diff --git a/doc/Jamfile.v2 b/doc/Jamfile.v2 index 92878ef4a..38b395e21 100644 --- a/doc/Jamfile.v2 +++ b/doc/Jamfile.v2 @@ -32,7 +32,7 @@ boostbook standalone # How far down sections get TOC's toc.section.depth=10 # Max depth in each TOC: - toc.max.depth=1 + toc.max.depth=4 # How far down we go with TOC's generate.section.toc.level=10 # Index on type: diff --git a/doc/background/error.qbk b/doc/background/error.qbk index 7fce12a01..a3a79b14c 100644 --- a/doc/background/error.qbk +++ b/doc/background/error.qbk @@ -41,7 +41,7 @@ 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] +[h4:zero_error 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 diff --git a/doc/background/implementation.qbk b/doc/background/implementation.qbk index dd9da941c..4fe0d6ec8 100644 --- a/doc/background/implementation.qbk +++ b/doc/background/implementation.qbk @@ -125,14 +125,14 @@ This implementation is believed to follow these proposals and to assist compatib ['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]. +[link math_toolkit.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] +See [link math_toolkit.error_handling error handling] for a detailed explanation of the mechanism, and -[link math_toolkit.dist.stat_tut.weg.error_eg error_handling example] +[link math_toolkit.stat_tut.weg.error_eg error_handling example] and [@../../../example/error_handling_example.cpp error_handling_example.cpp] @@ -146,7 +146,7 @@ However, for simplicity, this is not done for most examples.] Functions that are not mathematically defined, like the Cauchy mean, fail to compile by default. -A [link math_toolkit.policy.pol_ref.assert_undefined policy] +A [link math_toolkit.pol_ref.assert_undefined policy] allows control of this. If the policy is to permit undefined functions, then calling them @@ -267,7 +267,7 @@ the most common usage, lower = -1, mode = 0 and upper = 1 would be more suitable 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]. +[link math_toolkit.internals2.minimax our Remez code]. Rational rather than Polynomial approximations are used to ensure accuracy: polynomial approximations are often wonderful up to @@ -541,8 +541,8 @@ If you fail to do this you will get XEP warnings in the log file like [pre \[warning\]could not find any font family matching "Times New Roman"; replaced by Helvetica] -(html is the default so it is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html -using command line >bjam -a > math_docs.log). +(html is the default so it is generated at libs\math\doc\html\index.html +using command line >bjam -a > math_toolkit.docs.log). is provided in the xep.xml downloaded, but the Windows TrueType fonts are commented out. diff --git a/doc/background/lanczos.qbk b/doc/background/lanczos.qbk index ecc4d8983..0fa3bee29 100644 --- a/doc/background/lanczos.qbk +++ b/doc/background/lanczos.qbk @@ -50,8 +50,8 @@ 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 +[link math_toolkit.sf_gamma.igamma gamma_p gamma] and +[link math_toolkit.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. diff --git a/doc/background/references.qbk b/doc/background/references.qbk index 7982304bf..86f1a5eaf 100644 --- a/doc/background/references.qbk +++ b/doc/background/references.qbk @@ -103,7 +103,7 @@ discusses using Boost.Math in finance. Robert Demming & Daniel J. Duffy, Introduction to the C++ Boost Libraries - Volume I - Foundations and Volume II ISBN 978-94-91028-01-4, Advanced Libraries and Applications, ISBN 978-94-91028-02-1 (to be published in 2011). -discusses application of Boost.Math, especially in finance.] +discusses application of Boost.Math, especially in finance. [endsect] [/section:references References] [/ diff --git a/doc/complex/complex-tr1.qbk b/doc/complex/complex-tr1.qbk index a2bf78c14..df182be30 100644 --- a/doc/complex/complex-tr1.qbk +++ b/doc/complex/complex-tr1.qbk @@ -4,7 +4,7 @@ [def __ex '''ex'''] [def __te '''2ε'''] -[section:inverse_complex Complex Number Functions] +[mathpart inverse_complex..Complex Number 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 @@ -137,7 +137,7 @@ __formula [$../../images/atanh.png] [endsect] -[endsect] +[endmathpart] diff --git a/doc/concepts/concepts.qbk b/doc/concepts/concepts.qbk index da919dd2a..420892ba8 100644 --- a/doc/concepts/concepts.qbk +++ b/doc/concepts/concepts.qbk @@ -269,7 +269,7 @@ 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.high_precision.use_ntl a thin wrapper class] +For example, with [link math_toolkit.high_precision.use_ntl a thin wrapper class] one of the types provided with [@http://shoup.net/ntl/ NTL (RR)] can be used. But now that __multiprecision library is available, this has become the reference real number type. @@ -459,8 +459,8 @@ 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 +[link math_toolkit.dist_ref reference documentation], and +[link math_toolkit.stat_tut tutorial] of the statistical distributions. In the following table, /d/ is an object of type `DistributionType`, @@ -558,7 +558,7 @@ that instantiates every template in this library with type }}} // namespaces `real_concept` is an archetype for -[link math_toolkit.using_udt.concepts user defined real types], +[link math_toolkit.concepts user defined real types], it declares its standard library functions in its own namespace: these will only be found if they are called unqualified allowing argument dependent lookup to locate them. In addition @@ -566,11 +566,11 @@ 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 numeric_limits is not a conceptual requirement -for [link math_toolkit.using_udt.concepts RealType]s. +for [link math_toolkit.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 -[linkmath_toolkit.using_udt.high_precision.use_ntl "Using With NTL - a High-Precision Floating-Point Library"]. +[link math_toolkit.high_precision.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 @@ -595,7 +595,7 @@ Distribution Concept models statistical distributions. }}} // namespaces The class template `distribution_archetype` is a model of the -[link math_toolkit.using_udt.dist_concept Distribution concept]. +[link math_toolkit.dist_concept Distribution concept]. The class template `DistributionConcept` is a [@../../../../../libs/concept_check/index.html concept checking class] @@ -607,11 +607,11 @@ 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]. +[link math_toolkit.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]. +required by the [link math_toolkit.dist_concept Distribution concept]. These are checked by calls like v = pdf(dist, x); // (Result v is ignored). diff --git a/doc/constants/constants.qbk b/doc/constants/constants.qbk index 9080034fc..0d55cbd35 100644 --- a/doc/constants/constants.qbk +++ b/doc/constants/constants.qbk @@ -1,4 +1,6 @@ -[section:constants Mathematical Constants] + + +[mathpart constants..Mathematical Constants] [section:intro Introduction] @@ -8,7 +10,7 @@ Boost.Math provides a collection of mathematical constants. * Readable. For the very many jobs just using built-in like `double`, you can just write expressions like ``double area = pi * r * r;`` -(If that's all you want, jump direct to [link math_toolkit.constants.tutorial.non_templ use in non-template code]!) +(If that's all you want, jump direct to [link math_toolkit.tutorial.non_templ use in non-template code]!) * Effortless - avoiding a search of reference sources. * Usable with both builtin floating point types, and user-defined, possibly extended precision, types such as NTL, MPFR/GMP, mp_float: in the latter case the constants are computed to the necessary precision and then cached. @@ -19,7 +21,7 @@ chosen floating-point type * Less risk of inaccurate result from functions pow, trig and log at [@http://en.wikipedia.org/wiki/Corner_case corner cases]. * Less risk of [@http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html cancellation error]. * Portable - as possible between different systems using different floating-point precisions: -see [link math_toolkit.constants.tutorial.templ use in template code]. +see [link math_toolkit.tutorial.templ use in template code]. * Tested - by comparison with other published sources, or separately computed at long double precision. * Faster - can avoid (re-)calculation at runtime. * If the value returned is a builtin type then it's returned by value as a `constexpr` (C++11 feature, if available). @@ -64,7 +66,7 @@ we could have written instead: Likewise, constants that are suitable for use at `long double` precision are available in the namespace `boost::math::long_double_constants`. -You can see the full list of available constants at [link math_toolkit.constants.constants]. +You can see the full list of available constants at [link math_toolkit.constants]. Some examples of using constants are at [@../../../example/constants_eg1.cpp constants_eg1]. @@ -579,7 +581,7 @@ to diverge wildly because internal comparisons just fail. [h4 What is the Internal Format of the constants, and why?] -See [link math_toolkit.constants.tutorial tutorial] above for normal use, +See [link math_toolkit.tutorial tutorial] above for normal use, but this FAQ explains the internal details used for the constants. Constants are stored as 100 decimal digit values. @@ -605,7 +607,7 @@ So, for example, a constant like pi is internally defined as In this case the significand is 109 decimal digits, ensuring 100 decimal digits are exact, and exponent is zero. -See [link math_toolkit.constants.new_const defining new constants] to calculate new constants. +See [link math_toolkit.new_const defining new constants] to calculate new constants. A macro definition like this can be pasted into user code where convenient, or into `boost/math/constants.hpp` if it is to be added to the Boost.Math library. @@ -729,7 +731,7 @@ Some physical constants may be available in Boost.Units. [endsect] [/section:FAQ FAQ] -[endsect] [/section:constants Mathematical Constants] +[endmathpart] [/section:constants Mathematical Constants] [/ Copyright 2012 John Maddock and Paul A. Bristow. diff --git a/doc/distributions/background.qbk b/doc/distributions/background.qbk index 56270612b..09cb043bb 100644 --- a/doc/distributions/background.qbk +++ b/doc/distributions/background.qbk @@ -62,12 +62,12 @@ 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 +[link math_toolkit.pol_overview Policies]. It is strongly recommended that you read the tutorial -[link math_toolkit.policy.pol_tutorial.understand_dis_quant +[link math_toolkit.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] +[link math_toolkit.pol_ref.discrete_quant_ref reference docs] describe how to change the rounding policy for these distributions. ] diff --git a/doc/distributions/bernoulli.qbk b/doc/distributions/bernoulli.qbk index fc08b5cb4..987a2012c 100644 --- a/doc/distributions/bernoulli.qbk +++ b/doc/distributions/bernoulli.qbk @@ -61,7 +61,7 @@ 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] +All the [link math_toolkit.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, diff --git a/doc/distributions/beta.qbk b/doc/distributions/beta.qbk index 3a88f925b..55d5f5c74 100644 --- a/doc/distributions/beta.qbk +++ b/doc/distributions/beta.qbk @@ -180,7 +180,7 @@ Returns the value of [beta][space] that gives: [h4 Non-member Accessor Functions] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 @@ -217,8 +217,8 @@ 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; +[link math_toolkit.sf_beta.beta_function beta functions] __beta and +[link math_toolkit.sf_beta.ibeta_function incomplete beta functions] __ibeta and __ibetac; please refer to these functions for information on accuracy. [h4 Implementation] diff --git a/doc/distributions/binomial.qbk b/doc/distributions/binomial.qbk index 9b31a3750..53818c66e 100644 --- a/doc/distributions/binomial.qbk +++ b/doc/distributions/binomial.qbk @@ -133,7 +133,7 @@ want to be 95% sure that the true value is [*greater than] some value, 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.] +[link math_toolkit.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/ @@ -213,7 +213,7 @@ want to be 95% sure that the true value is [*less than] some value, 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.] +[link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.] [note In order to obtain a two sided bound on the success fraction, you @@ -227,7 +227,7 @@ 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.] +[link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.] ] @@ -280,11 +280,11 @@ 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.] +[link math_toolkit.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] +All the [link math_toolkit.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 @@ -328,7 +328,7 @@ the context of this distribution: [h4 Examples] -Various [link math_toolkit.dist.stat_tut.weg.binom_eg worked examples] +Various [link math_toolkit.stat_tut.weg.binom_eg worked examples] are available illustrating the use of the binomial distribution. [h4 Accuracy] @@ -374,7 +374,7 @@ 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]).]] + ([link math_toolkit.internals1.roots2 TOMS Algorithm 748]).]] [[quantile from the complement][Found numerically as above.]] [[mean][ `p * n` ]] [[variance][ `p * n * (1-p)` ]] diff --git a/doc/distributions/cauchy.qbk b/doc/distributions/cauchy.qbk index cad1483c2..e91ebe2d6 100644 --- a/doc/distributions/cauchy.qbk +++ b/doc/distributions/cauchy.qbk @@ -72,12 +72,12 @@ 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] +All the [link math_toolkit.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] +[/link math_toolkit.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. diff --git a/doc/distributions/chi_squared.qbk b/doc/distributions/chi_squared.qbk index ed4f208a9..77237e04c 100644 --- a/doc/distributions/chi_squared.qbk +++ b/doc/distributions/chi_squared.qbk @@ -102,7 +102,7 @@ See also section on Sample sizes required in [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. (We have followed the usual restriction of the mode to degrees of freedom >= 2, @@ -114,13 +114,13 @@ The domain of the random variable is \[0, +[infin]\]. [h4 Examples] -Various [link math_toolkit.dist.stat_tut.weg.cs_eg worked examples] +Various [link math_toolkit.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]: +[link math_toolkit.sf_gamma.igamma incomplete gamma functions]: please refer to the accuracy data for those functions. [h4 Implementation] diff --git a/doc/distributions/chi_squared_examples.qbk b/doc/distributions/chi_squared_examples.qbk index 40568d854..8b6e483a1 100644 --- a/doc/distributions/chi_squared_examples.qbk +++ b/doc/distributions/chi_squared_examples.qbk @@ -364,7 +364,7 @@ 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 +The class template [link math_toolkit.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 diff --git a/doc/distributions/dist_algorithms.qbk b/doc/distributions/dist_algorithms.qbk index 94e2e4abc..a29daeeb8 100644 --- a/doc/distributions/dist_algorithms.qbk +++ b/doc/distributions/dist_algorithms.qbk @@ -65,7 +65,7 @@ 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] +[link math_toolkit.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] diff --git a/doc/distributions/dist_reference.qbk b/doc/distributions/dist_reference.qbk index c8d9b015f..85055c5fb 100644 --- a/doc/distributions/dist_reference.qbk +++ b/doc/distributions/dist_reference.qbk @@ -110,7 +110,7 @@ package could be added reasonably easily, for example: InputIterator b, typename std::iterator_traits::value_type expected_mean); -Returns the probability that the data in the sequence [a,b) has the mean +Returns the probability that the data in the sequence \[a,b) has the mean /expected_mean/. [h4 Integration With Statistical Accumulators] diff --git a/doc/distributions/dist_tutorial.qbk b/doc/distributions/dist_tutorial.qbk index 9f9b11ff8..54f8da984 100644 --- a/doc/distributions/dist_tutorial.qbk +++ b/doc/distributions/dist_tutorial.qbk @@ -1,10 +1,10 @@ [/ def names all end in distrib to avoid clashes with names of functions] -[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 __binomial_distrib [link math_toolkit.dist_ref.dists.binomial_dist Binomial Distribution]] +[def __chi_squared_distrib [link math_toolkit.dist_ref.dists.chi_squared_dist Chi Squared Distribution]] +[def __normal_distrib [link math_toolkit.dist_ref.dists.normal_dist Normal Distribution]] +[def __F_distrib [link math_toolkit.dist_ref.dists.f_dist Fisher F Distribution]] +[def __students_t_distrib [link math_toolkit.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.]] @@ -41,14 +41,14 @@ or specific `using` declarations like `using boost::math::normal;` (*recommende Each kind of distribution in this library is a class type - an object. -[link math_toolkit.policy Policies] provide fine-grained control +[link 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 +[link math_toolkit.stat_tut.weg.nag_library the comparison to other statistics libraries.] ] [/tip] @@ -216,7 +216,7 @@ 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 +(See [link math_toolkit.implementation0.handling_of_floating_point_infin Handling of Floating-Point Infinity] for rationale). @@ -250,12 +250,12 @@ 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 +[link math_toolkit.pol_overview Policies]. It is strongly recommended that you read the tutorial -[link math_toolkit.policy.pol_tutorial.understand_dis_quant +[link math_toolkit.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] +[link math_toolkit.pol_ref.discrete_quant_ref reference docs] describe how to change the rounding policy for these distributions. @@ -269,7 +269,6 @@ degrees of freedom do have a genuine meaning. [endsect] [/ section:generic Generic operations common to all distributions are non-member functions] -[#complements] [section:complements Complements are supported too - and when to use them] Often you don't want the value of the CDF, but its complement, which is diff --git a/doc/distributions/error_handling_example.qbk b/doc/distributions/error_handling_example.qbk index 21c55f441..0a68607f1 100644 --- a/doc/distributions/error_handling_example.qbk +++ b/doc/distributions/error_handling_example.qbk @@ -1,6 +1,6 @@ [section:error_eg Error Handling Example] -See [link math_toolkit.main_overview.error_handling error handling documentation] +See [link math_toolkit.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. diff --git a/doc/distributions/exponential.qbk b/doc/distributions/exponential.qbk index 17d923a90..af83a2970 100644 --- a/doc/distributions/exponential.qbk +++ b/doc/distributions/exponential.qbk @@ -52,7 +52,7 @@ 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] +All the [link math_toolkit.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 \[0, +[infin]\]. diff --git a/doc/distributions/extreme_value.qbk b/doc/distributions/extreme_value.qbk index 5e81eb51e..affc509f4 100644 --- a/doc/distributions/extreme_value.qbk +++ b/doc/distributions/extreme_value.qbk @@ -76,7 +76,7 @@ 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] +All the [link math_toolkit.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 \[-[infin], +[infin]\]. diff --git a/doc/distributions/fisher.qbk b/doc/distributions/fisher.qbk index 401696bd7..f09388d2b 100644 --- a/doc/distributions/fisher.qbk +++ b/doc/distributions/fisher.qbk @@ -64,21 +64,21 @@ 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] +All the [link math_toolkit.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 +Various [link math_toolkit.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 its [link math_toolkit.special.sf_beta.ibeta_inv_function inverses], +[link math_toolkit.sf_beta.ibeta_function incomplete beta function] +and its [link math_toolkit.sf_beta.ibeta_inv_function inverses], refer to those functions for accuracy data. [h4 Implementation] diff --git a/doc/distributions/gamma.qbk b/doc/distributions/gamma.qbk index d74776cf5..11b2be3e9 100644 --- a/doc/distributions/gamma.qbk +++ b/doc/distributions/gamma.qbk @@ -94,7 +94,7 @@ 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 +All the [link math_toolkit.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 \[0,+[infin]\]. diff --git a/doc/distributions/geometric.qbk b/doc/distributions/geometric.qbk index 9e03a026c..5129b4516 100644 --- a/doc/distributions/geometric.qbk +++ b/doc/distributions/geometric.qbk @@ -161,7 +161,7 @@ want to be 95% sure that the true value is [*greater than] some value, p``[sub min]`` = geometric_distribution:: find_lower_bound_on_p(failures, 0.05); -[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.] +[link math_toolkit.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" @@ -195,7 +195,7 @@ want to be 95% sure that the true value is [*less than] some value, p``[sub max]`` = geometric_distribution::find_upper_bound_on_p( k, 0.05); -[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.] +[link math_toolkit.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" @@ -230,7 +230,7 @@ For example: Returns the smallest number of trials we must conduct to be 95% (1-0.05) 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.] +[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.] This function uses numeric inversion of the geometric distribution to obtain the result: another interpretation of the result is that it finds @@ -267,7 +267,7 @@ of observing more than k failures. [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 diff --git a/doc/distributions/geometric_example.qbk b/doc/distributions/geometric_example.qbk index 85b0f5b8d..940564802 100644 --- a/doc/distributions/geometric_example.qbk +++ b/doc/distributions/geometric_example.qbk @@ -7,7 +7,7 @@ See full source C++ of this example at [@../../example/geometric_examples.cpp geometric_examples.cpp] -[link math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.] +[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.] [endsect] [/section:geometric_eg Geometric Distribution Examples] diff --git a/doc/distributions/hypergeometric.qbk b/doc/distributions/hypergeometric.qbk index 24664c67f..881a1fe25 100644 --- a/doc/distributions/hypergeometric.qbk +++ b/doc/distributions/hypergeometric.qbk @@ -75,7 +75,7 @@ Returns the number of objects /n/ which are sampled from the population /N/. [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 the unsigned integers in the range @@ -93,12 +93,12 @@ the requested coverage will be present in the tails. This behaviour can be changed so that the quantile functions are rounded differently using -[link math_toolkit.policy.pol_overview Policies]. It is strongly +[link math_toolkit.pol_overview Policies]. It is strongly recommended that you read the tutorial -[link math_toolkit.policy.pol_tutorial.understand_dis_quant +[link math_toolkit.pol_tutorial.understand_dis_quant Understanding Quantiles of Discrete Distributions] before using the quantile function on the Hypergeometric distribution. The -[link math_toolkit.policy.pol_ref.discrete_quant_ref reference docs] +[link math_toolkit.pol_ref.discrete_quant_ref reference docs] describe how to change the rounding policy for these distributions. diff --git a/doc/distributions/inverse_chi_squared.qbk b/doc/distributions/inverse_chi_squared.qbk index 943db4295..4c1f9f983 100644 --- a/doc/distributions/inverse_chi_squared.qbk +++ b/doc/distributions/inverse_chi_squared.qbk @@ -118,7 +118,7 @@ Returns the scale [xi] 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 +All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. The domain of the random variate is \[0,+[infin]\]. diff --git a/doc/distributions/inverse_gamma.qbk b/doc/distributions/inverse_gamma.qbk index 177f2fd87..0f8afea4d 100644 --- a/doc/distributions/inverse_gamma.qbk +++ b/doc/distributions/inverse_gamma.qbk @@ -81,7 +81,7 @@ Returns the [beta] scale parameter of this inverse gamma distribution. [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] that are generic to all +All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. The domain of the random variate is \[0,+[infin]\]. diff --git a/doc/distributions/inverse_gaussian.qbk b/doc/distributions/inverse_gaussian.qbk index 4d41eadd3..612801fe6 100644 --- a/doc/distributions/inverse_gaussian.qbk +++ b/doc/distributions/inverse_gaussian.qbk @@ -108,7 +108,7 @@ Returns the scale [lambda] 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 +All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. The domain of the random variate is \[0,+[infin]). diff --git a/doc/distributions/laplace.qbk b/doc/distributions/laplace.qbk index 6f54d6384..7a55edae7 100644 --- a/doc/distributions/laplace.qbk +++ b/doc/distributions/laplace.qbk @@ -72,7 +72,7 @@ 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 +All the [link math_toolkit.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 \[-[infin],+[infin]\]. diff --git a/doc/distributions/logistic.qbk b/doc/distributions/logistic.qbk index e73343b35..e718483c1 100644 --- a/doc/distributions/logistic.qbk +++ b/doc/distributions/logistic.qbk @@ -55,7 +55,7 @@ Returns the scale of this distribution. [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 \[-\[max_value\], +\[min_value\]\]. diff --git a/doc/distributions/lognormal.qbk b/doc/distributions/lognormal.qbk index 0302889ee..bd275b92a 100644 --- a/doc/distributions/lognormal.qbk +++ b/doc/distributions/lognormal.qbk @@ -75,7 +75,7 @@ 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 +All the [link math_toolkit.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 \[0,+[infin]\]. @@ -84,7 +84,7 @@ The domain of the random variable is \[0,+[infin]\]. 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], +[link math_toolkit.sf_erf.error_function error function], and as such should have very low error rates. [h4 Implementation] diff --git a/doc/distributions/nc_beta.qbk b/doc/distributions/nc_beta.qbk index ef2f645d8..30b63ca7d 100644 --- a/doc/distributions/nc_beta.qbk +++ b/doc/distributions/nc_beta.qbk @@ -79,7 +79,7 @@ Returns the parameter /lambda/ from which this object was constructed. [h4 Non-member Accessors] -Most of the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +Most of the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] are supported: __cdf, __pdf, __quantile, __mean, __variance, __sd, __median, __mode, __hazard, __chf, __range and __support. @@ -195,7 +195,7 @@ and the relation: [equation nc_beta_ref1] Quantiles are computed using a specially modified version of -[link math_toolkit.toolkit.internals1.roots2 bracket_and_solve_root], +[link math_toolkit.internals1.roots2 bracket_and_solve_root], starting the search for the root at the mean of the distribution. (A Cornish-Fisher type expansion was also tried, but while this gets quite close to the root in many cases, when it is wrong it tends to diff --git a/doc/distributions/nc_chi_squared.qbk b/doc/distributions/nc_chi_squared.qbk index c83f84d1a..f2b6017c1 100644 --- a/doc/distributions/nc_chi_squared.qbk +++ b/doc/distributions/nc_chi_squared.qbk @@ -115,7 +115,7 @@ this function returns the non centrality parameter /lambda/ such that: [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 \[0, +[infin]\]. @@ -123,7 +123,7 @@ The domain of the random variable is \[0, +[infin]\]. [h4 Examples] There is a -[link math_toolkit.dist.stat_tut.weg.nccs_eg worked example] +[link math_toolkit.stat_tut.weg.nccs_eg worked example] for the noncentral chi-squared distribution. [h4 Accuracy] diff --git a/doc/distributions/nc_f.qbk b/doc/distributions/nc_f.qbk index 6da627c73..535a5da45 100644 --- a/doc/distributions/nc_f.qbk +++ b/doc/distributions/nc_f.qbk @@ -76,7 +76,7 @@ Returns the non-centrality parameter /lambda/ from which this object was constru [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 \[0, +[infin]\]. diff --git a/doc/distributions/nc_t.qbk b/doc/distributions/nc_t.qbk index 68707c014..0f9a23d3f 100644 --- a/doc/distributions/nc_t.qbk +++ b/doc/distributions/nc_t.qbk @@ -69,7 +69,7 @@ Returns the non-centrality parameter /delta/ from which this object was construc [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 \[-[infin], +[infin]\]. @@ -194,7 +194,7 @@ t-Distribution". C. van Eeden. International Statistical Review, 29, 4-31. N. Balkrishnan. 1995. John Wiley and Sons New York. The quantile is calculated via the usual -[link math_toolkit.toolkit.internals1.roots2 +[link math_toolkit.internals1.roots2 derivative-free root-finding techniques], with the initial guess taken as the quantile of a normal approximation to the noncentral T. diff --git a/doc/distributions/negative_binomial.qbk b/doc/distributions/negative_binomial.qbk index b6b945c7a..0c5a4be1a 100644 --- a/doc/distributions/negative_binomial.qbk +++ b/doc/distributions/negative_binomial.qbk @@ -156,7 +156,7 @@ want to be 95% sure that the true value is [*greater than] some value, 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.] +[link math_toolkit.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" @@ -192,7 +192,7 @@ want to be 95% sure that the true value is [*less than] some value, 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.] +[link math_toolkit.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" @@ -227,7 +227,7 @@ For example: 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.] +[link math_toolkit.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 @@ -264,7 +264,7 @@ of observing more than k failures. [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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 diff --git a/doc/distributions/non_members.qbk b/doc/distributions/non_members.qbk index 85f36ead2..653d24979 100644 --- a/doc/distributions/non_members.qbk +++ b/doc/distributions/non_members.qbk @@ -4,61 +4,61 @@ Properties that are common to all distributions are accessed via non-member getter functions: non-membership 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 +to the [link math_toolkit.dist_ref.nmp.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] +Or use the [link math_toolkit.dist_ref.nmp.function_index function index] to go straight to the function you want if you already know its name. -[h4 [#function_index]Function Index] +[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]. +* __cdf. +* __ccdf. +* __chf. +* __hazard. * __kurtosis. * __kurtosis_excess * __mean. -* [link math.dist.median 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]. +* __pdf. +* __range. +* __quantile. +* __quantile_c. * __skewness. -* [link math.dist.sd standard_deviation]. -* [link math.dist.support support]. +* __sd. +* __support. * __variance. -[h4 [#concept_index]Conceptual Index] +[h4:concept_index Conceptual Index] * __ccdf. * __cdf. * __chf. -* [link cdf_inv Inverse Cumulative Distribution Function]. -* [link survival_inv Inverse Survival Function]. +* [link math_toolkit.dist_ref.nmp.cdf_inv Inverse Cumulative Distribution Function]. +* [link math_toolkit.dist_ref.nmp.survival_inv Inverse Survival Function]. * __hazard -* [link lower_critical Lower Critical Value]. +* [link math_toolkit.dist_ref.nmp.lower_critical Lower Critical Value]. * __kurtosis. * __kurtosis_excess * __mean. -* [link math.dist.median median]. +* __median. * __mode. -* [link cdfPQ P]. -* [link percent Percent Point Function]. +* [link math_toolkit.dist_ref.nmp.cdfPQ P]. +* [link math_toolkit.dist_ref.nmp.percent Percent Point Function]. * __pdf. -* [link pmf Probability Mass Function]. -* [link math.dist.range range]. -* [link cdfPQ Q]. +* [link math_toolkit.dist_ref.nmp.pmf Probability Mass Function]. +* __range. +* [link math_toolkit.dist_ref.nmp.cdfPQ Q]. * __quantile. -* [link math.dist.quantile_c Quantile from the complement of the probability]. +* [link math_toolkit.dist_ref.nmp.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]. +* [link math_toolkit.dist_ref.nmp.survival Survival Function]. +* [link math_toolkit.dist_ref.nmp.support support]. +* [link math_toolkit.dist_ref.nmp.upper_critical Upper Critical Value]. * __variance. -[h4 [#math.dist.cdf]Cumulative Distribution Function] +[h4:cdf Cumulative Distribution Function] template RealType cdf(const ``['Distribution-Type]``& dist, const RealType& x); @@ -75,7 +75,7 @@ normal distribution: [$../graphs/cdf.png] -[h4 [#math.dist.ccdf]Complement of the Cumulative Distribution Function] +[h4:ccdf Complement of the Cumulative Distribution Function] template RealType cdf(const ``['Unspecified-Complement-Type]``& comp); @@ -105,7 +105,7 @@ normal distribution: See __why_complements for why the complement is useful and when it should be used. -[h4 [#math.dist.hazard]Hazard Function] +[h4:hazard Hazard Function] template RealType hazard(const ``['Distribution-Type]``& dist, const RealType& x); @@ -121,7 +121,7 @@ the defined range for the distribution. Some authors refer to this as the conditional failure density function rather than the hazard function.] -[h4 [#math.dist.chf]Cumulative Hazard Function] +[h4:chf Cumulative Hazard Function] template RealType chf(const ``['Distribution-Type]``& dist, const RealType& x); @@ -136,7 +136,7 @@ the defined range for the distribution. [caution Some authors refer to this as simply the "Hazard Function".] -[h4 [#math.dist.mean]mean] +[h4:mean mean] template RealType mean(const ``['Distribution-Type]``& dist); @@ -146,14 +146,14 @@ Returns the mean of the distribution /dist/. This function may return a __domain_error if the distribution does not have a defined mean (for example the Cauchy distribution). -[h4 [#math.dist.median]median] +[h4:median median] template RealType median(const ``['Distribution-Type]``& dist); Returns the median of the distribution /dist/. -[h4 [#math.dist.mode]mode] +[h4:mode mode] template RealType mode(const ``['Distribution-Type]``& dist); @@ -163,7 +163,7 @@ Returns the mode of the distribution /dist/. This function may return a __domain_error if the distribution does not have a defined mode. -[h4 [#math.dist.pdf]Probability Density Function] +[h4:pdf Probability Density Function] template RealType pdf(const ``['Distribution-Type]``& dist, const RealType& x); @@ -184,14 +184,14 @@ For example, for a standard normal distribution the pdf looks like this: [$../graphs/pdf.png] -[h4 [#math.dist.range]Range] +[h4: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] +[h4:quantile Quantile] template RealType quantile(const ``['Distribution-Type]``& dist, const RealType& p); @@ -211,8 +211,8 @@ distribution: [$../graphs/quantile.png] -[h4 [#math.dist.quantile_c]Quantile from the complement of the probability.] -[link complements complements] +[h4:quantile_c Quantile from the complement of the probability.] +See also [link math_toolkit.stat_tut.overview.complements complements]. template @@ -246,7 +246,7 @@ distribution: [$../graphs/survival_inv.png] -[h4 [#math.dist.sd]Standard Deviation] +[h4:sd Standard Deviation] template RealType standard_deviation(const ``['Distribution-Type]``& dist); @@ -256,7 +256,7 @@ 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] +[h4:support support] template std::pair support(const ``['Distribution-Type]``& dist); @@ -270,7 +270,7 @@ 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] +[h4:variance Variance] template RealType variance(const ``['Distribution-Type]``& dist); @@ -280,7 +280,7 @@ 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] +[h4:skewness Skewness] template RealType skewness(const ``['Distribution-Type]``& dist); @@ -290,7 +290,7 @@ 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] +[h4:kurtosis Kurtosis] template RealType kurtosis(const ``['Distribution-Type]``& dist); @@ -322,7 +322,7 @@ a defined kurtosis. 'Proper' kurtosis can have a value from zero to + infinity. -[h4 [#math.dist.kurtosis_excess]Kurtosis excess] +[h4:kurtosis_excess Kurtosis excess] template RealType kurtosis_excess(const ``['Distribution-Type]``& dist); @@ -347,50 +347,50 @@ Kurtosis excess can have a value from -2 to + infinity. The kurtosis excess of a normal distribution is zero. -[h4 [#cdfPQ]P and Q] +[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. +and its [link math_toolkit.dist_ref.nmp.ccdf complement] respectively. Lowercase p and q are sometimes used to refer to the values returned by these functions. -[h4 [#percent]Percent Point Function or Percentile] +[h4:percent Percent Point Function or Percentile] The percent point function, also known as the percentile, is the same as the __quantile. -[h4 [#cdf_inv]Inverse CDF Function.] +[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.] +[h4:survival_inv Inverse Survival Function.] The inverse of the survival function, is the same as computing the -[link math.dist.quantile_c quantile +[link math_toolkit.dist_ref.nmp.quantile_c quantile from the complement of the probability]. -[h4 [#pmf]Probability Mass Function] +[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.] +[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.] +[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 +calculating the [link math_toolkit.dist_ref.nmp.quantile_c quantile from the complement of the probability]. -[h4 [#survival]Survival Function] +[h4:survival Survival Function] Refer to the __ccdf. diff --git a/doc/distributions/normal.qbk b/doc/distributions/normal.qbk index 98d96d4a7..c78cb6f2e 100644 --- a/doc/distributions/normal.qbk +++ b/doc/distributions/normal.qbk @@ -74,7 +74,7 @@ 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 +All the [link math_toolkit.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 \[-[max_value], +[min_value]\]. @@ -86,7 +86,7 @@ if RealType permits. [h4 Accuracy] The normal distribution is implemented in terms of the -[link math_toolkit.special.sf_erf.error_function error function], +[link math_toolkit.sf_erf.error_function error function], and as such should have very low error rates. [h4 Implementation] diff --git a/doc/distributions/pareto.qbk b/doc/distributions/pareto.qbk index 4e7e2c417..7584d53e9 100644 --- a/doc/distributions/pareto.qbk +++ b/doc/distributions/pareto.qbk @@ -70,7 +70,7 @@ 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 +All the [link math_toolkit.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 \[scale, [infin]\]. diff --git a/doc/distributions/poisson.qbk b/doc/distributions/poisson.qbk index 4e63eef98..432862ede 100644 --- a/doc/distributions/poisson.qbk +++ b/doc/distributions/poisson.qbk @@ -57,7 +57,7 @@ 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 +All the [link math_toolkit.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 \[0, [infin]\]. diff --git a/doc/distributions/rayleigh.qbk b/doc/distributions/rayleigh.qbk index 1647191e1..910c2b7d7 100644 --- a/doc/distributions/rayleigh.qbk +++ b/doc/distributions/rayleigh.qbk @@ -72,7 +72,7 @@ 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 +All the [link math_toolkit.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 \[0, max_value\]. diff --git a/doc/distributions/skew_normal.qbk b/doc/distributions/skew_normal.qbk index fdd281fac..9dde567c5 100644 --- a/doc/distributions/skew_normal.qbk +++ b/doc/distributions/skew_normal.qbk @@ -112,7 +112,7 @@ by Adelchi Azzalini (2005-11-2). [h4 Non-member Accessors] -All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] +All the [link math_toolkit.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]]. @@ -145,7 +145,7 @@ __Mathematica was also used to generate some more accurate spot test data. The skew_normal distribution with shape = zero is implemented as a special case, equivalent to the normal distribution in terms of the -[link math_toolkit.special.sf_erf.error_function error function], +[link math_toolkit.sf_erf.error_function error function], and therefore should have excellent accuracy. The PDF and mean, variance, skewness and kurtosis are also accurately evaluated using diff --git a/doc/distributions/students_t.qbk b/doc/distributions/students_t.qbk index ec48c0a3d..157ba0fbb 100644 --- a/doc/distributions/students_t.qbk +++ b/doc/distributions/students_t.qbk @@ -105,21 +105,21 @@ 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 +All the [link math_toolkit.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 +Various [link math_toolkit.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 its inverses], +[link math_toolkit.sf_beta.ibeta_function incomplete beta function] +and [link math_toolkit.sf_beta.ibeta_inv_function its inverses], refer to accuracy data on those functions for more information. [h4 Implementation] diff --git a/doc/distributions/students_t_examples.qbk b/doc/distributions/students_t_examples.qbk index 2ad26a41d..ced395baf 100644 --- a/doc/distributions/students_t_examples.qbk +++ b/doc/distributions/students_t_examples.qbk @@ -759,13 +759,13 @@ 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]. +* [link math_toolkit.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 +* [link math_toolkit.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 +* [link math_toolkit.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] diff --git a/doc/distributions/triangular.qbk b/doc/distributions/triangular.qbk index 4b237f990..dc7ab02a4 100644 --- a/doc/distributions/triangular.qbk +++ b/doc/distributions/triangular.qbk @@ -102,7 +102,7 @@ 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 +All the [link math_toolkit.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\, @@ -151,7 +151,7 @@ Some 'known good' test values were obtained from * [@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] +* 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, diff --git a/doc/distributions/uniform.qbk b/doc/distributions/uniform.qbk index 5eb664802..04e1e7919 100644 --- a/doc/distributions/uniform.qbk +++ b/doc/distributions/uniform.qbk @@ -87,7 +87,7 @@ 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] +All the [link math_toolkit.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, diff --git a/doc/distributions/weibull.qbk b/doc/distributions/weibull.qbk index ffe6295ba..0d57776de 100644 --- a/doc/distributions/weibull.qbk +++ b/doc/distributions/weibull.qbk @@ -84,7 +84,7 @@ 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 +All the [link math_toolkit.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 \[0, [infin]\]. diff --git a/doc/gcd/math-gcd.qbk b/doc/gcd/math-gcd.qbk index 4b4bbfb2b..150ac9165 100644 --- a/doc/gcd/math-gcd.qbk +++ b/doc/gcd/math-gcd.qbk @@ -1,4 +1,5 @@ -[section:gcd_lcm Integer Utilities (Greatest Common Divisor and Least Common Multiple)] + +[mathpart gcd_lcm Integer Utilities (Greatest Common Divisor and Least Common Multiple)] [section Introduction] @@ -233,5 +234,5 @@ pool library. The code had updates by Helmut Zeisel. [endsect] -[endsect] +[endmathpart] diff --git a/doc/html/boostbook.css b/doc/html/boostbook.css deleted file mode 100644 index 2e15c0110..000000000 --- a/doc/html/boostbook.css +++ /dev/null @@ -1,646 +0,0 @@ -/*============================================================================= - Copyright (c) 2004 Joel de Guzman - http://spirit.sourceforge.net/ - - Distributed under the Boost Software License, Version 1.0. 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-Math Toolkit

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-Paul A. Bristow -

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-Hubert Holin -

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-Christopher Kormanyos -

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-Bruno Lalande -

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-John Maddock -

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-Johan Råde -

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-Benjamin Sobotta -

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-Gautam Sewani -

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-Thijs van den Berg -

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-Xiaogang Zhang -

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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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- Distributed under the 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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Table of Contents

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-
Overview
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Statistical Distributions and Functions
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Special Functions
-
Floating Point Utilities
-
TR1 and C99 external "C" - Functions
-
Mathematical Constants
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Complex Number Functions
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Quaternions
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Octonions
-
Integer Utilities (Greatest Common - Divisor and Least Common Multiple)
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Internals and Internal Details
-
Use with User-Defined Floating-Point - Types
-
Policies
-
Performance
-
Backgrounders
-
Library Status
-
Indexes
-
-
-

- This manual is also available in printer - friendly PDF format, and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, - Classification 519.2-dc22. -

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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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- 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, - if T is a User-defined type without NaN support), 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 numerical approximation for distribution dist - is available as usual as an accessor non-member function median using median(dist), - that may be evaluated (in the absence of an analytic formula) by calling -

-

- quantile(dist, 0.5) (this - is the mathematical definition of course). -

-

- 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, and it is supported and tested by the distribution. -

-

- The range for these distributions is set to infinity if supported by the - platform, (by testing std::numeric_limits<RealType>::has_infinity) - else the maximum value provided for the RealType - by Boost.Math. -

-

- Testing for has_infinity is obviously important for arbitrary precision types - where infinity makes much less sense than for IEEE754 floating-point. -

-

- So far we have not set support() function (only range) on the grounds that - the PDF is uninteresting/zero for infinities. -

-

- Users who require special handling of infinity (or other specific value) - can, of course, always intercept this before calling a distribution or function - and return their own choice of value, or other behavior. This will often - be simpler than trying to handle the aftermath of the error policy. -

-

- 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 in Boost.Multiprecision - like cpp_dec_float. - and others 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. -

-

- For normal C++ usage, the Boost.Math static - const constants are now thread-safe - so for built-in real-number types: float, - double and long - double are all thread safe. -

-

- For User_defined types, for example, cpp_dec_float, - the Boost.Math should also be thread-safe, (thought we are unsure how to - rigorously prove this). -

-

- (Thread safety has received attention in the C++11 Standard revision, so - hopefully all compilers will 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. The The - R Project for Statistical Computing provided the widest range of - distributions, but the usual Intel X86 distribution uses 64-but doubles, - so our use was limited to the 15 to 17 decimal digit accuracy. -

-

- 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. -

-
- - Testing - for Invalid Parameters to Functions and Constructors -
-

- After finding that some 'bad' parameters (like NaN) were not throwing a - domain_error exception as - they should, a function -

-

- check_out_of_range (in test_out_of_range.hpp) was devised by JM to check (using - Boost.Test's BOOST_CHECK_THROW macro) that bad parameters passed to constructors - and functions throw domain_error - exceptions. -

-

- Usage is check_out_of_range< DistributionType - >(list-of-params); - Where list-of-params is a list of valid - parameters from which the distribution can be constructed - ie the same number - of args are passed to the function, as are passed to the distribution constructor. -

-

- The values of the parameters are not important, but must be valid - to pass the contructor checks; the default values are suitable, but must - be explicitly provided, for example: -

-
check_out_of_range<extreme_value_distribution<RealType> >(1, 2);
-
-

- Checks made are: -

-
    -
  • - Infinity or NaN (if available) passed in place of each of the valid params. -
    • -
  • - Infinity or NaN (if available) as a random variable. -
    • -
  • - Out-of-range random variable passed to pdf and cdf (ie outside of "range(DistributionType)"). -
    • -
  • - Out-of-range probability passed to quantile function and complement. -
    • -
-

- but does not check finite but out-of-range - parameters to the constructor because these are specific to each distribution, - for example: -

-
BOOST_CHECK_THROW(pdf(pareto_distribution<RealType>(0, 1), 0), std::domain_error);
-BOOST_CHECK_THROW(pdf(pareto_distribution<RealType>(1, 0), 0), std::domain_error);
-
-

- checks scale and shape parameters are both > 0 by checking - that domain_error exception - is thrown if either are == 0. -

-

- (Use of check_out_of_range - function may mean that some previous tests are now redundant). -

-

- It was also noted that if more than one parameter is bad, then only the first - detected will be reported by the error message. -

-
- - Creating - and Managing the Equations -
-

- Equations that fit on a single line can most easily be produced by inline - Quickbook code using templates for Unicode Greek and Unicode Math symbols. - All Greek letter and small set of Math symbols is available at /boost-path/libs/math/doc/sf_and_dist/html4_symbols.qbk -

-

- Where equations need to use more than one line, real Math editors were used. -

-

- 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. -

-

- Conversion 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
-
-

- See also the section on "Using Python to run Inkscape" and "Using - inkscape to convert scalable vector SVG files to Portable Network graphic - PNG". -

-

- 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 SVGs using Batik - and a command such as: -

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

- Or using Inkscape (File, Export bitmap, Drawing tab, bitmap size (default - size, 100 dpi), Filename (default). png) -

-

- or Using Cygwin, a command such as: -

-
for file in *.svg; do
-  /cygdrive/c/progra~1/Inkscape/inkscape -d 120 -e $(cygpath -a -w $(basename $file .svg).png) $(cygpath -a -w $file);
-done
-

- Currently Inkscape seems to generate the better looking png's. -

-

- 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 >math_pdf.log
-

- Note that XEP will have to be configured to use and - embed whatever fonts are used by the SVG equations (almost certainly - editing the sample xep.xml provided by the XEP installation). If you fail - to do this you will get XEP warnings in the log file like -

-
[warning]could not find any font family matching "Times New Roman"; replaced by Helvetica
-

- (html is the default so it is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html - using command line >bjam -a > math_docs.log). -

-
<!-- Sample configuration for Windows TrueType fonts.  -->
-
-

- is provided in the xep.xml downloaded, but the Windows TrueType fonts are - commented out. -

-

- JM's XEP config file \xep\xep.xml 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> -->
-         <!-- actually called l_10646.ttf on Windows 2000 and Vista Sp1 -->
-         <font><font-data ttf="l_10646.ttf"></font>
-    </font-family>
-
-

- PAB had to alter his because the Lucida Sans Unicode font had a different - name. Other 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 produced in SVG format and then converted to PNG's using the - same process as the equations. -

-

- The programs /libs/math/doc/sf_and_dist/graphs/dist_graphs.cpp and - /libs/math/doc/sf_and_dist/graphs/sf_graphs.cpp generate - the SVG's directly using the Google - Summer of Code 2007 project of Jacob Voytko (whose work so far, considerably - enhanced and now reasonably mature and usable, by Paul A. Bristow, is at - .\boost-sandbox\SOC\2007\visualization). -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/backgrounders/lanczos.html b/doc/html/math_toolkit/backgrounders/lanczos.html deleted file mode 100644 index 49350afd7..000000000 --- a/doc/html/math_toolkit/backgrounders/lanczos.html +++ /dev/null @@ -1,566 +0,0 @@ - - - -The Lanczos Approximation - - - - - - - - -
-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 = D B 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 59. 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 60. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/backgrounders/refs.html b/doc/html/math_toolkit/backgrounders/refs.html deleted file mode 100644 index c7e9ee410..000000000 --- a/doc/html/math_toolkit/backgrounders/refs.html +++ /dev/null @@ -1,207 +0,0 @@ - - - -References - - - - - - - - -
-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 a replacement for the legendary Abramowitz and Stegun's - Handbook of Mathematical Functions (often called simply A&S), -

-

- 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. -

-

- NIST Handbook of Mathematical Functions Edited by: Frank W. J. Olver, University - of Maryland and National Institute of Standards and Technology, Maryland, - Daniel W. Lozier, National Institute of Standards and Technology, Maryland, - Ronald F. Boisvert, National Institute of Standards and Technology, Maryland, - Charles W. Clark, National Institute of Standards and Technology, Maryland - and University of Maryland. -

-

- ISBN: 978-0521140638 (paperback), 9780521192255 (hardback), July 2010, Cambridge - University Press. -

-

- 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. -

-

- Extreme Value - Distributions, Theory and Applications Samuel Kotz & Saralees - Nadarajah, ISBN 978-1-86094-224-2 & 1-86094-224-5 Oct 2000, Chapter 1.2 - discusses the various extreme value distributions. -

-

- pugh.pdf - (application/pdf Object) Pugh Msc Thesis on the Lanczos 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 -
-

- We found (and used to create cross-check spot values - as far as their accuracy - allowed). -

-

- 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. -

-

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

-

- http://www.adsciengineering.com/bpdcalc/ - 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. -

-

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

-

- DCFLIB - C++ version. -

-

- 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/ -

-

- 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. -

-

- _R R Development Core Team (2010). - R: A language and environment for statistical computing. R Foundation for - Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org. -

-

- For use of R, see: -

-

- Jim Albert, Bayesian Computation with R, ISBN 978-0-387-71384-7. -

-

- C++ - Statistical Distributions in Boost - QuantNetwork forum discusses - using Boost.Math in finance. -

-

- Quantnet - Boost and computational finance. Robert Demming & Daniel J. Duffy, - Introduction to the C++ Boost Libraries - Volume I - Foundations and Volume - II ISBN 978-94-91028-01-4, Advanced Libraries and Applications, ISBN 978-94-91028-02-1 - (to be published in 2011). discusses application of Boost.Math, especially - in finance.] -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/backgrounders/relative_error.html b/doc/html/math_toolkit/backgrounders/relative_error.html deleted file mode 100644 index ac8ffca5a..000000000 --- a/doc/html/math_toolkit/backgrounders/relative_error.html +++ /dev/null @@ -1,114 +0,0 @@ - - - -Relative Error - - - - - - - - -
-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 what http://en.wikipedia.org/wiki/William_Kahan - called http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/backgrounders/remez.html b/doc/html/math_toolkit/backgrounders/remez.html deleted file mode 100644 index d5106ffc8..000000000 --- a/doc/html/math_toolkit/backgrounders/remez.html +++ /dev/null @@ -1,534 +0,0 @@ - - - -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 its 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 1.3x10-3  to 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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Tutorial
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The Mathematical Constants
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Defining New Constants
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FAQs
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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-FAQs -

-
- - Why - are these Constants Chosen? -
-

- It is, of course, impossible to please everyone with a list like this. -

-

- Some of the criteria we have used are: -

-
    -
  • - Used in Boost.Math. -
    • -
  • - Commonly used. -
    • -
  • - Expensive to compute. -
    • -
  • - Requested by users. -
    • -
  • - Used - in science and mathematics. -
    • -
  • - No integer values (because so cheap to construct).
    (You can easily - define your own if found convenient, for example: FPT - one =static_cast<FPT>(42);). -
    • -
-
- - How - are constants named? -
-
    -
  • - Not macros, so no upper case. -
    • -
  • - All lower case (following C++ standard names). -
    • -
  • - No CamelCase. -
    • -
  • - Underscore as _ delimiter between words. -
    • -
  • - Numbers spelt as words rather than decimal digits (except following pow). -
    • -
  • - Abbreviation conventions: -
      -
    • - root for square root. -
      • -
    • - cbrt for cube root. -
      • -
    • - pow for pow function using decimal digits like pow23 for n2/3. -
      • -
    • - div for divided by or operator /. -
      • -
    • - minus for operator -, plus for operator +. -
      • -
    • - sqr for squared. -
      • -
    • - cubed for cubed n3. -
      • -
    • - words for greek, like π, ζ and Γ. -
      • -
    • - words like half, third, three_quarters, sixth for fractions. (Digit(s) - can get muddled). -
      • -
    • - log10 for log10 -
      • -
    • - ln for loge -
      • -
    -
    • -
-
- - How are - the constants derived? -
-

- The constants have all been calculated using high-precision software working - with up to 300-bit precision giving about 100 decimal digits. (The precision - can be arbitrarily chosen and is limited only by compute time). -

-
- - How - Accurate are the constants? -
-

- The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of - reasonably-foreseeable floating-point hardware (256-bit) and should meet - most high-precision computations. -

-
- - How are - the constants tested? -
-
    -
  1. - Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double - literals, with at least 35 decimal digits, enough to be accurate for - all long double implementations. The tolerance is usually twice long double - epsilon. -
    2. -
  2. - Comparison with calculation at long double precision. This often requires - a slightly higher tolerance than two epsilon because of computational - noise from round-off etc, especially when trig and other functions are - called. -
    3. -
  3. - Comparison with independent published values, for example, using The On-Line Encyclopedia of Integer Sequences - (OEIS) again using at least 35 decimal digits strings. -
    4. -
  4. - Comparison with independely calculated values using arbitrary precision - tools like Mathematica, - again using at least 35 decimal digits literal strings. -
    5. -
-
- - - - - -
[Warning]Warning

- We have not yet been able to check that - all constants are accurate at the full - arbitrary precision, at present 100 decimal digits. But certain key values - like e and pi appear to be accurate and internal - consistencies suggest that others are this accurate too. -

-
- - Why is - Portability important? -
-

- Code written using math constants is easily portable even when using different - floating-point types with differing precision. -

-

- It is a mistake to expect that results of computations will be identical, but you can achieve the best - accuracy possible for the floating-point type in use. -

-

- This has no extra cost to the user, but reduces irritating, and often confusing - and very hard-to-trace effects, caused by the intrinsically limited precision - of floating-point calculations. -

-

- A harmless symptom of this limit is a spurious least-significant digit; at - worst, slightly inaccurate constants sometimes cause iterating algorithms - to diverge wildly because internal comparisons just fail. -

-
- - What - is the Internal Format of the constants, and why? -
-

- See tutorial above - for normal use, but this FAQ explains the internal details used for the constants. -

-

- Constants are stored as 100 decimal digit values. However, some compilers - do not accept decimal digits strings as long as this. So the constant is - split into two parts, with the first containing at least 128-bit long double - precision (35 decimal digits), and for consistency should be in scientific - format with a signed exponent. -

-

- The second part is the value of the constant expressed as a string literal, - accurate to at least 100 decimal digits (in practice that means at least - 102 digits). Again for consistency use scientific format with a signed exponent. -

-

- For types with precision greater than a long double, then if T is constructible - T is constructible from a - const char* then it's directly constructed from the - string, otherwise we fall back on lexical_cast to convert to type T. (Using a string is necessary because - you can't use a numeric constant since even a long - double might not have enough digits). -

-

- So, for example, a constant like pi is internally defined as -

-
BOOST_DEFINE_MATH_CONSTANT(pi, 3.141592653589793238462643383279502884e+00, "3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00");
-
-

- In this case the significand is 109 decimal digits, ensuring 100 decimal - digits are exact, and exponent is zero. -

-

- See defining new constants - to calculate new constants. -

-

- A macro definition like this can be pasted into user code where convenient, - or into boost/math/constants.hpp if - it is to be added to the Boost.Math library. -

-
- - What - Floating-point Types could I use? -
-

- Apart from the built-in floating-point types float, - double, long - double, there are several arbitrary - precision floating-point classes available, but most are not licensed for - commercial use. -

-
- - Boost.Multiprecision - by Christopher Kormanyos -
-

- This work is based on an earlier work called e-float: Algorithm 910: A Portable - C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS, - {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469 - e_float - but is now re-factored and available under the Boost license in the Boost-sandbox - at multiprecision - where it is being refined and prepared for review. -

-
- - Boost.cpp_float - by John Maddock using Expression Templates -
-

- Big Number - which is a reworking of e_float - by Christopher Kormanyos to use expression templates for faster execution. -

-
- - NTL - class quad_float -
-

- NTL by Victor Shoup has fixed - and arbitrary high precision fixed and floating-point types. However none - of these are licenced for commercial use. -

-
#include <NTL/quad_float.h> // quad precision 106-bit, about 32 decimal digits.
-using NTL::to_quad_float; // Less precise than arbitrary precision NTL::RR.
-
-

- NTL class quad_float, which - gives a form of quadruple precision, 106-bit significand (but without an - extended exponent range.) With an IEC559/IEEE 754 compatible processor, for - example Intel X86 family, with 64-bit double, and 53-bit significand, using - the significands of two 64-bit doubles, - if std::numeric_limits<double>::digits10 is 16, then we get about twice - the precision, so std::numeric_limits<quad_float>::digits10() - should be 32. (the default std::numeric_limits<RR>::digits10() should be about 40). (which seems to agree - with experiments). We output constants (including some noisy bits, an approximation - to std::numeric_limits<RR>::max_digits10()) - by adding 2 extra decimal digits, so using quad_float::SetOutputPrecision(32 + - 2); -

-

- Apple Mac/Darwin uses a similar doubledouble 106-bit - for its built-in long double - type. -

-
- - - - - -
[Note]Note

- The precision of all doubledouble - floating-point types is rather odd and values given are only approximate. -

-

- New projects should use Boost.Multiprecision. -

-
- - NTL - class RR -
-

- Arbitrary precision floating point with NTL class RR, default is 150 bit - (about 50 decimal digits) used here with 300 bit to output 100 decimal digits, - enough for many practical non-'number-theoretic' C++ applications. -

-

- NTL A Library for doing Number Theory - is not licenced for commercial use. -

-

- This class is used in Boost.Math and is an option when using big_number projects - to calculate new math constants. -

-

- New projects should use Boost.Multiprecision. -

-
- - GMP - and MPFR -
-

- GMP and MPFR - have also been used to compute constants, but are licensed under the Lesser GPL license - and are not licensed for commercial use. -

-
- - What - happened to a previous collection of constants proposed for Boost? -
-

- A review concluded that the way in which the constants were presented did - not meet many peoples needs. None of the methods proposed met many users' - essential requirement to allow writing simply pi - rather than pi(). - Many science and engineering equations look difficult to read when because - function call brackets can be confused with the many other brackets often - needed. All the methods then proposed of avoiding the brackets failed to - meet all needs, often on grounds of complexity and lack of applicability - to various realistic scenarios. -

-

- So the simple namespace method, proposed on its own, but rejected at the - first review, has been added to allow users to have convenient access to - float, double and long double values, but combined with template struct and - functions to allow simultaneous use with other non-built-in floating-point - types. -

-
- - Why - do the constants (internally) have a struct rather than a simple function? -
-

- A function mechanism was provided by in previous versions of Boost.Math. -

-

- The new mechanism is to permit partial specialization. See Custom Specializing - a constant above. It should also allow use with other packages like ttmath Bignum C++ library. -

-
- - Where - can I find other high precision constants? -
-
    -
  1. - Constants with very high precision and good accuracy (>40 decimal - digits) from Simon Plouffe's web based collection http://pi.lacim.uqam.ca/eng/. -
    2. -
  2. - The On-Line Encyclopedia of Integer Sequences - (OEIS) -
    3. -
  3. - Checks using printed text optically scanned values and converted from: - D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, - ISBN 0 201 89683 4 (1997) -
    4. -
  4. - M. Abrahamovitz & I. E. Stegun, National Bureau of Standards, Handbook - of Mathematical Functions, a reference source for formulae now superceded - by -
    5. -
  5. - Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, - NIST Handbook of Mathemetical Functions, Cambridge University Press, - ISBN 978-0-521-14063-8, 2010. -
    6. -
  6. - John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 - 7. -
    7. -
  7. - Some values from Cephes Mathematical Library, Stephen L. Moshier and - CALC100 100 decimal digit Complex Variable Calculator Program, a DOS - utility. -
    8. -
  8. - Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at Number, - constants and computation. -
    9. -
-
- - Where - are Physical Constants? -
-

- Not here in this Boost.Math collection, because physical constants: -

-
    -
  • - Are measurements, not truely constants. -
    • -
  • - Are not truly constant and keeping changing as mensuration technology - improves. -
    • -
  • - Have a instrinsic uncertainty. -
    • -
  • - Mathematical constants are stored and represented at varying precision, - but should never be inaccurate. -
    • -
-

- Some physical constants may be available in Boost.Units. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/constants.html b/doc/html/math_toolkit/constants/constants.html deleted file mode 100644 index 8fc6dd928..000000000 --- a/doc/html/math_toolkit/constants/constants.html +++ /dev/null @@ -1,1480 +0,0 @@ - - - -The Mathematical Constants - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- This section lists the mathematical constants, their use(s) (and sometimes - rationale for their inclusion). -

