main(int, char* []) +{ + + // Check that can construct weibull distribution using the two convenience methods: + using namespace boost::math; + weibull myw1(2); // Using typedef + weibull_distribution<> myw2(2); // Using default RealType double. + + // Basic sanity-check spot values. + // (Parameter value, arbitrarily zero, only communicates the floating point type). + test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 % + test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 % +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_spots(0.0L); // Test long double. +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x0582)) + test_spots(boost::math::concepts::real_concept(0.)); // Test real concept. +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif + + return 0; +} // int test_main(int, char* []) + +/* + +Output: + +Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_weibull.exe" +Running 1 test case... +Tolerance for type float is 0.002 % +Tolerance for type float is 5.96046e-005 % +Tolerance for type double is 0.002 % +Tolerance for type double is 1.11022e-013 % +Tolerance for type long double is 0.002 % +Tolerance for type long double is 1.11022e-013 % +Tolerance for type class boost::math::concepts::real_concept is 0.002 % +Tolerance for type class boost::math::concepts::real_concept is 1.11022e-013 % +*** No errors detected + +*/ + + + + diff --git a/test/tgamma_delta_ratio_data.ipp b/test/tgamma_delta_ratio_data.ipp new file mode 100644 index 000000000..3a2969b84 --- /dev/null +++ b/test/tgamma_delta_ratio_data.ipp @@ -0,0 +1,570 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 560> tgamma_delta_ratio_data = { { + { SC_(0.2585242462158203125e2), SC_(0.5408298164866209845058619976043701171875e-7), SC_(0.9999998251530248995276873521622051469349e0), SC_(0.1000000174847005556584935896873247717521e1) }, + { SC_(0.2585242462158203125e2), SC_(0.676797071719192899763584136962890625e-7), SC_(0.999999781195646858662207729479648955493e0), SC_(0.1000000218804400836041852388787649318772e1) }, + { SC_(0.2585242462158203125e2), SC_(0.227188110102360951714217662811279296875e-6), SC_(0.9999992655149263979633607427932657111612e0), SC_(0.1000000734485611035140775476483228633931e1) }, + { SC_(0.2585242462158203125e2), SC_(0.43750014810939319431781768798828125e-6), SC_(0.9999985855896265447456778263434928930538e0), SC_(0.1000001414412366465930704463900488591982e1) }, + { SC_(0.2585242462158203125e2), SC_(0.53738921224066871218383312225341796875e-6), SC_(0.9999982626548644744952904509631607678305e0), SC_(0.1000001737348142509415721770085153528871e1) }, + { SC_(0.2585242462158203125e2), SC_(0.187765863302047364413738250732421875e-5), SC_(0.9999939296621942038545330388383612018463e0), SC_(0.1000006070374515974386401812531964569938e1) }, + { SC_(0.2585242462158203125e2), SC_(0.364947482012212276458740234375e-5), SC_(0.9999882015392083420820590401050831666868e0), SC_(0.1000011798599471698320496927282851435367e1) }, + { SC_(0.2585242462158203125e2), SC_(0.465787525172345340251922607421875e-5), SC_(0.9999849414855571510186069460633635722574e0), SC_(0.100001505874034945059522274595122297502e1) }, + { SC_(0.2585242462158203125e2), SC_(0.12453912859200499951839447021484375e-4), SC_(0.9999597380646515528354486024651847636645e0), SC_(0.1000040263550319943886221473682554015115e1) }, + { SC_(0.2585242462158203125e2), SC_(0.199610440176911652088165283203125e-4), SC_(0.9999354692330693579923518462608760820048e0), SC_(0.1000064534915704102352131840666481142235e1) }, + { SC_(0.2585242462158203125e2), SC_(0.33494274248369038105010986328125e-4), SC_(0.9998917208889694530058425389988593934263e0), SC_(0.1000108290792416164012804795838323764558e1) }, + { SC_(0.2585242462158203125e2), SC_(0.944348503253422677516937255859375e-4), SC_(0.9996947442966006592976441458929101277858e0), SC_(0.1000305348561075817062325093547661856711e1) }, + { SC_(0.2585242462158203125e2), SC_(0.1560666714794933795928955078125e-3), SC_(0.9994955727290581849404289655378810140685e0), SC_(0.1000504680885139037822534025743748797801e1) }, + { SC_(0.2585242462158203125e2), SC_(0.2901323023252189159393310546875e-3), SC_(0.9990624580199676173916703708565427843875e0), SC_(0.1000938418466905798031791435452101055018e1) }, + { SC_(0.2585242462158203125e2), SC_(0.755313201807439327239990234375e-3), SC_(0.9975610861022903221561899703141791348538e0), SC_(0.100244485418677879663798501597244544674e1) }, + { SC_(0.2585242462158203125e2), SC_(0.19461731426417827606201171875e-2), SC_(0.9937278186773194106390554970972851459813e0), SC_(0.100631161956687281471898918527812610804e1) }, + { SC_(0.2585242462158203125e2), SC_(0.38232556544244289398193359375e-2), SC_(0.9877154377124412300856863814182060327707e0), SC_(0.1012436766025790015381991545261255096474e1) }, + { SC_(0.2585242462158203125e2), SC_(0.779867731034755706787109375e-2), SC_(0.9751013656425511680030902634736420475346e0), SC_(0.1025531946310101796079629524884350857053e1) }, + { SC_(0.2585242462158203125e2), SC_(0.15350691974163055419921875e-1), SC_(0.9515790661841248294367778038596333142378e0), SC_(0.1050875058164998685409675651470553880948e1) }, + { SC_(0.2585242462158203125e2), SC_(0.307452343404293060302734375e-1), SC_(0.9053659020566855236992060373732306230212e0), SC_(0.1104484626687114609435978036501693747322e1) }, + { SC_(0.2585242462158203125e2), SC_(0.36175407469272613525390625e-1), SC_(0.8896041540632749956354535217058411588909e0), SC_(0.1124037454973942332969819033951835840808e1) }, + { SC_(0.2585242462158203125e2), SC_(0.10786493122577667236328125e0), SC_(0.7054284924577535378692687185388981829148e0), SC_(0.1416927796010049530382552725533678655455e1) }, + { SC_(0.2585242462158203125e2), SC_(0.2463240921497344970703125e0), SC_(0.4504341244929360176728122486115363051051e0), SC_(0.221477413672608805343931793057582616074e1) }, + { SC_(0.2585242462158203125e2), SC_(0.49527740478515625e0), SC_(0.2006872187401179782199716363143168099368e0), SC_(0.493490170967777908350210290285351802481e1) }, + { SC_(0.2585242462158203125e2), SC_(0.97858345508575439453125e0), SC_(0.4148819733420400314432511432493435069162e-1), SC_(0.2320967613276263777519155169613012005116e2) }, + { SC_(0.2585242462158203125e2), SC_(0.110986173152923583984375e1), SC_(0.2699620660586217424137708692874661231742e-1), SC_(0.3528517488424231302164184504667130625721e2) }, + { SC_(0.2585242462158203125e2), SC_(0.2970751285552978515625e1), SC_(0.5703385229085370029592211356722646834358e-4), SC_(0.1236967264821060149332720477406637969004e5) }, + { SC_(0.2585242462158203125e2), SC_(0.7192423343658447265625e1), SC_(0.312799338488474106328954065959738498016e-10), SC_(0.4040224153039980846285110727435371987011e10) }, + { SC_(0.2761920928955078125e2), SC_(0.5408298164866209845058619976043701171875e-7), SC_(0.9999998215100084504946489151623116996137e0), SC_(0.1000000178490023300344336544888751699239e1) }, + { SC_(0.2761920928955078125e2), SC_(0.676797071719192899763584136962890625e-7), SC_(0.9999997766367584992913505687442889847483e0), SC_(0.1000000223363291222972544528238080595588e1) }, + { SC_(0.2761920928955078125e2), SC_(0.227188110102360951714217662811279296875e-6), SC_(0.9999992502115994334380753664176258020911e0), SC_(0.100000074978896084660048277653220975194e1) }, + { SC_(0.2761920928955078125e2), SC_(0.43750014810939319431781768798828125e-6), SC_(0.9999985561197604324202048627296723864958e0), SC_(0.1000001443882317303562220288256332814481e1) }, + { SC_(0.2761920928955078125e2), SC_(0.53738921224066871218383312225341796875e-6), SC_(0.9999982264565157295217773527075480014079e0), SC_(0.100000177354661908492980625430723946754e1) }, + { SC_(0.2761920928955078125e2), SC_(0.187765863302047364413738250732421875e-5), SC_(0.9999938031843246022968565105662072274863e0), SC_(0.100000619685394617023653168345222076774e1) }, + { SC_(0.2761920928955078125e2), SC_(0.364947482012212276458740234375e-5), SC_(0.9999879557143788616314464477919030261072e0), SC_(0.1000012044430196635832390178516949941026e1) }, + { SC_(0.2761920928955078125e2), SC_(0.465787525172345340251922607421875e-5), SC_(0.9999846277369625852014465179641266929718e0), SC_(0.1000015372498547580600989775783392643319e1) }, + { SC_(0.2761920928955078125e2), SC_(0.12453912859200499951839447021484375e-4), SC_(0.9999588992062972129323604627467921024011e0), 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SC_(0.100014648417652299301561171051674700624e1) }, + { SC_(0.7978768157958984375e2), SC_(0.944348503253422677516937255859375e-4), SC_(0.999587113161598233468454734084678305589e0), SC_(0.1000413057271838561947236206167671415781e1) }, + { SC_(0.7978768157958984375e2), SC_(0.1560666714794933795928955078125e-3), SC_(0.9993177391847333530108166399937846934428e0), SC_(0.1000682726305481164972102002211593624753e1) }, + { SC_(0.7978768157958984375e2), SC_(0.2901323023252189159393310546875e-3), SC_(0.9987320295266112808968321744348581125643e0), SC_(0.100126957920068045138514617691027204372e1) }, + { SC_(0.7978768157958984375e2), SC_(0.755313201807439327239990234375e-3), SC_(0.996702393329728451276260054525472578427e0), SC_(0.1003308509638527465188853170750534124253e1) }, + { SC_(0.7978768157958984375e2), SC_(0.19461731426417827606201171875e-2), SC_(0.9915253014075514648177162925683418590611e0), SC_(0.1008547084790305965144114625881233228855e1) }, + { SC_(0.7978768157958984375e2), SC_(0.38232556544244289398193359375e-2), SC_(0.9834194646621488038141741559240478803343e0), SC_(0.1016859897102447874057820269275378350685e1) }, + { SC_(0.7978768157958984375e2), SC_(0.779867731034755706787109375e-2), SC_(0.96647031314981521958706391422552055032e0), SC_(0.1034692136253289961682530203701497931086e1) }, + { SC_(0.7978768157958984375e2), SC_(0.15350691974163055419921875e-1), SC_(0.9350722926704194403056098755684301267726e0), SC_(0.1069432851206020276222977029635706852091e1) }, + { SC_(0.7978768157958984375e2), SC_(0.307452343404293060302734375e-1), SC_(0.8741899273929740118512348989798936860313e0), SC_(0.1143902539794300345755398855610730787373e1) }, + { SC_(0.7978768157958984375e2), SC_(0.36175407469272613525390625e-1), SC_(0.853673444028947574453146920519234106748e0), SC_(0.1171388781196630028604025966953752030829e1) }, + { SC_(0.7978768157958984375e2), SC_(0.10786493122577667236328125e0), SC_(0.6238930467503061165450147922993730709957e0), SC_(0.1602603644554494464631490632648253985713e1) }, + { SC_(0.7978768157958984375e2), SC_(0.2463240921497344970703125e0), SC_(0.3404185101843088803843854355525436974705e0), SC_(0.2935313484497773990220628475801307630996e1) }, + { SC_(0.7978768157958984375e2), SC_(0.49527740478515625e0), SC_(0.1144707397438024439442613742493198994445e0), SC_(0.870887189584181010190648970619663964418e1) }, + { SC_(0.7978768157958984375e2), SC_(0.97858345508575439453125e0), SC_(0.137674647370524488408642123698494921405e-1), SC_(0.7176300487625859938764819984066740114382e2) }, + { SC_(0.7978768157958984375e2), SC_(0.110986173152923583984375e1), SC_(0.7740752638659382404316274553019359244794e-2), SC_(0.1271948689069195209595747138373138092093e3) }, + { SC_(0.7978768157958984375e2), SC_(0.2970751285552978515625e1), SC_(0.2157992909774354428566814255364545463342e-5), SC_(0.4145708927186640053679609302173996067288e6) }, + { SC_(0.7978768157958984375e2), SC_(0.7192423343658447265625e1), SC_(0.1594087689108406351642321400531650818267e-13), SC_(0.3263982781831842992979765044201168559318e14) } + } }; +#undef SC_ + diff --git a/test/tgamma_delta_ratio_int.ipp b/test/tgamma_delta_ratio_int.ipp new file mode 100644 index 000000000..ef7baaeee --- /dev/null +++ b/test/tgamma_delta_ratio_int.ipp @@ -0,0 +1,354 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 344> tgamma_delta_ratio_int = { { + { SC_(0.24302618503570556640625e1), SC_(0.1e1), SC_(0.4114782939349022594054088894256271572472e0), SC_(0.14302618503570556640625e1) }, + { SC_(0.301645793914794921875e2), SC_(0.1e1), SC_(0.3315146506841289877594755891064182591548e-1), SC_(0.291645793914794921875e2) }, + { SC_(0.301645793914794921875e2), SC_(0.3e1), SC_(0.3307223761155750711531487469609546246538e-4), SC_(0.2231320586281996185139958610577082254167e5) }, + { SC_(0.301645793914794921875e2), SC_(0.5e1), SC_(0.2918858268609190809990278724248604514024e-7), SC_(0.1469147518070480379794665014907786525186e8) }, + { SC_(0.301645793914794921875e2), SC_(0.7e1), SC_(0.2295219166275465213873613192776013762556e-10), SC_(0.8223734198694550134689130502530947232135e10) }, + { SC_(0.301645793914794921875e2), SC_(0.9e1), SC_(0.1618208430275734434924519990812495766662e-13), SC_(0.3857786609269072710128224389442555988969e13) }, + { SC_(0.301645793914794921875e2), SC_(0.11e2), SC_(0.1028721387871736550385318599923041671733e-16), SC_(0.1490824979713097171415993481052223628558e16) }, + { SC_(0.301645793914794921875e2), SC_(0.13e2), SC_(0.5926882460007524968122855042901912157197e-20), SC_(0.4648203922170334340207682876080006032116e18) }, + { SC_(0.301645793914794921875e2), SC_(0.15e2), SC_(0.3109028494792192917110790291054315941926e-23), SC_(0.1139409800082989383839979690424223084142e21) }, + { SC_(0.301645793914794921875e2), SC_(0.17e2), SC_(0.1491138039359247528148047903357857018565e-26), SC_(0.2124665771291534304735753615395767158197e23) }, + { SC_(0.301645793914794921875e2), SC_(0.19e2), SC_(0.6564083719175744994115607229861959803378e-30), SC_(0.2885559839005887499381495946913235780383e25) }, + { SC_(0.33848663330078125e2), SC_(0.1e1), SC_(0.2954326409431340851983450434701073702185e-1), SC_(0.32848663330078125e2) }, + { SC_(0.33848663330078125e2), SC_(0.3e1), SC_(0.2364827053120534463560561879334289442712e-4), SC_(0.3227344028825395710668999527115374803543e5) }, + { SC_(0.33848663330078125e2), SC_(0.5e1), SC_(0.1695614414470450244250500226271753919667e-7), SC_(0.2779046705870235738963729137775015086257e8) }, + { SC_(0.33848663330078125e2), SC_(0.7e1), SC_(0.1095310525613279305495592465256381394661e-10), SC_(0.2077891515483201757744317351815521823156e11) }, + { SC_(0.33848663330078125e2), SC_(0.9e1), SC_(0.6407340740919580652793383140467826508761e-14), SC_(0.1334639554269227650049353460252698351418e14) }, + { SC_(0.33848663330078125e2), SC_(0.11e2), SC_(0.3410234021200152276254976943763997265065e-17), SC_(0.7272585453555431260958160354377596176673e16) }, + { SC_(0.33848663330078125e2), SC_(0.13e2), SC_(0.1658471451363357954157478377453720607441e-20), SC_(0.3312774860859689395345508883625889572398e19) }, + { SC_(0.33848663330078125e2), SC_(0.15e2), SC_(0.7398454132291610784864741044066350592108e-24), SC_(0.123937789059580278543168468020516206455e22) }, + { SC_(0.33848663330078125e2), SC_(0.17e2), SC_(0.3038328895194104537239753148827028386063e-27), SC_(0.3727133034351634026738359532400494439921e24) }, + { SC_(0.33848663330078125e2), SC_(0.19e2), SC_(0.1152438286925339630084625472113528311441e-30), SC_(0.8771116810258575546832070468458856366617e26) }, + { SC_(0.3462689208984375e2), SC_(0.1e1), SC_(0.2887928831167915505519744344972123660605e-1), SC_(0.3362689208984375e2) }, + { SC_(0.3462689208984375e2), SC_(0.3e1), SC_(0.2213137918593517448219688061454782526597e-4), SC_(0.3469915936700885481513978447765111923218e5) }, + { SC_(0.3462689208984375e2), SC_(0.5e1), SC_(0.1522721097426024895896792330316918422863e-7), SC_(0.3148531028341027662367459391135879115051e8) }, + { SC_(0.3462689208984375e2), SC_(0.7e1), SC_(0.9458379866953813757703618778885035880068e-11), SC_(0.2490085216056195944036457961808452623757e11) }, + { SC_(0.3462689208984375e2), SC_(0.9e1), SC_(0.5330391348709183914871234647798016253296e-14), SC_(0.1699145729193855721519591364984158133864e14) }, + { SC_(0.3462689208984375e2), SC_(0.11e2), SC_(0.2737840710883982300048275891615376700953e-17), SC_(0.9886597038736954287739003770107517853574e16) }, + { SC_(0.3462689208984375e2), SC_(0.13e2), SC_(0.1286917838872088415509337429304986457503e-20), SC_(0.4837999869770538119582666222954992161451e19) }, + { SC_(0.3462689208984375e2), SC_(0.15e2), SC_(0.5556765445986870190361609497371887346956e-24), SC_(0.195862450405695992267877670894152909715e22) }, + { SC_(0.3462689208984375e2), SC_(0.17e2), SC_(0.2211687277609313503714629255037786857888e-27), SC_(0.643083442616985917719955843589721349925e24) }, + { SC_(0.3462689208984375e2), SC_(0.19e2), SC_(0.8140292584590354784620261881745875674798e-31), SC_(0.1670902155865215796953312290630024870723e27) }, + { SC_(0.38969058990478515625e2), SC_(0.1e1), SC_(0.2566138433684874120271097708825946441219e-1), SC_(0.37969058990478515625e2) }, + { SC_(0.38969058990478515625e2), SC_(0.3e1), SC_(0.1567112478553435406702123549078289771168e-4), SC_(0.504890624506973244356999863668988837162e5) }, + { SC_(0.38969058990478515625e2), SC_(0.5e1), SC_(0.8689906310391695151299331020686331039092e-8), SC_(0.5997424205505611045426742533651556939386e8) }, + { SC_(0.38969058990478515625e2), SC_(0.7e1), SC_(0.4394951869162062144764399567493282296601e-11), SC_(0.6321223889260369330291849413168960841557e11) }, + { SC_(0.38969058990478515625e2), SC_(0.9e1), SC_(0.2035525982765477083776753980340839550068e-14), SC_(0.5866813580649745354849775687876536135352e14) }, + { SC_(0.38969058990478515625e2), SC_(0.11e2), SC_(0.8665501322102869433082774775249219150551e-18), SC_(0.475351131137151882575690233470282894011e17) }, + { SC_(0.38969058990478515625e2), SC_(0.13e2), SC_(0.3402404204209133714893020990890855167026e-21), SC_(0.332917433408466146766602692025570579486e20) }, + { SC_(0.38969058990478515625e2), SC_(0.15e2), SC_(0.1236000924581999848584000411105138094148e-24), SC_(0.199246039489905250504763182886478199254e23) }, + { SC_(0.38969058990478515625e2), SC_(0.17e2), SC_(0.4166349010445106129065594246766652774957e-28), SC_(0.1005412674170810825848072062572878094182e26) }, + { SC_(0.38969058990478515625e2), SC_(0.19e2), SC_(0.1306678052162296296302493472338051227226e-31), SC_(0.4209988378715069570506244784182271066633e28) }, + { SC_(0.41507625579833984375e2), SC_(0.1e1), SC_(0.2409195867098306907239470727577262147923e-1), SC_(0.40507625579833984375e2) }, + { SC_(0.41507625579833984375e2), SC_(0.3e1), SC_(0.1302686404207614714166737816227289198388e-4), SC_(0.6162606769838344661754936382180858345237e5) }, + { SC_(0.41507625579833984375e2), SC_(0.5e1), SC_(0.6431633349331497141015690712861511428175e-8), SC_(0.8438545889861793470458005660841864423808e8) }, + { SC_(0.41507625579833984375e2), SC_(0.7e1), SC_(0.2910943566542678829029393631028762223452e-11), SC_(0.1033961398568079581975133990707297357778e12) }, + { SC_(0.41507625579833984375e2), SC_(0.9e1), SC_(0.1212136898003702865187467684297844048505e-14), SC_(0.1126245913457844277854195511466495940505e15) }, + { SC_(0.41507625579833984375e2), SC_(0.11e2), SC_(0.4659326979515309613033501220500527426672e-18), SC_(0.1082573300089022204314702243358580760484e18) }, + { SC_(0.41507625579833984375e2), SC_(0.13e2), SC_(0.1658384118269441042099831807050083163649e-21), SC_(0.91065236945021286603137668511485592393e20) }, + { SC_(0.41507625579833984375e2), SC_(0.15e2), SC_(0.5481194532589979478160202963103776716729e-25), SC_(0.6640129568173827446546480656426857387714e23) }, + { SC_(0.41507625579833984375e2), SC_(0.17e2), SC_(0.1686718935768932765905700917212028498685e-28), SC_(0.4150953075743104724995577406049683820437e26) }, + { SC_(0.41507625579833984375e2), SC_(0.19e2), SC_(0.4844596658382802531655518188204297321602e-32), SC_(0.2196272737681924152751776983696792699708e29) }, + { SC_(0.43424015045166015625e2), SC_(0.1e1), SC_(0.2302873188856175403626965536052891307432e-1), SC_(0.42424015045166015625e2) }, + { SC_(0.43424015045166015625e2), SC_(0.3e1), SC_(0.1141212965168497956273138117867728882609e-4), SC_(0.7104007410816445601492441497271101980004e5) }, + { SC_(0.43424015045166015625e2), SC_(0.5e1), SC_(0.5183530746499905821724585148177857891242e-8), SC_(0.1076135606728602311731733060546098971879e9) }, + { SC_(0.43424015045166015625e2), SC_(0.7e1), SC_(0.2165842442020049314291887953595061977946e-11), SC_(0.1466915836463339236767082142468157916416e12) }, + { SC_(0.43424015045166015625e2), SC_(0.9e1), SC_(0.8352634046179587916965001059409801159114e-15), SC_(0.1788811192909514734098773167283202538717e15) }, + { SC_(0.43424015045166015625e2), SC_(0.11e2), SC_(0.298233654616673156227132531689098798769e-18), SC_(0.1938607613675135351748920729783474442776e18) }, + { SC_(0.43424015045166015625e2), SC_(0.13e2), SC_(0.9887080919142083101358820841666548050753e-22), 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SC_(0.1852983963408025127050027652437224363032e-36), SC_(0.2370274853775499735793736259776542351768e37) }, + { SC_(0.274804931640625e3), SC_(0.1e1), SC_(0.3638944883666592336005387344183240775372e-2), SC_(0.273804931640625e3) }, + { SC_(0.274804931640625e3), SC_(0.3e1), SC_(0.4766500323634846786726418114758219069239e-7), SC_(0.2030256060269160057941917330026626586914e8) }, + { SC_(0.274804931640625e3), SC_(0.5e1), SC_(0.6154024546690911439951622301069331686792e-12), SC_(0.1483396562376533341738866147081245261408e13) }, + { SC_(0.274804931640625e3), SC_(0.7e1), SC_(0.7832477005907424889818045391405198637502e-17), SC_(0.1067856930955473531945270749186455957324e18) }, + { SC_(0.274804931640625e3), SC_(0.9e1), SC_(0.9827964089790961021148982989284943306226e-22), SC_(0.7573034896598354273614969311589563647814e22) }, + { SC_(0.274804931640625e3), SC_(0.11e2), SC_(0.121589512687081208208115625113120868527e-26), SC_(0.5290283392609316558714446432000706324911e27) }, + { SC_(0.274804931640625e3), SC_(0.13e2), SC_(0.1483336845938378842536785854889108110605e-31), SC_(0.3639906861316648922337742363934475695538e32) }, + { SC_(0.274804931640625e3), SC_(0.15e2), SC_(0.1784583808206143680516380935195677948469e-36), SC_(0.2466342921397522584830960161333299921649e37) }, + { SC_(0.28719293212890625e3), SC_(0.1e1), SC_(0.3481979840475847909058779449603208590017e-2), SC_(0.28619293212890625e3) }, + { SC_(0.28719293212890625e3), SC_(0.3e1), SC_(0.4177873173937715786615028868933817117241e-7), SC_(0.2319588437626001831290523114148527383804e8) }, + { SC_(0.28719293212890625e3), SC_(0.5e1), SC_(0.4944103935479594101728970184152086286306e-12), SC_(0.1853700117663054521198762979989883556474e13) }, + { SC_(0.28719293212890625e3), SC_(0.7e1), SC_(0.5771176926689689378272114553351095923104e-17), SC_(0.1460498293495902410425332026379780530262e18) }, + { SC_(0.28719293212890625e3), SC_(0.9e1), SC_(0.6645477608758422510175449153902904054049e-22), SC_(0.1134361728180550139843806606053711572247e23) }, + { SC_(0.28719293212890625e3), SC_(0.11e2), SC_(0.7549410333602551531197573935902332367997e-27), SC_(0.8684529178419808289808578708873987720728e27) }, + { SC_(0.28719293212890625e3), SC_(0.13e2), SC_(0.8461831166568759018593468368502015254964e-32), SC_(0.6552994599183951259948386008534097644671e32) }, + { SC_(0.28719293212890625e3), SC_(0.15e2), SC_(0.9358777379643248526116678325853529542453e-37), SC_(0.4872884452211613428565268037217322153101e37) }, + { SC_(0.2872945556640625e3), SC_(0.1e1), SC_(0.34807481739031415536832748605597495153e-2), SC_(0.2862945556640625e3) }, + { SC_(0.2872945556640625e3), SC_(0.3e1), SC_(0.4173456607681247504031635400359819979724e-7), SC_(0.2322068976483985302365908864885568618774e8) }, + { SC_(0.2872945556640625e3), SC_(0.5e1), SC_(0.493542599056095325492887019079469782537e-12), SC_(0.1857016860987350865488208769832611253273e13) }, + { SC_(0.2872945556640625e3), SC_(0.7e1), SC_(0.575704887226117173104059000343519557497e-17), SC_(0.1464171118451576900115689247386842164521e18) }, + { SC_(0.2872945556640625e3), SC_(0.9e1), SC_(0.6624639478608017189860202166580888604212e-22), SC_(0.1138043902274032262102313769657480009329e23) }, + { SC_(0.2872945556640625e3), SC_(0.11e2), SC_(0.7520584942137318140193092188213010036169e-27), SC_(0.871912063111857878002810464645372494142e27) }, + { SC_(0.2872945556640625e3), SC_(0.13e2), SC_(0.8423788942077451050455222863566182476902e-32), SC_(0.6583964742879698484706986690635311042633e32) }, + { SC_(0.2872945556640625e3), SC_(0.15e2), SC_(0.9310408458992971329277502791420183457416e-37), SC_(0.489956399730382818727393107483496092863e37) }, + { SC_(0.28788824462890625e3), SC_(0.1e1), SC_(0.3473570104569640055962131640759410960191e-2), SC_(0.28688824462890625e3) }, + { SC_(0.28788824462890625e3), SC_(0.3e1), SC_(0.4147778860826452996008042939379237489236e-7), SC_(0.2336595739914530255396130087319761514664e8) }, + { SC_(0.28788824462890625e3), SC_(0.5e1), SC_(0.4885092799811013921823782549177388581846e-12), SC_(0.1876488429114554752620527238325746098634e13) }, + { SC_(0.28788824462890625e3), SC_(0.7e1), SC_(0.5675297891024780138525697551464866109164e-17), SC_(0.1485786541228514073866161629333886138035e18) }, + { SC_(0.28788824462890625e3), SC_(0.9e1), SC_(0.6504343827584720578007790628197932503555e-22), SC_(0.115976844211059217259122774325677854357e23) }, + { SC_(0.28788824462890625e3), SC_(0.11e2), SC_(0.7354567247175934183847257682176508160719e-27), SC_(0.8923720939211117300320489938065609307863e27) }, + { SC_(0.28788824462890625e3), SC_(0.13e2), SC_(0.8205193867978273440011202644178884562983e-32), SC_(0.6767610329032719369025509390108872609635e32) }, + { SC_(0.28788824462890625e3), SC_(0.15e2), SC_(0.9033112602451792738749403271987921872272e-37), SC_(0.5058171670631253364671295649996237533421e37) }, + { SC_(0.289501678466796875e3), SC_(0.1e1), SC_(0.3454211406635041713966881609665888745256e-2), SC_(0.288501678466796875e3) }, + { SC_(0.289501678466796875e3), SC_(0.3e1), SC_(0.407905168480763018987021853172545228673e-7), SC_(0.2376380058306495433427585339813958853483e8) }, + { SC_(0.289501678466796875e3), SC_(0.5e1), SC_(0.4751385373606723027962691314425005391974e-12), SC_(0.1930231496923894088462545699714651073089e13) }, + { SC_(0.289501678466796875e3), SC_(0.7e1), SC_(0.5459747147595941934931644310213716724959e-17), SC_(0.154591661548185942785563127012845776958e18) }, + { SC_(0.289501678466796875e3), SC_(0.9e1), SC_(0.6189505505380654067781824225524105132565e-22), SC_(0.1220681936608413875283547987492080381155e23) }, + { SC_(0.289501678466796875e3), SC_(0.11e2), SC_(0.6923248423462699258122608833489787193542e-27), SC_(0.9501994073223823891184486514363262976698e27) }, + { SC_(0.289501678466796875e3), SC_(0.13e2), SC_(0.7641406071861718441256397914223803198376e-32), SC_(0.7290849633976240067942394862692220173534e32) }, + { SC_(0.289501678466796875e3), SC_(0.15e2), SC_(0.8323086212678717271324694441691502737994e-37), SC_(0.551375411474552213507390248281996458309e37) }, + { SC_(0.290340789794921875e3), SC_(0.1e1), SC_(0.3444228421043890893385637226929434835625e-2), SC_(0.289340789794921875e3) }, + { SC_(0.290340789794921875e3), SC_(0.3e1), SC_(0.404390772308938863697334002937411013455e-7), SC_(0.2397248344799799768131265409465413540602e8) }, + { SC_(0.290340789794921875e3), SC_(0.5e1), SC_(0.4683584166674361180389483263241938077867e-12), SC_(0.1958664738832016351565037051374224319343e13) }, + { SC_(0.290340789794921875e3), SC_(0.7e1), SC_(0.5351351314668937271340160893033367766944e-17), SC_(0.1578004983694902935556729547125288747567e18) }, + { SC_(0.290340789794921875e3), SC_(0.9e1), SC_(0.6032486205167581064858567225995659032751e-22), SC_(0.1253472175648567782767177252957740552768e23) }, + { SC_(0.290340789794921875e3), SC_(0.11e2), SC_(0.6709901097200500439708799760438557985054e-27), SC_(0.9816018026087682770199262928304104956695e27) }, + { SC_(0.290340789794921875e3), SC_(0.13e2), SC_(0.7364808408015348014875920182650010471344e-32), SC_(0.757750009443894137565133378045564307973e32) }, + { SC_(0.290340789794921875e3), SC_(0.15e2), SC_(0.7977567303667583794462380205536503660325e-37), SC_(0.5765560036226951637956770556326362477979e37) }, + { SC_(0.29069146728515625e3), SC_(0.1e1), SC_(0.3440073454302810870077806885983947303719e-2), SC_(0.28969146728515625e3) }, + { SC_(0.29069146728515625e3), SC_(0.3e1), SC_(0.4029340128900865341649997325071470796219e-7), SC_(0.2406005592837546327814379765186458826065e8) }, + { SC_(0.29069146728515625e3), SC_(0.5e1), SC_(0.4655593347102129154770919320328943988628e-12), SC_(0.197064624191144911113364924372333547971e13) }, + { SC_(0.29069146728515625e3), SC_(0.7e1), SC_(0.5306781300850624992667955983404647577365e-17), SC_(0.1591583373838064839072653291218129731538e18) }, + { SC_(0.29069146728515625e3), SC_(0.9e1), SC_(0.5968181020379241221614164786831856442513e-22), SC_(0.1267406078595540656088386383125599592152e23) }, + { SC_(0.29069146728515625e3), SC_(0.11e2), SC_(0.6622874185773110848120483926856882408505e-27), SC_(0.9950025958171749642234595698560649206203e27) }, + { SC_(0.29069146728515625e3), SC_(0.13e2), SC_(0.7252425880730941952837038289017818623456e-32), SC_(0.7700349214883739827198954590726953390616e32) }, + { SC_(0.29069146728515625e3), SC_(0.15e2), SC_(0.7837732137791901630412251772353086649926e-37), SC_(0.5873940084556647854075928357945884520601e37) }, + { SC_(0.291207244873046875e3), SC_(0.1e1), SC_(0.3433980498788601758552647643188802171892e-2), SC_(0.290207244873046875e3) }, + { SC_(0.291207244873046875e3), SC_(0.3e1), SC_(0.4008040930623875410783774854230681603441e-7), SC_(0.2418924493681085849559053713164757937193e8) }, + { SC_(0.291207244873046875e3), SC_(0.5e1), SC_(0.4614788198337042276507609088027282453366e-12), SC_(0.1988375146420684092396039255743712842102e13) }, + { SC_(0.291207244873046875e3), SC_(0.7e1), SC_(0.5241996237359538672911416208963837661703e-17), SC_(0.1611736435885476675751117506657723086633e18) }, + { SC_(0.291207244873046875e3), SC_(0.9e1), SC_(0.5874980228979750505195933021348442376062e-22), SC_(0.1288150308509289144242116755013782892893e23) }, + { SC_(0.291207244873046875e3), SC_(0.11e2), SC_(0.6497104124488598123554519089149390087186e-27), SC_(0.1015014875890825032948492161856198979056e28) }, + { SC_(0.291207244873046875e3), SC_(0.13e2), SC_(0.7090475834671838093657350842661830590173e-32), SC_(0.7884379604158280127712554109520427430725e32) }, + { SC_(0.291207244873046875e3), SC_(0.15e2), SC_(0.7636792489590183720543570028907531957389e-37), SC_(0.6036805286139762775881723279121968447932e37) }, + { SC_(0.294351806640625e3), SC_(0.1e1), SC_(0.33972952685862159057449623236968809744e-2), SC_(0.293351806640625e3) }, + { SC_(0.294351806640625e3), SC_(0.3e1), SC_(0.3881379282177214776636623282889873598749e-7), SC_(0.2498689343664573460409883409738540649414e8) }, + { SC_(0.294351806640625e3), SC_(0.5e1), SC_(0.4375088373240974671031998487110516857923e-12), SC_(0.2099244362868845821585772253776470963604e13) }, + { SC_(0.294351806640625e3), SC_(0.7e1), SC_(0.4866029054251045038186644286072296391012e-17), SC_(0.1739400555379920223170681407628313381261e18) }, + { SC_(0.294351806640625e3), SC_(0.9e1), SC_(0.534057883457178772917208250643286627359e-22), SC_(0.1421281681033299298280403078077015645206e23) }, + { SC_(0.294351806640625e3), SC_(0.11e2), SC_(0.5784500466126953910491646789719517325936e-27), SC_(0.1145149364382182705993366363639828841506e28) }, + { SC_(0.294351806640625e3), SC_(0.13e2), SC_(0.6183650292423694233820451668640331858573e-32), SC_(0.9097088412307421913798995051293929461461e32) }, + { SC_(0.294351806640625e3), SC_(0.15e2), SC_(0.6524731626576039436199662440884420606257e-37), SC_(0.7124547053047129902329409548948893697576e37) }, + { SC_(0.29787152099609375e3), SC_(0.1e1), SC_(0.3357152092472492109647570746796270004039e-2), SC_(0.29687152099609375e3) }, + { SC_(0.29787152099609375e3), SC_(0.3e1), SC_(0.3745857576153496103596756759785568454842e-7), SC_(0.2590028433522621662064011616166681051254e8) }, + { SC_(0.29787152099609375e3), SC_(0.5e1), SC_(0.4124278994306611649605113542682886173912e-12), SC_(0.2229149394453432862981100413241140703895e13) }, + { SC_(0.29787152099609375e3), SC_(0.7e1), SC_(0.4481254385408609056603897429430645730762e-17), SC_(0.1892483485866810077637251708385021969851e18) }, + { SC_(0.29787152099609375e3), SC_(0.9e1), SC_(0.4805556612415073905868174503601992803965e-22), SC_(0.1584682915857995126890517336563712708005e23) }, + { SC_(0.29787152099609375e3), SC_(0.11e2), SC_(0.5086482969169712932808559443461158258716e-27), SC_(0.1308665081238300833231561815310422586917e28) }, + { SC_(0.29787152099609375e3), SC_(0.13e2), SC_(0.5314446802305598153808326995941985254172e-32), SC_(0.1065733067256237279284265031964211394528e33) }, + { SC_(0.29787152099609375e3), SC_(0.15e2), SC_(0.5481525108002032281510522944958079149289e-37), SC_(0.8557747958232836453478265781438336038069e37) }, + { SC_(0.298941925048828125e3), SC_(0.1e1), SC_(0.3345131332236732968010854526498232035615e-2), SC_(0.297941925048828125e3) }, + { SC_(0.298941925048828125e3), SC_(0.3e1), SC_(0.3705896653431679485941827175484149138275e-7), SC_(0.2618241086285098069268428844225127249956e8) }, + { SC_(0.298941925048828125e3), SC_(0.5e1), SC_(0.4051450186662736366284562621407566077077e-12), SC_(0.2269904983065584876428858402786024561227e13) }, + { SC_(0.298941925048828125e3), SC_(0.7e1), SC_(0.4371220920128659338444328164220492742505e-17), SC_(0.1941268809835496579746004922828460068201e18) }, + { SC_(0.298941925048828125e3), SC_(0.9e1), SC_(0.4654869913634073404921977229252675829028e-22), SC_(0.1637581799972646839641245577548494961723e23) }, + { SC_(0.298941925048828125e3), SC_(0.11e2), SC_(0.489284975257170578658380454302158217284e-27), SC_(0.1362443397656620723242541241397155077558e28) }, + { SC_(0.298941925048828125e3), SC_(0.13e2), SC_(0.5076942804322773765508159231416954589399e-32), SC_(0.111786745604799935748770629080275328159e33) }, + { SC_(0.298941925048828125e3), SC_(0.15e2), SC_(0.5200735573667448543558639259673314877186e-37), SC_(0.9044325612703634083164448623285298649691e37) } + } }; +#undef