From bb1fbff202dd942cdca77b8914ed3d16cfa8168d Mon Sep 17 00:00:00 2001 From: John Maddock Date: Thu, 28 Dec 2006 11:04:39 +0000 Subject: [PATCH] Misc updates and fixes to the code, including a re-ordering of parameters to ensure TR1 compatibility. Updated test data. Ellint docs added. 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test/test_ellint_2.cpp | 5 +- test/test_ellint_3.cpp | 19 +- tools/ellint_f_data.cpp | 2 +- tools/ellint_pi2_data.cpp | 2 +- tools/ellint_pi3_data.cpp | 2 +- 78 files changed, 50943 insertions(+), 46 deletions(-) create mode 100644 doc/ellint_carlson.qbk create mode 100644 doc/ellint_introduction.qbk create mode 100644 doc/ellint_legendre.qbk create mode 100644 doc/equations/ellint.xml create mode 100644 doc/equations/ellint1.png create mode 100644 doc/equations/ellint1.tex create mode 100644 doc/equations/ellint10.png create mode 100644 doc/equations/ellint10.tex create mode 100644 doc/equations/ellint11.png create mode 100644 doc/equations/ellint11.tex create mode 100644 doc/equations/ellint12.png create mode 100644 doc/equations/ellint12.tex create mode 100644 doc/equations/ellint13.png create mode 100644 doc/equations/ellint13.tex create mode 100644 doc/equations/ellint14.png create mode 100644 doc/equations/ellint14.tex create mode 100644 doc/equations/ellint15.png create mode 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mode 100644 doc/equations/ellint6.tex create mode 100644 doc/equations/ellint7.png create mode 100644 doc/equations/ellint7.tex create mode 100644 doc/equations/ellint8.png create mode 100644 doc/equations/ellint8.tex create mode 100644 doc/equations/ellint9.png create mode 100644 doc/equations/ellint9.tex create mode 100644 doc/graphs/ellint_1.png create mode 100644 doc/graphs/ellint_1.ps create mode 100644 doc/graphs/ellint_1.rgd create mode 100644 doc/graphs/ellint_2.png create mode 100644 doc/graphs/ellint_2.ps create mode 100644 doc/graphs/ellint_2.rgd create mode 100644 doc/graphs/ellint_3.png create mode 100644 doc/graphs/ellint_3.ps create mode 100644 doc/graphs/ellint_3.rgd create mode 100644 doc/graphs/ellint_c.png diff --git a/doc/ellint_carlson.qbk b/doc/ellint_carlson.qbk new file mode 100644 index 000000000..6842078cf --- /dev/null +++ b/doc/ellint_carlson.qbk @@ -0,0 +1,173 @@ +[/ +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] + +[caution __caution ] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_rf(T x, T y, T z) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_rd(T x, T y, T z) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_rj(T x, T y, T z, T p) + + }} // namespaces + + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_rc(T x, T y) + + }} // 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: + +[$../graphs/ellint_c.png] + + template + T ellint_rf(T x, T y, T z) + +Returns Carlson's Elliptic Integral R[sub F]: + +[$../equations/ellint9.png] + +Requires that all of the arguments are non-negative, and at most +one may be zero. Otherwise returns the result of __domain_error. + + template + T ellint_rd(T x, T y, T z) + +Returns Carlson's elliptic integral R[sub D]: + +[$../equations/ellint10.png] + +Requires that x and y are non-negative, with at most one of them +zero, and that z >= 0. Otherwise returns the result of __domain_error. + + template + T ellint_rj(T x, T y, T z, T p) + +Returns Carlson's elliptic integral R[sub J]: + +[$../equations/ellint11.png] + +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. + +When ['p < 0] the function returns the Cauchy principle value using the relation: + +[$../equations/ellint17.png] + + template + T ellint_rc(T x, T y) + +Returns Carlson's elliptic integral R[sub C]: + +[$../equations/ellint12.png] + +Requires that ['x > 0] and that ['y != 0]. +Otherwise returns the result of __domain_error. + +When ['y < 0] the function returns the Cauchy principle value using the relation: + +[$../equations/ellint18.png] + +[heading Testing] + +There are two sets of tests: spot tests compare selected values with test data +given in: + +[:B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227 +Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms, +Volume 10, Number 1 / March, 1995, p13-26.] + +While random test data generated using NTL::RR at 1000-bit precision and our +implementation checks for rounding-errors and/or regressions. + +There are also sanity checks that use the inter-relations between the integrals +to verify their correctness: see the above Carlson paper for details. + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Carlson Elliptic Integrals +[[Significand Size] [Platform and