-
-

Table 52. Mathematical Constants

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- name -

- 		-

- formula -

- 		-

- Value (6 decimals) -

- 		-

- Uses and Rationale -

-
-

- Rational fractions -

- 		- - -
-

- half -

- 		-

- 1/2 -

- 		-

- 0.5 -

- 		-
-

- third -

- 		-

- 1/3 -

- 		-

- 0.333333 -

- 		-
-

- two_thirds -

- 		-

- 2/3 -

- 		-

- 0.66667 -

- 		-
-

- three_quarters -

- 		-

- 3/4 -

- 		-

- 0.75 -

- 		-
-

- two and related -

- 		- - -
-

- root_two -

- 		-

- √2 -

- 		-

- 1.41421 -

- 		-
-

- root_three -

- 		-

- √3 -

- 		-

- 1.73205 -

- 		-
-

- half_root_two -

- 		-

- √2 /2 -

- 		-

- 0.707106 -

- 		-
-

- ln_two -

- 		-

- ln(2) -

- 		-

- 0.693147 -

- 		-
-

- ln_ten -

- 		-

- ln(10) -

- 		-

- 2.30258 -

- 		-
-

- ln_ln_two -

- 		-

- ln(ln(2)) -

- 		-

- -0.366512 -

- 		-

- Gumbel distribution median -

-
-

- root_ln_four -

- 		-

- √ln(4) -

- 		-

- 1.177410 -

- 		-
-

- one_div_root_two -

- 		-

- 1/√2 -

- 		-

- 0.707106 -

- 		-
-

- π and related -

- 		- - -
-

- pi -

- 		-

- pi -

- 		-

- 3.14159 -

- 		-

- Ubiquitous. Archimedes constant π -

-
-

- half_pi -

- 		-

- π/2 -

- 		-

- 1.570796 -

- 		-
-

- third_pi -

- 		-

- π/3 -

- 		-

- 1.04719 -

- 		-
-

- sixth_pi -

- 		-

- π/6 -

- 		-

- 0.523598 -

- 		-
-

- two_pi -

- 		-

- 2π -

- 		-

- 6.28318 -

- 		-

- Many uses, most simply, circumference of a circle -

-
-

- two_thirds_pi -

- 		-

- 2/3 π -

- 		-

- 2.09439 -

- 		-

- volume - of a hemi-sphere = 4/3 π r³ -

-
-

- three_quarters_pi -

- 		-

- 3/4 π -

- 		-

- 2.35619 -

- 		-

- = 3/4 π r³ -

-
-

- four_thirds_pi -

- 		-

- 4/3 π -

- 		-

- 4.18879 -

- 		-

- volume - of a sphere = 4/3 π r³ -

-
-

- one_div_two_pi -

- 		-

- 1/(2π) -

- 		-

- 1.59155 -

- 		-

- Widely used -

-
-

- root_pi -

- 		-

- √π -

- 		-

- 1.77245 -

- 		-

- Widely used -

-
-

- root_half_pi -

- 		-

- √ π/2 -

- 		-

- 1.25331 -

- 		-

- Widely used -

-
-

- root_two_pi -

- 		-

- √ π*2 -

- 		-

- 2.50662 -

- 		-

- Widely used -

-
-

- one_div_root_pi -

- 		-

- 1/√π -

- 		-

- 0.564189 -

- 		-
-

- one_div_root_two_pi -

- 		-

- 1/√(2π) -

- 		-

- 0.398942 -

- 		-
-

- root_one_div_pi -

- 		-

- √(1/π -

- 		-

- 0.564189 -

- 		-
-

- pi_minus_three -

- 		-

- π-3 -

- 		-

- 1.41593 -

- 		-
-

- four_minus_pi -

- 		-

- 4 -π -

- 		-

- 0.858407 -

- 		-
-

- pow23_four_minus_pi -

- 		-

- 42/3 - π -

- 		-

- 0.795316 -

- 		-
-

- pi_pow_e -

- 		-

- πe -

- 		-

- 22.4591 -

- 		-
-

- pi_sqr -

- 		-

- π2 -

- 		-

- 9.86960 -

- 		-
-

- pi_sqr_div_six -

- 		-

- π2/6 -

- 		-

- 1.64493 -

- 		-
-

- pi_cubed -

- 		-

- π3 -

- 		-

- 31.00627 -

- 		-
-

- cbrt_pi -

- 		-

- √3 π -

- 		-

- 1.46459 -

- 		-
-

- one_div_cbrt_pi -

- 		-

- 1/√3 π -

- 		-

- 0.682784 -

- 		-
-

- Euler's e and related -

- 		- - -
-

- e -

- 		-

- e -

- 		-

- 2.71828 -

- 		-

- Euler's - constant e -

-
-

- exp_minus_half -

- 		-

- e -1/2 -

- 		-

- 0.606530 -

- 		-
-

- e_pow_pi -

- 		-

- e π -

- 		-

- 23.14069 -

- 		-
-

- root_e -

- 		-

- √ e -

- 		-

- 1.64872 -

- 		-
-

- log10_e -

- 		-

- log10(e) -

- 		-

- 0.434294 -

- 		-
-

- one_div_log10_e -

- 		-

- 1/log10(e) -

- 		-

- 2.30258 -

- 		-
-

- Trigonometric -

- 		- - -
-

- degree -

- 		-

- radians = π / 180 -

- 		-

- 0.017453 -

- 		-
-

- radian -

- 		-

- degrees = 180 / π -

- 		-

- 57.2957 -

- 		-
-

- sin_one -

- 		-

- sin(1) -

- 		-

- 0.841470 -

- 		-
-

- cos_one -

- 		-

- cos(1) -

- 		-

- 0.54030 -

- 		-
-

- sinh_one -

- 		-

- sinh(1) -

- 		-

- 1.17520 -

- 		-
-

- cosh_one -

- 		-

- cosh(1) -

- 		-

- 1.54308 -

- 		-
-

- Phi -

- 		-

- Phidias golden ratio -

- 		-

- Phidias - golden ratio -

- 		-
-

- phi -

- 		-

- (1 + √5) /2 -

- 		-

- 1.61803 -

- 		-

- finance -

-
-

- ln_phi -

- 		-

- ln(φ) -

- 		-

- 0.48121 -

- 		-
-

- one_div_ln_phi -

- 		-

- 1/ln(φ) -

- 		-

- 2.07808 -

- 		-
-

- Euler's Gamma -

- 		- - -
-

- euler -

- 		-

- euler -

- 		-

- 0.577215 -

- 		-

- Euler-Mascheroni - gamma constant -

-
-

- one_div_euler -

- 		-

- 1/euler -

- 		-

- 1.73245 -

- 		-
-

- euler_sqr -

- 		-

- euler2 -

- 		-

- 0.333177 -

- 		-
-

- Misc -

- 		- - -
-

- zeta_two -

- 		-

- ζ(2) -

- 		-

- 1.64493 -

- 		-

- Riemann - zeta function -

-
-

- zeta_three -

- 		-

- ζ(3) -

- 		-

- 1.20205 -

- 		-

- Riemann - zeta function -

-
-

- catalan -

- 		-

- K -

- 		-

- 0.915965 -

- 		-

- Catalan - (or Glaisher) combinatorial constant -

-
-

- glaisher -

- 		-

- A -

- 		-

- 1.28242 -

- 		-

- Decimal expansion - of Glaisher-Kinkelin constant -

-
-

- khinchin -

- 		-

- k -

- 		-

- 2.685452 -

- 		-

- Decimal expansion - of Khinchin constant -

-
-

- extreme_value_skewness -

- 		-

- 12√6 ζ(3)/ π3 -

- 		-

- 1.139547 -

- 		-

- Extreme value distribution -

-
-

- rayleigh_skewness -

- 		-

- 2√π(π-3)/(4 - π)3/2 -

- 		-

- 0.631110 -

- 		-

- Rayleigh distribution skewness -

-
-

- rayleigh_kurtosis_excess -

- 		-

- -(6π2-24π+16)/(4-π)2 -

- 		-

- 0.245089 -

- 		-

- Rayleigh - distribution kurtosis excess -

-
-

- rayleigh_kurtosis -

- 		-

- 3+(6π2-24π+16)/(4-π)2 -

- 		-

- 3.245089 -

- 		-

- Rayleigh distribution kurtosis -

-
-
-
- - - - - -
[Note]Note

- Integer values are not included in this - list of math constants, however interesting, because they can be so easily - and exactly constructed, even for UDT, for example: static_cast<cpp_float>(42). -

-
- - - - - -
[Tip]Tip

- If you know the approximate value of the constant, you can search for the - value to find Boost.Math chosen name in this table. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/intro.html b/doc/html/math_toolkit/constants/intro.html deleted file mode 100644 index 220d96e9a..000000000 --- a/doc/html/math_toolkit/constants/intro.html +++ /dev/null @@ -1,119 +0,0 @@ - - - -Introduction - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Boost.Math provides a collection of mathematical constants. -

-
- - Why - use Boost.Math mathematical constants? -
-
    -
  • - Readable. For the very many jobs just using built-in like double, you can just write expressions - like -
    double area = pi * r * r;
    - (If that's all you want, jump direct to use - in non-template code!) -
    • -
  • - Effortless - avoiding a search of reference sources. -
    • -
  • - Usable with both builtin floating point types, and user-defined, possibly - extended precision, types such as NTL, MPFR/GMP, mp_float: in the latter - case the constants are computed to the necessary precision and then cached. -
    • -
  • - Accurate - ensuring that the values are as accurate as possible for the - chosen floating-point type -
      -
    • - No loss of accuracy from repeated rounding of intermediate computations. -
      • -
    • - Result is computed with higher precision and only rounded once. -
      • -
    • - Less risk of inaccurate result from functions pow, trig and log - at corner - cases. -
      • -
    • - Less risk of cancellation - error. -
      • -
    -
    • -
  • - Portable - as possible between different systems using different floating-point - precisions: see use - in template code. -
    • -
  • - Tested - by comparison with other published sources, or separately computed - at long double precision. -
    • -
  • - Faster - can avoid (re-)calculation at runtime. -
      -
    • - If the value returned is a builtin type then it's returned by value - as a constexpr - (C++11 feature, if available). -
      • -
    • - If the value is computed and cached (or constructed from a string - representation and cached), then it's returned by constant reference.
      - This can be significant if: -
      • -
    • - Functions pow, trig or log are used. -
      • -
    • - Inside an inner loop. -
      • -
    • - Using a high-precision UDT like Boost.Multiprecision. -
      • -
    • - Compiler optimizations possible with built-in types, especially - double, are not available. -
      • -
    -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/new_const.html b/doc/html/math_toolkit/constants/new_const.html deleted file mode 100644 index 5c75ad065..000000000 --- a/doc/html/math_toolkit/constants/new_const.html +++ /dev/null @@ -1,259 +0,0 @@ - - - -Defining New Constants - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The library provides some helper code to assist in defining new constants; - the process for defining a constant called my_constant - goes like this: -

-

- 1. Define a function that calculates the value of the - constant. This should be a template function, and be placed in - boost/math/constants/calculate_constants.hpp if - the constant is to be added to this library, or else defined at the top of - your source file if not. -

-

- The function should look like this: -

-
namespace boost{ namespace math{ namespace constants{ namespace detail{
-
-template <class Real>
-template <int N>
-Real constant_my_constant<Real>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
-{
-  int required_precision = N ? N : tools::digits<Real>();
-  Real result = /* value computed to required_precision bits */ ;
-  return result;
-}
-
-}}}} // namespaces
-
-

- Then define a placeholder for the constant itself: -

-
namespace boost{ namespace math{ namespace constants{
-
-BOOST_DEFINE_MATH_CONSTANT(my_constant, 0.0, "0");
-
-}}}
-
-

- For example, to calculate π/2, add to boost/math/constants/calculate_constants.hpp -

-
template <class T>
-template<int N>
-inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
-{
-   BOOST_MATH_STD_USING
-   return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2);
-}
-
-

- Then to boost/math/constants/constants.hpp add: -

-
BOOST_DEFINE_MATH_CONSTANT(half_pi, 0.0, "0");  // Actual values are temporary, we'll replace them later.
-
-
- - - - - -
[Note]Note

- Previously defined constants like pi and e can be used, but by not simply calling pi<T>(); specifying the precision via the - policy pi<T, policies::policy<policies::digits2<N> > >() is essential to ensure full accuracy. -

-
- - - - - -
[Warning]Warning
-

- Newly defined constants can only be used once they are included in boost/math/constants/constants.hpp. So if you add template - <class - T, - class N> T constant_my_constant{...}, - then you cannot define constant_my_constant - until you add the temporary BOOST_DEFINE_MATH_CONSTANT(my_constant, 0.0, - "0"). - Failing to do this will result in surprising compile errors: -

-
error C2143: syntax error : missing ';' before '<'
-error C2433: 'constant_root_two_div_pi' : 'inline' not permitted on data declarations
-error C2888: 'T constant_root_two_div_pi' : symbol cannot be defined within namespace 'detail'
-error C2988: unrecognizable template declaration/definition
-
-

-

-
-

- 2. You will need an arbitrary precision type to use - to calculate the value. This library currently supports either - cpp_float, NTL::RR - or mpfr_class used via the - bindings in boost/math/bindings. - The default is to use NTL::RR unless - you define an alternate macro, for example, USE_MPFR - or USE_CPP_FLOAT at the start - of your program. -

-

- 3. It is necessary to link to the Boost.Regex library, and probably to your - chosen arbitrary precision type library. -

-

- 4. The complete program to generate the constant half_pi - using function calculate_half_pi - is then: -

-
#define USE_CPP_FLOAT // If required.
-#include <boost/math/constants/generate.hpp>
-
-int main()
-{
-   BOOST_CONSTANTS_GENERATE(half_pi);
-}
-
-

- The output from the program is a snippet of C++ code (actually a macro call) - that can be cut and pasted into boost/math/constants/constants.hpp or else into your own code, for example: -

-
BOOST_DEFINE_MATH_CONSTANT(half_pi, 1.570796326794896619231321691639751442e+00, "1.57079632679489661923132169163975144209858469968755291048747229615390820314310449931401741267105853399107404326e+00");
-
-

- This macro BOOST_DEFINE_MATH_CONSTANT inserts a C++ struct code snippet that - declares the float, double and long - double versions of the constant, plus - a decimal digit string representation correct to 100 decimal digits, and - all the meta-programming machinery needed to select between them. -

-

- The result of an expanded macro for Pi is shown below. -

-

-

-
// Preprocessed pi constant, annotated.
-
-namespace boost
-{
-  namespace math
-  {
-    namespace constants
-    {
-      namespace detail
-      {
-        template <class T> struct constant_pi
-        {
-          private:
-            // Default implementations from string of decimal digits:
-            static inline T get_from_string()
-            {
-            static const T result
-               = detail::convert_from_string<T>("3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00",
-               boost::is_convertible<const char*, T>());
-              return result;
-            }
-            template <int N> static T compute();
-
-          public:
-            // Default implementations from string of decimal digits:
-            static inline T get(const mpl::int_<construct_from_string>&)
-            {
-              constant_initializer<T, & constant_pi<T>::get_from_string >::do_nothing();
-              return get_from_string();
-            }
-            // Float, double and long double versions:
-            static inline T get(const mpl::int_<construct_from_float>)
-            {
-              return 3.141592653589793238462643383279502884e+00F;
-            }
-            static inline  T get(const mpl::int_<construct_from_double>&)
-            {
-              return 3.141592653589793238462643383279502884e+00;
-            }
-            static inline  T get(const mpl::int_<construct_from_long_double>&)
-            {
-              return 3.141592653589793238462643383279502884e+00L;
-            }
-            // For very high precision that is nonetheless can be calculated at compile time:
-            template <int N> static inline T get(const mpl::int_<N>& n)
-            {
-              constant_initializer2<T, N, & constant_pi<T>::template compute<N> >::do_nothing();
-              return compute<N>();
-            }
-            //For true arbitrary precision, which may well vary at runtime.
-            static inline T get(const mpl::int_<0>&)
-            {
-              return tools::digits<T>() > max_string_digits ? compute<0>() : get(mpl::int_<construct_from_string>());
-            }
-         }; // template <class T> struct constant_pi
-      } //  namespace detail
-
-      // The actual forwarding function (including policy to control precision).
-      template <class T, class Policy> inline T pi( )
-      {
-        return detail:: constant_pi<T>::get(typename construction_traits<T, Policy>::type());
-      }
-      // The actual forwarding function (using default policy to control precision).
-      template <class T> inline  T pi()
-      {
-        return pi<T, boost::math::policies::policy<> >()
-      }
-    } //     namespace constants
-
-    // Namespace specific versions, for the three built-in floats:
-    namespace float_constants
-    {
-      static const float pi = 3.141592653589793238462643383279502884e+00F;
-    }
-    namespace double_constants
-    {
-      static const double pi = 3.141592653589793238462643383279502884e+00;
-    }
-    namespace long_double_constants
-    {
-      static const long double pi = 3.141592653589793238462643383279502884e+00L;
-    }
-    namespace constants{;
-    } // namespace constants
-  } // namespace math
-} // namespace boost
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/tutorial.html b/doc/html/math_toolkit/constants/tutorial.html deleted file mode 100644 index 7b66d7e7d..000000000 --- a/doc/html/math_toolkit/constants/tutorial.html +++ /dev/null @@ -1,44 +0,0 @@ - - - -Tutorial - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Use in non-template - code
-
Use in template - code
-
Use With - User-Defined Types
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/tutorial/non_templ.html b/doc/html/math_toolkit/constants/tutorial/non_templ.html deleted file mode 100644 index cbfeeb0ad..000000000 --- a/doc/html/math_toolkit/constants/tutorial/non_templ.html +++ /dev/null @@ -1,78 +0,0 @@ - - - -Use in non-template code - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- When using the math constants at your chosen fixed precision in non-template - code, you can simply add a using - namespace declaration, for example, - using namespace - boost::math::double_constants, to make the constants - of the correct precision for your code visible in the current scope, and - then use each constant as a simple variable - sans brackets: -

-
#include <boost/math/constants/constants.hpp>
-
-double area(double r)
-{
-   using namespace boost::math::double_constants;
-   return pi * r * r;
-}
-
-

- Had our function been written as taking a float - rather than a double, we could - have written instead: -

-
#include <boost/math/constants/constants.hpp>
-
-float area(float r)
-{
-   using namespace boost::math::float_constants;
-   return pi * r * r;
-}
-
-

- Likewise, constants that are suitable for use at long - double precision are available in - the namespace boost::math::long_double_constants. -

-

- You can see the full list of available constants at math_toolkit.constants.constants. -

-

- Some examples of using constants are at constants_eg1. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/tutorial/templ.html b/doc/html/math_toolkit/constants/tutorial/templ.html deleted file mode 100644 index a82090c38..000000000 --- a/doc/html/math_toolkit/constants/tutorial/templ.html +++ /dev/null @@ -1,156 +0,0 @@ - - - -Use in template code - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- When using the constants inside a function template, we need to ensure - that we use a constant of the correct precision for our template parameters. - We can do this by calling the function-template versions, pi<FPType>(), - of the constants like this: -

-
#include <boost/math/constants/constants.hpp>
-
-template <class Real>
-Real area(Real r)
-{
-   using namespace boost::math::constants;
-   return pi<Real>() * r * r;
-}
-
-

- Although this syntax is a little less "cute" than the non-template - version, the code is no less efficient (at least for the built-in types - float, double - and long double) - : the function template versions of the constants are simple inline functions - that return a constant of the correct precision for the type used. In addition, - these functions are declared constexp - for those compilers that support this, allowing the result to be used in - constant-expressions provided the template argument is a literal type. -

-
- - - - - -
[Tip]Tip

- Keep in mind the difference between the variable version, just pi, and the template-function version: - the template-function requires both a <floating-point-type> - and function call () brackets, - for example: pi<double>(). - You cannot write double p = pi<>(), - nor double p - = pi(). -

-
- - - - - -
[Note]Note
-

- You can always use both variable and - template-function versions provided calls are fully - qualified, for example: -

-
double my_pi1 = boost::math::constants::pi<double>();
-double my_pi2 = boost::math::double_constants::pi;
-
-

-

-
-
- - - - - -
[Warning]Warning
-

- It may be tempting to simply define -

-
using namespace boost::math::double_constants;
-using namespace boost::math::constants;
-
-

- but if you do define two namespaces, this will, of course, create ambiguity! -

-
double my_pi = pi(); // error C2872: 'pi' : ambiguous symbol
-double my_pi2 = pi; // Context does not allow for disambiguation of overloaded function
-
-

- Although the mistake above is fairly obvious, it is also not too difficult - to do this accidentally, or worse, create it in someone elses code. -

-

- Therefore is it prudent to avoid this risk by localising - the scope of such definitions, as shown above. -

-
-
- - - - - -
[Tip]Tip
-

- Be very careful with the type provided as parameter. For example, providing - an integer instead of a floating-point - type can be disastrous (a C++ feature). -

-

-

-
cout << "Area = " << area(2) << endl; // Area = 12!!!
-

-

-

- You should get a compiler warning -

-
warning : 'return' : conversion from 'double' to 'int', possible loss of data
-
-

- Failure to heed this warning can lead to very wrong answers! -

-

- You can also avoid this by being explicit about the type of Area. -

-
cout << "Area = " << area<double>(2) << endl; // Area = 12.566371
-

-

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/constants/tutorial/user_def.html b/doc/html/math_toolkit/constants/tutorial/user_def.html deleted file mode 100644 index 8f76ddfbc..000000000 --- a/doc/html/math_toolkit/constants/tutorial/user_def.html +++ /dev/null @@ -1,327 +0,0 @@ - - - -Use With User-Defined Types - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The most common example of a high-precision user-defined type will probably - be Boost.Multiprecision. -

-

- The syntax for using the function-call constants with user-defined types - is the same as it is in the template class, which is to say we use: -

-
#include <boost/math/constants/constants.hpp>
-
-boost::math::constants::pi<UserDefinedType>();
-
-

- For example: -

-
boost::math::constants::pi<boost::multiprecision::cpp_dec_float_50>();
-
-

- giving π with a precision of 50 decimal digits. -

-

- However, since the precision of the user-defined type may be much greater - than that of the built-in floating point types, how the value returned - is created is as follows: -

-
    -
  • - If the precision of the type is known at compile time: -
      -
    • - If the precision is less than or equal to that of a float and the type is constructable - from a float then - our code returns a float - literal. If the user-defined type is a literal type then the - function call that returns the constant will be a constexp. -
      • -
    • - If the precision is less than or equal to that of a double and the type is constructable - from a double then - our code returns a double - literal. If the user-defined type is a literal type then the - function call that returns the constant will be a constexp. -
      • -
    • - If the precision is less than or equal to that of a long double - and the type is constructable from a long - double then our code returns - a long double - literal. If the user-defined type is a literal type then the - function call that returns the constant will be a constexp. -
      • -
    • - If the precision is less than or equal to that of a __float128 (and the compiler - supports such a type) and the type is constructable from a __float128 then our code returns - a __float128 - literal. If the user-defined type is a literal type then the - function call that returns the constant will be a constexp. -
      • -
    • - If the precision is less than 100 decimal digits, then the constant - will be constructed (just the once, then cached in a thread-safe - manner) from a string representation of the constant. In this - case the value is returned as a const reference to the cached - value. -
      • -
    • - Otherwise the value is computed (just once, then cached in a - thread-safe manner). In this case the value is returned as a - const reference to the cached value. -
      • -
    -
    • -
  • - If the precision is unknown at compile time then: -
      -
    • - If the runtime precision (obtained from a call to boost::math::tools::digits<T>()) - is less than 100 decimal digits, then the constant is constructed - "on the fly" from the string representation of the - constant. -
      • -
    • - Otherwise the value is constructed "on the fly" by - calculating then value of the constant using the current default - precision of the type. Note that this can make use of the constants - rather expensive. -
      • -
    -
    • -
-

- In addition, it is possible to pass a Policy - type as a second template argument, and use this to control the precision: -

-
#include <boost/math/constants/constants.hpp>
-
-typedef boost::math::policies::policy<boost::math::policies::digits2<80> > my_policy_type;
-boost::math::constants::pi<MyType, my_policy_type>();
-
-
- - - - - -
[Note]Note

- Boost.Math doesn't know how to control the internal precision of MyType, the policy just controls how - the selection process above is carried out, and the calculation precision - if the result is computed. -

-

- It is also possible to control which method is used to construct the constant - by specialising the traits class construction_traits: -

-
namespace boost{ namespace math{ namespace constant{
-
-template <class T, class Policy>
-struct construction_traits
-{
-   typedef mpl::int_<N> type;
-};
-
-}}} // namespaces
-
-

- Where N takes one of the following values: -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- N -

- 		-

- Meaning -

-
-

- 0 -

- 		-

- The precision is unavailable at compile time; either construct - from a decimal digit string or calculate on the fly depending - upon the runtime precision. -

-
-

- 1 -

- 		-

- Return a float precision constant. -

-
-

- 2 -

- 		-

- Return a double precision constant. -

-
-

- 3 -

- 		-

- Return a long double precision constant. -

-
-

- 4 -

- 		-

- Construct the result from the string representation, and cache - the result. -

-
-

- Any other value N -

- 		-

- Sets the compile time precision to N bits. -

-
-
- - Custom - Specializing a constant -
-

- In addition, for user-defined types that need special handling, it's possible - to [partially-] specialize the internal structure used by each constant. - For example, suppose we're using the C++ wrapper around MPFR mpfr_class: this has its own representation - of Pi which we may well wish to use in place of the above mechanism. We - can achieve this by specialising the class template boost::math::constants::detail::constant_pi: -

-
namespace boost{ namespace math{ namespace constants{ namespace detail{
-
-template<>
-struct constant_pi<mpfr_class>
-{
-   template<int N>
-   static mpfr_class get(const mpl::int_<N>&)
-   {
-      // The template param N is one of the values in the table above,
-      // we can either handle all cases in one as is the case here,
-      // or overload "get" for the different options.
-      mpfr_class result;
-      mpfr_const_pi(result.get_mpfr_t(), GMP_RNDN);
-      return result;
-   }
-};
-
-}}}} // namespaces
-
-
- - Diagnosing - what meta-programmed code is doing -
-

- Finally, since it can be tricky to diagnose what meta-programmed code is - doing, there is a diagnostic routine that prints information about how - this library will handle a specific type, it can be used like this: -

-
#include <boost/math/constants/info.hpp>
-
-int main()
-{
-   boost::math::constants::print_info_on_type<MyType>();
-}
-
-

- If you wish, you can also pass an optional std::ostream argument to the - print_info_on_type function. - Typical output for a user-defined type looks like this: -

-
Information on the Implementation and Handling of
-Mathematical Constants for Type class boost::math::concepts::real_concept
-
-Checking for std::numeric_limits<class boost::math::concepts::real_concept> specialisation: no
-boost::math::policies::precision<class boost::math::concepts::real_concept, Policy>
-reports that there is no compile type precision available.
-boost::math::tools::digits<class boost::math::concepts::real_concept>()
-reports that the current runtime precision is
-53 binary digits.
-No compile time precision is available, the construction method
-will be decided at runtime and results will not be cached
-- this may lead to poor runtime performance.
-Current runtime precision indicates that
-the constant will be constructed from a string on each call.
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist.html b/doc/html/math_toolkit/dist.html deleted file mode 100644 index dcbd4fb3c..000000000 --- a/doc/html/math_toolkit/dist.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Statistical Distributions and Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Statistical Distributions - Tutorial
-
Statistical Distributions - Reference
-
Extras/Future Directions
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref.html b/doc/html/math_toolkit/dist/dist_ref.html deleted file mode 100644 index ff56566e0..000000000 --- a/doc/html/math_toolkit/dist/dist_ref.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Statistical Distributions Reference - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Non-Member Properties
-
Distributions
-
Distribution - Algorithms
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dist_algorithms.html b/doc/html/math_toolkit/dist/dist_ref/dist_algorithms.html deleted file mode 100644 index eea839b69..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dist_algorithms.html +++ /dev/null @@ -1,135 +0,0 @@ - - - -Distribution Algorithms - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists.html b/doc/html/math_toolkit/dist/dist_ref/dists.html deleted file mode 100644 index 0c0e6f5e0..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists.html +++ /dev/null @@ -1,98 +0,0 @@ - - - -Distributions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Bernoulli - Distribution
-
Beta - Distribution
-
Binomial - Distribution
-
Cauchy-Lorentz - Distribution
-
Chi - Squared Distribution
-
Exponential - Distribution
-
Extreme - Value Distribution
-
F Distribution
-
Gamma - (and Erlang) Distribution
-
Geometric - Distribution
-
Hypergeometric - Distribution
-
Inverse - Chi Squared Distribution
-
Inverse - Gamma Distribution
-
Inverse - Gaussian (or Inverse Normal) Distribution
-
Laplace - Distribution
-
Logistic - Distribution
-
Log - Normal Distribution
-
Negative - Binomial Distribution
-
Noncentral - Beta Distribution
-
Noncentral - Chi-Squared Distribution
-
Noncentral - F Distribution
-
Noncentral - T Distribution
-
Normal - (Gaussian) Distribution
-
Pareto Distribution
-
Poisson - Distribution
-
Rayleigh - Distribution
-
Skew - Normal Distribution
-
Students - t Distribution
-
Triangular - Distribution
-
Uniform - Distribution
-
Weibull - Distribution
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html deleted file mode 100644 index 7d93f0cf4..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/bernoulli_dist.html +++ /dev/null @@ -1,350 +0,0 @@ - - - -Bernoulli Distribution - - - - - - - - -
-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: -

-

- -

-

- and the Cumulative - distribution function -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/beta_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/beta_dist.html deleted file mode 100644 index aaac26954..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/beta_dist.html +++ /dev/null @@ -1,618 +0,0 @@ - - - -Beta Distribution - - - - - - - - -
-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. -

-

- -

-

- If α = β = 1, then it is a __space uniform - distribution, equal to unity in the entire interval x = 0 to - 1. If α __space and β __space 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 __space or β __space from presumed-known mean and - variance. -

-

- The other pair estimates either α __space or β __space from the cdf and x. -

-

- It is also possible to estimate α __space and β __space 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 __space 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 __space and β __space are - integers. -

-

- With integer values of α __space and β __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 - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html deleted file mode 100644 index d7982fef6..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html +++ /dev/null @@ -1,911 +0,0 @@ - - - -Binomial Distribution - - - - - - - - -
-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 clopper_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: -

-

- -

-

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

-

- -

-
- - - - - -
[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 13. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html deleted file mode 100644 index d2e37caa4..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html +++ /dev/null @@ -1,300 +0,0 @@ - - - -Cauchy-Lorentz Distribution - - - - - - - - -
-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: -

-

- -

-

- While the following graph shows how the shape (scale) parameter alters - the distribution: -

-

- -

-
- - 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 its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html deleted file mode 100644 index 9037fd984..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html +++ /dev/null @@ -1,406 +0,0 @@ - - - -Chi Squared Distribution - - - - - - - - -
-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: -

-

- -

-
- - 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. -

-

- (We have followed the usual restriction of the mode to degrees of freedom - >= 2, but note that the maximum of the pdf is actually zero for degrees - of freedom from 2 down to 0, and provide an extended definition that - would avoid a discontinuity in the mode as alternative code in a comment). -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/exp_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/exp_dist.html deleted file mode 100644 index 099fedd47..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/exp_dist.html +++ /dev/null @@ -1,324 +0,0 @@ - - - -Exponential Distribution - - - - - - - - -
-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: -

-

- -

-
- - 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 -
-
-

- (See also the reference documentation for the related Extreme - Distributions.) -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html deleted file mode 100644 index 2507825e5..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html +++ /dev/null @@ -1,326 +0,0 @@ - - - -Extreme Value Distribution - - - - - - - - -
-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 relationship of the types of extreme value distributions, of which - this is but one, is discussed by Extreme - Value Distributions, Theory and Applications Samuel Kotz & Saralees - Nadarajah. -

-

- 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: -

-

- -

-

- And this graph illustrates how the PDF varies with the shape parameter: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/f_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/f_dist.html deleted file mode 100644 index 3ce0cfd54..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/f_dist.html +++ /dev/null @@ -1,433 +0,0 @@ - - - -F Distribution - - - - - - - - -
-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. -

-

- -

-
- - 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 its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html deleted file mode 100644 index 6d6ca922c..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html +++ /dev/null @@ -1,367 +0,0 @@ - - - -Gamma (and Erlang) Distribution - - - - - - - - -
-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_distribution<double> gamma;
-

-

-

- Instead if you want a double precision gamma distribution you can write -

-

-

-
boost::math::gamma_distribution<> my_gamma(1, 1);
-

-

-
-

- 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: -

-

- -

-

- -

-

- 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 its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html deleted file mode 100644 index 1e1d7f73d..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html +++ /dev/null @@ -1,855 +0,0 @@ - - - -Geometric Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/geometric.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class geometric_distribution;
-
-typedef geometric_distribution<> geometric;
-
-template <class RealType, class Policy>
-class geometric_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-   // Constructor from success_fraction:
-   geometric_distribution(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 geometric_distribution - represents a geometric - distribution: it is used when there are exactly two mutually - exclusive outcomes of a Bernoulli - trial: these outcomes are labelled "success" and "failure". -

-

- For Bernoulli - trials each with success fraction p, the - geometric distribution gives the probability of observing k - trials (failures, events, occurrences, or arrivals) before the first - success. -

-
- - - - - -
[Note]Note

- For this implementation, the set of trials includes - zero (unlike another definition where the set of trials - starts at one, sometimes named shifted). -

-

- The geometric distribution assumes that success_fraction p - is fixed for all k trials. -

-

- The probability that there are k failures before - the first success is -

-

-    Pr(Y=k) = (1-p)kp -

-

- For example, when throwing a 6-face dice the success probability p - = 1/6 = 0.1666 ̇  . Throwing repeatedly until a three - appears, the probability distribution of the number of times not-a-three - is thrown is geometric. -

-

- Geometric distribution has the Probability Density Function PDF: -

-

-    (1-p)kp -

-

- The following graph illustrates how the PDF and CDF vary for three examples - of the success fraction p, (when considering the - geometric distribution as a continuous function), -

-

- -

-

- -

-

- and as discrete. -

-

- -

-

- -

-
- - Related - Distributions -
-

- The geometric distribution is a special case of the Negative - Binomial Distribution with successes parameter r - = 1, so only one first and only success is required : thus by definition -    geometric(p) == negative_binomial(1, - p) -

-
negative_binomial_distribution(RealType r, RealType success_fraction);
-negative_binomial nb(1, success_fraction);
-geometric g(success_fraction);
-ASSERT(pdf(nb, 1) == pdf(g, 1));
-
-

- This implementation uses real numbers for the computation throughout - (because it uses the real-valued power - and exponential functions). So to obtain a conventional strictly-discrete - geometric distribution you must ensure that an integer value is provided - for the number of trials (random variable) k, and - take integer values (floor or ceil functions) from functions that return - a number of successes. -

-
- - - - - -
[Caution]Caution
-

- The geometric 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 geometric distribution. The reference - docs describe how to change the rounding policy for these distributions. -

-
-
- - Member - Functions -
-
- - Constructor -
-
geometric_distribution(RealType p);
-
-

- Constructor: p or success_fraction is the probability - of success of a single trial. -

-

- Requires: 0 <= - p <= - 1. -

-
- - Accessors -
-
RealType success_fraction() const; // successes / trials (0 <= p <= 1)
-
-

- Returns the success_fraction parameter p from which - this distribution was constructed. -

-
RealType successes() const; // required successes always one,
-// included for compatibility with negative binomial distribution
-// with successes r == 1.
-
-

- Returns unity. -

-

- The following functions are equivalent to those provided for the negative - binomial, with successes = 1, but are provided here for completeness. -

-

- The best method of calculation for the following functions is disputed: - see Binomial - Distribution and Negative - Binomial Distribution for more discussion. -

-
- - Lower - Bound on success_fraction Parameter p -
-
static RealType find_lower_bound_on_p(
-  RealType failures,
-  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 1st success. -

-
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 from n - trials the best estimate for the success fraction is simply 1/n, - but if you want to be 95% sure that the true value is greater - than some value, pmin, then: -

-
pmin = geometric_distribution<RealType>::
-   find_lower_bound_on_p(failures, 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 success_fraction Parameter p -
-
static RealType find_upper_bound_on_p(
-   RealType trials,
-   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. -

-
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 = geometric_distribution<RealType>::find_upper_bound_on_p(
-                    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: -

-
geometric_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
-
-

- Returns the smallest number of trials we must conduct to be 95% (1-0.05) - sure of seeing 10 failures that occur with frequency one half. -

-

- Worked - Example. -

-

- This function uses numeric inversion of the geometric distribution to - obtain the result: another interpretation of the result is that it finds - the number of trials (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: -

-
geometric_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 geometric distribution to - obtain the result: another interpretation of the result, is that it finds - the number of trials 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 14. Meaning of the non-member accessors.

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Meaning -

-
-

- Probability Density Function -

- 		-

- The probability of obtaining exactly - k failures from k trials with - success fraction p. For example: -

-

-

-
pdf(geometric(p), k)
-

-

-
-

- Cumulative Distribution Function -

- 		-

- The probability of obtaining k failures - or fewer from k trials with - success fraction p and success on the last trial. For example: -

-

-

-
cdf(geometric(p), k)
-

-

-
-

- Complement of the Cumulative - Distribution Function -

- 		-

- The probability of obtaining more than - k failures from k trials with - success fraction p and success on the last trial. For example: -

-

-

-
cdf(complement(geometric(p), k))
-

-

-
-

- Quantile -

- 		-

- The greatest number of failures - k expected to be observed from k - 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(geometric(p), P)
-

-

-
-

- Quantile from the complement - of the probability -

- 		-

- The smallest number of failures - k expected to be observed from k - 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(geometric(p), P))
-

-

-
-
-
- - Accuracy -
-

- This distribution is implemented using the pow and exp functions, so - most results are accurate within a few epsilon for the RealType. For - extreme values of double - p, for example 0.9999999999, accuracy can fall significantly, - for example to 10 decimal digits (from 16). -

-
- - Implementation -
-

- In the following table, p is the probability that - any one trial will be successful (the success fraction), k - is the number of failures, p is the probability - and q = 1-p, x is the given - probability to estimate the expected number of failures using the quantile. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- pdf = p * pow(q, k) -

-
-

- cdf -

- 		-

- cdf = 1 - qk=1 -

-
-

- cdf complement -

- 		-

- exp(log1p(-p) * (k+1)) -

-
-

- quantile -

- 		-

- k = log1p(-x) / log1p(-p) -1 -

-
-

- quantile from the complement -

- 		-

- k = log(x) / log1p(-p) -1 -

-
-

- mean -

- 		-

- (1-p)/p -

-
-

- variance -

- 		-

- (1-p)/p² -

-
-

- mode -

- 		-

- 0 -

-
-

- skewness -

- 		-

- (2-p)/√q -

-
-

- kurtosis -

- 		-

- 9+p²/q -

-
-

- kurtosis excess -

- 		-

- 6 +p²/q -

-
-

- parameter estimation member functions -

- 		-

- See Negative - Binomial Distribution -

-
-

- find_lower_bound_on_p -

- 		-

- See Negative - Binomial Distribution -

-
-

- find_upper_bound_on_p -

- 		-

- See Negative - Binomial Distribution -

-
-

- find_minimum_number_of_trials -

- 		-

- See Negative - Binomial Distribution -

-
-

- find_maximum_number_of_trials -

- 		-

- See Negative - Binomial Distribution -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html deleted file mode 100644 index 34e87ffed..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html +++ /dev/null @@ -1,334 +0,0 @@ - - - -Hypergeometric Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/hypergeometric.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class hypergeometric_distribution;
-
-template <class RealType, class Policy>
-class hypergeometric_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-   // Construct:
-   hypergeometric_distribution(unsigned r, unsigned n, unsigned N);
-   // Accessors:
-   unsigned total()const;
-   unsigned defective()const;
-   unsigned sample_count()const;
-};
-
-typedef hypergeometric_distribution<> hypergeometric;
-
-}} // namespaces
-
-

- The hypergeometric distribution describes the number of "events" - k from a sample n drawn from - a total population N without replacement. -

-

- Imagine we have a sample of N objects of which - r are "defective" and N-r are "not - defective" (the terms "success/failure" or "red/blue" - are also used). If we sample n items without - replacement then what is the probability that exactly k - items in the sample are defective? The answer is given by the pdf of - the hypergeometric distribution f(k; r, n, N), whilst the probability of k - defectives or fewer is given by F(k; r, n, N), where F(k) is the CDF - of the hypergeometric distribution. -

-
- - - - - -
[Note]Note

- Unlike almost all of the other distributions in this library, the hypergeometric - distribution is strictly discrete: it can not be extended to real valued - arguments of its parameters or random variable. -

-

- The following graph shows how the distribution changes as the proportion - of "defective" items changes, while keeping the population - and sample sizes constant: -

-

- -

-

- Note that since the distribution is symmetrical in parameters n - and r, if we change the sample size and keep the - population and proportion "defective" the same then we obtain - basically the same graphs: -

-

- -

-
- - Member - Functions -
-
hypergeometric_distribution(unsigned r, unsigned n, unsigned N);
-
-

- Constructs a hypergeometric distribution with with a population of N - objects, of which r are defective, and from which - n are sampled. -

-
unsigned total()const;
-
-

- Returns the total number of objects N. -

-
unsigned defective()const;
-
-

- Returns the number of objects r in population N - which are defective. -

-
unsigned sample_count()const;
-
-

- Returns the number of objects n which are sampled - from the population N. -

-
- - 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 the unsigned integers in the range - [max(0, n + r - N), min(n, r)]. A domain_error - is raised if the random variable is outside this range, or is not an - integral value. -

-
- - - - - -
[Caution]Caution
-

- 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 using Policies. - It is strongly recommended that you read the tutorial Understanding - Quantiles of Discrete Distributions before using the quantile - function on the Hypergeometric distribution. The reference - docs describe how to change the rounding policy for these distributions. -

-

- However, note that the implementation method of the quantile function - always returns an integral value, therefore attempting to use a Policy that requires (or produces) - a real valued result will result in a compile time error. -

-
-
- - Accuracy -
-

- For small N such that N < boost::math::max_factorial<RealType>::value - then table based lookup of the results gives an accuracy to a few epsilon. - boost::math::max_factorial<RealType>::value is 170 at double or long double - precision. -

-

- For larger N such that N < boost::math::prime(boost::math::max_prime) then only basic arithmetic is required - for the calculation and the accuracy is typically < 20 epsilon. This - takes care of N up to 104729. -

-

- For N > - boost::math::prime(boost::math::max_prime) - then accuracy quickly degrades, with 5 or 6 decimal digits being lost - for N = 110000. -

-

- In general for very large N, the user should expect to loose log10N decimal - digits of precision during the calculation, with the results becoming - meaningless for N >= 1015. -

-
- - Testing -
-

- There are three sets of tests: our implementation is tested against a - table of values produced by Mathematica's implementation of this distribution. - We also sanity check our implementation against some spot values computed - using the online calculator here http://stattrek.com/Tables/Hypergeometric.aspx. - Finally we test accuracy against some high precision test data using - this implementation and NTL::RR. -

-
- - Implementation -
-

- The PDF can be calculated directly using the formula: -

-

- -

-

- However, this can only be used directly when the largest of the factorials - is guaranteed not to overflow the floating point representation used. - This formula is used directly when N - < max_factorial<RealType>::value - in which case table lookup of the factorials gives a rapid and accurate - implementation method. -

-

- For larger N the method described in "An Accurate - Computation of the Hypergeometric Distribution Function", Trong - Wu, ACM Transactions on Mathematical Software, Vol. 19, No. 1, March - 1993, Pages 33-43 is used. The method relies on the fact that there is - an easy method for factorising a factorial into the product of prime - numbers: -

-

- -

-

- Where pi is the i'th prime number, and ei is a small positive integer or - zero, which can be calculated via: -

-

- -

-

- Further we can combine the factorials in the expression for the PDF to - yield the PDF directly as the product of prime numbers: -

-

- -

-

- With this time the exponents ei being either positive, negative or zero. - Indeed such a degree of cancellation occurs in the calculation of the - ei that many are zero, and typically most have a magnitude or no more - than 1 or 2. -

-

- Calculation of the product of the primes requires some care to prevent - numerical overflow, we use a novel recursive method which splits the - calculation into a series of sub-products, with a new sub-product started - each time the next multiplication would cause either overflow or underflow. - The sub-products are stored in a linked list on the program stack, and - combined in an order that will guarantee no overflow or unnecessary-underflow - once the last sub-product has been calculated. -

-

- This method can be used as long as N is smaller than the largest prime - number we have stored in our table of primes (currently 104729). The - method is relatively slow (calculating the exponents requires the most - time), but requires only a small number of arithmetic operations to calculate - the result (indeed there is no shorter method involving only basic arithmetic - once the exponents have been found), the method is therefore much more - accurate than the alternatives. -

-

- For much larger N, we can calculate the PDF from the factorials using - either lgamma, or by directly combining lanczos approximations to avoid - calculating via logarithms. We use the latter method, as it is usually - 1 or 2 decimal digits more accurate than computing via logarithms with - lgamma. However, in this area where N > 104729, the user should expect - to loose around log10N decimal digits during the calculation in the worst - case. -

-

- The CDF and its complement is calculated by directly summing the PDF's. - We start by deciding whether the CDF, or its complement, is likely to - be the smaller of the two and then calculate the PDF at k - (or k+1 if we're calculating the complement) and - calculate successive PDF values via the recurrence relations: -

-

- -

-

- Until we either reach the end of the distributions domain, or the next - PDF value to be summed would be too small to affect the result. -

-

- The quantile is calculated in a similar manner to the CDF: we first guess - which end of the distribution we're nearer to, and then sum PDFs starting - from the end of the distribution this time, until we have some value - k that gives the required CDF. -

-

- The median is simply the quantile at 0.5, and the remaining properties - are calculated via: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html deleted file mode 100644 index 6dbeb02d6..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html +++ /dev/null @@ -1,473 +0,0 @@ - - - -Inverse Chi Squared Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/inverse_chi_squared.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class inverse_chi_squared_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-
-   inverse_chi_squared_distribution(RealType df = 1); // Not explicitly scaled, default 1/df.
-   inverse_chi_squared_distribution(RealType df, RealType scale = 1/df);  // Scaled.
-
-   RealType degrees_of_freedom()const; // Default 1.
-   RealType scale()const; // Optional scale [xi] (variance), default 1/degrees_of_freedom.
-};
-
-}} // namespace boost // namespace math
-
-

- The inverse chi squared distribution is a continuous probability distribution - of the reciprocal of a variable distributed - according to the chi squared distribution. -

-

- The sources below give confusingly different formulae using different - symbols for the distribution pdf, but they are all the same, or related - by a change of variable, or choice of scale. -

-

- Two constructors are available to implement both the scaled and (implicitly) - unscaled versions. -

-

- The main version has an explicit scale parameter which implements the - scaled - inverse chi_squared distribution. -

-

- A second version has an implicit scale = 1/degrees of freedom and gives - the 1st definition in the Wikipedia - inverse chi_squared distribution. The 2nd Wikipedia inverse chi_squared - distribution definition can be implemented by explicitly specifying a - scale = 1. -

-

- Both definitions are also available in Wolfram Mathematica and in The R Project for Statistical Computing - (geoR) with default scale = 1/degrees of freedom. -

-

- See -

-
-

- The inverse_chi_squared distribution is used in Bayesian - statistics: the scaled inverse chi-square is conjugate prior - for the normal distribution with known mean, model parameter σ² (variance). -

-

- See conjugate - priors including a table of distributions and their priors. -

-

- See also Inverse - Gamma Distribution and Chi - Squared Distribution. -

-

- The inverse_chi_squared distribution is a special case of a inverse_gamma - distribution with ν (degrees_of_freedom) shape (α) and scale (β) where -

-

-    α= ν /2 and β = ½. -

-
- - - - - -
[Note]Note
-

- This distribution does provide the - typedef: -

-

-

-
typedef inverse_chi_squared_distribution<double> inverse_chi_squared;
-

-

-

- If you want a double precision - inverse_chi_squared distribution you can use -

-

-

-
boost::math::inverse_chi_squared_distribution<>
-

-

-

- or you can write inverse_chi_squared - my_invchisqr(2, 3); -

-
-

- For degrees of freedom parameter ν, the (unscaled) - inverse chi_squared distribution is defined by the probability density - function (PDF): -

-

-    f(x;ν) = 2-ν/2 x-ν/2-1 e-1/2x / Γ(ν/2) -

-

- and Cumulative Density Function (CDF) -

-

-    F(x;ν) = Γ(ν/2, 1/2x) / Γ(ν/2) -

-

- For degrees of freedom parameter ν and scale parameter ξ, the scaled inverse chi_squared distribution is defined - by the probability density function (PDF): -

-

-    f(x;ν, ξ) = (ξν/2)ν/2 e-νξ/2x x-1-ν/2 / Γ(ν/2) -

-

- and Cumulative Density Function (CDF) -

-

-    F(x;ν, ξ) = Γ(ν/2, νξ/2x) / Γ(ν/2) -

-

- The following graphs illustrate how the PDF and CDF of the inverse chi_squared - distribution varies for a few values of parameters ν and ξ: -

-

- -

-

- -

-
- - Member - Functions -
-
inverse_chi_squared_distribution(RealType df = 1); // Implicitly scaled 1/df.
-inverse_chi_squared_distribution(RealType df = 1, RealType scale); // Explicitly scaled.
-
-

- Constructs an inverse chi_squared distribution with ν degrees of freedom - df, and scale scale with default - value 1/df. -

-

- Requires that the degrees of freedom ν parameter is greater than zero, - otherwise calls domain_error. -

-
RealType degrees_of_freedom()const;
-
-

- Returns the degrees_of_freedom ν parameter of this distribution. -

-
RealType scale()const;
-
-

- Returns the scale ξ 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 variate is [0,+∞]. -

-
- - - - - -
[Note]Note

- Unlike some definitions, this implementation supports a random variate - equal to zero as a special case, returning zero for both pdf and cdf. -

-
- - Accuracy -
-

- The inverse gamma distribution is implemented in terms of the incomplete - gamma functions like the Inverse - Gamma Distribution that use 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. - But in general, gamma (and thus inverse gamma) results are often accurate - to a few epsilon, >14 decimal digits accuracy for 64-bit double. unless - iteration is involved, as for the estimation of degrees of freedom. -

-
- - Implementation -
-

- In the following table ν is the degrees of freedom parameter and ξ is the - scale parameter of the distribution, x is the random - variate, p is the probability and q = - 1-p its complement. Parameters α for shape and β for scale are - used for the inverse gamma function: α = ν/2 and β = ν * ξ/2. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- Using the relation: pdf = gamma_p_derivative(α, - β/ x, β) / x * x -

-
-

- cdf -

- 		-

- Using the relation: p = gamma_q(α, - β / x) -

-
-

- cdf complement -

- 		-

- Using the relation: q = gamma_p(α, - β / x) -

-
-

- quantile -

- 		-

- Using the relation: x = β  / gamma_q_inv(α, - p) -

-
-

- quantile from the complement -

- 		-

- Using the relation: x = α  / gamma_p_inv(α, - q) -

-
-

- mode -

- 		-

- ν * ξ / (ν + 2) -

-
-

- median -

- 		-

- no closed form analytic equation is known, but is evaluated - as quantile(0.5) -

-
-

- mean -

- 		-

- νξ / (ν - 2) for ν > 2, else a domain_error -

-
-

- variance -

- 		-

- 2 ν² ξ² / ((ν -2)² (ν -4)) for ν >4, else a domain_error -

-
-

- skewness -

- 		-

- 4 √2 √(ν-4) /(ν-6) for ν >6, else a domain_error -

-
-

- kurtosis_excess -

- 		-

- 12 * (5ν - 22) / ((ν - 6) * (ν - 8)) for ν >8, else a domain_error -

-
-

- kurtosis -

- 		-

- 3 + 12 * (5ν - 22) / ((ν - 6) * (ν-8)) for ν >8, else a domain_error -

-
-
- - References -
-
    -
  1. - Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern, - Donald B. Rubin, ISBN-13: 978-1584883883, Chapman & Hall; 2 edition - (29 July 2003). -
    2. -
  2. - Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, - Springer; 2nd ed. edition (10 Jun 2009) -
    3. -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html deleted file mode 100644 index 9d5f385b6..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html +++ /dev/null @@ -1,370 +0,0 @@ - - - -Inverse Gamma Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/inverse_gamma.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class inverse_gamma_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-
-   inverse_gamma_distribution(RealType shape, RealType scale = 1)
-
-   RealType shape()const;
-   RealType scale()const;
-};
-
-}} // namespaces
-
-

- The inverse_gamma distribution is a continuous probability distribution - of the reciprocal of a variable distributed according to the gamma distribution. -

-

- The inverse_gamma distribution is used in Bayesian statistics. -

-

- See inverse - gamma distribution. -

-

- R - inverse gamma distribution functions. -

-

- Wolfram - inverse gamma distribution. -

-

- See also Gamma - Distribution. -

-
- - - - - -
[Note]Note
-

- In spite of potential confusion with the inverse gamma function, this - distribution does provide the typedef: -

-

-

-
typedef inverse_gamma_distribution<double> gamma;
-

-

-

- If you want a double precision - gamma distribution you can use -

-

-

-
boost::math::inverse_gamma_distribution<>
-

-

-

- or you can write inverse_gamma - my_ig(2, 3); -

-
-

- For shape parameter α and scale parameter β, it is defined by the probability - density function (PDF): -

-

-    f(x;α, β) = βα * (1/x) α+1 exp(-β/x) / Γ(α) -

-

- and cumulative density function (CDF) -

-

-    F(x;α, β) = Γ(α, β/x) / Γ(α) -

-

- The following graphs illustrate how the PDF and CDF of the inverse gamma - distribution varies as the parameters vary: -

-

- -

-

- -

-
- - Member - Functions -
-
inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);
-
-

- Constructs an inverse gamma distribution with shape α and scale β. -

-

- Requires that the shape and scale parameters are greater than zero, otherwise - calls domain_error. -

-
RealType shape()const;
-
-

- Returns the α shape parameter of this inverse gamma distribution. -

-
RealType scale()const;
-
-

- Returns the β scale parameter of this inverse gamma 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 variate is [0,+∞]. -

-
- - - - - -
[Note]Note

- Unlike some definitions, this implementation supports a random variate - equal to zero as a special case, returning zero for pdf and cdf. -

-
- - Accuracy -
-

- The inverse gamma 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. - But in general, inverse_gamma results are accurate to a few epsilon, - >14 decimal digits accuracy for 64-bit double. -

-
- - Implementation -
-

- In the following table α is the shape parameter of the distribution, α   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 = gamma_p_derivative(α, - β/ x, β) / x * x -

-
-

- cdf -

- 		-

- Using the relation: p = gamma_q(α, - β / x) -

-
-

- cdf complement -

- 		-

- Using the relation: q = gamma_p(α, - β / x) -

-
-

- quantile -

- 		-

- Using the relation: x = β  / gamma_q_inv(α, - p) -

-
-

- quantile from the complement -

- 		-

- Using the relation: x = α  / gamma_p_inv(α, - q) -

-
-

- mode -

- 		-

- β / (α + 1) -

-
-

- median -

- 		-

- no analytic equation is known, but is evaluated as quantile(0.5) -

-
-

- mean -

- 		-

- β / (α - 1) for α > 1, else a domain_error -

-
-

- variance -

- 		-

- (β * β) / ((α - 1) * (α - 1) * (α - 2)) for α >2, else a domain_error -

-
-

- skewness -

- 		-

- 4 * sqrt (α -2) / (α -3) for α >3, else a domain_error -

-
-

- kurtosis_excess -

- 		-

- (30 * α - 66) / ((α-3)*(α - 4)) for α >4, else a domain_error -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html deleted file mode 100644 index da486dec7..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html +++ /dev/null @@ -1,443 +0,0 @@ - - - -Inverse Gaussian (or Inverse Normal) Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/inverse_gaussian.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class inverse_gaussian_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-
-   inverse_gaussian_distribution(RealType mean = 1, RealType scale = 1);
-
-   RealType mean()const; // mean default 1.
-   RealType scale()const; // Optional scale, default 1 (unscaled).
-   RealType shape()const; // Shape = scale/mean.
-};
-typedef inverse_gaussian_distribution<double> inverse_gaussian;
-
-}} // namespace boost // namespace math
-
-

- The Inverse Gaussian distribution distribution is a continuous probability - distribution. -

-

- The distribution is also called 'normal-inverse Gaussian distribution', - and 'normal Inverse' distribution. -

-

- It is also convenient to provide unity as default for both mean and scale. - This is the Standard form for all distributions. The Inverse Gaussian - distribution was first studied in relation to Brownian motion. In 1956 - M.C.K. Tweedie used the name Inverse Gaussian because there is an inverse - relationship between the time to cover a unit distance and distance covered - in unit time. The inverse Gaussian is one of family of distributions - that have been called the Tweedie - distributions. -

-

- (So inverse in the name may mislead: it does not relate to the inverse of a distribution). -

-

- The tails of the distribution decrease more slowly than the normal distribution. - It is therefore suitable to model phenomena where numerically large values - are more probable than is the case for the normal distribution. For stock - market returns and prices, a key characteristic is that it models that - extremely large variations from typical (crashes) can occur even when - almost all (normal) variations are small. -

-

- Examples are returns from financial assets and turbulent wind speeds. -

-

- The normal-inverse Gaussian distributions form a subclass of the generalised - hyperbolic distributions. -

-

- See distribution. - Weisstein, - Eric W. "Inverse Gaussian Distribution." From MathWorld--A - Wolfram Web Resource. -

-

- If you want a double precision - inverse_gaussian distribution you can use -

-

-

-
boost::math::inverse_gaussian_distribution<>
-

-

-

- or, more conveniently, you can write -

-
using boost::math::inverse_gaussian;
-inverse_gaussian my_ig(2, 3);
-
-

- For mean parameters μ and scale (also called precision) parameter λ, and - random variate x, the inverse_gaussian distribution is defined by the - probability density function (PDF): -

-

-    f(x;μ, λ) = √(λ/2πx3) e-λ(x-μ)²/2μ²x -

-

- and Cumulative Density Function (CDF): -

-

-    F(x;μ, λ) = Φ{√(λx) (xμ-1)} + e2μ/λ Φ{-√(λ/μ) (1+x/μ)} -

-

- where Φ is the standard normal distribution CDF. -

-

- The following graphs illustrate how the PDF and CDF of the inverse_gaussian - distribution varies for a few values of parameters μ and λ: -

-

- -

-

- -

-

- Tweedie also provided 3 other parameterisations where (μ and λ) are replaced - by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian - applications. These can be found on Seshadri, page 2 and are also discussed - by Chhikara and Folks on page 105. Another related parameterisation, - the __wald_distrib (where mean μ is unity) is also provided. -

-
- - Member - Functions -
-
inverse_gaussian_distribution(RealType df = 1, RealType scale = 1); // optionally scaled.
-
-

- Constructs an inverse_gaussian distribution with μ mean, and scale λ, with - both default values 1. -

-

- Requires that both the mean μ parameter and scale λ are greater than zero, - otherwise calls domain_error. -

-
RealType mean()const;
-
-

- Returns the mean μ parameter of this distribution. -

-
RealType scale()const;
-
-

- Returns the scale λ 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 variate is [0,+∞). -

-
- - - - - -
[Note]Note

- Unlike some definitions, this implementation supports a random variate - equal to zero as a special case, returning zero for both pdf and cdf. -

-
- - Accuracy -
-

- The inverse_gaussian distribution is implemented in terms of the exponential - function and standard normal distribution N0,1 Φ : - refer to the accuracy data for those functions for more information. - But in general, gamma (and thus inverse gamma) results are often accurate - to a few epsilon, >14 decimal digits accuracy for 64-bit double. -

-
- - Implementation -
-

- In the following table μ is the mean parameter and λ is the scale parameter - of the inverse_gaussian distribution, x is the random - variate, p is the probability and q = - 1-p its complement. Parameters μ for shape and λ for scale are - used for the inverse gaussian function. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- √(λ/ 2πx3) e-λ(x - μ)²/ 2μ²x -

-
-

- cdf -

- 		-

- Φ{√(λx) (xμ-1)} + e2μ/λ Φ{-√(λ/μ) (1+x/μ)} -

-
-

- cdf complement -

- 		-

- using complement of Φ above. -

-
-

- quantile -

- 		-

- No closed form known. Estimated using a guess refined by Newton-Raphson - iteration. -

-
-

- quantile from the complement -

- 		-

- No closed form known. Estimated using a guess refined by Newton-Raphson - iteration. -

-
-

- mode -

- 		-

- μ {√(1+9μ²/4λ²)² - 3μ/2λ} -

-
-

- median -

- 		-

- No closed form analytic equation is known, but is evaluated - as quantile(0.5) -

-
-

- mean -

- 		-

- μ -

-
-

- variance -

- 		-

- μ³/λ -

-
-

- skewness -

- 		-

- 3 √ (μ/λ) -

-
-

- kurtosis_excess -

- 		-

- 15μ/λ -

-
-

- kurtosis -

- 		-

- 12μ/λ -

-
-
- - References -
-
    -
  1. - Wald, A. (1947). Sequential analysis. Wiley, NY. -
    2. -
  2. - The Inverse Gaussian distribution : theory, methodology, and applications, - Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989). -
    3. -
  3. - The Inverse Gaussian distribution : statistical theory and applications, - Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998). -
    4. -
  4. - Numpy - and Scipy Documentation. -
    5. -
  5. - R - statmod invgauss functions. -
    6. -
  6. - R - SuppDists invGauss functions. (Note that these R implementations - names differ in case). -
    7. -
  7. - StatSci.org invgauss - help. -
    8. -
  8. - invgauss - R source. -
    9. -
  9. - pwald, - qwald. -
    10. -
  10. - Brighton - Webs wald. -
    11. -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html deleted file mode 100644 index 62a9b3b03..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html +++ /dev/null @@ -1,356 +0,0 @@ - - - -Laplace Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/laplace.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class laplace_distribution;
-
-typedef laplace_distribution<> laplace;
-
-template <class RealType, class Policy>
-class laplace_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-   // Construct:
-   laplace_distribution(RealType location = 0, RealType scale = 1);
-   // Accessors:
-   RealType location()const;
-   RealType scale()const;
-};
-
-}} // namespaces
-
-

- Laplace distribution is the distribution of differences between two independent - variates with identical exponential distributions (Abramowitz and Stegun - 1972, p. 930). It is also called the double exponential distribution. -

-

- For location parameter μ   and scale parameter σ   it is defined by the probability - density function: -

-

- -

-

- The location and scale parameters are equivalent to the mean and standard - deviation of the normal or Gaussian distribution. -

-

- The following graph illustrates the effect of the parameters μ   and σ   on the - PDF. Note that the range of the random variable remains [-∞,+∞] irrespective - of the value of the location parameter: -

-

- -

-
- - Member - Functions -
-
laplace_distribution(RealType location = 0, RealType scale = 1);
-
-

- Constructs a laplace distribution with location location - and scale scale. -

-

- The location parameter is the same as the mean of the random variate. -

-

- The scale parameter is proportional to the standard deviation 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 [-∞,+∞]. -

-
- - Accuracy -
-

- The laplace distribution is implemented in terms of the standard library - log and exp functions and as such should have very small errors. -

-
- - Implementation -
-

- In the following table μ is the location parameter of the distribution, - σ is its scale parameter, x is the random variate, - p is the probability and its complement q - = 1-p. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- Using the relation: pdf = e-abs(x-μ) / σ / (2 * σ) -

-
-

- cdf -

- 		-

- Using the relations: -

-

- x < μ : p = e(x-μ)/σ / σ -

-

- x >= μ : p = 1 - e(μ-x)/σ / σ - -

-
-

- cdf complement -

- 		-

- Using the relation: -

-

- -x < μ : q = e(-x-μ)/σ / σ -

-

- -x >= μ : q = 1 - e(μ+x)/σ / σ - -

-
-

- quantile -

- 		-

- Using the relations: -

-

- p < 0.5 : x = μ + σ * log(2*p) -

-

- p >= 0.5 : x = μ - σ * log(2-2*p) -

-
-

- quantile from the complement -

- 		-

- Using the relation: -

-

- q > 0.5: x = μ + σ*log(2-2*q) -

-

- q <=0.5: x = μ - σ*log( 2*q ) -

-
-

- mean -

- 		-

- μ -

-
-

- variance -

- 		-

- 2 * σ2 -

-
-

- mode -

- 		-

- μ -

-
-

- skewness -

- 		-

- 0 -

-
-

- kurtosis -

- 		-

- 6 -

-
-

- kurtosis excess -

- 		-

- 3 -

-
-
- - References -
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html deleted file mode 100644 index c672ed9ad..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html +++ /dev/null @@ -1,292 +0,0 @@ - - - -Logistic Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/logistic.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class logistic_distribution;
-
-template <class RealType, class Policy>
-class logistic_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-   // Construct:
-   logistic_distribution(RealType location = 0, RealType scale = 1);
-   // Accessors:
-   RealType location()const; // location.
-   RealType scale()const; // scale.
-
-};
-
-typedef logistic_distribution<> logistic;
-
-}} // namespaces
-
-

- The logistic distribution is a continous probability distribution. It - has two parameters - location and scale. The cumulative distribution - function of the logistic distribution appears in logistic regression - and feedforward neural networks. Among other applications, United State - Chess Federation and FIDE use it to calculate chess ratings. -