SC_ + diff --git a/test/tgamma_delta_ratio_int2.ipp b/test/tgamma_delta_ratio_int2.ipp new file mode 100644 index 000000000..eae1564e2 --- /dev/null +++ b/test/tgamma_delta_ratio_int2.ipp @@ -0,0 +1,200 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 190> tgamma_delta_ratio_int2 = { { + { SC_(0.11e2), SC_(0.1e2), SC_(0.1491552059147518383844493235618106619139e-11), SC_(0.36288e7) }, + { SC_(0.12e2), SC_(0.1e2), SC_(0.7812891738391762962994964567523415624059e-12), SC_(0.399168e8) }, + { SC_(0.12e2), SC_(0.11e2), SC_(0.3551314426541710437724983894328825283663e-13), SC_(0.399168e8) }, + { SC_(0.13e2), SC_(0.1e2), SC_(0.4261577311850052525269980673194590340396e-12), SC_(0.2395008e9) }, + { SC_(0.13e2), SC_(0.11e2), SC_(0.1852859700804370663160861162258517539303e-13), SC_(0.4790016e9) }, + { SC_(0.13e2), SC_(0.12e2), SC_(0.7720248753351544429836921509410489747094e-15), SC_(0.4790016e9) }, + { SC_(0.14e2), SC_(0.1e2), SC_(0.2408717611045681862109119510936072801093e-12), SC_(0.10378368e10) }, + { SC_(0.14e2), SC_(0.11e2), SC_(0.1003632337935700775878799796223363667122e-13), SC_(0.31135104e10) }, + { SC_(0.14e2), SC_(0.12e2), SC_(0.4014529351742803103515199184893454668489e-15), SC_(0.62270208e10) }, + { SC_(0.14e2), SC_(0.13e2), SC_(0.1544049750670308885967384301882097949419e-16), SC_(0.62270208e10) }, + { SC_(0.15e2), SC_(0.1e2), SC_(0.1405085273109981086230319714712709133971e-12), SC_(0.36324288e10) }, + { SC_(0.15e2), SC_(0.11e2), SC_(0.5620341092439924344921278858850836535885e-14), SC_(0.145297152e11) }, + { SC_(0.15e2), SC_(0.12e2), SC_(0.2161669650938432440354338022634937129186e-15), SC_(0.435891456e11) }, + { SC_(0.15e2), SC_(0.13e2), SC_(0.8006183892364564593904955639388656034024e-17), SC_(0.871782912e11) }, + { SC_(0.15e2), SC_(0.14e2), SC_(0.2859351390130201640680341299781662869294e-18), SC_(0.871782912e11) }, + { SC_(0.16e2), SC_(0.1e2), SC_(0.8430511638659886517381918288276254803827e-13), SC_(0.108972864e11) }, + { SC_(0.16e2), SC_(0.11e2), SC_(0.324250447640764866053150703395240569378e-14), SC_(0.54486432e11) }, + { SC_(0.16e2), SC_(0.12e2), SC_(0.1200927583854684689085743345908298405104e-15), SC_(0.217945728e12) }, + { SC_(0.16e2), SC_(0.13e2), SC_(0.4289027085195302461020511949672494303941e-17), SC_(0.653837184e12) }, + { SC_(0.16e2), SC_(0.14e2), SC_(0.1478974856963897400351900672300860104807e-18), SC_(0.1307674368e13) }, + { SC_(0.16e2), SC_(0.15e2), SC_(0.4929916189879658001173002241002867016024e-20), SC_(0.1307674368e13) }, + { SC_(0.17e2), SC_(0.1e2), SC_(0.5188007162252237856850411254323849110047e-13), SC_(0.290594304e11) }, + { SC_(0.17e2), SC_(0.11e2), SC_(0.1921484134167495502537189353453277448166e-14), SC_(0.1743565824e12) }, + { SC_(0.17e2), SC_(0.12e2), SC_(0.6862443336312483937632819119475990886306e-16), SC_(0.871782912e12) }, + { SC_(0.17e2), SC_(0.13e2), SC_(0.2366359771142235840563041075681376167692e-17), SC_(0.3487131648e13) }, + { SC_(0.17e2), SC_(0.14e2), SC_(0.7887865903807452801876803585604587225639e-19), SC_(0.10461394944e14) }, + { SC_(0.17e2), SC_(0.15e2), SC_(0.254447287219595251673445276954986684698e-20), SC_(0.20922789888e14) }, + { SC_(0.17e2), SC_(0.16e2), SC_(0.7951477725612351614795164904843333896814e-22), SC_(0.20922789888e14) }, + { SC_(0.18e2), SC_(0.1e2), SC_(0.3266523028084742354313221900870571661882e-13), SC_(0.705729024e11) }, + { SC_(0.18e2), SC_(0.11e2), SC_(0.1166615367173122269397579250310918450672e-14), SC_(0.4940103168e12) }, + { SC_(0.18e2), SC_(0.12e2), SC_(0.4022811610941800928957169828658339485076e-16), SC_(0.29640619008e13) }, + { SC_(0.18e2), SC_(0.13e2), SC_(0.1340937203647266976319056609552779828359e-17), SC_(0.14820309504e14) }, + { SC_(0.18e2), SC_(0.14e2), SC_(0.4325603882733119278448569708234773639867e-19), SC_(0.59281238016e14) }, + { SC_(0.18e2), SC_(0.15e2), SC_(0.1351751213354099774515178033823366762458e-20), SC_(0.177843714048e15) }, + { SC_(0.18e2), SC_(0.16e2), SC_(0.4096215798042726589439933435828384128662e-22), SC_(0.355687428096e15) }, + { SC_(0.18e2), SC_(0.17e2), SC_(0.1204769352365507820423509834067171802548e-23), SC_(0.355687428096e15) }, + { SC_(0.19e2), SC_(0.1e2), SC_(0.209990766091162008491564265055965321121e-13), SC_(0.1587890304e12) }, + { SC_(0.19e2), SC_(0.11e2), SC_(0.7241060899695241672122905691585011073137e-15), SC_(0.12703122432e13) }, + { SC_(0.19e2), SC_(0.12e2), SC_(0.2413686966565080557374301897195003691046e-16), SC_(0.88921857024e13) }, + { SC_(0.19e2), SC_(0.13e2), SC_(0.778608698891961470120742547482259255176e-18), SC_(0.533531142144e14) }, + { SC_(0.19e2), SC_(0.14e2), SC_(0.2433152184037379594127320460882060172425e-19), SC_(0.266765571072e15) }, + { SC_(0.19e2), SC_(0.15e2), SC_(0.7373188436476907860991880184491091431591e-21), SC_(0.1067062284288e16) }, + { SC_(0.19e2), SC_(0.16e2), SC_(0.2168584834257914076762317701320909244586e-22), SC_(0.3201186852864e16) }, + { SC_(0.19e2), SC_(0.17e2), SC_(0.6195956669308325933606622003774026413102e-24), SC_(0.6402373705728e16) }, + { SC_(0.19e2), SC_(0.18e2), SC_(0.1721099074807868314890728334381674003639e-25), SC_(0.6402373705728e16) }, + { SC_(0.2e2), SC_(0.1e2), SC_(0.1375801570942095917703352081401152103896e-13), SC_(0.3352212864e12) }, + { SC_(0.2e2), SC_(0.11e2), SC_(0.4586005236473653059011173604670507012987e-15), SC_(0.30169915776e13) }, + { SC_(0.2e2), SC_(0.12e2), SC_(0.1479356527894726793229410840216292584834e-16), SC_(0.241359326208e14) }, + { SC_(0.2e2), SC_(0.13e2), SC_(0.4622989149671021228841908875675914327607e-18), SC_(0.1689515283456e15) }, + { SC_(0.2e2), SC_(0.14e2), SC_(0.1400905802930612493588457235053307372002e-19), SC_(0.10137091700736e16) }, + { SC_(0.2e2), SC_(0.15e2), SC_(0.4120311185090036745848403632509727564713e-21), SC_(0.5068545850368e16) }, + { SC_(0.2e2), SC_(0.16e2), SC_(0.1177231767168581927385258180717065018489e-22), SC_(0.20274183401472e17) }, + { SC_(0.2e2), SC_(0.17e2), SC_(0.3270088242134949798292383835325180606915e-24), SC_(0.60822550204416e17) }, + { SC_(0.2e2), SC_(0.18e2), SC_(0.8838076330094458914303740095473461099769e-26), SC_(0.121645100408832e18) }, + { SC_(0.2e2), SC_(0.19e2), SC_(0.2325809560551173398500984235650910815729e-27), SC_(0.121645100408832e18) }, + { SC_(0.21e2), SC_(0.1e2), SC_(0.9172010472947306118022347209341014025973e-14), SC_(0.6704425728e12) }, + { SC_(0.21e2), SC_(0.11e2), SC_(0.2958713055789453586458821680432585169669e-15), SC_(0.6704425728e13) }, + { SC_(0.21e2), SC_(0.12e2), SC_(0.9245978299342042457683817751351828655215e-17), SC_(0.60339831552e14) }, + { SC_(0.21e2), SC_(0.13e2), SC_(0.2801811605861224987176914470106614744005e-18), SC_(0.482718652416e15) }, + { SC_(0.21e2), SC_(0.14e2), SC_(0.8240622370180073491696807265019455129425e-20), SC_(0.3379030566912e16) }, + { SC_(0.21e2), SC_(0.15e2), SC_(0.2354463534337163854770516361434130036979e-21), SC_(0.20274183401472e17) }, + { SC_(0.21e2), SC_(0.16e2), SC_(0.6540176484269899596584767670650361213829e-23), SC_(0.10137091700736e18) }, + { SC_(0.21e2), SC_(0.17e2), SC_(0.1767615266018891782860748019094692219954e-24), SC_(0.40548366802944e18) }, + { SC_(0.21e2), SC_(0.18e2), SC_(0.4651619121102346797001968471301821631458e-26), SC_(0.121645100408832e19) }, + { SC_(0.21e2), SC_(0.19e2), SC_(0.1192722851564704306923581659308159392681e-27), SC_(0.243290200817664e19) }, + { SC_(0.21e2), SC_(0.2e2), SC_(0.2981807128911760767308954148270398481704e-29), SC_(0.243290200817664e19) }, + { SC_(0.22e2), SC_(0.1e2), SC_(0.6213297417157852531563525528908428856304e-14), SC_(0.12799358208e13) }, + { SC_(0.22e2), SC_(0.11e2), SC_(0.1941655442861828916113601727783884017595e-15), SC_(0.140792940288e14) }, + { SC_(0.22e2), SC_(0.12e2), SC_(0.5883804372308572473071520387223890962409e-17), SC_(0.140792940288e15) }, + { SC_(0.22e2), SC_(0.13e2), SC_(0.1730530697737815433256329525654085577179e-18), SC_(0.1267136462592e16) }, + { SC_(0.22e2), SC_(0.14e2), SC_(0.4944373422108044095018084359011673077655e-20), SC_(0.10137091700736e17) }, + { SC_(0.22e2), SC_(0.15e2), SC_(0.1373437061696678915282801210836575854904e-21), SC_(0.70959641905152e17) }, + { SC_(0.22e2), SC_(0.16e2), SC_(0.3711992058639672744007570840098853661903e-23), SC_(0.425757851430912e18) }, + { SC_(0.22e2), SC_(0.17e2), SC_(0.9768400154314928273704133789733825426061e-25), SC_(0.212878925715456e19) }, + { SC_(0.22e2), SC_(0.18e2), SC_(0.2504717988285879044539521484547134724631e-26), SC_(0.851515702861824e19) }, + { SC_(0.22e2), SC_(0.19e2), SC_(0.6261794970714697611348803711367836811578e-28), SC_(0.2554547108585472e20) }, + { SC_(0.22e2), SC_(0.2e2), SC_(0.1527267066027975027158244807650691905263e-29), SC_(0.5109094217170944e20) }, + { SC_(0.22e2), SC_(0.21e2), SC_(0.3636350157209464350376773351549266441102e-31), SC_(0.5109094217170944e20) }, + { SC_(0.23e2), SC_(0.1e2), SC_(0.4271641974296023615449923801124544838709e-14), SC_(0.23465490048e13) }, + { SC_(0.23e2), SC_(0.11e2), SC_(0.129443696190788594407573448518925601173e-15), SC_(0.281585880576e14) }, + { SC_(0.23e2), SC_(0.12e2), SC_(0.3807167535023193953163924956438988269794e-17), SC_(0.3097444686336e15) }, + { SC_(0.23e2), SC_(0.13e2), SC_(0.1087762152863769700903978558982568077084e-18), SC_(0.3097444686336e16) }, + { SC_(0.23e2), SC_(0.14e2), SC_(0.3021561535732693613622162663840466880789e-20), SC_(0.27877002177024e17) }, + { SC_(0.23e2), SC_(0.15e2), SC_(0.8166382529007280036816655848217478056187e-22), SC_(0.223016017416192e18) }, + { SC_(0.23e2), SC_(0.16e2), SC_(0.2149048033949284220214909433741441593733e-23), SC_(0.1561112121913344e19) }, + { SC_(0.23e2), SC_(0.17e2), SC_(0.5510379574228933897986947266003696394188e-25), SC_(0.9366672731480064e19) }, + { SC_(0.23e2), SC_(0.18e2), SC_(0.1377594893557233474496736816500924098547e-26), SC_(0.4683336365740032e20) }, + { SC_(0.23e2), SC_(0.19e2), SC_(0.3359987545261545059748138576831522191578e-28), SC_(0.18733345462960128e21) }, + { SC_(0.23e2), SC_(0.2e2), SC_(0.7999970345860821570828901373408386170424e-30), SC_(0.56200036388880384e21) }, + { SC_(0.23e2), SC_(0.21e2), SC_(0.1860458219967632923448581714746136318703e-31), SC_(0.112400072777760768e22) }, + { SC_(0.23e2), SC_(0.22e2), SC_(0.4228314136290074826019503897150309815235e-33), SC_(0.112400072777760768e22) }, + { SC_(0.24e2), SC_(0.1e2), SC_(0.2977205012388137671374189315935288826979e-14), SC_(0.41515867008e13) }, + { SC_(0.24e2), SC_(0.11e2), SC_(0.8756485330553346092277027399809673020527e-16), SC_(0.539706271104e14) }, + { SC_(0.24e2), SC_(0.12e2), SC_(0.2501852951586670312079150685659906577293e-17), SC_(0.6476475253248e15) }, + { SC_(0.24e2), SC_(0.13e2), SC_(0.6949591532185195311330974126833073825815e-19), SC_(0.71241227785728e16) }, + { SC_(0.24e2), SC_(0.14e2), SC_(0.1878267981671674408467830845090019952923e-20), SC_(0.71241227785728e17) }, + { SC_(0.24e2), SC_(0.15e2), SC_(0.4942810478083353706494291697605315665587e-22), SC_(0.641171050071552e18) }, + { SC_(0.24e2), SC_(0.16e2), SC_(0.1267387302072654796536997871180850170663e-23), SC_(0.5129368400572416e19) }, + { SC_(0.24e2), SC_(0.17e2), SC_(0.3168468255181636991342494677952125426658e-25), SC_(0.35905578804006912e20) }, + { SC_(0.24e2), SC_(0.18e2), SC_(0.772797135410155363742071872671250104063e-27), SC_(0.215433472824041472e21) }, + { SC_(0.24e2), SC_(0.19e2), SC_(0.1839993179547988961290647315883928819198e-28), SC_(0.107716736412020736e22) }, + { SC_(0.24e2), SC_(0.2e2), SC_(0.4279053905925555723931737943916113533018e-30), SC_(0.430866945648082944e22) }, + { SC_(0.24e2), SC_(0.21e2), SC_(0.972512251346717209984485896344571257504e-32), SC_(0.1292600836944248832e23) }, + { SC_(0.24e2), SC_(0.22e2), SC_(0.2161138336326038244409968658543491683342e-33), SC_(0.2585201673888497664e23) }, + { SC_(0.24e2), SC_(0.23e2), SC_(0.4698126818100083140021670996833677572483e-35), SC_(0.2585201673888497664e23) }, + { SC_(0.25e2), SC_(0.1e2), SC_(0.2101556479332803062146486575954321524926e-14), SC_(0.71170057728e13) }, + { SC_(0.25e2), SC_(0.11e2), SC_(0.6004447083808008748989961645583775785504e-16), SC_(0.996380808192e14) }, + { SC_(0.25e2), SC_(0.12e2), SC_(0.1667901967724446874719433790439937718196e-17), SC_(0.12952950506496e16) }, + { SC_(0.25e2), SC_(0.13e2), SC_(0.4507843156012018580322794028216047887015e-19), SC_(0.155435406077952e17) }, + { SC_(0.25e2), SC_(0.14e2), SC_(0.1186274514740004889558630007425275759741e-20), SC_(0.1709789466857472e18) }, + { SC_(0.25e2), SC_(0.15e2), SC_(0.3041729524974371511688794890834040409592e-22), SC_(0.1709789466857472e19) }, + { SC_(0.25e2), SC_(0.16e2), SC_(0.760432381243592877922198722708510102398e-24), SC_(0.15388105201717248e20) }, + { SC_(0.25e2), SC_(0.17e2), SC_(0.1854713124984372872980972494411000249751e-25), SC_(0.123104841613737984e21) }, + { SC_(0.25e2), SC_(0.18e2), SC_(0.4415983630915173507097553558121429166074e-27), SC_(0.861733891296165888e21) }, + { SC_(0.25e2), SC_(0.19e2), SC_(0.1026972937422133373743617106539867247924e-28), SC_(0.5170403347776995328e22) }, + { SC_(0.25e2), SC_(0.2e2), SC_(0.233402940323212130396276615122697101801e-30), SC_(0.2585201673888497664e23) }, + { SC_(0.25e2), SC_(0.21e2), SC_(0.5186732007182491786583924780504380040021e-32), SC_(0.10340806695553990656e24) }, + { SC_(0.25e2), SC_(0.22e2), SC_(0.1127550436344019953605201039240082617396e-33), SC_(0.31022420086661971968e24) }, + { SC_(0.25e2), SC_(0.23e2), SC_(0.2399043481583021177883406466468260888076e-35), SC_(0.62044840173323943936e24) }, + { SC_(0.25e2), SC_(0.24e2), SC_(0.4998007253297960787257096805142210183493e-37), SC_(0.62044840173323943936e24) }, + { SC_(0.26e2), SC_(0.1e2), SC_(0.1501111770952002187247490411395943946376e-14), SC_(0.11861676288e14) }, + { SC_(0.26e2), SC_(0.11e2), SC_(0.4169754919311117186798584476099844295489e-16), SC_(0.17792514432e15) }, + { SC_(0.26e2), SC_(0.12e2), SC_(0.1126960789003004645080698507054011971754e-17), SC_(0.249095202048e16) }, + { SC_(0.26e2), SC_(0.13e2), SC_(0.2965686286850012223896575018563189399352e-19), SC_(0.3238237626624e17) }, + { SC_(0.26e2), SC_(0.14e2), SC_(0.760432381243592877922198722708510102398e-21), SC_(0.38858851519488e18) }, + { SC_(0.26e2), SC_(0.15e2), SC_(0.1901080953108982194805496806771275255995e-22), SC_(0.427447366714368e19) }, + { SC_(0.26e2), SC_(0.16e2), SC_(0.4636782812460932182452431236027500624378e-24), SC_(0.427447366714368e20) }, + { SC_(0.26e2), SC_(0.17e2), SC_(0.1103995907728793376774388389530357291519e-25), SC_(0.3847026300429312e21) }, + { SC_(0.26e2), SC_(0.18e2), SC_(0.2567432343555333434359042766349668119811e-27), SC_(0.30776210403434496e22) }, + { SC_(0.26e2), SC_(0.19e2), SC_(0.5835073508080303259906915378067427545024e-29), SC_(0.215433472824041472e23) }, + { SC_(0.26e2), SC_(0.2e2), SC_(0.1296683001795622946645981195126095010005e-30), SC_(0.1292600836944248832e24) }, + { SC_(0.26e2), SC_(0.21e2), SC_(0.281887609086004988401300259810020654349e-32), SC_(0.646300418472124416e24) }, + { SC_(0.26e2), SC_(0.22e2), SC_(0.5997608703957552944708516166170652220191e-34), SC_(0.2585201673888497664e25) }, + { SC_(0.26e2), SC_(0.23e2), SC_(0.1249501813324490196814274201285552545873e-35), SC_(0.7755605021665492992e25) }, + { SC_(0.26e2), SC_(0.24e2), SC_(0.2550003700662224891457702451603168460966e-37), SC_(0.15511210043330985984e26) }, + { SC_(0.26e2), SC_(0.25e2), SC_(0.5100007401324449782915404903206336921931e-39), SC_(0.15511210043330985984e26) }, + { SC_(0.27e2), SC_(0.1e2), SC_(0.1084136279020890468567631963785959516827e-14), SC_(0.19275223968e14) }, + { SC_(0.27e2), SC_(0.11e2), SC_(0.293009805140781207720981611834043112656e-16), SC_(0.308403583488e15) }, + { SC_(0.27e2), SC_(0.12e2), SC_(0.7710784345810031782131095048264292438315e-18), SC_(0.462605375232e16) }, + { SC_(0.27e2), SC_(0.13e2), SC_(0.1977124191233341482597716679042126266235e-19), SC_(0.6476475253248e17) }, + { SC_(0.27e2), SC_(0.14e2), SC_(0.4942810478083353706494291697605315665587e-21), SC_(0.84194178292224e18) }, + { SC_(0.27e2), SC_(0.15e2), SC_(0.1205563531239842367437632121367150162338e-22), SC_(0.1010330139506688e20) }, + { SC_(0.27e2), SC_(0.16e2), SC_(0.2870389360094862779613409812778928957948e-24), SC_(0.11113631534573568e21) }, + { SC_(0.27e2), SC_(0.17e2), SC_(0.6675324093243866929333511192509137111507e-26), SC_(0.11113631534573568e22) }, + { SC_(0.27e2), SC_(0.18e2), SC_(0.1517119112100878847575797998297531161706e-27), SC_(0.100022683811162112e23) }, + { SC_(0.27e2), SC_(0.19e2), SC_(0.3371375804668619661279551107327847026014e-29), SC_(0.800181470489296896e23) }, + { SC_(0.27e2), SC_(0.2e2), SC_(0.7329077836236129698433806755060537013074e-31), SC_(0.5601270293425078272e24) }, + { SC_(0.27e2), SC_(0.21e2), SC_(0.155937826302896376562421420320436957725e-32), SC_(0.33607621760550469632e25) }, + { SC_(0.27e2), SC_(0.22e2), SC_(0.324870471464367451171711292334243661927e-34), SC_(0.16803810880275234816e26) }, + { SC_(0.27e2), SC_(0.23e2), SC_(0.6630009621721784717790026374168237998511e-36), SC_(0.67215243521100939264e26) }, + { SC_(0.27e2), SC_(0.24e2), SC_(0.1326001924344356943558005274833647599702e-37), SC_(0.201645730563302817792e27) }, + { SC_(0.27e2), SC_(0.25e2), SC_(0.2600003773224229301094127989869897254318e-39), SC_(0.403291461126605635584e27) }, + { SC_(0.27e2), SC_(0.26e2), SC_(0.5000007256200440963642553826672879335227e-41), SC_(0.403291461126605635584e27) }, + { SC_(0.28e2), SC_(0.1e2), SC_(0.7911264738801092608466503519519164041712e-15), SC_(0.30613591008e14) }, + { SC_(0.28e2), SC_(0.11e2), SC_(0.2081911773368708581175395663031358958345e-16), SC_(0.520431047136e15) }, + { SC_(0.28e2), SC_(0.12e2), SC_(0.5338235316330022003013835033413740918834e-18), SC_(0.8326896754176e16) }, + { SC_(0.28e2), SC_(0.13e2), SC_(0.1334558829082505500753458758353435229708e-19), SC_(0.12490345131264e18) }, + { SC_(0.28e2), SC_(0.14e2), SC_(0.3255021534347574392081606727691305438313e-21), SC_(0.174864831837696e19) }, + { SC_(0.28e2), SC_(0.15e2), SC_(0.775005127225612950495620649450310818646e-23), SC_(0.2273242813890048e20) }, + { SC_(0.28e2), SC_(0.16e2), SC_(0.1802337505175844070920048021977467020107e-24), SC_(0.27278913766680576e21) }, + { SC_(0.28e2), SC_(0.17e2), SC_(0.4096221602672372888454654595403334136607e-26), SC_(0.300068051433486336e22) }, + { SC_(0.28e2), SC_(0.18e2), SC_(0.9102714672605273085454787989785186970237e-28), SC_(0.300068051433486336e23) }, + { SC_(0.28e2), SC_(0.19e2), SC_(0.197885101578375501857712782386634499353e-29), SC_(0.2700612462901377024e24) }, + { SC_(0.28e2), SC_(0.2e2), SC_(0.4210321310178202167185378348651797858574e-31), SC_(0.21604899703211016192e25) }, + { SC_(0.28e2), SC_(0.21e2), SC_(0.877150272953792118163620489302457887203e-33), SC_(0.151234297922477113344e26) }, + { SC_(0.28e2), SC_(0.22e2), SC_(0.1790102597864881873803307121025424259598e-34), SC_(0.907405787534862680064e26) }, + { SC_(0.28e2), SC_(0.23e2), SC_(0.3580205195729763747606614242050848519196e-36), SC_(0.453702893767431340032e27) }, + { SC_(0.28e2), SC_(0.24e2), SC_(0.7020010187705419112954145572648722586658e-38), SC_(0.1814811575069725360128e28) }, + { SC_(0.28e2), SC_(0.25e2), SC_(0.1350001959174119060183489533201677420511e-39), SC_(0.5444434725209176080384e28) }, + { SC_(0.28e2), SC_(0.26e2), SC_(0.2547173507875696339968848175852221548134e-41), SC_(0.10888869450418352160768e29) }, + { SC_(0.28e2), SC_(0.27e2), SC_(0.4716987977547585814757126251578188052101e-43), SC_(0.10888869450418352160768e29) }, + { SC_(0.29e2), SC_(0.1e2), SC_(0.5829352965432384027291107856487805083366e-15), SC_(0.47621141568e14) }, + { SC_(0.29e2), SC_(0.11e2), SC_(0.1494705888572406160843873809355847457273e-16), SC_(0.857180548224e15) }, + { SC_(0.29e2), SC_(0.12e2), SC_(0.3736764721431015402109684523389618643184e-18), SC_(0.14572069319808e17) }, + { SC_(0.29e2), SC_(0.13e2), SC_(0.9114060296173208297828498837535655227277e-20), SC_(0.233153109116928e18) }, + { SC_(0.29e2), SC_(0.14e2), SC_(0.2170014356231716261387737818460870292209e-21), SC_(0.349729663675392e19) }, + { SC_(0.29e2), SC_(0.15e2), SC_(0.50465450144923633985761344615369076563e-23), SC_(0.4896215291455488e20) }, + { SC_(0.29e2), SC_(0.16e2), SC_(0.114694204874826440876730328671293355825e-24), SC_(0.63650798788921344e21) }, + { SC_(0.29e2), SC_(0.17e2), SC_(0.2548760108329476463927340637139852351666e-26), SC_(0.763809585467056128e22) }, + { SC_(0.29e2), SC_(0.18e2), SC_(0.5540782844194514052015957906825765981884e-28), SC_(0.8401905440137617408e23) }, + { SC_(0.29e2), SC_(0.19e2), SC_(0.1178889966849896606811905937622503400401e-29), SC_(0.8401905440137617408e24) }, + { SC_(0.29e2), SC_(0.2e2), SC_(0.2456020764270617930858137370046882084168e-31), SC_(0.75617148961238556672e25) }, + { SC_(0.29e2), SC_(0.21e2), SC_(0.5012287274021669246649259938871187926874e-33), SC_(0.604937191689908453376e26) }, + { SC_(0.29e2), SC_(0.22e2), SC_(0.1002457454804333849329851987774237585375e-34), SC_(0.4234560341829359173632e27) }, + { SC_(0.29e2), SC_(0.23e2), SC_(0.1965602852557517351627160760341642324264e-36), SC_(0.25407362050976155041792e28) }, + { SC_(0.29e2), SC_(0.24e2), SC_(0.3780005485687533368513770692964696777431e-38), SC_(0.12703681025488077520896e29) }, + { SC_(0.29e2), SC_(0.25e2), SC_(0.7132085822051949751912774892386220334776e-40), SC_(0.50814724101952310083584e29) }, + { SC_(0.29e2), SC_(0.26e2), SC_(0.1320756633713324028131995350441892654588e-41), SC_(0.152444172305856930250752e30) }, + { SC_(0.29e2), SC_(0.27e2), SC_(0.2401375697660589142058173364439804826524e-43), SC_(0.304888344611713860501504e30) }, + { SC_(0.29e2), SC_(0.28e2), SC_(0.4288170888679623467961023865071080047364e-45), SC_(0.304888344611713860501504e30) } + } }; +#undef SC_ + diff --git a/test/tgamma_ratio_data.ipp b/test/tgamma_ratio_data.ipp new file mode 100644 index 000000000..a5f86c315 --- /dev/null +++ b/test/tgamma_ratio_data.ipp @@ -0,0 +1,410 @@ +// (C) Copyright John Maddock 2006-7. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define SC_(x) static_cast(BOOST_JOIN(x, L)) + static const boost::array, 400> tgamma_ratio_data = { { + { SC_(6.68193912506103515625), SC_(6.68193912506103515625), SC_(1) }, + { SC_(6.68193912506103515625), SC_(8.095367431640625), SC_(0.06544775114440229981894259276990196062764) }, + { SC_(6.68193912506103515625), SC_(8.50289630889892578125), SC_(0.02833082889188931171456297874515804727699) }, + { SC_(6.68193912506103515625), SC_(11.04233455657958984375), SC_(0.9977723749373989179839784596883914860086e-4) }, + { SC_(6.68193912506103515625), SC_(12.6096343994140625), SC_(0.2224848351207471598094573384811702572789e-5) }, + { SC_(6.68193912506103515625), SC_(15.36791515350341796875), SC_(0.1707401795086213382244662934709192347257e-8) }, + { SC_(6.68193912506103515625), SC_(16.792018890380859375), SC_(0.3420632447590243481697274715444998893408e-10) }, + { SC_(6.68193912506103515625), SC_(28.25031280517578125), SC_(0.1600680453581408921824330663433115251581e-25) }, + { SC_(6.68193912506103515625), SC_(28.2665882110595703125), SC_(0.1516395254249637529260062571562632237052e-25) }, + { SC_(6.68193912506103515625), SC_(32.353244781494140625), SC_(0.1435206894862598198928388181910160004221e-31) }, + { SC_(6.68193912506103515625), SC_(41.1067352294921875), SC_(0.3302103464979977987816033282184687805192e-45) }, + { SC_(6.68193912506103515625), SC_(42.080410003662109375), SC_(0.886133585915892288509502556440259057933e-47) }, + { SC_(6.68193912506103515625), SC_(45.4780120849609375), SC_(0.2445790945282646979483178649067515339092e-52) }, + { SC_(6.68193912506103515625), SC_(45.842041015625), SC_(0.6109956711604874877647044786673395753644e-53) }, + { SC_(6.68193912506103515625), SC_(47.9603271484375), SC_(0.1802664125152357631935884543327977871009e-56) }, + { SC_(6.68193912506103515625), SC_(48.314647674560546875), SC_(0.4585435903890089500439815393256424830955e-57) }, + { SC_(6.68193912506103515625), SC_(48.4493560791015625), SC_(0.2722991298511892486629788976421558265475e-57) }, + { SC_(6.68193912506103515625), SC_(48.50565338134765625), SC_(0.2189817526247552395152959212887295282992e-57) }, + { SC_(6.68193912506103515625), SC_(49.65830230712890625), SC_(0.2491707234921651072352951178714074480602e-59) }, + { SC_(6.68193912506103515625), SC_(49.830142974853515625), SC_(0.1275499540450052064566893368701946482382e-59) }, + { SC_(8.095367431640625), SC_(6.68193912506103515625), SC_(15.27936380569631379215981282038521263139) }, + { SC_(8.095367431640625), SC_(8.095367431640625), SC_(1) }, + { SC_(8.095367431640625), SC_(8.50289630889892578125), SC_(0.4328770415561089543785983031850532551013) }, + { SC_(8.095367431640625), SC_(11.04233455657958984375), SC_(0.001524532711194214483439552686740177905313) }, + { SC_(8.095367431640625), SC_(12.6096343994140625), SC_(0.3399426737060256217390151315994565352827e-4) }, + { SC_(8.095367431640625), SC_(15.36791515350341796875), SC_(0.2608801318962120302585268540253614111335e-7) }, + { SC_(8.095367431640625), SC_(16.792018890380859375), SC_(0.5226508761230075927654898387420647235484e-9) }, + { SC_(8.095367431640625), SC_(28.25031280517578125), SC_(0.2445737898693733797769611259209203584443e-24) }, + { SC_(8.095367431640625), SC_(28.2665882110595703125), SC_(0.2316955476291157102884553711299427658036e-24) }, + { SC_(8.095367431640625), SC_(32.353244781494140625), SC_(0.2192904828304937772446028970624034599691e-30) }, + { SC_(8.095367431640625), SC_(41.1067352294921875), SC_(0.5045404016547946090253729607520301559963e-44) }, + { SC_(8.095367431640625), SC_(42.080410003662109375), SC_(0.1353955743965516944489997901950715452755e-45) }, + { SC_(8.095367431640625), SC_(45.4780120849609375), SC_(0.3737012964565144972084775063603050427547e-51) }, + { SC_(8.095367431640625), SC_(45.842041015625), SC_(0.9335625143366679579486394992496451351018e-52) }, + { SC_(8.095367431640625), SC_(47.9603271484375), SC_(0.2754356098768014320479195593046571558735e-55) }, + { SC_(8.095367431640625), SC_(48.314647674560546875), SC_(0.7006254338323859447417411411637098417379e-56) }, + { SC_(8.095367431640625), SC_(48.4493560791015625), SC_(0.4160557468970861681934763285488463143071e-56) }, + { SC_(8.095367431640625), SC_(48.50565338134765625), SC_(0.3345901865162628968218063609001765927876e-56) }, + { SC_(8.095367431640625), SC_(49.65830230712890625), SC_(0.3807170133965351751944945486685393449817e-58) }, + { SC_(8.095367431640625), SC_(49.830142974853515625), SC_(0.1948882151253480684761778785767440869907e-58) }, + { SC_(8.50289630889892578125), SC_(6.68193912506103515625), SC_(35.29723764228744086812043153239973434675) }, + { SC_(8.50289630889892578125), SC_(8.095367431640625), SC_(2.310124825297257793166027233518450467265) }, + { SC_(8.50289630889892578125), SC_(8.50289630889892578125), SC_(1) }, + { SC_(8.50289630889892578125), SC_(11.04233455657958984375), SC_(0.003521860863107489503907274588697768037321) }, + { SC_(8.50289630889892578125), SC_(12.6096343994140625), SC_(0.7853100097062151498541705753484156861086e-4) }, + { SC_(8.50289630889892578125), SC_(15.36791515350341796875), SC_(0.6026656691202623868522041531664940403622e-7) }, + { SC_(8.50289630889892578125), SC_(16.792018890380859375), SC_(0.1207388763895121639743664392369700074862e-8) }, + { SC_(8.50289630889892578125), SC_(28.25031280517578125), SC_(0.564995983594274416859732458938454292755e-24) }, + { SC_(8.50289630889892578125), SC_(28.2665882110595703125), SC_(0.5352456364888634023094937531252471776606e-24) }, + { SC_(8.50289630889892578125), SC_(32.353244781494140625), SC_(0.5065883883381457468092824152960142285326e-30) }, + { SC_(8.50289630889892578125), SC_(41.1067352294921875), SC_(0.1165551307228190672944688371949229156942e-43) }, + { SC_(8.50289630889892578125), SC_(42.080410003662109375), SC_(0.3127806776488558533920147278994474194652e-45) }, + { 