Compiler] [R[sub F]] [R[sub D]] [R[sub J]] [R[sub C]]] +[[53] [Win32 / Visual C++ 8.0] [Peek=2.9 Mean=0.75] [Peek=2.6 Mean=0.9] [Peek=108 Mean=6.9] [Peek=2.4 Mean=0.6] ] +[[64] [Red Hat Linux / G++ 3.4] [Peek=2.5 Mean=0.75] [Peek=2.7 Mean=0.9] [Peek=105 Mean=8] [Peek=1.9 Mean=0.7] ] +[[113] [HP-UX / HP aCC 6] [Peek=5.3 Mean=1.6] [Peek=2.9 Mean=0.99] [Peek=180 Mean=12] [Peek=1.8 Mean=0.7] ] +] + +[heading Implementation] + +The key of Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] is the +duplication theorem: + +[$../equations/ellint13.png] + +By applying it repeatedly, ['x], ['y], ['z] get +closer and closer. When they are nearly equal, the special case equation + +[$../equations/ellint16.png] + +is used. More specifically, ['[R F]] is evaluated from a Taylor series +expansion to the fifth order. The calculations of the other three integrals +are analogous. + +For ['p < 0] in ['R[sub J](x, y, z, p)] and ['y < 0] in ['R[sub C](x, y)], +the integrals are singular and their Cauchy principal values are returned via the +relations: + +[$../equations/ellint17.png] + +[$../equations/ellint18.png] + +[endsect] diff --git a/doc/ellint_introduction.qbk b/doc/ellint_introduction.qbk new file mode 100644 index 000000000..70d191a99 --- /dev/null +++ b/doc/ellint_introduction.qbk @@ -0,0 +1,208 @@ +[/ +Copyright (c) 2006 Xiaogang Zhang +Use, modification and distribution are subject to the +Boost Software License, Version 1.0. (See accompanying file +LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +] + +[section:ellint_intro Elliptic Integral Overview] + +The main reference for the elliptic integrals is: + +[:M. Abramowitz and I. A. Stegun (Eds.) (1964) +Handbook of Mathematical Functions with Formulas, Graphs, and +Mathematical Tables, +National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.] + +Mathworld also contain a lot of useful background information: + +[:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W. +"Elliptic Integral." From MathWorld--A Wolfram Web Resource.]] + +As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia]. + +[h4 Notation] + +All variables are real numbers unless otherwise noted. + + + +[h4 [#ellint_def]Definition] + +[$../equations/ellint1.png] + +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) + +[$../equations/ellint2.png] + +Elliptic Integral of the Second Kind (Legendre form) + +[$../equations/ellint3.png] + +Elliptic Integral of the Third Kind (Legendre form) + +[$../equations/ellint4.png] + +where + +[$../equations/ellint5.png] + +[note ['[phi]] is called the amplitude. \n + ['k] is called the modulus. \n + ['[alpha]] is called the modular angle. \n + ['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:\n\n +k = sin[alpha]\n +m = k[super 2] = sin[super 2][alpha]\n\n +So that the integral of the third kind (for example) may be expressed as +either:\n\n +[Pi](n, [phi], k)\n +[Pi](n, [phi] \\ [alpha])\n +[Pi](n, [phi]| m)\n\n +To further complicate matters, some texts refer to the ['complement +of the parameter m], or 1 - m, where:\n\n +1 - m = 1 - k[super 2] = cos[super 2][alpha]\n\n +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) + +[$../equations/ellint6.png] + +Complete Elliptic Integral of the Second Kind (Legendre form) + +[$../equations/ellint7.png] + +Complete Elliptic Integral of the Third Kind (Legendre form) + +[$../equations/ellint8.png] + +Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78 Carlson78]] gives an alternative definition of +elliptic integral's canonical forms: + +Carlson's Elliptic Integral of the First Kind + +[$../equations/ellint9.png] + +where ['x], ['y], ['z] are nonnegative and at most one of them +may be zero. + +Carlson's Elliptic Integral of the Second Kind + +[$../equations/ellint10.png] + +where ['x], ['y] are nonnegative, at most one of them may be zero, +and ['z] must be positive. + +Carlson's Elliptic Integral of the Third Kind + +[$../equations/ellint11.png] + +where ['x], ['y], ['z] are nonnegative, at most one of them may be +zero, and ['p] must be nonzero. + +Carlson's Degenerate Elliptic Integral + +[$../equations/ellint12.png] + +where ['x] is nonnegative and ['y] is nonzero. + +[note ['R[sub C](x, y) = R[sub F](x, y, y)] \n + ['R[sub D](x, y, z) = R[sub J](x, y, z, z)]] + + + +[h4 [#ellint_theorem]Duplication Theorem] + +Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that + +[$../equations/ellint13.png] + +[h4 [#ellint_formula]Carlson's Formulas] + +The Legendre form and Carlson form of elliptic integrals are related +by equations: + +[$../equations/ellint14.png] + +In