-

- The following graph shows how the distribution changes as the parameters - change: -

-

- -

-
- - Member - Functions -
-
logistic_distribution(RealType u = 0, RealType s = 1);
-
-

- Constructs a logistic distribution with location u - and scale s. -

-

- Requires scale > - 0, otherwise a domain_error - is raised. -

-
RealType location()const;
-
-

- Returns the location of this distribution. -

-
RealType scale()const;
-
-

- Returns the scale 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 [-[max_value], +[min_value]]. However, - the pdf and cdf support inputs of +∞ and -∞ -as special cases if RealType - permits. -

-

- At p=1 and p=0, the - quantile function returns the result of +overflow_error - and -overflow_error, while the - complement quantile function returns the result of -overflow_error - and +overflow_error respectively. -

-
- - Accuracy -
-

- The logistic distribution is implemented in terms of the std::exp and the std::log - functions, so its accuracy is related to the accurate implementations - of those functions on a given platform. When calculating the quantile - with a non-zero position parameter catastrophic - cancellation errors can occur: in such cases, only a low absolute - error can be guarenteed. -

-
- - Implementation -
-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- Using the relation: pdf = e-(x-u)/s / (s*(1+e-(x-u)/s)2) -

-
-

- cdf -

- 		-

- Using the relation: p = 1/(1+e-(x-u)/s) -

-
-

- cdf complement -

- 		-

- Using the relation: q = 1/(1+e(x-u)/s) -

-
-

- quantile -

- 		-

- Using the relation: x = u - s*log(1/p-1) -

-
-

- quantile from the complement -

- 		-

- Using the relation: x = u + s*log(p/1-p) -

-
-

- mean -

- 		-

- u -

-
-

- mode -

- 		-

- The same as the mean. -

-
-

- skewness -

- 		-

- 0 -

-
-

- kurtosis excess -

- 		-

- 6/5 -

-
-

- variance -

- 		-

- (π*s)2 / 3 -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html deleted file mode 100644 index a86d04204..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html +++ /dev/null @@ -1,324 +0,0 @@ - - - -Log Normal Distribution - - - - - - - - -
-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: -

-

- -

-

- The next graph illustrates the effect of the scale parameter on the PDF: -

-

- -

-
- - 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 its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html deleted file mode 100644 index 73d00c397..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html +++ /dev/null @@ -1,408 +0,0 @@ - - - -Noncentral Beta Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/non_central_beta.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class non_central_beta_distribution;
-
-typedef non_central_beta_distribution<> non_central_beta;
-
-template <class RealType, class Policy>
-class non_central_beta_distribution
-{
-public:
-   typedef RealType  value_type;
-   typedef Policy    policy_type;
-
-   // Constructor:
-   non_central_beta_distribution(RealType alpha, RealType beta, RealType lambda);
-
-   // Accessor to shape parameters:
-   RealType alpha()const;
-   RealType beta()const;
-
-   // Accessor to non-centrality parameter lambda:
-   RealType non_centrality()const;
-};
-
-}} // namespaces
-
-

- The noncentral beta distribution is a generalization of the Beta - Distribution. -

-

- It is defined as the ratio X = χm2(λ) / (χm2(λ) + χn2) where χm2(λ) is a noncentral - χ2 -random variable with m degrees of freedom, and - χn2 -is a central χ2 random variable with n degrees of - freedom. -

-

- This gives a PDF that can be expressed as a Poisson mixture of beta distribution - PDFs: -

-

- -

-

- where P(i;λ/2) is the discrete Poisson probablity at i, - with mean λ/2, and Ix'(α, β) is the derivative of the incomplete beta function. - This leads to the usual form of the CDF as: -

-

- -

-

- The following graph illustrates how the distribution changes for different - values of λ: -

-

- -

-
- - Member - Functions -
-
non_central_beta_distribution(RealType a, RealType b, RealType lambda);
-
-

- Constructs a noncentral beta distribution with shape parameters a - and b and non-centrality parameter lambda. -

-

- Requires a > 0, b > 0 and lambda >= 0, otherwise calls domain_error. -

-
RealType alpha()const;
-
-

- Returns the parameter a from which this object was - constructed. -

-
RealType beta()const;
-
-

- Returns the parameter b from which this object was - constructed. -

-
RealType non_centrality()const;
-
-

- Returns the parameter lambda from which this object - was constructed. -

-
- - Non-member - Accessors -
-

- Most of the usual non-member - accessor functions are supported: Cumulative - Distribution Function, Probability - Density Function, Quantile, - mean, variance, - standard deviation, median, - mode, Hazard - Function, Cumulative Hazard Function, - range and support. -

-

- Mean and variance are implemented using hypergeometric pfq functions - and relations given in Wolfram - Noncentral Beta Distribution. -

-

- However, the following are not currently implemented: skewness, - kurtosis and kurtosis_excess. -

-

- The domain of the random variable is [0, 1]. -

-
- - Accuracy -
-

- The following table shows the peak errors (in units of epsilon) - found on various platforms with various floating point types. No comparison - to the R-2.5.1 Math library, - or to the FORTRAN implementations of AS226 or AS310 are given since these - appear to only guarantee absolute error: this would causes our test harness - to assign an "infinite" error to these - libraries for some of our test values when measuring relative - error. Unless otherwise specified any floating-point type - that is narrower than the one shown will have effectively - zero error. -

-
-

Table 16. Errors In CDF of the Noncentral Beta

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- α, β,λ < 200 -

- 		-

- α,β,λ > 200 -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=620 Mean=22 -

- 		-

- Peak=8670 Mean=1040 -

-
-

- 64 -

- 		-

- RedHat Linux IA32, gcc-4.1.1 -

- 		-

- Peak=825 Mean=50 -

- 		-

- Peak=2.5x104 Mean=4000 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-3.4.4 -

- 		-

- Peak=825 Mean=30 -

- 		-

- Peak=1.7x104 Mean=2500 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=420 Mean=50 -

- 		-

- Peak=9200 Mean=1200 -

-
-
-

- Error rates for the PDF, the complement of the CDF and for the quantile - functions are broadly similar. -

-
- - Tests -
-

- There are two sets of test data used to verify this implementation: firstly - we can compare with a few sample values generated by the R - library. Secondly, we have tables of test data, computed with - this implementation and using interval arithmetic - this data should - be accurate to at least 50 decimal digits - and is the used for our accuracy - tests. -

-
- - Implementation -
-

- The CDF and its complement are evaluated as follows: -

-

- First we determine which of the two values (the CDF or its complement) - is likely to be the smaller, the crossover point is taken to be the mean - of the distribution: for this we use the approximation due to: R. Chattamvelli - and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta - Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997), - pp. 146-156. -

-

- -

-

- Then either the CDF or its complement is computed using the relations: -

-

- -

-

- The summation is performed by starting at i = λ/2, and then recursing - in both directions, using the usual recurrence relations for the Poisson - PDF and incomplete beta functions. This is the "Method 2" described - by: -

-

- Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures - of continuous distributions: noncentral chisquare, noncentral t and the - distribution of the square of the sample multiple correlation coefficient", - Computational Statistics & Data Analysis 43 (2003) 249-267. -

-

- Specific applications of the above formulae to the noncentral beta distribution - can be found in: -

-

- Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities", - Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244. -

-

- H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing - Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No. - 2. (1990), pp. 311-312. -

-

- Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing - Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No. - 4. (1995), pp. 551-552. -

-

- Harry O. Posten, "An Effective Algorithm for the Noncentral Beta - Distribution Function", The American Statistician, Vol. 47, No. - 2. (May, 1993), pp. 129-131. -

-

- R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function", - The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234. -

-

- Of these, the Posten reference provides the most complete overview, and - includes the modification starting iteration at λ/2. -

-

- The main difference between this implementation and the above references - is the direct computation of the complement when most efficient to do - so, and the accumulation of the sum to -1 rather than subtracting the - result from 1 at the end: this can substantially reduce the number of - iterations required when the result is near 1. -

-

- The PDF is computed using the methodology of Benton and Krishnamoorthy - and the relation: -

-

- -

-

- Quantiles are computed using a specially modified version of bracket_and_solve_root, - starting the search for the root at the mean of the distribution. (A - Cornish-Fisher type expansion was also tried, but while this gets quite - close to the root in many cases, when it is wrong it tends to introduce - quite pathological behaviour: more investigation in this area is probably - warranted). -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html deleted file mode 100644 index 4badc3dea..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html +++ /dev/null @@ -1,506 +0,0 @@ - - - -Noncentral Chi-Squared Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/non_central_chi_squared.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class non_central_chi_squared_distribution;
-
-typedef non_central_chi_squared_distribution<> non_central_chi_squared;
-
-template <class RealType, class Policy>
-class non_central_chi_squared_distribution
-{
-public:
-   typedef RealType  value_type;
-   typedef Policy    policy_type;
-
-   // Constructor:
-   non_central_chi_squared_distribution(RealType v, RealType lambda);
-
-   // Accessor to degrees of freedom parameter v:
-   RealType degrees_of_freedom()const;
-
-   // Accessor to non centrality parameter lambda:
-   RealType non_centrality()const;
-
-   // Parameter finders:
-   static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p);
-   template <class A, class B, class C>
-   static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c);
-
-   static RealType find_non_centrality(RealType v, RealType x, RealType p);
-   template <class A, class B, class C>
-   static RealType find_non_centrality(const complemented3_type<A,B,C>& c);
-};
-
-}} // namespaces
-
-

- The noncentral chi-squared distribution is a generalization of the Chi Squared - Distribution. If Xi are ν independent, normally distributed random - variables with means μi and variances σi2, then the random variable -

-

- -

-

- is distributed according to the noncentral chi-squared distribution. -

-

- The noncentral chi-squared distribution has two parameters: ν which specifies - the number of degrees of freedom (i.e. the number of Xi), and λ which is - related to the mean of the random variables Xi by: -

-

- -

-

- (Note that some references define λ as one half of the above sum). -

-

- This leads to a PDF of: -

-

- -

-

- where f(x;k) is the central chi-squared distribution - PDF, and Iv(x) is a modified Bessel function of - the first kind. -

-

- The following graph illustrates how the distribution changes for different - values of λ: -

-

- -

-
- - Member - Functions -
-
non_central_chi_squared_distribution(RealType v, RealType lambda);
-
-

- Constructs a Chi-Squared distribution with v degrees - of freedom and non-centrality parameter lambda. -

-

- Requires v > 0 and lambda >= 0, otherwise calls domain_error. -

-
RealType degrees_of_freedom()const;
-
-

- Returns the parameter v from which this object was - constructed. -

-
RealType non_centrality()const;
-
-

- Returns the parameter lambda from which this object - was constructed. -

-
static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p);
-
-

- This function returns the number of degrees of freedom v - such that: cdf(non_central_chi_squared<RealType, - Policy>(v, lambda), - x) - == p -

-
template <class A, class B, class C>
-static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c);
-
-

- When called with argument boost::math::complement(lambda, x, q) this function returns the number of - degrees of freedom v such that: -

-

- cdf(complement(non_central_chi_squared<RealType, - Policy>(v, lambda), - x)) - == q. -

-
static RealType find_non_centrality(RealType v, RealType x, RealType p);
-
-

- This function returns the non centrality parameter lambda - such that: -

-

- cdf(non_central_chi_squared<RealType, - Policy>(v, lambda), - x) - == p -

-
template <class A, class B, class C>
-static RealType find_non_centrality(const complemented3_type<A,B,C>& c);
-
-

- When called with argument boost::math::complement(v, x, q) this function returns the non centrality - parameter lambda such that: -

-

- cdf(complement(non_central_chi_squared<RealType, - Policy>(v, lambda), - x)) - == q. -

-
- - 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 -
-

- There is a worked - example for the noncentral chi-squared distribution. -

-
- - 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 R-2.5.1 Math - library. Unless otherwise specified, any floating-point type - that is narrower than the one shown will have effectively - zero error. -

-
-

Table 17. Errors In CDF of the Noncentral Chi-Squared

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- ν,λ < 200 -

- 		-

- ν,λ > 200 -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=50 Mean=9.9 -

-

- R Peak=685 Mean=109 -

- 		-

- Peak=9780 Mean=718 -

-

- R Peak=3x108 Mean=2x107 -

-
-

- 64 -

- 		-

- RedHat Linux IA32, gcc-4.1.1 -

- 		-

- Peak=270 Mean=27 -

- 		-

- Peak=7900 Mean=900 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-3.4.4 -

- 		-

- Peak=107 Mean=17 -

- 		-

- Peak=5000 Mean=630 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=270 Mean=20 -

- 		-

- Peak=4600 Mean=560 -

-
-
-

- Error rates for the complement of the CDF and for the quantile functions - are broadly similar. Special mention should go to the mode - function: there is no closed form for this function, so it is evaluated - numerically by finding the maxima of the PDF: in principal this can not - produce an accuracy greater than the square root of the machine epsilon. -

-
- - Tests -
-

- There are two sets of test data used to verify this implementation: firstly - we can compare with published data, for example with Table 6 of "Self-Validating - Computations of Probabilities for Selected Central and Noncentral Univariate - Probability Functions", Morgan C. Wang and William J. Kennedy, Journal - of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994), - pp. 878-887. Secondly, we have tables of test data, computed with this - implementation and using interval arithmetic - this data should be accurate - to at least 50 decimal digits - and is the used for our accuracy tests. -

-
- - Implementation -
-

- The CDF and its complement are evaluated as follows: -

-

- First we determine which of the two values (the CDF or its complement) - is likely to be the smaller: for this we can use the relation due to - Temme (see "Asymptotic and Numerical Aspects of the Noncentral Chi-Square - Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No. - 5, 55-63, 1993) that: -

-

- F(ν,λ;ν+λ) ≈ 0.5 -

-

- and so compute the CDF when the random variable is less than ν+λ, and its - complement when the random variable is greater than ν+λ. If necessary the - computed result is then subtracted from 1 to give the desired result - (the CDF or its complement). -

-

- For small values of the non centrality parameter, the CDF is computed - using the method of Ding (see "Algorithm AS 275: Computing the Non-Central - #2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol. - 41, No. 2. (1992), pp. 478-482). This uses the following series representation: -

-

- -

-

- which requires just one call to gamma_p_derivative - with the subsequent terms being computed by recursion as shown above. -

-

- For larger values of the non-centrality parameter, Ding's method can - take an unreasonable number of terms before convergence is achieved. - Furthermore, the largest term is not the first term, so in extreme cases - the first term may be zero, leading to a zero result, even though the - true value may be non-zero. -

-

- Therefore, when the non-centrality parameter is greater than 200, the - method due to Krishnamoorthy (see "Computing discrete mixtures of - continuous distributions: noncentral chisquare, noncentral t and the - distribution of the square of the sample multiple correlation coefficient", - Denise Benton and K. Krishnamoorthy, Computational Statistics & Data - Analysis, 43, (2003), 249-267) is used. -

-

- This method uses the well known sum: -

-

- -

-

- Where Pa(x) is the incomplete gamma function. -

-

- The method starts at the λth term, which is where the Poisson weighting - function achieves its maximum value, although this is not necessarily - the largest overall term. Subsequent terms are calculated via the normal - recurrence relations for the incomplete gamma function, and iteration - proceeds both forwards and backwards until sufficient precision has been - achieved. It should be noted that recurrence in the forwards direction - of Pa(x) is numerically unstable. However, since we always start after - the largest term in the series, numeric instability is introduced more - slowly than the series converges. -

-

- Computation of the complement of the CDF uses an extension of Krishnamoorthy's - method, given that: -

-

- -

-

- we can again start at the λ'th term and proceed in both directions from - there until the required precision is achieved. This time it is backwards - recursion on the incomplete gamma function Qa(x) which is unstable. However, - as long as we start well before the largest term, - this is not an issue in practice. -

-

- The PDF is computed directly using the relation: -

-

- -

-

- Where f(x; v) is the PDF of the central Chi - Squared Distribution and Iv(x) is a modified - Bessel function, see cyl_bessel_i. - For small values of the non-centrality parameter the relation in terms - of cyl_bessel_i - is used. However, this method fails for large values of the non-centrality - parameter, so in that case the infinite sum is evaluated using the method - of Benton and Krishnamoorthy, and the usual recurrence relations for - successive terms. -

-

- The quantile functions are computed by numeric inversion of the CDF. -

-

- There is no closed - form for the mode of the noncentral chi-squared distribution: - it is computed numerically by finding the maximum of the PDF. Likewise, - the median is computed numerically via the quantile. -

-

- The remaining non-member functions use the following formulas: -

-

- -

-

- Some analytic properties of noncentral distributions (particularly unimodality, - and monotonicity of their modes) are surveyed and summarized by: -

-

- Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, - 141 (2003) 3-12. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html deleted file mode 100644 index b265d4b85..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html +++ /dev/null @@ -1,408 +0,0 @@ - - - -Noncentral F Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/non_central_f.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class non_central_f_distribution;
-
-typedef non_central_f_distribution<> non_central_f;
-
-template <class RealType, class Policy>
-class non_central_f_distribution
-{
-public:
-   typedef RealType  value_type;
-   typedef Policy    policy_type;
-
-   // Constructor:
-   non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
-
-   // Accessor to degrees_of_freedom parameters v1 & v2:
-   RealType degrees_of_freedom1()const;
-   RealType degrees_of_freedom2()const;
-
-   // Accessor to non-centrality parameter lambda:
-   RealType non_centrality()const;
-};
-
-}} // namespaces
-
-

- The noncentral F distribution is a generalization of the Fisher - F Distribution. It is defined as the ratio -

-
F = (X/v1) / (Y/v2)
-
-

- where X is a noncentral χ2 -random variable with v1 - degrees of freedom and non-centrality parameter λ, and Y is a central - χ2 random variable with v2 degrees of freedom. -

-

- This gives the following PDF: -

-

- -

-

- where Lab(c) is a generalised Laguerre polynomial and B(a,b) is the - beta - function, or -

-

- -

-

- The following graph illustrates how the distribution changes for different - values of λ: -

-

- -

-
- - Member - Functions -
-
non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
-
-

- Constructs a non-central beta distribution with parameters v1 - and v2 and non-centrality parameter lambda. -

-

- Requires v1 > 0, v2 > 0 and lambda >= 0, otherwise calls domain_error. -

-
RealType degrees_of_freedom1()const;
-
-

- Returns the parameter v1 from which this object - was constructed. -

-
RealType degrees_of_freedom2()const;
-
-

- Returns the parameter v2 from which this object - was constructed. -

-
RealType non_centrality()const;
-
-

- Returns the non-centrality parameter lambda from - which this object was constructed. -

-
- - 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 -
-

- This distribution is implemented in terms of the Noncentral - Beta Distribution: refer to that distribution for accuracy data. -

-
- - Tests -
-

- Since this distribution is implemented by adapting another distribution, - the tests consist of basic sanity checks computed by the R-2.5.1 - Math library statistical package and its pbeta and dbeta functions. -

-
- - Implementation -
-

- In the following table v1 and v2 - are the first and second degrees of freedom parameters of the distribution, - λ -is the non-centrality parameter, x is the random - variate, p is the probability, and q = - 1-p. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- Implemented in terms of the non-central beta PDF using the - relation: -

-

- f(x;v1,v2;λ) = (v1/v2) / ((1+y)*(1+y)) * g(y/(1+y);v1/2,v2/2;λ) -

-

- where g(x; a, b; λ) is the non central beta PDF, and: -

-

- y = x * v1 / v2 -

-
-

- cdf -

- 		-

- Using the relation: -

-

- p = By(v1/2, v2/2; λ) -

-

- where Bx(a, b; λ) is the noncentral beta distribution CDF and -

-

- y = x * v1 / v2 -

-
-

- cdf complement -

- 		-

- Using the relation: -

-

- q = 1 - By(v1/2, v2/2; λ) -

-

- where 1 - Bx(a, b; λ) is the complement of the noncentral beta - distribution CDF and -

-

- y = x * v1 / v2 -

-
-

- quantile -

- 		-

- Using the relation: -

-

- x = (bx / (1-bx)) * (v1 / v2) -

-

- where -

-

- bx = Qp-1(v1/2, v2/2; λ) -

-

- and -

-

- Qp-1(v1/2, v2/2; λ) -

-

- is the noncentral beta quantile. -

-
-

- quantile -

-

- from the complement -

- 		-

- Using the relation: -

-

- x = (bx / (1-bx)) * (v1 / v2) -

-

- where -

-

- bx = QCq-1(v1/2, v2/2; λ) -

-

- and -

-

- QCq-1(v1/2, v2/2; λ) -

-

- is the noncentral beta quantile from the complement. -

-
-

- mean -

- 		-

- v2 * (v1 + l) / (v1 * (v2 - 2)) -

-
-

- mode -

- 		-

- By numeric maximalisation of the PDF. -

-
-

- variance -

- 		-

- Refer to, Weisstein, - Eric W. "Noncentral F-Distribution." From MathWorld--A - Wolfram Web Resource. -

-
-

- skewness -

- 		-

- Refer to, Weisstein, - Eric W. "Noncentral F-Distribution." From MathWorld--A - Wolfram Web Resource., and to the Mathematica - documentation -

-
-

- kurtosis and kurtosis excess -

- 		-

- Refer to, Weisstein, - Eric W. "Noncentral F-Distribution." From MathWorld--A - Wolfram Web Resource., and to the Mathematica - documentation -

-
-

- Some analytic properties of noncentral distributions (particularly unimodality, - and monotonicity of their modes) are surveyed and summarized by: -

-

- Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, - 141 (2003) 3-12. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html deleted file mode 100644 index 59977f89b..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html +++ /dev/null @@ -1,407 +0,0 @@ - - - -Noncentral T Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/non_central_t.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class non_central_t_distribution;
-
-typedef non_central_t_distribution<> non_central_t;
-
-template <class RealType, class Policy>
-class non_central_t_distribution
-{
-public:
-   typedef RealType  value_type;
-   typedef Policy    policy_type;
-
-   // Constructor:
-   non_central_t_distribution(RealType v, RealType delta);
-
-   // Accessor to degrees_of_freedom parameter v:
-   RealType degrees_of_freedom()const;
-
-   // Accessor to non-centrality parameter delta:
-   RealType non_centrality()const;
-};
-
-}} // namespaces
-
-

- The noncentral T distribution is a generalization of the Students - t Distribution. Let X have a normal distribution with mean δ and - variance 1, and let ν S2 have a chi-squared distribution with degrees of - freedom ν. Assume that X and S2 are independent. The distribution of tν(δ)=X/S - is called a noncentral t distribution with degrees of freedom ν and noncentrality - parameter δ. -

-

- This gives the following PDF: -

-

- -

-

- where 1F1(a;b;x) is a confluent hypergeometric function. -

-

- The following graph illustrates how the distribution changes for different - values of ν and δ: -

-

- - -

-
- - Member - Functions -
-
non_central_t_distribution(RealType v, RealType delta);
-
-

- Constructs a non-central t distribution with degrees of freedom parameter - v and non-centrality parameter delta. -

-

- Requires v > 0 (including positive infinity) - and finite delta, otherwise calls domain_error. -

-
RealType degrees_of_freedom()const;
-
-

- Returns the parameter v from which this object was - constructed. -

-
RealType non_centrality()const;
-
-

- Returns the non-centrality parameter delta from - which this object was constructed. -

-
- - 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 [-∞, +∞]. -

-
- - 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 18. Errors In CDF of the Noncentral T Distribution

-
----- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size (bits) -

- 		-

- Platform and Compiler -

- 		-

- ν, δ < 600 -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=120 Mean=26 -

-
-

- 64 -

- 		-

- RedHat Linux IA32, gcc-4.1.1 -

- 		-

- Peak=121 Mean=26 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-3.4.4 -

- 		-

- Peak=122 Mean=25 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=115 Mean=24 -

-
-
-
- - - - - -
[Caution]Caution

- The complexity of the current algorithm is dependent upon δ2: consequently - the time taken to evaluate the CDF increases rapidly for δ > 500, - likewise the accuracy decreases rapidly for very large δ. -

-

- Accuracy for the quantile and PDF functions should be broadly similar. - The mode is determined numerically and cannot in - principal be more accurate than the square root of floating-point type - FPT epsilon, accessed using boost::math::tools::epsilon<FPT>(). For 64-bit double, - epsilon is about 1e-16, so the fractional accuracy is limited to 1e-8. -

-
- - Tests -
-

- There are two sets of tests of this distribution: -

-

- Basic sanity checks compare this implementation to the test values given - in "Computing discrete mixtures of continuous distributions: noncentral - chisquare, noncentral t and the distribution of the square of the sample - multiple correlation coefficient." Denise Benton, K. Krishnamoorthy, - Computational Statistics & Data Analysis 43 (2003) 249-267. -

-

- Accuracy checks use test data computed with this implementation and arbitary - precision interval arithmetic: this test data is believed to be accurate - to at least 50 decimal places. -

-

- The cases of large (or infinite) ν and/or large δ has received special treatment - to avoid catastrophic loss of accuracy. New tests have been added to - confirm the improvement achieved. -

-

- From Boost 1.52, degrees of freedom ν can be +∞ -when the normal distribution - located at δ -(equivalent to the central Student's t distribution) is used - in place for accuracy and speed. -

-
- - Implementation -
-

- The CDF is computed using a modification of the method described in "Computing - discrete mixtures of continuous distributions: noncentral chisquare, - noncentral t and the distribution of the square of the sample multiple - correlation coefficient." Denise Benton, K. Krishnamoorthy, Computational - Statistics & Data Analysis 43 (2003) 249-267. -

-

- This uses the following formula for the CDF: -

-

- -

-

- Where Ix(a,b) is the incomplete beta function, and Φ(x) is the normal - CDF at x. -

-

- Iteration starts at the largest of the Poisson weighting terms (at i - = δ2 / 2) and then proceeds in both directions as per Benton and Krishnamoorthy's - paper. -

-

- Alternatively, by considering what happens when t = ∞, we have x = 1, - and therefore Ix(a,b) = 1 and: -

-

- -

-

- From this we can easily show that: -

-

- -

-

- and therefore we have a means to compute either the probability or its - complement directly without the risk of cancellation error. The crossover - criterion for choosing whether to calculate the CDF or its complement - is the same as for the Noncentral - Beta Distribution. -

-

- The PDF can be computed by a very similar method using: -

-

- -

-

- Where Ix'(a,b) is the derivative of the incomplete beta function. -

-

- For both the PDF and CDF we switch to approximating the distribution - by a Student's t distribution centred on δ when ν is very large. The crossover - location appears to be when δ/(4ν) < ε, this location was estimated by - inspection of equation 2.6 in "A Comparison of Approximations To - Percentiles of the Noncentral t-Distribution". H. Sahai and M. M. - Ojeda, Revista Investigacion Operacional Vol 21, No 2, 2000, page 123. -

-

- Equation 2.6 is a Fisher-Cornish expansion by Eeden and Johnson. The - second term includes the ratio δ/(4ν), so when this term become negligible, - this and following terms can be ignored, leaving just Student's t distribution - centred on δ. -

-

- This was also confirmed by experimental testing. -

-

- See also -

-
    -
  • - "Some Approximations to the Percentage Points of the Noncentral - t-Distribution". C. van Eeden. International Statistical Review, - 29, 4-31. -
    • -
  • - "Continuous Univariate Distributions". N.L. Johnson, S. - Kotz and N. Balkrishnan. 1995. John Wiley and Sons New York. -
    • -
-

- The quantile is calculated via the usual derivative-free - root-finding techniques, with the initial guess taken as the quantile - of a normal approximation to the noncentral T. -

-

- There is no closed form for the mode, so this is computed via functional - maximisation of the PDF. -

-

- The remaining functions (mean, variance etc) are implemented using the - formulas given in Weisstein, Eric W. "Noncentral Student's t-Distribution." - From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html - and in the Mathematica - documentation. -

-

- Some analytic properties of noncentral distributions (particularly unimodality, - and monotonicity of their modes) are surveyed and summarized by: -

-

- Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, - 141 (2003) 3-12. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html deleted file mode 100644 index c90cf22c8..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html +++ /dev/null @@ -1,901 +0,0 @@ - - - -Negative Binomial Distribution - - - - - - - - -
-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: -

-

- -

-

- Alternatively, this graph shows how the shape of the PDF varies as the - number of successes changes: -

-

- -

-
- - 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. -

-

- The best method of calculation for the following functions is disputed: - see Binomial - Distribution for more discussion. -

-
- - 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 rth - 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 15. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/normal_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/normal_dist.html deleted file mode 100644 index a9aba5ded..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/normal_dist.html +++ /dev/null @@ -1,319 +0,0 @@ - - - -Normal (Gaussian) Distribution - - - - - - - - -
-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: -

-

- -

-

- The cumulative distribution function is given by -

-

-   -

-

- and illustrated by this graph -

-

- -

-
- - 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. -

-
-

- median -

- 		-

- The same as the mean. -

-
-

- skewness -

- 		-

- 0 -

-
-

- kurtosis -

- 		-

- 3 -

-
-

- kurtosis excess -

- 		-

- 0 -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/pareto.html b/doc/html/math_toolkit/dist/dist_ref/dists/pareto.html deleted file mode 100644 index d21f994f8..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/pareto.html +++ /dev/null @@ -1,346 +0,0 @@ - - - -Pareto Distribution - - - - - - - - -
-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 scale = 1, RealType shape = 1)
-   // Accessors:
-   RealType scale()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 scale parameter β   > 0. If x < β  , the - pdf is zero. -

-

- 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 scale parameter - β: -

-

- -

-

- And this graph illustrates how the PDF varies with the shape parameter - α: -

-

- -

-
- - Related - distributions -
-
- - Member - Functions -
-
pareto_distribution(RealType scale = 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 scale()const;
-
-

- Returns the scale 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 [scale, ∞]. -

-
- - Accuracy -
-

- The Pareto distribution is implemented in terms of the standard library - exp functions plus expm1 and so should - have very small errors, usually only a few epsilon. -

-

- If probability is near to unity (or the complement of a probability near - zero) see also why complements?. -

-
- - Implementation -
-

- In the following table α   is the shape parameter of the distribution, and - β   is its scale 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 = β / (1 - p)1/α -

-
-

- quantile from the complement -

- 		-

- Using the relation: x = β / (q)1/α -

-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html deleted file mode 100644 index d841b8c28..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html +++ /dev/null @@ -1,326 +0,0 @@ - - - -Poisson Distribution - - - - - - - - -
-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: -

-

- -

-
- - - - - -
[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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/rayleigh.html b/doc/html/math_toolkit/dist/dist_ref/dists/rayleigh.html deleted file mode 100644 index 36da6932c..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/rayleigh.html +++ /dev/null @@ -1,342 +0,0 @@ - - - -Rayleigh Distribution - - - - - - - - -
-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: -

-

- -

-

- and the Cumulative Distribution Function (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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/skew_normal_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/skew_normal_dist.html deleted file mode 100644 index 24e7057ea..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/skew_normal_dist.html +++ /dev/null @@ -1,496 +0,0 @@ - - - -Skew Normal Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/distributions/skew_normal.hpp>
-

-

-
namespace boost{ namespace math{
-
-template <class RealType = double,
-          class Policy   = policies::policy<> >
-class skew_normal_distribution;
-
-typedef skew_normal_distribution<> normal;
-
-template <class RealType, class Policy>
-class skew_normal_distribution
-{
-public:
-   typedef RealType value_type;
-   typedef Policy   policy_type;
-   // Constructor:
-   skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
-   // Accessors:
-   RealType location()const; // mean if normal.
-   RealType scale()const; // width, standard deviation if normal.
-   RealType shape()const; // The distribution is right skewed if shape > 0 and is left skewed if shape < 0.
-                          // The distribution is normal if shape is zero.
-};
-
-}} // namespaces
-
-

- The skew normal distribution is a variant of the most well known Gaussian - statistical distribution. -

-

- The skew normal distribution with shape zero resembles the Normal - Distribution, hence the latter can be regarded as a special case - of the more generic skew normal distribution. -

-

- If the standard (mean = 0, scale = 1) normal distribution probability - density function is -

-

-     -

-

- and the cumulative distribution function -

-

-     -

-

- then the PDF - of the skew - normal distribution with shape parameter α, defined by O'Hagan - and Leonhard (1976) is -

-

-     -

-

- Given location - ξ, scale - ω, and shape - α, it can be transformed, - to the form: -

-

-     -

-

- and CDF: -

-

-     -

-

- where T(h,a) is Owen's T function, and Φ(x) - is the normal distribution. -

-

- The variation the PDF and CDF with its parameters is illustrated in the - following graphs: -

-

- - -

-
- - Member - Functions -
-
skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
-
-

- Constructs a skew_normal distribution with location ξ, scale ω and shape - α. -

-

- Requires scale > 0, otherwise domain_error - is called. -

-
RealType location()const;
-
-

- returns the location ξ of this distribution, -

-
RealType scale()const;
-
-

- returns the scale ω of this distribution, -

-
RealType shape()const;
-
-

- returns the shape α of this distribution. -

-

- (Location and scale function match other similar distributions, allowing - the functions find_location - and find_scale to be - used generically). -

-
- - - - - -
[Note]Note
-

- While the shape parameter may be chosen arbitrarily (finite), the resulting - skewness of the distribution is in - fact limited to about (-1, 1); strictly, the interval is (-0.9952717, - 0.9952717). -

-

- A parameter δ is related to the shape α by δ = α / (1 + α²), and used in the - expression for skewness - -

-
-
- - References -
-
-
- - 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]]. Infinite values are not supported. -

-

- There are no closed-form - expression known for the mode and median, but these are computed - for the -

-
    -
  • - mode - by finding the maximum of the PDF. -
    • -
  • - median - by computing quantile(1/2). -
    • -
-

- The maximum of the PDF is sought through searching the root of f'(x)=0. -

-

- Both involve iterative methods that will have lower accuracy than other - estimates. -

-
- - Testing -
-

- The R Project for Statistical - Computing using library(sn) described at Skew-Normal - Probability Distribution, and at R - skew-normal(sn) package. -

-

- Package sn provides functions related to the skew-normal (SN) and the - skew-t (ST) probability distributions, both for the univariate and for - the the multivariate case, including regression models. -

-

- Wolfram - Mathematica was also used to generate some more accurate spot - test data. -

-
- - Accuracy -
-

- The skew_normal distribution with shape = zero is implemented as a special - case, equivalent to the normal distribution in terms of the error - function, and therefore should have excellent accuracy. -

-

- The PDF and mean, variance, skewness and kurtosis are also accurately - evaluated using analytical - expressions. The CDF requires Owen's - T function that is evaluated using a Boost C++ Owens - T implementation of the algorithms of M. Patefield and D. Tandy, - Journal of Statistical Software, 5(5), 1-25 (2000); the complicated accuracy - of this function is discussed in detail at Owens - T. -

-

- The median and mode are calculated by iterative root finding, and both - will be less accurate. -

-
- - Implementation -
-

- In the following table, ξ is the location of the distribution, and ω is its - scale, and α is its shape. -

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Implementation Notes -

-
-

- pdf -

- 		-

- Using: -

-
-

- cdf -

- 		-

- Using:
where T(h,a) - is Owen's T function, and Φ(x) is the normal - distribution. -

-
-

- cdf complement -

- 		-

- Using: complement of normal distribution + 2 * Owens_t -

-
-

- quantile -

- 		-

- Maximum of the pdf is sought through searching the root of - f'(x)=0 -

-
-

- quantile from the complement -

- 		-

- -quantile(SN(-location ξ, scale ω, -shapeα), p) -

-
-

- location -

- 		-

- location ξ -

-
-

- scale -

- 		-

- scale ω -

-
-

- shape -

- 		-

- shape α -

-
-

- median -

- 		-

- quantile(1/2) -

-
-

- mean -

- 		-

- -

-
-

- mode -

- 		-

- Maximum of the pdf is sought through searching the root of - f'(x)=0 -

-
-

- variance -

- 		-

- -

-
-

- skewness -

- 		-

- -

-
-

- kurtosis -

- 		-

- kurtosis excess-3 -

-
-

- kurtosis excess -

- 		-

- -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html deleted file mode 100644 index 6c1025231..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html +++ /dev/null @@ -1,425 +0,0 @@ - - - -Students t Distribution - - - - - - - - -
-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 (possibly - to hide that they were using statistics), 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: -

-

- -

-
- - 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 are 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 its - 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) -

-
-

- mode -

- 		-

- 0 -

-
-

- mean -

- 		-

- 0 -

-
-

- variance -

- 		-

- if (v > 2) v / (v - 2) else NaN -

-
-

- skewness -

- 		-

- if (v > 3) 0 else NaN -

-
-

- kurtosis -

- 		-

- if (v > 4) 3 * (v - 2) / (v - 4) else NaN -

-
-

- kurtosis excess -

- 		-

- if (v > 4) 6 / (df - 4) else NaN -

-
-

- If the moment index k is less than v, - then the moment is undefined. Evaluating the moment will throw a domain_error unless ignored by a policy, - when it will return std::numeric_limits<>::quiet_NaN(); -

-

- (For simplicity, we have not implemented the return of infinity in some - cases as suggested by Wikipedia - Student's t. See also https://svn.boost.org/trac/boost/ticket/7177.) -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html deleted file mode 100644 index 78f41b82d..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html +++ /dev/null @@ -1,420 +0,0 @@ - - - -Triangular Distribution - - - - - - - - -
-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: -

-

- -

-

- and cumulative distribution function -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html deleted file mode 100644 index e6a662ed6..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html +++ /dev/null @@ -1,363 +0,0 @@ - - - -Uniform Distribution - - - - - - - - -
-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: -

-

- -

-

- Likewise for the 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html b/doc/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html deleted file mode 100644 index c35c2799c..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html +++ /dev/null @@ -1,363 +0,0 @@ - - - -Weibull Distribution - - - - - - - - -
-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: -

-

- -

-

- While this graph illustrates how the PDF varies with the scale parameter - β: -

-

- -

-
- - 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. The relationship of the types of extreme value - distributions, of which the Weibull is but one, is discussed by Extreme Value - Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah. -

-
- - 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 - its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/dist_ref/nmp.html b/doc/html/math_toolkit/dist/dist_ref/nmp.html deleted file mode 100644 index 0b196023d..000000000 --- a/doc/html/math_toolkit/dist/dist_ref/nmp.html +++ /dev/null @@ -1,667 +0,0 @@ - - - -Non-Member Properties - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Properties that are common to all distributions are accessed via non-member - getter functions: non-membership 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 percentile, or fractile, 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 or Percentile -
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/future.html b/doc/html/math_toolkit/dist/future.html deleted file mode 100644 index f799c1798..000000000 --- a/doc/html/math_toolkit/dist/future.html +++ /dev/null @@ -1,140 +0,0 @@ - - - -Extras/Future Directions - - - - - - - - -
-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 is easy 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. -

-

- The Statistical - Distribution Explorer utility for Windows is a usage example. -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut.html b/doc/html/math_toolkit/dist/stat_tut.html deleted file mode 100644 index 9c5901bfd..000000000 --- a/doc/html/math_toolkit/dist/stat_tut.html +++ /dev/null @@ -1,50 +0,0 @@ - - - -Statistical Distributions Tutorial - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Overview of Distributions
-
Worked Examples
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/dist_params.html b/doc/html/math_toolkit/dist/stat_tut/dist_params.html deleted file mode 100644 index 051103c21..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/dist_params.html +++ /dev/null @@ -1,91 +0,0 @@ - - - -Discrete Probability Distributions - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview.html b/doc/html/math_toolkit/dist/stat_tut/overview.html deleted file mode 100644 index b1dbab77d..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview.html +++ /dev/null @@ -1,49 +0,0 @@ - - - -Overview of Distributions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Headers - and Namespaces
-
Distributions - are Objects
-
Generic - operations common to all distributions are non-member functions
-
Complements - are supported too - and when to use them
-
Parameters - can be calculated
-
Summary
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/complements.html b/doc/html/math_toolkit/dist/stat_tut/overview/complements.html deleted file mode 100644 index ffda58785..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/complements.html +++ /dev/null @@ -1,192 +0,0 @@ - - - -Complements are supported too - and when to use them - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Often you don't want the value of the CDF, but its complement, which - is to say 1-p rather than p. - It is tempting to calculate the CDF and subtract it from 1, but if p - is very close to 1 then cancellation - error will cause you to lose accuracy, perhaps totally. -

-

- See below "Why and when - to use 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, or percentiles or fractiles, 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. -

-

- With all distributions, even for more reasonable probability (unless - the value of p can be represented exactly in the floating-point type) - the loss of accuracy quickly becomes significant if you simply calculate - probability from 1 - p (because it will be mostly garbage digits for - p ~ 1). -

-

- So always avoid, for example, using a probability near to unity like - 0.99999 -

-

- quantile(my_distribution, - 0.99999) -

-

- and instead use -

-

- quantile(complement(my_distribution, - 0.00001)) -

-

- since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point - arithmetic. -

-

- This assumes that the 0.00001 value is either a constant, or can be - computed by some manner other than subtracting 0.99999 from 1. -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/generic.html b/doc/html/math_toolkit/dist/stat_tut/overview/generic.html deleted file mode 100644 index 3ffdef7a5..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/generic.html +++ /dev/null @@ -1,248 +0,0 @@ - - - -Generic operations common to all distributions are non-member functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 any 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 -

-

- The concept of a random - variable is closely linked to the term random - variate: a random variate is a particular value (outcome) of - a random variable. and distribution - parameters are conventionally distinguished (for example in - Wikipedia and Wolfram MathWorld) by placing a semi-colon or vertical - bar) after the random - variable (whose value you 'choose'), to separate the variate - from the parameter(s) that defines the shape of the distribution.
- For example, the binomial distribution probability distribution function - (PDF) is written as f(k| n, p) = Pr(K = k|n, p) - = probability of observing k successes out of n trials. K is the random variable, - k is the random - variate, the parameters are n (trials) and p (probability). -

-
-
- - - - - -
[Note]Note

- By convention, random - variate are lower case, usually k is integral, x if real, and - random variable - are upper case, K if integral, X if real. But this implementation treats - all as floating point values RealType, - so if you really want an integral result, you must round: see note - on Discrete Probability Distributions below for details. -

-

- 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 ranges of random - variate values that are permitted and are supported can be tested - by using two functions range - and support. -

-

- 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. -

-

- For most distributions, with probability distribution functions one might - describe as 'well-behaved', we have decided that it is most useful for - the supported range to exclude random - variate values like exact zero if the end point - is discontinuous. For example, the Weibull (scale 1, shape - 1) distribution smoothly heads for unity as the random variate x declines - towards zero. But at x = zero, the value of the pdf is suddenly exactly - zero, by definition. If you are plotting the PDF, or otherwise calculating, - zero is not the most useful value for the lower limit of supported, as - we discovered. So for this, and similar distributions, we have decided - it is most numerically useful to use the closest value to zero, min_value, - for the limit of the supported range. (The range - remains from zero, so you will still get pdf(weibull, 0) - == 0). - (Exponential and gamma distributions have similarly discontinuous functions). -

-

- 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). -

-
- - - - - -
[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. -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/headers.html b/doc/html/math_toolkit/dist/stat_tut/overview/headers.html deleted file mode 100644 index 2bd441a05..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/headers.html +++ /dev/null @@ -1,69 +0,0 @@ - - - -Headers and Namespaces - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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/my_distribution.hpp> - or the "include all the distributions" header: <boost/math/distributions.hpp>. -

-

- For example, to use the Students-t distribution include either <boost/math/students_t.hpp> - or <boost/math/distributions.hpp> -

-

- You also need to bring distribution names into scope, perhaps with a - using namespace - boost::math; - declaration, -

-

- or specific using declarations - like using boost::math::normal; (recommended). -

-
- - - - - -
[Caution]Caution

- Some math function names are also used in namespace std so including - <random> could cause ambiguity! -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/objects.html b/doc/html/math_toolkit/dist/stat_tut/overview/objects.html deleted file mode 100644 index b8b1b7279..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/objects.html +++ /dev/null @@ -1,126 +0,0 @@ - - - -Distributions are Objects - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Each kind of distribution in this library is a class type - an object. -

-

- 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; // Avoid potential ambiguity with names in std <random>
-// Safer to declare specific functions with using statement(s):
-
-using boost::math::beta_distribution;
-using boost::math::binomial_distribution;
-using boost::math::students_t;
-
-// 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. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/parameters.html b/doc/html/math_toolkit/dist/stat_tut/overview/parameters.html deleted file mode 100644 index 52adae4fa..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/parameters.html +++ /dev/null @@ -1,64 +0,0 @@ - - - -Parameters can be calculated - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/overview/summary.html b/doc/html/math_toolkit/dist/stat_tut/overview/summary.html deleted file mode 100644 index 80cbb41b4..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/overview/summary.html +++ /dev/null @@ -1,65 +0,0 @@ - - - -Summary - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/variates.html b/doc/html/math_toolkit/dist/stat_tut/variates.html deleted file mode 100644 index b7ec50c80..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/variates.html +++ /dev/null @@ -1,68 +0,0 @@ - - - -Random Variates and Distribution Parameters - - - - - - - - -
-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 its 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg.html b/doc/html/math_toolkit/dist/stat_tut/weg.html deleted file mode 100644 index d1e0922f6..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg.html +++ /dev/null @@ -1,66 +0,0 @@ - - - -Worked Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Distribution - Construction Examples
-
Student's t - Distribution Examples
-
Chi Squared - Distribution Examples
-
F Distribution - Examples
-
Binomial - Distribution Examples
-
Geometric - Distribution Examples
-
Negative - Binomial Distribution Examples
-
Normal - Distribution Examples
-
Inverse - Chi-Squared Distribution Bayes Example
-
Non Central - Chi Squared Example
-
Error Handling - Example
-
Find Location - and Scale Examples
-
Comparison - with C, R, FORTRAN-style Free Functions
-
Using the - Distributions from Within C#
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/binom_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg.html deleted file mode 100644 index 640bf145a..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg.html +++ /dev/null @@ -1,51 +0,0 @@ - - - -Binomial Distribution Examples - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html b/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html deleted file mode 100644 index feb995e9f..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_conf.html +++ /dev/null @@ -1,233 +0,0 @@ - - - -Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 α  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 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html deleted file mode 100644 index 8c5741c67..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binom_size_eg.html +++ /dev/null @@ -1,150 +0,0 @@ - - - -Estimating Sample Sizes for a Binomial Distribution. - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html b/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html deleted file mode 100644 index 2befdf452..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_coinflip_example.html +++ /dev/null @@ -1,320 +0,0 @@ - - - -Binomial Coin-Flipping Example - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html b/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html deleted file mode 100644 index 0d8a2065c..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/binom_eg/binomial_quiz_example.html +++ /dev/null @@ -1,551 +0,0 @@ - - - -Binomial Quiz Example - - - - - - - - -
-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;
-#include <exception>
-  using std::exception;
-
-

-

-

- The number of correct answers, X, is distributed as a binomial random - variable with binomial distribution parameters: questions n and success - fraction probability p. 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 = 1. / answers; // If a random guess, p = 1/4 = 0.25.
-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, pass_score) << endl;
-cout << "Probability of getting all " << questions << " 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 right is 0.000247132
-Probability of getting exactly all 16 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
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/c_sharp.html b/doc/html/math_toolkit/dist/stat_tut/weg/c_sharp.html deleted file mode 100644 index 01920101e..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/c_sharp.html +++ /dev/null @@ -1,49 +0,0 @@ - - - -Using the Distributions from Within C# - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The distributions in this library can be used from the C# programming - language when they are built using Microsofts Common Language Runtime - option. -

-

- An example of this kind of usage is given in the distribution_explorer - example: see boost-root/libs/math/dot_net_example - for the source code: the application consists of a C++ dll that contains - the actual distributions, and a C# GUI that allows you to explore their - properties. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/cs_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg.html deleted file mode 100644 index ca8bf5c48..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg.html +++ /dev/null @@ -1,45 +0,0 @@ - - - -Chi Squared Distribution Examples - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html b/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html deleted file mode 100644 index 7ba36381a..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html +++ /dev/null @@ -1,227 +0,0 @@ - - - -Confidence Intervals on the Standard Deviation - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html b/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html deleted file mode 100644 index 9b8da37f9..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_size.html +++ /dev/null @@ -1,172 +0,0 @@ - - - -Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html b/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html deleted file mode 100644 index fd476fbe7..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_test.html +++ /dev/null @@ -1,287 +0,0 @@ - - - -Chi-Square Test for the Standard Deviation - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html deleted file mode 100644 index 4a231d3bb..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/dist_construct_eg.html +++ /dev/null @@ -1,401 +0,0 @@ - - - -Distribution Construction Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The structure of distributions is rather different from some other statistical - libraries, for example, those written in less object-oriented language - like FORTRAN and C: these provide a few arguments to each free function. -

-

- Boost.Math 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. -

-

- For this demonstration, first we need some includes to access the negative - binomial distribution (and the binomial, beta and gamma distributions - too). -

-

- To demonstrate the use with a high precision User-defined floating-point - type cpp_dec_float we - also need an include from Boost.Multiprecision. -

-

-

-
#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.
-
-#include <boost/multiprecision/cpp_dec_float.hpp> // for cpp_dec_float_100
-
-

-

-

- 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 we might be tempted to write -

-

-

-
using namespace boost::math;
-
-

-

-

- but this might risk ambiguity with names in std - random so much - better is explicit using boost::math:: - statements, for example: -

-

-

-
using boost::math::negative_binomial_distribution;
-
-

-

-

- and we can still 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:  both arguments are double:
-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 you can use your own RealType, for example, boost::math::cpp_dec_float_50 - (an arbitrary 50 decimal digits precision type), then we can write: -

-

-

-
 using namespace boost::multiprecision;
-
- negative_binomial_distribution<cpp_dec_float_50>  mydist8(8, 0.25);
- // `integer` arguments are promoted to your RealType exactly, but
- // `double` argument are converted to RealType,
- // possibly losing precision, so don't write:
-
- negative_binomial_distribution<cpp_dec_float_50>  mydist20(8, 0.23456789012345678901234567890);
-// to avoid truncation of second parameter to `0.2345678901234567`.
-
- negative_binomial_distribution<cpp_dec_float_50>  mydist21(8, cpp_dec_float_50("0.23456789012345678901234567890") );
-
- // Ensure that all potentially significant digits are shown.
- std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
- cpp_dec_float_50 x("1.23456789012345678901234567890");
- std::cout << pdf(mydist8, x) << std::endl;
-
-

-

-
showing  0.00012630010495970320103876754721976419438231705359935
-
-
- - - - - -
[Warning]Warning

- When using multiprecision, it is all too easy to get accidental truncation! -

-

- For example, if you write -

-

-

-
std::cout << pdf(mydist8, 1.23456789012345678901234567890) << std::endl;
-
-

-

-

- showing 0.00012630010495970318465064569310967179576805651692929, which - is wrong at about the 17th decimal digit! -

-

- This is because the value provided is truncated to a double, - effectively double x - = 1.23456789012345678901234567890; -

-

- Then the now double x - is passed to function pdf, - and this truncated double - value is finally promoted to cpp_dec_float_50. -

-

- Another way of quietly getting the wrong answer is to write: -

-

-

-
std::cout << pdf(mydist8, cpp_dec_float_50(1.23456789012345678901234567890)) << std::endl;
-
-

-

-

- A correct way from a multi-digit string value is -

-

-

-
std::cout << pdf(mydist8, cpp_dec_float_50("1.23456789012345678901234567890")) << std::endl;
-
-

-

-
- - - - - -
[Tip]Tip

- Getting about 17 decimal digits followed by many zeros is often a sign - of accidental truncation. -

-
- - 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 distribution, 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 (zero) and standard - deviation (unity) 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.
-
-  }
-  catch(std::exception &ex)
-  {
-    std::cout << ex.what() << std::endl;
-  }
-
-  return 0;
-}  // int main()
-
-

-

-

- There is no useful output from this demonstration program, of course. -

-

- See distribution_construction.cpp - for full source code. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/error_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/error_eg.html deleted file mode 100644 index 5de65bfc9..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/error_eg.html +++ /dev/null @@ -1,207 +0,0 @@ - - - -Error Handling Example - - - - - - - - -
-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: -

-

-

-
   #define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
-
-// 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;
-
-#include <cstddef>
-   // using ::errno
-
-

-

-

- 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; // Clear/reset.
-      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;
-         if (errno != 0)
-         { // So errno has been set.
-           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!
-
-

- And finally let's build with: -

-
#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
-
-

- Which gives the output show errno: -

-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/f_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/f_eg.html deleted file mode 100644 index b4f8d5a54..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/f_eg.html +++ /dev/null @@ -1,322 +0,0 @@ - - - -F Distribution Examples - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/find_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/find_eg.html deleted file mode 100644 index 64fccf0a0..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/find_eg.html +++ /dev/null @@ -1,45 +0,0 @@ - - - -Find Location and Scale Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Find - Location (Mean) Example
-
Find - Scale (Standard Deviation) Example
-
Find - mean and standard deviation example
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html deleted file mode 100644 index b52d29571..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_location_eg.html +++ /dev/null @@ -1,217 +0,0 @@ - - - -Find Location (Mean) Example - - - - - - - - -
-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; // for mean
-#include <boost/math/distributions/find_scale.hpp>
-  using boost::math::find_scale; // for standard devation
-  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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html deleted file mode 100644 index cbb95154b..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_mean_and_sd_eg.html +++ /dev/null @@ -1,541 +0,0 @@ - - - -Find mean and standard deviation example - - - - - - - - -
-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;
-#include <stdexcept>
-  using std::exception;
-
-

-

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html deleted file mode 100644 index a06ddfcdc..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/find_eg/find_scale_eg.html +++ /dev/null @@ -1,232 +0,0 @@ - - - -Find Scale (Standard Deviation) Example - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/geometric_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/geometric_eg.html deleted file mode 100644 index 927fea3f1..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/geometric_eg.html +++ /dev/null @@ -1,504 +0,0 @@ - - - -Geometric Distribution Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- For this example, we will opt to #define two 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
-
-

-

-
- - - - - -
[Caution]Caution

- It is vital to #include distributions etc after - the above #defines -

-

- After that we need some includes to provide easy access to the negative - binomial distribution, and we need some std library iostream, of course. -

-

-

-
#include <boost/math/distributions/geometric.hpp>
-  // for geometric_distribution
-  using ::boost::math::geometric_distribution; // 
-  using ::boost::math::geometric; // 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 <boost/math/distributions/negative_binomial.hpp>
-  // for negative_binomial_distribution
-  using boost::math::negative_binomial; // typedef provides default type is double.
-
-#include <boost/math/distributions/normal.hpp>
-  // for negative_binomial_distribution
-  using boost::math::normal; // typedef provides default type is double.
-
-#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. -

-

- Simple try'n'catch blocks (see below) will ensure that you get a helpful - error message instead of an abrupt (and silent) program abort. -

-
- - Throwing - a dice -
-

- The Geometric distribution describes the probability (p) - of a number of failures to get the first success in k - Bernoulli trials. (A Bernoulli - trial is one with only two possible outcomes, success of failure, - and p is the probability of success). -

-

- Suppose an 'fair' 6-face dice is thrown repeatedly: -

-

-

-
double success_fraction = 1./6; // success_fraction (p) = 0.1666
-// (so failure_fraction is 1 - success_fraction = 5./6 = 1- 0.1666 = 0.8333)
-
-

-

-

- If the dice is thrown repeatedly until the first - time a three appears. The probablility distribution - of the number of times it is thrown not - getting a three (not-a-threes - number of failures to get a three) is a geometric - distribution with the success_fraction = 1/6 = 0.1666 ̇. -

-

- We therefore start by constructing a geometric distribution with the - one parameter success_fraction, the probability of success. -

-

-

-
geometric g6(success_fraction); // type double by default.
-
-

-

-

- To confirm, we can echo the success_fraction parameter of the distribution. -

-

-

-
cout << "success fraction of a six-sided dice is " << g6.success_fraction() << endl;
-
-

-

-

- So the probability of getting a three at the first throw (zero failures) - is -

-

-

-
cout << pdf(g6, 0) << endl; // 0.1667
-cout << cdf(g6, 0) << endl; // 0.1667
-
-

-

-

- Note that the cdf and pdf are identical because the is only one throw. - If we want the probability of getting the first three - on the 2nd throw: -

-

-

-
cout << pdf(g6, 1) << endl; // 0.1389
-
-

-

-

- If we want the probability of getting the first three - on the 1st or 2nd throw (allowing one failure): -

-

-

-
cout << "pdf(g6, 0) + pdf(g6, 1) = " << pdf(g6, 0) + pdf(g6, 1) << endl;
-
-

-

-

- Or more conveniently, and more generally, we can use the Cumulative Distribution - Function CDF. -

-

-

-
cout << "cdf(g6, 1) = " << cdf(g6, 1) << endl; // 0.3056
-
-

-

-

- If we allow many more (12) throws, the probability of getting our three - gets very high: -

-

-

-
cout << "cdf(g6, 12) = " << cdf(g6, 12) << endl; // 0.9065 or 90% probability.
-
-

-

-

- If we want to be much more confident, say 99%, we can estimate the number - of throws to be this sure using the inverse or quantile. -

-

-

-
cout << "quantile(g6, 0.99) = " << quantile(g6, 0.99) << endl; // 24.26
-
-

-

-

- Note that the value returned is not an integer: if you want an integer - result you should use either floor, round or ceil functions, or use the - policies mechanism. See Understanding - Quantiles of Discrete Distributions -

-

- The geometric distribution is related to the negative binomial    negative_binomial_distribution(RealType r, RealType - p); - with parameter r = 1. So we could get the same result - using the negative binomial, but using the geometric the results will - be faster, and may be more accurate. -

-

-

-
negative_binomial nb(1, success_fraction);
-cout << pdf(nb, 1) << endl; // 0.1389
-cout << cdf(nb, 1) << endl; // 0.3056
-
-

-

-

- We could also the complement to express the required probability as 1 - - 0.99 = 0.01 (and get the same result): -

-

-

-
cout << "quantile(complement(g6, 1 - p))  " << quantile(complement(g6, 0.01)) << endl; // 24.26
-
-

-

-

- Note too that Boost.Math geometric distribution is implemented as a continuous - function. Unlike other implementations (for example R) it uses - the number of failures as a real parameter, - not as an integer. If you want this integer behaviour, you may need to - enforce this by rounding the parameter you pass, probably rounding down, - to the nearest integer. For example, R returns the success fraction probability - for all values of failures from 0 to 0.999999 thus: -

-
   R> formatC(pgeom(0.0001,0.5, FALSE), digits=17) "               0.5"
-
-

- So in Boost.Math the equivalent is -

-

-

-
    geometric g05(0.5);  // Probability of success = 0.5 or 50%
-    // Output all potentially significant digits for the type, here double.
-
-#ifdef BOOST_NO_CXX11_NUMERIC_LIMITS
-  int max_digits10 = 2 + (boost::math::policies::digits<double, boost::math::policies::policy<> >() * 30103UL) / 100000UL;
-  cout << "BOOST_NO_CXX11_NUMERIC_LIMITS is defined" << endl;
-#else
-  int max_digits10 = std::numeric_limits<double>::max_digits10;
-#endif
-  cout << "Show all potentially significant decimal digits std::numeric_limits<double>::max_digits10 = "
-    << max_digits10 << endl;
-  cout.precision(max_digits10); // 
-
-    cout << cdf(g05, 0.0001) << endl; // returns 0.5000346561579232, not exact 0.5.
-
-

-

-

- To get the R discrete behaviour, you simply need to round with, for example, - the floor function. -

-

-

-
cout << cdf(g05, floor(0.0001)) << endl; // returns exactly 0.5
-
-

-

-
> formatC(pgeom(0.9999999,0.5, FALSE), digits=17) [1] "              0.25"
-> formatC(pgeom(1.999999,0.5, FALSE), digits=17)[1] "              0.25" k = 1
-> formatC(pgeom(1.9999999,0.5, FALSE), digits=17)[1] "0.12500000000000003" k = 2
-
-

- shows that R makes an arbitrary round-up decision at about 1e7 from the - next integer above. This may be convenient in practice, and could be - replicated in C++ if desired. -

-
- - Surveying - customers to find one with a faulty product -
-

- A company knows from warranty claims that 2% of their products will be - faulty, so the 'success_fraction' of finding a fault is 0.02. It wants - to interview a purchaser of faulty products to assess their 'user experience'. -

-

- To estimate how many customers they will probably need to contact in - order to find one who has suffered from the fault, we first construct - a geometric distribution with probability 0.02, and then chose a confidence, - say 80%, 95%, or 99% to finding a customer with a fault. Finally, we - probably want to round up the result to the integer above using the - ceil function. (We could - also use a policy, but that is hardly worthwhile for this simple application.) -

-

- (This also assumes that each customer only buys one product: if customers - bought more than one item, the probability of finding a customer with - a fault obviously improves.) -

-

-

-
cout.precision(5);
-geometric g(0.02); // On average, 2 in 100 products are faulty.
-double c = 0.95; // 95% confidence.
-cout << " quantile(g, " << c << ") = " << quantile(g, c) << endl;
-
-cout << "To be " << c * 100
-  << "% confident of finding we customer with a fault, need to survey "
-  <<  ceil(quantile(g, c)) << " customers." << endl; // 148
-c = 0.99; // Very confident.
-cout << "To be " << c * 100
-  << "% confident of finding we customer with a fault, need to survey "
-  <<  ceil(quantile(g, c)) << " customers." << endl; // 227
-c = 0.80; // Only reasonably confident.
-cout << "To be " << c * 100
-  << "% confident of finding we customer with a fault, need to survey "
-  <<  ceil(quantile(g, c)) << " customers." << endl; // 79
-
-

-

-
- - Basket - Ball Shooters -
-

- According to Wikipedia, average pro basket ball players get free - throws in the baskets 70 to 80 % of the time, but some get as - high as 95%, and others as low as 50%. Suppose we want to compare the - probabilities of failing to get a score only on the first or on the fifth - shot? To start we will consider the average shooter, say 75%. So we construct - a geometric distribution with success_fraction parameter 75/100 = 0.75. -

-

-

-
cout.precision(2);
-geometric gav(0.75); // Shooter averages 7.5 out of 10 in the basket.
-
-

-

-

- What is probability of getting 1st try in the basket, that is with no - failures? -

-

-

-
cout << "Probability of score on 1st try = " << pdf(gav, 0) << endl; // 0.75
-
-

-

-

- This is, of course, the success_fraction probability 75%. What is the - probability that the shooter only scores on the fifth shot? So there - are 5-1 = 4 failures before the first success. -