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SC_(0.2988730812496183533966920676743957277123e57) }, + { SC_(48.50565338134765625), SC_(8.50289630889892578125), SC_(0.1293752952120933719032874782784950114307e57) }, + { SC_(48.50565338134765625), SC_(11.04233455657958984375), SC_(0.4556417888604494171080971046736770628338e54) }, + { SC_(48.50565338134765625), SC_(12.6096343994140625), SC_(0.1015997143387534962938471142842497266745e53) }, + { SC_(48.50565338134765625), SC_(15.36791515350341796875), SC_(0.779700488566277306704754402527940807414e49) }, + { SC_(48.50565338134765625), SC_(16.792018890380859375), SC_(0.1562062777646958653353788960127206831866e48) }, + { SC_(48.50565338134765625), SC_(28.25031280517578125), SC_(73096522171156316264505015125427.87849809) }, + { SC_(48.50565338134765625), SC_(28.2665882110595703125), SC_(69247562231731518727938759809179.17837236) }, + { SC_(48.50565338134765625), SC_(32.353244781494140625), SC_(65540022292266205196135588.37445817652441) }, + { SC_(48.50565338134765625), SC_(41.1067352294921875), SC_(1507935444574.885075047791039699544486501) }, + { SC_(48.50565338134765625), SC_(42.080410003662109375), SC_(40466092507.45934103380971787300376014527) }, + { SC_(48.50565338134765625), SC_(45.4780120849609375), SC_(111689.257938935572585547796542841786287) }, + { SC_(48.50565338134765625), SC_(45.842041015625), SC_(27901.67051989409614876831141855717974302) }, + { SC_(48.50565338134765625), SC_(47.9603271484375), SC_(8.23202894097474578256346862985747732831) }, + { SC_(48.50565338134765625), SC_(48.314647674560546875), SC_(2.093980822113357920271431337630760310643) }, + { SC_(48.50565338134765625), SC_(48.4493560791015625), SC_(1.243478630467434834721499987219186699646) }, + { SC_(48.50565338134765625), SC_(48.50565338134765625), SC_(1) }, + { SC_(48.50565338134765625), SC_(49.65830230712890625), SC_(0.01137860668779746965294028255799791422986) }, + { SC_(48.50565338134765625), SC_(49.830142974853515625), SC_(0.005824684135375125377888394010247691105308) }, + { SC_(49.65830230712890625), SC_(6.68193912506103515625), SC_(0.4013312583375967416661578325917486444011e60) }, + { SC_(49.65830230712890625), SC_(8.095367431640625), SC_(0.2626622832214886218015717184826554078257e59) }, + { SC_(49.65830230712890625), SC_(8.50289630889892578125), SC_(0.113700472089290789897629737177398002534e59) }, + { SC_(49.65830230712890625), SC_(11.04233455657958984375), SC_(0.4004372427681186817057187355081811104948e56) }, + { SC_(49.65830230712890625), SC_(12.6096343994140625), SC_(0.8929011884004219495315315213522833289702e54) }, + { SC_(49.65830230712890625), SC_(15.36791515350341796875), SC_(0.6852337109098215178887399704270903577856e51) }, + { SC_(49.65830230712890625), SC_(16.792018890380859375), SC_(0.1372806724501805853779867836918351364192e50) }, + { SC_(49.65830230712890625), SC_(28.25031280517578125), SC_(6424031006322219535859735319895730.406516) }, + { SC_(49.65830230712890625), SC_(28.2665882110595703125), SC_(6085768155251669725746553272367876.464595) }, + { SC_(49.65830230712890625), SC_(32.353244781494140625), SC_(5759933890900014436468813901.114329886265) }, + { SC_(49.65830230712890625), SC_(41.1067352294921875), SC_(132523733876135.2881084287595872245597653) }, + { SC_(49.65830230712890625), SC_(42.080410003662109375), SC_(3556331070908.319456261711716187098696346) }, + { SC_(49.65830230712890625), SC_(45.4780120849609375), SC_(9815723.577009849317698980162780336496513) }, + { SC_(49.65830230712890625), SC_(45.842041015625), SC_(2452116.61545662911113776272285650291177) }, + { SC_(49.65830230712890625), SC_(47.9603271484375), SC_(723.4654617074386650606404106270002904964) }, + { SC_(49.65830230712890625), SC_(48.314647674560546875), SC_(184.0278761334604933216410145833973305409) }, + { SC_(49.65830230712890625), SC_(48.4493560791015625), SC_(109.2821524274104329582796424993272683682) }, + { SC_(49.65830230712890625), SC_(48.50565338134765625), SC_(87.88422233386534838722699682176254064215) }, + { SC_(49.65830230712890625), SC_(49.65830230712890625), SC_(1) }, + { SC_(49.65830230712890625), SC_(49.830142974853515625), SC_(0.5118978355778457700907478004290928982057) }, + { SC_(49.830142974853515625), SC_(6.68193912506103515625), SC_(0.7840065545199304670108533691898249387968e60) }, + { SC_(49.830142974853515625), SC_(8.095367431640625), SC_(0.5131146587580068328804320686790697457432e59) }, + { SC_(49.830142974853515625), SC_(8.50289630889892578125), SC_(0.2221155554622383892333093987944533873854e59) }, + { SC_(49.830142974853515625), SC_(11.04233455657958984375), SC_(0.782260081869838348283677686535832364175e56) }, + { SC_(49.830142974853515625), SC_(12.6096343994140625), SC_(0.1744295690159517988946627267959914710007e55) }, + { SC_(49.830142974853515625), SC_(15.36791515350341796875), SC_(0.1338614198546686499416667263462438494101e52) }, + { SC_(49.830142974853515625), SC_(16.792018890380859375), SC_(0.26817982595143034220545161257463255727e50) }, + { SC_(49.830142974853515625), SC_(28.25031280517578125), SC_(12549439672997599030300754159544767.03381) }, + { SC_(49.830142974853515625), SC_(28.2665882110595703125), SC_(11888638185746322677793115992561779.63174) }, + { SC_(49.830142974853515625), SC_(32.353244781494140625), SC_(11252116126644737085315595239.69631493006) }, + { SC_(49.830142974853515625), SC_(41.1067352294921875), SC_(258887076024727.6417888699288378265039062) }, + { SC_(49.830142974853515625), SC_(42.080410003662109375), SC_(6947345395383.094961087609307748110446687) }, + { SC_(49.830142974853515625), SC_(45.4780120849609375), SC_(19175161.3208709184277483875675091000146) }, + { SC_(49.830142974853515625), SC_(45.842041015625), SC_(4790246.109731262408906051742019966433475) }, + { SC_(49.830142974853515625), SC_(47.9603271484375), SC_(1413.300489717384632443829918755401512936) }, + { SC_(49.830142974853515625), SC_(48.314647674560546875), SC_(359.5011803980852094949309930259792159491) }, + { SC_(49.830142974853515625), SC_(48.4493560791015625), SC_(213.4843025934060293962890224642100962277) }, + { SC_(49.830142974853515625), SC_(48.50565338134765625), SC_(171.6831293780700953348759427887881234575) }, + { SC_(49.830142974853515625), SC_(49.65830230712890625), SC_(1.953514804123306623532978094346942918559) }, + { SC_(49.830142974853515625), SC_(49.830142974853515625), SC_(1) } + } }; +#undef SC_ + diff --git a/tools/bessel_data.cpp b/tools/bessel_data.cpp new file mode 100644 index 000000000..d239fa75b --- /dev/null +++ b/tools/bessel_data.cpp @@ -0,0 +1,357 @@ +// Copyright (c) 2007 John Maddock +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Computes test data for the various bessel functions using +// archived - deliberately naive - version of the code. +// We'll rely on the high precision of boost::math::ntl::RR to get us out of +// trouble and not worry about how long the calculations take. +// This provides a reasonably independent set of test data to +// compare against newly added asymptotic expansions etc. +// +#include + +#include +#include "ntl_rr_lanczos.hpp" + +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace boost::math::detail; +using namespace std; + +// Compute J(v, x) and Y(v, x) simultaneously by Steed's method, see +// Barnett et al, Computer Physics Communications, vol 8, 377 (1974) +template +int bessel_jy_bare(T v, T x, T* J, T* Y, int kind = need_j|need_y) +{ + // Jv1 = J_(v+1), Yv1 = Y_(v+1), fv = J_(v+1) / J_v + // Ju1 = J_(u+1), Yu1 = Y_(u+1), fu = J_(u+1) / J_u + T u, Jv, Ju, Yv, Yv1, Yu, Yu1, fv, fu; + T W, p, q, gamma, current, prev, next; + bool reflect = false; + int n, k, s; + + using namespace std; + using namespace boost::math::tools; + using namespace boost::math::constants; + + if (v < 0) + { + reflect = true; + v = -v; // v is non-negative from here + kind = need_j|need_y; // need both for reflection formula + } + n = real_cast(v + 0.5L); + u = v - n; // -1/2 <= u < 1/2 + + if (x < 0) + { + *J = *Y = policies::raise_domain_error("", + "Real argument x=%1% must be non-negative, complex number result not supported", x, policies::policy<>()); + return 1; + } + if (x == 0) + { + *J = *Y = policies::raise_overflow_error( + "", 0, policies::policy<>()); + return 1; + } + + // x is positive until reflection + W = T(2) / (x * pi()); // Wronskian + if (x <= 2) // x in (0, 2] + { + if(temme_jy(u, x, &Yu, &Yu1, policies::policy<>())) // Temme series + { + // domain error: + *J = *Y = Yu; + return 1; + } + prev = Yu; + current = Yu1; + for (k = 1; k <= n; k++) // forward recurrence for Y + { + next = 2 * (u + k) * current / x - prev; + prev = current; + current = next; + } + Yv = prev; + Yv1 = current; + CF1_jy(v, x, &fv, &s, policies::policy<>()); // continued fraction CF1 + Jv = W / (Yv * fv - Yv1); // Wronskian relation + } + else // x in (2, \infty) + { + // Get Y(u, x): + CF1_jy(v, x, &fv, &s, policies::policy<>()); + // tiny initial value to prevent overflow + T init = sqrt(tools::min_value()); + prev = fv * s * init; + current = s * init; + for (k = n; k > 0; k--) // backward recurrence for J + { + next = 2 * (u + k) * current / x - prev; + prev = current; + current = next; + } + T ratio = (s * init) / current; // scaling ratio + // can also call CF1() to get fu, not much difference in precision + fu = prev / current; + CF2_jy(u, x, &p, &q, policies::policy<>()); // continued fraction CF2 + T t = u / x - fu; // t = J'/J + gamma = (p - t) / q; + Ju = sign(current) * sqrt(W / (q + gamma * (p - t))); + + Jv = Ju * ratio; // normalization + + Yu = gamma * Ju; + Yu1 = Yu * (u/x - p - q/gamma); + + // compute Y: + prev = Yu; + current = Yu1; + for (k = 1; k <= n; k++) // forward recurrence for Y + { + next = 2 * (u + k) * current / x - prev; + prev = current; + current = next; + } + Yv = prev; + } + + if (reflect) + { + T z = (u + n % 2) * pi(); + *J = cos(z) * Jv - sin(z) * Yv; // reflection formula + *Y = sin(z) * Jv + cos(z) * Yv; + } + else + { + *J = Jv; + *Y = Yv; + } + + return 0; +} + +int progress = 0; + +template +T cyl_bessel_j_bare(T v, T x) +{ + T j, y; + bessel_jy_bare(v, x, &j, &y); + + std::cout << progress++ << ": J(" << v << ", " << x << ") = " << j << std::endl; + + if(fabs(j) > 1e30) + throw std::domain_error(""); + + return j; +} + +template +T cyl_bessel_i_bare(T v, T x) +{ + using namespace std; + if(x < 0) + { + // better have integer v: + if(floor(v) == v) + { + T r = cyl_bessel_i_bare(v, -x); + if(tools::real_cast(v) & 1) + r = -r; + return r; + } + else + return policies::raise_domain_error( + "", + "Got x = %1%, but we need x >= 0", x, policies::policy<>()); + } + if(x == 0) + { + return (v == 0) ? 1 : 0; + } + T I, K; + boost::math::detail::bessel_ik(v, x, &I, &K, 0xffff, policies::policy<>()); + + std::cout << progress++ << ": I(" << v << ", " << x << ") = " << I << std::endl; + + if(fabs(I) > 1e30) + throw std::domain_error(""); + + return I; +} + +template +T cyl_bessel_k_bare(T v, T x) +{ + using namespace std; + if(x < 0) + { + return policies::raise_domain_error( + "", + "Got x = %1%, but we need x > 0", x, policies::policy<>()); + } + if(x == 0) + { + return (v == 0) ? policies::raise_overflow_error("", 0, policies::policy<>()) + : policies::raise_domain_error( + "", + "Got x = %1%, but we need x > 0", x, policies::policy<>()); + } + T I, K; + bessel_ik(v, x, &I, &K, 0xFFFF, policies::policy<>()); + + std::cout << progress++ << ": K(" << v << ", " << x << ") = " << K << std::endl; + + if(fabs(K) > 1e30) + throw std::domain_error(""); + + return K; +} + +template +T cyl_neumann_bare(T v, T x) +{ + T j, y; + bessel_jy(v, x, &j, &y, 0xFFFF, policies::policy<>()); + + std::cout << progress++ << ": Y(" << v << ", " << x << ") = " << y << std::endl; + + if(fabs(y) > 1e30) + throw std::domain_error(""); + + return y; +} + +template +T sph_bessel_j_bare(T v, T x) +{ + std::cout << progress++ << ": j(" << v << ", " << x << ") = "; + if((v < 0) || (floor(v) != v)) + throw std::domain_error(""); + T r = sqrt(constants::pi() / (2 * x)) * cyl_bessel_j_bare(v+0.5, x); + std::cout << r << std::endl; + return r; +} + +template +T sph_bessel_y_bare(T v, T x) +{ + std::cout << progress++ << ": y(" << v << ", " << x << ") = "; + if((v < 0) || (floor(v) != v)) + throw std::domain_error(""); + T r = sqrt(constants::pi() / (2 * x)) * cyl_neumann_bare(v+0.5, x); + std::cout << r << std::endl; + return r; +} + +enum +{ + func_J = 0, + func_Y, + func_I, + func_K, + func_j, + func_y +}; + +int main(int argc, char* argv[]) +{ + std::cout << std::setprecision(17) << std::scientific; + std::cout << sph_bessel_j_bare(0., 0.1185395751953125e4) << std::endl; + std::cout << sph_bessel_j_bare(22., 0.6540834903717041015625) << std::endl; + + parameter_info arg1, arg2; + test_data data; + + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + int functype = 0; + std::string letter = "J"; + + if(argc == 2) + { + if(std::strcmp(argv[1], "--Y") == 0) + { + functype = func_Y; + letter = "Y"; + } + else if(std::strcmp(argv[1], "--I") == 0) + { + functype = func_I; + letter = "I"; + } + else if(std::strcmp(argv[1], "--K") == 0) + { + functype = func_K; + letter = "K"; + } + else if(std::strcmp(argv[1], "--j") == 0) + { + functype = func_j; + letter = "j"; + } + else if(std::strcmp(argv[1], "--y") == 0) + { + functype = func_y; + letter = "y"; + } + else + assert(0); + } + + bool cont; + std::string line; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the Bessel " << letter << " function\n\n"; + do{ + get_user_parameter_info(arg1, "v"); + get_user_parameter_info(arg2, "x"); + boost::math::ntl::RR (*fp)(boost::math::ntl::RR, boost::math::ntl::RR); + if(functype == func_J) + fp = cyl_bessel_j_bare; + else if(functype == func_I) + fp = cyl_bessel_i_bare; + else if(functype == func_K) + fp = cyl_bessel_k_bare; + else if(functype == func_Y) + fp = cyl_neumann_bare; + else if(functype == func_j) + fp = sph_bessel_j_bare; + else if(functype == func_y) + fp = sph_bessel_y_bare; + else + assert(0); + + data.insert(fp, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=bessel_j_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "bessel_j_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + + + + diff --git a/tools/beta_data.cpp b/tools/beta_data.cpp new file mode 100644 index 000000000..826a83d62 --- /dev/null +++ b/tools/beta_data.cpp @@ -0,0 +1,72 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +#include + +using namespace boost::math::tools; + +struct beta_data_generator +{ + boost::math::ntl::RR operator()(boost::math::ntl::RR a, boost::math::ntl::RR b) + { + if(a < b) + throw std::domain_error(""); + // very naively calculate spots: + boost::math::ntl::RR g1, g2, g3; + int s1, s2, s3; + g1 = boost::math::lgamma(a, &s1); + g2 = boost::math::lgamma(b, &s2); + g3 = boost::math::lgamma(a+b, &s3); + g1 += g2 - g3; + g1 = exp(g1); + g1 *= s1 * s2 * s3; + return g1; + } +}; + + +int main() +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the beta function:\n" + " beta(a, b)\n\n"; + + bool cont; + std::string line; + + do{ + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "b"); + data.insert(beta_data_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=beta_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "beta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "beta_data"); + + return 0; +} + diff --git a/tools/carlson_ellint_data.cpp b/tools/carlson_ellint_data.cpp new file mode 100644 index 000000000..d74a5bc01 --- /dev/null +++ b/tools/carlson_ellint_data.cpp @@ -0,0 +1,151 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +//#include +#include +#include +#include +#include +#include +#include + +float extern_val; +// confuse the compilers optimiser, and force a truncation to float precision: +float truncate_to_float(float const * pf) +{ + extern_val = *pf; + return *pf; +} + +std::tr1::tuple generate_rf_data(boost::math::ntl::RR n) +{ + static std::tr1::mt19937 r; + std::tr1::uniform_real ur(0, 1); + std::tr1::uniform_int ui(-100, 100); + float x = ur(r); + x = ldexp(x, ui(r)); + boost::math::ntl::RR xr(truncate_to_float(&x)); + float y = ur(r); + y = ldexp(y, ui(r)); + boost::math::ntl::RR yr(truncate_to_float(&y)); + float z = ur(r); + z = ldexp(z, ui(r)); + boost::math::ntl::RR zr(truncate_to_float(&z)); + + boost::math::ntl::RR result = boost::math::ellint_rf(xr, yr, zr); + return std::tr1::make_tuple(xr, yr, zr, result); +} + +std::tr1::tuple generate_rc_data(boost::math::ntl::RR n) +{ + static std::tr1::mt19937 r; + std::tr1::uniform_real ur(0, 1); + std::tr1::uniform_int ui(-100, 100); + float x = ur(r); + x = ldexp(x, ui(r)); + boost::math::ntl::RR xr(truncate_to_float(&x)); + float y = ur(r); + y = ldexp(y, ui(r)); + boost::math::ntl::RR yr(truncate_to_float(&y)); + + boost::math::ntl::RR result = boost::math::ellint_rc(xr, yr); + return std::tr1::make_tuple(xr, yr, result); +} + +std::tr1::tuple generate_rj_data(boost::math::ntl::RR n) +{ + static std::tr1::mt19937 r; + std::tr1::uniform_real ur(0, 1); + std::tr1::uniform_real nur(-1, 1); + std::tr1::uniform_int ui(-100, 100); + float x = ur(r); + x = ldexp(x, ui(r)); + boost::math::ntl::RR xr(truncate_to_float(&x)); + float y = ur(r); + y = ldexp(y, ui(r)); + boost::math::ntl::RR yr(truncate_to_float(&y)); + float z = ur(r); + z = ldexp(z, ui(r)); + boost::math::ntl::RR zr(truncate_to_float(&z)); + float p = nur(r); + p = ldexp(p, ui(r)); + boost::math::ntl::RR pr(truncate_to_float(&p)); + + boost::math::ellint_rj(x, y, z, p); + + boost::math::ntl::RR result = boost::math::ellint_rj(xr, yr, zr, pr); + return std::tr1::make_tuple(xr, yr, zr, pr, result); +} + +std::tr1::tuple generate_rd_data(boost::math::ntl::RR n) +{ + static std::tr1::mt19937 r; + std::tr1::uniform_real ur(0, 1); + std::tr1::uniform_int ui(-100, 100); + float x = ur(r); + x = ldexp(x, ui(r)); + boost::math::ntl::RR xr(truncate_to_float(&x)); + float y = ur(r); + y = ldexp(y, ui(r)); + boost::math::ntl::RR yr(truncate_to_float(&y)); + float z = ur(r); + z = ldexp(z, ui(r)); + boost::math::ntl::RR zr(truncate_to_float(&z)); + + boost::math::ntl::RR result = boost::math::ellint_rd(xr, yr, zr); + return std::tr1::make_tuple(xr, yr, zr, result); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + int count; + std::cout << "Number of points: "; + std::cin >> count; + + arg1 = make_periodic_param(boost::math::ntl::RR(0), boost::math::ntl::RR(1), count); + arg1.type |= dummy_param; + + // + // Change this next line to get the R variant you want: + // + data.insert(&generate_rd_data, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_rf_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_rf_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + + diff --git a/tools/cbrt_data.cpp b/tools/cbrt_data.cpp new file mode 100644 index 000000000..f98e28e63 --- /dev/null +++ b/tools/cbrt_data.cpp @@ -0,0 +1,68 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include + +#include + +using namespace boost::math::tools; +using namespace std; + +struct cube_data_generator +{ + boost::math::ntl::RR operator()(boost::math::ntl::RR z) + { + boost::math::ntl::RR result = z*z*z; + // if result is out of range of a float, + // don't include in test data as it messes up our results: + if(result > (std::numeric_limits::max)()) + throw std::domain_error(""); + if(result < (std::numeric_limits::min)()) + throw std::domain_error(""); + return result; + } +}; + +int main(int argc, char* argv[]) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the cbrt function:\n\n"; + + bool cont; + std::string line; + + do{ + if(0 == get_user_parameter_info(arg1, "z")) + return 1; + data.insert(cube_data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=cbrt_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "cbrt_data.ipp"; + std::ofstream ofs(line.c_str()); + ofs << std::scientific; + write_code(ofs, data, "cbrt_data"); + + return 0; +} + + + + diff --git a/tools/digamma_data.cpp b/tools/digamma_data.cpp new file mode 100644 index 000000000..7f2679222 --- /dev/null +++ b/tools/digamma_data.cpp @@ -0,0 +1,67 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include "ntl_rr_digamma.hpp" +#include +#include +#include + +#include + +using namespace boost::math::tools; +using namespace std; + +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the digamma function:\n" + " digamma(z)\n\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "z")) + return 1; + data.insert(&boost::math::digamma, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=digamma_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "digamma_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "digamma_data"); + + return 0; +} + + + diff --git a/tools/ellint_e_data.cpp b/tools/ellint_e_data.cpp new file mode 100644 index 000000000..f2b83c555 --- /dev/null +++ b/tools/ellint_e_data.cpp @@ -0,0 +1,64 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +template +T ellint_e_data(T k) +{ + return ellint_2(k); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "phi")) + return 1; + + data.insert(&ellint_e_data, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_e_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_e_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/ellint_f_data.cpp b/tools/ellint_f_data.cpp new file mode 100644 index 000000000..eb909e308 --- /dev/null +++ b/tools/ellint_f_data.cpp @@ -0,0 +1,76 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +float extern_val; +// confuse the compilers optimiser, and force a truncation to float precision: +float truncate_to_float(float const * pf) +{ + extern_val = *pf; + return *pf; +} + + + +template +T ellint_f_data(T phi, T k) +{ + return ellint_1(k, phi); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "phi")) + return 1; + if(0 == get_user_parameter_info(arg2, "k")) + return 1; + + data.insert(&ellint_f_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_f.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_f.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/ellint_k_data.cpp b/tools/ellint_k_data.cpp new file mode 100644 index 000000000..e239b650c --- /dev/null +++ b/tools/ellint_k_data.cpp @@ -0,0 +1,64 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +template +T ellint_k_data(T k) +{ + return ellint_1(k); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "phi")) + return 1; + + data.insert(&ellint_k_data, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_k_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_k_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/ellint_pi2_data.cpp b/tools/ellint_pi2_data.cpp new file mode 100644 index 000000000..9bde56feb --- /dev/null +++ b/tools/ellint_pi2_data.cpp @@ -0,0 +1,64 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) +#include +//#include +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + if(0 == get_user_parameter_info(arg2, "k")) + return 1; + + boost::math::ntl::RR (*fp)(boost::math::ntl::RR, boost::math::ntl::RR) = &ellint_3; + data.insert(fp, arg2, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_pi2_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_pi2_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + + diff --git a/tools/ellint_pi3_data.cpp b/tools/ellint_pi3_data.cpp new file mode 100644 index 000000000..f328a48da --- /dev/null +++ b/tools/ellint_pi3_data.cpp @@ -0,0 +1,79 @@ +// Copyright John Maddock 2006. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + + +#include +//#include +#include +#include +#include +#include +#include +#include + +float extern_val; +// confuse the compilers optimiser, and force a truncation to float precision: +float truncate_to_float(float const * pf) +{ + extern_val = *pf; + return *pf; +} + +std::tr1::tuple generate_data(boost::math::ntl::RR n, boost::math::ntl::RR phi) +{ + static std::tr1::mt19937 r; + std::tr1::uniform_real ui(0, 1); + float k = ui(r); + boost::math::ntl::RR kr(truncate_to_float(&k)); + boost::math::ntl::RR result = boost::math::ellint_3(kr, n, phi); + return std::tr1::make_tuple(kr, result); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + if(0 == get_user_parameter_info(arg2, "phi")) + return 1; + + data.insert(&generate_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=ellint_pi3_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ellint_pi3_data.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + + diff --git a/tools/erf_data.cpp b/tools/erf_data.cpp new file mode 100644 index 000000000..885be4809 --- /dev/null +++ b/tools/erf_data.cpp @@ -0,0 +1,225 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include // for inverses +#include +#include +#include + +#include + +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; +using namespace std; + +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +struct erf_data_generator +{ + std::tr1::tuple operator()(boost::math::ntl::RR z) + { + // very naively calculate spots using the gamma function at high precision: + int sign = 1; + if(z < 0) + { + sign = -1; + z = -z; + } + boost::math::ntl::RR g1, g2; + g1 = boost::math::tgamma_lower(boost::math::ntl::RR(0.5), z * z); + g1 /= sqrt(boost::math::constants::pi()); + g1 *= sign; + + if(z < 0.5) + { + g2 = 1 - (sign * g1); + } + else + { + g2 = boost::math::tgamma(boost::math::ntl::RR(0.5), z * z); + g2 /= sqrt(boost::math::constants::pi()); + } + if(sign < 1) + g2 = 2 - g2; + return std::tr1::make_tuple(g1, g2); + } +}; + +double double_factorial(int N) +{ + double result = 1; + while(N > 2) + { + N -= 2; + result *= N; + } + return result; +} + +void asymptotic_limit(int Bits) +{ + // + // The following block of code estimates how large z has + // to be before we can use the asymptotic expansion for + // erf/erfc and still get convergence: the series becomes + // divergent eventually so we have to be careful! + // + double result = (std::numeric_limits::max)(); + int terms = 0; + for(int n = 1; n < 15; ++n) + { + double lim = (Bits-n) * log(2.0) - log(sqrt(3.14)) + log(double_factorial(2*n+1)); + double x = 1; + while(x*x + (2*n+1)*log(x) <= lim) + x += 0.1; + if(x < result) + { + result = x; + terms = n; + } + } + + std::cout << "Erf asymptotic limit for " + << Bits << " bit numbers is " + << result << " after approximately " + << terms << " terms." << std::endl; + + result = (std::numeric_limits::max)(); + terms = 0; + for(int n = 1; n < 30; ++n) + { + double x = pow(double_factorial(2*n+1)/pow(2.0, n-Bits), 1 / (2.0*n)); + if(x < result) + { + result = x; + terms = n; + } + } + + std::cout << "Erfc asymptotic limit for " + << Bits << " bit numbers is " + << result << " after approximately " + << terms << " terms." << std::endl; +} + +std::tr1::tuple erfc_inv(boost::math::ntl::RR r) +{ + boost::math::ntl::RR x = exp(-r * r); + x = NTL::RoundToPrecision(x.value(), 64); + std::cout << x << " "; + boost::math::ntl::RR result = boost::math::erfc_inv(x); + std::cout << result << std::endl; + return std::tr1::make_tuple(x, result); +} + + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + if(argc >= 2) + { + if(strcmp(argv[1], "--limits") == 0) + { + asymptotic_limit(24); + asymptotic_limit(53); + asymptotic_limit(64); + asymptotic_limit(106); + asymptotic_limit(113); + return 0; + } + else if(strcmp(argv[1], "--erf_inv") == 0) + { + boost::math::ntl::RR (*f)(boost::math::ntl::RR); + f = boost::math::erf_inv; + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse erf function:\n"; + std::cout << "Enter the number of data points: "; + int points; + std::cin >> points; + data.insert(f, make_random_param(boost::math::ntl::RR(-1), boost::math::ntl::RR(1), points)); + } + else if(strcmp(argv[1], "--erfc_inv") == 0) + { + std::tr1::tuple (*f)(boost::math::ntl::RR); + f = erfc_inv; + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse erfc function:\n"; + std::cout << "Enter the maximum *result* expected from erfc_inv: "; + double max_val; + std::cin >> max_val; + std::cout << "Enter the number of data points: "; + int points; + std::cin >> points; + parameter_info arg = make_random_param(boost::math::ntl::RR(0), boost::math::ntl::RR(max_val), points); + arg.type |= dummy_param; + data.insert(f, arg); + } + } + else + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the erf and erfc functions:\n" + " erf(z) and erfc(z)\n\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "a")) + return 1; + data.insert(erf_data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + std::cout << "Enter name of test data file [default=erf_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "erf_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "erf_data"); + + return 0; +} + +/* Output for asymptotic limits: + +Erf asymptotic limit for 24 bit numbers is 2.8 after approximately 6 terms. +Erfc asymptotic limit for 24 bit numbers is 4.12064 after approximately 17 terms. +Erf asymptotic limit for 53 bit numbers is 4.3 after approximately 11 terms. +Erfc asymptotic limit for 53 bit numbers is 6.19035 after approximately 29 terms. +Erf asymptotic limit for 64 bit numbers is 4.8 after approximately 12 terms. +Erfc asymptotic limit for 64 bit numbers is 7.06004 after approximately 29 terms. +Erf asymptotic limit for 106 bit numbers is 6.5 after approximately 14 terms. +Erfc asymptotic limit for 106 bit numbers is 11.6626 after approximately 29 terms. +Erf asymptotic limit for 113 bit numbers is 6.8 after approximately 14 terms. +Erfc asymptotic limit for 113 bit numbers is 12.6802 after approximately 29 terms. +*/ + diff --git a/tools/factorial_tables.cpp b/tools/factorial_tables.cpp new file mode 100644 index 000000000..67234eed1 --- /dev/null +++ b/tools/factorial_tables.cpp @@ -0,0 +1,43 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include + +void write_table(unsigned max_exponent) +{ + boost::math::ntl::RR max = ldexp(boost::math::ntl::RR(1), max_exponent); + + std::vector factorials; + factorials.push_back(1); + + boost::math::ntl::RR f(1); + unsigned i = 1; + + while(f < max) + { + factorials.push_back(f); + ++i; + f *= i; + } + + // + // now write out the results to cout: + // + std::cout << std::scientific; + std::cout << " static const boost::array factorials = {\n"; + for(unsigned j = 0; j < factorials.size(); ++j) + std::cout << " " << factorials[j] << "L,\n"; + std::cout << " };\n\n"; +} + + +int main() +{ + boost::math::ntl::RR::SetPrecision(300); + boost::math::ntl::RR::SetOutputPrecision(40); + write_table(16384/*std::numeric_limits::max_exponent*/); +} diff --git a/tools/gamma_P_inva_data.cpp b/tools/gamma_P_inva_data.cpp new file mode 100644 index 000000000..66e410c1f --- /dev/null +++ b/tools/gamma_P_inva_data.cpp @@ -0,0 +1,75 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +#include +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; + +// +// Force trunctation to float precision of input values: +// we must ensure that the input values are exactly representable +// in whatever type we are testing, or the output values will all +// be thrown off: +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +struct gamma_inverse_generator_a +{ + std::tr1::tuple operator()(const boost::math::ntl::RR x, const boost::math::ntl::RR p) + { + boost::math::ntl::RR x1 = boost::math::gamma_p_inva(x, p); + boost::math::ntl::RR x2 = boost::math::gamma_q_inva(x, p); + std::cout << "Inverse for " << x << " " << p << std::endl; + return std::tr1::make_tuple(x1, x2); + } +}; + + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(100); + + bool cont; + std::string line; + + parameter_info arg1, arg2; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete gamma function:\n" + " gamma_p_inva(a, p) and gamma_q_inva(a, q)\n\n"; + + arg1 = make_power_param(boost::math::ntl::RR(0), -4, 24); + arg2 = make_random_param(boost::math::ntl::RR(0), boost::math::ntl::RR(1), 15); + data.insert(gamma_inverse_generator_a(), arg1, arg2); + + line = "igamma_inva_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "igamma_inva_data"); + + return 0; +} + diff --git a/tools/generate_rational_code.cpp b/tools/generate_rational_code.cpp new file mode 100644 index 000000000..090175f84 --- /dev/null +++ b/tools/generate_rational_code.cpp @@ -0,0 +1,738 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +int max_order = 20; +const char* path_prefix = "..\\..\\..\\boost\\math\\tools\\detail\\polynomial_"; +const char* path_prefix2 = "..\\..\\..\\boost\\math\\tools\\detail\\rational_"; + +const char* copyright_string = +"// (C) Copyright John Maddock 2007.