particular, + +[$../equations/ellint15.png] + +[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] + +# [#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/ellint_legendre.qbk b/doc/ellint_legendre.qbk new file mode 100644 index 000000000..ccdc4943d --- /dev/null +++ b/doc/ellint_legendre.qbk @@ -0,0 +1,287 @@ +[/ +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 Elliptic Integrals of the First Kind - Legendre Form] + +[caution __caution] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_1(T k, T phi); + + template + T ellint_1(T k); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the first kind +['F([phi], k)] and it's complete counterpart ['K(k) = F([pi]/2, k)]. + +[$../graphs/ellint_1.png] + + template + T ellint_1(T k, T phi); + +Returns the incomplete elliptic integral of the first kind ['F([phi], k)]: + +[$../equations/ellint2.png] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + + template + T ellint_1(T k); + +Returns the complete elliptic integral of the first kind ['K(k)]: + +[$../equations/ellint6.png] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Elliptic Integrals of the First Kind +[[Significand Size] [Platform and Compiler] [F([phi], k)] [K(k)] ] +[[53] [Win32 / Visual C++ 8.0] [Peek=3 Mean=0.8] [Peek=1.8 Mean=0.7] ] +[[64] [Red Hat Linux / G++ 3.4] [Peek=2.6 Mean=1.7] [Peek=2.2 Mean=1.8] ] +[[113] [HP-UX / HP aCC 6] [Peek=4.6 Mean=1.5] [Peek=3.7 Mean=1.5] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [@functions.wolfram.com +functions.wolfram.com], and random test data generated using +NTL::RR at 1000-bit precision and this implementation. + +[heading Implementation] + +These functions are implemented in terms of Carlson's integrals +using the relations: + +[$../equations/ellint19.png] + +and + +[$../equations/ellint20.png] + + +[endsect] + +[section Elliptic Integrals of the Second Kind - Legendre Form] + +[caution __caution] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_2(T k, T phi); + + template + T ellint_2(T k); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the second kind +['E([phi], k)] and it's complete counterpart ['E(k) = E([pi]/2, k)]. + +[$../graphs/ellint_2.png] + + template + T ellint_2(T k, T phi); + +Returns the incomplete elliptic integral of the second kind ['E([phi], k)]: + +[$../equations/ellint3.png] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + + template + T ellint_2(T k); + +Returns the complete elliptic integral of the first kind ['E(k)]: + +[$../equations/ellint7.png] + +Requires -1 <= k <= 1, otherwise returns the result of __domain_error. + +[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] [Peek=4.6 Mean=1.2] [Peek=3.5 Mean=1.0] ] +[[64] [Red Hat Linux / G++ 3.4] [Peek=4.3 Mean=1.1] [Peek=4.6 Mean=1.2] ] +[[113] [HP-UX / HP aCC 6] [Peek=5.8 Mean=2.2] [Peek=10.8 Mean=2.3] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [@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: + +[$../equations/ellint21.png] + +and + +[$../equations/ellint22.png] + + +[endsect] + +[section Elliptic Integrals of the Third Kind - Legendre Form] + +[caution __caution] + +[heading Synopsis] + +`` + #include +`` + + namespace boost { namespace math { + + template + T ellint_3(T k, T n, T phi); + + template + T ellint_3(T k, T n); + + }} // namespaces + +[heading Description] + +These two functions evaluate the incomplete elliptic integral of the third kind +['[Pi](n, [phi], k)] and it's complete counterpart ['[Pi](n, k) = E(n, [pi]/2, k)]. + +[$../graphs/ellint_3.png] + + template + T ellint_3(T k, T n, T phi); + +Returns the incomplete elliptic integral of the third kind ['[Pi](n, [phi], k)]: + +[$../equations/ellint4.png] + +Requires ['-1 <= k <= 1] and ['n < 1/sin[super 2]([phi])], otherwise +returns the result of __domain_error. + +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 it's "natural" range +of \[0, [pi]\/2\]. + + template + T ellint_3(T k, T n); + +Returns the complete elliptic integral of the first kind ['[Pi](n, k)]: + +[$../equations/ellint8.png] + +Requires ['-1 <= k <= 1] and ['n < 1], otherwise returns the result of __domain_error. + +[heading Accuracy] + +These functions are computed using only basic arithmetic operations, so +there isn't much variation in accuracy over differing platforms. +Note that only results for the widest floating point type on the +system are given as narrower types have __zero_error. All values +are relative errors in units of epsilon. + +[table Errors Rates in the Elliptic Integrals of the Third Kind +[[Significand Size] [Platform and Compiler] [[Pi](n, [phi], k)] [[Pi](n, k)] ] +[[53] [Win32 / Visual C++ 8.0] [Peek=29 Mean=2.2] [Peek=3 Mean=0.8] ] +[[64] [Red