-

-

-
cout << "Probability of score on 5th try = " << pdf(gav, 4) << endl; // 0.0029
-
-

-

-

- Now compare this with the poor and the best players success fraction. - We need to constructing new distributions with the different success - fractions, and then get the corresponding probability density functions - values: -

-

-

-
geometric gbest(0.95);
-cout << "Probability of score on 5th try = " << pdf(gbest, 4) << endl; // 5.9e-6
-geometric gmediocre(0.50);
-cout << "Probability of score on 5th try = " << pdf(gmediocre, 4) << endl; // 0.031
-
-

-

-

- So we can see the very much smaller chance (0.000006) of 4 failures by - the best shooters, compared to the 0.03 of the mediocre. -

-
- - Estimating - failures -
-

- Of course one man's failure is an other man's success. So a fault can - be defined as a 'success'. -

-

- If a fault occurs once after 100 flights, then one might naively say - that the risk of fault is obviously 1 in 100 = 1/100, a probability of - 0.01. -

-

- This is the best estimate we can make, but while it is the truth, it - is not the whole truth, for it hides the big uncertainty when estimating - from a single event. "One swallow doesn't make a summer." To - show the magnitude of the uncertainty, the geometric (or the negative - binomial) distribution can be used. -

-

- If we chose the popular 95% confidence in the limits, corresponding to - an alpha of 0.05, because we are calculating a two-sided interval, we - must divide alpha by two. -

-

-

-
double alpha = 0.05;
-double k = 100; // So frequency of occurence is 1/100.
-cout << "Probability is failure is " << 1/k << endl;
-double t = geometric::find_lower_bound_on_p(k, alpha/2);
-cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-  << t << endl; // 0.00025
-t = geometric::find_upper_bound_on_p(k, alpha/2);
-cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-  << t << endl; // 0.037
-
-

-

-

- So while we estimate the probability is 0.01, it might lie between 0.0003 - and 0.04. Even if we relax our confidence to alpha = 90%, the bounds - only contract to 0.0005 and 0.03. And if we require a high confidence, - they widen to 0.00005 to 0.05. -

-

-

-
alpha = 0.1; // 90% confidence.
-t = geometric::find_lower_bound_on_p(k, alpha/2);
-cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-  << t << endl; // 0.0005
-t = geometric::find_upper_bound_on_p(k, alpha/2);
-cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-  << t << endl; // 0.03
-
-alpha = 0.01; // 99% confidence.
-t = geometric::find_lower_bound_on_p(k, alpha/2);
-cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-  << t << endl; // 5e-005
-t = geometric::find_upper_bound_on_p(k, alpha/2);
-cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
-    << t << endl; // 0.052
-
-

-

-

- In real life, there will usually be more than one event (fault or success), - when the negative binomial, which has the neccessary extra parameter, - will be needed. -

-

- As noted above, using a catch block is always a good idea, even if you - hope not 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(g, -1)
-

- for example, we would get an unhelpful abort, but with a catch: -

-
Message from thrown exception was:
- Error in function boost::math::pdf(const exponential_distribution<double>&, double):
- Number of failures argument is -1, but must be >= 0 !
-
-

- See full source C++ of this example at geometric_examples.cpp -

-

- See - negative_binomial confidence interval example. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/inverse_chi_squared_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/inverse_chi_squared_eg.html deleted file mode 100644 index ae0d3737f..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/inverse_chi_squared_eg.html +++ /dev/null @@ -1,367 +0,0 @@ - - - -Inverse Chi-Squared Distribution Bayes Example - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The scaled-inversed-chi-squared distribution is the conjugate prior distribution - for the variance (σ2) parameter of a normal distribution with known expectation - (μ). As such it has widespread application in Bayesian statistics: -

-

- In Bayesian - inference, the strength of belief into certain parameter values - is itself described through a distribution. Parameters hence become themselves - modelled and interpreted as random variables. -

-

- In this worked example, we perform such a Bayesian analysis by using - the scaled-inverse-chi-squared distribution as prior and posterior distribution - for the variance parameter of a normal distribution. -

-

- For more general information on Bayesian type of analyses, see: -

-
    -
  • - Andrew Gelman, John B. Carlin, Hal E. Stern, Donald B. Rubin, Bayesian - Data Analysis, 2003, ISBN 978-1439840955. -
    • -
  • - Jim Albert, Bayesian Compution with R, Springer, 2009, ISBN 978-0387922973. -
    • -
-

- (As the scaled-inversed-chi-squared is another parameterization of the - inverse-gamma distribution, this example could also have used the inverse-gamma - distribution). -

-

- Consider precision machines which produce balls for a high-quality ball - bearing. Ideally each ball should have a diameter of precisely 3000 μm - (3 mm). Assume that machines generally produce balls of that size on - average (mean), but individual balls can vary slightly in either direction - following (approximately) a normal distribution. Depending on various - production conditions (e.g. raw material used for balls, workplace temperature - and humidity, maintenance frequency and quality) some machines produce - balls tighter distributed around the target of 3000 μm, while others produce - balls with a wider distribution. Therefore the variance parameter of - the normal distribution of the ball sizes varies from machine to machine. - An extensive survey by the precision machinery manufacturer, however, - has shown that most machines operate with a variance between 15 and 50, - and near 25 μm2 on average. -

-

- Using this information, we want to model the variance of the machines. - The variance is strictly positive, and therefore we look for a statistical - distribution with support in the positive domain of the real numbers. - Given the expectation of the normal distribution of the balls is known - (3000 μm), for reasons of conjugacy, it is customary practice in Bayesian - statistics to model the variance to be scaled-inverse-chi-squared distributed. -

-

- In a first step, we will try to use the survey information to model the - general knowledge about the variance parameter of machines measured by - the manufacturer. This will provide us with a generic prior distribution - that is applicable if nothing more specific is known about a particular - machine. -

-

- In a second step, we will then combine the prior-distribution information - in a Bayesian analysis with data on a specific single machine to derive - a posterior distribution for that machine. -

-
- - Step - one: Using the survey information. -
-

- Using the survey results, we try to find the parameter set of a scaled-inverse-chi-squared - distribution so that the properties of this distribution match the results. - Using the mathematical properties of the scaled-inverse-chi-squared distribution - as guideline, we see that that both the mean and mode of the scaled-inverse-chi-squared - distribution are approximately given by the scale parameter (s) of the - distribution. As the survey machines operated at a variance of 25 μm2 on - average, we hence set the scale parameter (sprior) of our prior distribution - equal to this value. Using some trial-and-error and calls to the global - quantile function, we also find that a value of 20 for the degrees-of-freedom - (νprior) parameter is adequate so that most of the prior distribution - mass is located between 15 and 50 (see figure below). -

-

- We first construct our prior distribution using these values, and then - list out a few quantiles: -

-

-

-
double priorDF = 20.0;
-double priorScale = 25.0;
-
-inverse_chi_squared prior(priorDF, priorScale);
-// Using an inverse_gamma distribution instead, we could equivalently write
-// inverse_gamma prior(priorDF / 2.0, priorScale * priorDF / 2.0);
-
-cout << "Prior distribution:" << endl << endl;
-cout << "  2.5% quantile: " << quantile(prior, 0.025) << endl;
-cout << "  50% quantile: " << quantile(prior, 0.5) << endl;
-cout << "  97.5% quantile: " << quantile(prior, 0.975) << endl << endl;
-
-

-

-

- This produces this output: -

-
Prior distribution:
-
-2.5% quantile: 14.6
-50% quantile: 25.9
-97.5% quantile: 52.1
-
-

- Based on this distribution, we can now calculate the probability of having - a machine working with an unusual work precision (variance) at <= - 15 or > 50. For this task, we use calls to the boost::math:: functions cdf - and complement, respectively, - and find a probability of about 0.031 (3.1%) for each case. -

-

-

-
cout << "  probability variance <= 15: " << boost::math::cdf(prior, 15.0) << endl;
-cout << "  probability variance <= 25: " << boost::math::cdf(prior, 25.0) << endl;
-cout << "  probability variance > 50: "
-  << boost::math::cdf(boost::math::complement(prior, 50.0))
-<< endl << endl;
-
-

-

-

- This produces this output: -

-
probability variance <= 15: 0.031
-probability variance <= 25: 0.458
-probability variance > 50: 0.0318
-
-

- Therefore, only 3.1% of all precision machines produce balls with a variance - of 15 or less (particularly precise machines), but also only 3.2% of - all machines produce balls with a variance of as high as 50 or more (particularly - imprecise machines). Moreover, slightly more than one-half (1 - 0.458 - = 54.2%) of the machines work at a variance greater than 25. -

-

- Notice here the distinction between a Bayesian - analysis and a frequentist - analysis: because we model the variance as random variable itself, we - can calculate and straightforwardly interpret probabilities for given - parameter values directly, while such an approach is not possible (and - interpretationally a strict must-not) in the frequentist - world. -

-
- - Step - 2: Investigate a single machine -
-

- In the second step, we investigate a single machine, which is suspected - to suffer from a major fault as the produced balls show fairly high size - variability. Based on the prior distribution of generic machinery performance - (derived above) and data on balls produced by the suspect machine, we - calculate the posterior distribution for that machine and use its properties - for guidance regarding continued machine operation or suspension. -

-

- It can be shown that if the prior distribution was chosen to be scaled-inverse-chi-square - distributed, then the posterior distribution is also scaled-inverse-chi-squared-distributed - (prior and posterior distributions are hence conjugate). For more details - regarding conjugacy and formula to derive the parameters set for the - posterior distribution see Conjugate - prior. -

-

- Given the prior distribution parameters and sample data (of size n), - the posterior distribution parameters are given by the two expressions: -

-

-    νposterior = νprior + n -

-

- which gives the posteriorDF below, and -

-

-    sposterior = (νpriorsprior + Σni=1(xi - μ)2) / (νprior + n) -

-

- which after some rearrangement gives the formula for the posteriorScale - below. -

-

- Machine-specific data consist of 100 balls which were accurately measured - and show the expected mean of 3000 μm and a sample variance of 55 (calculated - for a sample mean defined to be 3000 exactly). From these data, the prior - parameterization, and noting that the term Σni=1(xi - μ)2 equals the sample - variance multiplied by n - 1, it follows that the posterior distribution - of the variance parameter is scaled-inverse-chi-squared distribution - with degrees-of-freedom (νposterior) = 120 and scale (sposterior) = 49.54. -

-

-

-
int ballsSampleSize = 100;
-cout <<"balls sample size: " << ballsSampleSize << endl;
-double ballsSampleVariance = 55.0;
-cout <<"balls sample variance: " << ballsSampleVariance << endl;
-
-double posteriorDF = priorDF + ballsSampleSize;
-cout << "prior degrees-of-freedom: " << priorDF << endl;
-cout << "posterior degrees-of-freedom: " << posteriorDF << endl;
-
-double posteriorScale =
-  (priorDF * priorScale + (ballsSampleVariance * (ballsSampleSize - 1))) / posteriorDF;
-cout << "prior scale: " << priorScale  << endl;
-cout << "posterior scale: " << posteriorScale << endl;
-
-

-

-

- An interesting feature here is that one needs only to know a summary - statistics of the sample to parameterize the posterior distribution: - the 100 individual ball measurements are irrelevant, just knowledge of - the sample variance and number of measurements is sufficient. -

-

- That produces this output: -

-
balls sample size: 100
-balls sample variance: 55
-prior degrees-of-freedom: 20
-posterior degrees-of-freedom: 120
-prior scale: 25
-posterior scale: 49.5
-
-
-

- To compare the generic machinery performance with our suspect machine, - we calculate again the same quantiles and probabilities as above, and - find a distribution clearly shifted to greater values (see figure). -

-

- -

-

-

-
inverse_chi_squared posterior(posteriorDF, posteriorScale);
-
- cout << "Posterior distribution:" << endl << endl;
- cout << "  2.5% quantile: " << boost::math::quantile(posterior, 0.025) << endl;
- cout << "  50% quantile: " << boost::math::quantile(posterior, 0.5) << endl;
- cout << "  97.5% quantile: " << boost::math::quantile(posterior, 0.975) << endl << endl;
-
- cout << "  probability variance <= 15: " << boost::math::cdf(posterior, 15.0) << endl;
- cout << "  probability variance <= 25: " << boost::math::cdf(posterior, 25.0) << endl;
- cout << "  probability variance > 50: "
-   << boost::math::cdf(boost::math::complement(posterior, 50.0)) << endl;
-
-

-

-

- This produces this output: -

-
Posterior distribution:
-
-   2.5% quantile: 39.1
-   50% quantile: 49.8
-   97.5% quantile: 64.9
-
-   probability variance <= 15: 2.97e-031
-   probability variance <= 25: 8.85e-010
-   probability variance > 50: 0.489
-
-

- Indeed, the probability that the machine works at a low variance (<= - 15) is almost zero, and even the probability of working at average or - better performance is negligibly small (less than one-millionth of a - permille). On the other hand, with an almost near-half probability (49%), - the machine operates in the extreme high variance range of > 50 characteristic - for poorly performing machines. -

-

- Based on this information the operation of the machine is taken out of - use and serviced. -

-

- In summary, the Bayesian analysis allowed us to make exact probabilistic - statements about a parameter of interest, and hence provided us results - with straightforward interpretation. -

-

- A full sample output is: -

-
Inverse_chi_squared_distribution Bayes example:
-
-   Prior distribution:
-
-    2.5% quantile: 14.6
-    50% quantile: 25.9
-    97.5% quantile: 52.1
-
-    probability variance <= 15: 0.031
-    probability variance <= 25: 0.458
-    probability variance > 50: 0.0318
-
-  balls sample size: 100
-  balls sample variance: 55
-  prior degrees-of-freedom: 20
-  posterior degrees-of-freedom: 120
-  prior scale: 25
-  posterior scale: 49.5
-  Posterior distribution:
-
-    2.5% quantile: 39.1
-    50% quantile: 49.8
-    97.5% quantile: 64.9
-
-    probability variance <= 15: 2.97e-031
-    probability variance <= 25: 8.85e-010
-    probability variance > 50: 0.489
-
-
-

- (See also the reference documentation for the Inverse - chi squared Distribution.) -

-

- See the full source C++ of this example at ../../example/inverse_chi_squared_bayes_eg.cpp -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/nag_library.html b/doc/html/math_toolkit/dist/stat_tut/weg/nag_library.html deleted file mode 100644 index 052316ef1..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/nag_library.html +++ /dev/null @@ -1,106 +0,0 @@ - - - -Comparison with C, R, FORTRAN-style Free Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/nccs_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/nccs_eg.html deleted file mode 100644 index 39ce3a46a..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/nccs_eg.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Non Central Chi Squared Example - - - - - - - - -
-PrevUpHomeNext -
-
-
-
Tables - of the power function of the chi2 test.
-

- (See also the reference documentation for the Noncentral - Chi Squared Distribution.) -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/nccs_eg/nccs_power_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/nccs_eg/nccs_power_eg.html deleted file mode 100644 index 5a2e73116..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/nccs_eg/nccs_power_eg.html +++ /dev/null @@ -1,1301 +0,0 @@ - - - -Tables of the power function of the chi2 test. - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- This example computes a table of the power of the χ2 -test at the 5% significance - level, for various degrees of freedom and non-centrality parameters. - The table is deliberately the same as Table 6 from "The Non-Central - χ2 and F-Distributions and their applications.", P. B. Patnaik, - Biometrika, Vol. 36, No. 1/2 (June 1949), 202-232. -

-

- First we need some includes to access the non-central chi squared distribution - (and some basic std output of course). -

-

-

-
#include <boost/math/distributions/non_central_chi_squared.hpp>
-using boost::math::chi_squared;
-using boost::math::non_central_chi_squared;
-
-#include <iostream>
-using std::cout; using std::endl;
-using std::setprecision;
-
-int main()
-{
-
-

-

-

- Create a table of the power of the χ2 test at 5% significance level, - start with a table header: -

-

-

-
cout << "[table\n[[[nu]]";
-for(int lam = 2; lam <= 20; lam += 2)
-{
-   cout << "[[lambda]=" << lam << "]";
-}
-cout << "]\n";
-
-

-

-

- (Note: the enclosing [] brackets are to format as a table in Boost.Quickbook). -

-

- Enumerate the rows and columns and print the power of the test for - each table cell: -

-

-

-
for(int n = 2; n <= 20; ++n)
-{
-   cout << "[[" << n << "]";
-   for(int lam = 2; lam <= 20; lam += 2)
-   {
-
-

-

-

- Calculate the χ2 statistic for a 5% significance: -

-

-

-
double cs = quantile(complement(chi_squared(n), 0.05));
-
-

-

-

- The power of the test is given by the complement of the CDF of the - non-central χ2 distribution: -

-

-

-
double beta = cdf(complement(non_central_chi_squared(n, lam), cs));
-
-

-

-

- Then output the cell value: -

-

-

-
         cout << "[" << setprecision(3) << beta << "]";
-      }
-      cout << "]" << endl;
-   }
-   cout << "]" << endl;
-}
-
-

-

-

- The output from this program is a table in Boost.Quickbook format as - shown below. -

-

- We can interpret this as follows - for example if ν=10 and λ=10 then - the power of the test is 0.542 - so we have only a 54% chance of correctly - detecting that our null hypothesis is false, and a 46% chance of incurring - a type II error (failing to reject the null hypothesis when it is in - fact false): -

-
------------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- ν -

- 		-

- λ=2 -

- 		-

- λ=4 -

- 		-

- λ=6 -

- 		-

- λ=8 -

- 		-

- λ=10 -

- 		-

- λ=12 -

- 		-

- λ=14 -

- 		-

- λ=16 -

- 		-

- λ=18 -

- 		-

- λ=20 -

-
-

- 2 -

- 		-

- 0.226 -

- 		-

- 0.415 -

- 		-

- 0.584 -

- 		-

- 0.718 -

- 		-

- 0.815 -

- 		-

- 0.883 -

- 		-

- 0.928 -

- 		-

- 0.957 -

- 		-

- 0.974 -

- 		-

- 0.985 -

-
-

- 3 -

- 		-

- 0.192 -

- 		-

- 0.359 -

- 		-

- 0.518 -

- 		-

- 0.654 -

- 		-

- 0.761 -

- 		-

- 0.84 -

- 		-

- 0.896 -

- 		-

- 0.934 -

- 		-

- 0.959 -

- 		-

- 0.975 -

-
-

- 4 -

- 		-

- 0.171 -

- 		-

- 0.32 -

- 		-

- 0.47 -

- 		-

- 0.605 -

- 		-

- 0.716 -

- 		-

- 0.802 -

- 		-

- 0.866 -

- 		-

- 0.912 -

- 		-

- 0.943 -

- 		-

- 0.964 -

-
-

- 5 -

- 		-

- 0.157 -

- 		-

- 0.292 -

- 		-

- 0.433 -

- 		-

- 0.564 -

- 		-

- 0.677 -

- 		-

- 0.769 -

- 		-

- 0.839 -

- 		-

- 0.89 -

- 		-

- 0.927 -

- 		-

- 0.952 -

-
-

- 6 -

- 		-

- 0.146 -

- 		-

- 0.27 -

- 		-

- 0.403 -

- 		-

- 0.531 -

- 		-

- 0.644 -

- 		-

- 0.738 -

- 		-

- 0.813 -

- 		-

- 0.869 -

- 		-

- 0.911 -

- 		-

- 0.94 -

-
-

- 7 -

- 		-

- 0.138 -

- 		-

- 0.252 -

- 		-

- 0.378 -

- 		-

- 0.502 -

- 		-

- 0.614 -

- 		-

- 0.71 -

- 		-

- 0.788 -

- 		-

- 0.849 -

- 		-

- 0.895 -

- 		-

- 0.928 -

-
-

- 8 -

- 		-

- 0.131 -

- 		-

- 0.238 -

- 		-

- 0.357 -

- 		-

- 0.477 -

- 		-

- 0.588 -

- 		-

- 0.685 -

- 		-

- 0.765 -

- 		-

- 0.829 -

- 		-

- 0.879 -

- 		-

- 0.915 -

-
-

- 9 -

- 		-

- 0.125 -

- 		-

- 0.225 -

- 		-

- 0.339 -

- 		-

- 0.454 -

- 		-

- 0.564 -

- 		-

- 0.661 -

- 		-

- 0.744 -

- 		-

- 0.811 -

- 		-

- 0.863 -

- 		-

- 0.903 -

-
-

- 10 -

- 		-

- 0.121 -

- 		-

- 0.215 -

- 		-

- 0.323 -

- 		-

- 0.435 -

- 		-

- 0.542 -

- 		-

- 0.64 -

- 		-

- 0.723 -

- 		-

- 0.793 -

- 		-

- 0.848 -

- 		-

- 0.891 -

-
-

- 11 -

- 		-

- 0.117 -

- 		-

- 0.206 -

- 		-

- 0.309 -

- 		-

- 0.417 -

- 		-

- 0.523 -

- 		-

- 0.62 -

- 		-

- 0.704 -

- 		-

- 0.775 -

- 		-

- 0.833 -

- 		-

- 0.878 -

-
-

- 12 -

- 		-

- 0.113 -

- 		-

- 0.198 -

- 		-

- 0.297 -

- 		-

- 0.402 -

- 		-

- 0.505 -

- 		-

- 0.601 -

- 		-

- 0.686 -

- 		-

- 0.759 -

- 		-

- 0.818 -

- 		-

- 0.866 -

-
-

- 13 -

- 		-

- 0.11 -

- 		-

- 0.191 -

- 		-

- 0.286 -

- 		-

- 0.387 -

- 		-

- 0.488 -

- 		-

- 0.584 -

- 		-

- 0.669 -

- 		-

- 0.743 -

- 		-

- 0.804 -

- 		-

- 0.854 -

-
-

- 14 -

- 		-

- 0.108 -

- 		-

- 0.185 -

- 		-

- 0.276 -

- 		-

- 0.374 -

- 		-

- 0.473 -

- 		-

- 0.567 -

- 		-

- 0.653 -

- 		-

- 0.728 -

- 		-

- 0.791 -

- 		-

- 0.842 -

-
-

- 15 -

- 		-

- 0.105 -

- 		-

- 0.179 -

- 		-

- 0.267 -

- 		-

- 0.362 -

- 		-

- 0.459 -

- 		-

- 0.552 -

- 		-

- 0.638 -

- 		-

- 0.713 -

- 		-

- 0.777 -

- 		-

- 0.83 -

-
-

- 16 -

- 		-

- 0.103 -

- 		-

- 0.174 -

- 		-

- 0.259 -

- 		-

- 0.351 -

- 		-

- 0.446 -

- 		-

- 0.538 -

- 		-

- 0.623 -

- 		-

- 0.699 -

- 		-

- 0.764 -

- 		-

- 0.819 -

-
-

- 17 -

- 		-

- 0.101 -

- 		-

- 0.169 -

- 		-

- 0.251 -

- 		-

- 0.341 -

- 		-

- 0.434 -

- 		-

- 0.525 -

- 		-

- 0.609 -

- 		-

- 0.686 -

- 		-

- 0.752 -

- 		-

- 0.807 -

-
-

- 18 -

- 		-

- 0.0992 -

- 		-

- 0.165 -

- 		-

- 0.244 -

- 		-

- 0.332 -

- 		-

- 0.423 -

- 		-

- 0.512 -

- 		-

- 0.596 -

- 		-

- 0.673 -

- 		-

- 0.74 -

- 		-

- 0.796 -

-
-

- 19 -

- 		-

- 0.0976 -

- 		-

- 0.161 -

- 		-

- 0.238 -

- 		-

- 0.323 -

- 		-

- 0.412 -

- 		-

- 0.5 -

- 		-

- 0.584 -

- 		-

- 0.66 -

- 		-

- 0.728 -

- 		-

- 0.786 -

-
-

- 20 -

- 		-

- 0.0961 -

- 		-

- 0.158 -

- 		-

- 0.232 -

- 		-

- 0.315 -

- 		-

- 0.402 -

- 		-

- 0.489 -

- 		-

- 0.572 -

- 		-

- 0.648 -

- 		-

- 0.716 -

- 		-

- 0.775 -

-
-

- See nc_chi_sq_example.cpp - for the full C++ source code. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html deleted file mode 100644 index bc163e553..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg.html +++ /dev/null @@ -1,52 +0,0 @@ - - - -Negative Binomial Distribution Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html b/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html deleted file mode 100644 index 92f30f20a..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_conf.html +++ /dev/null @@ -1,233 +0,0 @@ - - - -Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html deleted file mode 100644 index 47b4bb1dc..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html +++ /dev/null @@ -1,213 +0,0 @@ - - - -Estimating Sample Sizes for the Negative Binomial. - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 (number of trials) 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. The values range - from a very low 0.5 or 50% confidence up to an extremely high confidence - of 99.999. -

-

- 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 = " << (int)failures;
-  cout << ",   Success fraction = " << fixed << setprecision(1) << 100 * p << "%" << endl;
-   // Print table header:
-   cout << "____________________________\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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html b/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html deleted file mode 100644 index a9a2d475a..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html +++ /dev/null @@ -1,603 +0,0 @@ - - - -Negative Binomial Sales Quota Example. - - - - - - - - -
-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 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html b/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html deleted file mode 100644 index 8a051d4d2..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html +++ /dev/null @@ -1,134 +0,0 @@ - - - -Negative Binomial Table Printing Example. - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/normal_example.html b/doc/html/math_toolkit/dist/stat_tut/weg/normal_example.html deleted file mode 100644 index 861bb61b9..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/normal_example.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Normal Distribution Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-
Some - Miscellaneous Examples of the Normal (Gaussian) Distribution
-

- (See also the reference documentation for the Normal - Distribution.) -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html b/doc/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html deleted file mode 100644 index 3e6b6e155..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/normal_example/normal_misc.html +++ /dev/null @@ -1,604 +0,0 @@ - - - -Some Miscellaneous Examples of the Normal (Gaussian) Distribution - - - - - - - - -
-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.
-// but std::numeric_limits<double>::max_digits10; on new Standard Libraries gives
-// 17, the maximum number of digits that can possibly be significant.
-// std::numeric_limits<double>::digits10; == 15 is number of guaranteed digits,
-// the other two digits being 'noisy'.
-
-// 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg.html deleted file mode 100644 index f4b9cc18e..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg.html +++ /dev/null @@ -1,50 +0,0 @@ - - - -Student's t Distribution Examples - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html deleted file mode 100644 index 96dd4d5e0..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/paired_st.html +++ /dev/null @@ -1,71 +0,0 @@ - - - -Comparing two paired samples with the Student's t distribution - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html deleted file mode 100644 index 94f2dc68d..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_intervals.html +++ /dev/null @@ -1,260 +0,0 @@ - - - -Calculating confidence intervals on the mean with the Students-t distribution - - - - - - - - -
-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, /α/ 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html deleted file mode 100644 index aa4d59153..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_size.html +++ /dev/null @@ -1,171 +0,0 @@ - - - -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 - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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)
-_______________________________________________________________
-    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
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html deleted file mode 100644 index 497e32e31..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/tut_mean_test.html +++ /dev/null @@ -1,326 +0,0 @@ - - - -Testing a sample mean for difference from a "true" mean - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html b/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html deleted file mode 100644 index d4a1c25b9..000000000 --- a/doc/html/math_toolkit/dist/stat_tut/weg/st_eg/two_sample_students_t.html +++ /dev/null @@ -1,350 +0,0 @@ - - - -Comparing the means of two samples with the Students-t test - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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" << "=  " << sp << "\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                                          =  20.145
-   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                              =  6.3426
-   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
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/extern_c.html b/doc/html/math_toolkit/extern_c.html deleted file mode 100644 index 4a7b0b5bf..000000000 --- a/doc/html/math_toolkit/extern_c.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -TR1 and C99 external "C" Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
C99 and TR1 C Functions Overview
-
C99 C Functions
-
TR1 C Functions Quick - Reference
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/extern_c/c99.html b/doc/html/math_toolkit/extern_c/c99.html deleted file mode 100644 index 751247e45..000000000 --- a/doc/html/math_toolkit/extern_c/c99.html +++ /dev/null @@ -1,469 +0,0 @@ - - - -C99 C Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Supported - C99 Functions -
-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-typedef unspecified float_t;
-typedef unspecified double_t;
-
-double acosh(double x);
-float acoshf(float x);
-long double acoshl(long double x);
-
-double asinh(double x);
-float asinhf(float x);
-long double asinhl(long double x);
-
-double atanh(double x);
-float atanhf(float x);
-long double atanhl(long double x);
-
-double cbrt(double x);
-float cbrtf(float x);
-long double cbrtl(long double x);
-
-double copysign(double x, double y);
-float copysignf(float x, float y);
-long double copysignl(long double x, long double y);
-
-double erf(double x);
-float erff(float x);
-long double erfl(long double x);
-
-double erfc(double x);
-float erfcf(float x);
-long double erfcl(long double x);
-
-double expm1(double x);
-float expm1f(float x);
-long double expm1l(long double x);
-
-double fmax(double x, double y);
-float fmaxf(float x, float y);
-long double fmaxl(long double x, long double y);
-
-double fmin(double x, double y);
-float fminf(float x, float y);
-long double fminl(long double x, long double y);
-
-double hypot(double x, double y);
-float hypotf(float x, float y);
-long double hypotl(long double x, long double y);
-
-double lgamma(double x);
-float lgammaf(float x);
-long double lgammal(long double x);
-
-long long llround(double x);
-long long llroundf(float x);
-long long llroundl(long double x);
-
-double log1p(double x);
-float log1pf(float x);
-long double log1pl(long double x);
-
-long lround(double x);
-long lroundf(float x);
-long lroundl(long double x);
-
-double nextafter(double x, double y);
-float nextafterf(float x, float y);
-long double nextafterl(long double x, long double y);
-
-double nexttoward(double x, long double y);
-float nexttowardf(float x, long double y);
-long double nexttowardl(long double x, long double y);
-
-double round(double x);
-float roundf(float x);
-long double roundl(long double x);
-
-double tgamma(double x);
-float tgammaf(float x);
-long double tgammal(long double x);
-
-double trunc(double x);
-float truncf(float x);
-long double truncl(long double x);
-
-}}}} // namespaces
-
-

- In addition sufficient additional overloads of the double - versions of the above functions are provided, so that calling the function - with any mixture of float, - double, long - double, or integer - arguments is supported, with the return type determined by the result - type calculation rules. -

-

- For example: -

-
acoshf(2.0f);  // float version, returns float.
-acosh(2.0f);   // also calls the float version and returns float.
-acosh(2.0);    // double version, returns double.
-acoshl(2.0L);  // long double version, returns a long double.
-acosh(2.0L);   // also calls the long double version.
-acosh(2);      // integer argument is treated as a double, returns double.
-
-
- - Quick - Reference -
-

- More detailed descriptions of these functions are available in the C99 standard. -

-
typedef unspecified float_t;
-typedef unspecified double_t;
-
-

- In this implementation float_t - is the same as type float, and - double_t the same as type - double unless the preprocessor - symbol FLT_EVAL_METHOD is defined, in which case these are set as follows: -

-
----- - - - - - - - - - - - - - - - - - - - - - - -
-

- FLT_EVAL_METHOD -

- 		-

- float_t -

- 		-

- double_t -

-
-

- 0 -

- 		-

- float -

- 		-

- double -

-
-

- 1 -

- 		-

- double -

- 		-

- double -

-
-

- 2 -

- 		-

- long double -

- 		-

- long double -

-
-
double acosh(double x);
-float acoshf(float x);
-long double acoshl(long double x);
-
-

- Returns the inverse hyperbolic cosine of x. -

-

- See also acosh - for the full template (header only) version of this function. -

-
double asinh(double x);
-float asinhf(float x);
-long double asinhl(long double x);
-
-

- Returns the inverse hyperbolic sine of x. -

-

- See also asinh - for the full template (header only) version of this function. -

-
double atanh(double x);
-float atanhf(float x);
-long double atanhl(long double x);
-
-

- Returns the inverse hyperbolic tangent of x. -

-

- See also atanh - for the full template (header only) version of this function. -

-
double cbrt(double x);
-float cbrtf(float x);
-long double cbrtl(long double x);
-
-

- Returns the cubed root of x. -

-

- See also cbrt for - the full template (header only) version of this function. -

-
double copysign(double x, double y);
-float copysignf(float x, float y);
-long double copysignl(long double x, long double y);
-
-

- Returns a value with the magnitude of x and the sign - of y. -

-
double erf(double x);
-float erff(float x);
-long double erfl(long double x);
-
-

- Returns the error function of x: -

-

- -

-

- See also erf - for the full template (header only) version of this function. -

-
double erfc(double x);
-float erfcf(float x);
-long double erfcl(long double x);
-
-

- Returns the complementary error function of x 1-erf(x) - without the loss of precision implied by the subtraction. -

-

- See also erfc - for the full template (header only) version of this function. -

-
double expm1(double x);
-float expm1f(float x);
-long double expm1l(long double x);
-
-

- Returns exp(x)-1 - without the loss of precision implied by the subtraction. -

-

- See also expm1 for - the full template (header only) version of this function. -

-
double fmax(double x, double y);
-float fmaxf(float x, float y);
-long double fmaxl(long double x, long double y);
-
-

- Returns the larger (most positive) of x and y. -

-
double fmin(double x, double y);
-float fminf(float x, float y);
-long double fminl(long double x, long double y);
-
-

- Returns the smaller (most negative) of x and y. -

-
double hypot(double x, double y);
-float hypotf(float x, float y);
-long double hypotl(long double x, long double y);
-
-

- Returns sqrt(x*x - + y*y) - without the danger of numeric overflow implied by that formulation. -

-

- See also hypot for - the full template (header only) version of this function. -

-
double lgamma(double x);
-float lgammaf(float x);
-long double lgammal(long double x);
-
-

- Returns the log of the gamma function of x. -

-

- -

-

- See also lgamma - for the full template (header only) version of this function. -

-
long long llround(double x);
-long long llroundf(float x);
-long long llroundl(long double x);
-
-

- Returns the value x rounded to the nearest integer as - a long long: - equivalent to floor(x + 0.5) -

-

- See also round for - the full template (header only) version of this function. -

-
double log1p(double x);
-float log1pf(float x);
-long double log1pl(long double x);
-
-

- Returns the log(x+1) without the loss of precision implied by - that formulation. -

-

- See also log1p for - the full template (header only) version of this function. -

-
long lround(double x);
-long lroundf(float x);
-long lroundl(long double x);
-
-

- Returns the value x rounded to the nearest integer as - a long: equivalent to floor(x + 0.5) -

-

- See also round for - the full template (header only) version of this function. -

-
double nextafter(double x, double y);
-float nextafterf(float x, float y);
-long double nextafterl(long double x, long double y);
-
-

- Returns the next representable floating point number after x - in the direction of y, or x if - x == - y. -

-
double nexttoward(double x, long double y);
-float nexttowardf(float x, long double y);
-long double nexttowardl(long double x, long double y);
-
-

- As nextafter, but with y - always expressed as a long double. -

-
double round(double x);
-float roundf(float x);
-long double roundl(long double x);
-
-

- Returns the value x rounded to the nearest integer: - equivalent to floor(x + 0.5) -

-

- See also round for - the full template (header only) version of this function. -

-
double tgamma(double x);
-float tgammaf(float x);
-long double tgammal(long double x);
-
-

- Returns the gamma function of x: -

-

- -

-

- See also tgamma - for the full template (header only) version of this function. -

-
double trunc(double x);
-float truncf(float x);
-long double truncl(long double x);
-
-

- Returns x truncated to the nearest integer. -

-

- See also trunc for - the full template (header only) version of this function. -

-

- See also C99 - ISO Standard -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/extern_c/tr1.html b/doc/html/math_toolkit/extern_c/tr1.html deleted file mode 100644 index ca36c035b..000000000 --- a/doc/html/math_toolkit/extern_c/tr1.html +++ /dev/null @@ -1,611 +0,0 @@ - - - -C99 and TR1 C Functions Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Many of the special functions included in this library are also a part of - the either the C99 - Standard ISO/IEC 9899:1999 or the Technical - Report on C++ Library Extensions. Therefore this library includes - a thin wrapper header boost/math/tr1.hpp - that provides compatibility with these two standards. -

-

- There are various pros and cons to using the library in this way: -

-

- Pros: -

-
    -
  • - The header to include is lightweight (i.e. fast to compile). -
    • -
  • - The functions have extern "C" linkage, and so are usable from - other languages (not just C and C++). -
    • -
  • - C99 and C++ TR1 Standard compatibility. -
    • -
-

- Cons: -

-
    -
  • - You will need to compile and link to the external Boost.Math libraries. -
    • -
  • - Limited to support for the types, float, - double and long double. -
    • -
  • - Error handling is handled via setting ::errno and returning NaN's and - infinities: this may be less flexible than an C++ exception based approach. -
    • -
-
- - - - - -
[Note]Note

- The separate libraries are required only - if you choose to use boost/math/tr1.hpp rather than some other Boost.Math - header, the rest of Boost.Math remains header-only. -

-

- The separate libraries required in order to use tr1.hpp can be compiled using - bjam from within the libs/math/build directory, or from the Boost root directory - using the usual Boost-wide install procedure. Alternatively the source files - are located in libs/math/src and each have the same name as the function - they implement. The various libraries are named as follows: -

-
----- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Name -

- 		-

- Type -

- 		-

- Functions -

-
-

- boost_math_c99f-<suffix> -

- 		-

- float -

- 		-

- C99 Functions -

-
-

- boost_math_c99-<suffix> -

- 		-

- double -

- 		-

- C99 Functions -

-
-

- boost_math_c99l-<suffix> -

- 		-

- long double -

- 		-

- C99 Functions -

-
-

- boost_math_tr1f-<suffix> -

- 		-

- float -

- 		-

- TR1 Functions -

-
-

- boost_math_tr1-<suffix> -

- 		-

- double -

- 		-

- TR1 Functions -

-
-

- boost_math_tr1l-<suffix> -

- 		-

- long double -

- 		-

- TR1 Functions -

-
-

- Where <suffix> encodes the compiler and build options - used to build the libraries: for example "libboost_math_tr1-vc80-mt-gd.lib" - would be the statically linked TR1 library to use with Visual C++ 8.0, in - multithreading debug mode, with the DLL VC++ runtime, where as "boost_math_tr1-vc80-mt.lib" - would be import library for the TR1 DLL to be used with Visual C++ 8.0 with - the release multithreaded DLL VC++ runtime. Refer to the getting started - guide for a full - explanation of the <suffix> meanings. -

-
- - - - - -
[Note]Note
-

- Visual C++ users will typically have the correct library variant to link - against selected for them by boost/math/tr1.hpp based on your compiler - settings. -

-

- Users will need to define BOOST_MATH_TR1_DYN_LINK when building their code - if they want to link against the DLL versions of these libraries rather - than the static versions. -

-

- Users can disable auto-linking by defining BOOST_MATH_TR1_NO_LIB when building: - this is typically only used when linking against a customised build of - the libraries. -

-
-
- - - - - -
[Note]Note

- Linux and Unix users will generally only have one variant of these libraries - installed, and can generally just link against -lboost_math_tr1 etc. -

-
- - Usage - Recomendations -
-

- This library now presents the user with a choice: -

-
    -
  • - To include the header only versions of the functions and have an easier - time linking, but a longer compile time. -
    • -
  • - To include the TR1 headers and link against an external library. -
    • -
-

- Which option you choose depends largely on how you prefer to work and how - your system is set up. -

-

- For example a casual user who just needs the acosh function, would probably - be better off including <boost/math/special_functions/acosh.hpp> - and using boost::math::acosh(x) in their - code. -

-

- However, for large scale software development where compile times are significant, - and where the Boost libraries are already built and installed on the system, - then including <boost/math/tr1.hpp> and using boost::math::tr1::acosh(x) will - speed up compile times, reduce object files sizes (since there are no templates - being instantiated any more), and also speed up debugging runtimes - since - the externally compiled libraries can be compiler optimised, rather than - built using full settings - the difference in performance between release - and debug builds can be as much as 20 times, so for complex applications - this can be a big win. -

-
- - Supported - C99 Functions -
-

- See also the quick reference guide - for these functions. -

-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-typedef unspecified float_t;
-typedef unspecified double_t;
-
-double acosh(double x);
-float acoshf(float x);
-long double acoshl(long double x);
-
-double asinh(double x);
-float asinhf(float x);
-long double asinhl(long double x);
-
-double atanh(double x);
-float atanhf(float x);
-long double atanhl(long double x);
-
-double cbrt(double x);
-float cbrtf(float x);
-long double cbrtl(long double x);
-
-double copysign(double x, double y);
-float copysignf(float x, float y);
-long double copysignl(long double x, long double y);
-
-double erf(double x);
-float erff(float x);
-long double erfl(long double x);
-
-double erfc(double x);
-float erfcf(float x);
-long double erfcl(long double x);
-
-double expm1(double x);
-float expm1f(float x);
-long double expm1l(long double x);
-
-double fmax(double x, double y);
-float fmaxf(float x, float y);
-long double fmaxl(long double x, long double y);
-
-double fmin(double x, double y);
-float fminf(float x, float y);
-long double fminl(long double x, long double y);
-
-double hypot(double x, double y);
-float hypotf(float x, float y);
-long double hypotl(long double x, long double y);
-
-double lgamma(double x);
-float lgammaf(float x);
-long double lgammal(long double x);
-
-long long llround(double x);
-long long llroundf(float x);
-long long llroundl(long double x);
-
-double log1p(double x);
-float log1pf(float x);
-long double log1pl(long double x);
-
-long lround(double x);
-long lroundf(float x);
-long lroundl(long double x);
-
-double nextafter(double x, double y);
-float nextafterf(float x, float y);
-long double nextafterl(long double x, long double y);
-
-double nexttoward(double x, long double y);
-float nexttowardf(float x, long double y);
-long double nexttowardl(long double x, long double y);
-
-double round(double x);
-float roundf(float x);
-long double roundl(long double x);
-
-double tgamma(double x);
-float tgammaf(float x);
-long double tgammal(long double x);
-
-double trunc(double x);
-float truncf(float x);
-long double truncl(long double x);
-
-}}}} // namespaces
-
-
- - Supported - TR1 Functions -
-

- See also the quick reference guide - for these functions. -

-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-// [5.2.1.1] associated Laguerre polynomials:
-double assoc_laguerre(unsigned n, unsigned m, double x);
-float assoc_laguerref(unsigned n, unsigned m, float x);
-long double assoc_laguerrel(unsigned n, unsigned m, long double x);
-
-// [5.2.1.2] associated Legendre functions:
-double assoc_legendre(unsigned l, unsigned m, double x);
-float assoc_legendref(unsigned l, unsigned m, float x);
-long double assoc_legendrel(unsigned l, unsigned m, long double x);
-
-// [5.2.1.3] beta function:
-double beta(double x, double y);
-float betaf(float x, float y);
-long double betal(long double x, long double y);
-
-// [5.2.1.4] (complete) elliptic integral of the first kind:
-double comp_ellint_1(double k);
-float comp_ellint_1f(float k);
-long double comp_ellint_1l(long double k);
-
-// [5.2.1.5] (complete) elliptic integral of the second kind:
-double comp_ellint_2(double k);
-float comp_ellint_2f(float k);
-long double comp_ellint_2l(long double k);
-
-// [5.2.1.6] (complete) elliptic integral of the third kind:
-double comp_ellint_3(double k, double nu);
-float comp_ellint_3f(float k, float nu);
-long double comp_ellint_3l(long double k, long double nu);
-
-// [5.2.1.8] regular modified cylindrical Bessel functions:
-double cyl_bessel_i(double nu, double x);
-float cyl_bessel_if(float nu, float x);
-long double cyl_bessel_il(long double nu, long double x);
-
-// [5.2.1.9] cylindrical Bessel functions (of the first kind):
-double cyl_bessel_j(double nu, double x);
-float cyl_bessel_jf(float nu, float x);
-long double cyl_bessel_jl(long double nu, long double x);
-
-// [5.2.1.10] irregular modified cylindrical Bessel functions:
-double cyl_bessel_k(double nu, double x);
-float cyl_bessel_kf(float nu, float x);
-long double cyl_bessel_kl(long double nu, long double x);
-
-// [5.2.1.11] cylindrical Neumann functions;
-// cylindrical Bessel functions (of the second kind):
-double cyl_neumann(double nu, double x);
-float cyl_neumannf(float nu, float x);
-long double cyl_neumannl(long double nu, long double x);
-
-// [5.2.1.12] (incomplete) elliptic integral of the first kind:
-double ellint_1(double k, double phi);
-float ellint_1f(float k, float phi);
-long double ellint_1l(long double k, long double phi);
-
-// [5.2.1.13] (incomplete) elliptic integral of the second kind:
-double ellint_2(double k, double phi);
-float ellint_2f(float k, float phi);
-long double ellint_2l(long double k, long double phi);
-
-// [5.2.1.14] (incomplete) elliptic integral of the third kind:
-double ellint_3(double k, double nu, double phi);
-float ellint_3f(float k, float nu, float phi);
-long double ellint_3l(long double k, long double nu, long double phi);
-
-// [5.2.1.15] exponential integral:
-double expint(double x);
-float expintf(float x);
-long double expintl(long double x);
-
-// [5.2.1.16] Hermite polynomials:
-double hermite(unsigned n, double x);
-float hermitef(unsigned n, float x);
-long double hermitel(unsigned n, long double x);
-
-// [5.2.1.18] Laguerre polynomials:
-double laguerre(unsigned n, double x);
-float laguerref(unsigned n, float x);
-long double laguerrel(unsigned n, long double x);
-
-// [5.2.1.19] Legendre polynomials:
-double legendre(unsigned l, double x);
-float legendref(unsigned l, float x);
-long double legendrel(unsigned l, long double x);
-
-// [5.2.1.20] Riemann zeta function:
-double riemann_zeta(double);
-float riemann_zetaf(float);
-long double riemann_zetal(long double);
-
-// [5.2.1.21] spherical Bessel functions (of the first kind):
-double sph_bessel(unsigned n, double x);
-float sph_besself(unsigned n, float x);
-long double sph_bessell(unsigned n, long double x);
-
-// [5.2.1.22] spherical associated Legendre functions:
-double sph_legendre(unsigned l, unsigned m, double theta);
-float sph_legendref(unsigned l, unsigned m, float theta);
-long double sph_legendrel(unsigned l, unsigned m, long double theta);
-
-// [5.2.1.23] spherical Neumann functions;
-// spherical Bessel functions (of the second kind):
-double sph_neumann(unsigned n, double x);
-float sph_neumannf(unsigned n, float x);
-long double sph_neumannl(unsigned n, long double x);
-
-}}}} // namespaces
-
-

- In addition sufficient additional overloads of the double - versions of the above functions are provided, so that calling the function - with any mixture of float, - double, long - double, or integer - arguments is supported, with the return type determined by the result - type calculation rules. -

-
- - Currently - Unsupported C99 Functions -
-
double exp2(double x);
-float exp2f(float x);
-long double exp2l(long double x);
-
-double fdim(double x, double y);
-float fdimf(float x, float y);
-long double fdiml(long double x, long double y);
-
-double fma(double x, double y, double z);
-float fmaf(float x, float y, float z);
-long double fmal(long double x, long double y, long double z);
-
-int ilogb(double x);
-int ilogbf(float x);
-int ilogbl(long double x);
-
-long long llrint(double x);
-long long llrintf(float x);
-long long llrintl(long double x);
-
-double log2(double x);
-float log2f(float x);
-long double log2l(long double x);
-
-double logb(double x);
-float logbf(float x);
-long double logbl(long double x);
-
-long lrint(double x);
-long lrintf(float x);
-long lrintl(long double x);
-
-double nan(const char *str);
-float nanf(const char *str);
-long double nanl(const char *str);
-
-double nearbyint(double x);
-float nearbyintf(float x);
-long double nearbyintl(long double x);
-
-double remainder(double x, double y);
-float remainderf(float x, float y);
-long double remainderl(long double x, long double y);
-
-double remquo(double x, double y, int *pquo);
-float remquof(float x, float y, int *pquo);
-long double remquol(long double x, long double y, int *pquo);
-
-double rint(double x);
-float rintf(float x);
-long double rintl(long double x);
-
-double scalbln(double x, long ex);
-float scalblnf(float x, long ex);
-long double scalblnl(long double x, long ex);
-
-double scalbn(double x, int ex);
-float scalbnf(float x, int ex);
-long double scalbnl(long double x, int ex);
-
-
- - Currently - Unsupported TR1 Functions -
-
// [5.2.1.7] confluent hypergeometric functions:
-double conf_hyperg(double a, double c, double x);
-float conf_hypergf(float a, float c, float x);
-long double conf_hypergl(long double a, long double c, long double x);
-
-// [5.2.1.17] hypergeometric functions:
-double hyperg(double a, double b, double c, double x);
-float hypergf(float a, float b, float c, float x);
-long double hypergl(long double a, long double b, long double c,
-long double x);
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/extern_c/tr1_ref.html b/doc/html/math_toolkit/extern_c/tr1_ref.html deleted file mode 100644 index 8b1339192..000000000 --- a/doc/html/math_toolkit/extern_c/tr1_ref.html +++ /dev/null @@ -1,530 +0,0 @@ - - - -TR1 C Functions Quick Reference - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Supported - TR1 Functions -
-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-// [5.2.1.1] associated Laguerre polynomials:
-double assoc_laguerre(unsigned n, unsigned m, double x);
-float assoc_laguerref(unsigned n, unsigned m, float x);
-long double assoc_laguerrel(unsigned n, unsigned m, long double x);
-
-// [5.2.1.2] associated Legendre functions:
-double assoc_legendre(unsigned l, unsigned m, double x);
-float assoc_legendref(unsigned l, unsigned m, float x);
-long double assoc_legendrel(unsigned l, unsigned m, long double x);
-
-// [5.2.1.3] beta function:
-double beta(double x, double y);
-float betaf(float x, float y);
-long double betal(long double x, long double y);
-
-// [5.2.1.4] (complete) elliptic integral of the first kind:
-double comp_ellint_1(double k);
-float comp_ellint_1f(float k);
-long double comp_ellint_1l(long double k);
-
-// [5.2.1.5] (complete) elliptic integral of the second kind:
-double comp_ellint_2(double k);
-float comp_ellint_2f(float k);
-long double comp_ellint_2l(long double k);
-
-// [5.2.1.6] (complete) elliptic integral of the third kind:
-double comp_ellint_3(double k, double nu);
-float comp_ellint_3f(float k, float nu);
-long double comp_ellint_3l(long double k, long double nu);
-
-// [5.2.1.8] regular modified cylindrical Bessel functions:
-double cyl_bessel_i(double nu, double x);
-float cyl_bessel_if(float nu, float x);
-long double cyl_bessel_il(long double nu, long double x);
-
-// [5.2.1.9] cylindrical Bessel functions (of the first kind):
-double cyl_bessel_j(double nu, double x);
-float cyl_bessel_jf(float nu, float x);
-long double cyl_bessel_jl(long double nu, long double x);
-
-// [5.2.1.10] irregular modified cylindrical Bessel functions:
-double cyl_bessel_k(double nu, double x);
-float cyl_bessel_kf(float nu, float x);
-long double cyl_bessel_kl(long double nu, long double x);
-
-// [5.2.1.11] cylindrical Neumann functions;
-// cylindrical Bessel functions (of the second kind):
-double cyl_neumann(double nu, double x);
-float cyl_neumannf(float nu, float x);
-long double cyl_neumannl(long double nu, long double x);
-
-// [5.2.1.12] (incomplete) elliptic integral of the first kind:
-double ellint_1(double k, double phi);
-float ellint_1f(float k, float phi);
-long double ellint_1l(long double k, long double phi);
-
-// [5.2.1.13] (incomplete) elliptic integral of the second kind:
-double ellint_2(double k, double phi);
-float ellint_2f(float k, float phi);
-long double ellint_2l(long double k, long double phi);
-
-// [5.2.1.14] (incomplete) elliptic integral of the third kind:
-double ellint_3(double k, double nu, double phi);
-float ellint_3f(float k, float nu, float phi);
-long double ellint_3l(long double k, long double nu, long double phi);
-
-// [5.2.1.15] exponential integral:
-double expint(double x);
-float expintf(float x);
-long double expintl(long double x);
-
-// [5.2.1.16] Hermite polynomials:
-double hermite(unsigned n, double x);
-float hermitef(unsigned n, float x);
-long double hermitel(unsigned n, long double x);
-
-// [5.2.1.18] Laguerre polynomials:
-double laguerre(unsigned n, double x);
-float laguerref(unsigned n, float x);
-long double laguerrel(unsigned n, long double x);
-
-// [5.2.1.19] Legendre polynomials:
-double legendre(unsigned l, double x);
-float legendref(unsigned l, float x);
-long double legendrel(unsigned l, long double x);
-
-// [5.2.1.20] Riemann zeta function:
-double riemann_zeta(double);
-float riemann_zetaf(float);
-long double riemann_zetal(long double);
-
-// [5.2.1.21] spherical Bessel functions (of the first kind):
-double sph_bessel(unsigned n, double x);
-float sph_besself(unsigned n, float x);
-long double sph_bessell(unsigned n, long double x);
-
-// [5.2.1.22] spherical associated Legendre functions:
-double sph_legendre(unsigned l, unsigned m, double theta);
-float sph_legendref(unsigned l, unsigned m, float theta);
-long double sph_legendrel(unsigned l, unsigned m, long double theta);
-
-// [5.2.1.23] spherical Neumann functions;
-// spherical Bessel functions (of the second kind):
-double sph_neumann(unsigned n, double x);
-float sph_neumannf(unsigned n, float x);
-long double sph_neumannl(unsigned n, long double x);
-
-}}}} // namespaces
-
-

- In addition sufficient additional overloads of the double - versions of the above functions are provided, so that calling the function - with any mixture of float, - double, long - double, or integer - arguments is supported, with the return type determined by the result - type calculation rules. -

-

- For example: -

-
expintf(2.0f);  // float version, returns float.
-expint(2.0f);   // also calls the float version and returns float.
-expint(2.0);    // double version, returns double.
-expintl(2.0L);  // long double version, returns a long double.
-expint(2.0L);   // also calls the long double version.
-expint(2);      // integer argument is treated as a double, returns double.
-
-
- - Quick - Reference -
-
// [5.2.1.1] associated Laguerre polynomials:
-double assoc_laguerre(unsigned n, unsigned m, double x);
-float assoc_laguerref(unsigned n, unsigned m, float x);
-long double assoc_laguerrel(unsigned n, unsigned m, long double x);
-
-

- The assoc_laguerre functions return: -

-

- -

-

- See also laguerre - for the full template (header only) version of this function. -

-
// [5.2.1.2] associated Legendre functions:
-double assoc_legendre(unsigned l, unsigned m, double x);
-float assoc_legendref(unsigned l, unsigned m, float x);
-long double assoc_legendrel(unsigned l, unsigned m, long double x);
-
-

- The assoc_legendre functions return: -

-

- -

-

- See also legendre_p - for the full template (header only) version of this function. -

-
// [5.2.1.3] beta function:
-double beta(double x, double y);
-float betaf(float x, float y);
-long double betal(long double x, long double y);
-
-

- Returns the beta function of x and y: -

-

- -

-

- See also beta - for the full template (header only) version of this function. -

-
// [5.2.1.4] (complete) elliptic integral of the first kind:
-double comp_ellint_1(double k);
-float comp_ellint_1f(float k);
-long double comp_ellint_1l(long double k);
-
-

- Returns the complete elliptic integral of the first kind of k: -

-

- -

-

- See also ellint_1 - for the full template (header only) version of this function. -

-
// [5.2.1.5] (complete) elliptic integral of the second kind:
-double comp_ellint_2(double k);
-float comp_ellint_2f(float k);
-long double comp_ellint_2l(long double k);
-
-

- Returns the complete elliptic integral of the second kind of k: -

-

- -

-

- See also ellint_2 - for the full template (header only) version of this function. -

-
// [5.2.1.6] (complete) elliptic integral of the third kind:
-double comp_ellint_3(double k, double nu);
-float comp_ellint_3f(float k, float nu);
-long double comp_ellint_3l(long double k, long double nu);
-
-

- Returns the complete elliptic integral of the third kind of k - and nu: -

-

- -

-

- See also ellint_3 - for the full template (header only) version of this function. -

-
// [5.2.1.8] regular modified cylindrical Bessel functions:
-double cyl_bessel_i(double nu, double x);
-float cyl_bessel_if(float nu, float x);
-long double cyl_bessel_il(long double nu, long double x);
-
-

- Returns the modified bessel function of the first kind of nu - and x: -

-

- -

-

- See also cyl_bessel_i - for the full template (header only) version of this function. -

-
// [5.2.1.9] cylindrical Bessel functions (of the first kind):
-double cyl_bessel_j(double nu, double x);
-float cyl_bessel_jf(float nu, float x);
-long double cyl_bessel_jl(long double nu, long double x);
-
-

- Returns the bessel function of the first kind of nu - and x: -

-

- -

-

- See also cyl_bessel_j - for the full template (header only) version of this function. -

-
// [5.2.1.10] irregular modified cylindrical Bessel functions:
-double cyl_bessel_k(double nu, double x);
-float cyl_bessel_kf(float nu, float x);
-long double cyl_bessel_kl(long double nu, long double x);
-
-

- Returns the modified bessel function of the second kind of nu - and x: -

-

- -

-

- See also cyl_bessel_k - for the full template (header only) version of this function. -

-
// [5.2.1.11] cylindrical Neumann functions;
-// cylindrical Bessel functions (of the second kind):
-double cyl_neumann(double nu, double x);
-float cyl_neumannf(float nu, float x);
-long double cyl_neumannl(long double nu, long double x);
-
-

- Returns the bessel function of the second kind (Neumann function) of nu - and x: -

-

- -

-

- See also cyl_neumann - for the full template (header only) version of this function. -

-
// [5.2.1.12] (incomplete) elliptic integral of the first kind:
-double ellint_1(double k, double phi);
-float ellint_1f(float k, float phi);
-long double ellint_1l(long double k, long double phi);
-
-

- Returns the incomplete elliptic integral of the first kind of k - and phi: -

-

- -

-

- See also ellint_1 - for the full template (header only) version of this function. -

-
// [5.2.1.13] (incomplete) elliptic integral of the second kind:
-double ellint_2(double k, double phi);
-float ellint_2f(float k, float phi);
-long double ellint_2l(long double k, long double phi);
-
-

- Returns the incomplete elliptic integral of the second kind of k - and phi: -

-

- -

-

- See also ellint_2 - for the full template (header only) version of this function. -

-
// [5.2.1.14] (incomplete) elliptic integral of the third kind:
-double ellint_3(double k, double nu, double phi);
-float ellint_3f(float k, float nu, float phi);
-long double ellint_3l(long double k, long double nu, long double phi);
-
-

- Returns the incomplete elliptic integral of the third kind of k, - nu and phi: -

-

- -

-

- See also ellint_3 - for the full template (header only) version of this function. -

-
// [5.2.1.15] exponential integral:
-double expint(double x);
-float expintf(float x);
-long double expintl(long double x);
-
-

- Returns the exponential integral Ei of x: -

-

- -

-

- See also expint - for the full template (header only) version of this function. -

-
// [5.2.1.16] Hermite polynomials:
-double hermite(unsigned n, double x);
-float hermitef(unsigned n, float x);
-long double hermitel(unsigned n, long double x);
-
-

- Returns the n'th Hermite polynomial of x: -

-

- -

-

- See also hermite - for the full template (header only) version of this function. -

-
// [5.2.1.18] Laguerre polynomials:
-double laguerre(unsigned n, double x);
-float laguerref(unsigned n, float x);
-long double laguerrel(unsigned n, long double x);
-
-

- Returns the n'th Laguerre polynomial of x: -

-

- -

-

- See also laguerre - for the full template (header only) version of this function. -

-
// [5.2.1.19] Legendre polynomials:
-double legendre(unsigned l, double x);
-float legendref(unsigned l, float x);
-long double legendrel(unsigned l, long double x);
-
-

- Returns the l'th Legendre polynomial of x: -

-

- -

-

- See also legendre_p - for the full template (header only) version of this function. -

-
// [5.2.1.20] Riemann zeta function:
-double riemann_zeta(double);
-float riemann_zetaf(float);
-long double riemann_zetal(long double);
-
-

- Returns the Riemann Zeta function of x: -

-

- -

-

- See also zeta for - the full template (header only) version of this function. -

-
// [5.2.1.21] spherical Bessel functions (of the first kind):
-double sph_bessel(unsigned n, double x);
-float sph_besself(unsigned n, float x);
-long double sph_bessell(unsigned n, long double x);
-
-

- Returns the spherical Bessel function of the first kind of x - jn(x): -

-

- -

-

- See also sph_bessel - for the full template (header only) version of this function. -

-
// [5.2.1.22] spherical associated Legendre functions:
-double sph_legendre(unsigned l, unsigned m, double theta);
-float sph_legendref(unsigned l, unsigned m, float theta);
-long double sph_legendrel(unsigned l, unsigned m, long double theta);
-
-

- Returns the spherical associated Legendre function of l, - m and theta: -

-

- -

-

- See also spherical_harmonic - for the full template (header only) version of this function. -

-
// [5.2.1.23] spherical Neumann functions;
-// spherical Bessel functions (of the second kind):
-double sph_neumann(unsigned n, double x);
-float sph_neumannf(unsigned n, float x);
-long double sph_neumannl(unsigned n, long double x);
-
-

- Returns the spherical Neumann function of x yn(x): -

-

- -

-

- See also sph_bessel - for the full template (header only) version of this function. -

-
- - Currently - Unsupported TR1 Functions -
-
// [5.2.1.7] confluent hypergeometric functions:
-double conf_hyperg(double a, double c, double x);
-float conf_hypergf(float a, float c, float x);
-long double conf_hypergl(long double a, long double c, long double x);
-
-// [5.2.1.17] hypergeometric functions:
-double hyperg(double a, double b, double c, double x);
-float hypergf(float a, float b, float c, float x);
-long double hypergl(long double a, long double b, long double c,
-long double x);
-
-
- - - - - -
[Note]Note

- These two functions are not implemented as they are not believed to be - numerically stable. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/gcd_lcm.html b/doc/html/math_toolkit/gcd_lcm.html deleted file mode 100644 index a35da6245..000000000 --- a/doc/html/math_toolkit/gcd_lcm.html +++ /dev/null @@ -1,54 +0,0 @@ - - - -Integer Utilities (Greatest Common Divisor and Least Common Multiple) - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/gcd_lcm/compile_time.html b/doc/html/math_toolkit/gcd_lcm/compile_time.html deleted file mode 100644 index 9f0950a06..000000000 --- a/doc/html/math_toolkit/gcd_lcm/compile_time.html +++ /dev/null @@ -1,98 +0,0 @@ - - - -Compile time GCD and LCM determination - - - - - - - - -
-PrevUpHomeNext -
-
-
-