\n" +"// Use, modification and distribution are subject to the\n" +"// Boost Software License, Version 1.0. (See accompanying file\n" +"// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)\n" +"//\n" +"// This file is machine generated, do not edit by hand\n\n"; + + +void print_polynomials(int max_order) +{ + for(int i = 2; i <= max_order; ++i) + { + std::stringstream filename; + filename << path_prefix << "horner1_" << i << ".hpp"; + std::ofstream ofs(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Polynomial evaluation using Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]);\n" + "}\n\n"; + + for(int order = 2; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " return static_cast("; + + for(int bracket = 2; bracket < order; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x + a[" << order - 2 << "]" ; + for(int item = order - 3; item >= 0; --item) + { + ofs << ") * x + a[" << item << "]"; + } + + ofs << ");\n" + "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + + filename.str(""); + filename << path_prefix << "horner2_" << i << ".hpp"; + ofs.close(); + ofs.open(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Polynomial evaluation using second order Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<2>*)\n" + "{\n" + " return static_cast(a[1] * x + a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<3>*)\n" + "{\n" + " return static_cast((a[2] * x + a[1]) * x + a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<4>*)\n" + "{\n" + " return static_cast(((a[3] * x + a[2]) * x + a[1]) * x + a[0]);\n" + "}\n\n"; + + for(int order = 5; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " V x2 = x * x;\n" + " return static_cast("; + + if(order & 1) + { + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x2 + a[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << " + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 2 << "] * x2 + a[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << ") * x"; + } + else + { + for(int bracket = 0; bracket < (order - 1) / 2; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x2 + a[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << ") * x + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 2 << "] * x2 + a[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + } + ofs << ");\n" + "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + + + filename.str(""); + filename << path_prefix << "horner3_" << i << ".hpp"; + ofs.close(); + ofs.open(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Unrolled polynomial evaluation using second order Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_POLY_EVAL_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<2>*)\n" + "{\n" + " return static_cast(a[1] * x + a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<3>*)\n" + "{\n" + " return static_cast((a[2] * x + a[1]) * x + a[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<4>*)\n" + "{\n" + " return static_cast(((a[3] * x + a[2]) * x + a[1]) * x + a[0]);\n" + "}\n\n"; + + for(int order = 5; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_polynomial_c_imp(const T* a, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " V x2 = x * x;\n" + " V t[2];\n"; + + if(order & 1) + { + ofs << " t[0] = static_cast(a[" << order - 1 << "] * x2 + a[" << order - 3 << "]);\n" ; + ofs << " t[1] = static_cast(a[" << order - 2 << "] * x2 + a[" << order - 4 << "]);\n" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << " t[0] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[1] *= x2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[1] += static_cast(a[" << item - 1 << "]);\n"; + } + ofs << + " t[1] *= x;\n" + " return t[0] + t[1];\n"; + } + else + { + ofs << " t[0] = a[" << order - 1 << "] * x2 + a[" << order - 3 << "];\n" ; + ofs << " t[1] = a[" << order - 2 << "] * x2 + a[" << order - 4 << "];\n" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << " t[0] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[1] *= x2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[1] += static_cast(a[" << item - 1 << "]);\n"; + } + ofs << " t[0] *= x;\n"; + ofs << " return t[0] + t[1];\n"; + } + ofs << "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + } +} + +void print_rationals(int max_order) +{ + for(int i = 2; i <= max_order; ++i) + { + std::stringstream filename; + filename << path_prefix2 << "horner1_" << i << ".hpp"; + std::ofstream ofs(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Polynomial evaluation using Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_POLY_RAT_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_POLY_RAT_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T*, const U*, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]) / static_cast(b[0]);\n" + "}\n\n"; + + for(int order = 2; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " if(x <= 1)\n" + " return static_cast(("; + + for(int bracket = 2; bracket < order; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x + a[" << order - 2 << "]" ; + for(int item = order - 3; item >= 0; --item) + { + ofs << ") * x + a[" << item << "]"; + } + + ofs << ") / ("; + for(int bracket = 2; bracket < order; ++bracket) + ofs << "("; + ofs << "b[" << order - 1 << "] * x + b[" << order - 2 << "]" ; + for(int item = order - 3; item >= 0; --item) + { + ofs << ") * x + b[" << item << "]"; + } + + ofs << "));\n else\n {\n V z = 1 / x;\n return static_cast(("; + + for(int bracket = order - 1; bracket > 1; --bracket) + ofs << "("; + ofs << "a[" << 0 << "] * z + a[" << 1 << "]" ; + for(int item = 2; item <= order - 1; ++item) + { + ofs << ") * z + a[" << item << "]"; + } + + ofs << ") / ("; + for(int bracket = 2; bracket < order; ++bracket) + ofs << "("; + ofs << "b[" << 0 << "] * z + b[" << 1 << "]" ; + for(int item = 2; item <= order - 1; ++item) + { + ofs << ") * z + b[" << item << "]"; + } + + ofs << "));\n }\n"; + + ofs << "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + + filename.str(""); + filename << path_prefix2 << "horner2_" << i << ".hpp"; + ofs.close(); + ofs.open(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Polynomial evaluation using second order Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_RAT_EVAL_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_RAT_EVAL_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]) / static_cast(b[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<2>*)\n" + "{\n" + " return static_cast((a[1] * x + a[0]) / (b[1] * x + b[0]));\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<3>*)\n" + "{\n" + " return static_cast(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<4>*)\n" + "{\n" + " return static_cast((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));\n" + "}\n\n"; + + for(int order = 5; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " if(x <= 1)\n {\n" + " V x2 = x * x;\n" + " return static_cast(("; + + if(order & 1) + { + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x2 + a[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << " + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 2 << "] * x2 + a[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << ") * x"; + } + else + { + for(int bracket = 0; bracket < (order - 1) / 2; ++bracket) + ofs << "("; + ofs << "a[" << order - 1 << "] * x2 + a[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + ofs << ") * x + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << order - 2 << "] * x2 + a[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + a[" << item << "]"; + } + } + ofs << ") / ("; + if(order & 1) + { + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << order - 1 << "] * x2 + b[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + b[" << item << "]"; + } + ofs << " + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << order - 2 << "] * x2 + b[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + b[" << item << "]"; + } + ofs << ") * x"; + } + else + { + for(int bracket = 0; bracket < (order - 1) / 2; ++bracket) + ofs << "("; + ofs << "b[" << order - 1 << "] * x2 + b[" << order - 3 << "]" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << ") * x2 + b[" << item << "]"; + } + ofs << ") * x + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << order - 2 << "] * x2 + b[" << order - 4 << "]" ; + for(int item = order - 6; item >= 0; item -= 2) + { + ofs << ") * x2 + b[" << item << "]"; + } + } + + ofs << "));\n }\n else\n {\n V z = 1 / x;\n V z2 = 1 / (x * x);\n return static_cast(("; + + if(order & 1) + { + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << 0 << "] * z2 + a[" << 2 << "]" ; + for(int item = 4; item < order; item += 2) + { + ofs << ") * z2 + a[" << item << "]"; + } + ofs << " + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << 1 << "] * z2 + a[" << 3 << "]" ; + for(int item = 5; item < order; item += 2) + { + ofs << ") * z2 + a[" << item << "]"; + } + ofs << ") * z"; + } + else + { + for(int bracket = 0; bracket < (order - 1) / 2; ++bracket) + ofs << "("; + ofs << "a[" << 0 << "] * z2 + a[" << 2 << "]" ; + for(int item = 4; item < order; item += 2) + { + ofs << ") * z2 + a[" << item << "]"; + } + ofs << ") * z + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "a[" << 1 << "] * z2 + a[" << 3 << "]" ; + for(int item = 5; item < order; item += 2) + { + ofs << ") * z2 + a[" << item << "]"; + } + } + + ofs << ") / ("; + + if(order & 1) + { + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << 0 << "] * z2 + b[" << 2 << "]" ; + for(int item = 4; item < order; item += 2) + { + ofs << ") * z2 + b[" << item << "]"; + } + ofs << " + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << 1 << "] * z2 + b[" << 3 << "]" ; + for(int item = 5; item < order; item += 2) + { + ofs << ") * z2 + b[" << item << "]"; + } + ofs << ") * z"; + } + else + { + for(int bracket = 0; bracket < (order - 1) / 2; ++bracket) + ofs << "("; + ofs << "b[" << 0 << "] * z2 + b[" << 2 << "]" ; + for(int item = 4; item < order; item += 2) + { + ofs << ") * z2 + b[" << item << "]"; + } + ofs << ") * z + "; + for(int bracket = 0; bracket < (order - 1) / 2 - 1; ++bracket) + ofs << "("; + ofs << "b[" << 1 << "] * z2 + b[" << 3 << "]" ; + for(int item = 5; item < order; item += 2) + { + ofs << ") * z2 + b[" << item << "]"; + } + } + ofs << "));\n }\n"; + + ofs << "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + + + filename.str(""); + filename << path_prefix2 << "horner3_" << i << ".hpp"; + ofs.close(); + ofs.open(filename.str().c_str()); + if(ofs.bad()) + break; + // + // Output the boilerplate at the top of the header: + // + ofs << copyright_string << + "// Polynomial evaluation using second order Horners rule\n" + "#ifndef BOOST_MATH_TOOLS_RAT_EVAL_" << i << "_HPP\n" + "#define BOOST_MATH_TOOLS_RAT_EVAL_" << i << "_HPP\n\n" + "namespace boost{ namespace math{ namespace tools{ namespace detail{\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const mpl::int_<0>*)\n" + "{\n" + " return static_cast(0);\n" + "}\n" + "\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const mpl::int_<1>*)\n" + "{\n" + " return static_cast(a[0]) / static_cast(b[0]);\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<2>*)\n" + "{\n" + " return static_cast((a[1] * x + a[0]) / (b[1] * x + b[0]));\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<3>*)\n" + "{\n" + " return static_cast(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));\n" + "}\n\n" + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<4>*)\n" + "{\n" + " return static_cast((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));\n" + "}\n\n"; + + for(int order = 5; order <= i; ++order) + { + ofs << + "template \n" + "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const mpl::int_<" << order << ">*)\n" + "{\n" + " if(x <= 1)\n {\n" + " V x2 = x * x;\n" + " V t[4];\n"; + + if(order & 1) + { + ofs << " t[0] = a[" << order - 1 << "] * x2 + a[" << order - 3 << "];\n" ; + ofs << " t[1] = a[" << order - 2 << "] * x2 + a[" << order - 4 << "];\n" ; + ofs << " t[2] = b[" << order - 1 << "] * x2 + b[" << order - 3 << "];\n" ; + ofs << " t[3] = b[" << order - 2 << "] * x2 + b[" << order - 4 << "];\n" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << " t[0] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[1] *= x2;\n"; + ofs << " t[2] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[3] *= x2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[1] += static_cast(a[" << item - 1 << "]);\n"; + ofs << " t[2] += static_cast(b[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[3] += static_cast(b[" << item - 1 << "]);\n"; + } + ofs << " t[1] *= x;\n"; + ofs << " t[3] *= x;\n"; + } + else + { + ofs << " t[0] = a[" << order - 1 << "] * x2 + a[" << order - 3 << "];\n" ; + ofs << " t[1] = a[" << order - 2 << "] * x2 + a[" << order - 4 << "];\n" ; + ofs << " t[2] = b[" << order - 1 << "] * x2 + b[" << order - 3 << "];\n" ; + ofs << " t[3] = b[" << order - 2 << "] * x2 + b[" << order - 4 << "];\n" ; + for(int item = order - 5; item >= 0; item -= 2) + { + ofs << " t[0] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[1] *= x2;\n"; + ofs << " t[2] *= x2;\n"; + if(item - 1 >= 0) + ofs << " t[3] *= x2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[1] += static_cast(a[" << item - 1 << "]);\n"; + ofs << " t[2] += static_cast(b[" << item << "]);\n"; + if(item - 1 >= 0) + ofs << " t[3] += static_cast(b[" << item - 1 << "]);\n"; + } + ofs << " t[0] *= x;\n"; + ofs << " t[2] *= x;\n"; + } + ofs << " return (t[0] + t[1]) / (t[2] + t[3]);\n"; + + ofs << " }\n else\n {\n V z = 1 / x;\n V z2 = 1 / (x * x);\n V t[4];\n"; + + if(order & 1) + { + ofs << " t[0] = a[" << 0 << "] * z2 + a[" << 2 << "];\n" ; + ofs << " t[1] = a[" << 1 << "] * z2 + a[" << 3 << "];\n" ; + ofs << " t[2] = b[" << 0 << "] * z2 + b[" << 2 << "];\n" ; + ofs << " t[3] = b[" << 1 << "] * z2 + b[" << 3 << "];\n" ; + for(int item = 4; item < order; item += 2) + { + ofs << " t[0] *= z2;\n"; + if(item + 1 < order) + ofs << " t[1] *= z2;\n"; + ofs << " t[2] *= z2;\n"; + if(item + 1 < order) + ofs << " t[3] *= z2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item + 1 < order) + ofs << " t[1] += static_cast(a[" << item + 1 << "]);\n"; + ofs << " t[2] += static_cast(b[" << item << "]);\n"; + if(item + 1 < order) + ofs << " t[3] += static_cast(b[" << item + 1 << "]);\n"; + } + ofs << " t[1] *= z;\n"; + ofs << " t[3] *= z;\n"; + } + else + { + ofs << " t[0] = a[" << 0 << "] * z2 + a[" << 2 << "];\n" ; + ofs << " t[1] = a[" << 1 << "] * z2 + a[" << 3 << "];\n" ; + ofs << " t[2] = b[" << 0 << "] * z2 + b[" << 2 << "];\n" ; + ofs << " t[3] = b[" << 1 << "] * z2 + b[" << 3 << "];\n" ; + for(int item = 4; item < order; item += 2) + { + ofs << " t[0] *= z2;\n"; + if(item + 1 < order) + ofs << " t[1] *= z2;\n"; + ofs << " t[2] *= z2;\n"; + if(item + 1 < order) + ofs << " t[3] *= z2;\n"; + ofs << " t[0] += static_cast(a[" << item << "]);\n"; + if(item + 1 < order) + ofs << " t[1] += static_cast(a[" << item + 1 << "]);\n"; + ofs << " t[2] += static_cast(b[" << item << "]);\n"; + if(item + 1 < order) + ofs << " t[3] += static_cast(b[" << item + 1 << "]);\n"; + } + ofs << " t[0] *= z;\n"; + ofs << " t[2] *= z;\n"; + } + ofs << " return (t[0] + t[1]) / (t[2] + t[3]);\n }\n"; + + ofs << "}\n\n"; + } + // + // And finally the boilerplate at the end of the header: + // + ofs << "\n}}}} // namespaces\n\n#endif // include guard\n\n"; + } +} + +int main() +{ + for(int i = 2; i <= max_order; ++i) + { + print_polynomials(i); + print_rationals(i); + } + return 0; +} + + + diff --git a/tools/generate_rational_test.cpp b/tools/generate_rational_test.cpp new file mode 100644 index 000000000..d3719faed --- /dev/null +++ b/tools/generate_rational_test.cpp @@ -0,0 +1,526 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_POLY_METHOD 0 +#define BOOST_MATH_RATIONAL_METHOD 0 + +#include +#include +#include +#include +#include + +int main() +{ + using namespace boost::math; + using namespace boost::math::tools; + + static const unsigned max_order = 20; + + boost::math::ntl::RR::SetPrecision(500); + boost::math::ntl::RR::SetOutputPrecision(40); + + std::tr1::mt19937 rnd; + std::tr1::variate_generator< + std::tr1::mt19937, + std::tr1::uniform_int<> > gen(rnd, std::tr1::uniform_int<>(1, 12)); + + for(unsigned i = 1; i < max_order; ++i) + { + std::vector coef; + for(unsigned j = 0; j < i; ++j) + { + coef.push_back(gen()); + } + std::cout << std::scientific; + std::cout << +" //\n" +" // Polynomials of order " << i-1 << "\n" +" //\n" +" static const U n" << i << "c[" << i << "] = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + std::cout << + " static const boost::array n" << i << "a = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + + boost::math::ntl::RR r1 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + boost::math::ntl::RR r2 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + boost::math::ntl::RR r3 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + boost::math::ntl::RR r4 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + boost::math::ntl::RR r5 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(6.5), i); + boost::math::ntl::RR r6 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(10247.25), i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(6.5), " << i << "),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(10247.25), " << i << "),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(6.5)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(10247.25)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(6.5)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(10247.25)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + r1 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + r5 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(6.5), i); + r6 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(10247.25), i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(6.5f), " << i << "),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(10247.25f), " << i << "),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(6.5f)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(10247.25f)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(6.5f)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(10247.25f)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + if(i > 1) + { + r1 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + r5 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(6.5), i); + r6 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(10247.25), i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(6.5f), " << i << "),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(10247.25f), " << i << "),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(6.5f)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(10247.25f)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n"; + if(fabs(r5) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(6.5f)),\n" + " static_cast(" << r5 << "L),\n" + " tolerance);\n"; + if(fabs(r6) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(10247.25f)),\n" + " static_cast(" << r6 << "L),\n" + " tolerance);\n\n"; + } + + r1 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + r5 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(6.5), i); + r6 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(10247.25), i); + + coef.clear(); + for(unsigned j = 0; j < i; ++j) + { + coef.push_back(gen()); + } + std::cout << +" //\n" +" // Rational functions of order " << i-1 << "\n" +" //\n" +" static const U d" << i << "c[" << i << "] = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + std::cout << + " static const boost::array d" << i << "a = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + + boost::math::ntl::RR r1d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + boost::math::ntl::RR r2d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + boost::math::ntl::RR r3d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + boost::math::ntl::RR r4d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + boost::math::ntl::RR r5d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(6.5), i); + boost::math::ntl::RR r6d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(10247.25), i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n"; + if(fabs(r5/r5d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(6.5f), " << i << "),\n" + " static_cast(" << r5/r5d << "L),\n" + " tolerance);\n"; + if(fabs(r6/r6d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(10247.25f), " << i << "),\n" + " static_cast(" << r6/r6d << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n"; + if(fabs(r5/r5d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(6.5f)),\n" + " static_cast(" << r5/r5d << "L),\n" + " tolerance);\n"; + if(fabs(r6/r6d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(10247.25f)),\n" + " static_cast(" << r6/r6d << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n"; + if(fabs(r5/r5d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(6.5f)),\n" + " static_cast(" << r5/r5d << "L),\n" + " tolerance);\n"; + if(fabs(r6/r6d) < tools::max_value()) + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(10247.25f)),\n" + " static_cast(" << r6/r6d << "L),\n" + " tolerance);\n\n"; + } + + return 0; +} + + + diff --git a/tools/hermite_data.cpp b/tools/hermite_data.cpp new file mode 100644 index 000000000..1df52aa1b --- /dev/null +++ b/tools/hermite_data.cpp @@ -0,0 +1,73 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + + +template +std::tr1::tuple hermite_data(T n, T x) +{ + n = floor(n); + T r1 = hermite(boost::math::tools::real_cast(n), x); + return std::tr1::make_tuple(n, x, r1); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2, arg3; + test_data data; + + std::cout << boost::math::hermite(10, static_cast(1e300)) << std::endl; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + if(0 == get_user_parameter_info(arg2, "x")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + + data.insert(&hermite_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=hermite.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "hermite.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/ibeta_data.cpp b/tools/ibeta_data.cpp new file mode 100644 index 000000000..e4c5f76a2 --- /dev/null +++ b/tools/ibeta_data.cpp @@ -0,0 +1,310 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include + +// speed up beta function computation with a 90-decimal digit approximation: +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +template +struct ibeta_fraction1_t +{ + typedef std::pair result_type; + + ibeta_fraction1_t(T a_, T b_, T x_) : a(a_), b(b_), x(x_), k(1) {} + + result_type operator()() + { + T aN; + if(k & 1) + { + int m = (k - 1) / 2; + aN = -(a + m) * (a + b + m) * x; + aN /= a + 2*m; + aN /= a + 2*m + 1; + } + else + { + int m = k / 2; + aN = m * (b - m) *x; + aN /= a + 2*m - 1; + aN /= a + 2*m; + } + ++k; + return std::make_pair(aN, T(1)); + } + +private: + T a, b, x; + int k; +}; + +// +// This function caches previous calls to beta +// just so we can speed things up a bit: +// +template +T get_beta(T a, T b) +{ + static std::map, T> m; + + if(a < b) + std::swap(a, b); + + std::pair p(a, b); + std::map, T>::const_iterator i = m.find(p); + if(i != m.end()) + return i->second; + + T r = beta(a, b); + p.first = a; + p.second = b; + m[p] = r; + + return r; +} + +// +// compute the continued fraction: +// +template +T get_ibeta_fraction1(T a, T b, T x) +{ + ibeta_fraction1_t f(a, b, x); + T fract = boost::math::tools::continued_fraction_a(f, boost::math::policies::digits >()); + T denom = (a * (fract + 1)); + T num = pow(x, a) * pow(1 - x, b); + if(num == 0) + return 0; + else if(denom == 0) + return -1; + return num / denom; +} +// +// calculate the incomplete beta from the fraction: +// +template +std::pair ibeta_fraction1(T a, T b, T x) +{ + T bet = get_beta(a, b); + if(x > ((a+1)/(a+b+2))) + { + T fract = get_ibeta_fraction1(b, a, 1-x); + if(fract/bet > 0.75) + { + fract = get_ibeta_fraction1(a, b, x); + return std::make_pair(fract, bet - fract); + } + return std::make_pair(bet - fract, fract); + } + T fract = get_ibeta_fraction1(a, b, x); + if(fract/bet > 0.75) + { + fract = get_ibeta_fraction1(b, a, 1-x); + return std::make_pair(bet - fract, fract); + } + return std::make_pair(fract, bet - fract); + +} +// +// calculate the regularised incomplete beta from the fraction: +// +template +std::pair ibeta_fraction1_regular(T a, T b, T x) +{ + T bet = get_beta(a, b); + if(x > ((a+1)/(a+b+2))) + { + T fract = get_ibeta_fraction1(b, a, 1-x); + if(fract == 0) + bet = 1; // normalise so we don't get 0/0 + else if(bet == 0) + return std::make_pair(T(-1), T(-1)); // Yikes!! + if(fract / bet > 0.75) + { + fract = get_ibeta_fraction1(a, b, x); + return std::make_pair(fract / bet, 1 - (fract / bet)); + } + return std::make_pair(1 - (fract / bet), fract / bet); + } + T fract = get_ibeta_fraction1(a, b, x); + if(fract / bet > 0.75) + { + fract = get_ibeta_fraction1(b, a, 1-x); + return std::make_pair(1 - (fract / bet), fract / bet); + } + return std::make_pair(fract / bet, 1 - (fract / bet)); +} + +// +// we absolutely must trunctate the input values to float +// precision: we have to be certain that the input values +// can be represented exactly in whatever width floating +// point type we are testing, otherwise the output will +// necessarily be off. +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +std::tr1::mt19937 rnd; +std::tr1::uniform_real ur_a(1.0F, 5.0F); +std::tr1::variate_generator > gen(rnd, ur_a); +std::tr1::uniform_real ur_a2(0.0F, 100.0F); +std::tr1::variate_generator > gen2(rnd, ur_a2); + +struct beta_data_generator +{ + std::tr1::tuple operator()(boost::math::ntl::RR ap, boost::math::ntl::RR bp, boost::math::ntl::RR x_) + { + float a = truncate_to_float(real_cast(gen() * pow(boost::math::ntl::RR(10), ap))); + float b = truncate_to_float(real_cast(gen() * pow(boost::math::ntl::RR(10), bp))); + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::endl; + std::pair ib_full = ibeta_fraction1(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + std::pair ib_reg = ibeta_fraction1_regular(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +// medium sized values: +struct beta_data_generator_medium +{ + std::tr1::tuple operator()(boost::math::ntl::RR x_) + { + boost::math::ntl::RR a = gen2(); + boost::math::ntl::RR b = gen2(); + boost::math::ntl::RR x = x_; + a = ConvPrec(a.value(), 22); + b = ConvPrec(b.value(), 22); + x = ConvPrec(x.value(), 22); + std::cout << a << " " << b << " " << x << std::endl; + //boost::math::ntl::RR exp_beta = boost::math::beta(a, b, x); + std::pair ib_full = ibeta_fraction1(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + /*exp_beta = boost::math::tools::relative_error(ib_full.first, exp_beta); + if(exp_beta > 1e-40) + { + std::cout << exp_beta << std::endl; + }*/ + std::pair ib_reg = ibeta_fraction1_regular(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +struct beta_data_generator_small +{ + std::tr1::tuple operator()(boost::math::ntl::RR x_) + { + float a = truncate_to_float(gen2()/10); + float b = truncate_to_float(gen2()/10); + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::endl; + std::pair ib_full = ibeta_fraction1(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + std::pair ib_reg = ibeta_fraction1_regular(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +struct beta_data_generator_int +{ + std::tr1::tuple operator()(boost::math::ntl::RR a, boost::math::ntl::RR b, boost::math::ntl::RR x_) + { + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::endl; + std::pair ib_full = ibeta_fraction1(a, b, boost::math::ntl::RR(x)); + std::pair ib_reg = ibeta_fraction1_regular(a, b, boost::math::ntl::RR(x)); + return std::tr1::make_tuple(a, b, x, ib_full.first, ib_full.second, ib_reg.first, ib_reg.second); + } +}; + +int test_main(int, char* []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2, arg3, arg4, arg5; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the incomplete beta functions:\n" + " beta(a, b, x) and ibeta(a, b, x)\n\n" + "This is not an interactive program be prepared for a long wait!!!