Hat Linux / G++ 3.4] [Peek=14 Mean=1.3] [Peek=2.3 Mean=0.8] ] +[[113] [HP-UX / HP aCC 6] [Peek=10 Mean=1.4] [Peek=4.2 Mean=1.1] ] +] + + +[heading Testing] + +The tests use a mixture of spot test values calculated using the online +calculator at [@functions.wolfram.com +functions.wolfram.com], and random test data generated using +NTL::RR at 1000-bit precision and this implementation. + +[heading Implementation] + +The implementation for [Pi](n, [phi], k) first siphons off the special cases: + +['[Pi](0, [phi], k) = F([phi], k)] + +['[Pi](n, [pi]/2, k) = [Pi](n, k)] + +and + +[$../equations/ellint23.png] + +Then if n < 0 the relations (A&S 17.7.15/16): + +[$../equations/ellint24.png] + +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: + +[$../equations/ellint25.png] + +and + +[$../equations/ellint26.png] + +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. Likewise the identities 17.7.7/8 +in A&S for reducing n to the range \[0,1\] appear to be no longer applicable. + +[endsect] diff --git a/doc/equations/ellint.xml b/doc/equations/ellint.xml new file mode 100644 index 0000000000000000000000000000000000000000..9bdcd88be3ec4213bf2b175e554bab217aba1f02 GIT binary patch literal 47010 zcmezW&xXOCp@JcoA%{VMp^PDwp@^ZFA(J7WA&vXE z0<|H4A(X+1!H2<3H99ITE26YBr1_iL$AX`fq3K*=wCYLjS zpdLdxgE501Lq0yj9GUP(-G-NPj&}B$t z(1WUi*%uA=ImiVe44z=WE5P+YLRTGZ8X*@MGFUQLGFTwF3e|n;40a5$(2xfC5MnkY z2J;w-!SMnTC(a&A6kmeO2E||oLn=ceLkiRuMFw34U5rpqW+(!yDq%=vNMTT5C}+q7 z#|a{KoEZ`siWy266c}6?3K>el;Ri`sAQ4bHDgnm>db(3!FksMQuwXD|PypKuNmcp` zsIE!^r^QOB8)31d08Pt~kWivZh$}HDfKz=jI0Ys$R5I8yC@~~4?0xeP@N`3&W75twKRLn2s5DvAt3wg^KUAqR;E5FZ_5w-1}DG-&<>#T6(#(bXXH zi8B?Z1Ee~W!H%Jl!G=MfA(sKfhU7q)3T!k;S3ZLsgEmwbh)tX>kokpBT_84oT@W{6 z#tFKcAnJ$>C6Mcj8A=(Fu*Mp)*&vl5R|b*bVn`Tz5v2p-Zo+N>xr12Wf!q&LPdL;- zDnS^e8sy6gqTCG90m2|%pm^7T>H@Lx$0ke{$S(5I8+mqBL0t&4mpB)K{6e~2AYE$U 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Fit Data 4] +Options1=0 +Options2=0 +Options3=0 +Options4=0 +Options5=0 +Options6=0 +X Position=14 +Y Position=5 +Style=1 +Font="00000000000000000010Arial" +Correlation Coefficient=0 +Accuracy=8 +SF Or DP=0 +Display Accuracy=0 +Display Trailing Zeros=0 +[End Best Fit Data 4] + +[Best Fit Data 5] +Options1=0 +Options2=0 +Options3=0 +Options4=0 +Options5=0 +Options6=0 +X Position=17 +Y Position=5 +Style=1 +Font="00000000000000000010Arial" +Correlation Coefficient=0 +Accuracy=8 +SF Or DP=0 +Display Accuracy=0 +Display Trailing Zeros=0 +[End Best Fit Data 5] + +[Best Fit Data 6] +Options1=0 +Options2=0 +Options3=0 +Options4=0 +Options5=0 +Options6=0 +X Position=20 +Y Position=5 +Style=1 +Font="00000000000000000010Arial" +Correlation Coefficient=0 +Accuracy=8 +SF Or DP=0 +Display Accuracy=0 +Display Trailing Zeros=0 +[End Best Fit Data 6] + +[Equations] +Equation 1="" +Equation 2="" +Equation 3="" +Equation 4="" +Equation 5="" +Equation 6="" +Equation 7="" +Equation 8="" +Equation 9="" +Equation 10="" +Equation 11="" +Equation 12="" +Angle Measurement=0 +[End Equations] + +[Equation Options] +Equation 1="0100000000" +Equation 2="0100000000" +Equation 3="0100000000" +Equation 4="0100000000" +Equation 5="0100000000" +Equation 6="0100000000" +Equation 7="0100000000" +Equation 8="0100000000" +Equation 9="0100000000" +Equation 10="0100000000" +Equation 11="0100000000" +Equation 12="0100000000" +[End Equation Options] + +[Print Setup & Margins] +Paper Size=9 +Orientation=2 +Left=2.5 +Right=2.5 +Top=2.5 +Bottom=2.5 +[End Print Setup & Margins] + +[Datasheet] +Column Width 1=1500 +Column Width 2=1500 +Column Width 3=1500 +Column Width 4=1500 +Column Width 5=1500 +Column Width 6=1500 +Column Width 7=1500 +Data Titles 1="phi = 0.1" +Data Titles 2="phi = 0.75" +Data Titles 3="phi = 1.25" +Data Titles 4="phi = Pi/2" +Data Titles 5="Y5 Data" +Data Titles 6="Y6 Data" +[End Datasheet] + +[Other Graph Options] +Zoom Factor=1 +Auto Redraw=1 +[End Other Graph Options] + +[End] diff 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@@ [library Math Toolkit [quickbook 1.3] - [copyright 2006 John Maddock and Paul A. Bristow] + [copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang] [purpose Various Special Functions, Statistical Distributions, and Numerical Tools] [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.]] + [authors [Maddock, John], [Bristow, Paul A.], [Zhang, Xiaogang]] [category math] [/last-revision $Date$] ] @@ -241,6 +241,15 @@ external graphing application. [include laguerre.qbk] [include hermite.qbk] [include spherical_harmonic.qbk] + +[section Elliptic Integrals] + +[include ellint_introduction.qbk] +[include ellint_carlson.qbk] +[include