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

-
typedef unspecified static_gcd_type;
-
-template < static_gcd_type Value1, static_gcd_type Value2 >
-struct boost::math::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
-{
-};
-
-template < static_gcd_type Value1, static_gcd_type Value2 >
-struct boost::math::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
-{
-};
-
-

- The type static_gcd_type - is the widest unsigned-integer-type that is supported for use in integral-constant-expressions - by the compiler. Usually this the same type as boost::uintmax_t, - but may fall back to being unsigned - long for some older compilers. -

-

- The boost::math::static_gcd and boost::math::static_lcm class templates take - two value-based template parameters of the static_gcd_type - type and inherit from the type boost::mpl::integral_c. Inherited from the base class, - they have a member value that 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 static_gcd_type. -

-

- - Example -

-
#include <boost/math/common_factor.hpp>
-#include <algorithm>
-#include <iterator>
-#include <iostream>
-
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/gcd_lcm/credits.html b/doc/html/math_toolkit/gcd_lcm/credits.html deleted file mode 100644 index 94eafd9ca..000000000 --- a/doc/html/math_toolkit/gcd_lcm/credits.html +++ /dev/null @@ -1,42 +0,0 @@ - - - -Credits - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/gcd_lcm/demo.html b/doc/html/math_toolkit/gcd_lcm/demo.html deleted file mode 100644 index d1224b023..000000000 --- a/doc/html/math_toolkit/gcd_lcm/demo.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Demonstration Program - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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 -
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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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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  • - 17 Dec 2005: Converted documentation to Quickbook Format. -
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    • -
  • - 7 Nov 2001: Initial version -
    • -
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- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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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. -

-
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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-
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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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-
-
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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 );
-
-typedef see-below static_gcd_type;
-
-template < static_gcd_type Value1, static_gcd_type Value2 >
-   struct static_gcd;
-template < static_gcd_type Value1, static_gcd_type Value2 >
-   struct static_lcm;
-
-}
-}
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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- 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. -

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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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

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

-

- Formula: acos -

-
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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-
-

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

-

- Formula: acosh -

-
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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-std::complex<T> asin(const std::complex<T>& z);
-
-

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

-

- Formula: asin -

-
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Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

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-std::complex<T> asinh(const std::complex<T>& z);
-
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- Effects: returns the inverse hyperbolic - sine of the complex number z. -

-

- Formula: asinh -

-
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-PrevUpHomeNext -
- - diff --git a/doc/html/math_toolkit/inverse_complex/atan.html b/doc/html/math_toolkit/inverse_complex/atan.html deleted file mode 100644 index 835ce435e..000000000 --- a/doc/html/math_toolkit/inverse_complex/atan.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -atan - - - - - - - - -
-PrevUpHomeNext -
-
-

-atan -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/inverse_complex/atanh.html b/doc/html/math_toolkit/inverse_complex/atanh.html deleted file mode 100644 index 923f3e7b4..000000000 --- a/doc/html/math_toolkit/inverse_complex/atanh.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -atanh - - - - - - - - -
-PrevUpHomeNext -
-
-

-atanh -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/inverse_complex/history.html b/doc/html/math_toolkit/inverse_complex/history.html deleted file mode 100644 index cf173e3d9..000000000 --- a/doc/html/math_toolkit/inverse_complex/history.html +++ /dev/null @@ -1,44 +0,0 @@ - - - -History - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/inverse_complex/implementation.html b/doc/html/math_toolkit/inverse_complex/implementation.html deleted file mode 100644 index 49a0c46be..000000000 --- a/doc/html/math_toolkit/inverse_complex/implementation.html +++ /dev/null @@ -1,54 +0,0 @@ - - - -Implementation and Accuracy - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 9.5ε 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview.html b/doc/html/math_toolkit/main_overview.html deleted file mode 100644 index 3bcf23fcc..000000000 --- a/doc/html/math_toolkit/main_overview.html +++ /dev/null @@ -1,64 +0,0 @@ - - - -Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
About the Math Toolkit
-
Navigation
-
Document Conventions
-
Other Hints and tips
-
Directory and - File Structure
-
Namespaces
-
Calculation of - the Type of the Result
-
Error Handling
-
Compilers
-
Configuration - Macros
-
Policies
-
Thread Safety
-
Performance
-
If and How to Build - a Boost.Math Library, and its Examples and Tests
-
History and What's - New
-
C99 and C++ TR1 C-style - Functions
-
Frequently Asked Questions - FAQ
-
Contact Info and - Support
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/building.html b/doc/html/math_toolkit/main_overview/building.html deleted file mode 100644 index bd939cef1..000000000 --- a/doc/html/math_toolkit/main_overview/building.html +++ /dev/null @@ -1,154 +0,0 @@ - - - -If and How to Build a Boost.Math Library, and its Examples and Tests - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Building - a Library (shared, dynamic .dll or static .lib) -
-

- The first thing you need to ask yourself is "Do I need to build anything - at all?" as the bulk of this library is header only: meaning you can - use it just by #including the necessary header(s). -

-

- For most simple uses, including a header (or few) is best for compile time - and program size. -

-

- Refer to C99 and C++ TR1 C-style Functions - for pros and cons of using the TR1 components as opposed to the header only - ones. -

-

- The only time you need to build - the library is if you want to use the extern - "C" functions declared in - <boost/math/tr1.hpp>. To build this using Boost.Build, from - a commandline boost-root directory issue a command like: -

-
bjam toolset=gcc --with-math install
-
-

- that will do the job on Linux, while: -

-
bjam toolset=msvc --with-math --build-type=complete stage
-
-

- will work better on Windows (leaving libraries built in sub-folder /stage - below your Boost root directory). Either way you should consult the getting - started guide for more information. -

-

- You can also build the libraries from your favourite IDE or command line - tool: each extern "C" - function declared in <boost/math/tr1.hpp> has its own source file with the same - name in libs/math/src/tr1. Just - select the sources corresponding to the functions you are using and build - them into a library, or else add them directly to your project. Note that - the directory libs/math/src/tr1 will - need to be in your compiler's #include path as well as the boost-root directory - (MSVC Tools, Options, Projects and Solutions, VC++ Directories, Include files). -

-
- - - - - -
[Note]Note

- If you are using a Windows compiler that supports auto-linking and you - have built the sources yourself (or added them directly to your project) - then you will need to prevent <boost/math/tr1.hpp> - from trying to auto-link to the binaries that Boost.Build generates. You - can do this by defining either BOOST_MATH_NO_LIB or BOOST_ALL_NO_LIB at - project level (so the defines get passed to each compiler invocation). -

-

- Optionally the sources in libs/math/src/tr1 - have support for using libs/math/src/tr1/pch.hpp as a precompiled header if - your compiler supports precompiled headers. Note that normally - this header is a do-nothing include: to activate the header so that it #includes - everything required by all the sources you will need to define BOOST_BUILD_PCH_ENABLED - on the command line, both when building the pre-compiled header and when - building the sources. Boost.Build will do this automatically when appropriate. -

-
- - Building - the Examples -
-

- The examples are all located in libs/math/example, they can all be built without - reference to any external libraries, either with Boost.Build using the supplied - Jamfile, or from your compiler's command line. The only requirement is that - the Boost headers are in your compilers #include search path. -

-
- - Building - the Tests -
-

- The tests are located in libs/math/test and are best built using Boost.Build - and the supplied Jamfile. If you plan to build them separately from your - favourite IDE then you will need to add libs/math/test to the list of your compiler's search - paths. -

-

- You will also need to build and link to the Boost.Regex library for many - of the tests: this can built from the command line by following the getting - started guide, using a command such as: -

-
bjam toolset=gcc --with-regex install
-
-

- or -

-
bjam toolset=msvc --with-regex --build-type=complete stage
-
-

- depending on whether you are on Linux or Windows. -

-

- Many of the tests have optional precompiled header support using the header - libs/math/test/pch.hpp. Note that normally this header is - a do-nothing include: to activate the header so that it #includes everything - required by all the sources you will need to define BOOST_BUILD_PCH_ENABLED - on the command line, both when building the pre-compiled header and when - building the sources. Boost.Build will do this automatically when appropriate. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/compilers_overview.html b/doc/html/math_toolkit/main_overview/compilers_overview.html deleted file mode 100644 index 1702ae543..000000000 --- a/doc/html/math_toolkit/main_overview/compilers_overview.html +++ /dev/null @@ -1,673 +0,0 @@ - - - -Compilers - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 9. 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. -

-
-

- Windows 7/Netbeans 7.2 -

- 		-

- Clang 3.1 -

- 		-

- Yes -

- 		-

- Spot examples OK. Expect all tests to compile and run OK. -

-
-

- 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 -

- 		-

- Clang 3.2 -

- 		-

- Yes -

- 		-

- All tests OK. -

-
-

- 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. -

-
-

- Apple Mac OS X, -

- 		-

- Clang 3.2 -

- 		-

- Yes -

- 		-

- All tests expected to be OK. -

-
-

- IBM AIX -

- 		-

- IBM xlc 5.3 -

- 		-

- Yes -

- 		-

- All tests pass except for our fpclassify tests which fail due to - a bug in std::numeric_limits, the bug effects - the test code, not fpclassify itself. The IBM compiler group are - aware of the problem. -

-
-
-
-

Table 10. Unsupported Compilers

-
---- - - - - - - - - - - - - - - -
-

- Platform -

- 		-

- Compiler -

-
-

- Windows -

- 		-

- Borland C++ 5.9.2 (Borland Developer Studio 2007) -

-
-

- Windows -

- 		-

- MSVC 6 and 7 -

-
-
-

- If your 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/config_macros.html b/doc/html/math_toolkit/main_overview/config_macros.html deleted file mode 100644 index b7c9605cf..000000000 --- a/doc/html/math_toolkit/main_overview/config_macros.html +++ /dev/null @@ -1,405 +0,0 @@ - - - -Configuration Macros - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Almost all configuration details are set up automatically by <boost\math\tools\config.hpp>. -

-

- In normal use, only policy configuration macros are likely to be used. See - policy reference. -

-

- For reference, information on Boost.Math macros used internally are described - briefly below. -

-
-

Table 11. Boost.Math Macros

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- MACRO -

- 		-

- Notes -

-
-

- BOOST_MATH_NO_REAL_CONCEPT_TESTS -

- 		-

- Do not try to use real concept tests (hardware or software does - not support real_concept type). -

-
-

- BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS -

- 		-

- Do not produce or use long double functions: this macro gets set - when the platform's long double or standard library long double - support is absent or buggy. -

-
-

- BOOST_MATH_CONTROL_FP -

- 		-

- Controls FP hardware exceptions - our tests don't support hardware - exceptions on MSVC. May get set to something like: _control87(MCW_EM,MCW_EM). -

-
-

- BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS -

- 		-

- This macro is used by our test cases, it is set when an assignment - of a function template to a function pointer requires explicit - template arguments to be provided on the function name. -

-
-

- BOOST_MATH_USE_C99 -

- 		-

- Use C99 math functions. -

-
-

- BOOST_NO_NATIVE_LONG_DOUBLE_FP_CLASSIFY -

- 		-

- define if no native (or buggy) fpclassify(long - double) - even though the other C99 functions are present. -

-
-

- BOOST_MATH_EXPLICIT_TEMPLATE_TYPE -

- 		-

- Workaround helper macro for broken compilers - composed from other - Boost.Config macros, do not edit. -

-
-

- BOOST_MATH_EXPLICIT_TEMPLATE_NON_TYPE -

- 		-

- Workaround helper macro for broken compilers - composed from other - Boost.Config macros, do not edit. -

-
-

- BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_TYPE_SPEC -

- 		-

- Workaround helper macro for broken compilers - composed from other - Boost.Config macros, do not edit. -

-
-

- BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_NON_TYPE_SPEC -

- 		-

- Workaround helper macro for broken compilers - composed from other - Boost.Config macros, do not edit. -

-
-

- BOOST_MATH_SMALL_CONSTANT(x) -

- 		-

- Helper macro used in our test cases to set underflowing constants - set to zero if this would cause compiler issues. -

-
-

- BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS -

- 		-

- Set if constants too large for a float, will cause "bad" - values to be stored in the data, rather than infinity or a suitably - large value. -

-
-

- BOOST_MATH_STD_USING -

- 		-

- Provides using statements - for many std:: (abs to sqrt) and boost::math (rounds, modf) functions. - This allows these functions to be called unqualified so that if - argument-dependent - Argument Dependent Lookup fails to find a suitable overload, - then the std:: versions will also be considered. -

-
-

- BOOST_FPU_EXCEPTION_GUARD -

- 		-

- Used at the entrypoint to each special function to reset all FPU - exception flags prior to internal calculations, and then merge - the old and new exception flags on function exit. Used as a workaround - on platforms or hardware that behave strangely if any FPU exception - flags are set when calling standard library functions. -

-
-

- BOOST_MATH_INSTRUMENT -

- 		-

- Define to output diagnostics for math functions. This is rather - 'global' to Boost.Math and so coarse-grained that it will probably - produce copious output! (Especially because full precision values - are output). Designed primarily for internal use and development. -

-
-

- BOOST_MATH_INSTRUMENT_CODE(x) -

- 		-

- Output selected named variable, for example BOOST_MATH_INSTRUMENT_CODE("guess = " - << guess); Used by BOOST_MATH_INSTRUMENT -

-
-

- BOOST_MATH_INSTRUMENT_VARIABLE(name) -

- 		-

- Output selected variable, for example BOOST_MATH_INSTRUMENT_VARIABLE(result); Used by BOOST_MATH_INSTRUMENT -

-
-

- BOOST_MATH_INSTRUMENT_FPU -

- 		-

- Output the state of the FPU's control flags. -

-
-
-
-

Table 12. Boost.Math Tuning

-
---- - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Macros for Tuning performance options for specific compilers -

- 		-

- Notes -

-
-

- BOOST_MATH_POLY_METHOD -

- 		-

- See the performance tuning - section. -

-
-

- BOOST_MATH_RATIONAL_METHOD -

- 		-

- See the performance tuning - section. -

-
-

- BOOST_MATH_MAX_POLY_ORDER -

- 		-

- See the performance tuning - section. -

-
-

- BOOST_MATH_INT_TABLE_TYPE -

- 		-

- See the performance tuning - section. -

-
-

- BOOST_MATH_INT_VALUE_SUFFIX -

- 		-

- Helper macro for appending the correct suffix to integer constants - which may actually be stored as reals depending on the value of - BOOST_MATH_INT_TABLE_TYPE. -

-
-
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/contact.html b/doc/html/math_toolkit/main_overview/contact.html deleted file mode 100644 index 9424ac236..000000000 --- a/doc/html/math_toolkit/main_overview/contact.html +++ /dev/null @@ -1,55 +0,0 @@ - - - -Contact Info and Support - - - - - - - - -
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- The main support for this library is via the Boost mailing lists: -

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- You can also find JM at john - at - johnmaddock.co.uk and PAB at pbristow - - at - hetp.u-net.com. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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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- This documentation aims to use of the following naming and formatting conventions. -

-
    -
  • - C++ Code is in fixed width font - and is syntax-highlighted in color, for example double. -
    • -
  • - Other code is in block teletype fixed-width font. -
    • -
  • - Replaceable text that you will need - to supply is in italics. -
    • -
  • - If a name refers to a free function, it is specified like this: free_function(); - that is, it is in code font and its name is - followed by () to indicate - that it is a free function. -
    • -
  • - If a name refers to a class template, it is specified like this: class_template<>; - that is, it is in code font and its name is followed by <> to indicate that it is a class - template. -
    • -
  • - If a name refers to a function-like macro, it is specified like this: - MACRO(); - that is, it is uppercase in code font and its name is followed by () to indicate that it is a function-like - macro. Object-like macros appear without the trailing (). -
    • -
  • - Names that refer to concepts in the generic programming - sense (like template parameter names) are specified in CamelCase. -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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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- - boost/math -
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-
-
/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, - GNU Multiple Precision Arithmetic Library, - GNU MPFR library 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. -

-
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- - boost/libs -
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/doc/
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- 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 many .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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/error_handling.html b/doc/html/math_toolkit/main_overview/error_handling.html deleted file mode 100644 index e295c510f..000000000 --- a/doc/html/math_toolkit/main_overview/error_handling.html +++ /dev/null @@ -1,1112 +0,0 @@ - - - -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? -
    • -
-
- - - - - -
[Warning]Warning

- The default error actions are to throw an exception with an informative - error message. If you do not try to catch the exception, you will not see - the message! -

-

- 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. -

-
Rounding Error
-

- Occurs when the argument to one of the rounding functions trunc, - round and - modf can not - be represented as an integer type, is outside the range of the result - type. -

-
Evaluation Error
-

- Occurs if no method of evaluation is known, or when an internal error - occurred 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. -

-
Indeterminate Result Error
-

- Occurs when the result of a function is not defined for the values - that were passed to it. -

-
-
-

- 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 Rounding Errors

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- Action -

- 		-

- Behaviour -

-
-

- throw_on_error -

- 		-

- Throws boost::math::rounding_error -

-
-

- errno_on_error -

- 		-

- Sets ::errno - to ERANGE and returns - the largest representable value of the target integer type (or - the most negative value if the argument to the function was less - than zero). -

-
-

- ignore_error -

- 		-

- Returns the largest representable value of - the target integer type (or the most negative value if the argument - to the function was less than zero). -

-
-

- user_error -

- 		-

- Returns the result of boost::math::policies::user_rounding_error: - this - function must be defined by the user. -

-
-
-
-

Table 7. 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 - the closest approximation found. -

-
-

- ignore_error -

- 		-

- Returns the closest approximation found. -

-
-

- user_error -

- 		-

- Returns the result of boost::math::policies::user_evaluation_error: - this - function must be defined by the user. -

-
-
-
-

Table 8. Possible Actions for Indeterminate Result Errors

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- Action -

- 		-

- Behaviour -

-
-

- throw_on_error -

- 		-

- Throws std::domain_error -

-
-

- errno_on_error -

- 		-

- Sets ::errno - to EDOM and returns - the same value as ignore_error. -

-
-

- ignore_error -

- 		-

- Returns a default result that depends on - the function where the error occurred. -

-
-

- user_error -

- 		-

- Returns the result of boost::math::policies::user_indeterminate_result_error: - this - function must be defined by the user. -

-
-
-

- All these error conditions are in namespace boost::math::policies, made available, - for example, a by namespace declaration using namespace - boost::math::policies; - or individual using declarations using - boost::math::policies::overflow_error;. -

-
- - 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. -
    • -
  • - If there is more than one error, only the first detected will be reported - in the throw message. -
    • -
-
- - 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. -

-
- - Indeterminate Result Errors -
-

- When the result of a special function is indeterminate for the value that - was passed to it, then the function returns the result of: -

-
boost::math::policies::raise_overflow_error<T>(FunctionName, Message, Val, Default, 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 for which the - result is indeterminate, Default is an alternative default result that must - be returned for ignore_error - and errno_on_erro policies, - and Policy is the current policy - in use for the function that was called. -

-

- The default policy for this function is ignore_error: - note that this error type is reserved for situations where the result is - mathematically undefined or indeterminate, but there is none the less a convention - for what the result should be: for example the C99 standard specifies that - the result of 00 is 1, even though the result is actually mathematically indeterminate. -

-
- - Rounding Errors -
-

- When one of the rounding functions round, - trunc or modf - is called with an argument that has no integer representation, or is too - large to be represented in the result type then the value returned is the - result of a call to: -

-
boost::math::policies::raise_rounding_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 argument, and Policy - is the current policy in use for the called function. -

-

- The default behaviour of this function is to throw a boost::math::rounding_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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/faq.html b/doc/html/math_toolkit/main_overview/faq.html deleted file mode 100644 index 0bbb03afb..000000000 --- a/doc/html/math_toolkit/main_overview/faq.html +++ /dev/null @@ -1,219 +0,0 @@ - - - -Frequently Asked Questions FAQ - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  1. - I'm a FORTRAN/NAG/SPSS/SAS/Cephes/MathCad/R user and I don't - see where the functions like dnorm(mean, sd) are in Boost.Math? -
    Nearly all are provided, and many more like mean, skewness, quantiles, - complements ... but Boost.Math makes full use of C++, and it looks a - bit different. But do not panic! See section on construction and the - many examples. Briefly, the distribution is constructed with the parameters - (like location and scale) (things after the | in representation like - P(X=k|n, p) or ; in a common represention of pdf f(x; μσ2). Functions like - pdf, cdf are called with the name of that distribution and the random - variate often called x or k. For example, normal - my_norm(0, 1); pdf(my_norm, 2.0); -
    -
    2. -
  2. - I'm a user of New - SAS Functions for Computing Probabilities.
    You will find - the interface more familar, but to be able to select a distribution (perhaps - using a string) see the Extras/Future Directions section, and /boost/libs/math/dot_net_example/boost_math.cpp - for an example that is used to create a C# utility (that you might also - find useful): see Statistical - Distribution Explorer
    . -
    3. -
  3. - I'm allegic to reading manuals and prefer to learn from examples.
    - Fear not - you are not alone! Many examples are available for functions - and distributions. Some are referenced directly from the text. Others - can be found at \boost_latest_release\libs\math\example. If you are a - Visual Studio user, you should be able to create projects from each of - these, making sure that the Boost library is in the include directories - list. -
    4. -
  4. - How do I make sure that the Boost library is in the Visual - Studio include directories list?
    You can add an include - path, for example, your Boost place /boost-latest_release, for example - X:/boost_1_45_0/ - if you have a separate partition X for Boost releases. Or you can use - an environment variable BOOST_ROOT set to your Boost place, and include - that. Visual Studio before 2010 provided Tools, Options, VC++ Directories - to control directories: Visual Studio 2010 instead provides property - sheets to assist. You may find it convenient to create a new one adding - \boost-latest_release; to the existing include items in $(IncludePath). -
    5. -
  5. - I'm a FORTRAN/NAG/SPSS/SAS/Cephes/MathCad/R user and I don't - see where the properties like mean, median, mode, variance, skewness - of distributions are in Boost.Math?
    They are all available - (if defined for the parameters with which you constructed the distribution) - via 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. -
    6. -
  6. - I am a C programmer. Can I user Boost.Math with C?
    - Yes you can, including all the special functions, and TR1 functions like - isnan. They appear as C functions, by being declared as "extern - C". -
    7. -
  7. - I am a C# (Basic? F# FORTRAN? Other CLI?) programmer. Can I - use Boost.Math with C#?
    Yes you can, including all - the special functions, and TR1 functions like isnan. But you must build the Boost.Math as a dynamic library (.dll) and - compile with the /CLI option. See the boost/math/dot_net_example - folder which contains an example that builds a simple statistical distribution - app with a GUI. See Statistical - Distribution Explorer
    -
    8. -
  8. - What these "policies" things for?
    - Policies are a powerful (if necessarily complex) fine-grain mechanism - that allow you to customise the behaviour of the Boost.Math library according - to your precise needs. See Policies. - But if, very probably, the default behaviour suits you, you don't need - to know more. -
    9. -
  9. - I am a C user and expect to see global C-style::errno - set for overflow/errors etc?
    You can achieve what you - want - see error - handling policies and user - error handling and many examples. -
    10. -
  10. - I am a C user and expect to silently return a max value for - overflow?
    You (and C++ users too) can return whatever - you want on overflow - see overflow_error - and error - handling policies and several examples. -
    11. -
  11. - I don't want any error message for overflow etc? -
    You can control exactly what happens for all the abnormal conditions, - including the values returned. See domain_error, - overflow_error error - handling policies user - error handling etc and examples. -
    12. -
  12. - My environment doesn't allow and/or I don't want exceptions. - Can I still user Boost.Math?
    Yes but you must customise - the error handling: see user - error handling and changing - policies defaults . -
    13. -
  13. - The docs are several hundreds of pages long! Can I read the - docs off-line or on paper?
    Yes - you can download the - Boost current release of most documentation as a zip of pdfs (including - Boost.Math) from Sourceforge, for example https://sourceforge.net/projects/boost/files/boost-docs/1.45.0/boost_pdf_1_45_0.tar.gz/download. - And you can print any pages you need (or even print all pages - but be - warned that there are several hundred!). Both html and pdf versions are - highly hyperlinked. The entire Boost.Math pdf can be searched with Adobe - Reader, Edit, Find ... This can often find what you seek, a partial substitute - for a full index. -
    14. -
  14. - I want a compact version for an embedded application. Can I - use float precision?
    Yes - by selecting RealType template - parameter as float: for example normal_distribution<float> your_normal(mean, - sd); (But double may still be used internally, so space saving may be - less that you hope for). You can also change the promotion policy, but - accuracy might be much reduced. -
    15. -
  15. - I seem to get somewhat different results compared to other - programs. Why? We hope Boost.Math to be more accurate: our - priority is accuracy (over speed). See the section on accuracy. But for - evaluations that require iterations there are parameters which can change - the required accuracy. You might be able to squeeze a little more accuracy - at the cost of runtime. -
    16. -
  16. - Will my program run more slowly compared to other math functions - and statistical libraries? Probably, thought not always, and - not by too much: our priority is accuracy. For most functions, making - sure you have the latest compiler version with all optimisations switched - on is the key to speed. For evaluations that require iteration, you may - be able to gain a little more speed at the expense of accuracy. See detailed - suggestions and results on performance. -
    17. -
  17. - How do I handle infinity and NaNs portably?
    - See nonfinite fp_facets - for Facets for Floating-Point Infinities and NaNs. -
    18. -
  18. - Where are the pre-built libraries?
    Good news - - you probably don't need any! - just #include - <boost/math/distribution_you_want>. - But in the unlikely event that you do, see building - libraries. -
    19. -
  19. - I don't see the function or distribution that I want. -
    You could try an email to ask the authors - but no promises! -
    20. -
  20. - I need more decimal digits for values/computations. -
    You can use Boost.Math with Boost.Multiprecision: - typically cpp_dec_float - is a useful user-defined type to provide a fixed number of decimal digits, - usually 50 or 100. -
    21. -
  21. - Why can't I write something really simple like cpp_int - one(1); cpp_dec_float_50 - two(2); one - * two; Because cpp_int - might be bigger than cpp_dec_float - can hold, - so you must make an explicit conversion. - See mixed - multiprecision arithmetic and conversion. -
    22. -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/hints.html b/doc/html/math_toolkit/main_overview/hints.html deleted file mode 100644 index 0692e4ce5..000000000 --- a/doc/html/math_toolkit/main_overview/hints.html +++ /dev/null @@ -1,103 +0,0 @@ - - - -Other Hints and tips - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - If you have a feature request, or if it appears that the implementation - is in error, please search first in the Boost - Trac. -
    • -
  • - Trac entries may - indicate that updates or corrections that solve your problem are in - Boost-trunk - where changes are being assembled and tested ready for the next release. - You may, at your own risk, download new versions from there. -
    • -
  • - If you do not understand why things work the way they do, see the rationale - section. -
    • -
  • - If you do not find your idea/feature/complaint, please reach the author - preferably through the Boost development list, or email the author(s) - direct. -
    • -
-
- - Admonishments -
-
- - - - - -
[Note]Note

- In addition, notes such as this one specify non-essential information that - provides additional background or rationale. -

-
- - - - - -
[Tip]Tip

- These blocks contain information that you may find helpful while coding. -

-
- - - - - -
[Important]Important

- These contain information that is imperative to understanding a concept. - Failure to follow suggestions in these blocks will probably result in undesired - behavior. Read all of these you find. -

-
- - - - - -
[Warning]Warning

- Failure to heed this will lead to incorrect, and very likely undesired, - results. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/history1.html b/doc/html/math_toolkit/main_overview/history1.html deleted file mode 100644 index f43157b34..000000000 --- a/doc/html/math_toolkit/main_overview/history1.html +++ /dev/null @@ -1,628 +0,0 @@ - - - -History and What's New - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Currently open bug reports can be viewed here. -

-

- All bug reports including closed ones can be viewed here. -

-
- - Boost-1.54 -
-
    -
  • - Added many references to Boost.Multiprecision and cpp_dec_float_50 - as an example of a User-defined Type (UDT). -
    • -
  • - Added Clang to list of supported compilers. -
    • -
  • - Fixed constants to use a thread-safe cache of computed values when used - at arbitrary precision. -
    • -
  • - Added finding zeros of Bessel functions cyl_bessel_j_zero, - cyl_neumann_zero, air_ai_zeros and air_bi_zeros(by - Christopher Kormanyos). -
    • -
  • - More accuracy improvements to the Bessel J and Y functions from Rocco - Romeo. -
    • -
  • - Fixed nasty cyclic dependency bug that caused some headers to not compile - #7999. -
    • -
  • - Fixed bug in tgamma - that caused spurious overflow for arguments between 142.5 and 143. -
    • -
  • - Fixed bug in raise_rounding_error that caused it to return an incorrect - result when throwing an exception is turned off #7905. -
    • -
  • - Added minimal __float128 support. -
    • -
  • - Fixed bug in edge-cases of poisson quantile #8308. -
    • -
  • - Adjusted heuristics used in Halley iteration to cope with inverting the - incomplete beta in tricky regions where the derivative is flatlining. - Example is computing the quantile of the Fisher F distribution for probabilities - smaller than machine epsilon. See ticket #8314. -
    • -
-
- - Boost-1.53 -
-
    -
  • - Fixed issues #7325, - #7415 - and #7416, - #7183, - #7649, - #7694, - #4445, - #7492, - #7891, - #7429. -
    • -
  • - Fixed mistake in calculating pooled standard deviation in two-sample - students t example #7402. -
    • -
  • - Improve complex acos/asin/atan, see #7290, - #7291. -
    • -
  • - Improve accuracy in some corner cases of cyl_bessel_j - and gamma_p/gamma_q thanks - to suggestions from Rocco Romeo. -
    • -
  • - Improve accuracy of Bessel J and Y for integer orders thanks to suggestions - from Rocco Romeo. -
    • -
-
- - Boost-1.52 -
-
    -
  • - Corrected moments for small degrees of freedom #7177 - (reported by Thomas Mang). -
    • -
  • - Added Airy functions - and Jacobi Elliptic functions. -
    • -
  • - Corrected failure to detect bad parameters in many distributions #6934 (reported - by Florian Schoppmann) by adding a function check_out_of_range to test - many possible bad parameters. This test revealed several distributions - where the checks for bad parameters were ineffective, and these have - been rectified. -
    • -
  • - Fixed issue in Hankel functions that causes incorrect values to be returned - for x < 0 and ν odd, see #7135. -
    • -
  • - Fixed issues #6517, - #6362, - #7053, - #2693, - #6937, - #7099. -
    • -
  • - Permitted infinite degrees of freedom #7259 - implemented using the normal distribution (requested by Thomas Mang). -
    • -
  • - Much enhanced accuracy for large degrees of freedom ν and/or large non-centrality - δ -by switching to use the Students t distribution (or Normal distribution - for infinite degrees of freedom) centered at delta, when δ / (4 * ν) < - epsilon for the floating-point type in use. #7259. - It was found that the incomplete beta was suffering from serious cancellation - errors when degrees of freedom was very large. (That has now been fixed - in our code, but any code based on Didonato and Morris's original papers - (probably every implementation out there actually) will have the same - issue). -
    • -
-
- - Boost-1.51 -
-

- See Boost-1.52 - some items were added but not listed in time for the release. -

-
- - Boost-1.50 -
-
    -
  • - Promoted math constants to be 1st class citizens, including convenient - access to the most widely used built-in float, double, long double via - three namespaces. -
    • -
  • - Added the Owen's T function and Skew Normal distribution written by Benjamin - Sobotta: see Owens T - and skew_normal_distrib. -
    • -
  • - Added Hankel functions cyl_hankel_1, - cyl_hankel_2, - sph_hankel_1 - and sph_hankel_2. -
    • -
  • - Corrected issue #6627 - nonfinite_num_put formatting of 0.0 is incorrect based on a patch - submitted by K R Walker. -
    • -
  • - Changed constant initialization mechanism so that it is thread safe even - for user-defined types, also so that user defined types get the full - precision of the constant, even when long - double does not. So for example - 128-bit rational approximations will work with UDT's and do the right - thing, even though long double may be only 64 or 80 bits. -
    • -
  • - Fixed issue in bessel_jy - which causes Y8.5(4π) to yield a NaN. -
    • -
-
- - Boost-1.49 -
-
    -
  • - Deprecated wrongly named twothirds - math constant in favour of two_thirds - (with underscore separator). (issue #6199). -
    • -
  • - Refactored test data and some special function code to improve support - for arbitary precision and/or expression-template-enabled types. -
    • -
  • - Added new faster zeta function evaluation method. -
    • -
-

- Fixed issues: -

-
    -
  • - Corrected CDF complement for Laplace distribution (issue #6151). -
    • -
  • - Corrected branch cuts on the complex inverse trig functions, to handle - signed zeros (issue #6171). -
    • -
  • - Fixed bug in bessel_yn - which caused incorrect overflow errors to be raised for negative n - (issue #6367). -
    • -
  • - Also fixed minor/cosmetic/configuration issues #6120, - #6191, - #5982, - #6130, - #6234, - #6307, - #6192. -
    • -
-
- - Boost-1.48 -
-
    -
  • - Added new series evaluation methods to the cyclic Bessel I, J, K and - Y functions. Also taken great care to avoid spurious over and underflow - of these functions. Fixes issue #5560 -
    • -
  • - Added an example of using Inverse Chi-Squared distribution for Bayesian - statistics, provided by Thomas Mang. -
    • -
  • - Added tests to use improved version of lexical_cast which handles C99 - nonfinites without using globale facets. -
    • -
  • - Corrected wrong out-of-bound uniform distribution CDF complement values - #5733. -
    • -
  • - Enabled long double support on OpenBSD (issue #6014). -
    • -
  • - Changed nextafter and related functions to behave in the same way as - other implementations - so that nextafter(+INF, 0) is a finite value - (issue #5832). -
    • -
  • - Changed tuple include configuration to fix issue when using in conjunction - with Boost.Tr1 (issue #5934). -
    • -
  • - Changed class eps_tolerance to behave correctly when both ends of the - range are zero (issue #6001). -
    • -
  • - Fixed missing include guards on prime.hpp (issue #5927). -
    • -
  • - Removed unused/undocumented constants from constants.hpp (issue #5982). -
    • -
  • - Fixed missing std:: prefix in nonfinite_num_facets.hpp (issue #5914). -
    • -
  • - Minor patches for Cray compiler compatibility. -
    • -
-
- - Boost-1.47 -
-
    -
  • - Added changesign function to sign.hpp to facilitate addition of nonfinite - facets. -
    • -
  • - Addition of nonfinite facets from Johan Rade, with tests, examples of - use for C99 format infinity and NaN, and documentation. -
    • -
  • - Added tests and documentation of changesign from Johan Rade. -
    • -
-
- - Boost-1.46.1 -
-
-
- - Boost-1.46.0 -
-
    -
  • - Added Wald, Inverse Gaussian and geometric distributions. -
    • -
  • - Added information about configuration macros. -
    • -
  • - Added support for mpreal as a real-numbered type. -
    • -
-
- - Boost-1.45.0 -
-
    -
  • - Added warnings about potential ambiguity with std random library in distribution - and function names. -
    • -
  • - Added inverse gamma distribution and inverse chi_square and scaled inverse - chi_square. -
    • -
  • - Editorial revision of documentation, and added FAQ. -
    • -
-
- - Boost-1.44.0 -
-
    -
  • - Fixed incorrect range and support for Rayleigh distribution. -
    • -
  • - Fixed numerical error in the quantile of the Student's T distribution: - the function was returning garbage values for non-integer degrees of - freedom between 2 and 3. -
    • -
-
- - Boost-1.41.0 -
-
  • - Significantly improved performance for the incomplete gamma function - and its inverse. -
-
- - Boost-1.40.0 -
-
    -
  • - Added support for MPFR as a bignum type. -
    • -
  • - Added some full specializations of the policy classes to reduce compile - times. -
    • -
  • - Added logistic and hypergeometric distributions, from Gautam Sewani's - Google Summer of Code project. -
    • -
  • - Added Laplace distribution submitted by Thijs van den Berg. -
    • -
  • - Updated performance test code to include new distributions, and improved - the performance of the non-central distributions. -
    • -
  • - Added SSE2 optimised Lanczos - approximation code, from Gautam Sewani's Google Summer of Code - project. -
    • -
  • - Fixed bug in cyl_bessel_i that used an incorrect approximation for ν = - 0.5, also effects the non-central Chi Square Distribution when ν = 3, see - bug report #2877. -
    • -
  • - Fixed minor bugs #2873. -
    • -
-
- - Boost-1.38.0 -
-
    -
  • - Added Johan Råde's optimised floating point classification routines. -
    • -
  • - Fixed code so that it compiles in GCC's -pedantic mode (bug report #1451). -
    • -
-
- - Boost-1.37.0 -
-
  • - Improved accuracy and testing of the inverse hypergeometric functions. -
-
- - Boost-1.36.0 -
-
    -
  • - Added Noncentral Chi Squared Distribution. -
    • -
  • - Added Noncentral Beta Distribution. -
    • -
  • - Added Noncentral F Distribution. -
    • -
  • - Added Noncentral T Distribution. -
    • -
  • - Added Exponential Integral Functions. -
    • -
  • - Added Zeta Function. -
    • -
  • - Added Rounding and Truncation functions. -
    • -
  • - Added Compile time powers of runtime bases. -
    • -
  • - Added SSE2 optimizations for Lanczos evaluation. -
    • -
-
- - Boost-1.35.0: - 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: . -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/intro.html b/doc/html/math_toolkit/main_overview/intro.html deleted file mode 100644 index b0bbe1784..000000000 --- a/doc/html/math_toolkit/main_overview/intro.html +++ /dev/null @@ -1,126 +0,0 @@ - - - -About the Math Toolkit - - - - - - - - -
-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, including Boost.Multiprecision, - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/namespaces.html b/doc/html/math_toolkit/main_overview/namespaces.html deleted file mode 100644 index 67c1439c9..000000000 --- a/doc/html/math_toolkit/main_overview/namespaces.html +++ /dev/null @@ -1,87 +0,0 @@ - - - -Namespaces - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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);
-
-
- - - - - -
[Warning]Warning

- Some distribution names are also used in std random library, so to avoid - the risk of ambiguity it is better to make explicit using declarations, - for example: using boost::math::students_t_distribution -

-

- 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. -

-
- - - - - -
[Tip]Tip

- Many code snippets assume implicit namespace(s), for example, std:: - or boost::math. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/navigation.html b/doc/html/math_toolkit/main_overview/navigation.html deleted file mode 100644 index e00567743..000000000 --- a/doc/html/math_toolkit/main_overview/navigation.html +++ /dev/null @@ -1,94 +0,0 @@ - - - -Navigation - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- -

-

- Boost.Math documentation is provided in both HTML and PDF formats. -

-
    -
  • - Tutorials are listed in the Table - of Contents and include many examples that should help you - get started quickly. -
    • -
  • - Source code of the many Examples - will often be your quickest starting point. -
    • -
  • - Index (general) includes all entries. -
    • -
  • - Specific Indexes list only functions, - class signatures, macros and typedefs. -
    • -
-
- - Using the - Indexes -
-

- The main index will usually help, especially if you know a word describing - what it does, without needing to know the exact name chosen for the function. -

-
- - - - - -
[Tip]Tip

- When using the index, keep in mind that clicking on an index term only - takes you to the section containing the - index entry. This section may be several pages long, so you may need to - use the find facility of your browser - or PDF reader to get to the index term itself. -

-
- - - - - -
[Tip]Tip

- A PDF reader usually allows a global find; - this can be really useful if the term you expect to be indexed is not the - one chosen by the authors. You might find searching the PDF version and - viewing the HTML version will locate an elusive item. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/perf_over.html b/doc/html/math_toolkit/main_overview/perf_over.html deleted file mode 100644 index 592afea22..000000000 --- a/doc/html/math_toolkit/main_overview/perf_over.html +++ /dev/null @@ -1,73 +0,0 @@ - - - -Performance - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- By and large the performance of this library should be acceptable for most - needs. However, you should note that this 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 FORTRAN 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/pol_overview.html b/doc/html/math_toolkit/main_overview/pol_overview.html deleted file mode 100644 index 286c72eb0..000000000 --- a/doc/html/math_toolkit/main_overview/pol_overview.html +++ /dev/null @@ -1,109 +0,0 @@ - - - -Policies - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/result_type.html b/doc/html/math_toolkit/main_overview/result_type.html deleted file mode 100644 index 9fdd5f03c..000000000 --- a/doc/html/math_toolkit/main_overview/result_type.html +++ /dev/null @@ -1,143 +0,0 @@ - - - -Calculation of the Type of the Result - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/main_overview/threads.html b/doc/html/math_toolkit/main_overview/threads.html deleted file mode 100644 index 7855f6672..000000000 --- a/doc/html/math_toolkit/main_overview/threads.html +++ /dev/null @@ -1,42 +0,0 @@ - - - -Thread Safety - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The library is fully thread safe and re-entrant for all functions regards - of the data type they are instantiated on. Thread safety limitations relating - to user defined types present in previous releases (prior to 1.50.0) have - been removed. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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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-
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- - diff --git a/doc/html/math_toolkit/main_overview/tr1.html b/doc/html/math_toolkit/main_overview/tr1.html deleted file mode 100644 index b38ae9b7a..000000000 --- a/doc/html/math_toolkit/main_overview/tr1.html +++ /dev/null @@ -1,612 +0,0 @@ - - - -C99 and C++ TR1 C-style Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Many of the special functions included in this library are also a part of - the either the C99 - Standard ISO/IEC 9899:1999 or the Technical - Report on C++ Library Extensions. Therefore this library includes - a thin wrapper header boost/math/tr1.hpp - that provides compatibility with these two standards. -

-

- There are various pros and cons to using the library in this way: -

-

- Pros: -

-
    -
  • - The header to include is lightweight (i.e. fast to compile). -
    • -
  • - The functions have extern "C" linkage, and so are usable from - other languages (not just C and C++). -
    • -
  • - C99 and C++ TR1 Standard compatibility. -
    • -
-

- Cons: -

-
    -
  • - You will need to compile and link to the external Boost.Math libraries. -
    • -
  • - Limited to support for the types, float, - double and long double. -
    • -
  • - Error handling is handled via setting ::errno and returning NaN's and - infinities: this may be less flexible than an C++ exception based approach. -
    • -
-
- - - - - -
[Note]Note

- The separate libraries are required only - if you choose to use boost/math/tr1.hpp rather than some other Boost.Math - header, the rest of Boost.Math remains header-only. -

-

- The separate libraries required in order to use tr1.hpp can be compiled using - bjam from within the libs/math/build directory, or from the Boost root directory - using the usual Boost-wide install procedure. Alternatively the source files - are located in libs/math/src and each have the same name as the function - they implement. The various libraries are named as follows: -

-
----- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Name -

- 		-

- Type -

- 		-

- Functions -

-
-

- boost_math_c99f-<suffix> -

- 		-

- float -

- 		-

- C99 Functions -

-
-

- boost_math_c99-<suffix> -

- 		-

- double -

- 		-

- C99 Functions -

-
-

- boost_math_c99l-<suffix> -

- 		-

- long double -

- 		-

- C99 Functions -

-
-

- boost_math_tr1f-<suffix> -

- 		-

- float -

- 		-

- TR1 Functions -

-
-

- boost_math_tr1-<suffix> -

- 		-

- double -

- 		-

- TR1 Functions -

-
-

- boost_math_tr1l-<suffix> -

- 		-

- long double -

- 		-

- TR1 Functions -

-
-

- Where <suffix> encodes the compiler and build options - used to build the libraries: for example "libboost_math_tr1-vc80-mt-gd.lib" - would be the statically linked TR1 library to use with Visual C++ 8.0, in - multithreading debug mode, with the DLL VC++ runtime, where as "boost_math_tr1-vc80-mt.lib" - would be import library for the TR1 DLL to be used with Visual C++ 8.0 with - the release multithreaded DLL VC++ runtime. Refer to the getting started - guide for a full - explanation of the <suffix> meanings. -

-
- - - - - -
[Note]Note
-

- Visual C++ users will typically have the correct library variant to link - against selected for them by boost/math/tr1.hpp based on your compiler - settings. -

-

- Users will need to define BOOST_MATH_TR1_DYN_LINK when building their code - if they want to link against the DLL versions of these libraries rather - than the static versions. -

-

- Users can disable auto-linking by defining BOOST_MATH_TR1_NO_LIB when building: - this is typically only used when linking against a customised build of - the libraries. -

-
-
- - - - - -
[Note]Note

- Linux and Unix users will generally only have one variant of these libraries - installed, and can generally just link against -lboost_math_tr1 etc. -

-
- - Usage - Recomendations -
-

- This library now presents the user with a choice: -

-
    -
  • - To include the header only versions of the functions and have an easier - time linking, but a longer compile time. -
    • -
  • - To include the TR1 headers and link against an external library. -
    • -
-

- Which option you choose depends largely on how you prefer to work and how - your system is set up. -

-

- For example a casual user who just needs the acosh function, would probably - be better off including <boost/math/special_functions/acosh.hpp> - and using boost::math::acosh(x) in their - code. -

-

- However, for large scale software development where compile times are significant, - and where the Boost libraries are already built and installed on the system, - then including <boost/math/tr1.hpp> and using boost::math::tr1::acosh(x) will - speed up compile times, reduce object files sizes (since there are no templates - being instantiated any more), and also speed up debugging runtimes - since - the externally compiled libraries can be compiler optimised, rather than - built using full settings - the difference in performance between release - and debug builds can be as much as 20 times, so for complex applications - this can be a big win. -

-
- - Supported - C99 Functions -
-

- See also the quick reference guide - for these functions. -

-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-typedef unspecified float_t;
-typedef unspecified double_t;
-
-double acosh(double x);
-float acoshf(float x);
-long double acoshl(long double x);
-
-double asinh(double x);
-float asinhf(float x);
-long double asinhl(long double x);
-
-double atanh(double x);
-float atanhf(float x);
-long double atanhl(long double x);
-
-double cbrt(double x);
-float cbrtf(float x);
-long double cbrtl(long double x);
-
-double copysign(double x, double y);
-float copysignf(float x, float y);
-long double copysignl(long double x, long double y);
-
-double erf(double x);
-float erff(float x);
-long double erfl(long double x);
-
-double erfc(double x);
-float erfcf(float x);
-long double erfcl(long double x);
-
-double expm1(double x);
-float expm1f(float x);
-long double expm1l(long double x);
-
-double fmax(double x, double y);
-float fmaxf(float x, float y);
-long double fmaxl(long double x, long double y);
-
-double fmin(double x, double y);
-float fminf(float x, float y);
-long double fminl(long double x, long double y);
-
-double hypot(double x, double y);
-float hypotf(float x, float y);
-long double hypotl(long double x, long double y);
-
-double lgamma(double x);
-float lgammaf(float x);
-long double lgammal(long double x);
-
-long long llround(double x);
-long long llroundf(float x);
-long long llroundl(long double x);
-
-double log1p(double x);
-float log1pf(float x);
-long double log1pl(long double x);
-
-long lround(double x);
-long lroundf(float x);
-long lroundl(long double x);
-
-double nextafter(double x, double y);
-float nextafterf(float x, float y);
-long double nextafterl(long double x, long double y);
-
-double nexttoward(double x, long double y);
-float nexttowardf(float x, long double y);
-long double nexttowardl(long double x, long double y);
-
-double round(double x);
-float roundf(float x);
-long double roundl(long double x);
-
-double tgamma(double x);
-float tgammaf(float x);
-long double tgammal(long double x);
-
-double trunc(double x);
-float truncf(float x);
-long double truncl(long double x);
-
-}}}} // namespaces
-
-
- - Supported - TR1 Functions -
-

- See also the quick reference guide - for these functions. -

-
namespace boost{ namespace math{ namespace tr1{ extern "C"{
-
-// [5.2.1.1] associated Laguerre polynomials:
-double assoc_laguerre(unsigned n, unsigned m, double x);
-float assoc_laguerref(unsigned n, unsigned m, float x);
-long double assoc_laguerrel(unsigned n, unsigned m, long double x);
-
-// [5.2.1.2] associated Legendre functions:
-double assoc_legendre(unsigned l, unsigned m, double x);
-float assoc_legendref(unsigned l, unsigned m, float x);
-long double assoc_legendrel(unsigned l, unsigned m, long double x);
-
-// [5.2.1.3] beta function:
-double beta(double x, double y);
-float betaf(float x, float y);
-long double betal(long double x, long double y);
-
-// [5.2.1.4] (complete) elliptic integral of the first kind:
-double comp_ellint_1(double k);
-float comp_ellint_1f(float k);
-long double comp_ellint_1l(long double k);
-
-// [5.2.1.5] (complete) elliptic integral of the second kind:
-double comp_ellint_2(double k);
-float comp_ellint_2f(float k);
-long double comp_ellint_2l(long double k);
-
-// [5.2.1.6] (complete) elliptic integral of the third kind:
-double comp_ellint_3(double k, double nu);
-float comp_ellint_3f(float k, float nu);
-long double comp_ellint_3l(long double k, long double nu);
-
-// [5.2.1.8] regular modified cylindrical Bessel functions:
-double cyl_bessel_i(double nu, double x);
-float cyl_bessel_if(float nu, float x);
-long double cyl_bessel_il(long double nu, long double x);
-
-// [5.2.1.9] cylindrical Bessel functions (of the first kind):
-double cyl_bessel_j(double nu, double x);
-float cyl_bessel_jf(float nu, float x);
-long double cyl_bessel_jl(long double nu, long double x);
-
-// [5.2.1.10] irregular modified cylindrical Bessel functions:
-double cyl_bessel_k(double nu, double x);
-float cyl_bessel_kf(float nu, float x);
-long double cyl_bessel_kl(long double nu, long double x);
-
-// [5.2.1.11] cylindrical Neumann functions;
-// cylindrical Bessel functions (of the second kind):
-double cyl_neumann(double nu, double x);
-float cyl_neumannf(float nu, float x);
-long double cyl_neumannl(long double nu, long double x);
-
-// [5.2.1.12] (incomplete) elliptic integral of the first kind:
-double ellint_1(double k, double phi);
-float ellint_1f(float k, float phi);
-long double ellint_1l(long double k, long double phi);
-
-// [5.2.1.13] (incomplete) elliptic integral of the second kind:
-double ellint_2(double k, double phi);
-float ellint_2f(float k, float phi);
-long double ellint_2l(long double k, long double phi);
-
-// [5.2.1.14] (incomplete) elliptic integral of the third kind:
-double ellint_3(double k, double nu, double phi);
-float ellint_3f(float k, float nu, float phi);
-long double ellint_3l(long double k, long double nu, long double phi);
-
-// [5.2.1.15] exponential integral:
-double expint(double x);
-float expintf(float x);
-long double expintl(long double x);
-
-// [5.2.1.16] Hermite polynomials:
-double hermite(unsigned n, double x);
-float hermitef(unsigned n, float x);
-long double hermitel(unsigned n, long double x);
-
-// [5.2.1.18] Laguerre polynomials:
-double laguerre(unsigned n, double x);
-float laguerref(unsigned n, float x);
-long double laguerrel(unsigned n, long double x);
-
-// [5.2.1.19] Legendre polynomials:
-double legendre(unsigned l, double x);
-float legendref(unsigned l, float x);
-long double legendrel(unsigned l, long double x);
-
-// [5.2.1.20] Riemann zeta function:
-double riemann_zeta(double);
-float riemann_zetaf(float);
-long double riemann_zetal(long double);
-
-// [5.2.1.21] spherical Bessel functions (of the first kind):
-double sph_bessel(unsigned n, double x);
-float sph_besself(unsigned n, float x);
-long double sph_bessell(unsigned n, long double x);
-
-// [5.2.1.22] spherical associated Legendre functions:
-double sph_legendre(unsigned l, unsigned m, double theta);
-float sph_legendref(unsigned l, unsigned m, float theta);
-long double sph_legendrel(unsigned l, unsigned m, long double theta);
-
-// [5.2.1.23] spherical Neumann functions;
-// spherical Bessel functions (of the second kind):
-double sph_neumann(unsigned n, double x);
-float sph_neumannf(unsigned n, float x);
-long double sph_neumannl(unsigned n, long double x);
-
-}}}} // namespaces
-
-

- In addition sufficient additional overloads of the double - versions of the above functions are provided, so that calling the function - with any mixture of float, - double, long - double, or integer - arguments is supported, with the return type determined by the result - type calculation rules. -

-
- - Currently - Unsupported C99 Functions -
-
double exp2(double x);
-float exp2f(float x);
-long double exp2l(long double x);
-
-double fdim(double x, double y);
-float fdimf(float x, float y);
-long double fdiml(long double x, long double y);
-
-double fma(double x, double y, double z);
-float fmaf(float x, float y, float z);
-long double fmal(long double x, long double y, long double z);
-
-int ilogb(double x);
-int ilogbf(float x);
-int ilogbl(long double x);
-
-long long llrint(double x);
-long long llrintf(float x);
-long long llrintl(long double x);
-
-double log2(double x);
-float log2f(float x);
-long double log2l(long double x);
-
-double logb(double x);
-float logbf(float x);
-long double logbl(long double x);
-
-long lrint(double x);
-long lrintf(float x);
-long lrintl(long double x);
-
-double nan(const char *str);
-float nanf(const char *str);
-long double nanl(const char *str);
-
-double nearbyint(double x);
-float nearbyintf(float x);
-long double nearbyintl(long double x);
-
-double remainder(double x, double y);
-float remainderf(float x, float y);
-long double remainderl(long double x, long double y);
-
-double remquo(double x, double y, int *pquo);
-float remquof(float x, float y, int *pquo);
-long double remquol(long double x, long double y, int *pquo);
-
-double rint(double x);
-float rintf(float x);
-long double rintl(long double x);
-
-double scalbln(double x, long ex);
-float scalblnf(float x, long ex);
-long double scalblnl(long double x, long ex);
-
-double scalbn(double x, int ex);
-float scalbnf(float x, int ex);
-long double scalbnl(long double x, int ex);
-
-
- - Currently - Unsupported TR1 Functions -
-
// [5.2.1.7] confluent hypergeometric functions:
-double conf_hyperg(double a, double c, double x);
-float conf_hypergf(float a, float c, float x);
-long double conf_hypergl(long double a, long double c, long double x);
-
-// [5.2.1.17] hypergeometric functions:
-double hyperg(double a, double b, double c, double x);
-float hypergf(float a, float b, float c, float x);
-long double hypergl(long double a, long double b, long double c,
-long double x);
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions.html b/doc/html/math_toolkit/octonions.html deleted file mode 100644 index d6c7d6ca9..000000000 --- a/doc/html/math_toolkit/octonions.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -Octonions - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/acknowledgements.html b/doc/html/math_toolkit/octonions/acknowledgements.html deleted file mode 100644 index a2cef28a6..000000000 --- a/doc/html/math_toolkit/octonions/acknowledgements.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Acknowledgements - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/header.html b/doc/html/math_toolkit/octonions/header.html deleted file mode 100644 index 952888a65..000000000 --- a/doc/html/math_toolkit/octonions/header.html +++ /dev/null @@ -1,39 +0,0 @@ - - - -Header File - - - - - - - - -
-PrevUpHomeNext -
-
-
-

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

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/history.html b/doc/html/math_toolkit/octonions/history.html deleted file mode 100644 index 9975c6690..000000000 --- a/doc/html/math_toolkit/octonions/history.html +++ /dev/null @@ -1,102 +0,0 @@ - - - -History - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/mem_fun.html b/doc/html/math_toolkit/octonions/mem_fun.html deleted file mode 100644 index 73ee7b2da..000000000 --- a/doc/html/math_toolkit/octonions/mem_fun.html +++ /dev/null @@ -1,252 +0,0 @@ - - - -Octonion Member Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/non_mem.html b/doc/html/math_toolkit/octonions/non_mem.html deleted file mode 100644 index b742300cb..000000000 --- a/doc/html/math_toolkit/octonions/non_mem.html +++ /dev/null @@ -1,213 +0,0 @@ - - - -Octonion Non-Member Operators - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/oct_create.html b/doc/html/math_toolkit/octonions/oct_create.html deleted file mode 100644 index 1c546f6a2..000000000 --- a/doc/html/math_toolkit/octonions/oct_create.html +++ /dev/null @@ -1,75 +0,0 @@ - - - -Octonion Creation Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
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 The R Project for Statistical - Computing 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/oct_specialization.html b/doc/html/math_toolkit/octonions/oct_specialization.html deleted file mode 100644 index 1900aa302..000000000 --- a/doc/html/math_toolkit/octonions/oct_specialization.html +++ /dev/null @@ -1,232 +0,0 @@ - - - -Octonion Specializations - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/oct_trans.html b/doc/html/math_toolkit/octonions/oct_trans.html deleted file mode 100644 index 8cda4e864..000000000 --- a/doc/html/math_toolkit/octonions/oct_trans.html +++ /dev/null @@ -1,141 +0,0 @@ - - - -Octonions Transcendentals - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/oct_typedefs.html b/doc/html/math_toolkit/octonions/oct_typedefs.html deleted file mode 100644 index fd80d92a7..000000000 --- a/doc/html/math_toolkit/octonions/oct_typedefs.html +++ /dev/null @@ -1,63 +0,0 @@ - - - -Octonion Member Typedefs - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/oct_value_ops.html b/doc/html/math_toolkit/octonions/oct_value_ops.html deleted file mode 100644 index e0e5ee9eb..000000000 --- a/doc/html/math_toolkit/octonions/oct_value_ops.html +++ /dev/null @@ -1,98 +0,0 @@ - - - -Octonion Value Operations - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/octonion.html b/doc/html/math_toolkit/octonions/octonion.html deleted file mode 100644 index 8652af82a..000000000 --- a/doc/html/math_toolkit/octonions/octonion.html +++ /dev/null @@ -1,104 +0,0 @@ - - - -Template Class octonion - - - - - - - - -
-PrevUpHomeNext -
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/overview.html b/doc/html/math_toolkit/octonions/overview.html deleted file mode 100644 index 1910c1c0a..000000000 --- a/doc/html/math_toolkit/octonions/overview.html +++ /dev/null @@ -1,76 +0,0 @@ - - - -Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/synopsis.html b/doc/html/math_toolkit/octonions/synopsis.html deleted file mode 100644 index f7e410e64..000000000 --- a/doc/html/math_toolkit/octonions/synopsis.html +++ /dev/null @@ -1,131 +0,0 @@ - - - -Synopsis - - - - - - - - -
-PrevUpHomeNext -
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/tests.html b/doc/html/math_toolkit/octonions/tests.html deleted file mode 100644 index eaa035dd1..000000000 --- a/doc/html/math_toolkit/octonions/tests.html +++ /dev/null @@ -1,51 +0,0 @@ - - - -Test Program - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/octonions/todo.html b/doc/html/math_toolkit/octonions/todo.html deleted file mode 100644 index 6c3513dd7..000000000 --- a/doc/html/math_toolkit/octonions/todo.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -To Do - - - - - - - - -
-PrevUpHomeNext -
-
-

-To Do -

-
    -
  • - Improve testing. -
    • -
  • - Rewrite input operatore using Spirit (creates a dependency). -
    • -
  • - Put in place an Expression Template mechanism (perhaps borrowing from - uBlas). -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf.html b/doc/html/math_toolkit/perf.html deleted file mode 100644 index a7c0d765d..000000000 --- a/doc/html/math_toolkit/perf.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Performance - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/comp_compilers.html b/doc/html/math_toolkit/perf/comp_compilers.html deleted file mode 100644 index cff02e1ed..000000000 --- a/doc/html/math_toolkit/perf/comp_compilers.html +++ /dev/null @@ -1,369 +0,0 @@ - - - -Comparing Compilers - - - - - - - - -
-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 54. 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)

-

-

- 		-

-

-

1.00

-

-

(1.483e-007s)

-

-

- 		-

-

-

3.24

-

-

(1.336e-007s)

-

-

-
-

- erf_inv -

- 		-

-

-

1.00

-

-

(4.439e-008s)

-

-

- 		-

-

-

1.00

-

-

(4.888e-007s)

-

-

- 		-

-

-

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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/comparisons.html b/doc/html/math_toolkit/perf/comparisons.html deleted file mode 100644 index c8abc7ff0..000000000 --- a/doc/html/math_toolkit/perf/comparisons.html +++ /dev/null @@ -1,3128 +0,0 @@ - - - -Comparisons to Other Open Source Libraries - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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-1.13 and Cephes -
-

- All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, - Windows Vista machine, with the test program compiled with Microsoft Visual - C++ 2009 using the /Ox option. -

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Function -

- 		-

- Boost -

- 		-

- GSL-1.9 -

- 		-

- Cephes -

-
-

- cbrt -

- 		-

-

-

1.00

-

-

(4.873e-007s)

-

-

- 		-

- N/A -

- 		-

-

-

1.00

-

-

(6.699e-007s)

-

-

-
-

- log1p -

- 		-

-

-

1.00

-

-

(1.664e-007s)

-

-

- 		-

-

-

1.00

-

-

(2.677e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.189e-007s)

-

-

-
-

- expm1 -

- 		-

-

-

1.00

-

-

(8.760e-008s)

-

-

- 		-

-

-

1.00

-

-

(1.248e-007s)

-

-

- 		-

-

-

1.00

-

-

(8.169e-008s)

-

-

-
-

- tgamma -

- 		-

-

-

1.80

-

-

(2.997e-007s)

-

-

- 		-

-

-

1.54

-

-

(2.569e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.666e-007s)

-

-

-
-

- lgamma -

- 		-

-

-

2.20

-

-

(3.045e-007s)

-

-

- 		-

-

-

4.14

-

-

(5.713e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.381e-007s)

-

-

-
-

- erf - and erfc -

- 		-

-

-

1.00

-

-

(1.483e-007s)

-

-

- 		-

-

-

1.00

-

-

(7.052e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.722e-007s)

-

-

-
-

- gamma_p - and gamma_q -

- 		-

-

-

1.00

-

-

(6.182e-007s)

-

-

- 		-

-

-

3.57

-

-

(2.209e-006s)

-

-

- 		-

-

-

4.29

-

-

(2.651e-006s)

-

-

-
-

- gamma_p_inv - and gamma_q_inv -

- 		-

-

-

1.00

-

-

(1.943e-006s)