\n\n"; + + arg1 = make_periodic_param(boost::math::ntl::RR(-5), boost::math::ntl::RR(6), 11); + arg2 = make_periodic_param(boost::math::ntl::RR(-5), boost::math::ntl::RR(6), 11); + arg3 = make_random_param(boost::math::ntl::RR(0.0001), boost::math::ntl::RR(1), 10); + arg4 = make_random_param(boost::math::ntl::RR(0.0001), boost::math::ntl::RR(1), 100 /*500*/); + arg5 = make_periodic_param(boost::math::ntl::RR(1), boost::math::ntl::RR(41), 10); + + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3.type |= dummy_param; + arg4.type |= dummy_param; + arg5.type |= dummy_param; + + // comment out all but one of the following when running + // or this program will take forever to complete! + //data.insert(beta_data_generator(), arg1, arg2, arg3); + //data.insert(beta_data_generator_medium(), arg4); + //data.insert(beta_data_generator_small(), arg4); + data.insert(beta_data_generator_int(), arg5, arg5, arg3); + + test_data::const_iterator i, j; + i = data.begin(); + j = data.end(); + while(i != j) + { + boost::math::ntl::RR v1 = beta((*i)[0], (*i)[1], (*i)[2]); + boost::math::ntl::RR v2 = relative_error(v1, (*i)[3]); + std::string s = boost::lexical_cast((*i)[3]); + boost::math::ntl::RR v3 = boost::lexical_cast(s); + boost::math::ntl::RR v4 = relative_error(v3, (*i)[3]); + if(v2 > 1e-40) + { + std::cout << v2 << std::endl; + } + if(v4 > 1e-60) + { + std::cout << v4 << std::endl; + } + ++ i; + } + + std::cout << "Enter name of test data file [default=ibeta_data.ipp]"; + std::string line; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "ibeta_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "ibeta_data"); + + return 0; +} + + diff --git a/tools/ibeta_inv_data.cpp b/tools/ibeta_inv_data.cpp new file mode 100644 index 000000000..b13fd9400 --- /dev/null +++ b/tools/ibeta_inv_data.cpp @@ -0,0 +1,93 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#if 1 +#include +#include +#include +#include +#include +#include +#include + +#include +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; + +// +// Force trunctation to float precision of input values: +// we must ensure that the input values are exactly representable +// in whatever type we are testing, or the output values will all +// be thrown off: +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +std::tr1::mt19937 rnd; +std::tr1::uniform_real ur_a(1.0F, 5.0F); +std::tr1::variate_generator > gen(rnd, ur_a); +std::tr1::uniform_real ur_a2(0.0F, 100.0F); +std::tr1::variate_generator > gen2(rnd, ur_a2); + +struct ibeta_inv_data_generator +{ + std::tr1::tuple operator()(boost::math::ntl::RR ap, boost::math::ntl::RR bp, boost::math::ntl::RR x_) + { + float a = truncate_to_float(real_cast(gen() * pow(boost::math::ntl::RR(10), ap))); + float b = truncate_to_float(real_cast(gen() * pow(boost::math::ntl::RR(10), bp))); + float x = truncate_to_float(real_cast(x_)); + std::cout << a << " " << b << " " << x << std::flush; + boost::math::ntl::RR inv = boost::math::ibeta_inv(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + std::cout << " " << inv << std::flush; + boost::math::ntl::RR invc = boost::math::ibetac_inv(boost::math::ntl::RR(a), boost::math::ntl::RR(b), boost::math::ntl::RR(x)); + std::cout << " " << invc << std::endl; + return std::tr1::make_tuple(a, b, x, inv, invc); + } +}; + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(100); + + bool cont; + std::string line; + + parameter_info arg1, arg2, arg3; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete beta function:\n" + " ibeta_inv(a, p) and ibetac_inv(a, q)\n\n"; + + arg1 = make_periodic_param(boost::math::ntl::RR(-5), boost::math::ntl::RR(6), 11); + arg2 = make_periodic_param(boost::math::ntl::RR(-5), boost::math::ntl::RR(6), 11); + arg3 = make_random_param(boost::math::ntl::RR(0.0001), boost::math::ntl::RR(1), 10); + + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3.type |= dummy_param; + + data.insert(ibeta_inv_data_generator(), arg1, arg2, arg3); + + line = "ibeta_inv_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "ibeta_inv_data"); + + return 0; +} + +#endif diff --git a/tools/ibeta_invab_data.cpp b/tools/ibeta_invab_data.cpp new file mode 100644 index 000000000..9ac7ea0cc --- /dev/null +++ b/tools/ibeta_invab_data.cpp @@ -0,0 +1,96 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +#include +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; + +// +// Force trunctation to float precision of input values: +// we must ensure that the input values are exactly representable +// in whatever type we are testing, or the output values will all +// be thrown off: +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +std::tr1::mt19937 rnd; +std::tr1::uniform_real ur_a(1.0F, 5.0F); +std::tr1::variate_generator > gen(rnd, ur_a); +std::tr1::uniform_real ur_a2(0.0F, 100.0F); +std::tr1::variate_generator > gen2(rnd, ur_a2); + +struct ibeta_inv_data_generator +{ + std::tr1::tuple operator() + (boost::math::ntl::RR bp, boost::math::ntl::RR x_, boost::math::ntl::RR p_) + { + float b = truncate_to_float(real_cast(gen() * pow(boost::math::ntl::RR(10), bp))); + float x = truncate_to_float(real_cast(x_)); + float p = truncate_to_float(real_cast(p_)); + std::cout << b << " " << x << " " << p << std::flush; + boost::math::ntl::RR inv = boost::math::ibeta_inva(boost::math::ntl::RR(b), boost::math::ntl::RR(x), boost::math::ntl::RR(p)); + std::cout << " " << inv << std::flush; + boost::math::ntl::RR invc = boost::math::ibetac_inva(boost::math::ntl::RR(b), boost::math::ntl::RR(x), boost::math::ntl::RR(p)); + std::cout << " " << invc << std::endl; + boost::math::ntl::RR invb = boost::math::ibeta_invb(boost::math::ntl::RR(b), boost::math::ntl::RR(x), boost::math::ntl::RR(p)); + std::cout << " " << invb << std::flush; + boost::math::ntl::RR invbc = boost::math::ibetac_invb(boost::math::ntl::RR(b), boost::math::ntl::RR(x), boost::math::ntl::RR(p)); + std::cout << " " << invbc << std::endl; + return std::tr1::make_tuple(b, x, p, inv, invc, invb, invbc); + } +}; + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(100); + + bool cont; + std::string line; + + parameter_info arg1, arg2, arg3; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete beta function:\n" + " ibeta_inva(a, p) and ibetac_inva(a, q)\n\n"; + + arg1 = make_periodic_param(boost::math::ntl::RR(-5), boost::math::ntl::RR(6), 11); + arg2 = make_random_param(boost::math::ntl::RR(0.0001), boost::math::ntl::RR(1), 10); + arg3 = make_random_param(boost::math::ntl::RR(0.0001), boost::math::ntl::RR(1), 10); + + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3.type |= dummy_param; + + data.insert(ibeta_inv_data_generator(), arg1, arg2, arg3); + + line = "ibeta_inva_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "ibeta_inva_data"); + + return 0; +} + diff --git a/tools/igamma_data.cpp b/tools/igamma_data.cpp new file mode 100644 index 000000000..273870291 --- /dev/null +++ b/tools/igamma_data.cpp @@ -0,0 +1,183 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +#include +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; + +// +// Force trunctation to float precision of input values: +// we must ensure that the input values are exactly representable +// in whatever type we are testing, or the output values will all +// be thrown off: +// +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(boost::math::ntl::RR r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +// +// Our generator takes two arguments, but the second is interpreted +// as an instruction not a value, the second argument won't be placed +// in the output matrix by class test_data, so we add our algorithmically +// derived second argument to the output. +// +struct igamma_data_generator +{ + std::tr1::tuple operator()(boost::math::ntl::RR a, boost::math::ntl::RR x) + { + // very naively calculate spots: + boost::math::ntl::RR z; + switch((int)real_cast(x)) + { + case 1: + z = truncate_to_float((std::min)(boost::math::ntl::RR(1), a/100)); + break; + case 2: + z = truncate_to_float(a / 2); + break; + case 3: + z = truncate_to_float((std::max)(0.9*a, a - 2)); + break; + case 4: + z = a; + break; + case 5: + z = truncate_to_float((std::min)(1.1*a, a + 2)); + break; + case 6: + z = truncate_to_float(a * 2); + break; + case 7: + z = truncate_to_float((std::max)(boost::math::ntl::RR(100), a*100)); + break; + default: + BOOST_ASSERT(0 == "Can't get here!!"); + } + + //boost::math::ntl::RR g = boost::math::tgamma(a); + boost::math::ntl::RR lg = boost::math::tgamma_lower(a, z); + boost::math::ntl::RR ug = boost::math::tgamma(a, z); + boost::math::ntl::RR rlg = boost::math::gamma_p(a, z); + boost::math::ntl::RR rug = boost::math::gamma_q(a, z); + + return std::tr1::make_tuple(z, ug, rug, lg, rlg); + } +}; + +struct gamma_inverse_generator +{ + std::tr1::tuple operator()(const boost::math::ntl::RR a, const boost::math::ntl::RR p) + { + boost::math::ntl::RR x1 = boost::math::gamma_p_inv(a, p); + boost::math::ntl::RR x2 = boost::math::gamma_q_inv(a, p); + std::cout << "Inverse for " << a << " " << p << std::endl; + return std::tr1::make_tuple(x1, x2); + } +}; + +struct gamma_inverse_generator_a +{ + std::tr1::tuple operator()(const boost::math::ntl::RR x, const boost::math::ntl::RR p) + { + boost::math::ntl::RR x1 = boost::math::gamma_p_inva(x, p); + boost::math::ntl::RR x2 = boost::math::gamma_q_inva(x, p); + std::cout << "Inverse for " << x << " " << p << std::endl; + return std::tr1::make_tuple(x1, x2); + } +}; + + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(100); + + bool cont; + std::string line; + + parameter_info arg1, arg2; + test_data data; + + if((argc >= 2) && (std::strcmp(argv[1], "-inverse") == 0)) + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete gamma function:\n" + " gamma_p_inv(a, p) and gamma_q_inv(a, q)\n\n"; + do{ + get_user_parameter_info(arg1, "a"); + get_user_parameter_info(arg2, "p"); + data.insert(gamma_inverse_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else if((argc >= 2) && (std::strcmp(argv[1], "-inverse_a") == 0)) + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the inverse incomplete gamma function:\n" + " gamma_p_inva(a, p) and gamma_q_inva(a, q)\n\n"; + do{ + get_user_parameter_info(arg1, "x"); + get_user_parameter_info(arg2, "p"); + data.insert(gamma_inverse_generator_a(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else + { + arg2 = make_periodic_param(boost::math::ntl::RR(1), boost::math::ntl::RR(8), 7); + arg2.type |= boost::math::tools::dummy_param; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the incomplete gamma function:\n" + " gamma(a, z)\n\n"; + + do{ + get_user_parameter_info(arg1, "a"); + data.insert(igamma_data_generator(), arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + std::cout << "Enter name of test data file [default=igamma_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "igamma_data.ipp"; + std::ofstream ofs(line.c_str()); + write_code(ofs, data, "igamma_data"); + + return 0; +} + diff --git a/tools/igamma_temme_large_coef.cpp b/tools/igamma_temme_large_coef.cpp new file mode 100644 index 000000000..d84663890 --- /dev/null +++ b/tools/igamma_temme_large_coef.cpp @@ -0,0 +1,210 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace std; +using namespace NTL; +using namespace boost::math; + +// +// This program calculates the coefficients of the polynomials +// used for the regularized incomplete gamma functions gamma_p +// and gamma_q when parameter a is large, and sigma is small +// (where sigma = fabs(1 - x/a) ). +// +// See "The Asymptotic Expansion of the Incomplete Gamma Functions" +// N. M. Temme. +// Siam J. Math Anal. Vol 10 No 4, July 1979, p757. +// Coeffient calculation is described from Eq 3.8 (p762) onwards. +// + +// +// Alpha: +// +RR alpha(unsigned k) +{ + static map data; + if(data.empty()) + { + data[1] = 1; + } + + map::const_iterator pos = data.find(k); + if(pos != data.end()) + return (*pos).second; + // + // OK try and calculate the value: + // + RR result = alpha(k-1); + for(unsigned j = 2; j <= k-1; ++j) + { + result -= j * alpha(j) * alpha(k-j+1); + } + result /= (k+1); + data[k] = result; + return result; +} + +boost::math::ntl::RR gamma(unsigned k) +{ + static map data; + + map::const_iterator pos = data.find(k); + if(pos != data.end()) + return (*pos).second; + + boost::math::ntl::RR result = (k&1) ? -1 : 1; + + for(unsigned i = 1; i <= (2 * k + 1); i += 2) + result *= i; + result *= alpha(2 * k + 1); + data[k] = result; + return result; +} + +boost::math::ntl::RR Coeff(unsigned n, unsigned k) +{ + map > data; + if(data.empty()) + data[0][0] = boost::math::ntl::RR(-1) / 3; + + map >::const_iterator p1 = data.find(n); + if(p1 != data.end()) + { + map::const_iterator p2 = p1->second.find(k); + if(p2 != p1->second.end()) + { + return p2->second; + } + } + + // + // If we don't have the value, calculate it: + // + if(k == 0) + { + // special case: + boost::math::ntl::RR result = (n+2) * alpha(n+2); + data[n][k] = result; + return result; + } + // general case: + boost::math::ntl::RR result = gamma(k) * Coeff(n, 0) + (n+2) * Coeff(n+2, k-1); + data[n][k] = result; + return result; +} + +void calculate_terms(double sigma, double a, unsigned bits) +{ + cout << endl << endl; + cout << "Sigma: " << sigma << endl; + cout << "A: " << a << endl; + double lambda = 1 - sigma; + cout << "Lambda: " << lambda << endl; + double y = a * (-sigma - log1p(-sigma)); + cout << "Y: " << y << endl; + double z = -sqrt(2 * (-sigma - log1p(-sigma))); + cout << "Z: " << z << endl; + double dom = erfc(sqrt(y)) / 2; + cout << "Erfc term: " << dom << endl; + double lead = exp(-y) / sqrt(2 * constants::pi() * a); + cout << "Remainder factor: " << lead << endl; + double eps = ldexp(1.0, 1 - static_cast(bits)); + double target = dom * eps / lead; + cout << "Target smallest term: " << target << endl; + + unsigned max_n = 0; + + for(unsigned n = 0; n < 10000; ++n) + { + double term = tools::real_cast(Coeff(n, 0) * pow(z, (double)n)); + if(fabs(term) < target) + { + max_n = n-1; + break; + } + } + cout << "Max n required: " << max_n << endl; + + unsigned max_k; + for(unsigned k = 1; k < 10000; ++k) + { + double term = tools::real_cast(Coeff(0, k) * pow(a, -((double)k))); + if(fabs(term) < target) + { + max_k = k-1; + break; + } + } + cout << "Max k required: " << max_k << endl << endl; + + bool code = false; + cout << "Print code [0|1]? "; + cin >> code; + + int prec = 2 + (static_cast(bits) * 3010LL)/10000; + RR::SetOutputPrecision(prec); + + if(code) + { + cout << " T workspace[" << max_k+1 << "];\n\n"; + for(unsigned k = 0; k <= max_k; ++k) + { + cout << + " static const T C" << k << "[] = {\n"; + for(unsigned n = 0; n < 10000; ++n) + { + double term = tools::real_cast(Coeff(n, k) * pow(a, -((double)k)) * pow(z, (double)n)); + if(fabs(term) < target) + { + break; + } + cout << " " << Coeff(n, k) << "L,\n"; + } + cout << + " };\n" + " workspace[" << k << "] = tools::evaluate_polynomial(C" << k << ", z);\n\n"; + } + cout << " T result = tools::evaluate_polynomial(workspace, 1/a);\n\n"; + } +} + + +int main() +{ + RR::SetOutputPrecision(50); + RR::SetPrecision(1000); + + bool cont; + do{ + cont = false; + double sigma; + cout << "Enter max value for sigma (sigma = |1 - x/a|): "; + cin >> sigma; + double a; + cout << "Enter min value for a: "; + cin >> a; + unsigned precision; + cout << "Enter number of bits precision required: "; + cin >> precision; + + calculate_terms(sigma, a, precision); + + cout << "Try again[0|1]: "; + cin >> cont; + + }while(cont); + + + return 0; +} + diff --git a/tools/laguerre_data.cpp b/tools/laguerre_data.cpp new file mode 100644 index 000000000..bacc1aae3 --- /dev/null +++ b/tools/laguerre_data.cpp @@ -0,0 +1,106 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + + +template +std::tr1::tuple laguerre2_data(T n, T x) +{ + n = floor(n); + T r1 = laguerre(boost::math::tools::real_cast(n), x); + return std::tr1::make_tuple(n, x, r1); +} + +template +std::tr1::tuple laguerre3_data(T n, T m, T x) +{ + n = floor(n); + m = floor(m); + T r1 = laguerre(real_cast(n), real_cast(m), x); + return std::tr1::make_tuple(n, m, x, r1); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2, arg3; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + if(strcmp(argv[1], "--laguerre2") == 0) + { + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + if(0 == get_user_parameter_info(arg2, "x")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + + data.insert(&laguerre2_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else if(strcmp(argv[1], "--laguerre3") == 0) + { + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + if(0 == get_user_parameter_info(arg2, "m")) + return 1; + if(0 == get_user_parameter_info(arg3, "x")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3.type |= dummy_param; + + data.insert(&laguerre3_data, arg1, arg2, arg3); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + + std::cout << "Enter name of test data file [default=laguerre.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "laguerre.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/lanczos_generator.cpp b/tools/lanczos_generator.cpp new file mode 100644 index 000000000..f5bf59b63 --- /dev/null +++ b/tools/lanczos_generator.cpp @@ -0,0 +1,4368 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +// +// this is a sort of recursive include, since this file +// is used to create the contents of gamma.hpp. +// However, we will be using the "generic" version of gamma +// which doesn't use a lanczos approximation. +// +#include + +#include +#include +#include + +using boost::numeric::ublas::matrix; +using boost::numeric::ublas::vector; + +struct lanczos_spot_data +{ + int N; + double g; + double err; +}; + +lanczos_spot_data sweet_spots[] = { +// a few selected spots from Pugh's thesis: +3, 2.03209, 6.4e-7, +4, 3.655180, 8.5e-8, +9, 8.406094, 6.9e-15, +10, 9.656578, 2.1e-16, +11, 10.900511, 6.1e-18, +12, 12.066012, 1.1e-19, +13, 13.144565, 5.2e-21, +14, 13.726821, 4.0e-22, +15, 14.977863, 1.2e-23, +20, 20.298892, 7.8e-31, +21, 21.508926, 2.1e-32, +22, 22.618910, 1.8e-34, +23, 23.118012, 5.2e-35, + +// some more we've found, these are all the points where the first +// negleted term from the Lanczos series changes sign, there is one +// point just above that point, and one just below: + +3, 0.58613894134759903, 0.00036580426080686315, +3, 0.58613894879817963, 0.00036580416320668711, +3, 1.249951496720314, 4.2305536798953929e-005, +3, 1.2499515041708946, 4.2305534015367749e-005, +3, 1.8384982049465179, 1.164778449938813e-005, +3, 1.8384982123970985, 1.1647786529958048e-005, +3, 2.3276706635951996, 5.3732582230468998e-006, +3, 2.3276706710457802, 5.3732600294516588e-006, +3, 2.6038404405117035, 7.4060771171390379e-007, +3, 2.6038404479622841, 7.406058543248471e-007, +4, 0.62704569846391678, 0.00012800533557793425, +4, 0.62704570591449738, 0.00012800531551557324, +4, 1.3344274088740349, 9.0166140418663663e-006, +4, 1.3344274163246155, 9.0166116338646694e-006, +4, 1.9880444705486298, 1.4955557778725745e-006, +4, 1.9880444779992104, 1.4955557079994109e-006, +4, 2.5835030898451805, 4.5877676859600527e-007, +4, 2.5835030972957611, 4.5877667876325684e-007, +4, 3.1086832508444786, 2.1441725587707906e-007, +4, 3.1086832582950592, 2.1441723688775765e-007, +4, 3.5476611852645874, 1.0813760990515544e-007, +4, 3.547661192715168, 1.0813766234070327e-007, +4, 3.6492579728364944, 1.047642251124206e-007, +4, 3.649257980287075, 1.0476415796074403e-007, +5, 0.65435918420553207, 5.5274429026631278e-005, +5, 0.65435919165611267, 5.527441185903385e-005, +5, 1.389187254011631, 2.6307833416554052e-006, +5, 1.3891872614622116, 2.6307826816540255e-006, +5, 2.0801975876092911, 2.9014694764756466e-007, +5, 2.0801975950598717, 2.9014696142909125e-007, +5, 2.7272208333015442, 5.9176544536206063e-008, +5, 2.7272208407521248, 5.9176528019009275e-008, +5, 3.3269794136285782, 1.9379184595472579e-008, +5, 3.3269794210791588, 1.9379189057179851e-008, +5, 3.8722873777151108, 9.0123694745032944e-009, +5, 3.8722873851656914, 9.0123661007867372e-009, +5, 4.3516943231225014, 4.9816097608842378e-009, +5, 4.351694330573082, 4.9816087871575021e-009, +6, 0.67425807565450668, 2.7371903086326379e-005, +6, 0.67425808310508728, 2.7371893712684938e-005, +6, 1.4284561350941658, 9.4138428977455083e-007, +6, 1.4284561425447464, 9.4138417896969612e-007, +6, 2.1447814926505089, 7.4109049995920916e-008, +6, 2.1447815001010895, 7.4109044064461768e-008, +6, 2.8244904428720474, 1.0708333024058614e-008, +6, 2.824490450322628, 1.0708333661155651e-008, +6, 3.4669468700885773, 2.4940116660002578e-009, +6, 3.4669468775391579, 2.4940118560592828e-009, +6, 4.0696377158164978, 8.5267622527507512e-010, +6, 4.0696377232670784, 8.5267624048985822e-010, +6, 4.6278460025787354, 3.9323405231792876e-010, +6, 4.6278460100293159, 3.9323393550021793e-010, +6, 5.1345816180109978, 2.1943462473419721e-010, +6, 5.1345816254615784, 2.1943458105218707e-010, +6, 5.5917193591594696, 1.242916896389048e-010, +6, 5.5917193666100502, 1.2429175679293363e-010, +7, 0.68959406018257141, 1.4934177419305624e-005, +7, 0.68959406763315201, 1.4934171696393048e-005, +7, 1.4584139510989189, 3.8891482410396904e-007, +7, 1.4584139585494995, 3.8891475436265016e-007, +7, 2.1934122517704964, 2.2977989848570384e-008, +7, 2.193412259221077, 2.2977982819591052e-008, +7, 2.8964230641722679, 2.4742239281355561e-009, +7, 2.8964230716228485, 2.4742234478616864e-009, +7, 3.5678729563951492, 4.2816138965285379e-010, +7, 3.5678729638457298, 4.2816128067858218e-010, +7, 4.2069913893938065, 1.0931003472338519e-010, +7, 4.2069913968443871, 1.0931004659777642e-010, +7, 4.8118086308240891, 3.8518213120095777e-011, +7, 4.8118086382746696, 3.8518212172324096e-011, +7, 5.3790060579776764, 1.7613602961109952e-011, +7, 5.379006065428257, 1.7613603275796266e-011, +7, 5.9039439857006073, 9.7098281187573355e-012, +7, 5.9039439931511879, 9.7098271522398387e-012, +7, 6.3838147521018982, 5.7878801570305524e-012, +7, 6.3838147595524788, 5.7878771973838702e-012, +7, 6.7833591029047966, 2.829092368392454e-012, +7, 6.7833591103553772, 2.8290901754246269e-012, +8, 0.70188850909471512, 8.7610752024780369e-006, +8, 0.70188851654529572, 8.7610704217796693e-006, +8, 1.4822541922330856, 1.7881701068328982e-007, +8, 1.4822541996836662, 1.7881694190606741e-007, +8, 2.231784276664257, 8.2272386690167009e-009, +8, 2.2317842841148376, 8.2272367814664617e-009, +8, 2.9525627419352531, 6.8591536474980462e-010, +8, 2.9525627493858337, 6.8591515664364383e-010, +8, 3.6455128863453865, 9.1549235855884194e-011, +8, 3.6455128937959671, 9.1549217088498416e-011, +8, 4.3106606528162956, 1.8015466799942756e-011, +8, 4.3106606602668762, 1.8015467001952356e-011, +8, 4.9472371935844421, 4.918149362493385e-012, +8, 4.9472372010350227, 4.9181495178774722e-012, +8, 5.553666315972805, 1.7720036810370511e-012, +8, 5.5536663234233856, 1.7720035779693502e-012, +8, 6.1274929493665695, 8.0403304333886761e-013, +8, 6.1274929568171501, 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3.4143034915050112e-65, +48, 35.805177047848701, 3.4143040521491458e-65, +48, 36.410855405032635, 1.5004193222265229e-65, +48, 36.410855412483215, 1.5004189889736636e-65, +48, 37.011237367987633, 6.8526723104762059e-66, +48, 37.011237375438213, 6.8526715926507402e-66, +}; + +// +// A bunch of helper functions used for calculating the coefficients: +// +boost::math::ntl::RR factorial(unsigned n) +{ + static boost::array, 201> result; + static bool init = false; + if(!init) + { + unsigned k = 1; + boost::math::ntl::RR fact = 1; + do{ + result[k-1][0] = fact; + fact *= k++; + } + while(k < result.size()); + init = true; + } + //static const int array_size = result.size(); + if(n < result.size()) + return result[n][0]; + + unsigned i = (unsigned)result.size()-1; + boost::math::ntl::RR r = result[i][0]; + while(i < n) + r *= ++i; + return r; +} + +template +T binomial(int n, int k, T) +{ + T result; + if(k < 0) + { + result = 0; + } + else if(k > n) + { + result = 1; + } + else + { + result = factorial(n); + result /= factorial(k); + result /= factorial(n-k); + } + return result; +} +// +// Functions for creating the matrices that generate the coefficents. +// See http://my.fit.edu/~gabdo/gamma.txt and http://www.rskey.org/gamma.htm +// +template +matrix make_B(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if(i == 0) + result(i, j) = 1; + else if(j >= i) + { + T r = binomial(i+j-1, j-i, T()); + if((j-i) %2) + r = -r; + result(i, j) = r; + } + else + result(i, j) = 0; + } + } + return result; +} + +template +matrix make_C(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if((i==0) && (j == 0)) + result(i, j) = 0.5; + else if(j > i) + result(i, j) = 0; + else + { + T r = 0; + for(unsigned k = 0; k <= i; ++k) + { + r += binomial(2*i, 2*k, T()) * binomial(k, k+j-i, T()); + } + if((i-j)%2) + r = -r; + result(i, j) = r; + } + } + } + return result; +} + +template +matrix make_D(unsigned n, T) +{ + matrix result(n, n); + for(unsigned i = 0; i < n; ++i) + { + for(unsigned j = 0; j < n; ++j) + { + if(i != j) + result(i, j) = 0; + else if(i == 0) + result(i, j) = 1; + else if(i == 1) + result(i, j) = -1; + else + { + T r = result(i-1, i-1) * (2 * (2*i - 1)); + r /= (i-1); + result(i, j) = r; + } + } + } + return result; +} + +template +matrix make_F(unsigned n, T g) +{ + using namespace std; + matrix result(n, 1); + for(unsigned i = 0; i < n; ++i) + { + T r = factorial(2 * i); + r /= factorial(i); + r *= exp(double(i) + g + 0.5); + r /= ldexp(1.0, (2*i)-1); + r /= pow(g + i + 0.5, (long)i); + r /= sqrt(g + i + 0.5); + result(i, 0) = r; + } + return result; +} + +template +struct lanczos_info +{ + int n; // number of coefficients + T r; // the arbitrary parameter + std::vector c; // the coefficients + T err; // error found +}; +// +// Create the coefficients, caching previous matrix values as appropriate: +// +template +lanczos_info generate_lanczos(unsigned n, T g) +{ + static unsigned last_n = 0; + static matrix last_prefix; + + matrix B, C, D; + matrix F, P, E, T1, T2; + + if(n == last_n) + { + F = make_F(n, g); + P = prod(last_prefix, F); + } + else + { + B = make_B(n, g); + C = make_C(n, g); + D = make_D(n, g); + F = make_F(n, g); + + T1 = prod(D, B); + T2 = prod(T1, C); + P = prod(T2, F); + + last_n = n; + last_prefix = T2; + } + + lanczos_info result; + result.n = n; + result.r = g; + for(unsigned i = 0; i < n; ++i) + result.c.push_back(P(i, 0)); + + return result; +} +// +// Returns the factorials and half factorials: +// +template +std::vector > const & get_test_data() +{ + // returns factorials and half factorials in the range [0, 100]: + static std::vector > data; + if(data.empty()) + { + T fact = 1; + int k = 1; + std::vector item; + do + { + data.push_back(item); + data.back().push_back(k-1); + data.back().push_back(fact); + data.back().push_back(log(fact)); + fact = fact * T(k++); + }while(k < 100); + + fact = 0.5; + T srpi = sqrt(boost::math::constants::pi()); + T mul = 1.5; + do{ + data.push_back(item); + data.back().push_back(mul-1); + data.back().push_back(fact*srpi); + data.back().push_back(log(fact*srpi)); + fact *= mul; + mul += 1; + }while(mul < 100); + } + + return data; +} + +struct sort_on_0 +{ + template + bool operator ()(const T& a, const T& b) + { + return ((a[0]) < (b[0])); + } +}; +// +// Get test data for points near 0, 1, 2 etc. +template +std::vector > const & get_test_data_near_x(T x) +{ + // + // this *must* be called with a *very* high precision + // type T in order to stand any chance of long double + // precision results: + // + using namespace std; + using namespace boost::math; + static std::vector > data; + static T last_x(-10000); + if(last_x != x) + data.clear(); + if(!data.size()) + { + last_x = x; + std::vector item; + for(int i = -4; i > -20; --i) + { + double val = ldexp(2.0, i); + data.push_back(item); + data.back().push_back(val+x); + data.back().push_back(tgamma(T(val)+x)); + data.back().push_back(lgamma(T(val)+x)); + data.push_back(item); + data.back().push_back(-val+x); + data.back().push_back(tgamma(T(-val)+x)); + data.back().push_back(lgamma(T(-val)+x)); + } + T v = x-0.5; + T interval = T(1)/8; + while(v <= x+0.5) + { + if((floor(v) != v) || (v > 0)) + { + data.push_back(item); + data.back().push_back(v); + data.back().push_back(tgamma(v)); + data.back().push_back(lgamma(v)); + } + v += interval; + } + // + // sort data: + // + std::sort(data.begin(), data.end(), sort_on_0()); + } + return data; +} + +template +std::vector > const & get_test_data_near_1() +{ + return get_test_data_near_x(T(1)); +} + +template +std::vector > const & get_test_data_near_2() +{ + return get_test_data_near_x(T(2)); +} + +// +// Converts Lanczos approximation into rational form via +// polynomial arithmetic: +// +template +struct lanczos_rational +{ + std::vector num, denom; + int N; + T g; + + T gamma(T z) + { + using namespace std; + BOOST_ASSERT(num.size() == denom.size()); + BOOST_ASSERT(num.size() == N); + T zgh = z + g - T(0.5); + T prefix = pow(zgh, T(z - 0.5)) / exp(zgh); + T s1, s2; + if(z < 1) + { + s1 = num[N-1]; + s2 = denom[N-1]; + for(unsigned i = N-2; i >= 0; --i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += num[i]; + } + s1 /= s2; + return prefix * s1; + } + else + { + z = 1/z; + s1 = num[0]; + s2 = denom[0]; + for(int i = 1; i < N; ++i) + { + s1 *= z; + s2 *= z; + s1 += num[i]; + s2 += denom[i]; + } + s1 /= s2; + return prefix * s1; + } + } + T factorial(T z) + { + return this->gamma(z+1); + } + template + lanczos_rational(const lanczos_info& info) + { + U l_denom_coef[2] = { 0, 1 }; + boost::math::tools::polynomial l_denom(l_denom_coef, 1), l_num(info.c[1]); + for(unsigned i = 2; i < info.c.size(); ++i) + { + l_denom_coef[0] = i - 1; + boost::math::tools::polynomial bot(l_denom_coef, 1), top(info.c[i]); + l_num *= bot; + top *= l_denom; + l_denom *= bot; + l_num += top; + } + l_num += l_denom * info.c[0]; + for(unsigned i = 0; i < info.c.size(); ++i) + { + num.push_back(boost::math::tools::real_cast(l_num[i])); + denom.push_back(boost::math::tools::real_cast(l_denom[i])); + } + //std::cout << l_num << std::endl; + //std::cout << l_denom << std::endl; + BOOST_ASSERT(num.size() == l_num.degree()+1); + BOOST_ASSERT(denom.size() == l_denom.degree()+1); + g = boost::math::tools::real_cast(info.r); + N = info.n; + /* + for(unsigned i = 0; i < num.size(); ++i) + std::cout << num[i] << " "; + std::cout << std::endl; + for(unsigned i = 0; i < denom.size(); ++i) + std::cout << denom[i] << " "; + std::cout << std::endl; + */ + } +}; +// +// Code to test an approximation against the factorials and +// half factorials, returns the max error found: +// +template +T get_max_error(const lanczos_info& dat, R) +{ + T max_error = 0; + std::vector > const & tests = get_test_data(); + lanczos_rational rational(dat); + + for(unsigned i = 0; i < tests.size(); ++i) + { + R input = boost::math::tools::real_cast(tests[i][0]); + if(std::numeric_limits::is_specialized && (boost::math::tools::real_cast(tests[i][1]) > (std::numeric_limits::max)())) + continue; + + R gamr = rational.factorial(input); + if(std::numeric_limits::is_specialized && (gamr > (std::numeric_limits::max)())) + continue; + T gam = gamr; + T exp = tests[i][1]; + T err = boost::math::tools::relative_error(gam, exp); + if(err > max_error) + max_error = err; + } + + return max_error; +} +// +// This is what prints the "best" approximation out as code: +// +template +void print_code(const lanczos_info& l, const char* name) +{ + using namespace std; + lanczos_info l2(l); + T factor = exp(l.r); + for(unsigned k = 0; k < l2.c.size(); ++k) + l2.c[k] /= factor; + + lanczos_rational rat(l); + T max_term = 0; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + if(rat.denom[i] > max_term) + max_term = rat.denom[i]; + } + const char* denom_type; + const char* cast_type; + const char* suffix_type; + if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint16_t"; + cast_type = "static_cast"; + suffix_type = "u"; + } + else if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint32_t"; + cast_type = "static_cast"; + suffix_type = "u"; + } +#ifdef BOOST_HAS_LONG_LONG + else if(max_term < (std::numeric_limits::max)()) + { + denom_type = "boost::uint64_t"; + cast_type = ""; + suffix_type = "uLL"; + } +#endif + else + { + denom_type = "T"; + cast_type = "static_cast"; + suffix_type = "L"; + } + + std::cout << + "//\n" + "// Lanczos Coefficients for N=" << l.n << " G=" << l.r << "\n" + "// Max experimental error (with "; + if(std::strlen(name) == 0) + std::cout << "arbitary"; + else + std::cout << name; + std::cout << " precision arithmetic) " << l.err << + "\n// Generated with compiler: " << BOOST_COMPILER << " on " << BOOST_PLATFORM << " at " << __DATE__ << "\n" + "//\n" + "struct lanczos" << l.n << name << "\n" + "{\n" + " template \n" + " static T lanczos_sum(const T& z)\n" + " {\n" + " static const T num[" << rat.num.size() << "] = {\n"; + + for(unsigned i = 0; i < rat.num.size(); ++i) + { + std::cout << " static_cast(" << rat.num[i] << "L)"; + if(i != rat.num.size() - 1) + std::cout << ","; + std::cout << "\n"; + } + std::cout << + " };\n" + " static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n"; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << ")"; + if(i != rat.denom.size() - 1) + std::cout << ","; + std::cout << "\n"; + } + std::cout << + " };\n" + " return boost::math::tools::evaluate_rational(num, denom, z);\n" + " }\n\n" + " template \n" + " static T lanczos_sum_expG_scaled(const T& z)\n" + " {\n" + " static const T num[" << rat.num.size() << "] = {\n"; + + for(unsigned i = 0; i < rat.num.size(); ++i) + { + std::cout << " static_cast(" << (rat.num[i]/factor) << "L)"; + if(i != rat.num.size() - 1) + std::cout << ","; + std::cout << "\n"; + } + std::cout << + " };\n" + " static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n"; + for(unsigned i = 0; i < rat.denom.size(); ++i) + { + std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << ")"; + if(i != rat.denom.size() - 1) + std::cout << ","; + std::cout << "\n"; + } + std::cout << + " };\n" + " return boost::math::tools::evaluate_rational(num, denom, z);\n" + " }\n\n"; + + std::cout << + "\n template\n" + " static T lanczos_sum_near_1(const T& dz)\n" + " {\n" + " static const T d[" << l2.n-1 << "] = {\n"; + + factor = sqrt((l.r + 0.5)/boost::math::constants::e()) / exp(l.r); + + for(int i = 1; i < l.n; ++i) + { + std::cout << " static_cast(" << l.c[i]*factor << "L)"; + if(i != l.n - 1) + std::cout << ","; + std::cout << "\n"; + } + std::cout << + " };\n" + " T result = 