ellint_legendre.qbk] + +[endsect] + [include fpclassify.qbk] [include powers.qbk] [include error_handling.qbk] diff --git a/include/boost/math/special_functions/ellint_1.hpp b/include/boost/math/special_functions/ellint_1.hpp index 91918a1d8..316a1be68 100644 --- a/include/boost/math/special_functions/ellint_1.hpp +++ b/include/boost/math/special_functions/ellint_1.hpp @@ -124,7 +124,7 @@ inline T ellint_1(T k) // Elliptic integral (Legendre form) of the first kind template -inline T ellint_1(T phi, T k) +inline T ellint_1(T k, T phi) { typedef typename tools::evaluation::type>::type value_type; return tools::checked_narrowing_cast::type>(detail::ellint_f_imp(static_cast(phi), static_cast(k)), BOOST_CURRENT_FUNCTION); diff --git a/include/boost/math/special_functions/ellint_2.hpp b/include/boost/math/special_functions/ellint_2.hpp index 35b926371..9520954f3 100644 --- a/include/boost/math/special_functions/ellint_2.hpp +++ b/include/boost/math/special_functions/ellint_2.hpp @@ -125,7 +125,7 @@ inline T ellint_2(T k) // Elliptic integral (Legendre form) of the second kind template -inline T ellint_2(T phi, T k) +inline T ellint_2(T k, T phi) { typedef typename tools::evaluation::type>::type value_type; return tools::checked_narrowing_cast::type>(detail::ellint_e_imp(static_cast(phi), static_cast(k)), BOOST_CURRENT_FUNCTION); diff --git a/include/boost/math/special_functions/ellint_3.hpp b/include/boost/math/special_functions/ellint_3.hpp index 33e57fa84..597b680c7 100644 --- a/include/boost/math/special_functions/ellint_3.hpp +++ b/include/boost/math/special_functions/ellint_3.hpp @@ -1,4 +1,5 @@ // 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) @@ -47,6 +48,7 @@ T ellint_pi_imp(T v, T phi, T k, T vc) } T sphi = sin(fabs(phi)); + if(v > 1 / (sphi * sphi)) { // Complex result is a domain error: @@ -119,10 +121,11 @@ T ellint_pi_imp(T v, T phi, T k, T vc) // If v > 1 we can use the identity in A&S 17.7.7/8 // to shift to 0 <= v <= 1. Unfortunately this // identity appears only to function correctly when - // 0 <= phi <= PI/2, but it's when phi is outside that + // 0 <= phi <= pi/2, but it's when phi is outside that // range that we really need it: That's when - // Carlson's formula fails, and the periodicity + // Carlson's formula fails, and what's more the periodicity // reduction used below on phi doesn't work when v > 1. + // // So we're stuck... the code is archived here in case // some bright spart can figure out the fix. // @@ -131,8 +134,19 @@ T ellint_pi_imp(T v, T phi, T k, T vc) T Nm1 = (v - k2) / v; T p1 = sqrt((-vc) * (1 - k2 / v)); T delta = sqrt(1 - k2 * sphi * sphi); + // + // These next two terms have a large amount of cancellation + // so it's not clear if this relation is useable even if + // the issues with phi > pi/2 can be fixed: + // T result = -ellint_pi_imp(N, phi, k, Nm1); result += ellint_f_imp(phi, k); + // + // This log term gives the complex result when + // n > 1/sin^2(phi) + // However that case is dealt with as an error above, + // so we should always get a real result here: + // result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1); return result; } @@ -170,10 +184,15 @@ T ellint_pi_imp(T v, T phi, T k, T vc) } if((m > 0) && (v > 1)) + { + // + // The region with v > 1 and phi outside [0, pi/2] is + // currently unsupported: + // return tools::domain_error( BOOST_CURRENT_FUNCTION, "Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v); - + } T sinp = sin(rphi); T cosp = cos(rphi); x = cosp * cosp; @@ -248,7 +267,7 @@ T ellint_pi_imp(T v, T k, T vc) } // namespace detail template -inline T ellint_3(T v, T phi, T k) +inline T ellint_3(T k, T v, T phi) { typedef typename tools::evaluation::type>::type value_type; return tools::checked_narrowing_cast::type>( @@ -260,7 +279,7 @@ inline T ellint_3(T v, T phi, T k) } template -inline T ellint_3(T v, T k) +inline T ellint_3(T k, T v) { typedef typename tools::evaluation::type>::type value_type; return tools::checked_narrowing_cast::type>( diff --git a/include/boost/math/special_functions/ellint_rj.hpp b/include/boost/math/special_functions/ellint_rj.hpp index 1ec74d537..43726eb04 100644 --- a/include/boost/math/special_functions/ellint_rj.hpp +++ b/include/boost/math/special_functions/ellint_rj.hpp @@ -68,25 +68,23 @@ T ellint_rj_imp(T x, T y, T z, T p) // for p < 0, the integral is singular, return Cauchy principal value if (p < 0) { + // + // We must ensure that (z - y) * (y - x) is positive. + // Since the integral is symmetrical in x, y and z + // we can just permute the values: + // + if(x > y) + std::swap(x, y); + if(y > z) + std::swap(y, z); + if(x > y) + std::swap(x, y); + T