-

-

- 		-

- N/A -

- 		-

- +INF [1] -

-
-

- ibeta - and ibetac -

- 		-

-

-

1.00

-

-

(1.670e-006s)

-

-

- 		-

-

-

1.16

-

-

(1.935e-006s)

-

-

- 		-

-

-

1.16

-

-

(1.935e-006s)

-

-

-
-

- ibeta_inv - and ibetac_inv -

- 		-

-

-

1.00

-

-

(6.075e-006s)

-

-

- 		-

- N/A -

- 		-

-

-

2.45

-

-

(1.489e-005s)

-

-

-
-

- cyl_bessel_j -

- 		-

-

-

17.89[2]

-

-

(4.248e-005s)

-

-

- 		-

-

-

1.00

-

-

(5.214e-006s)

-

-

- 		-

-

-

1.00

-

-

(2.374e-006s)

-

-

-
-

- cyl_bessel_i -

- 		-

-

-

1.00

-

-

(5.924e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.487e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.823e-006s)

-

-

-
-

- cyl_bessel_k -

- 		-

-

-

1.00

-

-

(2.783e-006s)

-

-

- 		-

-

-

1.00

-

-

(3.927e-006s)

-

-

- 		-

- N/A -

-
-

- cyl_neumann -

- 		-

-

-

1.00

-

-

(4.465e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.230e-005s)

-

-

- 		-

-

-

1.00

-

-

(4.977e-006s)

-

-

-
-

[1] - Cephes gets stuck in an infinite loop while trying to execute - our test cases. -

-

[2] - The performance here is dominated by a few cases where the parameters - grow very large: faster asymptotic expansions are available, - but are of limited (or even frankly terrible) precision. The - same issue effects all of our Bessel function implementations, - but doesn't necessarily show in the current performance data. - More investigation is needed here. -

-
-
- - Comparison - to the R and DCDFLIB Statistical Libraries on Windows -
-

- All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, - Windows Vista machine, with the test program compiled with Microsoft Visual - C++ 2009, and R-2.9.2 compiled in "standalone mode" with MinGW-4.3 - (R-2.9.2 appears not to be buildable with Visual C++). -

-
-

Table 57. A Comparison to the R Statistical Library on Windows XP

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Statistical Function -

- 		-

- Boost -

- 		-

- R -

- 		-

- DCDFLIB -

-
-

- Beta - Distribution CDF -

- 		-

-

-

1.08

-

-

(1.385e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.278e-006s)

-

-

- 		-

-

-

1.06

-

-

(1.349e-006s)

-

-

-
-

- Beta - Distribution Quantile -

- 		-

-

-

1.00

-

-

(4.975e-006s)

-

-

- 		-

-

-

67.66[1]

-

-

(3.366e-004s)

-

-

- 		-

-

-

4.23

-

-

(2.103e-005s)

-

-

-
-

- Binomial - Distribution CDF -

- 		-

-

-

1.06

-

-

(4.503e-007s)

-

-

- 		-

-

-

1.81

-

-

(7.680e-007s)

-

-

- 		-

-

-

1.00

-

-

(4.239e-007s)

-

-

-
-

- Binomial - Distribution Quantile -

- 		-

-

-

1.00

-

-

(3.254e-006s)

-

-

- 		-

-

-

1.15

-

-

(3.746e-006s)

-

-

- 		-

-

-

7.25

-

-

(2.358e-005s)

-

-

-
-

- Cauchy - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.134e-007s)

-

-

- 		-

-

-

1.08

-

-

(1.227e-007s)

-

-

- 		-

- NA -

-
-

- Cauchy - Distribution Quantile -

- 		-

-

-

1.00

-

-

(1.203e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.203e-007s)

-

-

- 		-

- NA -

-
-

- Chi - Squared Distribution CDF -

- 		-

-

-

1.21

-

-

(5.021e-007s)

-

-

- 		-

-

-

2.83

-

-

(1.176e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.155e-007s)

-

-

-
-

- Chi - Squared Distribution Quantile -

- 		-

-

-

1.00

-

-

(1.930e-006s)

-

-

- 		-

-

-

2.72

-

-

(5.243e-006s)

-

-

- 		-

-

-

5.73

-

-

(1.106e-005s)

-

-

-
-

- Exponential - Distribution CDF -

- 		-

-

-

1.00

-

-

(3.798e-008s)

-

-

- 		-

-

-

5.89

-

-

(2.236e-007s)

-

-

- 		-

- NA -

-
-

- Exponential - Distribution Quantile -

- 		-

-

-

1.41

-

-

(9.006e-008s)

-

-

- 		-

-

-

1.00

-

-

(6.380e-008s)

-

-

- 		-

- NA -

-
-

- Fisher - F Distribution CDF -

- 		-

-

-

1.00

-

-

(9.556e-007s)

-

-

- 		-

-

-

1.34

-

-

(1.283e-006s)

-

-

- 		-

-

-

1.24

-

-

(1.183e-006s)

-

-

-
-

- Fisher - F Distribution Quantile -

- 		-

-

-

1.00

-

-

(6.987e-006s)

-

-

- 		-

-

-

1.33

-

-

(9.325e-006s)

-

-

- 		-

-

-

3.16

-

-

(2.205e-005s)

-

-

-
-

- Gamma - Distribution CDF -

- 		-

-

-

1.52

-

-

(6.240e-007s)

-

-

- 		-

-

-

3.11

-

-

(1.279e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.111e-007s)

-

-

-
-

- Gamma - Distribution Quantile -

- 		-

-

-

1.24

-

-

(2.179e-006s)

-

-

- 		-

-

-

6.25

-

-

(1.102e-005s)

-

-

- 		-

-

-

1.00

-

-

(1.764e-006s)

-

-

-
-

- hypergeometric - Distribution CDF -

- 		-

-

-

3.60[2]

-

-

(5.987e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.665e-007s)

-

-

- 		-

- NA -

-
-

- hypergeometric - Distribution Quantile -

- 		-

-

-

1.00

-

-

(5.684e-007s)

-

-

- 		-

-

-

3.53

-

-

(2.004e-006s)

-

-

- 		-

- NA -

-
-

- Logistic - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.714e-007s)

-

-

- 		-

-

-

5.24

-

-

(8.984e-007s)

-

-

- 		-

- NA -

-
-

- Logistic - Distribution Quantile -

- 		-

-

-

1.02

-

-

(2.084e-007s)

-

-

- 		-

-

-

1.00

-

-

(2.043e-007s)

-

-

- 		-

- NA -

-
-

- Log-normal - Distribution CDF -

- 		-

-

-

1.00

-

-

(3.579e-007s)

-

-

- 		-

-

-

1.49

-

-

(5.332e-007s)

-

-

- 		-

- NA -

-
-

- Log-normal - Distribution Quantile -

- 		-

-

-

1.00

-

-

(9.622e-007s)

-

-

- 		-

-

-

1.57

-

-

(1.507e-006s)

-

-

- 		-

- NA -

-
-

- Negative - Binomial Distribution CDF -

- 		-

-

-

1.00

-

-

(6.227e-007s)

-

-

- 		-

-

-

2.25

-

-

(1.403e-006s)

-

-

- 		-

-

-

2.21

-

-

(1.378e-006s)

-

-

-
-

- Negative - Binomial Distribution Quantile -

- 		-

-

-

1.00

-

-

(8.594e-006s)

-

-

- 		-

-

-

43.43[3]

-

-

(3.732e-004s)

-

-

- 		-

-

-

3.48

-

-

(2.994e-005s)

-

-

-
-

- Noncentral - Chi Squared Distribution CDF -

- 		-

-

-

2.16

-

-

(3.926e-006s)

-

-

- 		-

-

-

79.93

-

-

(1.450e-004s)

-

-

- 		-

-

-

1.00

-

-

(1.814e-006s)

-

-

-
-

- Noncentral - Chi Squared Distribution Quantile -

- 		-

-

-

5.00

-

-

(3.393e-004s)

-

-

- 		-

-

-

393.90[4]

-

-

(2.673e-002s)

-

-

- 		-

-

-

1.00

-

-

(6.786e-005s)

-

-

-
-

- Noncentral - F Distribution CDF -

- 		-

-

-

1.59

-

-

(1.128e-005s)

-

-

- 		-

-

-

1.00

-

-

(7.087e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.274e-006s)

-

-

-
-

- Noncentral - F Distribution Quantile -

- 		-

-

-

1.00

-

-

(4.750e-004s)

-

-

- 		-

-

-

1.62

-

-

(7.681e-004s)

-

-

- 		-

-

-

1.00

-

-

(4.274e-006s)

-

-

-
-

- noncentral - T distribution CDF -

- 		-

-

-

3.41

-

-

(1.852e-005s)

-

-

- 		-

-

-

1.00

-

-

(5.436e-006s)

-

-

- 		-

- NA -

-
-

- noncentral - T distribution Quantile -

- 		-

-

-

1.31

-

-

(5.768e-004s)

-

-

- 		-

-

-

1.00[5]

-

-

(4.411e-004s)

-

-

- 		-

- NA -

-
-

- Normal - Distribution CDF -

- 		-

-

-

1.00

-

-

(8.373e-008s)

-

-

- 		-

-

-

1.68

-

-

(1.409e-007s)

-

-

- 		-

-

-

6.01

-

-

(5.029e-007s)

-

-

-
-

- Normal - Distribution Quantile -

- 		-

-

-

1.29

-

-

(1.521e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.182e-007s)

-

-

- 		-

-

-

10.85

-

-

(1.283e-006s)

-

-

-
-

- Poisson - Distribution CDF -

- 		-

-

-

1.18

-

-

(5.193e-007s)

-

-

- 		-

-

-

2.98

-

-

(1.314e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.410e-007s)

-

-

-
-

- Poisson - Distribution -

- 		-

-

-

1.00

-

-

(1.203e-006s)

-

-

- 		-

-

-

2.20

-

-

(2.642e-006s)

-

-

- 		-

-

-

7.86

-

-

(9.457e-006s)

-

-

-
-

- Students - t Distribution CDF -

- 		-

-

-

1.00

-

-

(8.655e-007s)

-

-

- 		-

-

-

1.06

-

-

(9.166e-007s)

-

-

- 		-

-

-

1.04

-

-

(8.999e-007s)

-

-

-
-

- Students - t Distribution Quantile -

- 		-

-

-

1.00

-

-

(2.294e-006s)

-

-

- 		-

-

-

1.36

-

-

(3.131e-006s)

-

-

- 		-

-

-

4.82

-

-

(1.106e-005s)

-

-

-
-

- Weibull - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.865e-007s)

-

-

- 		-

-

-

2.33

-

-

(4.341e-007s)

-

-

- 		-

- NA -

-
-

- Weibull - Distribution Quantile -

- 		-

-

-

1.00

-

-

(3.608e-007s)

-

-

- 		-

-

-

1.22

-

-

(4.410e-007s)

-

-

- 		-

- NA -

-
-

[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] - This result is somewhat misleading: for small values of the parameters - there is virtually no difference between the two libraries, but - for large values the Boost implementation is much - slower, albeit with much improved precision. -

-

[3] - 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 -

-

[4] - 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. -

-

[5] - 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. -

-
-
-
- - Comparison - to the R Statistical Library on Linux -
-

- All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, - Ubuntu Linux 9 machine, with the test program and R-2.9.2 compiled with GNU - G++ 4.3.3 using -O3 -DNDEBUG=1. -

-
-

Table 58. A Comparison to the R Statistical Library on Linux

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Statistical Function -

- 		-

- Boost -

- 		-

- R -

- 		-

- DCDFLIB -

-
-

- Beta - Distribution CDF -

- 		-

-

-

2.09

-

-

(3.189e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.526e-006s)

-

-

- 		-

-

-

1.19

-

-

(1.822e-006s)

-

-

-
-

- Beta - Distribution Quantile -

- 		-

-

-

1.00

-

-

(1.185e-005s)

-

-

- 		-

-

-

30.51[1]

-

-

(3.616e-004s)

-

-

- 		-

-

-

2.52

-

-

(2.989e-005s)

-

-

-
-

- Binomial - Distribution CDF -

- 		-

-

-

4.41

-

-

(9.175e-007s)

-

-

- 		-

-

-

3.59

-

-

(7.476e-007s)

-

-

- 		-

-

-

1.00

-

-

(2.081e-007s)

-

-

-
-

- Binomial - Distribution Quantile -

- 		-

-

-

1.57

-

-

(6.925e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.407e-006s)

-

-

- 		-

-

-

7.43

-

-

(3.274e-005s)

-

-

-
-

- Cauchy - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.594e-007s)

-

-

- 		-

-

-

1.04

-

-

(1.654e-007s)

-

-

- 		-

- NA -

-
-

- Cauchy - Distribution Quantile -

- 		-

-

-

1.21

-

-

(1.752e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.448e-007s)

-

-

- 		-

- NA -

-
-

- Chi - Squared Distribution CDF -

- 		-

-

-

2.61

-

-

(1.376e-006s)

-

-

- 		-

-

-

2.36

-

-

(1.243e-006s)

-

-

- 		-

-

-

1.00

-

-

(5.270e-007s)

-

-

-
-

- Chi - Squared Distribution Quantile -

- 		-

-

-

1.00

-

-

(4.252e-006s)

-

-

- 		-

-

-

1.34

-

-

(5.700e-006s)

-

-

- 		-

-

-

3.47

-

-

(1.477e-005s)

-

-

-
-

- Exponential - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.342e-007s)

-

-

- 		-

-

-

1.25

-

-

(1.677e-007s)

-

-

- 		-

- NA -

-
-

- Exponential - Distribution Quantile -

- 		-

-

-

1.00

-

-

(8.827e-008s)

-

-

- 		-

-

-

1.07

-

-

(9.470e-008s)

-

-

- 		-

- NA -

-
-

- Fisher - F Distribution CDF -

- 		-

-

-

1.62

-

-

(2.324e-006s)

-

-

- 		-

-

-

1.19

-

-

(1.711e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.437e-006s)

-

-

-
-

- Fisher - F Distribution Quantile -

- 		-

-

-

1.53

-

-

(1.577e-005s)

-

-

- 		-

-

-

1.00

-

-

(1.033e-005s)

-

-

- 		-

-

-

2.63

-

-

(2.719e-005s)

-

-

-
-

- Gamma - Distribution CDF -

- 		-

-

-

3.18

-

-

(1.582e-006s)

-

-

- 		-

-

-

2.63

-

-

(1.309e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.980e-007s)

-

-

-
-

- Gamma - Distribution Quantile -

- 		-

-

-

2.19

-

-

(4.770e-006s)

-

-

- 		-

-

-

6.94

-

-

(1.513e-005s)

-

-

- 		-

-

-

1.00

-

-

(2.179e-006s)

-

-

-
-

- hypergeometric - Distribution CDF -

- 		-

-

-

2.20[2]

-

-

(3.522e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.601e-007s)

-

-

- 		-

- NA -

-
-

- hypergeometric - Distribution Quantile -

- 		-

-

-

1.00

-

-

(8.279e-007s)

-

-

- 		-

-

-

2.57

-

-

(2.125e-006s)

-

-

- 		-

- NA -

-
-

- Logistic - Distribution CDF -

- 		-

-

-

1.00

-

-

(9.398e-008s)

-

-

- 		-

-

-

2.75

-

-

(2.588e-007s)

-

-

- 		-

- NA -

-
-

- Logistic - Distribution Quantile -

- 		-

-

-

1.00

-

-

(9.893e-008s)

-

-

- 		-

-

-

1.30

-

-

(1.285e-007s)

-

-

- 		-

- NA -

-
-

- Log-normal - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.831e-007s)

-

-

- 		-

-

-

1.39

-

-

(2.539e-007s)

-

-

- 		-

- NA -

-
-

- Log-normal - Distribution Quantile -

- 		-

-

-

1.10

-

-

(5.551e-007s)

-

-

- 		-

-

-

1.00

-

-

(5.037e-007s)

-

-

- 		-

- NA -

-
-

- Negative - Binomial Distribution CDF -

- 		-

-

-

1.08

-

-

(1.563e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.444e-006s)

-

-

- 		-

-

-

1.00

-

-

(1.444e-006s)

-

-

-
-

- Negative - Binomial Distribution Quantile -

- 		-

-

-

1.00

-

-

(1.700e-005s)

-

-

- 		-

-

-

25.92[3]

-

-

(4.407e-004s)

-

-

- 		-

-

-

1.93

-

-

(3.274e-005s)

-

-

-
-

- Noncentral - Chi Squared Distribution CDF -

- 		-

-

-

5.06

-

-

(2.841e-005s)

-

-

- 		-

-

-

25.01

-

-

(1.405e-004s)

-

-

- 		-

-

-

1.00

-

-

(5.617e-006s)

-

-

-
-

- Noncentral - Chi Squared Distribution Quantile -

- 		-

-

-

8.47

-

-

(1.879e-003s)

-

-

- 		-

-

-

144.91[4]

-

-

(3.214e-002s)

-

-

- 		-

-

-

1.00

-

-

(2.218e-004s)

-

-

-
-

- Noncentral - F Distribution CDF -

- 		-

-

-

10.33

-

-

(5.868e-005s)

-

-

- 		-

-

-

1.42

-

-

(8.058e-006s)

-

-

- 		-

-

-

1.00

-

-

(5.682e-006s)

-

-

-
-

- Noncentral - F Distribution Quantile -

- 		-

-

-

5.64

-

-

(7.869e-004s)

-

-

- 		-

-

-

6.63

-

-

(9.256e-004s)

-

-

- 		-

-

-

1.00

-

-

(1.396e-004s)

-

-

-
-

- noncentral - T distribution CDF -

- 		-

-

-

4.91

-

-

(3.357e-005s)

-

-

- 		-

-

-

1.00

-

-

(6.844e-006s)

-

-

- 		-

- NA -

-
-

- noncentral - T distribution Quantile -

- 		-

-

-

1.57

-

-

(9.265e-004s)

-

-

- 		-

-

-

1.00[5]

-

-

(5.916e-004s)

-

-

- 		-

- NA -

-
-

- Normal - Distribution CDF -

- 		-

-

-

1.00

-

-

(1.074e-007s)

-

-

- 		-

-

-

1.16

-

-

(1.245e-007s)

-

-

- 		-

-

-

5.36

-

-

(5.762e-007s)

-

-

-
-

- Normal - Distribution Quantile -

- 		-

-

-

1.28

-

-

(1.902e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.490e-007s)

-

-

- 		-

-

-

10.35

-

-

(1.542e-006s)

-

-

-
-

- Poisson - Distribution CDF -

- 		-

-

-

2.43

-

-

(1.198e-006s)

-

-

- 		-

-

-

2.25

-

-

(1.110e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.937e-007s)

-

-

-
-

- Poisson - Distribution -

- 		-

-

-

1.11

-

-

(3.032e-006s)

-

-

- 		-

-

-

1.00

-

-

(2.724e-006s)

-

-

- 		-

-

-

4.07

-

-

(1.110e-005s)

-

-

-
-

- Students - t Distribution CDF -

- 		-

-

-

2.17

-

-

(2.020e-006s)

-

-

- 		-

-

-

1.00

-

-

(9.321e-007s)

-

-

- 		-

-

-

1.10

-

-

(1.021e-006s)

-

-

-
-

- Students - t Distribution Quantile -

- 		-

-

-

1.18

-

-

(3.972e-006s)

-

-

- 		-

-

-

1.00

-

-

(3.364e-006s)

-

-

- 		-

-

-

3.89

-

-

(1.308e-005s)

-

-

-
-

- Weibull - Distribution CDF -

- 		-

-

-

1.00

-

-

(3.662e-007s)

-

-

- 		-

-

-

1.04

-

-

(3.808e-007s)

-

-

- 		-

- NA -

-
-

- Weibull - Distribution Quantile -

- 		-

-

-

1.00

-

-

(4.112e-007s)

-

-

- 		-

-

-

1.05

-

-

(4.317e-007s)

-

-

- 		-

- NA -

-
-

[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] - This result is somewhat misleading: for small values of the parameters - there is virtually no difference between the two libraries, but - for large values the Boost implementation is much - slower, albeit with much improved precision. -

-

[3] - 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 -

-

[4] - 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. -

-

[5] - 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. -

-
-
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/getting_best.html b/doc/html/math_toolkit/perf/getting_best.html deleted file mode 100644 index 507742f7b..000000000 --- a/doc/html/math_toolkit/perf/getting_best.html +++ /dev/null @@ -1,286 +0,0 @@ - - - -Getting the Best Performance from this Library - - - - - - - - -
-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 53. 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

-

-

(1.483e-007s)

-

-

-
-

- erf_inv -

- 		-

-

-

19.28

-

-

(1.215e-006s)

-

-

- 		-

-

-

1.00

-

-

(4.888e-007s)

-

-

-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/interp.html b/doc/html/math_toolkit/perf/interp.html deleted file mode 100644 index acdc08032..000000000 --- a/doc/html/math_toolkit/perf/interp.html +++ /dev/null @@ -1,73 +0,0 @@ - - - -Interpreting these Results - - - - - - - - -
-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! -

-
- - - - - -
[Note]Note

- Since these tests were run, most compilers have improved their code optimisation, - and processor speeds have improved too, so these results are known to be - out of date. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/perf_over.html b/doc/html/math_toolkit/perf/perf_over.html deleted file mode 100644 index c7126187a..000000000 --- a/doc/html/math_toolkit/perf/perf_over.html +++ /dev/null @@ -1,73 +0,0 @@ - - - -Performance Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- By and large the performance of this library should be acceptable for most - needs. However, you should note that this 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 FORTRAN 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/perf_test_app.html b/doc/html/math_toolkit/perf/perf_test_app.html deleted file mode 100644 index 7fcce1771..000000000 --- a/doc/html/math_toolkit/perf/perf_test_app.html +++ /dev/null @@ -1,55 +0,0 @@ - - - -The Performance Test Application - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/perf/tuning.html b/doc/html/math_toolkit/perf/tuning.html deleted file mode 100644 index 20e81ae90..000000000 --- a/doc/html/math_toolkit/perf/tuning.html +++ /dev/null @@ -1,893 +0,0 @@ - - - -Performance Tuning Macros - - - - - - - - -
-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 55. A Comparison of Polynomial Evaluation Methods

-
------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Compiler/platform -

- 		-

- Method 0 -

- 		-

- Method 1 -

- 		-

- Method 2 -

- 		-

- Method 3 -

-
-

- Microsoft C++ 9.0, Polynomial evaluation -

- 		-

-

-

1.26

-

-

(7.421e-008s)

-

-

- 		-

-

-

1.22

-

-

(7.226e-008s)

-

-

- 		-

-

-

1.00

-

-

(5.901e-008s)

-

-

- 		-

-

-

1.04

-

-

(6.115e-008s)

-

-

-
-

- Microsoft C++ 9.0, Rational evaluation -

- 		-

-

-

1.00

-

-

(1.008e-007s)

-

-

- 		-

-

-

1.00

-

-

(1.008e-007s)

-

-

- 		-

-

-

1.43

-

-

(1.445e-007s)

-

-

- 		-

-

-

1.40

-

-

(1.409e-007s)

-

-

-
-

- Intel C++ 11.1 (Windows), Polynomial evaluation -

- 		-

-

-

1.18

-

-

(6.517e-008s)

-

-

- 		-

-

-

1.18

-

-

(6.505e-008s)

-

-

- 		-

-

-

1.00

-

-

(5.516e-008s)

-

-

- 		-

-

-

1.00

-

-

(5.516e-008s)

-

-

-
-

- Intel C++ 11.1 (Windows), Rational evaluation -

- 		-

-

-

1.00

-

-

(8.947e-008s)

-

-

- 		-

-

-

1.02

-

-

(9.130e-008s)

-

-

- 		-

-

-

1.49

-

-

(1.333e-007s)

-

-

- 		-

-

-

1.04

-

-

(9.325e-008s)

-

-

-
-

- 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 56. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy.html b/doc/html/math_toolkit/policy.html deleted file mode 100644 index eaee1f9c0..000000000 --- a/doc/html/math_toolkit/policy.html +++ /dev/null @@ -1,41 +0,0 @@ - - - -Policies - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Policy Overview
-
Policy Tutorial
-
Policy Reference
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_overview.html b/doc/html/math_toolkit/policy/pol_overview.html deleted file mode 100644 index 15f044c77..000000000 --- a/doc/html/math_toolkit/policy/pol_overview.html +++ /dev/null @@ -1,109 +0,0 @@ - - - -Policy Overview - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref.html b/doc/html/math_toolkit/policy/pol_ref.html deleted file mode 100644 index a31f4dad5..000000000 --- a/doc/html/math_toolkit/policy/pol_ref.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -Policy Reference - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Error - Handling Policies
-
Internal - Floating-point 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/assert_undefined.html b/doc/html/math_toolkit/policy/pol_ref/assert_undefined.html deleted file mode 100644 index 1702947ae..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/assert_undefined.html +++ /dev/null @@ -1,92 +0,0 @@ - - - -Mathematically Undefined Function Policies - - - - - - - - -
-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 meaningful 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> > >());
-
-

- policy<assert_undefined<false> - behaviour can also be obtained by defining the macro -

-
#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false
-
-

- at the head of the file - see Using - Macros to Change the Policy Defaults. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html b/doc/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html deleted file mode 100644 index b5ab7cef1..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/discrete_quant_ref.html +++ /dev/null @@ -1,258 +0,0 @@ - - - -Discrete Quantile Policies - - - - - - - - -
-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 boost::math::negative_binomial_distribution;
-
-using namespace boost::math::policies;
-
-typedef negative_binomial_distribution<
-      double,
-      policy<discrete_quantile<real> >
-   > dist_type;
-
-// Lower 5% quantile:
-double x = quantile(dist_type(20, 0.3), 0.05);
-// Upper 95% 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 boost::math::negative_binomial;
-
-// Use the default rounding policy integer_round_outwards.
-// Lower quantile rounded down:
-double x = quantile(negative_binomial(20, 0.3), 0.05); // rounded up 27 from 27.3898
-// Upper quantile rounded up:
-double y = quantile(complement(negative_binomial(20, 0.3), 0.05)); // rounded down to 69 from 68.1584
-
-

-

-

- 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 boost::math::negative_binomial_distribution;
-
-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); // 28 rounded up from 27.3898
-// Upper quantile rounded down:
-double y = quantile(complement(dist_type(20, 0.3), 0.05)); // 68 rounded down from 68.1584
-
-

-

-

- 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 boost::math::negative_binomial_distribution;
-
-using namespace boost::math::policies;
-
-typedef negative_binomial_distribution<
-      double,
-      policy<discrete_quantile<integer_round_nearest> >
-   > dist_type;
-
-// Lower quantile rounded (down) to nearest:
-double x = quantile(dist_type(20, 0.3), 0.05); // 27
-// Upper quantile rounded (down) to nearest:
-double y = quantile(complement(dist_type(20, 0.3), 0.05)); // 68
-
-

-

-

- Results in x = - 27 (rounded from 27.3898) and y = 68 (rounded from 68.1584). -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/error_handling_policies.html b/doc/html/math_toolkit/policy/pol_ref/error_handling_policies.html deleted file mode 100644 index 8fbc05355..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/error_handling_policies.html +++ /dev/null @@ -1,773 +0,0 @@ - - - -Error Handling Policies - - - - - - - - -
-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 -

-
-

- Indeterminate Result Error -

- 		-

- std::domain_error -

-
-
- - errno_on_error -
-

- Will set global ::errno - ::errno - to one of the following values depending upon the error type (often EDOM - = 33 and ERANGE = 34), 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 -

-
-

- Indeterminate Result Error -

- 		-

- EDOM -

-
-
- - ignore_error -
-

- Will return 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 (perhaps NaN) as to the result: which may be significantly - in error. -

-
-

- Indeterminate Result Error -

- 		-

- Depends on the function where the error occurred -

-
-
- - 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_rounding_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);
-
-template <class T>
-T user_indeterminate_result_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 six 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 -

-
-

- Rounding Error -

- 		-

- boost::math::policies::rounding_error<action> -

- 		-

- Raised When one of the rounding functions round, - trunc - or modf - is called with an argument that has no integer representation, - or is too large to be represented in the result type -

-

- Defaults to boost::math::policies::rounding_error<throw_on_error> -

-

- When the action is throw_on_error then throws - boost::math::rounding_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 -

-
-

- Indeterminate Result Error -

- 		-

- boost::math::policies::indeterminate_result_error<action> -

- 		-

- Raised when the result of a function is not defined for the values - that were passed to it. -

-

- Defaults to boost::math::policies::indeterminate_result_error<ignore_error> -

-

- When the action is throw_on_error then throws - std::domain_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 boost::math::tgamma;
-
-//using namespace boost::math::policies; may also be convenient.
-using boost::math::policies::policy;
-using boost::math::policies::evaluation_error;
-using boost::math::policies::domain_error;
-using boost::math::policies::overflow_error;
-using boost::math::policies::domain_error;
-using boost::math::policies::pole_error;
-using boost::math::policies::errno_on_error;
-
-// Define a policy:
-typedef policy<
-  domain_error<errno_on_error>,
-  pole_error<errno_on_error>,
-  overflow_error<errno_on_error>,
-  evaluation_error<errno_on_error>
-> my_policy;
-
-double my_value = 0.; // 
-
-// Call the function applying my_policy:
-double t1 = tgamma(my_value, my_policy());
-
-// Alternatively (and equivalently) we could use helpful function
-// make_policy and define everything at the call site:
-double t2 = tgamma(my_value,
-  make_policy(
-    domain_error<errno_on_error>(),
-    pole_error<errno_on_error>(),
-    overflow_error<errno_on_error>(),
-    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 boost::math::normal_distribution;
-
-using namespace boost::math::policies;
-
-// Define a specific policy:
-typedef policy<
-      overflow_error<ignore_error>
-      > my_policy;
-
-// Define the distribution, using my_policy:
-typedef normal_distribution<double, my_policy> my_norm;
-
-// Construct a my_norm distribution, using default mean and standard deviation,
-// and get a 0.05 or 5% quantile:
-double q = quantile(my_norm(), 0.05); // = -1.64485
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/internal_promotion.html b/doc/html/math_toolkit/policy/pol_ref/internal_promotion.html deleted file mode 100644 index ee3e0859e..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/internal_promotion.html +++ /dev/null @@ -1,148 +0,0 @@ - - - -Internal Floating-point Promotion Policies - - - - - - - - -
-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 control whether - internal promotion to a higher precision floating-point type 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 boost::math::tgamma;
-
-// Define a new policy *not* internally promoting RealType to double:
-typedef policy<
-      promote_double<false>
-      > my_policy;
-
-// Call the function, applying the new policy:
-double t1 = tgamma(some_value, my_policy());
-
-// Alternatively we could use helper function make_policy,
-// and concisely 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 boost::math::normal_distribution;
-
-using namespace boost::math::policies;
-
-// Define a policy:
-typedef policy<
-      promote_float<false>
-      > my_policy;
-
-// Define the new normal distribution using my_policy:
-typedef normal_distribution<float, my_policy> my_norm;
-
-// Get a quantile:
-float q = quantile(my_norm(), 0.05f);
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/iteration_pol.html b/doc/html/math_toolkit/policy/pol_ref/iteration_pol.html deleted file mode 100644 index 74c02aa72..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/iteration_pol.html +++ /dev/null @@ -1,58 +0,0 @@ - - - -Iteration Limits Policies - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/namespace_pol.html b/doc/html/math_toolkit/policy/pol_ref/namespace_pol.html deleted file mode 100644 index 88b8bbae5..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/namespace_pol.html +++ /dev/null @@ -1,156 +0,0 @@ - - - -Setting Polices at Namespace Scope - - - - - - - - -
-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>
-//using boost::math::tgamma;
-// Need not declare using boost::math::tgamma here,
-// because will define tgamma in myspace using macro below.
-
-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,
-// despite the large value of factorial 30 = 265252859812191058636308480000000
-// unlike default policy boost::math::tgamma;
-
-

-

-

- 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; // May be convenient in myspace.
-
-  // Define a policy called my_policy to use.
-  using boost::math::policies::policy;
-
-// In this case we want all the distribution accessor functions to compile,
-// even if they are mathematically undefined, so
-// make the policy assert_undefined.
-  using boost::math::policies::assert_undefined;
-
-typedef policy<assert_undefined<false> > my_policy;
-
-// Finally apply this policy to type double.
-BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy)
-} // namespace myspace
-
-// Now we can use myspace::cauchy etc, which will use policy
-// myspace::mypolicy:
-//
-// This compiles but throws a domain error exception at runtime.
-// Caution! If you omit the try'n'catch blocks, 
-// it will just silently terminate, giving no clues as to why! 
-// So try'n'catch blocks are very strongly recommended.
-
-void test_cauchy()
-{
-   try
-   {
-      double d = mean(myspace::cauchy());  // Cauchy does not have a mean!
-   }
-   catch(const std::domain_error& e)
-   {
-      cout << e.what() << 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/pol_ref_ref.html b/doc/html/math_toolkit/policy/pol_ref/pol_ref_ref.html deleted file mode 100644 index 545536ce0..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/pol_ref_ref.html +++ /dev/null @@ -1,251 +0,0 @@ - - - -Policy Class Reference - - - - - - - - -
-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,
-          class A12 = default_policy,
-          class A13 = 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 rounding_error_type;
-   typedef computed-from-template-arguments evaluation_error_type;
-   typedef computed-from-template-arguments indeterminate_result_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,
-          class A12 = default_policy,
-          class A13 = 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<...>::rounding_error_type
-
-

- Specifies how rounding errors are handled, will be an instance of boost::math::policies::rounding_error<> - with the template argument to rounding_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<...>::indeterminate_error_type
-
-

- Specifies how indeterminate result errors are handled, will be an instance - of boost::math::policies::indeterminate_result_error<> - with the template argument to indeterminate_result_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,
-          class A12 = default_policy,
-          class A13 = 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/policy_defaults.html b/doc/html/math_toolkit/policy/pol_ref/policy_defaults.html deleted file mode 100644 index 9921181db..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/policy_defaults.html +++ /dev/null @@ -1,251 +0,0 @@ - - - -Using Macros to Change the Policy Defaults - - - - - - - - -
-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-hoc - 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_ROUNDING_ERROR_POLICY -
-

- Defines what happens when a rounding 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_INDETERMINATE_RESULT_ERROR_POLICY -
-

- Defines what happens when the result is indeterminate, but where there - is none the less a convention for the result. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_ref/precision_pol.html b/doc/html/math_toolkit/policy/pol_ref/precision_pol.html deleted file mode 100644 index 77b2df8ee..000000000 --- a/doc/html/math_toolkit/policy/pol_ref/precision_pol.html +++ /dev/null @@ -1,113 +0,0 @@ - - - -Precision Policies - - - - - - - - -
-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. One - might want to trade precision for evaluation speed. -

-
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 boost::math::tgamma;
-using boost::math::policies::policy;
-using boost::math::policies::digits10;
-
-typedef policy<digits10<5> > my_pol_5; // Define a new, non-default, policy
-// to calculate tgamma to accuracy of approximately 5 decimal digits.
-
-

-

-

- Or again using helper function make_policy: -

-

-

-
#include <boost/math/special_functions/gamma.hpp>
-using boost::math::tgamma;
-
-using namespace boost::math::policies;
-
-double t = tgamma(12, policy<digits10<5> >());  // Concise make_policy.
-
-

-

-

- And for a quantile of a distribution to approximately 25-bit precision: -

-

-

-
#include <boost/math/distributions/normal.hpp>
-using boost::math::normal_distribution;
-
-using namespace boost::math::policies;
-
-const int bits = 25; // approximate precision.
-
-double q = quantile(
-      normal_distribution<double, policy<digits2<bits> > >(),
-      0.05); // 5% quantile.
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial.html b/doc/html/math_toolkit/policy/pol_tutorial.html deleted file mode 100644 index b3d31ddfb..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -Policy Tutorial - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html b/doc/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html deleted file mode 100644 index 1acdfe677..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/ad_hoc_dist_policies.html +++ /dev/null @@ -1,100 +0,0 @@ - - - -Setting Policies for Distributions on an Ad Hoc Basis - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- All of the statistical distributions in this library are class templates - that accept two template parameters: real type (float, double ...) and - policy (how to handle exceptional events), 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>
-using boost::math::binomial_distribution;
-
-// Begin by defining a policy type, that gives the behaviour we want:
-
-//using namespace boost::math::policies; or explicitly
-using boost::math::policies::policy;
-
-using boost::math::policies::promote_float;
-using boost::math::policies::discrete_quantile;
-using boost::math::policies::integer_round_nearest;
-
-typedef policy<
-   promote_float<false>, // Do not promote to double.
-   discrete_quantile<integer_round_nearest> // Round result to nearest integer.
-> mypolicy;
-//
-// Then define a new distribution that uses it:
-typedef boost::math::binomial_distribution<float, mypolicy> mybinom;
-
-//  And now use it to get the quantile:
-
-int main()
-{
-   cout << "quantile(mybinom(200, 0.25), 0.05) is: " <<
-      quantile(mybinom(200, 0.25), 0.05) << endl;
-}
-
-

-

-

- Which outputs: -

-
quantile is: 40
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html b/doc/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html deleted file mode 100644 index f884925e4..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/ad_hoc_sf_policies.html +++ /dev/null @@ -1,172 +0,0 @@ - - - -Changing the Policy on an Ad Hoc Basis for the Special Functions - - - - - - - - -
-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>
-using boost::math::tgamma;
-
-// Define the policy to use:
-using namespace boost::math::policies; // may be convenient, or
-
-using boost::math::policies::policy;
-// Types of error whose action can be altered by policies:.
-using boost::math::policies::evaluation_error;
-using boost::math::policies::domain_error;
-using boost::math::policies::overflow_error;
-using boost::math::policies::domain_error;
-using boost::math::policies::pole_error;
-// Actions on error (in enum error_policy_type):
-using boost::math::policies::errno_on_error;
-using boost::math::policies::ignore_error;
-using boost::math::policies::throw_on_error;
-using boost::math::policies::user_error;
-
-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:
-
-// http://msdn.microsoft.com/en-us/library/t3ayayh1.aspx 
-// Microsoft errno declared in STDLIB.H as "extern int errno;" 
-
-int main()
-{
-   errno = 0; // Reset.
-   cout << "Result of tgamma(30000) is: "
-      << tgamma(30000, c_policy()) << endl; // Too big parameter
-   cout << "errno = " << errno << endl; // errno 34 Numerical result out of range.
-   cout << "Result of tgamma(-10) is: "
-      << boost::math::tgamma(-10, c_policy()) << endl; // Negative parameter.
-   cout << "errno = " << errno << endl; // error 33 Numerical argument out of domain.
-} // int main()
-
-

-

-

- 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>
-using boost::math::tgamma;
-
-int main()
-{
-   // using namespace boost::math::policies; // or
-   using boost::math::policies::errno_on_error;
-   using boost::math::policies::make_policy;
-   using boost::math::policies::pole_error;
-   using boost::math::policies::domain_error;
-   using boost::math::policies::overflow_error;
-   using boost::math::policies::evaluation_error;
-
-   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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html b/doc/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html deleted file mode 100644 index e6a60f685..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/changing_policy_defaults.html +++ /dev/null @@ -1,270 +0,0 @@ - - - -Changing the Policy Defaults - - - - - - - - -
-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: -

-

-

-
   #define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error
-
-// 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;
-
-#include <cstddef>
-   // using ::errno
-
-

-

-

- 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; // Clear/reset.
-      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;
-         if (errno != 0)
-         { // So errno has been set.
-           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!
-
-

- And finally let's build with: -

-
#define BOOST_MATH_DOMAIN_ERROR_POLICY errno_on_error
-
-

- Which gives the output show errno: -

-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/namespace_policies.html b/doc/html/math_toolkit/policy/pol_tutorial/namespace_policies.html deleted file mode 100644 index 334ed6295..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/namespace_policies.html +++ /dev/null @@ -1,408 +0,0 @@ - - - -Setting Policies at Namespace or Translation Unit Scope - - - - - - - - -
-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 (for example, a C namespace for a C-style policy), 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 for our function: -

-

-

-
#include <boost/math/special_functions.hpp>
-//using boost::math::tgamma; // Not needed because using C::tgamma.
-
-

-

-

- Open up the "C" namespace that we'll use for our functions, and - define the policy type we want: in this case a C-style one that sets ::errno - and returns a standard value, rather than throwing exceptions. -

-

- Any policies we don't specify here will inherit the defaults. -

-

-

-
namespace C
-{ // To hold our C-style policy.
-  //using namespace boost::math::policies; or explicitly:
-  using boost::math::policies::policy;
-
-  using boost::math::policies::domain_error;
-  using boost::math::policies::pole_error;
-  using boost::math::policies::overflow_error;
-  using boost::math::policies::evaluation_error;
-  using boost::math::policies::errno_on_error;
-
-  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 c_policy 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;
-   cout << "Result of tgamma(30000) is: "
-      << C::tgamma(30000) << endl; // Note using C::tgamma
-   cout << "errno = " << errno << endl; // errno = 34
-   cout << "Result of tgamma(-10) is: "
-      << C::tgamma(-10) << endl;
-   cout << "errno = " << errno << endl; // errno = 33, overwriting previous value of 34.
-}
-
-

-

-

- 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 to set project wide - build macros (possibly fragile and easy to forget). -

-

- 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>
-// using boost::math::tgamma; // Would create an ambiguity between
-// 'double boost::math::tgamma<int>(T)' and
-// 'double 'anonymous-namespace'::tgamma<int>(RT)'.
-
-namespace
-{ // unnamed
-
-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)
-
-

-

-

- So that when we call tgamma(z), - we really end up calling boost::math::tgamma(z, - anonymous-namespace::c_policy()). -

-

-

-
} // close unnamed namespace
-
-int main()
-{
-   errno = 0;
-   cout << "Result of tgamma(30000) is: "
-      << tgamma(30000) << endl;
-      // tgamma in unnamed namespace in this translation unit (file) only.
-   cout << "errno = " << errno << endl;
-   cout << "Result of tgamma(-10) is: "
-      << tgamma(-10) << endl;
-   cout << "errno = " << errno << endl;
-   // Default tgamma policy would throw an exception, and abort.
-}
-
-

-

-

- Handling policies for 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 for all the distributions: -

-

-

-
#include <boost/math/distributions.hpp>
-
-

-

-

- Open up an appropriate namespace, calling it my_distributions, - 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;
-  // using boost::math::policies::errno_on_error; // etc.
-
-  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 type double - and policy types my_policy - as arguments: -

-

-

-
BOOST_MATH_DECLARE_DISTRIBUTIONS(double, my_policy)
-
-} // close namespace my_namespace
-
-

-

-

- We now have a set of typedefs defined in namespace my_distributions 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_distributions::normal we really end up using boost::math::normal_distribution<double, my_policy>: -

-

-

-
int main()
-{
-   // Construct distribution with something we know will overflow
-  // (using double rather than if promoted to long double):
-   my_distributions::normal norm(10, 2);
-
-   errno = 0;
-   cout << "Result of quantile(norm, 0) is: "
-      << quantile(norm, 0) << endl; // -infinity.
-   cout << "errno = " << errno << endl;
-   errno = 0;
-   cout << "Result of quantile(norm, 1) is: "
-      << quantile(norm, 1) << endl; // +infinity.
-   cout << "errno = " << errno << endl;
-
-   // Now try a discrete distribution.
-   my_distributions::binomial binom(20, 0.25);
-   cout << "Result of quantile(binom, 0.05) is: "
-      << quantile(binom, 0.05) << endl; // To check we get integer results.
-   cout << "Result of quantile(complement(binom, 0.05)) is: "
-      << quantile(complement(binom, 0.05)) << 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 project wide build - macros (possibly fragile and easy to forget). -

-
- - - - - -
[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> // All distributions.
-// using boost::math::normal; // Would create an ambguity between
-// boost::math::normal_distribution<RealType> boost::math::normal and
-// 'anonymous-namespace'::normal'.
-
-namespace
-{ // anonymous or unnnamed (rather than named as in policy_eg_6.cpp).
-
-  using namespace boost::math::policies;
-   // using boost::math::policies::errno_on_error; // etc.
-  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()
-{
-   // Construct distribution with something we know will overflow.
-   normal norm(10, 2); // using 'anonymous-namespace'::normal
-   errno = 0;
-   cout << "Result of quantile(norm, 0) is: "
-      << quantile(norm, 0) << endl;
-   cout << "errno = " << errno << endl;
-   errno = 0;
-   cout << "Result of quantile(norm, 1) is: "
-      << quantile(norm, 1) << endl;
-   cout << "errno = " << errno << endl;
-   //
-   // Now try a discrete distribution:
-   binomial binom(20, 0.25);
-   cout << "Result of quantile(binom, 0.05) is: "
-      << quantile(binom, 0.05) << endl;
-   cout << "Result of quantile(complement(binom, 0.05)) is: "
-      << quantile(complement(binom, 0.05)) << endl;
-}
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html b/doc/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html deleted file mode 100644 index 5951857aa..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/policy_tut_defaults.html +++ /dev/null @@ -1,132 +0,0 @@ - - - -Policies Have Sensible Defaults - - - - - - - - -
-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. -

-
Rounding Error
-

- Throws a boost::math::rounding_error exception. -

-
Internal Evaluation Error
-

- Throws a boost::math::evaluation_error exception. -

-
Indeterminate Result Error
-

- Returns a result that depends on the function where the error occurred. -

-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/policy_usage.html b/doc/html/math_toolkit/policy/pol_tutorial/policy_usage.html deleted file mode 100644 index 3742e642e..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/policy_usage.html +++ /dev/null @@ -1,63 +0,0 @@ - - - -So How are Policies Used Anyway? - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html b/doc/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html deleted file mode 100644 index c1328d22e..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html +++ /dev/null @@ -1,422 +0,0 @@ - - - -Understanding Quantiles of Discrete Distributions - - - - - - - - -
-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 (and some using statements for conciseness): -

-

-

-
#include <iostream>
-using std::cout; using std::endl;
-using std::left; using std::fixed; using std::right; using std::scientific;
-#include <iomanip>
-using std::setw;
-using std::setprecision;
-
-#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: -

-

-

-
// Avoid 
-// using namespace std; // and 
-// using namespace boost::math;
-// to avoid potential ambiguity of names, like binomial.
-// using namespace boost::math::policies; is small risk, but
-// the necessary items are brought into scope thus:
-
-using boost::math::binomial_distribution;
-using boost::math::policies::policy;
-using boost::math::policies::discrete_quantile;
-
-using boost::math::policies::integer_round_outwards;
-using boost::math::policies::integer_round_down;
-using boost::math::policies::integer_round_up;
-using boost::math::policies::integer_round_nearest;
-using boost::math::policies::integer_round_inwards;
-using boost::math::policies::real;
-
-using boost::math::binomial_distribution; // Not std::binomial_distribution.
-
-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()
-{
-   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";
-
-   cout << setw(25) << right
-      << "Policy"<< setw(18) << right
-      << "Lower Quantile" << setw(18) << right
-      << "Upper Quantile" << endl;
-
-   // Test integer_round_outwards:
-   cout << setw(25) << right
-      << "integer_round_outwards"
-      << setw(18) << right
-      << quantile(binom_round_outwards(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_round_outwards(50, 0.5), 0.95)
-      << endl;
-
-   // Test integer_round_inwards:
-   cout << setw(25) << right
-      << "integer_round_inwards"
-      << setw(18) << right
-      << quantile(binom_round_inwards(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_round_inwards(50, 0.5), 0.95)
-      << endl;
-
-   // Test integer_round_down:
-   cout << setw(25) << right
-      << "integer_round_down"
-      << setw(18) << right
-      << quantile(binom_round_down(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_round_down(50, 0.5), 0.95)
-      << endl;
-
-   // Test integer_round_up:
-   cout << setw(25) << right
-      << "integer_round_up"
-      << setw(18) << right
-      << quantile(binom_round_up(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_round_up(50, 0.5), 0.95)
-      << endl;
-
-   // Test integer_round_nearest:
-   cout << setw(25) << right
-      << "integer_round_nearest"
-      << setw(18) << right
-      << quantile(binom_round_nearest(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_round_nearest(50, 0.5), 0.95)
-      << endl;
-
-   // Test real:
-   cout << setw(25) << right
-      << "real"
-      << setw(18) << right
-      << quantile(binom_real_quantile(50, 0.5), 0.05)
-      << setw(18) << right
-      << quantile(binom_real_quantile(50, 0.5), 0.95)
-      << endl;
-} // int main()
-
-

-

-

- Which produces the program output: -

-
policy_eg_10.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\policy_eg_10.exe
-  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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html b/doc/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html deleted file mode 100644 index 64eff0317..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/user_def_err_pol.html +++ /dev/null @@ -1,523 +0,0 @@ - - - -Calling User Defined Error Handlers - - - - - - - - -
-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 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);
-template <class T, class TargetType>
-T user_rounding_error(const char* function, const char* message, const T& val, const TargetType& t);
-template <class T>
-T user_indeterminate_result_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 our user-defined error handlers that we want to use. We - only provide our special domain and pole error handlers; other errors like - overflow and underflow use the default. -

-

-

-
#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)
-    { // Ignoring function, message and val for this example, perhaps unhelpfully.
-       cerr << "Domain Error!" << endl;
-       return std::numeric_limits<T>::quiet_NaN();
-    }
-
-    template <class T>
-    T user_pole_error(const char* function, const char* message, const T& val)
-    { // Ignoring function, message and val for this example, perhaps unhelpfully.
-       cerr << "Pole Error!" << endl;
-       return std::numeric_limits<T>::quiet_NaN();
-    }
-  } // namespace policies
-}} // namespace boost{ namespace math
-
-

-

-

- 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()
-{
-   cout << "Result of erf_inv(-10) is: "
-      << erf_inv(-10) << endl;
-   cout << "Result of tgamma(-10) is: "
-      << tgamma(-10) << endl;
-}
-
-

-

-

- Which outputs: -

-
Domain Error!
-  Pole Error!
-  Result of erf_inv(-10) is: 1.#QNAN
-  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 real type T, like float or double. -

-
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 <boost/math/special_functions.hpp>
-
-

-

-

- Next we'll implement our own 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 in case function or message - are empty: -

-

-

-
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, perhaps - double: -

-

-

-
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 potentially significant digits of val: -

-

-

-
msg += ": \n";
-int prec = 2 + (std::numeric_limits<T>::digits * 30103UL) / 100000UL;
-// int prec = std::numeric_limits<T>::max_digits10; //  For C++0X Standard Library
-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 occurred 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;
-   // int prec = std::numeric_limits<T>::max_digits10; // For C++0X Standard Library
-   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,
-   // so we just return a NaN for the purposes of the example:
-   return std::numeric_limits<T>::quiet_NaN();
-}
-
-} // policies
-}} // boost::math
-
-

-

-

- 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
-{ // unnamed.
-
-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)
-
-} // 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:
-   cout << "Result of erf_inv(-10) is: "
-      << erf_inv(-10) << std::endl << endl;
-   // Raise a pole error:
-   cout << "Result of tgamma(-10) is: "
-      << tgamma(-10) << std::endl << endl;
-   // Raise an overflow error:
-   cout << "Result of tgamma(3000) is: "
-      << tgamma(3000) << std::endl << endl;
-   // Raise an underflow error:
-   cout << "Result of tgamma(-190.5) is: "
-      << tgamma(-190.5) << std::endl << 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!
-} // int main()
-
-

-

-

- 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 is dividing by infinity and so underflows. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html b/doc/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html deleted file mode 100644 index 0d4b05ce6..000000000 --- a/doc/html/math_toolkit/policy/pol_tutorial/what_is_a_policy.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -So Just What is a Policy Anyway? - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions.html b/doc/html/math_toolkit/quaternions.html deleted file mode 100644 index b7f8afdf3..000000000 --- a/doc/html/math_toolkit/quaternions.html +++ /dev/null @@ -1,57 +0,0 @@ - - - -Quaternions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
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
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/acknowledgement.html b/doc/html/math_toolkit/quaternions/acknowledgement.html deleted file mode 100644 index 76e6a8bc4..000000000 --- a/doc/html/math_toolkit/quaternions/acknowledgement.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Acknowledgements - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/create.html b/doc/html/math_toolkit/quaternions/create.html deleted file mode 100644 index f791f26ae..000000000 --- a/doc/html/math_toolkit/quaternions/create.html +++ /dev/null @@ -1,113 +0,0 @@ - - - -Quaternion Creation Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
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 The R Project - for Statistical Computing 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 The R - Project for Statistical Computing 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/exp.html b/doc/html/math_toolkit/quaternions/exp.html deleted file mode 100644 index 8ba454f75..000000000 --- a/doc/html/math_toolkit/quaternions/exp.html +++ /dev/null @@ -1,49 +0,0 @@ - - - -The Quaternionic Exponential - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Please refer to the following PDF's: -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/header.html b/doc/html/math_toolkit/quaternions/header.html deleted file mode 100644 index d2c6acdd9..000000000 --- a/doc/html/math_toolkit/quaternions/header.html +++ /dev/null @@ -1,39 +0,0 @@ - - - -Header File - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The interface and implementation are both supplied by the header file quaternion.hpp. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/history.html b/doc/html/math_toolkit/quaternions/history.html deleted file mode 100644 index 3407e8dc8..000000000 --- a/doc/html/math_toolkit/quaternions/history.html +++ /dev/null @@ -1,102 +0,0 @@ - - - -History - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/mem_fun.html b/doc/html/math_toolkit/quaternions/mem_fun.html deleted file mode 100644 index 455991bc2..000000000 --- a/doc/html/math_toolkit/quaternions/mem_fun.html +++ /dev/null @@ -1,229 +0,0 @@ - - - -Quaternion Member Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/mem_typedef.html b/doc/html/math_toolkit/quaternions/mem_typedef.html deleted file mode 100644 index 72699666b..000000000 --- a/doc/html/math_toolkit/quaternions/mem_typedef.html +++ /dev/null @@ -1,63 +0,0 @@ - - - -Quaternion Member Typedefs - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/non_mem.html b/doc/html/math_toolkit/quaternions/non_mem.html deleted file mode 100644 index 6385db247..000000000 --- a/doc/html/math_toolkit/quaternions/non_mem.html +++ /dev/null @@ -1,215 +0,0 @@ - - - -Quaternion Non-Member Operators - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/overview.html b/doc/html/math_toolkit/quaternions/overview.html deleted file mode 100644 index 42eb60ade..000000000 --- a/doc/html/math_toolkit/quaternions/overview.html +++ /dev/null @@ -1,86 +0,0 @@ - - - -Overview - - - - - - - - -
-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 The R Project for Statistical - Computing), 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/quat.html b/doc/html/math_toolkit/quaternions/quat.html deleted file mode 100644 index b6a2b4b94..000000000 --- a/doc/html/math_toolkit/quaternions/quat.html +++ /dev/null @@ -1,89 +0,0 @@ - - - -Template Class quaternion - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/spec.html b/doc/html/math_toolkit/quaternions/spec.html deleted file mode 100644 index b21acc04a..000000000 --- a/doc/html/math_toolkit/quaternions/spec.html +++ /dev/null @@ -1,182 +0,0 @@ - - - -Quaternion Specializations - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/synopsis.html b/doc/html/math_toolkit/quaternions/synopsis.html deleted file mode 100644 index 2524a0a28..000000000 --- a/doc/html/math_toolkit/quaternions/synopsis.html +++ /dev/null @@ -1,118 +0,0 @@ - - - -Synopsis - - - - - - - - -
-PrevUpHomeNext -
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/tests.html b/doc/html/math_toolkit/quaternions/tests.html deleted file mode 100644 index 9754e56dd..000000000 --- a/doc/html/math_toolkit/quaternions/tests.html +++ /dev/null @@ -1,52 +0,0 @@ - - - -Test Program - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 math_toolkit_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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/todo.html b/doc/html/math_toolkit/quaternions/todo.html deleted file mode 100644 index 6552951d7..000000000 --- a/doc/html/math_toolkit/quaternions/todo.html +++ /dev/null @@ -1,52 +0,0 @@ - - - -To Do - - - - - - - - -
-PrevUpHomeNext -
-
-

-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 example - implementation to an efficient one). -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/trans.html b/doc/html/math_toolkit/quaternions/trans.html deleted file mode 100644 index 3d419b3e1..000000000 --- a/doc/html/math_toolkit/quaternions/trans.html +++ /dev/null @@ -1,137 +0,0 @@ - - - -Quaternion Transcendentals - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/quaternions/value_op.html b/doc/html/math_toolkit/quaternions/value_op.html deleted file mode 100644 index 739feede8..000000000 --- a/doc/html/math_toolkit/quaternions/value_op.html +++ /dev/null @@ -1,99 +0,0 @@ - - - -Quaternion Value Operations - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special.html b/doc/html/math_toolkit/special.html deleted file mode 100644 index 89de390e3..000000000 --- a/doc/html/math_toolkit/special.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -Special Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Gamma Functions
-
Factorials and Binomial - Coefficients
-
Beta Functions
-
Error Functions
-
Polynomials
-
Bessel Functions
-
Hankel Functions
-
Airy Functions
-
Elliptic Integrals
-
Jacobi Elliptic Functions
-
Zeta Functions
-
Exponential Integrals
-
Basic Functions
-
Sinus Cardinal and Hyperbolic - Sinus Cardinal Functions
-
Inverse Hyperbolic Functions
-
Owen's T function
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/airy.html b/doc/html/math_toolkit/special/airy.html deleted file mode 100644 index 682f9e5c9..000000000 --- a/doc/html/math_toolkit/special/airy.html +++ /dev/null @@ -1,42 +0,0 @@ - - - -Airy Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Airy Ai Function
-
Airy Bi Function
-
Airy Ai' Function
-
Airy Bi' Function
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/airy/ai.html b/doc/html/math_toolkit/special/airy/ai.html deleted file mode 100644 index 0b6840ee9..000000000 --- a/doc/html/math_toolkit/special/airy/ai.html +++ /dev/null @@ -1,119 +0,0 @@ - - - -Airy Ai Function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/airy.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T>
- calculated-result-type airy_ai(T x);
-
- template <class T, class Policy>
- calculated-result-type airy_ai(T x, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- The function airy_ai - calculates the Airy function Ai which is the first solution to the differential - equation: -

-

- -

-

- See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram - Web Resource. http://mathworld.wolfram.com/AiryFunctions.html; -

-

- 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 how this function changes as x - changes: for negative x the function is cyclic, while - for positive x the value tends to zero: -

-

- -

-
- - Accuracy -
-

- This function is implemented entirely in terms of the Bessel functions - cyl_bessel_j - and cyl_bessel_k - - refer to those functions for detailed accuracy information. -

-

- In general though, the relative error is low (less than 100 ε) for x - > 0 while only the absolute error is low for x - < 0. -

-
- - Testing -
-

- Since this function is implemented in terms of other special functions, - there are only a few basic sanity checks, using test values from Wolfram Airy Functions. -

-
- - Implementation -
-

- This function is implemented in terms of the Bessel functions using the - relations: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/airy/aip.html b/doc/html/math_toolkit/special/airy/aip.html deleted file mode 100644 index 9073184e8..000000000 --- a/doc/html/math_toolkit/special/airy/aip.html +++ /dev/null @@ -1,115 +0,0 @@ - - - -Airy Ai' Function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/airy.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T>
- calculated-result-type airy_ai_prime(T x);
-
- template <class T, class Policy>
- calculated-result-type airy_ai_prime(T x, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- The function airy_ai_prime - calculates the Airy function Ai' which is the derivative of the first solution - to the differential equation: -

-

- -

-

- 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 how this function changes as x - changes: for negative x the function is cyclic, while - for positive x the value tends to zero: -

-

- -

-
- - Accuracy -
-

- This function is implemented entirely in terms of the Bessel functions - cyl_bessel_j - and cyl_bessel_k - - refer to those functions for detailed accuracy information. -

-

- In general though, the relative error is low (less than 100 ε) for x - > 0 while only the absolute error is low for x - < 0. -

-
- - Testing -
-

- Since this function is implemented in terms of other special functions, - there are only a few basic sanity checks, using test values from functions.wolfram.com. -

-
- - Implementation -
-

- This function is implemented in terms of the Bessel functions using the - relations: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/airy/bi.html b/doc/html/math_toolkit/special/airy/bi.html deleted file mode 100644 index 069c5ee2a..000000000 --- a/doc/html/math_toolkit/special/airy/bi.html +++ /dev/null @@ -1,115 +0,0 @@ - - - -Airy Bi Function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/airy.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T>
- calculated-result-type airy_bi(T x);
-
- template <class T, class Policy>
- calculated-result-type airy_bi(T x, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- The function airy_bi - calculates the Airy function Bi which is the second solution to the differential - equation: -

-

- -

-

- 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 how this function changes as x - changes: for negative x the function is cyclic, while - for positive x the value tends to infinity: -

-

- -

-
- - Accuracy -
-

- This function is implemented entirely in terms of the Bessel functions - cyl_bessel_i - and cyl_bessel_j - - refer to those functions for detailed accuracy information. -

-

- In general though, the relative error is low (less than 100 ε) for x - > 0 while only the absolute error is low for x - < 0. -

-
- - Testing -
-

- Since this function is implemented in terms of other special functions, - there are only a few basic sanity checks, using test values from functions.wolfram.com. -

-
- - Implementation -
-

- This function is implemented in terms of the Bessel functions using the - relations: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/airy/bip.html b/doc/html/math_toolkit/special/airy/bip.html deleted file mode 100644 index 96c6a3d36..000000000 --- a/doc/html/math_toolkit/special/airy/bip.html +++ /dev/null @@ -1,115 +0,0 @@ - - - -Airy Bi' Function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/airy.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T>
- calculated-result-type airy_bi_prime(T x);
-
- template <class T, class Policy>
- calculated-result-type airy_bi_prime(T x, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- The function airy_bi_prime - calculates the Airy function Bi' which is the derivative of the second - solution to the differential equation: -

-

- -

-

- 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 how this function changes as x - changes: for negative x the function is cyclic, while - for positive x the value tends to infinity: -

-

- -

-
- - Accuracy -
-

- This function is implemented entirely in terms of the Bessel functions - cyl_bessel_i - and cyl_bessel_j - - refer to those functions for detailed accuracy information. -

-

- In general though, the relative error is low (less than 100 ε) for x - > 0 while only the absolute error is low for x - < 0. -

-
- - Testing -
-

- Since this function is implemented in terms of other special functions, - there are only a few basic sanity checks, using test values from functions.wolfram.com. -