0;\n" + " for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n" + " {\n" + " result += (-d[k-1]*dz)/(k*dz + k*k);\n" + " }\n" + " return result;\n" + " }\n"; + + std::cout << + "\n template\n" + " static T lanczos_sum_near_2(const T& dz)\n" + " {\n" + " static const T d[" << l2.n-1 << "] = {\n"; + + factor = pow(boost::math::constants::e()/(l.r+1.5), T(1.5)) * exp(l.r); + //pow((l.r + 1.5)/boost::math::constants::e(), 1.5) / exp(l.r); + + for(int i = 1; i < l.n; ++i) + { + std::cout << " static_cast(" << l.c[i]/factor << "L),\n"; + } + std::cout << + " };\n" + " T result = 0;\n" + " T z = dz + 2;\n" + " for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n" + " {\n" + " result += (-d[k-1]*dz)/(z + k*z + k*k - 1);\n" + " }\n" + " return result;\n" + " }\n\n" + " static double g(){ return " << l.r << "; }\n" + "};\n\n"; +} +// +// Print out the test values: +// +void print_test_values(const std::vector >& v, const char* name, int offset = 1) +{ + std::cout << "#define SC_(x) static_cast(BOOST_JOIN(x, L))\n"; + std::cout << + " static const boost::array, " << v.size() << "> " << name << " = {\n"; + for(unsigned i = 0; i < v.size(); ++i) + { + std::cout << " SC_(" << (v[i][0] + offset) << "), SC_(" << v[i][1] << "), SC_(" << v[i][2] << "),\n"; + } + std::cout << " };\n#undef SC_\n\n"; +} +// +// Get the error for a specific approximation, and print out it's code: +// +void calculate_lanczos_spot(int n, boost::math::ntl::RR r, const char* suffix = "") +{ + lanczos_info info = generate_lanczos(n, r); + // note error is calculated at high precision: + info.err = get_max_error(info, r); + print_code(info, suffix); +} +// +// Scan the sweet-spots for the best approximation: +// +template +void find_best_lanczos(const char* name, T eps, int max_scan = 100) +{ + using namespace std; + + lanczos_info best; + best.err = 100; // best case had better be better than this! + for(int i = 0; i < sizeof(sweet_spots)/sizeof(sweet_spots[0]); ++i) + { + if((sweet_spots[i].err < eps*10) && (sweet_spots[i].N < max_scan)) + { + lanczos_info info = generate_lanczos(sweet_spots[i].N, boost::math::ntl::RR(sweet_spots[i].g)); + boost::math::ntl::RR err = get_max_error(info, eps); + if(err/eps < 1000) + { + std::cout << sweet_spots[i].N << " " << sweet_spots[i].g << " " << err/eps << std::endl; + } + if(err < best.err) + { + best = info; + best.err = err; + } + } + } + std::cout << std::endl; + + print_code(best, name); +} +int test_main(int argc, char*argv []) +{ + bool test_double(false), test_long(false), test_float(false), test_quad(false), spots(false), test_data(false); + + if(argc < 2) + { + std::cout << + "Useage:\n" + " -float test type float for the best approximation\n" + " -double test type double for the best approximation\n" + " -long-double test type long double for the best approximation\n" + " -spots print out the best cases found in previous runs\n" + " -data print out the test data\n" << std::flush; + return 0; + } + + for(int i = 1; i < argc; ++i) + { + if(std::strcmp(argv[i], "-float") == 0) + test_float = true; + else if(std::strcmp(argv[i], "-double") == 0) + test_double = true; + else if(std::strcmp(argv[i], "-long-double") == 0) + test_long = true; + else if(std::strcmp(argv[i], "-quad-float") == 0) + test_quad = true; + else if(std::strcmp(argv[i], "-spots") == 0) + spots = true; + else if(std::strcmp(argv[i], "-data") == 0) + test_data = true; + } + + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + if(spots) + { + // these are optimal N and R from Pugh: + calculate_lanczos_spot(61, 63.192152); + calculate_lanczos_spot(31, 32.080670); + calculate_lanczos_spot(6, 5.581); + calculate_lanczos_spot(11, 10.900511); + calculate_lanczos_spot(13, 13.144565); + calculate_lanczos_spot(22, 22.61891); + // these are the best cases we've already determined + // from previous runs: + calculate_lanczos_spot(6, 1.428456135094165802001953125); + calculate_lanczos_spot(13, 6.024680040776729583740234375); + calculate_lanczos_spot(17, 12.2252227365970611572265625); + boost::math::ntl::RR::SetOutputPrecision(90); + calculate_lanczos_spot(24, 20.3209821879863739013671875); + } + if(test_float) + { + find_best_lanczos("float", std::numeric_limits::epsilon()); + } + if(test_double) + { + find_best_lanczos("double", std::numeric_limits::epsilon()); + } + + if(test_long) + { + find_best_lanczos("long_double", std::numeric_limits::epsilon()); + } + if(test_quad) + { + //find_best_lanczos("quad_float", pow(NTL::quad_float(2), NTL::quad_float(-105))); + } + if(test_data) + { + std::cout << "Test Data follows:\n\n"; + + boost::math::ntl::RR::SetPrecision(1000); + + std::vector > const & tests = get_test_data(); + print_test_values(tests, "factorials"); + print_test_values(get_test_data_near_1(), "near_1", 0); + print_test_values(get_test_data_near_2(), "near_2", 0); + print_test_values(get_test_data_near_x(boost::math::ntl::RR(0)), "near_0", 0); + print_test_values(get_test_data_near_x(boost::math::ntl::RR(-10)), "near_m10", 0); + print_test_values(get_test_data_near_x(boost::math::ntl::RR(-55)), "near_m55", 0); + } + return 0; +} + + + diff --git a/tools/legendre_data.cpp b/tools/legendre_data.cpp new file mode 100644 index 000000000..0bc2b08d3 --- /dev/null +++ b/tools/legendre_data.cpp @@ -0,0 +1,105 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + + +template +std::tr1::tuple legendre_p_data(T n, T x) +{ + n = floor(n); + T r1 = legendre_p(boost::math::tools::real_cast(n), x); + T r2 = legendre_q(boost::math::tools::real_cast(n), x); + return std::tr1::make_tuple(n, x, r1, r2); +} + +template +std::tr1::tuple assoc_legendre_p_data(T n, T x) +{ + static tr1::mt19937 r; + int l = real_cast(floor(n)); + tr1::uniform_int<> ui((std::max)(-l, -40), (std::min)(l, 40)); + int m = ui(r); + T r1 = legendre_p(l, m, x); + return std::tr1::make_tuple(n, m, x, r1); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + if(strcmp(argv[1], "--legendre2") == 0) + { + do{ + if(0 == get_user_parameter_info(arg1, "l")) + return 1; + if(0 == get_user_parameter_info(arg2, "x")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + + data.insert(&legendre_p_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else if(strcmp(argv[1], "--legendre3") == 0) + { + do{ + if(0 == get_user_parameter_info(arg1, "l")) + return 1; + if(0 == get_user_parameter_info(arg2, "x")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + + data.insert(&assoc_legendre_p_data, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + + std::cout << "Enter name of test data file [default=legendre_p.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "legendre_p.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/log1p_expm1_data.cpp b/tools/log1p_expm1_data.cpp new file mode 100644 index 000000000..92a88ec7d --- /dev/null +++ b/tools/log1p_expm1_data.cpp @@ -0,0 +1,63 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include + +#include +#include +#include + +using namespace boost::math::tools; +using namespace std; + +struct data_generator +{ + std::tr1::tuple operator()(boost::math::ntl::RR z) + { + return std::tr1::make_tuple(boost::math::log1p(z), boost::math::expm1(z)); + } +}; + +int main(int argc, char* argv[]) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1; + test_data data; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the log1p and expm1 functions:\n\n"; + + bool cont; + std::string line; + + do{ + if(0 == get_user_parameter_info(arg1, "z")) + return 1; + data.insert(data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=log1p_expm1_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "log1p_expm1_data.ipp"; + std::ofstream ofs(line.c_str()); + ofs << std::scientific; + write_code(ofs, data, "log1p_expm1_data"); + + return 0; +} + + + + diff --git a/tools/ntl_rr_digamma.hpp b/tools/ntl_rr_digamma.hpp new file mode 100644 index 000000000..faf859210 --- /dev/null +++ b/tools/ntl_rr_digamma.hpp @@ -0,0 +1,292 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_NTL_DIGAMMA +#define BOOST_MATH_NTL_DIGAMMA + +#include +#include +#include +#include + +namespace boost{ namespace math{ + +boost::math::ntl::RR digamma_imp(boost::math::ntl::RR x) +{ + static const boost::math::ntl::RR P[61] = { + boost::lexical_cast("0.6660133691143982067148122682345055274952e81"), + boost::lexical_cast("0.6365271516829242456324234577164675383137e81"), + boost::lexical_cast("0.2991038873096202943405966144203628966976e81"), + boost::lexical_cast("0.9211116495503170498076013367421231351115e80"), + boost::lexical_cast("0.2090792764676090716286400360584443891749e80"), + boost::lexical_cast("0.3730037777359591428226035156377978092809e79"), + boost::lexical_cast("0.5446396536956682043376492370432031543834e78"), + boost::lexical_cast("0.6692523966335177847425047827449069256345e77"), + boost::lexical_cast("0.7062543624100864681625612653756619116848e76"), + boost::lexical_cast("0.6499914905966283735005256964443226879158e75"), + boost::lexical_cast("0.5280364564853225211197557708655426736091e74"), + boost::lexical_cast("0.3823205608981176913075543599005095206953e73"), + boost::lexical_cast("0.2486733714214237704739129972671154532415e72"), + boost::lexical_cast("0.1462562139602039577983434547171318011675e71"), + boost::lexical_cast("0.7821169065036815012381267259559910324285e69"), + boost::lexical_cast("0.3820552182348155468636157988764435365078e68"), + boost::lexical_cast("0.1711618296983598244658239925535632505062e67"), + boost::lexical_cast("0.7056661618357643731419080738521475204245e65"), + boost::lexical_cast("0.2685246896473614017356264531791459936036e64"), + boost::lexical_cast("0.9455168125599643085283071944864977592391e62"), + boost::lexical_cast("0.3087541626972538362237309145177486236219e61"), + boost::lexical_cast("0.9367928873352980208052601301625005737407e59"), + boost::lexical_cast("0.2645306130689794942883818547314327466007e58"), + boost::lexical_cast("0.6961815141171454309161007351079576190079e56"), + boost::lexical_cast("0.1709637824471794552313802669803885946843e55"), + boost::lexical_cast("0.3921553258481531526663112728778759311158e53"), + boost::lexical_cast("0.8409006354449988687714450897575728228696e51"), + boost::lexical_cast("0.1686755204461325935742097669030363344927e50"), + boost::lexical_cast("0.3166653542877070999007425197729038754254e48"), + boost::lexical_cast("0.5566029092358215049069560272835654229637e46"), + boost::lexical_cast("0.9161766287916328133080586672953875116242e44"), + boost::lexical_cast("1412317772330871298317974693514430627922000"), + boost::lexical_cast("20387991986727877473732570146112459874790"), + boost::lexical_cast("275557928713904105182512535678580359839.3"), + boost::lexical_cast("3485719851040516559072031256589598330.723"), + boost::lexical_cast("41247046743564028399938106707656877.40859"), + boost::lexical_cast("456274078125709314602601667471879.0147312"), + boost::lexical_cast("4714450683242899367025707077155.310613012"), + boost::lexical_cast("45453933537925041680009544258.75073849996"), + boost::lexical_cast("408437900487067278846361972.302331241052"), + boost::lexical_cast("3415719344386166273085838.705771571751035"), + boost::lexical_cast("26541502879185876562320.93134691487351145"), + boost::lexical_cast("191261415065918713661.1571433274648417668"), + boost::lexical_cast("1275349770108718421.645275944284937551702"), + boost::lexical_cast("7849171120971773.318910987434906905704272"), + boost::lexical_cast("44455946386549.80866460312682983576538056"), + boost::lexical_cast("230920362395.3198137186361608905136598046"), + boost::lexical_cast("1095700096.240863858624279930600654130254"), + boost::lexical_cast("4727085.467506050153744334085516289728134"), + boost::lexical_cast("18440.75118859447173303252421991479005424"), + boost::lexical_cast("64.62515887799460295677071749181651317052"), + boost::lexical_cast("0.201851568864688406206528472883512147547"), + boost::lexical_cast("0.0005565091674187978029138500039504078098143"), + boost::lexical_cast("0.1338097668312907986354698683493366559613e-5"), + boost::lexical_cast("0.276308225077464312820179030238305271638e-8"), + boost::lexical_cast("0.4801582970473168520375942100071070575043e-11"), + boost::lexical_cast("0.6829184144212920949740376186058541800175e-14"), + boost::lexical_cast("0.7634080076358511276617829524639455399182e-17"), + boost::lexical_cast("0.6290035083727140966418512608156646142409e-20"), + boost::lexical_cast("0.339652245667538733044036638506893821352e-23"), + boost::lexical_cast("0.9017518064256388530773585529891677854909e-27") + }; + static const boost::math::ntl::RR Q[61] = { + boost::lexical_cast("0"), + boost::lexical_cast("0.1386831185456898357379390197203894063459e81"), + boost::lexical_cast("0.6467076379487574703291056110838151259438e81"), + boost::lexical_cast("0.1394967823848615838336194279565285465161e82"), + boost::lexical_cast("0.1872927317344192945218570366455046340458e82"), + boost::lexical_cast("0.1772461045338946243584650759986310355937e82"), + boost::lexical_cast("0.1267294892200258648315971144069595555118e82"), + boost::lexical_cast("0.7157764838362416821508872117623058626589e81"), + boost::lexical_cast("0.329447266909948668265277828268378274513e81"), + boost::lexical_cast("0.1264376077317689779509250183194342571207e81"), + boost::lexical_cast("0.4118230304191980787640446056583623228873e80"), + boost::lexical_cast("0.1154393529762694616405952270558316515261e80"), + boost::lexical_cast("0.281655612889423906125295485693696744275e79"), + boost::lexical_cast("0.6037483524928743102724159846414025482077e78"), + boost::lexical_cast("0.1145927995397835468123576831800276999614e78"), + boost::lexical_cast("0.1938624296151985600348534009382865995154e77"), + boost::lexical_cast("0.293980925856227626211879961219188406675e76"), + boost::lexical_cast("0.4015574518216966910319562902099567437832e75"), + boost::lexical_cast("0.4961475457509727343545565970423431880907e74"), + boost::lexical_cast("0.5565482348278933960215521991000378896338e73"), + boost::lexical_cast("0.5686112924615820754631098622770303094938e72"), + boost::lexical_cast("0.5305988545844796293285410303747469932856e71"), + boost::lexical_cast("0.4533363413802585060568537458067343491358e70"), + boost::lexical_cast("0.3553932059473516064068322757331575565718e69"), + boost::lexical_cast("0.2561198565218704414618802902533972354203e68"), + boost::lexical_cast("0.1699519313292900324098102065697454295572e67"), + boost::lexical_cast("0.1039830160862334505389615281373574959236e66"), + boost::lexical_cast("0.5873082967977428281000961954715372504986e64"), + boost::lexical_cast("0.3065255179030575882202133042549783442446e63"), + boost::lexical_cast("0.1479494813481364701208655943688307245459e62"), + boost::lexical_cast("0.6608150467921598615495180659808895663164e60"), + boost::lexical_cast("0.2732535313770902021791888953487787496976e59"), + boost::lexical_cast("0.1046402297662493314531194338414508049069e58"), + boost::lexical_cast("0.3711375077192882936085049147920021549622e56"), + boost::lexical_cast("0.1219154482883895482637944309702972234576e55"), + boost::lexical_cast("0.3708359374149458741391374452286837880162e53"), + boost::lexical_cast("0.1044095509971707189716913168889769471468e52"), + boost::lexical_cast("0.271951506225063286130946773813524945052e50"), + boost::lexical_cast("0.6548016291215163843464133978454065823866e48"), + boost::lexical_cast("0.1456062447610542135403751730809295219344e47"), + boost::lexical_cast("0.2986690175077969760978388356833006028929e45"), + boost::lexical_cast("5643149706574013350061247429006443326844000"), + boost::lexical_cast("98047545414467090421964387960743688053480"), + boost::lexical_cast("1563378767746846395507385099301468978550"), + boost::lexical_cast("22823360528584500077862274918382796495"), + boost::lexical_cast("304215527004115213046601295970388750"), + boost::lexical_cast("3690289075895685793844344966820325"), + boost::lexical_cast("40584512015702371433911456606050"), + boost::lexical_cast("402834190897282802772754873905"), + boost::lexical_cast("3589522158493606918146495750"), + boost::lexical_cast("28530557707503483723634725"), + boost::lexical_cast("200714561335055753000730"), + boost::lexical_cast("1237953783437761888641"), + boost::lexical_cast("6614698701445762950"), + boost::lexical_cast("30155495647727505"), + boost::lexical_cast("114953256021450"), + boost::lexical_cast("356398020013"), + boost::lexical_cast("863113950"), + boost::lexical_cast("1531345"), + boost::lexical_cast("1770"), + boost::lexical_cast("1") + }; + static const boost::math::ntl::RR PD[60] = { + boost::lexical_cast("0.6365271516829242456324234577164675383137e81"), + 2*boost::lexical_cast("0.2991038873096202943405966144203628966976e81"), + 3*boost::lexical_cast("0.9211116495503170498076013367421231351115e80"), + 4*boost::lexical_cast("0.2090792764676090716286400360584443891749e80"), + 5*boost::lexical_cast("0.3730037777359591428226035156377978092809e79"), + 6*boost::lexical_cast("0.5446396536956682043376492370432031543834e78"), + 7*boost::lexical_cast("0.6692523966335177847425047827449069256345e77"), + 8*boost::lexical_cast("0.7062543624100864681625612653756619116848e76"), + 9*boost::lexical_cast("0.6499914905966283735005256964443226879158e75"), + 10*boost::lexical_cast("0.5280364564853225211197557708655426736091e74"), + 11*boost::lexical_cast("0.3823205608981176913075543599005095206953e73"), + 12*boost::lexical_cast("0.2486733714214237704739129972671154532415e72"), + 13*boost::lexical_cast("0.1462562139602039577983434547171318011675e71"), + 14*boost::lexical_cast("0.7821169065036815012381267259559910324285e69"), + 15*boost::lexical_cast("0.3820552182348155468636157988764435365078e68"), + 16*boost::lexical_cast("0.1711618296983598244658239925535632505062e67"), + 17*boost::lexical_cast("0.7056661618357643731419080738521475204245e65"), + 18*boost::lexical_cast("0.2685246896473614017356264531791459936036e64"), + 19*boost::lexical_cast("0.9455168125599643085283071944864977592391e62"), + 20*boost::lexical_cast("0.3087541626972538362237309145177486236219e61"), + 21*boost::lexical_cast("0.9367928873352980208052601301625005737407e59"), + 22*boost::lexical_cast("0.2645306130689794942883818547314327466007e58"), + 23*boost::lexical_cast("0.6961815141171454309161007351079576190079e56"), + 24*boost::lexical_cast("0.1709637824471794552313802669803885946843e55"), + 25*boost::lexical_cast("0.3921553258481531526663112728778759311158e53"), + 26*boost::lexical_cast("0.8409006354449988687714450897575728228696e51"), + 27*boost::lexical_cast("0.1686755204461325935742097669030363344927e50"), + 28*boost::lexical_cast("0.3166653542877070999007425197729038754254e48"), + 29*boost::lexical_cast("0.5566029092358215049069560272835654229637e46"), + 30*boost::lexical_cast("0.9161766287916328133080586672953875116242e44"), + 31*boost::lexical_cast("1412317772330871298317974693514430627922000"), + 32*boost::lexical_cast("20387991986727877473732570146112459874790"), + 33*boost::lexical_cast("275557928713904105182512535678580359839.3"), + 34*boost::lexical_cast("3485719851040516559072031256589598330.723"), + 35*boost::lexical_cast("41247046743564028399938106707656877.40859"), + 36*boost::lexical_cast("456274078125709314602601667471879.0147312"), + 37*boost::lexical_cast("4714450683242899367025707077155.310613012"), + 38*boost::lexical_cast("45453933537925041680009544258.75073849996"), + 39*boost::lexical_cast("408437900487067278846361972.302331241052"), + 40*boost::lexical_cast("3415719344386166273085838.705771571751035"), + 41*boost::lexical_cast("26541502879185876562320.93134691487351145"), + 42*boost::lexical_cast("191261415065918713661.1571433274648417668"), + 43*boost::lexical_cast("1275349770108718421.645275944284937551702"), + 44*boost::lexical_cast("7849171120971773.318910987434906905704272"), + 45*boost::lexical_cast("44455946386549.80866460312682983576538056"), + 46*boost::lexical_cast("230920362395.3198137186361608905136598046"), + 47*boost::lexical_cast("1095700096.240863858624279930600654130254"), + 48*boost::lexical_cast("4727085.467506050153744334085516289728134"), + 49*boost::lexical_cast("18440.75118859447173303252421991479005424"), + 50*boost::lexical_cast("64.62515887799460295677071749181651317052"), + 51*boost::lexical_cast("0.201851568864688406206528472883512147547"), + 52*boost::lexical_cast("0.0005565091674187978029138500039504078098143"), + 53*boost::lexical_cast("0.1338097668312907986354698683493366559613e-5"), + 54*boost::lexical_cast("0.276308225077464312820179030238305271638e-8"), + 55*boost::lexical_cast("0.4801582970473168520375942100071070575043e-11"), + 56*boost::lexical_cast("0.6829184144212920949740376186058541800175e-14"), + 57*boost::lexical_cast("0.7634080076358511276617829524639455399182e-17"), + 58*boost::lexical_cast("0.6290035083727140966418512608156646142409e-20"), + 59*boost::lexical_cast("0.339652245667538733044036638506893821352e-23"), + 60*boost::lexical_cast("0.9017518064256388530773585529891677854909e-27") + }; + static const boost::math::ntl::RR QD[60] = { + boost::lexical_cast("0.1386831185456898357379390197203894063459e81"), + 2*boost::lexical_cast("0.6467076379487574703291056110838151259438e81"), + 3*boost::lexical_cast("0.1394967823848615838336194279565285465161e82"), + 4*boost::lexical_cast("0.1872927317344192945218570366455046340458e82"), + 5*boost::lexical_cast("0.1772461045338946243584650759986310355937e82"), + 6*boost::lexical_cast("0.1267294892200258648315971144069595555118e82"), + 7*boost::lexical_cast("0.7157764838362416821508872117623058626589e81"), + 8*boost::lexical_cast("0.329447266909948668265277828268378274513e81"), + 9*boost::lexical_cast("0.1264376077317689779509250183194342571207e81"), + 10*boost::lexical_cast("0.4118230304191980787640446056583623228873e80"), + 11*boost::lexical_cast("0.1154393529762694616405952270558316515261e80"), + 12*boost::lexical_cast("0.281655612889423906125295485693696744275e79"), + 13*boost::lexical_cast("0.6037483524928743102724159846414025482077e78"), + 14*boost::lexical_cast("0.1145927995397835468123576831800276999614e78"), + 15*boost::lexical_cast("0.1938624296151985600348534009382865995154e77"), + 16*boost::lexical_cast("0.293980925856227626211879961219188406675e76"), + 17*boost::lexical_cast("0.4015574518216966910319562902099567437832e75"), + 18*boost::lexical_cast("0.4961475457509727343545565970423431880907e74"), + 19*boost::lexical_cast("0.5565482348278933960215521991000378896338e73"), + 20*boost::lexical_cast("0.5686112924615820754631098622770303094938e72"), + 21*boost::lexical_cast("0.5305988545844796293285410303747469932856e71"), + 22*boost::lexical_cast("0.4533363413802585060568537458067343491358e70"), + 23*boost::lexical_cast("0.3553932059473516064068322757331575565718e69"), + 24*boost::lexical_cast("0.2561198565218704414618802902533972354203e68"), + 25*boost::lexical_cast("0.1699519313292900324098102065697454295572e67"), + 26*boost::lexical_cast("0.1039830160862334505389615281373574959236e66"), + 27*boost::lexical_cast("0.5873082967977428281000961954715372504986e64"), + 28*boost::lexical_cast("0.3065255179030575882202133042549783442446e63"), + 29*boost::lexical_cast("0.1479494813481364701208655943688307245459e62"), + 30*boost::lexical_cast("0.6608150467921598615495180659808895663164e60"), + 31*boost::lexical_cast("0.2732535313770902021791888953487787496976e59"), + 32*boost::lexical_cast("0.1046402297662493314531194338414508049069e58"), + 33*boost::lexical_cast("0.3711375077192882936085049147920021549622e56"), + 34*boost::lexical_cast("0.1219154482883895482637944309702972234576e55"), + 35*boost::lexical_cast("0.3708359374149458741391374452286837880162e53"), + 36*boost::lexical_cast("0.1044095509971707189716913168889769471468e52"), + 37*boost::lexical_cast("0.271951506225063286130946773813524945052e50"), + 38*boost::lexical_cast("0.6548016291215163843464133978454065823866e48"), + 39*boost::lexical_cast("0.1456062447610542135403751730809295219344e47"), + 40*boost::lexical_cast("0.2986690175077969760978388356833006028929e45"), + 41*boost::lexical_cast("5643149706574013350061247429006443326844000"), + 42*boost::lexical_cast("98047545414467090421964387960743688053480"), + 43*boost::lexical_cast("1563378767746846395507385099301468978550"), + 44*boost::lexical_cast("22823360528584500077862274918382796495"), + 45*boost::lexical_cast("304215527004115213046601295970388750"), + 46*boost::lexical_cast("3690289075895685793844344966820325"), + 47*boost::lexical_cast("40584512015702371433911456606050"), + 48*boost::lexical_cast("402834190897282802772754873905"), + 49*boost::lexical_cast("3589522158493606918146495750"), + 50*boost::lexical_cast("28530557707503483723634725"), + 51*boost::lexical_cast("200714561335055753000730"), + 52*boost::lexical_cast("1237953783437761888641"), + 53*boost::lexical_cast("6614698701445762950"), + 54*boost::lexical_cast("30155495647727505"), + 55*boost::lexical_cast("114953256021450"), + 56*boost::lexical_cast("356398020013"), + 57*boost::lexical_cast("863113950"), + 58*boost::lexical_cast("1531345"), + 59*boost::lexical_cast("1770"), + 60*boost::lexical_cast("1") + }; + static const double g = 63.192152; + + boost::math::ntl::RR zgh = x + g - 0.5; + + boost::math::ntl::RR result = (x - 0.5) / zgh; + result += log(zgh); + result += tools::evaluate_polynomial(PD, x) / tools::evaluate_polynomial(P, x); + result -= tools::evaluate_polynomial(QD, x) / tools::evaluate_polynomial(Q, x); + result -= 1; + + return result; +} + +boost::math::ntl::RR digamma(boost::math::ntl::RR x) +{ + if(x < 0) + { + return digamma_imp(1-x) + constants::pi() / tan(constants::pi() * (1-x)); + } + return digamma_imp(x); +} + +}} + +#endif // include guard diff --git a/tools/ntl_rr_lanczos.hpp b/tools/ntl_rr_lanczos.hpp new file mode 100644 index 000000000..750d87c22 --- /dev/null +++ b/tools/ntl_rr_lanczos.hpp @@ -0,0 +1,902 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_LARGE_LANCZOS_HPP +#define BOOST_LARGE_LANCZOS_HPP + +#include +#include +#include + +struct lanczos13UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[13] = { + boost::lexical_cast("44012138428004.60895436261759919070125699"), + boost::lexical_cast("41590453358593.20051581730723108131357995"), + boost::lexical_cast("18013842787117.99677796276038389462742949"), + boost::lexical_cast("4728736263475.388896889723995205703970787"), + boost::lexical_cast("837910083628.4046470415724300225777912264"), + boost::lexical_cast("105583707273.4299344907359855510105321192"), + boost::lexical_cast("9701363618.494999493386608345339104922694"), + boost::lexical_cast("654914397.5482052641016767125048538245644"), + boost::lexical_cast("32238322.94213356530668889463945849409184"), + boost::lexical_cast("1128514.219497091438040721811544858643121"), + boost::lexical_cast("26665.79378459858944762533958798805525125"), + boost::lexical_cast("381.8801248632926870394389468349331394196"), + boost::lexical_cast("2.506628274631000502415763426076722427007"), + }; + static const boost::uint32_t denom[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 13); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[13] = { + boost::lexical_cast("86091529.53418537217994842267760536134841"), + boost::lexical_cast("81354505.17858011242874285785316135398567"), + boost::lexical_cast("35236626.38815461910817650960734605416521"), + boost::lexical_cast("9249814.988024471294683815872977672237195"), + boost::lexical_cast("1639024.216687146960253839656643518985826"), + boost::lexical_cast("206530.8157641225032631778026076868855623"), + boost::lexical_cast("18976.70193530288915698282139308582105936"), + boost::lexical_cast("1281.068909912559479885759622791374106059"), + boost::lexical_cast("63.06093343420234536146194868906771599354"), + boost::lexical_cast("2.207470909792527638222674678171050209691"), + boost::lexical_cast("0.05216058694613505427476207805814960742102"), + boost::lexical_cast("0.0007469903808915448316510079585999893674101"), + boost::lexical_cast("0.4903180573459871862552197089738373164184e-5"), + }; + static const boost::uint32_t denom[13] = { + static_cast(0u), + static_cast(39916800u), + static_cast(120543840u), + static_cast(150917976u), + static_cast(105258076u), + static_cast(45995730u), + static_cast(13339535u), + static_cast(2637558u), + static_cast(357423u), + static_cast(32670u), + static_cast(1925u), + static_cast(66u), + static_cast(1u), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 13); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[12] = { + boost::lexical_cast("4.832115561461656947793029596285626840312"), + boost::lexical_cast("-19.86441536140337740383120735104359034688"), + boost::lexical_cast("33.9927422807443239927197864963170585331"), + boost::lexical_cast("-31.41520692249765980987427413991250886138"), + boost::lexical_cast("17.0270866009599345679868972409543597821"), + boost::lexical_cast("-5.5077216950865501362506920516723682167"), + boost::lexical_cast("1.037811741948214855286817963800439373362"), + boost::lexical_cast("-0.106640468537356182313660880481398642811"), + boost::lexical_cast("0.005276450526660653288757565778182586742831"), + boost::lexical_cast("-0.0001000935625597121545867453746252064770029"), + boost::lexical_cast("0.462590910138598083940803704521211569234e-6"), + boost::lexical_cast("-0.1735307814426389420248044907765671743012e-9"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& dz) + { + static const T d[12] = { + boost::lexical_cast("26.96979819614830698367887026728396466395"), + boost::lexical_cast("-110.8705424709385114023884328797900204863"), + boost::lexical_cast("189.7258846119231466417015694690434770085"), + boost::lexical_cast("-175.3397202971107486383321670769397356553"), + boost::lexical_cast("95.03437648691551457087250340903980824948"), + boost::lexical_cast("-30.7406022781665264273675797983497141978"), + boost::lexical_cast("5.792405601630517993355102578874590410552"), + boost::lexical_cast("-0.5951993240669148697377539518639997795831"), + boost::lexical_cast("0.02944979359164017509944724739946255067671"), + boost::lexical_cast("-0.0005586586555377030921194246330399163602684"), + boost::lexical_cast("0.2581888478270733025288922038673392636029e-5"), + boost::lexical_cast("-0.9685385411006641478305219367315965391289e-9"), + }; + T result = 0; + T z = dz + 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 13.1445650000000000545696821063756942749; } +}; + + +// +// Lanczos Coefficients for N=22 G=22.61891 +// Max experimental error (with arbitary precision arithmetic) 2.9524e-38 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006 +// +struct