q = -p; T pmy = (z - y) * (y - x) / (y + q); // p - y - if(pmy < 0) - { - // - // TODO, FIXME if you can.... (JM Dec. 2006) - // - // The logic breaks down here, we can go into - // an infinite recursion unless we bail out right away!! - // - // This may be fixable by permuting x, y, and z, but may not - // be worth the hassle, fix this if you care about this use case! - // - return tools::domain_error( - BOOST_CURRENT_FUNCTION, - "Unable to compute Cauchy principle value, p had the value %1% " - "and (z-y)(y-x) was negative. Identity formula could not be applied!", - p); - } + + BOOST_ASSERT(pmy >= 0); + T p = pmy + y; value = ellint_rj(x, y, z, p); value *= pmy; diff --git a/test/ellint_e2_data.ipp b/test/ellint_e2_data.ipp index 97ff64361..15a0bd265 100644 --- a/test/ellint_e2_data.ipp +++ b/test/ellint_e2_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_e_data.ipp b/test/ellint_e_data.ipp index 092f5e2b7..fb8815a01 100644 --- a/test/ellint_e_data.ipp +++ b/test/ellint_e_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_f_data.ipp b/test/ellint_f_data.ipp index d47fe2cfa..93a9ff6bb 100644 --- a/test/ellint_f_data.ipp +++ b/test/ellint_f_data.ipp @@ -1,3 +1,13 @@ +// 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), diff --git a/test/ellint_pi2_data.ipp b/test/ellint_pi2_data.ipp index dbd68756e..102eb352d 100644 --- a/test/ellint_pi2_data.ipp +++ b/test/ellint_pi2_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_pi3_data.ipp b/test/ellint_pi3_data.ipp index d2ef113cc..fda8b075f 100644 --- a/test/ellint_pi3_data.ipp +++ b/test/ellint_pi3_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_pi3_large_data.ipp b/test/ellint_pi3_large_data.ipp index c1d87126c..5967d21ce 100644 --- a/test/ellint_pi3_large_data.ipp +++ b/test/ellint_pi3_large_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_rc_data.ipp b/test/ellint_rc_data.ipp index f4ad003aa..da7c7f5f0 100644 --- a/test/ellint_rc_data.ipp +++ b/test/ellint_rc_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_rd_data.ipp b/test/ellint_rd_data.ipp index 85221ced2..de544bcd5 100644 --- a/test/ellint_rd_data.ipp +++ b/test/ellint_rd_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_rf_data.ipp b/test/ellint_rf_data.ipp index 1d42e951b..dc5ea4c1a 100644 --- a/test/ellint_rf_data.ipp +++ b/test/ellint_rf_data.ipp @@ -1,3 +1,12 @@ +// 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), diff --git a/test/ellint_rj_data.ipp b/test/ellint_rj_data.ipp index f49c14170..500c028db 100644 --- a/test/ellint_rj_data.ipp +++ b/test/ellint_rj_data.ipp @@ -1,109 +1,242 @@ +// 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, 395> ellint_rj_data = { + 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), 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SC_(-0.71769535185590157312e20), SC_(-0.12181377246911709254263035815242321871232107633234e-40), + SC_(0.222246520174005267545980928e28), SC_(0.24442432318827852328354708908776139753005196553204e-21), SC_(0.1358339280940150480054651703124000152939184472259e-28), SC_(0.1929096832275390625e3), SC_(0.93043265128569252665328167060574294046977922061101e-14), SC_(0.228460490382675067103346688e28), SC_(0.1059283606822132483824577235054573273384526110874e-22), SC_(0.7495925e5), SC_(0.18313684375e6), SC_(0.45341966440085175811927852865624259192364499711347e-18), SC_(0.2295552972647520995079356416e28), SC_(0.8116024169921875e3), SC_(0.406449488413065216e18), SC_(0.1210972363736206571586395807697363987873694080033e-25), SC_(0.11680274510678021465134575907548276246526274420346e-21), + SC_(0.2314228013625886776106156032e28), SC_(0.14528588632178353334811760078082443214952945709229e-14), SC_(0.54849175282200488156594176e26), SC_(0.5798830216008354909718036651611328125e-7), SC_(0.54921077785694851775062771637993351556303331064265e-22), + SC_(0.3699745104553947847137951744e28), SC_(0.34977231725807742732787449097997501262533496350712e-30), SC_(0.120393285632e12), SC_(0.954876e8), SC_(0.22448886572410705870109454870839959584090293621426e-22), SC_(0.3838393489343095451259240448e28), SC_(0.103355013339542251670339584e27), SC_(0.98872197265625e4), SC_(0.57959118635033995205500453873348476463434053584933e-17), SC_(0.12041009164150948897691954156849133755234146202352e-26), SC_(0.5654821947040481368071995392e28), SC_(0.47785181184e14), SC_(0.40587309056e11), SC_(-0.37375441752374172210693359375e-3), SC_(0.48193946518009700351168874164523132159351681991985e-24), + SC_(0.6775909979186304065686470656e28), SC_(0.8062150656e10), SC_(0.348559752335036776448e21), SC_(-0.8760775059456393216e19), SC_(-0.10282120977675613763363596767906302856