-
- - Implementation -
-

- This function is implemented in terms of the Bessel functions using the - relations: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel.html b/doc/html/math_toolkit/special/bessel.html deleted file mode 100644 index 9833fa4b5..000000000 --- a/doc/html/math_toolkit/special/bessel.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Bessel Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Bessel Function - Overview
-
Bessel Functions - of the First and Second Kinds
-
Finding Zeros - of 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel/bessel.html b/doc/html/math_toolkit/special/bessel/bessel.html deleted file mode 100644 index a78344180..000000000 --- a/doc/html/math_toolkit/special/bessel/bessel.html +++ /dev/null @@ -1,690 +0,0 @@ - - - -Bessel Functions of the First and Second Kinds - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

- #include <boost/math/special_functions/bessel.hpp> -

-
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: -

-

- -

-

- The following graph shows the behaviour of Yv: this is also cyclic for - large x, but tends to -∞   for small x: -

-

- -

-
- - 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 39. 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=59 Mean=10 -

-

- 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 40. Errors Rates in cyl_neumann

-
------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- Y0   and Y1 -

- 		-

- Yn (integer orders) -

- 		-

- Yv (fractional orders) -

-
-

- 53 -

- 		-

- Win32 / Visual C++ 8.0 -

- 		-

- Peak=4.7 Mean=1.7 -

-

- GSL Peak=34 Mean=9 -

-

- Cephes Peak=330 - Mean=54 -

- 		-

- Peak=117 Mean=10 -

-

- GSL Peak=500 Mean=54 -

-

- Cephes Peak=923 - Mean=83 -

- 		-

- Peak=800 Mean=40 -

-

- 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 smaller than machine epsilon then the following - approximations for Y0(x), Y1(x), Y2(x) and Yn(x) can be used (see: http://functions.wolfram.com/03.03.06.0037.01, - http://functions.wolfram.com/03.03.06.0038.01, - http://functions.wolfram.com/03.03.06.0039.01 - and http://functions.wolfram.com/03.03.06.0040.01): -

-

- -

-

- -

-

- -

-

- -

-

- When x is small compared to v - and v is not an integer, then the following series - approximation can be used for Yv(x), this is also an area where other approximations - are often too slow to converge to be used (see http://functions.wolfram.com/03.03.06.0034.01): -

-

- -

-

- When x is small compared to v, - Jvx   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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel/bessel0.html b/doc/html/math_toolkit/special/bessel/bessel0.html deleted file mode 100644 index 6d3bf2adb..000000000 --- a/doc/html/math_toolkit/special/bessel/bessel0.html +++ /dev/null @@ -1,838 +0,0 @@ - - - -Finding Zeros of Bessel Functions of the First and Second Kinds - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

- #include <boost/math/special_functions/bessel.hpp> -

-

- Functions for obtaining both a single zero or root of the Bessel function, - and placing multiple zeros into a container like std::vector - by providing an output iterator. -

-

- The signature of the single value functions are: -

-
template <class T>
-T cyl_bessel_j_zero(
-         T v,            // Floating-point value for Jv.
-         int m);         // 1-based index of zero.
-
-template <class T>
-T cyl_neumann_zero(
-         T v,            // Floating-point value for Jv.
-         int m);         // 1-based index of zero.
-
-

- and for multiple zeros: -

-
template <class T, class OutputIterator>
-OutputIterator cyl_bessel_j_zero(
-                     T v,                       // Floating-point value for Jv.
-                     int start_index,           // 1-based index of first zero.
-                     unsigned number_of_zeros,  // How many zeros to generate.
-                     OutputIterator out_it);    // Destination for zeros.
-
-template <class T, class OutputIterator>
-OutputIterator cyl_neumann_zero(
-                     T v,                       // Floating-point value for Jv.
-                     int start_index,           // 1-based index of zero.
-                     unsigned number_of_zeros,  // How many zeros to generate
-                     OutputIterator out_it);    // Destination for zeros.
-
-

- There are also versions which allow control of the Policies - for error handling and precision. -

-
 template <class T>
- T cyl_bessel_j_zero(
-          T v,            // Floating-point value for Jv.
-          int m,          // 1-based index of zero.
-          const Policy&); // Policy to use.
-
- template <class T>
- T cyl_neumann_zero(
-          T v,            // Floating-point value for Jv.
-          int m,          // 1-based index of zero.
-          const Policy&); // Policy to use.
-
-
-template <class T, class OutputIterator>
-OutputIterator cyl_bessel_j_zero(
-                     T v,                       // Floating-point value for Jv.
-                     int start_index,           // 1-based index of first zero.
-                     unsigned number_of_zeros,  // How many zeros to generate.
-                     OutputIterator out_it,     // Destination for zeros.
-                     const Policy& pol);        // Policy to use.
-
-template <class T, class OutputIterator>
-OutputIterator cyl_neumann_zero(
-                     T v,                       // Floating-point value for Jv.
-                     int start_index,           // 1-based index of zero.
-                     unsigned number_of_zeros,  // How many zeros to generate.
-                     OutputIterator out_it,     // Destination for zeros.
-                     const Policy& pol);        // Policy to use.
-
-
- - Description -
-

- Every real order ν cylindrical Bessel and Neumann functions have an infinite - number of zeros on the positive real axis. The real zeros on the positive - real axis can be found by solving for the roots of -

-

-   Jν(jν, m) = 0 -

-

-   Yν(yν, m) = 0 -

-

- Here, jν, m represents the mth - root of the cylindrical Bessel function of order ν, - and yν, m represents the mth root - of the cylindrical Neumann function of order ν. -

-

- The zeros or roots (values of x - where the function crosses the horizontal y - = 0 - axis) of the Bessel and Neumann functions are computed by two functions, - cyl_bessel_j_zero and - cyl_neumann_zero. -

-

- In each case the index or rank of the zero returned is 1-based, which is - to say: -

-
cyl_bessel_j_zero(v, 1);
-
-

- returns the first zero of Bessel J. -

-

- Passing an start_index <= 0 results - in a std::domain_error being raised. -

-

- For certain parameters, however, the zero'th root is defined and it has - a value of zero. For example, the zero'th root of J[sub v](x) is defined and it has a value of zero - for all values of v > - 0 and for negative integer values - of v = - -n. - Similar cases are described in the implementation details below. -

-

- The order v of J can be positive, negative and zero - for the cyl_bessel_j and - cyl_neumann functions, - but not infinite nor NaN. -

-

- -

-

- -

-
- - Examples - of finding Bessel and Neumann zeros -
-

- This example demonstrates calculating zeros of the Bessel and Neumann functions. - It also shows how Boost.Math and Boost.Multiprecision can be combined to - provide a many decimal digit precision. For 50 decimal digit precision - we need to include -

-

-

-
#include <boost/multiprecision/cpp_dec_float.hpp>
-
-

-

-

- and a typedef for float_type may be convenient (allowing - a quick switch to re-compute at built-in double - or other precision) -

-

-

-
typedef boost::multiprecision::cpp_dec_float_50 float_type;
-
-

-

-

- To use the functions for finding zeros of the functions we need -

-

-

-
#include <boost/math/special_functions/bessel.hpp>
-
-

-

-

- This file includes the forward declaration signatures for the zero-finding - functions: -

-

-

-
//  #include <boost/math/special_functions/math_fwd.hpp>
-
-

-

-

- but more details are in the full documentation, for example at Boost.Math - Bessel functions. -

-

- This example shows obtaining both a single zero of the Bessel function, - and then placing multiple zeros into a container like std::vector - by providing an iterator. -

-
- - - - - -
[Tip]Tip

- It is always wise to place code using Boost.Math inside try'n'catch blocks; - this will ensure that helpful error messages are shown when exceptional - conditions arise. -

-

- First, evaluate a single Bessel zero. -

-

- The precision is controlled by the float-point type of template parameter - T of v - so this example has double - precision, at least 15 but up to 17 decimal digits (for the common 64-bit - double). -

-

-

-
//    double root = boost::math::cyl_bessel_j_zero(0.0, 1);
-//    // Displaying with default precision of 6 decimal digits:
-//    std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40483
-//    // And with all the guaranteed (15) digits:
-//    std::cout.precision(std::numeric_limits<double>::digits10);
-//    std::cout << "boost::math::cyl_bessel_j_zero(0.0, 1) " << root << std::endl; // 2.40482555769577
-
-

-

-

- But note that because the parameter v - controls the precision of the result, v - must be a floating-point type. So if you - provide an integer type, say 0, rather than 0.0, then it will fail to compile - thus: -

-
root = boost::math::cyl_bessel_j_zero(0, 1);
-
-

- with this error message -

-
error C2338: Order must be a floating-point type.
-
-

-

-

- Optionally, we can use a policy to ignore errors, C-style, returning some - value, perhaps infinity or NaN, or the best that can be done. (See user error - handling). -

-

- To create a (possibly unwise!) policy ignore_all_policy - that ignores all errors: -

-

-

-
typedef boost::math::policies::policy<
-  boost::math::policies::domain_error<boost::math::policies::ignore_error>,
-  boost::math::policies::overflow_error<boost::math::policies::ignore_error>,
-  boost::math::policies::underflow_error<boost::math::policies::ignore_error>,
-  boost::math::policies::denorm_error<boost::math::policies::ignore_error>,
-  boost::math::policies::pole_error<boost::math::policies::ignore_error>,
-  boost::math::policies::evaluation_error<boost::math::policies::ignore_error>
-            > ignore_all_policy;
-
-

-

-

- Examples of use of this ignore_all_policy - are -

-

-

-
double inf = std::numeric_limits<double>::infinity();
-double nan = std::numeric_limits<double>::quiet_NaN();
-
-double dodgy_root = boost::math::cyl_bessel_j_zero(-1.0, 1, ignore_all_policy());
-std::cout << "boost::math::cyl_bessel_j_zero(-1.0, 1) " << dodgy_root << std::endl; // 1.#QNAN
-double inf_root = boost::math::cyl_bessel_j_zero(inf, 1, ignore_all_policy());
-std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl; // 1.#QNAN
-double nan_root = boost::math::cyl_bessel_j_zero(nan, 1, ignore_all_policy());
-std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl; // 1.#QNAN
-
-

-

-

- Another version of cyl_bessel_j_zero - allows calculation of multiple zeros with one call, placing the results - in a container, often std::vector. - For example, generate and display the first five double - roots of Jv for integral order 2, as column J2(x) in - table 1 of Wolfram - Bessel Function Zeros. -

-

-

-
unsigned int n_roots = 5U;
-std::vector<double> roots;
-boost::math::cyl_bessel_j_zero(2.0, 1, n_roots, std::back_inserter(roots));
-std::copy(roots.begin(),
-          roots.end(),
-          std::ostream_iterator<double>(std::cout, "\n"));
-
-

-

-

- Or we can use Boost.Multiprecision to generate 50 decimal digit roots of - Jv for non-integral order v= 71/19 == 3.736842, - expressed as an exact-integer fraction to generate the most accurate value - possible for all floating-point types. -

-

- We set the precision of the output stream, and show trailing zeros to display - a fixed 50 decimal digits. -

-

-

-
std::cout.precision(std::numeric_limits<float_type>::digits10); // 50 decimal digits.
-std::cout << std::showpoint << std::endl; // Show trailing zeros.
-
-float_type x = float_type(71) / 19;
-float_type r = boost::math::cyl_bessel_j_zero(x, 1); // 1st root.
-std::cout << "x = " << x << ", r = " << r << std::endl;
-
-r = boost::math::cyl_bessel_j_zero(x, 20U); // 20th root.
-std::cout << "x = " << x << ", r = " << r << std::endl;
-
-std::vector<float_type> zeros;
-boost::math::cyl_bessel_j_zero(x, 1, 3, std::back_inserter(zeros));
-
-std::cout << "cyl_bessel_j_zeros" << std::endl;
-// Print the roots to the output stream.
-std::copy(zeros.begin(), zeros.end(),
-          std::ostream_iterator<float_type>(std::cout, "\n"));
-
-

-

-
- - Using - Output Iterator to sum zeros of Bessel Functions -
-

- This example demonstrates summing zeros of the Bessel functions. To use - the functions for finding zeros of the functions we need -

-

-

-
#include <boost/math/special_functions/bessel.hpp>
-
-

-

-

- We use the cyl_bessel_j_zero - output iterator parameter out_it - to create a sum of 1/zeros2 by defining a custom output - iterator: -

-

-

-
template <class T>
-struct output_summation_iterator
-{
-   output_summation_iterator(T* p) : p_sum(p)
-   {}
-   output_summation_iterator& operator*()
-   { return *this; }
-    output_summation_iterator& operator++()
-   { return *this; }
-   output_summation_iterator& operator++(int)
-   { return *this; }
-   output_summation_iterator& operator = (T const& val)
-   {
-     *p_sum += 1./ (val * val); // Summing 1/zero^2.
-     return *this;
-   }
-private:
-   T* p_sum;
-};
-
-

-

-

- The sum is calculated for many values, converging on the analytical exact - value of 1/8. -

-

-

-
using boost::math::cyl_bessel_j_zero;
-double nu = 1.;
-double sum = 0;
-output_summation_iterator<double> it(&sum);  // sum of 1/zeros^2
-cyl_bessel_j_zero(nu, 1, 10000, it);
-
-double s = 1/(4 * (nu + 1)); // 0.125 = 1/8 is exact analytical solution.
-std::cout << std::setprecision(6) << "nu = " << nu << ", sum = " << sum
-  << ", exact = " << s << std::endl;
-// nu = 1.00000, sum = 0.124990, exact = 0.125000
-
-

-

-
- - Calculating - zeros of the Neumann function. -
-

- This example also shows how Boost.Math and Boost.Multiprecision can be - combined to provide a many decimal digit precision. For 50 decimal digit - precision we need to include -

-

-

-
#include <boost/multiprecision/cpp_dec_float.hpp>
-
-

-

-

- and a typedef for float_type may be convenient (allowing - a quick switch to re-compute at built-in double - or other precision) -

-

-

-
typedef boost::multiprecision::cpp_dec_float_50 float_type;
-
-

-

-

- To use the functions for finding zeros of the cyl_neumann - function we need: -

-

-

-
#include <boost/math/special_functions/bessel.hpp>
-
-

-

-

- The Neumann (Bessel Y) function zeros are evaluated very similarly: -

-

-

-
using boost::math::cyl_neumann_zero;
-double zn = cyl_neumann_zero(2., 1);
-std::cout << "cyl_neumann_zero(2., 1) = " << zn << std::endl;
-
-std::vector<float> nzeros(3); // Space for 3 zeros.
-cyl_neumann_zero<float>(2.F, 1, nzeros.size(), nzeros.begin());
-
-std::cout << "cyl_neumann_zero<float>(2.F, 1, ";
-// Print the zeros to the output stream.
-std::copy(nzeros.begin(), nzeros.end(),
-          std::ostream_iterator<float>(std::cout, ", "));
-
-std::cout << "\n""cyl_neumann_zero(static_cast<float_type>(220)/100, 1) = "
-  << cyl_neumann_zero(static_cast<float_type>(220)/100, 1) << std::endl;
-// 3.6154383428745996706772556069431792744372398748422
-
-

-

-
- - Error - messages from 'bad' input -
-

- Another example demonstrates calculating zeros of the Bessel functions - showing the error messages from 'bad' input is handled by throwing exceptions. -

-

- To use the functions for finding zeros of the functions we need: -

-

-

-
#include <boost/math/special_functions/bessel.hpp>
-#include <boost/math/special_functions/airy.hpp>
-
-

-

-
- - - - - -
[Tip]Tip

- It is always wise to place all code using Boost.Math inside try'n'catch - blocks; this will ensure that helpful error messages can be shown when - exceptional conditions arise. -

-

- Examples below show messages from several 'bad' arguments that throw a - domain_error exception. -

-

-

-
try
-{ // Try a zero order v.
-  float dodgy_root = boost::math::cyl_bessel_j_zero(0.F, 0);
-  std::cout << "boost::math::cyl_bessel_j_zero(0.F, 0) " << dodgy_root << std::endl;
-  // Thrown exception Error in function boost::math::cyl_bessel_j_zero<double>(double, int):
-  // Requested the 0'th zero of J0, but the rank must be > 0 !
-}
-catch (std::exception& ex)
-{
-  std::cout << "Thrown exception " << ex.what() << std::endl;
-}
-
-

-

-
- - - - - -
[Note]Note

- The type shown in the error message is the type after - promotion, using precision - policy and internal - promotion policy, from float - to double in this case. -

-

- In this example the promotion goes: -

-
    -
  1. - Arguments are float and - int. -
    2. -
  2. - Treat int "as if" - it were a double, so arguments - are float and double. -
    3. -
  3. - Common type is double - - so that's the precision we want (and the type that will be returned). -
    4. -
  4. - Evaluate internally as double - for full float precision. -
    5. -
-

- See full code for other examples that promote from double - to long double. -

-

- Other examples of 'bad' inputs like infinity and NaN are below. Some compiler - warnings indicate that 'bad' values are detected at compile time. -

-

-

-
try
-{ // order v = inf
-   std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << std::endl;
-   double inf = std::numeric_limits<double>::infinity();
-   double inf_root = boost::math::cyl_bessel_j_zero(inf, 1);
-   std::cout << "boost::math::cyl_bessel_j_zero(inf, 1) " << inf_root << std::endl;
-   // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, unsigned):
-   // Order argument is 1.#INF, but must be finite >= 0 !
-}
-catch (std::exception& ex)
-{
-  std::cout << "Thrown exception " << ex.what() << std::endl;
-}
-
-try
-{ // order v = NaN, rank m = 1
-   std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << std::endl;
-   double nan = std::numeric_limits<double>::quiet_NaN();
-   double nan_root = boost::math::cyl_bessel_j_zero(nan, 1);
-   std::cout << "boost::math::cyl_bessel_j_zero(nan, 1) " << nan_root << std::endl;
-   // Throw exception Error in function boost::math::cyl_bessel_j_zero<long double>(long double, unsigned):
-   // Order argument is 1.#QNAN, but must be finite >= 0 !
-}
-catch (std::exception& ex)
-{
-  std::cout << "Thrown exception " << ex.what() << std::endl;
-}
-
-

-

-

- The output from other examples are shown appended to the full code listing. -

-

- The full code (and output) for these examples is at Bessel - zeros, Bessel - zeros iterator, Neumann - zeros, Bessel - error messages. -

-

- - Implementation -

-

- Various methods are used to compute initial estimates for jν, - m and yν, m ; these are described in detail - below. -

-

- After finding the initial estimate of a given root, its precision is subsequently - refined to the desired level using Newton-Raphson iteration from Boost.Math's - root-finding with - derivatives utilities combined with the functions cyl_bessel_j - and cyl_neumann. -

-

- Newton iteration requires both Jν(x) or Yν(x) - as well as its derivative. The derivatives of Jν(x) - and Yν(x) with respect to x are - given by M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, - NBS (1964). In particular, -

-

-   d/dx Jν(x) = Jν-1(x) - - ν Jν(x) / x -

-

-   d/dx Yν(x) = Yν-1(x) - - ν Yν(x) / x -

-

- Enumeration of the rank of a root (in other words the index of a root) - begins with one and counts up, in other words m,=1,2,3,… - The value of the first root is always greater than zero. -

-

- For certain special parameters, cylindrical Bessel functions and cylindrical - Neumann functions have a root at the origin. For example, Jν(x) - has a root at the origin for every positive order ν > 0, - and for every negative integer order ν = -n with n - ∈ ℕ + and n ≠ 0. -

-

- In addition, Yν(x) has a root at the origin for every - negative half-integer order ν = -n/2, with n - ∈ ℕ + and and n ≠ 0. -

-

- For these special parameter values, the origin with a value of x - = 0 is provided as the 0th root generated - by cyl_bessel_j_zero() - and cyl_neumann_zero(). -

-

- When calculating initial estimates for the roots of Bessel functions, a - distinction is made between positive order and negative order, and different - methods are used for these. In addition, different algorithms are used - for the first root m = 1 and for subsequent roots - with higher rank m ≥ 2. Furthermore, estimates of the - roots for Bessel functions with order above and below a cutoff at ν = - 2.2 are calculated with different methods. -

-

- Calculations of the estimates of jν,1 and yν,1 - with 0 ≤ ν < 2.2 use empirically tabulated values. - The coefficients for these have been generated by a computer algebra system. -

-

- Calculations of the estimates of jν,1 and yν,1 - with ν≥ 2.2 use Eqs.9.5.14 and 9.5.15 in M. Abramowitz - and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964). -

-

- In particular, -

-

-   jν,1 ≅ ν + 1.85575 ν⅓ + 1.033150 ν-⅓ - 0.00397 ν-1 - 0.0908 ν-5/3 + 0.043 ν-7/3 + - … -

-

- and -

-

-   yν,1 ≅ ν + 0.93157 ν⅓ + 0.26035 ν-⅓ + 0.01198 ν-1 - 0.0060 ν-5/3 - 0.001 ν-7/3 + - … -

-

- Calculations of the estimates of jν, m and yν, - m with rank m > 2 and 0 ≤ ν < - 2.2 use McMahon's approximation, as described in M. Abramowitz - and I. A. Stegan, Section 9.5 and 9.5.12. In particular, -

-

-   jν,m, yν,m ≅ β - (μ-1) / 8β -

-

-       - 4(μ-1)(7μ - 31) / 3(8β)3 -

-

-       -32(μ-1)(83μ² - 982μ + 3779) / 15(8β)5 -

-

-       -64(μ-1)(6949μ3 - 153855μ² + 1585743μ- 6277237) / 105(8a)7 -

-

-       - …     (5) -

-

- where μ = 4ν2 and β = (m + ½ν - ¼)π for - jν,m and β = (m + ½ν -¾)π for yν,m. -

-

- Calculations of the estimates of jν, m and yν, - m with ν ≥ 2.2 use one term in the asymptotic - expansion given in Eq.9.5.22 and top line of Eq.9.5.26 combined with Eq. - 9.3.39, all in M. Abramowitz and I. A. Stegun, Handbook of Mathematical - Functions, NBS (1964) explicit and easy-to-understand treatment for asymptotic - expansion of zeros. The latter two equations are expressed for argument - (x) greater than one. (Olver also gives the series - form of the equations in §10.21(vi) - McMahon's Asymptotic Expansions for Large Zeros - using slightly - different variable names). -

-

- In summary, -

-

-   jν, m ∼ νx(-ζ) + f1(-ζ/ν) -

-

- where -ζ = ν-2/3am and am is the - absolute value of the mth root of Ai(x) - on the negative real axis. -

-

- Here x = x(-ζ) is the inverse of the function -

-

-   ⅔(-ζ)3/2 = √(x² - 1) - cos⁻¹(1/x)     (7) -

-

- Furthermore, -

-

-   f1(-ζ) = ½x(-ζ) {h(-ζ)}² ⋅ b0(-ζ) -

-

- where -

-

-   h(-ζ) = {4(-ζ) / (x² - 1)}4 -

-

- and -

-

-   b0(-ζ) = -5/(48ζ²) + 1/(-ζ)½ ⋅ { 5/(24(x2-1)3/2) + 1/(8(x2-1)½)} -

-

- When solving for x(-ζ) in Eq. 7 above, the right-hand-side - is expanded to order 2 in a Taylor series for large x. - This results in -

-

-   ⅔(-ζ)3/2 ≈ x + 1/2x - π/2 -

-

- The positive root of the resulting quadratic equation is used to find an - initial estimate x(-ζ). This initial estimate is subsequently - refined with several steps of Newton-Raphson iteration in Eq. 7. -

-

- Estimates of the roots of cylindrical Bessel functions of negative order - on the positive real axis are found using interlacing relations. For example, - the mth root of the cylindrical Bessel function j-ν,m - is bracketed by the mth root and the (m+1)th - root of the Bessel function of corresponding positive integer order. In - other words, -

-

-   jnν,m < j-ν,m < jnν,m+1 -

-

- where m > 1 and nν represents - the integral floor of the absolute value of |-ν|. -

-

- Similar bracketing relations are used to find estimates of the roots of - Neumann functions of negative order, whereby a discontinuity at every negative - half-integer order needs to be handled. -

-

- Bracketing relations do not hold for the first root of cylindrical Bessel - functions and cylindrical Neumann functions with negative order. Therefore, - iterative algorithms combined with root-finding via bisection are used - to localize j-ν,1 and y-ν,1. -

-

- - Testing -

-

- The precision of evaluation of zeros was tested at 50 decimal digits using - cpp_dec_float_50 and found - identical with spot values computed by Wolfram - Alpha. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel/bessel_over.html b/doc/html/math_toolkit/special/bessel/bessel_over.html deleted file mode 100644 index 5d8cedd16..000000000 --- a/doc/html/math_toolkit/special/bessel/bessel_over.html +++ /dev/null @@ -1,205 +0,0 @@ - - - -Bessel Function Overview - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel/mbessel.html b/doc/html/math_toolkit/special/bessel/mbessel.html deleted file mode 100644 index beef1f7c8..000000000 --- a/doc/html/math_toolkit/special/bessel/mbessel.html +++ /dev/null @@ -1,481 +0,0 @@ - - - -Modified Bessel Functions of the First and Second Kinds - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

- #include <boost/math/special_functions/bessel.hpp> -

-
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. -

-

- -

-

- The following graph illustrates the exponential decay of Kv. -

-

- -

-
- - 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 41. 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 42. 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. -

-

- When x is small compared to v, - Ivx   is best computed directly from the series: -

-

- -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/bessel/sph_bessel.html b/doc/html/math_toolkit/special/bessel/sph_bessel.html deleted file mode 100644 index c77622c78..000000000 --- a/doc/html/math_toolkit/special/bessel/sph_bessel.html +++ /dev/null @@ -1,151 +0,0 @@ - - - -Spherical Bessel Functions of the First and Second Kinds - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

- #include <boost/math/special_functions/bessel.hpp> -

-
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: -

-

- -

-

- Likewise yv   is also cyclic for large x, but tends to -∞   -for small x: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint.html b/doc/html/math_toolkit/special/ellint.html deleted file mode 100644 index 88292bb53..000000000 --- a/doc/html/math_toolkit/special/ellint.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Elliptic Integrals - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint/ellint_1.html b/doc/html/math_toolkit/special/ellint/ellint_1.html deleted file mode 100644 index 23a4d65d6..000000000 --- a/doc/html/math_toolkit/special/ellint/ellint_1.html +++ /dev/null @@ -1,260 +0,0 @@ - - - -Elliptic Integrals of the First Kind - Legendre Form - - - - - - - - -
-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). -

-

- -

-

- 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 44. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint/ellint_2.html b/doc/html/math_toolkit/special/ellint/ellint_2.html deleted file mode 100644 index 2465e4e70..000000000 --- a/doc/html/math_toolkit/special/ellint/ellint_2.html +++ /dev/null @@ -1,260 +0,0 @@ - - - -Elliptic Integrals of the Second Kind - Legendre Form - - - - - - - - -
-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). -

-

- -

-

- 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 second 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 45. 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 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint/ellint_3.html b/doc/html/math_toolkit/special/ellint/ellint_3.html deleted file mode 100644 index fa6683eb0..000000000 --- a/doc/html/math_toolkit/special/ellint/ellint_3.html +++ /dev/null @@ -1,314 +0,0 @@ - - - -Elliptic Integrals of the Third Kind - Legendre Form - - - - - - - - -
-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). -

-

- -

-

- 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. -

-
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). -

-

- 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 46. 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) ; n <= 1 -

-

- Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1 [1] -

-

- 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 -

-

- -

-
-
-

[1] - I haven't been able to find a literature reference for this relation, - but it appears to be the convention used by Mathematica. Intuitively - the first 2 * m * Π(n, k) terms cancel out as the - derivative alternates between +∞ and -∞. -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint/ellint_carlson.html b/doc/html/math_toolkit/special/ellint/ellint_carlson.html deleted file mode 100644 index 3c23f4a12..000000000 --- a/doc/html/math_toolkit/special/ellint/ellint_carlson.html +++ /dev/null @@ -1,434 +0,0 @@ - - - -Elliptic Integrals - Carlson Form - - - - - - - - -
-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: -

-

- -

-

- 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 43. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/ellint/ellint_intro.html b/doc/html/math_toolkit/special/ellint/ellint_intro.html deleted file mode 100644 index fb9eea139..000000000 --- a/doc/html/math_toolkit/special/ellint/ellint_intro.html +++ /dev/null @@ -1,400 +0,0 @@ - - - -Elliptic Integral Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/expint.html b/doc/html/math_toolkit/special/expint.html deleted file mode 100644 index 8c4267def..000000000 --- a/doc/html/math_toolkit/special/expint.html +++ /dev/null @@ -1,42 +0,0 @@ - - - -Exponential Integrals - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Exponential Integral - En
-
Exponential Integral - Ei
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/expint/expint_i.html b/doc/html/math_toolkit/special/expint/expint_i.html deleted file mode 100644 index 83fa28ad5..000000000 --- a/doc/html/math_toolkit/special/expint/expint_i.html +++ /dev/null @@ -1,293 +0,0 @@ - - - -Exponential Integral Ei - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/expint.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type expint(T z);
-
-template <class T, class Policy>
-calculated-result-type expint(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 expint(T z);
-
-template <class T, class Policy>
-calculated-result-type expint(T z, const Policy&);
-
-

- Returns the exponential - integral of z: -

-

- -

-

- -

-
- - 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 Cody's SPECFUN implementation and the GSL-1.9 - library. Unless otherwise specified any floating point type that is narrower - than the one shown will have effectively zero - error. -

-
-

Table 50. Errors In the Function expint(z)

-
----- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- Error -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=2.4 Mean=0.6 -

-

- GSL Peak=8.9 Mean=0.7 -

-

- SPECFUN (Cody) Peak=2.5 Mean=0.6 -

-
-

- 64 -

- 		-

- RedHat Linux IA_EM64, gcc-4.1 -

- 		-

- Peak=5.1 Mean=0.8 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-4.1 -

- 		-

- Peak=5.0 Mean=0.8 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=1.9 Mean=0.63 -

-
-
-

- It should be noted that all three libraries tested above offer sub-epsilon - precision over most of their range. -

-

- GSL has the greatest difficulty near the positive root of En, while Cody's - SPECFUN along with this implementation increase their error rates very - slightly over the range [4,6]. -

-
- - 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 -
-

- For x < 0 this function just calls zeta(1, - -x): which in turn is implemented in terms of rational approximations when - the type of x has 113 or fewer bits of precision. -

-

- For x > 0 the generic version is implemented using the infinte series: -

-

- -

-

- However, when the precision of the argument type is known at compile time - and is 113 bits or less, then rational approximations devised - by JM are used. -

-

- For 0 < z < 6 a root-preserving approximation of the form: -

-

- -

-

- is used, where z0 is the positive root of the function, and R(z/3 - 1) is - a minimax rational approximation rescaled so that it is evaluated over - [-1,1]. Note that while the rational approximation over [0,6] converges - rapidly to the minimax solution it is rather ill-conditioned in practice. - Cody and Thacher [5] experienced the same issue and converted the polynomials into - Chebeshev form to ensure stable computation. By experiment we found that - the polynomials are just as stable in polynomial as Chebyshev form, provided - they are computed over the interval [-1,1]. -

-

- Over the a series of intervals [a,b] and [b,INF] the rational approximation - takes the form: -

-

- -

-

- where c is a constant, and R(t) is a minimax solution - optimised for low absolute error compared to c. Variable - t is 1/z when - the range in infinite and 2z/(b-a) - - (2a/(b-a) + 1) otherwise: this has the effect of scaling - z to the interval [-1,1]. As before rational approximations over arbitrary - intervals were found to be ill-conditioned: Cody and Thacher solved this - issue by converting the polynomials to their J-Fraction equivalent. However, - as long as the interval of evaluation was [-1,1] and the number of terms - carefully chosen, it was found that the polynomials could - be evaluated to suitable precision: error rates are typically 2 to 3 epsilon - which is comparible to the error rate that Cody and Thacher achieved using - J-Fractions, but marginally more efficient given that fewer divisions are - involved. -

-
-
-

[5] - W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations - for the exponential integral E1(x), Math. Comp. 22 (1968), 641-649, and - W. J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential - integral Ei(x), Math. Comp. 23 (1969), 289-303. -

-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/expint/expint_n.html b/doc/html/math_toolkit/special/expint/expint_n.html deleted file mode 100644 index 14183a0cd..000000000 --- a/doc/html/math_toolkit/special/expint/expint_n.html +++ /dev/null @@ -1,288 +0,0 @@ - - - -Exponential Integral En - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/expint.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type expint(unsigned n, T z);
-
-template <class T, class Policy>
-calculated-result-type expint(unsigned n, 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 expint(unsigned n, T z);
-
-template <class T, class Policy>
-calculated-result-type expint(unsigned n, T z, const Policy&);
-
-

- Returns the exponential - integral En of z: -

-

- -

-

- -

-
- - 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 Cephes library. - Unless otherwise specified any floating point type that is narrower than - the one shown will have effectively zero error. -

-
-

Table 49. Errors In the Function expint(n, z)

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- En -

- 		-

- E1 -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=7.1 Mean=1.8 -

-

- Cephes Peak=5.1 - Mean=1.3 -

- 		-

- Peak=0.99 Mean=0.5 -

-

- Cephes Peak=3.1 - Mean=1.1 -

-
-

- 64 -

- 		-

- RedHat Linux IA_EM64, gcc-4.1 -

- 		-

- Peak=9.9 Mean=2.1 -

- 		-

- Peak=0.97 Mean=0.4 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-4.1 -

- 		-

- Peak=9.9 Mean=2.1 -

- 		-

- Peak=0.97 Mean=0.4 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=23.3 Mean=3.7 -

- 		-

- Peak=1.6 Mean=0.5 -

-
-
-
- - 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 -
-

- The generic version of this function uses the continued fraction: -

-

- -

-

- for large x and the infinite series: -

-

- -

-

- for small x. -

-

- Where the precision of x is known at compile time - and is 113 bits or fewer in precision, then rational approximations devised - by JM are used for the n - == 1 - case. -

-

- For x < - 1 the approximating form is a minimax - approximation: -

-

- -

-

- and for x > - 1 a Chebyshev interpolated approximation - of the form: -

-

- -

-

- is used. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials.html b/doc/html/math_toolkit/special/factorials.html deleted file mode 100644 index f7c2102fc..000000000 --- a/doc/html/math_toolkit/special/factorials.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Factorials and Binomial Coefficients - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Factorial
-
Double - Factorial
-
Rising - Factorial
-
Falling - Factorial
-
Binomial - Coefficients
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials/sf_binomial.html b/doc/html/math_toolkit/special/factorials/sf_binomial.html deleted file mode 100644 index db466d7f4..000000000 --- a/doc/html/math_toolkit/special/factorials/sf_binomial.html +++ /dev/null @@ -1,139 +0,0 @@ - - - -Binomial Coefficients - - - - - - - - -
-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. -

-
- - - - - -
[Important]Important
-

- The functions described above are templates where the template argument - T can not be deduced from the arguments passed to the function. Therefore - if you write something like: -

-

- boost::math::binomial_coefficient(10, 2); -

-

- You will get a compiler error, ususally indicating that there is no such - function to be found. Instead you need to specifiy the return type explicity - and write: -

-

- boost::math::binomial_coefficient<double>(10, 2); -

-

- So that the return type is known. Further, the template argument must - be a real-valued type such as float - or double and not an integer - type - that would overflow far too easily! -

-
-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials/sf_double_factorial.html b/doc/html/math_toolkit/special/factorials/sf_double_factorial.html deleted file mode 100644 index 8b1d1450e..000000000 --- a/doc/html/math_toolkit/special/factorials/sf_double_factorial.html +++ /dev/null @@ -1,159 +0,0 @@ - - - -Double Factorial - - - - - - - - -
-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. -

-
- - - - - -
[Important]Important
-

- The functions described above are templates where the template argument - T can not be deduced from the arguments passed to the function. Therefore - if you write something like: -

-

- boost::math::double_factorial(2); -

-

- You will get a (possibly perplexing) compiler error, ususally indicating - that there is no such function to be found. Instead you need to specifiy - the return type explicity and write: -

-

- boost::math::double_factorial<double>(2); -

-

- So that the return type is known. Further, the template argument must - be a real-valued type such as float - or double and not an integer - type - that would overflow far too easily! -

-

- The source code static_assert - and comment just after the will be: -

-

-

-
BOOST_STATIC_ASSERT(!boost::is_integral<T>::value);
-// factorial<unsigned int>(n) is not implemented
-// because it would overflow integral type T for too small n
-// to be useful. Use instead a floating-point type,
-// and convert to an unsigned type if essential, for example:
-// unsigned int nfac = static_cast<unsigned int>(factorial<double>(n));
-// See factorial documentation for more detail.
-
-

-

-
-
- - - - - -
[Note]Note

- The argument to double_factorial - is type unsigned even though - technically -1!! is defined. -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials/sf_factorial.html b/doc/html/math_toolkit/special/factorials/sf_factorial.html deleted file mode 100644 index 809b4dd47..000000000 --- a/doc/html/math_toolkit/special/factorials/sf_factorial.html +++ /dev/null @@ -1,191 +0,0 @@ - - - -Factorial - - - - - - - - -
-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 -
-
- - - - - -
[Important]Important
-

- The functions described below are templates where the template argument - T CANNOT be deduced from the arguments passed to the function. Therefore - if you write something like: -

-

- boost::math::factorial(2); -

-

- You will get a (perhaps perplexing) compiler error, ususally indicating - that there is no such function to be found. Instead you need to specify - the return type explicity and write: -

-

- boost::math::factorial<double>(2); -

-

- So that the return type is known. -

-

- Furthermore, the template argument must be a real-valued type such as - float or double - and not an integer type - that would overflow far too easily for quite - small values of parameter i! -

-

- The source code static_assert - and comment just after the will be: -

-

-

-
BOOST_STATIC_ASSERT(!boost::is_integral<T>::value);
-// factorial<unsigned int>(n) is not implemented
-// because it would overflow integral type T for too small n
-// to be useful. Use instead a floating-point type,
-// and convert to an unsigned type if essential, for example:
-// unsigned int nfac = static_cast<unsigned int>(factorial<double>(n));
-// See factorial documentation for more detail.
-
-

-

-
-
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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials/sf_falling_factorial.html b/doc/html/math_toolkit/special/factorials/sf_falling_factorial.html deleted file mode 100644 index 04a22214a..000000000 --- a/doc/html/math_toolkit/special/factorials/sf_falling_factorial.html +++ /dev/null @@ -1,104 +0,0 @@ - - - -Falling Factorial - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/factorials/sf_rising_factorial.html b/doc/html/math_toolkit/special/factorials/sf_rising_factorial.html deleted file mode 100644 index 18c5d3f96..000000000 --- a/doc/html/math_toolkit/special/factorials/sf_rising_factorial.html +++ /dev/null @@ -1,108 +0,0 @@ - - - -Rising Factorial - - - - - - - - -
-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-1) -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/hankel.html b/doc/html/math_toolkit/special/hankel.html deleted file mode 100644 index ae11cd1f2..000000000 --- a/doc/html/math_toolkit/special/hankel.html +++ /dev/null @@ -1,42 +0,0 @@ - - - -Hankel Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Cyclic Hankel - Functions
-
Spherical Hankel - Functions
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/hankel/cyl_hankel.html b/doc/html/math_toolkit/special/hankel/cyl_hankel.html deleted file mode 100644 index ac060c57c..000000000 --- a/doc/html/math_toolkit/special/hankel/cyl_hankel.html +++ /dev/null @@ -1,153 +0,0 @@ - - - -Cyclic Hankel Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-
template <class T1, class T2>
-std::complex<calculated-result-type> cyl_hankel_1(T1 v, T2 x);
-
-template <class T1, class T2, class Policy>
-std::complex<calculated-result-type> cyl_hankel_1(T1 v, T2 x, const Policy&);
-
-template <class T1, class T2>
-std::complex<calculated-result-type> cyl_hankel_2(T1 v, T2 x);
-
-template <class T1, class T2, class Policy>
-std::complex<calculated-result-type> cyl_hankel_2(T1 v, T2 x, const Policy&);
-
-
- - Description -
-

- The functions cyl_hankel_1 - and cyl_hankel_2 - return the result of the Hankel - functions of the first and second kind respectively: -

-

- cyl_hankel_1(v, x) = Hv(1)(x) = Jv(x) + i Yv(x) -

-

- cyl_hankel_2(v, x) = Hv(2)(x) = Jv(x) - i Yv(x) -

-

- where: -

-

- Jv(x) is the Bessel function of the first kind, and - Yv(x) is the Bessel function of the second kind. -

-

- 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. -

-

- Note that while the arguments to these functions are real values, the results - are complex. That means that the functions can only be instantiated on - types float, double and long - double. The functions have also - been extended to operate over the whole range of v - and x (unlike cyl_bessel_j - and cyl_neumann). -

-
- - Performance -
-

- These functions are generally more efficient than two separate calls to - the underlying Bessel functions as internally Bessel J and Y can be computed - simultaneously. -

-
- - Testing -
-

- There are just a few spot tests to exercise all the special case handling - - the bulk of the testing is done on the Bessel functions upon which these - are based. -

-
- - Accuracy -
-

- Refer to cyl_bessel_j - and cyl_neumann. -

-
- - Implementation -
-

- For x < 0 the following reflection formulae are - used: -

-

- - -

-

- - -

-

- - -

-

- Otherwise the implementation is trivially in terms of the Bessel J and - Y functions. -

-

- Note however, that the Hankel functions compute the Bessel J and Y functions - simultaneously, and therefore a single Hankel function call is more efficient - than two Bessel function calls. The one exception is when v - is a small positive integer, in which case the usual Bessel function routines - for integer order are used. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/hankel/sph_hankel.html b/doc/html/math_toolkit/special/hankel/sph_hankel.html deleted file mode 100644 index cbb31013a..000000000 --- a/doc/html/math_toolkit/special/hankel/sph_hankel.html +++ /dev/null @@ -1,114 +0,0 @@ - - - -Spherical Hankel Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-
template <class T1, class T2>
-std::complex<calculated-result-type> sph_hankel_1(T1 v, T2 x);
-
-template <class T1, class T2, class Policy>
-std::complex<calculated-result-type> sph_hankel_1(T1 v, T2 x, const Policy&);
-
-template <class T1, class T2>
-std::complex<calculated-result-type> sph_hankel_2(T1 v, T2 x);
-
-template <class T1, class T2, class Policy>
-std::complex<calculated-result-type> sph_hankel_2(T1 v, T2 x, const Policy&);
-
-
- - Description -
-

- The functions sph_hankel_1 - and sph_hankel_2 - return the result of the spherical - Hankel functions of the first and second kind respectively: -

-

- -

-

- -

-

- 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. -

-

- Note that while the arguments to these functions are real values, the results - are complex. That means that the functions can only be instantiated on - types float, double and long - double. The functions have also - been extended to operate over the whole range of v - and x (unlike cyl_bessel_j - and cyl_neumann). -

-
- - Testing -
-

- There are just a few spot tests to exercise all the special case handling - - the bulk of the testing is done on the Bessel functions upon which these - are based. -

-
- - Accuracy -
-

- Refer to cyl_bessel_j - and cyl_neumann. -

-
- - Implementation -
-

- These functions are trivially implemented in terms of cyl_hankel_1 - and cyl_hankel_2. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/inv_hyper.html b/doc/html/math_toolkit/special/inv_hyper.html deleted file mode 100644 index 8fc927d4a..000000000 --- a/doc/html/math_toolkit/special/inv_hyper.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Inverse Hyperbolic Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Inverse - Hyperbolic Functions Overview
-
acosh
-
asinh
-
atanh
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/inv_hyper/acosh.html b/doc/html/math_toolkit/special/inv_hyper/acosh.html deleted file mode 100644 index 49f8029bc..000000000 --- a/doc/html/math_toolkit/special/inv_hyper/acosh.html +++ /dev/null @@ -1,123 +0,0 @@ - - - -acosh - - - - - - - - -
-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. -

-

- 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. -

-

- -

-
- - Accuracy -
-

- Generally accuracy is to within 1 or 2 epsilon across all supported platforms. -

-
- - Testing -
-

- This function is tested using a combination of random test values designed - to give full function coverage computed at high precision using the "naive" - formula: -

-

- -

-

- along with a selection of sanity check values computed using functions.wolfram.com - to at least 50 decimal digits. -

-
- - Implementation -
-

- For sufficiently large x, we can use the approximation: -

-

- -

-

- For x sufficiently close to 1 we can use the approximation: -

-

- -

-

- Otherwise for x close to 1 we can use the following rearrangement of the - primary definition to preserve accuracy: -

-

- -

-

- Otherwise the primary - definition is used: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/inv_hyper/asinh.html b/doc/html/math_toolkit/special/inv_hyper/asinh.html deleted file mode 100644 index 1968b393d..000000000 --- a/doc/html/math_toolkit/special/inv_hyper/asinh.html +++ /dev/null @@ -1,118 +0,0 @@ - - - -asinh - - - - - - - - -
-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. -

-

- 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. -

-
- - Accuracy -
-

- Generally accuracy is to within 1 or 2 epsilon across all supported platforms. -

-
- - Testing -
-

- This function is tested using a combination of random test values designed - to give full function coverage computed at high precision using the "naive" - formula: -

-

- -

-

- along with a selection of sanity check values computed using functions.wolfram.com - to at least 50 decimal digits. -

-
- - Implementation -
-

- For sufficiently large x we can use the approximation: -

-

- -

-

- While for very small x we can use the approximation: -

-

- -

-

- For 0.5 > x > ε the following rearrangement of the primary definition - is used: -

-

- -

-

- Otherwise evalution is via the primary - definition: -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/inv_hyper/atanh.html b/doc/html/math_toolkit/special/inv_hyper/atanh.html deleted file mode 100644 index d12fe427f..000000000 --- a/doc/html/math_toolkit/special/inv_hyper/atanh.html +++ /dev/null @@ -1,128 +0,0 @@ - - - -atanh - - - - - - - - -
-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. -

-

- 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. -

-

- -

-
- - Accuracy -
-

- Generally accuracy is to within 1 or 2 epsilon across all supported platforms. -

-
- - Testing -
-

- This function is tested using a combination of random test values designed - to give full function coverage computed at high precision using the "naive" - formula: -

-

- -

-

- along with a selection of sanity check values computed using functions.wolfram.com - to at least 50 decimal digits. -

-
- - Implementation -
-

- For sufficiently small x we can use the approximation: -

-

- -

-

- Otherwise the primary - definition: -

-

- -

-

- or its equivalent form: -

-

- -

-

- is used. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/inv_hyper/inv_hyper_over.html b/doc/html/math_toolkit/special/inv_hyper/inv_hyper_over.html deleted file mode 100644 index cebc5f193..000000000 --- a/doc/html/math_toolkit/special/inv_hyper/inv_hyper_over.html +++ /dev/null @@ -1,106 +0,0 @@ - - - -Inverse Hyperbolic Functions Overview - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi.html b/doc/html/math_toolkit/special/jacobi.html deleted file mode 100644 index bde0ed14a..000000000 --- a/doc/html/math_toolkit/special/jacobi.html +++ /dev/null @@ -1,66 +0,0 @@ - - - -Jacobi Elliptic Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Overvew of the - Jacobi Elliptic Functions
-
Jacobi - Elliptic SN, CN and DN
-
Jacobi Elliptic - Function cd
-
Jacobi Elliptic - Function cn
-
Jacobi Elliptic - Function cs
-
Jacobi Elliptic - Function dc
-
Jacobi Elliptic - Function dn
-
Jacobi Elliptic - Function ds
-
Jacobi Elliptic - Function nc
-
Jacobi Elliptic - Function nd
-
Jacobi Elliptic - Function ns
-
Jacobi Elliptic - Function sc
-
Jacobi Elliptic - Function sd
-
Jacobi Elliptic - Function sn
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jac_over.html b/doc/html/math_toolkit/special/jacobi/jac_over.html deleted file mode 100644 index e4ca14dd9..000000000 --- a/doc/html/math_toolkit/special/jacobi/jac_over.html +++ /dev/null @@ -1,118 +0,0 @@ - - - -Overvew of the Jacobi Elliptic Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- There are twelve Jacobi Elliptic functions, of which the three copolar - functions sn, cn and dn - are the most important as the other nine can be computed from these three - [2] [3] [4]. -

-

- These functions each take two arguments: a parameter, and a variable as - described below. -

-

- Like all elliptic functions these can be parameterised in a number of ways: -

-
    -
  • - In terms of a parameter m. -
    • -
  • - In terms of the elliptic modulus k where m - = k2. -
    • -
  • - In terms of the modular angle α, where m = sin2α. -
    • -
-

- In our implementation, these functions all take the elliptic modulus k - as the parameter. -

-

- In addition the variable u is sometimes expressed - as an amplitude φ, in our implementation we always use u. -

-

- Finally note that our functions all take the elliptic modulus as the first - argument - this is for alignment with the Elliptic Integrals. -

-

- There are twenve functions for computing the twelve individual Jacobi elliptic - functions: jacobi_cd, - jacobi_cn, - jacobi_cs, - jacobi_dc, - jacobi_dn, - jacobi_ds, - jacobi_nc, - jacobi_nd, - jacobi_ns, - jacobi_sc, - jacobi_sd - and jacobi_sn. -

-

- They are all called as for example: -

-
jacobi_cs(k, u);
-
-

- Note however that these individual functions are all really thin wrappers - around the function jacobi_elliptic - which calculates the three copolar functions sn, - cn and dn in a single function - call. Thus if you need more than one of these functions for a given set - of arguments, it's most efficient to use jacobi_elliptic. -

-
-
- - - -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_cd.html b/doc/html/math_toolkit/special/jacobi/jacobi_cd.html deleted file mode 100644 index 78af379fd..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_cd.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function cd - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_cd(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_cd(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function cd. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- cd(u, k) = cn(u, k) / dn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_cn.html b/doc/html/math_toolkit/special/jacobi/jacobi_cn.html deleted file mode 100644 index 02e2c11b8..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_cn.html +++ /dev/null @@ -1,75 +0,0 @@ - - - -Jacobi Elliptic Function cn - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_cn(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_cn(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function cn. -

-

- 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 a trivial wrapper around jacobi_elliptic. -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_cs.html b/doc/html/math_toolkit/special/jacobi/jacobi_cs.html deleted file mode 100644 index f36e0064e..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_cs.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function cs - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_cs(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_cs(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function cs. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- cs(u, k) = cn(u, k) / sn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_dc.html b/doc/html/math_toolkit/special/jacobi/jacobi_dc.html deleted file mode 100644 index d9c0f5236..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_dc.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function dc - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_dc(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_dc(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function dc. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- dc(u, k) = dn(u, k) / cn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_dn.html b/doc/html/math_toolkit/special/jacobi/jacobi_dn.html deleted file mode 100644 index cfbb4afe4..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_dn.html +++ /dev/null @@ -1,75 +0,0 @@ - - - -Jacobi Elliptic Function dn - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_dn(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_dn(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function dn. -

-

- 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 a trivial wrapper around jacobi_elliptic. -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_ds.html b/doc/html/math_toolkit/special/jacobi/jacobi_ds.html deleted file mode 100644 index befeffeb5..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_ds.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function ds - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_ds(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_ds(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function ds. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- ds(u, k) = dn(u, k) / sn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_elliptic.html b/doc/html/math_toolkit/special/jacobi/jacobi_elliptic.html deleted file mode 100644 index a5605fe49..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_elliptic.html +++ /dev/null @@ -1,295 +0,0 @@ - - - -Jacobi Elliptic SN, CN and DN - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U, class V>
- calculated-result-type jacobi_elliptic(T k, U u, V* pcn, V* pdn);
-
- template <class T, class U, class V, class Policy>
- calculated-result-type jacobi_elliptic(T k, U u, V* pcn, V* pdn, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- The function jacobi_elliptic - calculates the three copolar Jacobi elliptic functions sn(u, - k), cn(u, k) and dn(u, k). - The returned value is sn(u, k), and if provided, - *pcn is set to cn(u, k), and - *pdn is set to dn(u, k). -

-

- The functions are defined as follows, given: -

-

- -

-

- The the angle φ is called the amplitude and: -

-

- -

-
- - - - - -
[Note]Note
-

- φ is called the amplitude. -

-

- k is called the modulus. -

-
-
- - - - - -
[Caution]Caution
-

- Rather like other elliptic functions, the Jacobi functions are expressed - in a variety of different ways. In particular, the parameter k - (the modulus) may also be expressed using a modular angle α, or a parameter - m. These are related by: -

-

- k = sinα -

-

- m = k2 = sin2α -

-

- So that the function sn (for example) may be expressed - as either: -

-

- sn(u, k) -

-

- sn(u \ α) -

-

- sn(u| 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, and makes - this the first argument to the functions: this is for alignment with - the elliptic integrals which match the requirements of the Technical - Report on C++ Library Extensions. However, you should be extra - careful when using these 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. -

-

- The following graphs illustrate how these functions change as k - changes: for small k these are sine waves, while as - k tends to 1 they become hyperbolic functions: -

-

- -

-

- -

-

- -

-
- - Accuracy -
-

- These functions are computed using only basic arithmetic operations and - trigomometric functions, so there isn't much variation in accuracy over - differing platforms. Typically errors are trivially small for small angles, - and as is typical for cyclic functions, grow as the angle increases. 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 47. Errors Rates in the Jacobi Elliptic Functions

-
------ - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- u < 1 -

- 		-

- Large u -

-
-

- 53 -

- 		-

- Win32 / Visual C++ 8.0 -

- 		-

- Peak=2 Mean=0.5 -

- 		-

- Peak=44000 Mean=2500 -

-
-

- 64 -

- 		-

- Ubuntu Linux / G++ 4.7 -

- 		-

- Peak=2.0 Mean=0.5 -

- 		-

- Peak=25000 Mean=1500 -

-
-
-
- - 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 MPFR at 1000-bit precision and this - implementation. -

-
- - Implementation -
-

- For k > 1 we apply the relations: -

-

- -

-

- Then filter off the special cases: -

-

- sn(0, k) = 0 and cn(0, k) = dn(0, k) = 1. -

-

- sn(u, 0) = sin(u), cn(u, 0) = cos(u) and dn(u, 0) = 1. -

-

- sn(u, 1) = tanh(u), cn(u, 1) = dn(u, 1) = 1 / cosh(u). -

-

- And for k4 < ε we have: -

-

- -

-

- Otherwise the values are calculated using the method of arithmetic - geometric means. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_nc.html b/doc/html/math_toolkit/special/jacobi/jacobi_nc.html deleted file mode 100644 index 9b7ae149a..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_nc.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function nc - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_nc(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_nc(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function nc. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- nc(u, k) = 1 / cn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_nd.html b/doc/html/math_toolkit/special/jacobi/jacobi_nd.html deleted file mode 100644 index ebafd7b30..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_nd.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function nd - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_nd(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_nd(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function nd. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- nd(u, k) = 1 / dn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_ns.html b/doc/html/math_toolkit/special/jacobi/jacobi_ns.html deleted file mode 100644 index d83309bd4..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_ns.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function ns - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_ns(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_ns(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function ns. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- ns(u, k) = 1 / sn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_sc.html b/doc/html/math_toolkit/special/jacobi/jacobi_sc.html deleted file mode 100644 index d829a8aa5..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_sc.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function sc - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_sc(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_sc(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function sc. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- sc(u, k) = sn(u, k) / cn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_sd.html b/doc/html/math_toolkit/special/jacobi/jacobi_sd.html deleted file mode 100644 index c916ffd22..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_sd.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Jacobi Elliptic Function sd - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_sd(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_sd(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function sd. -

-

- 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 a trivial wrapper around jacobi_elliptic, - with: -

-

- sd(u, k) = sn(u, k) / dn(u, k) -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/jacobi/jacobi_sn.html b/doc/html/math_toolkit/special/jacobi/jacobi_sn.html deleted file mode 100644 index 95da3835b..000000000 --- a/doc/html/math_toolkit/special/jacobi/jacobi_sn.html +++ /dev/null @@ -1,75 +0,0 @@ - - - -Jacobi Elliptic Function sn - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/jacobi_elliptic.hpp>
-
-

-

-
namespace boost { namespace math {
-
- template <class T, class U>
- calculated-result-type jacobi_sn(T k, U u);
-
- template <class T, class U, class Policy>
- calculated-result-type jacobi_sn(T k, U u, const Policy& pol);
-
-}} // namespaces
-
-
- - Description -
-

- This function returns the Jacobi elliptic function sn. -

-

- 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 a trivial wrapper around jacobi_elliptic. -

-

- -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/owens_t.html b/doc/html/math_toolkit/special/owens_t.html deleted file mode 100644 index 94c6dcfeb..000000000 --- a/doc/html/math_toolkit/special/owens_t.html +++ /dev/null @@ -1,222 +0,0 @@ - - - -Owen's T function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/owens_t.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type owens_t(T h, T a);
-
-template <class T, class Policy>
-calculated-result-type owens_t(T h, T a, const Policy&);
-
-}} // namespaces
-
-
- - Description -
-

- Returns the Owens_t - function of h and a. -

-

- 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. -

-

-     -

-

- plot_owens_t -

-

- The function owens_t(h, a) gives the probability of the event (X - > h and 0 < Y < a * X), where X - and Y are independent standard normal random variables. -

-

- For h and a > 0, T(h,a), gives the volume of an uncorrelated bivariate - normal distribution with zero means and unit variances over the area between - y = ax and y = 0 and to the right - of x = h. -

-

- That is the area shaded in the figure below (Owens 1956). -

-

- -

-

- and is also illustrated by a 3D plot. -

-

- plot_owens_3d_xyp -

-

- This function is used in the computation of the Skew - Normal Distribution. It is also used in the computation of bivariate - and multivariate normal distribution probabilities. 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. -

-

- Owen's original paper (page 1077) provides some additional corner cases. -

-

- T(h, 0) = 0 -

-

- T(0, a) = ½π arctan(a) -

-

- T(h, 1) = ½ G(h) [1 - G(h)] -

-

- T(h, ∞) = G(|h|) -

-

- where G(h) is the univariate normal with zero mean and unit variance integral - from -∞ to h. -

-
- - Accuracy -
-

- Over the built-in types and range tested, errors are less than 10 * std::numeric_limits<RealType>::epsilon(). -

-
- - Testing -
-

- Test data was generated by Patefield and Tandy algorithms T1 and T4, and - also the suggested reference routine T7. -

-
    -
  • - T1 was rejected if the result was too small compared to atan(a) - (ie cancellation), -
    • -
  • - T4 was rejected if there was no convergence, -
    • -
  • - Both were rejected if they didn't agree. -
    • -
-

- Over the built-in types and range tested, errors are less than 10 std::numeric_limits<RealType>::epsilon(). -

-

- However, that there was a whole domain (large h, small - a) where it was not possible to generate any reliable - test values (all the methods got rejected for one reason or another). -

-

- There are also two sets of sanity tests: spot values are computed using - Wolfram - Mathematica and The R Project - for Statistical Computing. -

-
- - Implementation -
-

- The function was proposed and evaluated by Donald. - B. Owen, Tables for computing bivariate normal probabilities, Ann. Math. - Statist., 27, 1075-1090 (1956). -

-

- The algorithms of Patefield, M. and Tandy, D. "Fast and accurate Calculation - of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25 - (2000) are adapted for C++ with arbitrary RealType. -

-

- The Patefield-Tandy algorithm provides six methods of evalualution (T1 to - T6); the best method is selected according to the values of a - and h. See the original paper and the source in owens_t.hpp - for details. -

-

- The Patefield-Tandy algorithm is accurate to approximately 20 decimal places, - so for types with greater precision we use: -

-
    -
  • - A modified version of T1 which folds the calculation of atan(h) - into the T1 series (to avoid subtracting two values similar in magnitude), - and then accelerates the resulting alternating series using method 1 - from H. Cohen, F. Rodriguez Villegas, D. Zagier, "Convergence acceleration - of alternating series", Bonn, (1991). The result is valid everywhere, - but doesn't always converge, or may become too divergent in the first - few terms to sum accurately. This is used for ah < 1. -
    • -
  • - A modified version of T2 which is accelerated in the same manner as T1. - This is used for h > 1. -
    • -
  • - A version of T4 only when both T1 and T2 have failed to produce an accurate - answer. -
    • -
  • - Fallback to the Patefiled Tandy algorithm when all the above methods - fail: this happens not at all for our test data at 100 decimal digits - precision. However, there is a difficult area when a - is very close to 1 and the precision increases which may cause this to - happen in very exceptional circumstances. -
    • -
-

- Using the above algorithm and a 100-decimal digit type, results accurate - to 80 decimal places were obtained in the difficult area where a - is close to 1, and greater than 95 decimal places elsewhere. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers.html b/doc/html/math_toolkit/special/powers.html deleted file mode 100644 index a4aaeda97..000000000 --- a/doc/html/math_toolkit/special/powers.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Basic Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
sin_pi
-
cos_pi
-
log1p
-
expm1
-
cbrt
-
sqrt1pm1
-
powm1
-
hypot
-
Compile Time Power - of a Runtime Base
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/cbrt.html b/doc/html/math_toolkit/special/powers/cbrt.html deleted file mode 100644 index 89aebdb8a..000000000 --- a/doc/html/math_toolkit/special/powers/cbrt.html +++ /dev/null @@ -1,90 +0,0 @@ - - - -cbrt - - - - - - - - -
-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. -

-

- The following graph illustrates the behaviour of cbrt: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/cos_pi.html b/doc/html/math_toolkit/special/powers/cos_pi.html deleted file mode 100644 index a20318b46..000000000 --- a/doc/html/math_toolkit/special/powers/cos_pi.html +++ /dev/null @@ -1,69 +0,0 @@ - - - -cos_pi - - - - - - - - -
-PrevUpHomeNext -
-
-

-cos_pi -

-

-

-
#include <boost/math/special_functions/cos_pi.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type cos_pi(T x);
-
-template <class T, class Policy>
-calculated-result-type cos_pi(T x, const Policy&);
-
-}} // namespaces
-
-

- Returns the cosine of πx. -

-

- 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 performs exact all-integer arithmetic argument reduction - before computing the cosine of πx. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/ct_pow.html b/doc/html/math_toolkit/special/powers/ct_pow.html deleted file mode 100644 index 740badaaa..000000000 --- a/doc/html/math_toolkit/special/powers/ct_pow.html +++ /dev/null @@ -1,256 +0,0 @@ - - - -Compile Time Power of a Runtime Base - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The pow function effectively - computes the compile-time integral power of a run-time base. -