lanczos22UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[22] = { + boost::lexical_cast("46198410803245094237463011094.12173081986"), + boost::lexical_cast("43735859291852324413622037436.321513777"), + boost::lexical_cast("19716607234435171720534556386.97481377748"), + boost::lexical_cast("5629401471315018442177955161.245623932129"), + boost::lexical_cast("1142024910634417138386281569.245580222392"), + boost::lexical_cast("175048529315951173131586747.695329230778"), + boost::lexical_cast("21044290245653709191654675.41581372963167"), + boost::lexical_cast("2033001410561031998451380.335553678782601"), + boost::lexical_cast("160394318862140953773928.8736211601848891"), + boost::lexical_cast("10444944438396359705707.48957290388740896"), + boost::lexical_cast("565075825801617290121.1466393747967538948"), + boost::lexical_cast("25475874292116227538.99448534450411942597"), + boost::lexical_cast("957135055846602154.6720835535232270205725"), + boost::lexical_cast("29874506304047462.23662392445173880821515"), + boost::lexical_cast("769651310384737.2749087590725764959689181"), + boost::lexical_cast("16193289100889.15989633624378404096011797"), + boost::lexical_cast("273781151680.6807433264462376754578933261"), + boost::lexical_cast("3630485900.32917021712188739762161583295"), + boost::lexical_cast("36374352.05577334277856865691538582936484"), + boost::lexical_cast("258945.7742115532455441786924971194951043"), + boost::lexical_cast("1167.501919472435718934219997431551246996"), + boost::lexical_cast("2.50662827463100050241576528481104525333"), + }; + static const boost::uint64_t denom[22] = { + (0uLL), + (2432902008176640000uLL), + (8752948036761600000uLL), + (13803759753640704000uLL), + (12870931245150988800uLL), + (8037811822645051776uLL), + (3599979517947607200uLL), + (1206647803780373360uLL), + (311333643161390640uLL), + (63030812099294896uLL), + (10142299865511450uLL), + (1307535010540395uLL), + (135585182899530uLL), + (11310276995381uLL), + (756111184500uLL), + (40171771630uLL), + (1672280820uLL), + (53327946uLL), + (1256850uLL), + (20615uLL), + (210uLL), + (1uLL), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 22); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[22] = { + boost::lexical_cast("6939996264376682180.277485395074954356211"), + boost::lexical_cast("6570067992110214451.87201438870245659384"), + boost::lexical_cast("2961859037444440551.986724631496417064121"), + boost::lexical_cast("845657339772791245.3541226499766163431651"), + boost::lexical_cast("171556737035449095.2475716923888737881837"), + boost::lexical_cast("26296059072490867.7822441885603400926007"), + boost::lexical_cast("3161305619652108.433798300149816829198706"), + boost::lexical_cast("305400596026022.4774396904484542582526472"), + boost::lexical_cast("24094681058862.55120507202622377623528108"), + boost::lexical_cast("1569055604375.919477574824168939428328839"), + boost::lexical_cast("84886558909.02047889339710230696942513159"), + boost::lexical_cast("3827024985.166751989686050643579753162298"), + boost::lexical_cast("143782298.9273215199098728674282885500522"), + boost::lexical_cast("4487794.24541641841336786238909171265944"), + boost::lexical_cast("115618.2025760830513505888216285273541959"), + boost::lexical_cast("2432.580773108508276957461757328744780439"), + boost::lexical_cast("41.12782532742893597168530008461874360191"), + boost::lexical_cast("0.5453771709477689805460179187388702295792"), + boost::lexical_cast("0.005464211062612080347167337964166505282809"), + boost::lexical_cast("0.388992321263586767037090706042788910953e-4"), + boost::lexical_cast("0.1753839324538447655939518484052327068859e-6"), + boost::lexical_cast("0.3765495513732730583386223384116545391759e-9"), + }; + static const boost::uint64_t denom[22] = { + (0uLL), + (2432902008176640000uLL), + (8752948036761600000uLL), + (13803759753640704000uLL), + (12870931245150988800uLL), + (8037811822645051776uLL), + (3599979517947607200uLL), + (1206647803780373360uLL), + (311333643161390640uLL), + (63030812099294896uLL), + (10142299865511450uLL), + (1307535010540395uLL), + (135585182899530uLL), + (11310276995381uLL), + (756111184500uLL), + (40171771630uLL), + (1672280820uLL), + (53327946uLL), + (1256850uLL), + (20615uLL), + (210uLL), + (1uLL), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 22); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[21] = { + boost::lexical_cast("8.318998691953337183034781139546384476554"), + boost::lexical_cast("-63.15415991415959158214140353299240638675"), + boost::lexical_cast("217.3108224383632868591462242669081540163"), + boost::lexical_cast("-448.5134281386108366899784093610397354889"), + boost::lexical_cast("619.2903759363285456927248474593012711346"), + boost::lexical_cast("-604.1630177420625418522025080080444177046"), + boost::lexical_cast("428.8166750424646119935047118287362193314"), + boost::lexical_cast("-224.6988753721310913866347429589434550302"), + boost::lexical_cast("87.32181627555510833499451817622786940961"), + boost::lexical_cast("-25.07866854821128965662498003029199058098"), + boost::lexical_cast("5.264398125689025351448861011657789005392"), + boost::lexical_cast("-0.792518936256495243383586076579921559914"), + boost::lexical_cast("0.08317448364744713773350272460937904691566"), + boost::lexical_cast("-0.005845345166274053157781068150827567998882"), + boost::lexical_cast("0.0002599412126352082483326238522490030412391"), + boost::lexical_cast("-0.6748102079670763884917431338234783496303e-5"), + boost::lexical_cast("0.908824383434109002762325095643458603605e-7"), + boost::lexical_cast("-0.5299325929309389890892469299969669579725e-9"), + boost::lexical_cast("0.994306085859549890267983602248532869362e-12"), + boost::lexical_cast("-0.3499893692975262747371544905820891835298e-15"), + boost::lexical_cast("0.7260746353663365145454867069182884694961e-20"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& dz) + { + static const T d[21] = { + boost::lexical_cast("75.39272007105208086018421070699575462226"), + boost::lexical_cast("-572.3481967049935412452681346759966390319"), + boost::lexical_cast("1969.426202741555335078065370698955484358"), + boost::lexical_cast("-4064.74968778032030891520063865996757519"), + boost::lexical_cast("5612.452614138013929794736248384309574814"), + boost::lexical_cast("-5475.357667500026172903620177988213902339"), + boost::lexical_cast("3886.243614216111328329547926490398103492"), + boost::lexical_cast("-2036.382026072125407192448069428134470564"), + boost::lexical_cast("791.3727954936062108045551843636692287652"), + boost::lexical_cast("-227.2808432388436552794021219198885223122"), + boost::lexical_cast("47.70974355562144229897637024320739257284"), + boost::lexical_cast("-7.182373807798293545187073539819697141572"), + boost::lexical_cast("0.7537866989631514559601547530490976100468"), + boost::lexical_cast("-0.05297470142240154822658739758236594717787"), + boost::lexical_cast("0.00235577330936380542539812701472320434133"), + boost::lexical_cast("-0.6115613067659273118098229498679502138802e-4"), + boost::lexical_cast("0.8236417010170941915758315020695551724181e-6"), + boost::lexical_cast("-0.4802628430993048190311242611330072198089e-8"), + boost::lexical_cast("0.9011113376981524418952720279739624707342e-11"), + boost::lexical_cast("-0.3171854152689711198382455703658589996796e-14"), + boost::lexical_cast("0.6580207998808093935798753964580596673177e-19"), + }; + T result = 0; + T z = dz + 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 22.61890999999999962710717227309942245483; } +}; + +// +// Lanczos Coefficients for N=31 G=32.08067 +// Max experimental error (with arbitary precision arithmetic) 0.162e-52 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at May 9 2006 +// +struct lanczos31UDT +{ + template + static T lanczos_sum(const T& z) + { + static const T num[31] = { + boost::lexical_cast("0.2579646553333513328235723061836959833277e46"), + boost::lexical_cast("0.2444796504337453845497419271639377138264e46"), + boost::lexical_cast("0.1119885499016017172212179730662673475329e46"), + boost::lexical_cast("0.3301983829072723658949204487793889113715e45"), + boost::lexical_cast("0.7041171040503851585152895336505379417066e44"), + boost::lexical_cast("0.1156687509001223855125097826246939403504e44"), + boost::lexical_cast("1522559363393940883866575697565974893306000"), + boost::lexical_cast("164914363507650839510801418717701057005700"), + boost::lexical_cast("14978522943127593263654178827041568394060"), + boost::lexical_cast("1156707153701375383907746879648168666774"), + boost::lexical_cast("76739431129980851159755403434593664173.2"), + boost::lexical_cast("4407916278928188620282281495575981079.306"), + boost::lexical_cast("220487883931812802092792125175269667.3004"), + boost::lexical_cast("9644828280794966468052381443992828.433924"), + boost::lexical_cast("369996467042247229310044531282837.6549068"), + boost::lexical_cast("12468380890717344610932904378961.13494291"), + boost::lexical_cast("369289245210898235894444657859.0529720075"), + boost::lexical_cast("9607992460262594951559461829.34885209022"), + boost::lexical_cast("219225935074853412540086410.981421315799"), + boost::lexical_cast("4374309943598658046326340.720767382079549"), + boost::lexical_cast("76008779092264509404014.10530947173485581"), + boost::lexical_cast("1143503533822162444712.335663112617754987"), + boost::lexical_cast("14779233719977576920.37884890049671578409"), + boost::lexical_cast("162409028440678302.9992838032166348069916"), + boost::lexical_cast("1496561553388385.733407609544964535634135"), + boost::lexical_cast("11347624460661.81008311053190661436107043"), + boost::lexical_cast("68944915931.32004991941950530448472223832"), + boost::lexical_cast("322701221.6391432296123937035480931903651"), + boost::lexical_cast("1092364.213992634267819050120261755371294"), + boost::lexical_cast("2380.151399852411512711176940867823024864"), + boost::lexical_cast("2.506628274631000502415765284811045253007"), + }; + static const T denom[31] = { + boost::lexical_cast("0"), + boost::lexical_cast("0.8841761993739701954543616e31"), + boost::lexical_cast("0.3502799997985980526649278464e32"), + boost::lexical_cast("0.622621928420356134910574592e32"), + boost::lexical_cast("66951000306085302338993639424000"), + boost::lexical_cast("49361465831621147825759587123200"), + boost::lexical_cast("26751280755793398822580822142976"), + boost::lexical_cast("11139316913434780466101123891200"), + boost::lexical_cast("3674201658710345201899117607040"), + boost::lexical_cast("981347603630155088295475765440"), + boost::lexical_cast("215760462268683520394805979744"), + boost::lexical_cast("39539238727270799376544542000"), + boost::lexical_cast("6097272817323042122728617800"), + boost::lexical_cast("796974693974455191377937300"), + boost::lexical_cast("88776380550648116217781890"), + boost::lexical_cast("8459574446076318147830625"), + boost::lexical_cast("691254538651580660999025"), + boost::lexical_cast("48487623689430693038025"), + boost::lexical_cast("2918939500751087661105"), + boost::lexical_cast("150566737512021319125"), + boost::lexical_cast("6634460278534540725"), + boost::lexical_cast("248526574856284725"), + boost::lexical_cast("7860403394108265"), + boost::lexical_cast("207912996295875"), + boost::lexical_cast("4539323721075"), + boost::lexical_cast("80328850875"), + boost::lexical_cast("1122686019"), + boost::lexical_cast("11921175"), + boost::lexical_cast("90335"), + boost::lexical_cast("435"), + boost::lexical_cast("1"), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 31); + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + static const T num[31] = { + boost::lexical_cast("30137154810677525966583148469478.52374216"), + boost::lexical_cast("28561746428637727032849890123131.36314653"), + boost::lexical_cast("13083250730789213354063781611435.74046294"), + boost::lexical_cast("3857598154697777600846539129354.783647"), + boost::lexical_cast("822596651552555685068015316144.0952185852"), + boost::lexical_cast("135131964033213842052904200372.039133532"), + boost::lexical_cast("17787555889683709693655685146.19771358863"), + boost::lexical_cast("1926639793777927562221423874.149673297196"), + boost::lexical_cast("174989113988888477076973808.6991839697774"), + boost::lexical_cast("13513425905835560387095425.01158383184045"), + boost::lexical_cast("896521313378762433091075.1446749283094845"), + boost::lexical_cast("51496223433749515758124.71524415105430686"), + boost::lexical_cast("2575886794780078381228.37205955912263407"), + boost::lexical_cast("112677328855422964200.4155776009524490958"), + boost::lexical_cast("4322545967487943330.625233358130724324796"), + boost::lexical_cast("145663957202380774.0362027607207590519723"), + boost::lexical_cast("4314283729473470.686566233465428332496534"), + boost::lexical_cast("112246988185485.8877916434026906290603878"), + boost::lexical_cast("2561143864972.040563435178307062626388193"), + boost::lexical_cast("51103611767.9626550674442537989885239605"), + boost::lexical_cast("887985348.0369447209508500133077232094491"), + boost::lexical_cast("13359172.3954672607019822025834072685839"), + boost::lexical_cast("172660.8841147568768783928167105965064459"), + boost::lexical_cast("1897.370795407433013556725714874693719617"), + boost::lexical_cast("17.48383210090980598861217644749573257178"), + boost::lexical_cast("0.1325705316732132940835251054350153028901"), + boost::lexical_cast("0.0008054605783673449641889260501816356090452"), + boost::lexical_cast("0.377001130700104515644336869896819162464e-5"), + boost::lexical_cast("0.1276172868883867038813825443204454996531e-7"), + boost::lexical_cast("0.2780651912081116274907381023821492811093e-10"), + boost::lexical_cast("0.2928410648650955854121639682890739211234e-13"), + }; + static const T denom[31] = { + boost::lexical_cast("0"), + boost::lexical_cast("0.8841761993739701954543616e31"), + boost::lexical_cast("0.3502799997985980526649278464e32"), + boost::lexical_cast("0.622621928420356134910574592e32"), + boost::lexical_cast("66951000306085302338993639424000"), + boost::lexical_cast("49361465831621147825759587123200"), + boost::lexical_cast("26751280755793398822580822142976"), + boost::lexical_cast("11139316913434780466101123891200"), + boost::lexical_cast("3674201658710345201899117607040"), + boost::lexical_cast("981347603630155088295475765440"), + boost::lexical_cast("215760462268683520394805979744"), + boost::lexical_cast("39539238727270799376544542000"), + boost::lexical_cast("6097272817323042122728617800"), + boost::lexical_cast("796974693974455191377937300"), + boost::lexical_cast("88776380550648116217781890"), + boost::lexical_cast("8459574446076318147830625"), + boost::lexical_cast("691254538651580660999025"), + boost::lexical_cast("48487623689430693038025"), + boost::lexical_cast("2918939500751087661105"), + boost::lexical_cast("150566737512021319125"), + boost::lexical_cast("6634460278534540725"), + boost::lexical_cast("248526574856284725"), + boost::lexical_cast("7860403394108265"), + boost::lexical_cast("207912996295875"), + boost::lexical_cast("4539323721075"), + boost::lexical_cast("80328850875"), + boost::lexical_cast("1122686019"), + boost::lexical_cast("11921175"), + boost::lexical_cast("90335"), + boost::lexical_cast("435"), + boost::lexical_cast("1"), + }; + return boost::math::tools::evaluate_rational(num, denom, z, 31); + } + + + template + static T lanczos_sum_near_1(const T& dz) + { + static const T d[30] = { + boost::lexical_cast("11.80038544942943603508206880307972596807"), + boost::lexical_cast("-130.6355975335626214564236363322099481079"), + boost::lexical_cast("676.2177719145993049893392276809256538927"), + boost::lexical_cast("-2174.724497783850503069990936574060452057"), + boost::lexical_cast("4869.877180638131076410069103742986502022"), + boost::lexical_cast("-8065.744271864238179992762265472478229172"), + boost::lexical_cast("10245.03825618572106228191509520638651539"), + boost::lexical_cast("-10212.83902362683215459850403668669647192"), + boost::lexical_cast("8110.289185383288952562767679576754140336"), + boost::lexical_cast("-5179.310892558291062401828964000776095156"), + boost::lexical_cast("2673.987492589052370230989109591011091273"), + boost::lexical_cast("-1118.342574651205183051884250033505609141"), + boost::lexical_cast("378.5812742511620662650096436471920295596"), + boost::lexical_cast("-103.3725999812126067084828735543906768961"), + boost::lexical_cast("22.62913974335996321848099677797888917792"), + boost::lexical_cast("-3.936414819950859548507275533569588041446"), + boost::lexical_cast("0.5376818198843817355682124535902641644854"), + boost::lexical_cast("-0.0567827903603478957483409124122554243201"), + boost::lexical_cast("0.004545544993648879420352693271088478106482"), + boost::lexical_cast("-0.0002689795568951033950042375135970897959935"), + boost::lexical_cast("0.1139493459006846530734617710847103572122e-4"), + boost::lexical_cast("-0.3316581197839213921885210451302820192794e-6"), + boost::lexical_cast("0.6285613334898374028443777562554713906213e-8"), + boost::lexical_cast("-0.7222145115734409070310317999856424167091e-10"), + boost::lexical_cast("0.4562976983547274766890241815002584238219e-12"), + boost::lexical_cast("-0.1380593023819058919640038942493212141072e-14"), + boost::lexical_cast("0.1629663871586410129307496385264268190679e-17"), + boost::lexical_cast("-0.5429994291916548849493889660077076739993e-21"), + boost::lexical_cast("0.2922682842441892106795386303084661338957e-25"), + boost::lexical_cast("-0.8456967065309046044689041041336866118459e-31"), + }; + T result = 0; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& dz) + { + static const T d[30] = { + boost::lexical_cast("147.9979641587472136175636384176549713358"), + boost::lexical_cast("-1638.404318611773924210055619836375434296"), + boost::lexical_cast("8480.981744216135641122944743711911653273"), + boost::lexical_cast("-27274.93942104458448200467097634494071176"), + boost::lexical_cast("61076.98019918759324489193232276937262854"), + boost::lexical_cast("-101158.8762737154296509560513952101409264"), + boost::lexical_cast("128491.1252383947174824913796141607174379"), + boost::lexical_cast("-128087.2892038336581928787480535905496026"), + boost::lexical_cast("101717.5492545853663296795562084430123258"), + boost::lexical_cast("-64957.8330410311808907869707511362206858"), + boost::lexical_cast("33536.59139229792478811870738772305570317"), + boost::lexical_cast("-14026.01847115365926835980820243003785821"), + boost::lexical_cast("4748.087094096186515212209389240715050212"), + boost::lexical_cast("-1296.477510211815971152205100242259733245"), + boost::lexical_cast("283.8099337545793198947620951499958085157"), + boost::lexical_cast("-49.36969067101255103452092297769364837753"), + boost::lexical_cast("6.743492833270653628580811118017061581404"), + boost::lexical_cast("-0.7121578704864048548351804794951487823626"), + boost::lexical_cast("0.0570092738016915476694118877057948681298"), + boost::lexical_cast("-0.003373485297696102660302960722607722438643"), + boost::lexical_cast("0.0001429128843527532859999752593761934089751"), + boost::lexical_cast("-0.41595867130858508233493767243236888636e-5"), + boost::lexical_cast("0.7883284669307241040059778207492255409785e-7"), + boost::lexical_cast("-0.905786322462384670803148223703187214379e-9"), + boost::lexical_cast("0.5722790216999820323272452464661250331451e-11"), + boost::lexical_cast("-0.1731510870832349779315841757234562309727e-13"), + boost::lexical_cast("0.2043890314358438601429048378015983874378e-16"), + boost::lexical_cast("-0.6810185176079344204740000170500311171065e-20"), + boost::lexical_cast("0.3665567641131713928114853776588342403919e-24"), + boost::lexical_cast("-0.1060655106553177007425710511436497259484e-29"), + }; + T result = 0; + T z = dz + 2; + for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 32.08066999999999779902282170951366424561; } +}; + +// +// Lanczos Coefficients for N=61 G=63.192152 +// Max experimental error (with 1000-bit precision arithmetic) 3.740e-113 +// Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 12 2006 +// +struct lanczos61UDT +{ + template + static T lanczos_sum(const T& z) + { + using namespace boost; + static const T d[61] = { + boost::lexical_cast("2.50662827463100050241576528481104525300698674060993831662992357634229365460784197494659584"), + boost::lexical_cast("13349415823254323512107320481.3495396037261649201426994438803767191136434970492309775123879"), + boost::lexical_cast("-300542621510568204264185787475.230003734889859348050696493467253861933279360152095861484548"), + boost::lexical_cast("3273919938390136737194044982676.40271056035622723775417608127544182097346526115858803376474"), + boost::lexical_cast("-22989594065095806099337396006399.5874206181563663855129141706748733174902067950115092492439"), + boost::lexical_cast("116970582893952893160414263796102.542775878583510989850142808618916073286745084692189044738"), + boost::lexical_cast("-459561969036479455224850813196807.283291532483532558959779434457349912822256480548436066098"), + boost::lexical_cast("1450959909778264914956547227964788.89797179379520834974601372820249784303794436366366810477"), + boost::lexical_cast("-3782846865486775046285288437885921.41537699732805465141128848354901016102326190612528503251"), + boost::lexical_cast("8305043213936355459145388670886540.09976337905520168067329932809302445437208115570138102767"), + boost::lexical_cast("-15580988484396722546934484726970745.4927787160273626078250810989811865283255762028143642311"), + boost::lexical_cast("25262722284076250779006793435537600.0822901485517345545978818780090308947301031347345640449"), + boost::lexical_cast("-35714428027687018805443603728757116.5304655170478705341887572982734901197345415291580897698"), + boost::lexical_cast("44334726194692443174715432419157343.2294160783772787096321009453791271387235388689346602833"), + boost::lexical_cast("-48599573547617297831555162417695106.187829304963846482633791012658974681648157963911491985"), + boost::lexical_cast("47258466493366798944386359199482189.0753349196625125615316002614813737880755896979754845101"), + boost::lexical_cast("-40913448215392412059728312039233342.142914753896559359297977982314043378636755884088383226"), + boost::lexical_cast("31626312914486892948769164616982902.7262756989418188077611392594232674722318027323102462687"), + boost::lexical_cast("-21878079174441332123064991795834438.4699982361692990285700077902601657354101259411789722708"), + boost::lexical_cast("13567268503974326527361474986354265.3136632133935430378937191911532112778452274286122946396"), + boost::lexical_cast("-7551494211746723529747611556474669.62996644923557605747803028485900789337467673523741066527"), + boost::lexical_cast("3775516572689476384052312341432597.70584966904950490541958869730702790312581801585742038997"), + boost::lexical_cast("-1696271471453637244930364711513292.79902955514107737992185368006225264329876265486853482449"), + boost::lexical_cast("684857608019352767999083000986166.20765273693720041519286231015176745354062413008561259139"), + boost::lexical_cast("-248397566275708464679881624417990.410438107634139924805871051723444048539177890346227250473"), + boost::lexical_cast("80880368999557992138783568858556.1512378233079327986518410244522800950609595592170022878937"), + boost::lexical_cast("-23618197945394013802495450485616.9025005749893350650829964098117490779655546610665927669729"), + boost::lexical_cast("6176884636893816103087134481332.06708966653493024119556843727320635285468825056891248447124"), + boost::lexical_cast("-1444348683723439589948246285262.64080678953468490544615312565485170860503207005915261691108"), + boost::lexical_cast("301342031656979076702313946827.961658905182634508217626783081241074250132289461989777865387"), + boost::lexical_cast("-55959656587719766738301589651.3940625826610668990368881615587469329021742236397809951765678"), + boost::lexical_cast("9223339169004064297247180402.36227016155682738556103138196079389248843082157924368301293963"), + boost::lexical_cast("-1344882881571942601385730283.42710150182526891377514071881534880944872423492272147871101373"), + boost::lexical_cast("172841913316760599352601139.54409257740173055624405575900164401527761357324625574736896079"), + boost::lexical_cast("-19496120443876233531343952.3802212016691702737346568192063937387825469602063310488794471653"), + boost::lexical_cast("1920907372583710284097959.44121420322495784420169085871802458519363819782779653621724219067"), + boost::lexical_cast("-164429314798240461613359.399597503536657962383155875723527581699785846599051112719962464604"), + boost::lexical_cast("12154026644351189572525.1452249886865981747374191977801688548318519692423556934568426042152"), + boost::lexical_cast("-770443988366210815096.519382051977221101156336663806705367929328924137169970381042234329058"), + boost::lexical_cast("41558909851418707920.4696085656889424895313728719601503526476333404973280596225722152966128"), + boost::lexical_cast("-1890879946549708819.24562220042687554209318172044783707920086716716717574156283898330017796"), + boost::lexical_cast("71844996557297623.9583461685535340440524052492427928388171299145330229958643439878608673403"), + boost::lexical_cast("-2253785109518255.55600197759875781765803818232939130127735487613049577235879610065545755637"), + boost::lexical_cast("57616883849355.997562563968344493719962252675875692642406455612671495250543228005045106721"), + boost::lexical_cast("-1182495730353.08218118278997948852215670614084013289033222774171548915344541249351599628436"), + boost::lexical_cast("19148649358.6196967288062261380599423925174178776792840639099120170800869284300966978300613"), + boost::lexical_cast("-239779605.891370259668403359614360511661030470269478602533200704639655585967442891496784613"), + boost::lexical_cast("2267583.00284368310957842936892685032434505866445291643236133553754152047677944820353796872"), + boost::lexical_cast("-15749.490806784673108773558070497383604733010677027764233749920147549999361110299643477893"), + boost::lexical_cast("77.7059495149052727171505425584459982871343274332635726864135949842508025564999785370162956"), + boost::lexical_cast("-0.261619987273930331397625130282851629108569607193781378836014468617185550622160348688297247"), + boost::lexical_cast("0.000572252321659691600529444769356185993188551770817110673186068921175991328434642504616377475"), + boost::lexical_cast("-0.765167220661540041663007112207436426423746402583423562585653954743978584117929356523307954e-6"), + boost::lexical_cast("0.579179571056209077507916813937971472839851499147582627425979879366849876944438724610663401e-9"), + boost::lexical_cast("-0.224804733043915149719206760378355636826808754704148660354494460792713189958510735070096991e-12"), + boost::lexical_cast("0.392711975389579343321746945135488409914483448652884894759297584020979857734289645857714768e-16"), + boost::lexical_cast("-0.258603588346412049542768766878162221817684639789901440429511261589010049357907537684380983e-20"), + boost::lexical_cast("0.499992460848751668441190360024540741752242879565548017176883304716370989218399797418478685e-25"), + boost::lexical_cast("-0.196211614533318174187346267877390498735734213905206562766348625767919122511096089367364025e-30"), + boost::lexical_cast("0.874722648949676363732094858062907290148733370978226751462386623191111439121706262759209573e-37"), + boost::lexical_cast("-0.163907874717737848669759890242660846846105433791283903651924563157080252845636658802930428e-44"), + }; + T result = d[0]; + for(int k = 1; k < sizeof(d)/sizeof(d[0]); ++k) + { + result += d[k]/(z+(k-1)); + } + return result; + } + + template + static T lanczos_sum_expG_scaled(const T& z) + { + using namespace boost; + static const T d[61] = { + boost::lexical_cast("0.901751806425638853077358552989167785490911341809902155556127108480303870921448984935411583e-27"), + boost::lexical_cast("4.80241125306810017699523302110401965428995345115391817406006361151407344955277298373661032"), + boost::lexical_cast("-108.119283021710869401330097315436214587270846871451487282117128515476478251641970487922552"), + boost::lexical_cast("1177.78262074811362219818923738088833932279000985161077740440010901595132448469513438139561"), + boost::lexical_cast("-8270.43570321334374279057432173814835581983913167617217749736484999327758232081395983082867"), + boost::lexical_cast("42079.807161997077661752306902088979258826568702655699995911391774839958572703348502730394"), + boost::lexical_cast("-165326.003834443330215001219988296482004968548294447320869281647211603153902436231468280089"), + boost::lexical_cast("521978.361504895300685499370463597042533432134369277742485307843747923127933979566742421213"), + boost::lexical_cast("-1360867.51629992863544553419296636395576666570468519805449755596254912681418267100657262281"), + boost::lexical_cast("2987713.73338656161102517003716335104823650191612448011720936412226357385029800040631754755"), + boost::lexical_cast("-5605212.64915921452169919008770165304171481697939254152852673009005154549681704553438450709"), + boost::lexical_cast("9088186.58332916818449459635132673652700922052988327069535513580836143146727832380184335474"), + boost::lexical_cast("-12848155.5543636394746355365819800465364996596310467415907815393346205151090486190421959769"), + boost::lexical_cast("15949281.2867656960575878805158849857756293807220033618942867694361569866468996967761600898"), + boost::lexical_cast("-17483546.9948295433308250581770557182576171673272450149400973735206019559576269174369907171"), + boost::lexical_cast("17001087.8599749419660906448951424280111036786456594738278573653160553115681287326064596964"), + boost::lexical_cast("-14718487.0643665950346574802384331125115747311674609017568623694516187494204567579595827859"), + boost::lexical_cast("11377468.7255609717716845971105161298889777425898291183866813269222281486121330837791392732"), + boost::lexical_cast("-7870571.64253038587947746661946939286858490057774448573157856145556080330153403858747655263"), + boost::lexical_cast("4880783.08440908743641723492059912671377140680710345996273343885045364205895751515063844239"), + boost::lexical_cast("-2716626.7992639625103140035635916455652302249897918893040695025407382316653674141983775542"), + boost::lexical_cast("1358230.46602865696544327299659410214201327791319846880787515156343361311278133805428800255"), + boost::lexical_cast("-610228.440751458395860905749312275043435828322076830117123636938979942213530882048883969802"), + boost::lexical_cast("246375.416501158654327780901087115642493055617468601787093268312234390446439555559050129729"), + boost::lexical_cast("-89360.2599028475206119333931211015869139511677735549267100272095432140508089207221096740632"), + boost::lexical_cast("29096.4637987498328341260960356772198979319790332957125131055960448759586930781530063775634"), + boost::lexical_cast("-8496.57401431514433694413130585404918350686834939156759654375188338156288564260152505382438"), + boost::lexical_cast("2222.11523574301594407443285016240908726891841242444092960094015874546135316534057865883047"), + boost::lexical_cast("-519.599993280949289705514822058693289933461756514489674453254304308040888101533569480646682"), + boost::lexical_cast("108.406868361306987817730701109400305482972790224573776407166683184990131682003417239181112"), + boost::lexical_cast("-20.1313142142558596796857948064047373605439974799116521459977609253211918146595346493447238"), + boost::lexical_cast("3.31806787671783168020012913552384112429614503798293169239082032849759155847394955909684383"), + boost::lexical_cast("-0.483817477111537877685595212919784447924875428848331771524426361483392903320495411973587861"), + boost::lexical_cast("0.0621793463102927384924303843912913542297042029136293808338022462765755471002366206711862637"), + boost::lexical_cast("-0.00701366932085103924241526535768453911099671087892444015581511551813219720807206445462785293"), + boost::lexical_cast("0.000691040514756294308758606917671220770856269030526647010461217455799229645004351524024364997"), + boost::lexical_cast("-0.591529398871361458428147660822525365922497109038495896497692806150033516658042357799869656e-4"), + boost::lexical_cast("0.437237367535177689875119370170434437030440227275583289093139147244747901678407875809020739e-5"), + boost::lexical_cast("-0.277164853397051135996651958345647824709602266382721185838782221179129726200661453504250697e-6"), + boost::lexical_cast("0.149506899012035980148891401548317536032574502641368034781671941165064546410613201579653674e-7"), + boost::lexical_cast("-0.68023824066463262779882895193964639544061678698791279217407325790147925675797085217462974e-9"), + boost::lexical_cast("0.258460163734186329938721529982859244969655253624066115559707985878606277800329299821882688e-10"), + boost::lexical_cast("-0.810792256024669306744649981276512583535251727474303382740940985102669076169168931092026491e-12"), + boost::lexical_cast("0.207274966207031327521921078048021807442500113231320959236850963529304158700096495799022922e-13"), + boost::lexical_cast("-0.425399199286327802950259994834798737777721414442095221716122926637623478450472871269742436e-15"), + boost::lexical_cast("0.688866766744198529064607574117697940084548375790020728788313274612845280173338912495478431e-17"), + boost::lexical_cast("-0.862599751805643281578607291655858333628582704771553874199104377131082877406179933909898885e-19"), + boost::lexical_cast("0.815756005678735657200275584442908437977926312650210429668619446123450972547018343768177988e-21"), + boost::lexical_cast("-0.566583084099007858124915716926967268295318152203932871370429534546567151650626184750291695e-23"), + boost::lexical_cast("0.279544761599725082805446124351997692260093135930731230328454667675190245860598623539891708e-25"), + boost::lexical_cast("-0.941169851584987983984201821679114408126982142904386301937192011680047611188837432096199601e-28"), + boost::lexical_cast("0.205866011331040736302780507155525142187875191518455173304638008169488993406425201915370746e-30"), + boost::lexical_cast("-0.27526655245712584371295491216289353976964567057707464008951584303679019796521332324352501e-33"), + boost::lexical_cast("0.208358067979444305082929004102609366169534624348056112144990933897581971394396210379638792e-36"), + boost::lexical_cast("-0.808728107661779323263133007119729988596844663194254976820030366188579170595441991680169012e-40"), + boost::lexical_cast("0.141276924383478964519776436955079978012672985961918248012931336621229652792338950573694356e-43"), + boost::lexical_cast("-0.930318449287651389310440021745842417218125582685428432576258687100661462527604238849332053e-48"), + boost::lexical_cast("0.179870748819321661641184169834635133045197146966203370650788171790610563029431722308057539e-52"), + boost::lexical_cast("-0.705865243912790337263229413370018392321238639017433365017168104310561824133229343750737083e-58"), + boost::lexical_cast("0.3146787805734405996448268100558028857930560442377698646099945108125281507396722995748398e-64"), + boost::lexical_cast("-0.589653534231618730406843260601322236697428143603814281282790370329151249078338470962782338e-72"), + }; + T