864535909313e-33), SC_(0.7376334675354954747676196864e28), SC_(0.3832167254355459644063744e25), SC_(0.86986000872677090266227306614155168063007295131683e-15), SC_(-0.635832114613491743743541248e28), SC_(-0.24621535270065684155147722097385679585890775903462e-40), SC_(0.9551179238333844741448269824e28), SC_(0.7608356306944e13), SC_(0.5558017730712890625e1), SC_(0.41698188855306519252274162436284849815711030154158e-27), SC_(0.15612097421897443717100059952264109969304027375674e-18), + SC_(0.10531197197128523245005307904e29), SC_(0.12791422998544384e17), SC_(0.660148875e6), SC_(0.33461133902897741824e20), SC_(0.40438261456319118306216960428142150932574137900723e-32), SC_(0.12457122281020492087329554432e29), SC_(0.7671683806842035634176e22), SC_(0.232967583002618994214572012424468994140625e-7), SC_(-0.24723743603662803294991612279574155763839371502399e-16), SC_(0.22166650205242874655754051451333910848954470186131e-19), SC_(0.15703756321543097476213374976e29), SC_(0.3366045475006103515625e1), SC_(0.15573768317699432373046875e0), SC_(0.6116611973854022065889826509770647605297401467399e-30), SC_(0.11358697102558723782406276684855724930886990175612e-11), + SC_(0.16696330462519916007687979008e29), SC_(0.1930155264e10), SC_(0.63629325734928786425753617162284769825930652586976e-20), SC_(0.12920223388671875e4), SC_(0.2308199605723093565210069675443489489234691184839e-19), + SC_(0.17764058704888363507915948032e29), SC_(0.79766535274496e14), SC_(0.39754495142903122994659328e26), SC_(-0.14620739417686359499606065152e29), SC_(-0.57422745573841589040276380059216580638301950767727e-41), SC_(0.31748377778186208813596540928e29), SC_(0.1675580742552227722297344e25), SC_(0.42063803333440515664792681748951699777781240829747e-29), SC_(0.429656843192433391960064e24), SC_(0.23926759355712048744385470425532188141681573695193e-37), SC_(0.3649235853244794839343562752e29), SC_(0.12279541379302346893022276017590817959179716467588e-21), SC_(0.40871462586513926278727773609006357347084303910378e-20), SC_(0.87854579712e12), SC_(0.67492676307713588597128960931053684956956100221764e-24), + SC_(0.36533776047685956616782348288e29), SC_(0.82729172633504849398988767461560200899839401245117e-13), SC_(0.8131970819162112e17), SC_(-0.1192539684780058450996875762939453125e-5), SC_(0.13274939233001620432487252088510733166654023704738e-22), + SC_(0.63201895715548519849695117312e29), SC_(0.659935944174255143387136e24), SC_(0.235792458057403564453125e0), SC_(-0.3216055136681984e16), SC_(-0.18080738259064818364416597919348501783180949800115e-37), SC_(0.6372859541648241998931427328e29), SC_(0.242449971847236156463623046875e-2), SC_(0.27421585230570622547519032963236895739100873470306e-15), SC_(-0.303756275177001953125e2), SC_(-0.2116865223447144892600764830285901409868915835424e-14), + SC_(0.66201188727981103266043789312e29), SC_(0.733696256e9), SC_(0.350810660994547712e18), SC_(-0.2088726080984315558453090488910675048828125e-8), SC_(0.1518448361705534157919668106359692557429828526005e-25), + SC_(0.67397727780311165361354440704e29), SC_(0.2701057007003141150867456e25), SC_(0.32037437278282544783625240825131186284124851226807e-14), SC_(-0.14639644477145580259147550350462552160024642944336e-12), SC_(0.26795251743869315367874488984214665026594244506073e-20), SC_(0.74469943825056746033385570304e29), SC_(0.25381451890023692523876122624e29), SC_(0.353061568e9), SC_(0.3667164349963059066794812679290771484375e-7), SC_(0.70123509979767730264405232269736730186653622643256e-31), + SC_(0.83503273667541398709050802176e29), SC_(0.64436475218886656e17), SC_(0.1095445250048e15), SC_(-0.27828965472666412863213985278623730405131117028594e-26), SC_(0.18517871294794141505416209832668190646081405513643e-27), SC_(0.154135841378084202520367857664e30), SC_(0.11531852972032e14), SC_(0.28385876039913134187331286456979634325037409325887e-20), SC_(0.64800658718743622607072501400418684802673396916362e-20), SC_(0.31602768754325763081308012038930598073631539942634e-10), + SC_(0.205367992548434895604069433344e30), SC_(0.18794387499923534284127922718010683666989280027337e-18), SC_(0.644523545719734272e18), SC_(0.21339260514789921879362708770529188928577468686854e-21), SC_(0.77709407338464422363910627214199294770600429104781e-13), + SC_(0.288644753185258968910800093184e30), SC_(0.3712882e7), SC_(0.67304340158588786789888913154097233038086756096163e-21), SC_(0.228604896e9), 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/test_carlson.cpp b/test/test_carlson.cpp index e0c12e818..a41b790b5 100644 --- a/test/test_carlson.cpp +++ b/test/test_carlson.cpp @@ -1,4 +1,4 @@ -// 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) @@ -90,14 +90,14 @@ void expected_results() ".