result = d[0]; + for(int k = 1; k < sizeof(d)/sizeof(d[0]); ++k) + { + result += d[k]/(z+(k-1)); + } + return result; + } + + template + static T lanczos_sum_near_1(const T& dz) + { + using namespace boost; + static const T d[60] = { + boost::lexical_cast("23.2463658527729692390378860713647146932236940604550445351214987229819352880524561852919518"), + boost::lexical_cast("-523.358012551815715084547614110229469295755088686612838322817729744722233637819564673967396"), + boost::lexical_cast("5701.12892340421080714956066268650092612647620400476183901625272640935853188559347587495571"), + boost::lexical_cast("-40033.5506451901904954336453419007623117537868026994808919201793803506999271787018654246699"), + boost::lexical_cast("203689.884259074923009439144410340256983393397995558814367995938668111650624499963153485034"), + boost::lexical_cast("-800270.648969745331278757692597096167418585957703057412758177038340791380011708874081291202"), + boost::lexical_cast("2526668.23380061659863999395867315313385499515711742092815402701950519696944982260718031476"), + boost::lexical_cast("-6587362.57559198722630391278043503867973853468105110382293763174847657538179665571836023631"), + boost::lexical_cast("14462211.3454541602975917764900442754186801975533106565506542322063393991552960595701762805"), + boost::lexical_cast("-27132375.1879227404375395522940895789625516798992585980380939378508607160857820002128106898"), + boost::lexical_cast("43991923.8735251977856804364757478459275087361742168436524951824945035673768875988985478116"), + boost::lexical_cast("-62192284.0030124679010201921841372967696262036115679150017649233887633598058364494608060812"), + boost::lexical_cast("77203473.0770033513405070667417251568915937590689150831268228886281254637715669678358204991"), + boost::lexical_cast("-84630180.2217173903516348977915150565994784278120192219937728967986198118628659866582594874"), + boost::lexical_cast("82294807.2253549409847505891112074804416229757832871133388349982640444405470371147991706317"), + boost::lexical_cast("-71245738.2484649177928765605893043553453557808557887270209768163561363857395639001251515788"), + boost::lexical_cast("55073334.3180266913441333534260714059077572215147571872597651029894142803987981342430068594"), + boost::lexical_cast("-38097984.1648990787690036742690550656061009857688125101191167768314179751258568264424911474"), + boost::lexical_cast("23625729.5032184580395130592017474282828236643586203914515183078852982915252442161768527976"), + boost::lexical_cast("-13149998.4348054726172055622442157883429575511528431835657668083088902165366619827169829685"), + boost::lexical_cast("6574597.77221556423683199818131482663205682902023554728024972161230111356285973623550338976"), + boost::lexical_cast("-2953848.1483469149918109110050192571921018042012905892107136410603990336401921230407043408"), + boost::lexical_cast("1192595.29584357246380113611351829515963605337523874715861849584306265512819543347806085356"), + boost::lexical_cast("-432553.812019608638388918135375154289816441900572658692369491476137741687213006403648722272"), + boost::lexical_cast("140843.215385933866391177743292449477205328233960902455943995092958295858485718885800427129"), + boost::lexical_cast("-41128.186992679630058614841985110676526199977321524879849001760603476646382839182691529968"), + boost::lexical_cast("10756.2849191854701631989789887757784944313743544315113894758328432005767448056040879337769"), + boost::lexical_cast("-2515.15559672041299884426826962296210458052543246529646213159169885444118227871246315458787"), + boost::lexical_cast("524.750087004805200600237632074908875763734578390662349666321453103782638818305404274166951"), + boost::lexical_cast("-97.4468596421732493988298219295878130651986131393383646674144877163795143930682205035917965"), + boost::lexical_cast("16.0613108128210806736384551896802799172445298357754547684100294231532127326987175444453058"), + boost::lexical_cast("-2.34194813526540240672426202485306862230641838409943369059203455578340880900483887447559874"), + boost::lexical_cast("0.300982934748016059399829007219431333744032924923669397318820178988611410275964499475465815"), + boost::lexical_cast("-0.033950095985367909789000959795708551814461844488183964432565731809399824963680858993718525"), + boost::lexical_cast("0.00334502394288921146242772614150438404658527112198421937945605441697314216921393987758378122"), + boost::lexical_cast("-0.000286333429067523984607730553301991502191011265745476190940771685897687956262049750683013485"), + boost::lexical_cast("0.211647426149364947402896718485536530479491688838087899435991994237067890628274492042231115e-4"), + boost::lexical_cast("-0.134163345121302758109675190598169832775248626443483098532368562186356128620805552609175683e-5"), + boost::lexical_cast("0.723697303042715085329782938306424498336642078597508935450663080894255773653328980495047891e-7"), + boost::lexical_cast("-0.329273487343139063533251321553223583999676337945788660475231347828772272134656322947906888e-8"), + boost::lexical_cast("0.12510922551028971731767784013117088894558604838820475961392154031378891971216135267744134e-9"), + boost::lexical_cast("-0.392468958215589939603666430583400537413757786057185505426804034745840192914621891690369903e-11"), + boost::lexical_cast("0.100332717101049934370760667782927946803279422001380028508200697081188326364078428184546051e-12"), + boost::lexical_cast("-0.205917088291197705194762747639836655808855850989058813560983717575008725393428497910009756e-14"), + boost::lexical_cast("0.333450178247893143608439314203175490705783992567107481617660357577257627854979230819461489e-16"), + boost::lexical_cast("-0.417546693906616047110563550428133589051498362676394888715581845170969319500638944065594319e-18"), + boost::lexical_cast("0.394871691642184410859178529844325390739857256666676534513661579365702353214518478078730801e-20"), + boost::lexical_cast("-0.274258012587811199757875927548699264063511843669070634471054184977334027224611843434000922e-22"), + boost::lexical_cast("0.135315354265459854889496635967343027244391821142592599244505313738163473730636430399785048e-24"), + boost::lexical_cast("-0.455579032003288910408487905303223613382276173706542364543918076752861628464036586507967767e-27"), + boost::lexical_cast("0.99650703862462739161520123768147312466695159780582506041370833824093136783202694548427718e-30"), + boost::lexical_cast("-0.1332444609228706921659395801935919548447859029572115502899861345555006827214220195650058e-32"), + boost::lexical_cast("0.100856999148765307000182397631280249632761913433008379786888200467467364474581430670889392e-35"), + boost::lexical_cast("-0.39146979455613683472384690509165395074425354524713697428673406058157887065953366609738731e-39"), + boost::lexical_cast("0.683859606707931248105140296850112494069265272540298100341919970496564103098283709868586478e-43"), + boost::lexical_cast("-0.450326344248604222735147147805963966503893913752040066400766411031387063854141246972061792e-47"), + boost::lexical_cast("0.870675378039492904184581895322153006461319724931909799151743284769901586333730037761678891e-52"), + boost::lexical_cast("-0.341678395249272265744518787745356400350877656459401143889000625280131819505857966769964401e-57"), + boost::lexical_cast("0.152322191370871666358069530949353871960316638394428595988162174042653299702098929238880862e-63"), + boost::lexical_cast("-0.285425405297633795767452984791738825078111150078605004958179057245980222485147999495352632e-71"), + }; + T result = 0; + for(int k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(k*dz + k*k); + } + return result; + } + + template + static T lanczos_sum_near_2(const T& dz) + { + using namespace boost; + static const T d[60] = { + boost::lexical_cast("557.56438192770795764344217888434355281097193198928944200046501607026919782564033547346298"), + boost::lexical_cast("-12552.748616427645475141433405567201788681683808077269330800392600825597799119572762385222"), + boost::lexical_cast("136741.650054039199076788077149441364242294724343897779563222338447737802381279007988884806"), + boost::lexical_cast("-960205.223613240309942047656967301131022760634321049075674684679438471862998829007639437133"), + boost::lexical_cast("4885504.47588736223774859617054275229642041717942140469884121916073195308537421162982679289"), + boost::lexical_cast("-19194501.738192166918904824982935279260356575935661514109550613809352009246483412530545583"), + boost::lexical_cast("60602169.8633537742937457094837494059849674261357199218329545854990149896822944801504450843"), + boost::lexical_cast("-157997975.522506767297528502540724511908584668874487506510120462561270100749019845014382885"), + boost::lexical_cast("346876323.86092543685419723290495817330608574729543216092477261152247521712190505658568876"), + boost::lexical_cast("-650770365.471136883718747607976242475416651908858429752332176373467422603953536408709972919"), + boost::lexical_cast("1055146856.05909309330903130910708372830487315684258450293308627289334336871273080305128138"), + boost::lexical_cast("-1491682726.25614447429071368736790697283307005456720192465860871846879804207692411263924608"), + boost::lexical_cast("1851726287.94866167094858600116562210167031458934987154557042242638980748286188183300900268"), + boost::lexical_cast("-2029855953.68334371445800569238095379629407314338521720558391277508374519526853827142679839"), + boost::lexical_cast("1973842002.53354868177824629525448788555435194808657489238517523691040148611221295436287925"), + boost::lexical_cast("-1708829941.98209573247426625323314413060108441455314880934710595647408841619484540679859098"), + boost::lexical_cast("1320934627.12433688699625456833930317624783222321555050330381730035733198249283009357314954"), + boost::lexical_cast("-913780636.858542526294419197161614811332299249415125108737474024007693329922089123296358727"), + boost::lexical_cast("566663423.929632170286007468016419798879660054391183200464733820209439185545886930103546787"), + boost::lexical_cast("-315402880.436816230388857961460509181823167373029384218959199936902955049832392362044305869"), + boost::lexical_cast("157691811.550465734461741500275930418786875005567018699867961482249002625886064187146134966"), + boost::lexical_cast("-70848085.5705405970640618473551954585013808128304384354476488268600720054598122945113512731"), + boost::lexical_cast("28604413.4050137708444142264980840059788755325900041515286382001704951527733220637586013815"), + boost::lexical_cast("-10374808.7067303054787164054055989420809074792801592763124972441820101840292558840131568633"), + boost::lexical_cast("3378126.32016207486657791623723515804931231041318964254116390764473281291389374196880720069"), + boost::lexical_cast("-986460.090390653140964189383080344920103075349535669020623874668558777188889544398718979744"), + boost::lexical_cast("257989.631187387317948158483575125380011593209850756066176921901006833159795100137743395985"), + boost::lexical_cast("-60326.0391159227288325790327830741260824763549807922845004854215660451399733578621565837087"), + boost::lexical_cast("12586.1375399649496159880821645216260841794563919652590583420570326276086323953958907053394"), + boost::lexical_cast("-2337.26417330316922535871922886167444795452055677161063205953141105726549966801856628447293"), + boost::lexical_cast("385.230745012608736644117458716226876976056390433401632749144285378123105481506733917763829"), + boost::lexical_cast("-56.1716559403731491675970177460841997333796694857076749852739159067307309470690838101179615"), + boost::lexical_cast("7.21907953468550196348585224042498727840087634483369357697580053424523903859773769748599575"), + boost::lexical_cast("-0.814293485887386870805786409956942772883474224091975496298369877683530509729332902182019049"), + boost::lexical_cast("0.0802304419995150047616460464220884371214157889148846405799324851793571580868840034085001373"), + boost::lexical_cast("-0.00686771095380619535195996193943858680694970000948742557733102777115482017857981277171196115"), + boost::lexical_cast("0.000507636621977556438232617777542864427109623356049335590894564220687567763620803789858345916"), + boost::lexical_cast("-0.32179095465362720747836116655088181481893063531138957363431280817392443948706754917605911e-4"), + boost::lexical_cast("0.173578890579848508947329833426585354230744194615295570820295052665075101971588563893718407e-5"), + boost::lexical_cast("-0.789762880006288893888161070734302768702358633525134582027140158619195373770299678322596835e-7"), + boost::lexical_cast("0.300074637200885066788470310738617992259402710843493097610337134266720909870967550606601658e-8"), + boost::lexical_cast("-0.941337297619721713151110244242536308266701344868601679868536153775533330272973088246835359e-10"), + boost::lexical_cast("0.24064815013182536657310186836079333949814111498828401548170442715552017773994482539471456e-11"), + boost::lexical_cast("-0.493892399304811910466345686492277504628763169549384435563232052965821874553923373100791477e-13"), + boost::lexical_cast("0.799780678476644196901221989475355609743387528732994566453160178199295104357319723742820593e-15"), + boost::lexical_cast("-0.100148627870893347527249092785257443532967736956154251497569191947184705954310833302770086e-16"), + boost::lexical_cast("0.947100256812658897084619699699028861352615460106539259289295071616221848196411749449858071e-19"), + boost::lexical_cast("-0.657808193528898116367845405906343884364280888644748907819280236995018351085371701094007759e-21"), + boost::lexical_cast("0.324554050057463845012469010247790763753999056976705084126950591088538742509983426730851614e-23"), + boost::lexical_cast("-0.10927068902162908990029309141242256163212535730975970310918370355165185052827948996110107e-25"), + boost::lexical_cast("0.239012340507870646690121104637467232366271566488184795459093215303237974655782634371712486e-28"), + boost::lexical_cast("-0.31958700972990573259359660326375143524864710953063781494908314884519046349402409667329667e-31"), + boost::lexical_cast("0.241905641292988284384362036555782113275737930713192053073501265726048089991747342247551645e-34"), + boost::lexical_cast("-0.93894080230619233745797029179332447129464915420290457418429337322820997038069119047864035e-38"), + boost::lexical_cast("0.164023814025085488413251990798690797467351995518990067783355251949198292596815470576539877e-41"), + boost::lexical_cast("-0.108010831192689925518484618970761942019888832176355541674171850211917230280206410356465451e-45"), + boost::lexical_cast("0.208831600642796805563854019033577205240227465154130766898180386564934443551840379116390645e-50"), + boost::lexical_cast("-0.819516067465171848863933747691434138146789031214932473898084756489529673230665363014007306e-56"), + boost::lexical_cast("0.365344970579318347488211604761724311582675708113250505307342682118101409913523622073678179e-62"), + boost::lexical_cast("-0.684593199208628857931267904308244537968349564351534581234005234847904343404822808648361291e-70"), + }; + T result = 0; + T z = dz + 2; + for(int k = 1; k <= sizeof(d)/sizeof(d[0]); ++k) + { + result += (-d[k-1]*dz)/(z + k*z + k*k - 1); + } + return result; + } + + static double g(){ return 63.19215200000000010049916454590857028961181640625; } +}; + +namespace boost{ namespace math{ namespace lanczos{ + +template +struct lanczos +{ +#ifdef L13 + typedef lanczos13UDT type; +#elif defined(L22) + typedef lanczos22UDT type; +#elif defined(L31) + typedef lanczos31UDT type; +#else + typedef lanczos61UDT type; +#endif +}; + +}}} + +#endif + + diff --git a/tools/process_perf_results.cpp b/tools/process_perf_results.cpp new file mode 100644 index 000000000..816f422c9 --- /dev/null +++ b/tools/process_perf_results.cpp @@ -0,0 +1,127 @@ +// Copyright John Maddock 2007. + +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +std::map results; + +std::map extra_text; + +void load_file(std::string& s, std::istream& is) +{ + s.erase(); + if(is.bad()) return; + s.reserve(is.rdbuf()->in_avail()); + char c; + while(is.get(c)) + { + if(s.capacity() == s.size()) + s.reserve(s.capacity() * 3); + s.append(1, c); + } +} + +int main(int argc, const char* argv[]) +{ + // + // Set any additional text that should accumpany specific results: + // + extra_text["msvc-dist-beta-R-quantile"] = "[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]"; + extra_text["msvc-dist-nbinom-R-quantile"] = "[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in \"typical\" cases]"; + extra_text["gcc-4_2-dist-beta-R-quantile"] = "[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]"; + extra_text["gcc-4_2-dist-nbinom-R-quantile"] = "[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in \"typical\" cases]"; + boost::regex e("^Testing\\s+(\\S+)\\s+(\\S+)"); + std::string f; + for(int i = 1; i < argc-1; ++i) + { + std::ifstream is(argv[i]); + load_file(f, is); + boost::sregex_iterator a(f.begin(), f.end(), e), b; + while(a != b) + { + results[(*a).str(1)] = boost::lexical_cast((*a).str(2)); + ++a; + } + } + // + // Load quickbook file: + // + std::ifstream is(argv[argc-1]); + std::ofstream os(std::string(argv[argc-1]).append(".bak").c_str()); + e.assign("\\[perf\\s+([^\\s.]+)(?:\\[[^\\]]*\\]|[^\\]])*\\]"); + std::string newfile; + while(is.good()) + { + std::getline(is, f); + os << f << std::endl; + boost::sregex_iterator i(f.begin(), f.end(), e), j; + double min = (std::numeric_limits::max)(); + while(i != j) + { + std::cout << (*i).str() << std::endl << (*i).str(1) << std::endl; + std::string item = (*i).str(1); + if(results.find(item) != results.end()) + { + double r = results[item]; + if(r < min) + min = r; + } + ++i; + } + // + // Now perform the substitutions: + // + std::string newstr; + std::string tail; + i = boost::sregex_iterator(f.begin(), f.end(), e); + while(i != j) + { + std::string item = (*i).str(1); + newstr.append(i->prefix()); + if(results.find(item) != results.end()) + { + double v = results[item]; + double r = v / min; + newstr += std::string((*i)[0].first, (*i)[1].second); + newstr += "..[para "; + if(r < 1.01) + newstr += "*"; + newstr += (boost::format("%.2f") % r).str(); + if(r < 1.01) + newstr += "*"; + if(extra_text.find(item) != extra_text.end()) + { + newstr += extra_text[item]; + } + newstr += "][para ("; + newstr += (boost::format("%.3e") % results[item]).str(); + newstr += "s)]]"; + } + else + { + newstr.append(i->str()); + std::cerr << "Item " << item << " not found!!" << std::endl; + } + tail = i->suffix(); + ++i; + } + if(newstr.size()) + newfile.append(newstr).append(tail); + else + newfile.append(f); + newfile.append("\n"); + } + is.close(); + std::ofstream ns(argv[argc-1]); + ns << newfile; +} + diff --git a/tools/rational_tests.cpp b/tools/rational_tests.cpp new file mode 100644 index 000000000..ebb1bdb78 --- /dev/null +++ b/tools/rational_tests.cpp @@ -0,0 +1,366 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +int main() +{ + using namespace boost::math; + using namespace boost::math::tools; + + boost::math::ntl::RR::SetPrecision(500); + boost::math::ntl::RR::SetOutputPrecision(40); + + std::tr1::mt19937 rnd; + std::tr1::variate_generator< + std::tr1::mt19937, + std::tr1::uniform_int<> > gen(rnd, std::tr1::uniform_int<>(1, 12)); + + for(unsigned i = 1; i < 12; ++i) + { + std::vector coef; + for(unsigned j = 0; j < i; ++j) + { + coef.push_back(gen()); + } + std::cout << +" //\n" +" // Polynomials of order " << i-1 << "\n" +" //\n" +" static const U n" << i << "c[" << i << "] = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + std::cout << + " static const boost::array n" << i << "a = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + + boost::math::ntl::RR r1 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + boost::math::ntl::RR r2 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + boost::math::ntl::RR r3 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + boost::math::ntl::RR r4 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + r1 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_even_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_even_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + if(i > 1) + { + r1 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_odd_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3 << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_odd_polynomial(n" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4 << "L),\n" + " tolerance);\n\n"; + } + + r1 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + r2 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + r3 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + r4 = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + + coef.clear(); + for(unsigned j = 0; j < i; ++j) + { + coef.push_back(gen()); + } + std::cout << +" //\n" +" // Rational functions of order " << i-1 << "\n" +" //\n" +" static const U d" << i << "c[" << i << "] = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + std::cout << + " static const boost::array d" << i << "a = { "; + for(unsigned j = 0; j < i; ++j) + { + if(j) + std::cout << ", "; + std::cout << coef[j]; + } + std::cout << " };\n"; + + boost::math::ntl::RR r1d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.125), i); + boost::math::ntl::RR r2d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.25), i); + boost::math::ntl::RR r3d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(0.75), i); + boost::math::ntl::RR r4d = boost::math::tools::evaluate_polynomial(&coef[0], boost::math::ntl::RR(1) - boost::math::ntl::RR(1) / 64, i); + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.125), " << i << "),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.25), " << i << "),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.75), " << i << "),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(1.0f - 1.0f/64.0f), " << i << "),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.125)),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.25)),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(0.75)),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "c, d" << i << "c, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n\n"; + + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.125)),\n" + " static_cast(" << r1/r1d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.25)),\n" + " static_cast(" << r2/r2d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(0.75)),\n" + " static_cast(" << r3/r3d << "L),\n" + " tolerance);\n"; + std::cout << + " BOOST_CHECK_CLOSE(\n" + " boost::math::tools::evaluate_rational(n" << i << "a, d" << i << "a, static_cast(1.0f - 1.0f/64.0f)),\n" + " static_cast(" << r4/r4d << "L),\n" + " tolerance);\n\n"; + } + + return 0; +} + + + diff --git a/tools/spherical_harmonic_data.cpp b/tools/spherical_harmonic_data.cpp new file mode 100644 index 000000000..e66fc228e --- /dev/null +++ b/tools/spherical_harmonic_data.cpp @@ -0,0 +1,91 @@ +// (C) Copyright John Maddock 2007. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include +#include +#include + +using namespace boost::math::tools; +using namespace boost::math; +using namespace std; + +float extern_val; +// confuse the compilers optimiser, and force a truncation to float precision: +float truncate_to_float(float const * pf) +{ + extern_val = *pf; + return *pf; +} + + + +template +std::tr1::tuple spherical_harmonic_data(T i) +{ + static tr1::mt19937 r; + + int n = real_cast(floor(i)); + tr1::uniform_int<> ui(0, (std::min)(n, 40)); + int m = ui(r); + + std::tr1::uniform_real ur(-2*constants::pi(), 2*constants::pi()); + float _theta = ur(r); + float _phi = ur(r); + T theta = truncate_to_float(&_theta); + T phi = truncate_to_float(&_phi); + + T r1 = spherical_harmonic_r(n, m, theta, phi); + T r2 = spherical_harmonic_i(n, m, theta, phi); + return std::tr1::make_tuple(n, m, theta, phi, r1, r2); +} + +int test_main(int argc, char*argv []) +{ + using namespace boost::math::tools; + + boost::math::ntl::RR::SetOutputPrecision(50); + boost::math::ntl::RR::SetPrecision(1000); + + parameter_info arg1, arg2, arg3; + test_data data; + + bool cont; + std::string line; + + if(argc < 1) + return 1; + + do{ + if(0 == get_user_parameter_info(arg1, "n")) + return 1; + arg1.type |= dummy_param; + arg2.type |= dummy_param; + arg3 = arg2; + + data.insert(&spherical_harmonic_data, arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=spherical_harmonic.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "spherical_harmonic.ipp"; + std::ofstream ofs(line.c_str()); + line.erase(line.find('.')); + ofs << std::scientific; + write_code(ofs, data, line.c_str()); + + return 0; +} + diff --git a/tools/tgamma_ratio_data.cpp b/tools/tgamma_ratio_data.cpp new file mode 100644 index 000000000..ed1082cdd --- /dev/null +++ b/tools/tgamma_ratio_data.cpp @@ -0,0 +1,99 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +#include + +#include "ntl_rr_lanczos.hpp" + +using namespace boost::math::tools; +using namespace std; + +std::tr1::tuple + tgamma_ratio(const boost::math::ntl::RR& a, const boost::math::ntl::RR& delta) +{ + if(delta > a) + throw std::domain_error(""); + boost::math::ntl::RR tg = boost::math::tgamma(a); + boost::math::ntl::RR r1 = tg / boost::math::tgamma(a + delta); + boost::math::ntl::RR r2 = tg / boost::math::tgamma(a - delta); + if((r1 > (std::numeric_limits::max)()) || (r2 > (std::numeric_limits::max)())) + throw std::domain_error(""); + + return std::tr1::make_tuple(r1, r2); +} + +boost::math::ntl::RR tgamma_ratio2(const boost::math::ntl::RR& a, const boost::math::ntl::RR& b) +{ + return boost::math::tgamma(a) / boost::math::tgamma(b); +} + + +int test_main(int argc, char*argv []) +{ + boost::math::ntl::RR::SetPrecision(1000); + boost::math::ntl::RR::SetOutputPrecision(40); + + parameter_info arg1, arg2; + test_data data; + + bool cont; + std::string line; + + if((argc >= 2) && (strcmp(argv[1], "--ratio") == 0)) + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the function tgamma_ratio(a, b)\n\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "a")) + return 1; + if(0 == get_user_parameter_info(arg2, "b")) + return 1; + data.insert(&tgamma_ratio2, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + else + { + std::cout << "Welcome.\n" + "This program will generate spot tests for the function tgamma_delta_ratio(a, delta)\n\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "a")) + return 1; + if(0 == get_user_parameter_info(arg2, "delta")) + return 1; + data.insert(&tgamma_ratio, arg1, arg2); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + } + + std::cout << "Enter name of test data file [default=tgamma_ratio_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "tgamma_ratio_data.ipp"; + std::ofstream ofs(line.c_str()); + ofs << std::scientific; + write_code(ofs, data, "tgamma_ratio_data"); + + return 0; +} + + diff --git a/vc71_fix/Jamfile.v2 b/vc71_fix/Jamfile.v2 new file mode 100644 index 000000000..3ed51cf46 --- /dev/null +++ b/vc71_fix/Jamfile.v2 @@ -0,0 +1,6 @@ +# Copyright 2007 John Maddock and Paul A. Bristow. +# Distributed under the Boost Software License, Version 1.0. +# (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +lib vc_fix : instantiate_all.cpp : static ; + diff --git a/vc71_fix/instantiate_all.cpp b/vc71_fix/instantiate_all.cpp new file mode 100644 index 000000000..450ca188e --- /dev/null +++ b/vc71_fix/instantiate_all.cpp @@ -0,0 +1,29 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +// +// MSVC-7.1 has a problem with our tests: sometimes when a +// function is used via a function pointer, it does *not* +// instantiate the template, leading to unresolved externals +// at link time. Therefore we create a small library that +// instantiates "everything", and link all our tests against +// it for msvc-7.1 only. Note that due to some BBv2 limitations +// we can not place this in a sub-folder of the test directory +// as that would lead to recursive project dependencies... +// + +#include "../test/compile_test/instantiate.hpp" +#include + +void some_proc() +{ + instantiate(float(0)); + instantiate(double(0)); + instantiate(static_cast(0)); + instantiate(static_cast(0)); +} + + +