*", // platform largest_type, // test type(s) ".*RJ.*", // test data group - ".*", 150, 50); // test function + ".*", 180, 50); // test function add_expected_result( ".*", // compiler ".*", // stdlib ".*", // platform "real_concept", // test type(s) ".*RJ.*", // test data group - ".*", 150, 50); // test function + ".*", 180, 50); // test function add_expected_result( ".*", // compiler ".*", // stdlib diff --git a/test/test_ellint_1.cpp b/test/test_ellint_1.cpp index 6d8f2b121..1f62d6ccb 100644 --- a/test/test_ellint_1.cpp +++ b/test/test_ellint_1.cpp @@ -1,4 +1,5 @@ // 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 @@ -97,8 +98,8 @@ void do_test_ellint_f(T& data, const char* type_name, const char* test) result = boost::math::tools::test( data, boost::lambda::bind(fp2, - boost::lambda::ret(boost::lambda::_1[0]), - boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[0])), boost::lambda::ret(boost::lambda::_1[2])); handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ellint_1", test); diff --git a/test/test_ellint_2.cpp b/test/test_ellint_2.cpp index b87a7e1e7..741cfaac3 100644 --- a/test/test_ellint_2.cpp +++ b/test/test_ellint_2.cpp @@ -1,4 +1,5 @@ // 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) @@ -96,8 +97,8 @@ void do_test_ellint_e2(const T& data, const char* type_name, const char* test) result = boost::math::tools::test( data, boost::lambda::bind(fp2, - boost::lambda::ret(boost::lambda::_1[0]), - boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[0])), boost::lambda::ret(boost::lambda::_1[2])); handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ellint_2", test); diff --git a/test/test_ellint_3.cpp b/test/test_ellint_3.cpp index 39c616295..98eb2ef8b 100644 --- a/test/test_ellint_3.cpp +++ b/test/test_ellint_3.cpp @@ -1,4 +1,5 @@ // 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) @@ -112,9 +113,9 @@ void do_test_ellint_pi3(T& data, const char* type_name, const char* test) result = boost::math::tools::test( data, boost::lambda::bind(fp2, + boost::lambda::ret(boost::lambda::_1[2]), boost::lambda::ret(boost::lambda::_1[0]), - boost::lambda::ret(boost::lambda::_1[1]), - boost::lambda::ret(boost::lambda::_1[2])), + boost::lambda::ret(boost::lambda::_1[1])), boost::lambda::ret(boost::lambda::_1[3])); handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ellint_3", test); @@ -137,8 +138,8 @@ void do_test_ellint_pi2(T& data, const char* type_name, const char* test) result = boost::math::tools::test( data, boost::lambda::bind(fp2, - boost::lambda::ret(boost::lambda::_1[0]), - boost::lambda::ret(boost::lambda::_1[1])), + boost::lambda::ret(boost::lambda::_1[1]), + boost::lambda::ret(boost::lambda::_1[0])), boost::lambda::ret(boost::lambda::_1[2])); handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::ellint_3", test); @@ -211,11 +212,11 @@ void test_spots(T, const char* type_name) 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), T(0), T(1.0001)), std::domain_error); - BOOST_CHECK_THROW(boost::math::ellint_3(T(20), T(1.5), T(0.5)), std::domain_error); - BOOST_CHECK_THROW(boost::math::ellint_3(T(-1), T(1.0001)), std::domain_error); - BOOST_CHECK_THROW(boost::math::ellint_3(T(1), T(0.5)), std::domain_error); - BOOST_CHECK_THROW(boost::math::ellint_3(T(2), T(0.5)), std::domain_error); + 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* []) diff --git a/tools/ellint_f_data.cpp b/tools/ellint_f_data.cpp index bae3e075d..aca70597d 100644 --- a/tools/ellint_f_data.cpp +++ b/tools/ellint_f_data.cpp @@ -23,7 +23,7 @@ float truncate_to_float(float const * pf) template T ellint_f_data(T phi, T k) { - return ellint_1(phi, k); + return ellint_1(k, phi); } int test_main(int argc, char*argv []) diff --git a/tools/ellint_pi2_data.cpp b/tools/ellint_pi2_data.cpp index 9cb355c0d..8287e748c 100644 --- a/tools/ellint_pi2_data.cpp +++ b/tools/ellint_pi2_data.cpp @@ -38,7 +38,7 @@ int test_main(int argc, char*argv []) return 1; NTL::RR (*fp)(NTL::RR, NTL::RR) = &ellint_3; - data.insert(fp, arg1, arg2); + data.insert(fp, arg2, arg1); std::cout << "Any more data [y/n]?"; std::getline(std::cin, line); diff --git a/tools/ellint_pi3_data.cpp b/tools/ellint_pi3_data.cpp index f8168c087..39a79d30b 100644 --- a/tools/ellint_pi3_data.cpp +++ b/tools/ellint_pi3_data.cpp @@ -25,7 +25,7 @@ std::tr1::tuple generate_data(NTL::RR n, NTL::RR phi) std::tr1::uniform_real ui(0, 1); float k = ui(r); NTL::RR kr(truncate_to_float(&k)); - NTL::RR result = boost::math::ellint_3(n, phi, kr); + NTL::RR result = boost::math::ellint_3(kr, n, phi); return std::tr1::make_tuple(kr, result); }