-
- - Synopsis -
-

- #include <boost/math/special_functions/pow.hpp> -

-
namespace boost { namespace math {
-
-template <int N, typename T>
-calculated-result-type pow(T base);
-
-template <int N, typename T, class Policy>
-calculated-result-type pow(T base, const Policy& policy);
-
-}}
-
-
- - Rationale - and Usage -
-

- Computing the power of a number with an exponent that is known at compile - time is a common need for programmers. In such cases, the usual method - is to avoid the overhead implied by the pow, - powf and powl C functions by hardcoding an expression - such as: -

-
// Hand-written 8th power of a 'base' variable
-double result = base*base*base*base*base*base*base*base;
-
-

- However, this kind of expression is not really readable (knowing the value - of the exponent involves counting the number of occurrences of base), - error-prone (it's easy to forget an occurrence), syntactically bulky, and - non-optimal in terms of performance. -

-

- The pow function of Boost.Math helps writing this kind expression along - with solving all the problems listed above: -

-
// 8th power of a 'base' variable using math::pow
-double result = pow<8>(base);
-
-

- The expression is now shorter, easier to read, safer, and even faster. - Indeed, pow will compute - the expression such that only log2(N) products are made for a power of - N. For instance in the example above, the resulting expression will be - the same as if we had written this, with only one computation of each identical - subexpression: -

-
// Internal effect of pow<8>(base)
-double result = ((base*base)*(base*base))*((base*base)*(base*base));
-
-

- Only 3 different products were actually computed. -

-
- - Return - Type -
-

- The return type of these functions is computed using the result - type calculation rules. For example: -

-
    -
  • - If T is a float, the return - type is a float. -
    • -
  • - If T is a long double, - the return type is a long double. -
    • -
  • - Otherwise, the return type is a double. -
    • -
-
- - Policies -
-

- 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. -

-
- - Error - Handling -
-

- Two cases of errors can occur when using pow: -

-
-

- The default overflow error policy is throw_on_error. - A call like pow<-2>(0) will thus throw a std::overflow_error - exception. As shown in the link given above, other error handling policies - can be used: -

-
    -
  • - 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. -
    • -
-

- The default indeterminate result error policy is ignore_error, - which for this function returns 1 since it's the most commonly chosen result - for a power of 0. Here again, other error handling policies can be used: -

-
    -
  • - throw_on_error: Throws - std::domain_error -
    • -
  • - errno_on_error: Sets - ::errno - to EDOM and returns - 1. -
    • -
  • - user_error: Returns - the result of boost::math::policies::user_indeterminate_result_error: - this function must be defined by the user. -
    • -
-

- Here is an example of error handling customization where we want to specify - the result that has to be returned in case of error. We will thus use the - user_error policy, by passing - as second argument an instance of an overflow_error policy templated with - user_error: -

-
// First we open the boost::math::policies namespace and define the `user_overflow_error`
-// by making it return the value we want in case of error (-1 here)
-
-namespace boost { namespace math { namespace policies {
-template <class T>
-T user_overflow_error(const char*, const char*, const T&)
-{ return -1; }
-}}}
-
-
-// Then we invoke pow and indicate that we want to use the user_error policy
-using boost::math::policies;
-double result = pow<-5>(base, policy<overflow_error<user_error> >());
-
-// We can now test the returned value and treat the special case if needed:
-if (result == -1)
-{
-    // there was an error, do something...
-}
-
-

- Another way is to redefine the default overflow_error - policy by using the BOOST_MATH_OVERFLOW_ERROR_POLICY macro. Once the user_overflow_error function is defined - as above, we can achieve the same result like this: -

-
// Redefine the default error_overflow policy
-#define BOOST_MATH_OVERFLOW_ERROR_POLICY user_error
-#include <boost/math/special_functions/pow.hpp>
-
-// From this point, passing a policy in argument is no longer needed, a call like this one
-// will return -1 in case of error:
-
-double result = pow<-5>(base);
-
-
- - Acknowledgements -
-

- Bruno Lalande submitted this addition to Boost.Math. -

-

- Thanks to Joaquín López Muñoz and Scott McMurray for their help in -improving the implementation. -

-
- - References -
-

- D.E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical - Algorithms, 2nd ed., Addison-Wesley, Reading, MA, 1981 -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/expm1.html b/doc/html/math_toolkit/special/powers/expm1.html deleted file mode 100644 index 7959bf43f..000000000 --- a/doc/html/math_toolkit/special/powers/expm1.html +++ /dev/null @@ -1,99 +0,0 @@ - - - -expm1 - - - - - - - - -
-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. -

-

- The following graph illustrates the behaviour of expm1: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/hypot.html b/doc/html/math_toolkit/special/powers/hypot.html deleted file mode 100644 index 032e34997..000000000 --- a/doc/html/math_toolkit/special/powers/hypot.html +++ /dev/null @@ -1,80 +0,0 @@ - - - -hypot - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/log1p.html b/doc/html/math_toolkit/special/powers/log1p.html deleted file mode 100644 index 1dcd6d89f..000000000 --- a/doc/html/math_toolkit/special/powers/log1p.html +++ /dev/null @@ -1,112 +0,0 @@ - - - -log1p - - - - - - - - -
-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. -

-

- The following graph illustrates the behaviour of log1p: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/powm1.html b/doc/html/math_toolkit/special/powers/powm1.html deleted file mode 100644 index 2517dc0af..000000000 --- a/doc/html/math_toolkit/special/powers/powm1.html +++ /dev/null @@ -1,92 +0,0 @@ - - - -powm1 - - - - - - - - -
-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. -

-

- The following graph illustrates the behaviour of powm1: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/sin_pi.html b/doc/html/math_toolkit/special/powers/sin_pi.html deleted file mode 100644 index 08f12dedc..000000000 --- a/doc/html/math_toolkit/special/powers/sin_pi.html +++ /dev/null @@ -1,69 +0,0 @@ - - - -sin_pi - - - - - - - - -
-PrevUpHomeNext -
-
-

-sin_pi -

-

-

-
#include <boost/math/special_functions/sin_pi.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type sin_pi(T x);
-
-template <class T, class Policy>
-calculated-result-type sin_pi(T x, const Policy&);
-
-}} // namespaces
-
-

- Returns the sine of πx. -

-

- 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 performs exact all-integer arithmetic argument reduction - before computing the sine of πx. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/powers/sqrt1pm1.html b/doc/html/math_toolkit/special/powers/sqrt1pm1.html deleted file mode 100644 index 3e2d667f7..000000000 --- a/doc/html/math_toolkit/special/powers/sqrt1pm1.html +++ /dev/null @@ -1,95 +0,0 @@ - - - -sqrt1pm1 - - - - - - - - -
-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. -

-

- The following graph illustrates the behaviour of sqrt1pm1: -

-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_beta.html b/doc/html/math_toolkit/special/sf_beta.html deleted file mode 100644 index 845d64172..000000000 --- a/doc/html/math_toolkit/special/sf_beta.html +++ /dev/null @@ -1,45 +0,0 @@ - - - -Beta Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Beta
-
Incomplete - Beta Functions
-
The - Incomplete Beta Function Inverses
-
Derivative - of the Incomplete Beta Function
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_beta/beta_derivative.html b/doc/html/math_toolkit/special/sf_beta/beta_derivative.html deleted file mode 100644 index 77f309b93..000000000 --- a/doc/html/math_toolkit/special/sf_beta/beta_derivative.html +++ /dev/null @@ -1,95 +0,0 @@ - - - -Derivative of the Incomplete Beta Function - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_beta/beta_function.html b/doc/html/math_toolkit/special/sf_beta/beta_function.html deleted file mode 100644 index 71ad4973b..000000000 --- a/doc/html/math_toolkit/special/sf_beta/beta_function.html +++ /dev/null @@ -1,320 +0,0 @@ - - - -Beta - - - - - - - - -
-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: -

-

- -

-

- -

-

- 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 25. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_beta/ibeta_function.html b/doc/html/math_toolkit/special/sf_beta/ibeta_function.html deleted file mode 100644 index 3ebd29f2a..000000000 --- a/doc/html/math_toolkit/special/sf_beta/ibeta_function.html +++ /dev/null @@ -1,973 +0,0 @@ - - - -Incomplete Beta Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/beta.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 0 <= x <= 1. -

-

- The normalized functions ibeta - and ibetac - require a,b >= 0, and in addition that not both - a and b are zero. -

-

- The functions beta - and betac - require a,b > 0. -

-

- 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: -

-

- -

-

- -

-
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 26. 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 27. 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 28. 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 29. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_beta/ibeta_inv_function.html b/doc/html/math_toolkit/special/sf_beta/ibeta_inv_function.html deleted file mode 100644 index ca330646a..000000000 --- a/doc/html/math_toolkit/special/sf_beta/ibeta_inv_function.html +++ /dev/null @@ -1,525 +0,0 @@ - - - -The Incomplete Beta Function Inverses - - - - - - - - -
-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 ibetac_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 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: -

-

- -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_erf.html b/doc/html/math_toolkit/special/sf_erf.html deleted file mode 100644 index 9c75d9805..000000000 --- a/doc/html/math_toolkit/special/sf_erf.html +++ /dev/null @@ -1,41 +0,0 @@ - - - -Error Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Error Functions
-
Error Function - Inverses
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_erf/error_function.html b/doc/html/math_toolkit/special/sf_erf/error_function.html deleted file mode 100644 index 483c5cb67..000000000 --- a/doc/html/math_toolkit/special/sf_erf/error_function.html +++ /dev/null @@ -1,623 +0,0 @@ - - - -Error Functions - - - - - - - - -
-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: -

-

- -

-

- -

-
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: -

-

- -

-

- -

-
- - 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 30. 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 31. 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 is an odd function and - therefore erf is calculated using: -

-
erf(z) = z * (C + R(z*z));
-
-

- where the rational approximation R(z*z) is optimised for absolute error: - as long as its absolute error is small enough compared to the constant - C, then any round-off error incurred during the computation of R(z*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 - B)) / z;
-
-

- Again R(z - B) 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 - B) is small compared - to c then c + R(z - - B) 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. The constant B - is chosen so that the left hand end of the range of the rational approximation - is 0. -

-

- For large z over a range - [a, +∞] the above approximation is modified to: -

-
erfc(z) = exp(-z*z) * (C + R(1 / z)) / z;
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_erf/error_inv.html b/doc/html/math_toolkit/special/sf_erf/error_inv.html deleted file mode 100644 index 0d98d9c5a..000000000 --- a/doc/html/math_toolkit/special/sf_erf/error_inv.html +++ /dev/null @@ -1,197 +0,0 @@ - - - -Error Function Inverses - - - - - - - - -
-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);
-
-

- -

-
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);
-
-

- -

-
- - 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma.html b/doc/html/math_toolkit/special/sf_gamma.html deleted file mode 100644 index 6e3bb328f..000000000 --- a/doc/html/math_toolkit/special/sf_gamma.html +++ /dev/null @@ -1,49 +0,0 @@ - - - -Gamma Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Gamma
-
Log Gamma
-
Digamma
-
Ratios - of Gamma Functions
-
Incomplete Gamma - Functions
-
Incomplete - Gamma Function Inverses
-
Derivative - of the Incomplete Gamma Function
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/digamma.html b/doc/html/math_toolkit/special/sf_gamma/digamma.html deleted file mode 100644 index aa4f41392..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/digamma.html +++ /dev/null @@ -1,380 +0,0 @@ - - - -Digamma - - - - - - - - -
-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: -

-

- -

-

- -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/gamma_derivatives.html b/doc/html/math_toolkit/special/sf_gamma/gamma_derivatives.html deleted file mode 100644 index 5bd638a2a..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/gamma_derivatives.html +++ /dev/null @@ -1,100 +0,0 @@ - - - -Derivative of the Incomplete Gamma Function - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/gamma_ratios.html b/doc/html/math_toolkit/special/sf_gamma/gamma_ratios.html deleted file mode 100644 index b2e9e6655..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/gamma_ratios.html +++ /dev/null @@ -1,349 +0,0 @@ - - - -Ratios of Gamma Functions - - - - - - - - -
-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 19. 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 20. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/igamma.html b/doc/html/math_toolkit/special/sf_gamma/igamma.html deleted file mode 100644 index 240e1bea3..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/igamma.html +++ /dev/null @@ -1,1041 +0,0 @@ - - - -Incomplete Gamma Functions - - - - - - - - -
-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: -

-

- -

-
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: -

-

- -

-
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 21. 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 22. 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 23. 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 24. 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 - - (1(3x)) > 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 - (1(3x)) < 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 - * 0.75 < 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 0.75. 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 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) -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/igamma_inv.html b/doc/html/math_toolkit/special/sf_gamma/igamma_inv.html deleted file mode 100644 index 34868d5a9..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/igamma_inv.html +++ /dev/null @@ -1,251 +0,0 @@ - - - -Incomplete Gamma Function Inverses - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/lgamma.html b/doc/html/math_toolkit/special/sf_gamma/lgamma.html deleted file mode 100644 index 994c8aec1..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/lgamma.html +++ /dev/null @@ -1,473 +0,0 @@ - - - -Log Gamma - - - - - - - - -
-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. -

-

- -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_gamma/tgamma.html b/doc/html/math_toolkit/special/sf_gamma/tgamma.html deleted file mode 100644 index ab6e53382..000000000 --- a/doc/html/math_toolkit/special/sf_gamma/tgamma.html +++ /dev/null @@ -1,460 +0,0 @@ - - - -Gamma - - - - - - - - -
-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: -

-

- -

-

- -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_poly.html b/doc/html/math_toolkit/special/sf_poly.html deleted file mode 100644 index 18b879b20..000000000 --- a/doc/html/math_toolkit/special/sf_poly.html +++ /dev/null @@ -1,44 +0,0 @@ - - - -Polynomials - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Legendre (and - Associated) Polynomials
-
Laguerre (and - Associated) Polynomials
-
Hermite Polynomials
-
Spherical Harmonics
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_poly/hermite.html b/doc/html/math_toolkit/special/sf_poly/hermite.html deleted file mode 100644 index 64e2a7137..000000000 --- a/doc/html/math_toolkit/special/sf_poly/hermite.html +++ /dev/null @@ -1,275 +0,0 @@ - - - -Hermite Polynomials - - - - - - - - -
-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: -

-

- -

-
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 37. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_poly/laguerre.html b/doc/html/math_toolkit/special/sf_poly/laguerre.html deleted file mode 100644 index 82282e07c..000000000 --- a/doc/html/math_toolkit/special/sf_poly/laguerre.html +++ /dev/null @@ -1,449 +0,0 @@ - - - -Laguerre (and Associated) Polynomials - - - - - - - - -
-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: -

-

- -

-
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 35. 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 36. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_poly/legendre.html b/doc/html/math_toolkit/special/sf_poly/legendre.html deleted file mode 100644 index edbfc62b6..000000000 --- a/doc/html/math_toolkit/special/sf_poly/legendre.html +++ /dev/null @@ -1,703 +0,0 @@ - - - -Legendre (and Associated) Polynomials - - - - - - - - -
-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: -

-

- -

-
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: -

-

- -

-
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));
-v.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]));
-// Double check values:
-for(unsigned l = 1; l < 10; ++l)
-   assert(v[l] == legendre_p(l, x));
-
-

- 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));
-v.push_back(legendre_p(1 + m, m, x));
-for(unsigned l = 1; l < 10; ++l)
-   v.push_back(legendre_next(l + 10, m, x, v[l], v[l-1]));
-// Double check values:
-for(unsigned l = 1; l < 10; ++l)
-   assert(v[l] == legendre_p(10 + l, m, x));
-
-

- 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 32. 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 33. 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 34. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sf_poly/sph_harm.html b/doc/html/math_toolkit/special/sf_poly/sph_harm.html deleted file mode 100644 index 549059a7c..000000000 --- a/doc/html/math_toolkit/special/sf_poly/sph_harm.html +++ /dev/null @@ -1,302 +0,0 @@ - - - -Spherical Harmonics - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/spherical_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 38. 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sinc.html b/doc/html/math_toolkit/special/sinc.html deleted file mode 100644 index be987c8d3..000000000 --- a/doc/html/math_toolkit/special/sinc.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Sinus Cardinal and Hyperbolic Sinus Cardinal Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Sinus Cardinal - and Hyperbolic Sinus Cardinal Functions Overview
-
sinc_pi
-
sinhc_pi
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sinc/sinc_overview.html b/doc/html/math_toolkit/special/sinc/sinc_overview.html deleted file mode 100644 index c8e0d27e6..000000000 --- a/doc/html/math_toolkit/special/sinc/sinc_overview.html +++ /dev/null @@ -1,73 +0,0 @@ - - - -Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sinc/sinc_pi.html b/doc/html/math_toolkit/special/sinc/sinc_pi.html deleted file mode 100644 index ef0784a8e..000000000 --- a/doc/html/math_toolkit/special/sinc/sinc_pi.html +++ /dev/null @@ -1,72 +0,0 @@ - - - -sinc_pi - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/sinc/sinhc_pi.html b/doc/html/math_toolkit/special/sinc/sinhc_pi.html deleted file mode 100644 index ff565799b..000000000 --- a/doc/html/math_toolkit/special/sinc/sinhc_pi.html +++ /dev/null @@ -1,76 +0,0 @@ - - - -sinhc_pi - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/zetas.html b/doc/html/math_toolkit/special/zetas.html deleted file mode 100644 index 9fce1e5c5..000000000 --- a/doc/html/math_toolkit/special/zetas.html +++ /dev/null @@ -1,37 +0,0 @@ - - - -Zeta Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
Riemann Zeta Function
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/special/zetas/zeta.html b/doc/html/math_toolkit/special/zetas/zeta.html deleted file mode 100644 index 2285994fe..000000000 --- a/doc/html/math_toolkit/special/zetas/zeta.html +++ /dev/null @@ -1,305 +0,0 @@ - - - -Riemann Zeta Function - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/zeta.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class T>
-calculated-result-type zeta(T z);
-
-template <class T, class Policy>
-calculated-result-type zeta(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 zeta(T z);
-
-template <class T, class Policy>
-calculated-result-type zeta(T z, const Policy&);
-
-

- Returns the zeta - function of z: -

-

- -

-

- -

-

- -

-
- - 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 and - Cephes libraries. Unless - otherwise specified any floating point type that is narrower than the one - shown will have effectively zero error. -

-
-

Table 48. Errors In the Function zeta(z)

-
------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Significand Size -

- 		-

- Platform and Compiler -

- 		-

- z > 0 -

- 		-

- z < 0 -

-
-

- 53 -

- 		-

- Win32, Visual C++ 8 -

- 		-

- Peak=0.99 Mean=0.1 -

-

- GSL Peak=8.7 Mean=1.0 -

-

- Cephes Peak=2.1 - Mean=1.1 -

- 		-

- Peak=7.1 Mean=3.0 -

-

- GSL Peak=137 Mean=14 -

-

- Cephes Peak=5084 - Mean=470 -

-
-

- 64 -

- 		-

- RedHat Linux IA_EM64, gcc-4.1 -

- 		-

- Peak=0.99 Mean=0.5 -

- 		-

- Peak=570 Mean=60 -

-
-

- 64 -

- 		-

- Redhat Linux IA64, gcc-4.1 -

- 		-

- Peak=0.99 Mean=0.5 -

- 		-

- Peak=559 Mean=56 -

-
-

- 113 -

- 		-

- HPUX IA64, aCC A.06.06 -

- 		-

- Peak=1.0 Mean=0.4 -

- 		-

- Peak=1018 Mean=79 -

-
-
-
- - 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: -

-

- -

-

- The generic versions of these functions are implemented using the series: -

-

- -

-

- 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 0 < z < 1 the approximating form is: -

-

- -

-

- For a rational approximation R(1-z) and a constant C. -

-

- For 1 < z < 4 the approximating form is: -

-

- -

-

- For a rational approximation R(n-z) and a constant C and integer n. -

-

- For z > 4 the approximating form is: -

-

- ζ(z) = 1 + eR(z - n) -

-

- For a rational approximation R(z-n) and integer n, note that the accuracy - required for R(z-n) is not full machine precision, but an absolute error - of: ε/R(0). This saves us quite a few digits when dealing with large z, - especially when ε is small. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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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History and What's New
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Credits and Acknowledgements
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- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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 -
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-
-

- 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. -

-

- Bruno Lalande submitted the "compile time power of a runtime base" - code. -

-

- Johan Råde wrote the optimised floating-point classification and manipulation - code, and nonfinite facets to permit C99 output of infinities and NaNs. (nonfinite - facets were not added until Boost 1.47 but had been in use with Boost.Spirit). - This library was based on a suggestion from Robert Ramey, author of Boost.Serialization. - Paul A. Bristow expressed the need for better handling of Input - & Output of NaN and infinity for the C++ Standard Library and - suggested following the C99 format. -

-

- Antony Polukhin improved lexical cast avoiding stringstream so that it was - no longer necessary to use a globale C99 facet to handle nonfinites. -

-

- Håkan Ardö, Boris Gubenko, John Maddock, Markus Schöpflin and Olivier Verdier - tested the floating-point library and Martin Bonner, Peter Dimov and John - Maddock provided valuable advice. -

-

- Gautam Sewani coded the logistic distribution as part of a Google Summer - of Code project 2008. -

-

- M. A. (Thijs) van den Berg coded the Laplace distribution. (Thijs has also - threatened to implement some multivariate distributions). -

-

- Thomas Mang requested the inverse gamma in chi squared distributions for - Bayesian applications and helped in their implementation, and provided a - nice example of their use. -

-

- Professor Nico Temme for advice on the inverse incomplete beta function. -

-

- Victor Shoup for NTL, without which - it would have much more 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. -

-

- Plots of the functions and distributions were prepared in W3C - standard Scalable Vector Graphic (SVG) - format using a program created by Jacob Voytko during a Google - Summer of Code (2007). From 2012, the latest versions of all Internet - Browsers have support for rendering SVG (with varying quality). Older versions, - especially (Microsoft Internet Explorer (before IE 9) lack native SVG support - but can be made to work with Adobe's - free SVG viewer plugin). The SVG files can be converted to JPEG or - PNG using Inkscape. -

-

- 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. -

-

- Thanks to Mark Coleman and Georgi Boshnakov for spot test values from Wolfram Mathematica, - and of course, to Eric Weisstein for nurturing Wolfram - MathWorld, an invaluable resource. -

-

- The Skew-normal distribution and Owen's t function were written by Benjamin - Sobotta. -

-

- We thank Thomas Mang for persuading us to allow t distributions to have infinite - degrees of freedom and contributing to some long discussions about how to - improve accuracy for large non-centrality and/or large degrees of freedom. -

-

- Christopher Kormanyos wrote the e_float multiprecision library TOMS - Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function - Calculations which formed the basis for the Boost.Multiprecision - library which now can be used to allow most functions and distributions to - be computed up to a precision of the users' choice, no longer restricted - to built-in floating-point types like double. (And thanks to Topher Cooper - for bring Christopher's e_float to our attention). -

-

- Christopher Kormanyos wrote some examples for using Boost.Multiprecision, - and added methods for finding zeros of Bessel Functions. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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 -
-
-
-

- Currently open bug reports can be viewed here. -

-

- All bug reports including closed ones can be viewed here. -

-
- - Boost-1.54 -
-
    -
  • - Added many references to Boost.Multiprecision and cpp_dec_float_50 - as an example of a User-defined Type (UDT). -
    • -
  • - Added Clang to list of supported compilers. -
    • -
  • - Fixed constants to use a thread-safe cache of computed values when used - at arbitrary precision. -
    • -
  • - Added finding zeros of Bessel functions cyl_bessel_j_zero, - cyl_neumann_zero, air_ai_zeros and air_bi_zeros(by - Christopher Kormanyos). -
    • -
  • - More accuracy improvements to the Bessel J and Y functions from Rocco - Romeo. -
    • -
  • - Fixed nasty cyclic dependency bug that caused some headers to not compile - #7999. -
    • -
  • - Fixed bug in tgamma - that caused spurious overflow for arguments between 142.5 and 143. -
    • -
  • - Fixed bug in raise_rounding_error that caused it to return an incorrect - result when throwing an exception is turned off #7905. -
    • -
  • - Added minimal __float128 support. -
    • -
  • - Fixed bug in edge-cases of poisson quantile #8308. -
    • -
  • - Adjusted heuristics used in Halley iteration to cope with inverting the - incomplete beta in tricky regions where the derivative is flatlining. - Example is computing the quantile of the Fisher F distribution for probabilities - smaller than machine epsilon. See ticket #8314. -
    • -
-
- - Boost-1.53 -
-
    -
  • - Fixed issues #7325, - #7415 - and #7416, - #7183, - #7649, - #7694, - #4445, - #7492, - #7891, - #7429. -
    • -
  • - Fixed mistake in calculating pooled standard deviation in two-sample - students t example #7402. -
    • -
  • - Improve complex acos/asin/atan, see #7290, - #7291. -
    • -
  • - Improve accuracy in some corner cases of cyl_bessel_j - and gamma_p/gamma_q thanks - to suggestions from Rocco Romeo. -
    • -
  • - Improve accuracy of Bessel J and Y for integer orders thanks to suggestions - from Rocco Romeo. -
    • -
-
- - Boost-1.52 -
-
    -
  • - Corrected moments for small degrees of freedom #7177 - (reported by Thomas Mang). -
    • -
  • - Added Airy functions - and Jacobi Elliptic functions. -
    • -
  • - Corrected failure to detect bad parameters in many distributions #6934 (reported - by Florian Schoppmann) by adding a function check_out_of_range to test - many possible bad parameters. This test revealed several distributions - where the checks for bad parameters were ineffective, and these have - been rectified. -
    • -
  • - Fixed issue in Hankel functions that causes incorrect values to be returned - for x < 0 and ν odd, see #7135. -
    • -
  • - Fixed issues #6517, - #6362, - #7053, - #2693, - #6937, - #7099. -
    • -
  • - Permitted infinite degrees of freedom #7259 - implemented using the normal distribution (requested by Thomas Mang). -
    • -
  • - Much enhanced accuracy for large degrees of freedom ν and/or large non-centrality - δ -by switching to use the Students t distribution (or Normal distribution - for infinite degrees of freedom) centered at delta, when δ / (4 * ν) < - epsilon for the floating-point type in use. #7259. - It was found that the incomplete beta was suffering from serious cancellation - errors when degrees of freedom was very large. (That has now been fixed - in our code, but any code based on Didonato and Morris's original papers - (probably every implementation out there actually) will have the same - issue). -
    • -
-
- - Boost-1.51 -
-

- See Boost-1.52 - some items were added but not listed in time for the release. -

-
- - Boost-1.50 -
-
    -
  • - Promoted math constants to be 1st class citizens, including convenient - access to the most widely used built-in float, double, long double via - three namespaces. -
    • -
  • - Added the Owen's T function and Skew Normal distribution written by Benjamin - Sobotta: see Owens T - and skew_normal_distrib. -
    • -
  • - Added Hankel functions cyl_hankel_1, - cyl_hankel_2, - sph_hankel_1 - and sph_hankel_2. -
    • -
  • - Corrected issue #6627 - nonfinite_num_put formatting of 0.0 is incorrect based on a patch - submitted by K R Walker. -
    • -
  • - Changed constant initialization mechanism so that it is thread safe even - for user-defined types, also so that user defined types get the full - precision of the constant, even when long - double does not. So for example - 128-bit rational approximations will work with UDT's and do the right - thing, even though long double may be only 64 or 80 bits. -
    • -
  • - Fixed issue in bessel_jy - which causes Y8.5(4π) to yield a NaN. -
    • -
-
- - Boost-1.49 -
-
    -
  • - Deprecated wrongly named twothirds - math constant in favour of two_thirds - (with underscore separator). (issue #6199). -
    • -
  • - Refactored test data and some special function code to improve support - for arbitary precision and/or expression-template-enabled types. -
    • -
  • - Added new faster zeta function evaluation method. -
    • -
-

- Fixed issues: -

-
    -
  • - Corrected CDF complement for Laplace distribution (issue #6151). -
    • -
  • - Corrected branch cuts on the complex inverse trig functions, to handle - signed zeros (issue #6171). -
    • -
  • - Fixed bug in bessel_yn - which caused incorrect overflow errors to be raised for negative n - (issue #6367). -
    • -
  • - Also fixed minor/cosmetic/configuration issues #6120, - #6191, - #5982, - #6130, - #6234, - #6307, - #6192. -
    • -
-
- - Boost-1.48 -
-
    -
  • - Added new series evaluation methods to the cyclic Bessel I, J, K and - Y functions. Also taken great care to avoid spurious over and underflow - of these functions. Fixes issue #5560 -
    • -
  • - Added an example of using Inverse Chi-Squared distribution for Bayesian - statistics, provided by Thomas Mang. -
    • -
  • - Added tests to use improved version of lexical_cast which handles C99 - nonfinites without using globale facets. -
    • -
  • - Corrected wrong out-of-bound uniform distribution CDF complement values - #5733. -
    • -
  • - Enabled long double support on OpenBSD (issue #6014). -
    • -
  • - Changed nextafter and related functions to behave in the same way as - other implementations - so that nextafter(+INF, 0) is a finite value - (issue #5832). -
    • -
  • - Changed tuple include configuration to fix issue when using in conjunction - with Boost.Tr1 (issue #5934). -
    • -
  • - Changed class eps_tolerance to behave correctly when both ends of the - range are zero (issue #6001). -
    • -
  • - Fixed missing include guards on prime.hpp (issue #5927). -
    • -
  • - Removed unused/undocumented constants from constants.hpp (issue #5982). -
    • -
  • - Fixed missing std:: prefix in nonfinite_num_facets.hpp (issue #5914). -
    • -
  • - Minor patches for Cray compiler compatibility. -
    • -
-
- - Boost-1.47 -
-
    -
  • - Added changesign function to sign.hpp to facilitate addition of nonfinite - facets. -
    • -
  • - Addition of nonfinite facets from Johan Rade, with tests, examples of - use for C99 format infinity and NaN, and documentation. -
    • -
  • - Added tests and documentation of changesign from Johan Rade. -
    • -
-
- - Boost-1.46.1 -
-
-
- - Boost-1.46.0 -
-
    -
  • - Added Wald, Inverse Gaussian and geometric distributions. -
    • -
  • - Added information about configuration macros. -
    • -
  • - Added support for mpreal as a real-numbered type. -
    • -
-
- - Boost-1.45.0 -
-
    -
  • - Added warnings about potential ambiguity with std random library in distribution - and function names. -
    • -
  • - Added inverse gamma distribution and inverse chi_square and scaled inverse - chi_square. -
    • -
  • - Editorial revision of documentation, and added FAQ. -
    • -
-
- - Boost-1.44.0 -
-
    -
  • - Fixed incorrect range and support for Rayleigh distribution. -
    • -
  • - Fixed numerical error in the quantile of the Student's T distribution: - the function was returning garbage values for non-integer degrees of - freedom between 2 and 3. -
    • -
-
- - Boost-1.41.0 -
-
  • - Significantly improved performance for the incomplete gamma function - and its inverse. -
-
- - Boost-1.40.0 -
-
    -
  • - Added support for MPFR as a bignum type. -
    • -
  • - Added some full specializations of the policy classes to reduce compile - times. -
    • -
  • - Added logistic and hypergeometric distributions, from Gautam Sewani's - Google Summer of Code project. -
    • -
  • - Added Laplace distribution submitted by Thijs van den Berg. -
    • -
  • - Updated performance test code to include new distributions, and improved - the performance of the non-central distributions. -
    • -
  • - Added SSE2 optimised Lanczos - approximation code, from Gautam Sewani's Google Summer of Code - project. -
    • -
  • - Fixed bug in cyl_bessel_i that used an incorrect approximation for ν = - 0.5, also effects the non-central Chi Square Distribution when ν = 3, see - bug report #2877. -
    • -
  • - Fixed minor bugs #2873. -
    • -
-
- - Boost-1.38.0 -
-
    -
  • - Added Johan Råde's optimised floating point classification routines. -
    • -
  • - Fixed code so that it compiles in GCC's -pedantic mode (bug report #1451). -
    • -
-
- - Boost-1.37.0 -
-
  • - Improved accuracy and testing of the inverse hypergeometric functions. -
-
- - Boost-1.36.0 -
-
    -
  • - Added Noncentral Chi Squared Distribution. -
    • -
  • - Added Noncentral Beta Distribution. -
    • -
  • - Added Noncentral F Distribution. -
    • -
  • - Added Noncentral T Distribution. -
    • -
  • - Added Exponential Integral Functions. -
    • -
  • - Added Zeta Function. -
    • -
  • - Added Rounding and Truncation functions. -
    • -
  • - Added Compile time powers of runtime bases. -
    • -
  • - Added SSE2 optimizations for Lanczos evaluation. -
    • -
-
- - Boost-1.35.0: - 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: . -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/status/issues.html b/doc/html/math_toolkit/status/issues.html deleted file mode 100644 index a11df7f45..000000000 --- a/doc/html/math_toolkit/status/issues.html +++ /dev/null @@ -1,1251 +0,0 @@ - - - -Known Issues, and TODO List - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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. -

-
- - Derivatives - of Bessel functions (and their zeros) -
-

- Potentially, there could be native support for cyl_bessel_j_prime() and cyl_neumann_prime(). One could also imagine supporting the - zeros thereof, but they might be slower to calculate since root bracketing - might be needed instead of Newton iteration (for the lack of 2nd derivatives). -

-

- Since Boost.Math's Bessel functions are so excellent, the quick way to cyl_bessel_j_prime() - and cyl_neumann_prime() - would be via relationship with cyl_bessel_j() and cyl_neumann(). -

-
- - 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). -
    • -
-
- - Owen's - T Function -
-

- There is a problem area at arbitrary precision when a - is very close to 1. However, note that the value for T(h, 1) - is well known and easy to compute, and if we replaced the ak - terms in series T1, T2 or T4 by (ak - 1) then we would - have the difference between T(h, a) and T(h, - 1). Unfortunately this doesn't improve the convergence of those - series in that area. It certainly looks as though a new series in terms of - (1-a)k is both possible and desirable in this area, - but it remains elusive at present. -

-
- - Jocobi elliptic - functions -
-

- These are useful in engineering applications - we have had a request to add - these. -

-
- - Statistical - distributions -
-
  • - Student's t Perhaps switch to normal distribution as a better approximation - for very large degrees of freedom? -
-
- - Feature - Requests -
-

- The following table lists distributions that are found in other packages - but which are not yet present here, the more frequently the distribution - is found, the higher the priority for implementing it: -

-
-------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
-

- Distribution -

- 		-

- R -

- 		-

- Mathematica 6 -

- 		-

- NIST -

- 		-

- Regress+ -

- 		-

- Matlab -

-
-

- Geometric -

- 		-

- X -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- X -

-
-

- Multinomial -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

-
-

- Tukey Lambda -

- 		-

- X -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

-
-

- Half Normal / Folded Normal -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Chi -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Gumbel -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Discrete Uniform -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- X -

-
-

- Log Series -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Nakagami (generalised Chi) -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- X -

-
-

- Log Logistic -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

-
-

- Tukey (Studentized range) -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Wilcoxon rank sum -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Wincoxon signed rank -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Non-central Beta -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Maxwell -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Beta-Binomial -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Beta-negative Binomial -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Zipf -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Birnbaum-Saunders / Fatigue Life -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

-
-

- Double Exponential -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

-
-

- Power Normal -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

-
-

- Power Lognormal -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

- 		-

- - -

-
-

- Cosine -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Double Gamma -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Double Weibul -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Hyperbolic Secant -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Semicircular -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Bradford -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Birr / Fisk -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Reciprocal -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- X -

- 		-

- - -

-
-

- Kolmogorov Distribution -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

- 		-

- - -

-
-

- Also asked for more than once: -

-
    -
  • - Add support for interpolated distributions, possibly combine with numeric - integration and differentiation. -
    • -
  • - Add support for bivariate and multivariate distributions: most especially - the normal. -
    • -
  • - Add support for the log of the cdf and pdf: this is mainly a performance - optimisation since we can avoid some special function calls for some - distributions by returning the log of the result. -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit.html b/doc/html/math_toolkit/toolkit.html deleted file mode 100644 index 21a5dbe6a..000000000 --- a/doc/html/math_toolkit/toolkit.html +++ /dev/null @@ -1,41 +0,0 @@ - - - -Internals and Internal Details - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Overview
-
Utilities & internals
-
Testing and Development
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1.html b/doc/html/math_toolkit/toolkit/internals1.html deleted file mode 100644 index 74d7da8f4..000000000 --- a/doc/html/math_toolkit/toolkit/internals1.html +++ /dev/null @@ -1,51 +0,0 @@ - - - -Utilities & internals - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Series - Evaluation
-
Continued Fraction - Evaluation
-
Polynomial - and Rational Function Evaluation
-
Root Finding - With Derivatives: Newton-Raphson, Halley & Schroeder
-
Root Finding - Without Derivatives: Bisection, Bracket and TOMS748
-
Locating Function - Minima: Brent's algorithm
-
Tuples
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/cf.html b/doc/html/math_toolkit/toolkit/internals1/cf.html deleted file mode 100644 index a229ecf5c..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/cf.html +++ /dev/null @@ -1,277 +0,0 @@ - - - -Continued Fraction Evaluation - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/tools/fraction.hpp>
-
-

-

-
namespace boost{ namespace math{ namespace tools{
-
-template <class Gen, class U>
-typename detail::fraction_traits<Gen>::result_type
-   continued_fraction_b(Gen& g, const U& tolerance, boost::uintmax_t& max_terms)
-
-template <class Gen, class U>
-typename detail::fraction_traits<Gen>::result_type
-   continued_fraction_b(Gen& g, const U& tolerance)
-
-template <class Gen, class U>
-typename detail::fraction_traits<Gen>::result_type
-   continued_fraction_a(Gen& g, const U& tolerance, boost::uintmax_t& max_terms)
-
-template <class Gen, class U>
-typename detail::fraction_traits<Gen>::result_type
-   continued_fraction_a(Gen& g, const U& tolerance)
-
-//
-// These interfaces are present for legacy reasons, and are now deprecated:
-//
-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 tolerance - parameter is the precision desired in the result, evaluation of the fraction - will continue until the last term evaluated leaves the relative error in - the result less than tolerance. The deprecated interfaces - take a number of digits precision here, internally they just convert this - to a tolerance and forward call. -

-

- 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>::epsilon());
-
-

- 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 its 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>::epsilon());
-}
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/minima.html b/doc/html/math_toolkit/toolkit/internals1/minima.html deleted file mode 100644 index 031494fad..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/minima.html +++ /dev/null @@ -1,119 +0,0 @@ - - - -Locating Function Minima: Brent's algorithm - - - - - - - - -
-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 (for 64-bit double, - sqrt(1e-16)≅1e-8), 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/rational.html b/doc/html/math_toolkit/toolkit/internals1/rational.html deleted file mode 100644 index 315eaf758..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/rational.html +++ /dev/null @@ -1,228 +0,0 @@ - - - -Polynomial and Rational Function Evaluation - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/roots.html b/doc/html/math_toolkit/toolkit/internals1/roots.html deleted file mode 100644 index ba092b865..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/roots.html +++ /dev/null @@ -1,417 +0,0 @@ - - - -Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder - - - - - - - - -
-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 using derivatives: -

-
-

- 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 boost::math::tuple: -

-

- For the second order iterative methods (Newton - Raphson) the boost::math::tuple - should have two elements containing - the evaluation of the function and its first derivative. -

-

- For the third order methods (Halley - and Schroeder) the boost::math::tuple - should have three elements containing - the evaluation of the function and its first and second derivatives. -

-
-
T guess
-

- The initial starting value. A good guess is crucial to quick convergence! -

-
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: -

-
    -
  • - Default max_iter = (std::numeric_limits<boost::uintmax_t>::max)() is effectively 'iterate for ever'!. -
    • -
  • - 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 functor 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. -
    • -
  • - If the function is 'Really Well Behaved' (monotonic and has only one - root) the bracket bounds min and max may as well be set to the widest - limits like zero and numeric_limits<T>::max(). -
    • -
  • - But if the function more complex and may have more than one root or - a pole, the choice of bounds is protection against jumping out to seek - the 'wrong' root. -
    • -
  • - 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. -
    • -
  • - To get the binary digits of accuracy, use policies::get_max_root_iterations<Policy>()). -
    • -
  • - If you need some diagnostic output to see what is going on, you can - #define BOOST_MATH_INSTRUMENT - before the #include <boost/math/tools/roots.hpp>, and also ensure that display of - all the possibly significant digits with cout.precision(std::numeric_limits<double>::max_digits10): but be warned, this may produce copious - output! -
    • -
  • - 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 -
-

- Let's 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 by defining a function object (functor) that returns the evaluation - of the function to solve, along with its first derivative f'(x): -

-
template <class T>
-struct cbrt_functor
-{
-   cbrt_functor(T const& target) : a(target)
-   { // Constructor stores value to be 'cube-rooted'.
-   }
-   boost::math::tuple<T, T> operator()(T const& z)
-   { // z is estimate so far.
-      return boost::math::make_tuple(
-      z*z*z - a, // return both f(x)
-      3 * z*z);  // and f'(x)
-   }
-private:
-   T a; // to be 'cube-rooted'.
-};
-
-

- 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; // for frexp, ldexp, numeric_limits.
-   using namespace boost::math::tools;
-
-   int exp;
-   frexp(z, &exp); // Get exponent of z (ignore mantissa).
-   T min = ldexp(0.5, exp/3);
-   T max = ldexp(2.0, exp/3);
-   T guess = ldexp(1.0, exp/3); // Rough guess is to divide the exponent by three.
-   int digits = std::numeric_limits<T>::digits; // Maximum possible binary digits accuracy for type T.
-   return 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 it is 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 a probably optimization by computing - z², and reusing it, omits error handling, and does not handle negative values - of z correctly. (These are left as an exercise for the reader!) -

-

- The boost::math::cbrt function also includes these and - other improvements. -

-

- Now let's adapt the functor slightly to return the second derivative as - well: -

-
template <class T>
-struct cbrt_functor
-{
-   cbrt_functor(T const& target) : a(target){}
-   boost::math::tuple<T, T, T> operator()(T const& z)
-   {
-      return boost::math::make_tuple(
-      z*z*z - a,
-      3 * z*z,
-      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;
-   using namespace boost::math::tools;
-
-   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 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/roots2.html b/doc/html/math_toolkit/toolkit/internals1/roots2.html deleted file mode 100644 index 0a80f9caf..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/roots2.html +++ /dev/null @@ -1,580 +0,0 @@ - - - -Root Finding Without Derivatives: Bisection, Bracket and TOMS748 - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/tools/roots.hpp>
-
-

-

-
  namespace boost{ namespace math{
-  namespace tools{
-  // Bisection
-  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&);
-
-	// Bracket and Solve Root
-  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&);
-
-// TOMS 748 algorithm
-  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 any derivatives of f(x). -

-

- The bracket_and_solve_root - functions use TOMS Algorithm 748 that is asymptotically the most efficient - known, and have been shown to be optimal for a certain classes of smooth - functions. Variants with and without __policies are provided. -

-

- 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. -

-
- - Bisection -
-
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 updated value of max_iter - when the function returns), rather than because the termination condition - tol was satisfied. -

-
- - Bracket - and solve -
-
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. -

-
- - Algorithm - TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions -
-
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 toms748_solve 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. -

-

- toms748_solve 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 updated value of max_iter - against its previous value passed as parameter), 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;
-};
-
-

- eps_tolerance 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: enclosing zeros of continuous functions 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/series_evaluation.html b/doc/html/math_toolkit/toolkit/internals1/series_evaluation.html deleted file mode 100644 index c2b9335df..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/series_evaluation.html +++ /dev/null @@ -1,192 +0,0 @@ - - - -Series Evaluation - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/tools/series.hpp>
-
-

-

-
namespace boost{ namespace math{ namespace tools{
-
-template <class Functor, class U, class V>
-inline typename Functor::result_type sum_series(Functor& func, const U& tolerance, boost::uintmax_t& max_terms, const V& init_value);
-
-template <class Functor, class U, class V>
-inline typename Functor::result_type sum_series(Functor& func, const U& tolerance, boost::uintmax_t& max_terms);
-
-//
-// The following interfaces are now deprecated:
-//   
-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 precision required, summation will stop when - the next term is less than tolerance times the result. - The deprecated versions of sum_series take an integer number of bits here - - internally they just convert this to a tolerance and forward the call. -

-

- The 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);
-   // Set a limit on how many iterations we permit:
-   boost::uintmax_t max_iter = 1000;
-   // Add it up, with enough precision for full machine precision:
-   return tools::sum_series(s, std::numeric_limits<T>::epsilon(), max_iter);
-}
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals1/tuples.html b/doc/html/math_toolkit/toolkit/internals1/tuples.html deleted file mode 100644 index 070dbb572..000000000 --- a/doc/html/math_toolkit/toolkit/internals1/tuples.html +++ /dev/null @@ -1,78 +0,0 @@ - - - -Tuples - - - - - - - - -
-PrevUpHomeNext -
-
-

-Tuples -

-
- - Synopsis -
-

-

-
#include <boost/math/tools/tuple.hpp>
-
-

-

-
- - Description -
-

- This header defines the type boost::math::tuple, - the associated free functions ignore, - tie, make_tuple, - get, and associated types - tuple_size and tuple_element. -

-

- These types and functions are aliases for: -

-
    -
  • - std::tuple etc when available, otherwise: -
    • -
  • - std::tr1::tuple etc when available, otherwise: -
    • -
  • - boost::fusion::tuple etc if the compiler supports it, otherwise: -
    • -
  • - boost::tuple. -
    • -
-

- So this boost::math::tuple is strongly recommended for maximum - portability. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals2.html b/doc/html/math_toolkit/toolkit/internals2.html deleted file mode 100644 index cbd4b0754..000000000 --- a/doc/html/math_toolkit/toolkit/internals2.html +++ /dev/null @@ -1,45 +0,0 @@ - - - -Testing and Development - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Polynomials
-
Minimax Approximations - and the Remez Algorithm
-
Relative - Error and Testing
-
Graphing, - Profiling, and Generating Test Data for Special Functions
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals2/error_test.html b/doc/html/math_toolkit/toolkit/internals2/error_test.html deleted file mode 100644 index d8c8b380f..000000000 --- a/doc/html/math_toolkit/toolkit/internals2/error_test.html +++ /dev/null @@ -1,218 +0,0 @@ - - - -Relative Error and Testing - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals2/minimax.html b/doc/html/math_toolkit/toolkit/internals2/minimax.html deleted file mode 100644 index 071b935e4..000000000 --- a/doc/html/math_toolkit/toolkit/internals2/minimax.html +++ /dev/null @@ -1,278 +0,0 @@ - - - -Minimax Approximations and the Remez Algorithm - - - - - - - - -
-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 principles 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(S * (x + X)) + Y - where X is the x-offset, S - is the x-scale 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. -

-
x-scale val
-

- Sets the x-scale to val: the approximation will - be generated for f(S * (x + X)) + Y - where S is the x-scale, X - is the x-offset and Y is the y-offset. Defaults - to one. 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(S * (x + X)) + Y - where X is the x-offset, S - is the x-scale 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals2/polynomials.html b/doc/html/math_toolkit/toolkit/internals2/polynomials.html deleted file mode 100644 index aaa41411a..000000000 --- a/doc/html/math_toolkit/toolkit/internals2/polynomials.html +++ /dev/null @@ -1,128 +0,0 @@ - - - -Polynomials - - - - - - - - -
-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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals2/test_data.html b/doc/html/math_toolkit/toolkit/internals2/test_data.html deleted file mode 100644 index 4d214c816..000000000 --- a/doc/html/math_toolkit/toolkit/internals2/test_data.html +++ /dev/null @@ -1,541 +0,0 @@ - - - -Graphing, Profiling, and Generating Test Data for Special Functions - - - - - - - - -
-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: -

-

- -

-
- - 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 boost::math::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 boost::math::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/math/tools/tuple.hpp>
-
-template <class T>
-struct profile_gamma_fraction
-{
-   typedef boost::math::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 boost::math::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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/toolkit/internals_overview.html b/doc/html/math_toolkit/toolkit/internals_overview.html deleted file mode 100644 index 5c0d80c1c..000000000 --- a/doc/html/math_toolkit/toolkit/internals_overview.html +++ /dev/null @@ -1,50 +0,0 @@ - - - -Overview - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt.html b/doc/html/math_toolkit/using_udt.html deleted file mode 100644 index 053a71060..000000000 --- a/doc/html/math_toolkit/using_udt.html +++ /dev/null @@ -1,47 +0,0 @@ - - - -Use with User-Defined Floating-Point Types - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Using Boost.Math - with High-Precision Floating-Point Libraries
-
Conceptual Requirements - for Real Number Types
-
Conceptual Requirements - for Distribution Types
-
Conceptual Archetypes - for Reals and Distributions
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/archetypes.html b/doc/html/math_toolkit/using_udt/archetypes.html deleted file mode 100644 index 5df8d2679..000000000 --- a/doc/html/math_toolkit/using_udt/archetypes.html +++ /dev/null @@ -1,200 +0,0 @@ - - - -Conceptual Archetypes for Reals and Distributions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- There are a few concept archetypes available: -

-
    -
  • - Real concept for floating-point types. -
    • -
  • - distribution Concept for statistical distributions. -
    • -
-
- - Real - concept -
-

- std_real_concept is an archetype - for theReal types, including the built-in float, double, long double. -

-

-

-
#include <boost/concepts/std_real_concept.hpp>
-

-

-
namespace boost{
-namespace math{
-namespace concepts
-{
-  class std_real_concept;
-}
-}} // namespaces
-
-

- 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. -

-

- - Testing - the real concept -

-

- 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 its 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 its standard library functions in its 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 numeric_limits is not a conceptual requirement for - RealTypes. -

-

- 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 [linkmath_toolkit.using_udt.high_precision.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. -

-

- - Distribution - Concept -

-

- Distribution Concept models statistical distributions. -

-

-

-
#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. -

-

- - Testing - the distribution concept -

-

- 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/concepts.html b/doc/html/math_toolkit/using_udt/concepts.html deleted file mode 100644 index 1addbf0bc..000000000 --- a/doc/html/math_toolkit/using_udt/concepts.html +++ /dev/null @@ -1,1364 +0,0 @@ - - - -Conceptual Requirements for Real Number Types - - - - - - - - -
-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. But now that Boost.Multiprecision - library is available, this has become the reference real number type. -

-

- Submissions of binding to other extended precision types would also still - be 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/bindings/rr.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 -

-
-

- fmod(cr1) -

- 		-

- RealType -

-
-

- round(cr1) -

- 		-

- RealType -

-
-

- iround(cr1) -

- 		-

- int -

-
-

- trunc(cr1) -

- 		-

- RealType -

-
-

- itrunc(cr1) -

- 		-

- int -

-
-

- 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, log10, - lround, llround, - ltrunc, lltrunc - and modf are not currently - used, but may be if further special functions are added. -

-

- Note that the round, trunc and modf - functions are not part of the current C++ standard: they are part of the - additions added to C99 which will likely be in the next C++ standard. There - are Boost versions of these provided as a backup, and the functions are always - called unqualified so that argument-dependent-lookup can take place. -

-

- 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 boost/math/bindings/detail/big_lanczos.hpp: - in the former case 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/dist_concept.html b/doc/html/math_toolkit/using_udt/dist_concept.html deleted file mode 100644 index 197cdf1ed..000000000 --- a/doc/html/math_toolkit/using_udt/dist_concept.html +++ /dev/null @@ -1,390 +0,0 @@ - - - -Conceptual Requirements for Distribution Types - - - - - - - - -
-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 (or percentile) of the distribution. -

-
-

- quantile(complement(cd, cr)) -

- 		-

- RealType -

- 		-

- Returns the quantile (or percentile) 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 © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision.html b/doc/html/math_toolkit/using_udt/high_precision.html deleted file mode 100644 index e15b4d574..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision.html +++ /dev/null @@ -1,103 +0,0 @@ - - - -Using Boost.Math with High-Precision Floating-Point Libraries - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Why - use a high-precision library rather than built-in floating-point types?
-
Using - Boost.Multiprecision
-
Using - with GCC's __float128 datatype
-
Using - With MPFR or GMP - High-Precision Floating-Point Library
-
Using - e_float Library
-
Using - NTL Library
-
Using - without expression templates for Boost.Test and others
-
-

- The special functions, distributions, constants and tools in this library - can be used with a number of high-precision libraries, including: -

-
-

- The last four have some license restrictions; only Boost.Multiprecision - when using the cpp_float - backend can provide an unrestricted Boost - license. -

-

- At present, the price of a free license is slightly lower speed. -

-

- Of course, the main cost of higher precision is very much decreased (usually - at least hundred-fold) computation speed, and big increases in memory use. -

-

- Some libraries offer true arbitrary - precision arithmetic where the precision is limited only by avilable - memory and compute time, but most are used at some arbitrarily-fixed precision, - say 100 decimal digits. -

-

- Boost.Multiprecision - can operate in both ways, but the most popular choice is likely to be about - a hundred decimal digits, though examples of computing tens of thousands - of digits have been demonstrated. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/e_float.html b/doc/html/math_toolkit/using_udt/high_precision/e_float.html deleted file mode 100644 index 56d353ca7..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/e_float.html +++ /dev/null @@ -1,60 +0,0 @@ - - - -Using e_float Library - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Boost.Multiprecision - was a development from the e_float - (TOMS Algorithm 910) library by Christopher Kormanyos. -

-

- e_float can still be used with Boost.Math library via the header: -

-
<boost/math/bindings/e_float.hpp>
-
-

- And the type boost::math::ef::e_float: this type is a thin wrapper - class around ::e_float which provides the necessary syntactic sugar to - make everything "just work". -

-

- There is also a concept checking test program for e_float support here. -

-

- New projects are recommended to use Boost.Multiprecision - with cpp_float backend - instead. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/float128.html b/doc/html/math_toolkit/using_udt/high_precision/float128.html deleted file mode 100644 index cb1c315e4..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/float128.html +++ /dev/null @@ -1,56 +0,0 @@ - - - -Using with GCC's __float128 datatype - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- At present support for GCC's native __float128 - datatype is extremely limited: the numeric constants will all work with - that type, and that's about it. If you want to use the distributions or - special functions then you will need to provide your own wrapper header - that: -

-
    -
  • - Provides std::numeric_limits<__float128> support. -
    • -
  • - Provides overloads of the standard library math function for type __float128 - and which forward to the libquadmath equivalents. -
    • -
-

- Ultimately these facilities should be provided by GCC and libstdc++. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/use_mpfr.html b/doc/html/math_toolkit/using_udt/high_precision/use_mpfr.html deleted file mode 100644 index ce95da1ad..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/use_mpfr.html +++ /dev/null @@ -1,106 +0,0 @@ - - - -Using With MPFR or GMP - High-Precision Floating-Point Library - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- The special functions and tools in this library can be used with MPFR (an arbitrary precision number type - based on the GNU Multiple Precision Arithmetic - Library), either via the bindings in boost/math/bindings/mpfr.hpp, - or via boost/math/bindings/mpreal.hpp. -

-

- New projects are recommended to use Boost.Multiprecision - with GMP/MPFR backend instead. -

-

- In order to use these bindings you will need to have installed MPFR - plus its dependency the GMP library. - You will also need one of the two supported C++ wrappers for MPFR: gmpfrxx (or mpfr_class), - or mpfr-C++ (mpreal). -

-

- Unfortunately neither mpfr_class - nor mpreal quite satisfy - our conceptual requirements, so there is a very thin set of additional - interfaces and some helper traits defined in boost/math/bindings/mpfr.hpp - and boost/math/bindings/mpreal.hpp - that you should use in place of including 'gmpfrxx.h' or 'mpreal.h' directly. - The classes mpfr_class - or mpreal are then usable - unchanged once this header is included, so for example mpfr_class's - performance-enhancing expression templates are preserved and fully supported - by this library: -

-
#include <boost/math/bindings/mpfr.hpp>
-#include <boost/math/special_functions/gamma.hpp>
-
-int main()
-{
-   mpfr_class::set_dprec(500); // 500 bit precision
-   //
-   // Note that the argument to tgamma is
-   // an expression template - that's just fine here.
-   //
-   mpfr_class v = boost::math::tgamma(sqrt(mpfr_class(2)));
-   std::cout << std::setprecision(50) << v << std::endl;
-}
-
-

- Alternatively use with mpreal - would look like: -

-
#include <boost/math/bindings/mpreal.hpp>
-#include <boost/math/special_functions/gamma.hpp>
-
-int main()
-{
-   mpfr::mpreal::set_precision(500); // 500 bit precision
-   mpfr::mpreal v = boost::math::tgamma(sqrt(mpfr::mpreal(2)));
-   std::cout << std::setprecision(50) << v << std::endl;
-}
-
-

- For those functions that are based upon the Lanczos - approximation, the bindings defines a series of approximations with - up to 61 terms and accuracy up to approximately 3e-113. This therefore - sets the upper limit for accuracy to the majority of functions defined - this library when used with either mpfr_class - or mpreal. -

-

- There is a concept checking test program for mpfr support here - and here. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/use_multiprecision.html b/doc/html/math_toolkit/using_udt/high_precision/use_multiprecision.html deleted file mode 100644 index c1e147b15..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/use_multiprecision.html +++ /dev/null @@ -1,403 +0,0 @@ - - - -Using Boost.Multiprecision - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- All new projects are recommended to use Boost.Multiprecision. -

-
- - Using - Boost.Multiprecision cpp_float - for numerical calculations with high precision. -
-

- The Boost.Multiprecision library can be used for computations requiring - precision exceeding that of standard built-in types such as float, double - and long double. For extended-precision calculations, Boost.Multiprecision - supplies a template data type called cpp_dec_float. The number of decimal - digits of precision is fixed at compile-time via template parameter. -

-

- To use these floating-point types and constants, we need some includes: -

-

-

-
#include <boost/math/constants/constants.hpp>
-
-#include <boost/multiprecision/cpp_dec_float.hpp>
-// using boost::multiprecision::cpp_dec_float
-
-#include <iostream>
-#include <limits>
-
-

-

-

- So now we can demonstrate with some trivial calculations: -

-

-

-
int main()
-{
-
-

-

-

- Using typedef cpp_dec_float_50 - hides the complexity of multiprecision to allow us to define variables - with 50 decimal digit precision just like built-in double. -

-

-

-
using boost::multiprecision::cpp_dec_float_50;
-
-cpp_dec_float_50 seventh = cpp_dec_float_50(1) / 7;
-
-

-

-

- By default, output would only show the standard 6 decimal digits, so set - precision to show all 50 significant digits. -

-

-

-
std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
-std::cout << seventh << std::endl;
-
-

-

-

- which outputs: -

-
0.14285714285714285714285714285714285714285714285714
-
-

- We can also use constants, guaranteed to be initialized with the very last - bit of precision. -

-

-

-
cpp_dec_float_50 circumference = boost::math::constants::pi<cpp_dec_float_50>() * 2 * seventh;
-
-std::cout << circumference << std::endl;
-
-

-

-

- which outputs -

-
0.89759790102565521098932668093700082405633411410717
-
-
- - Using - Boost.Multiprecision to generate a high-precision array of sin coefficents - for use with FFT. -
-

- The Boost.Multiprecision library can be used for computations requiring - precision exceeding that of standard built-in types such as float, double - and long double. For extended-precision calculations, Boost.Multiprecision - supplies a template data type called cpp_dec_float. The number of decimal - digits of precision is fixed at compile-time via template parameter. -

-

- To use these floating-point types and constants, we need some includes: -

-

-

-
#include <boost/math/constants/constants.hpp>
-// using boost::math::constants::pi;
-
-#include <boost/multiprecision/cpp_dec_float.hpp>
-// using boost::multiprecision::cpp_dec_float
-
-#include <iostream>
-#include <limits>
-#include <vector>
-#include <algorithm>
-#include <iomanip>
-#include <iterator>
-#include <fstream>
-
-

-

-

- Define a text string which is a C++ comment with the program licence, copyright - etc. You could of course, tailor this to your needs, including copyright - claim. There are versions of array - provided by Boost/array in boost::array or the C++11 std::array, but since - not all platforms provide C++11 support, this program provides the Boost - version as fallback. -

-

-

-
static const char* prolog =
-{
-  "// Use, modification and distribution are subject to the\n"
-  "// Boost Software License, Version 1.0.\n"
-  "// (See accompanying file LICENSE_1_0.txt\n"
-  "// or copy at ""http://www.boost.org/LICENSE_1_0.txt)\n\n"
-
-  "// Copyright ???? 2013.\n\n"
-
-  "// Use boost/array if std::array (C++11 feature) is not available.\n"
-  "#ifdef  BOOST_NO_CXX11_HDR_ARRAY\n"
-  "#include <boost/array/array.hpp>\n"
-  "#else\n"
-  "#include <array>\n"
-  "#endif\n\n"
-};
-
-
-using boost::multiprecision::cpp_dec_float_50;
-using boost::math::constants::pi;
-// VS 2010 (wrongly) requires these at file scope, not local scope in main.
-// This program also requires -std=c++11 option to compile using Clang and GCC.
-
-int main()
-{
-
-

-

-

- One often needs to compute tables of numbers in mathematical software. -

-

- A fast Fourier transform (FFT), for example, may use a table of the values - of sin((π/2n) in its implementation details. In order to maximize the precision - in the FFT implementation, the precision of the tabulated trigonometric - values should exceed that of the built-in floating-point type used in the - FFT. -

-

- The sample below computes a table of the values of sin(π/2n) in the range - 1 <= n <= 31. -

-

- This program makes use of, among other program elements, the data type - boost::multiprecision::cpp_dec_float_50 for a precision of 50 - decimal digits from Boost.Multiprecision, the value of constant π retrieved - from Boost.Math, guaranteed to be initialized with the very last bit of - precision for the type, here cpp_dec_float_50, - and a C++11 lambda function combined with std::for_each(). -

-

- define the number of values in the array. -

-

-

-
std::size_t size = 32U;
-cpp_dec_float_50 p = pi<cpp_dec_float_50>();
-cpp_dec_float_50 p2 = boost::math::constants::pi<cpp_dec_float_50>();
-
-std::vector <cpp_dec_float_50> sin_values (size);
-unsigned n = 1U;
-// Generate the sine values.
-std::for_each
-(
-  sin_values.begin (),
-  sin_values.end (),
-  [&n](cpp_dec_float_50& y)
-  {
-    y = sin( pi<cpp_dec_float_50>() / pow(cpp_dec_float_50 (2), n));
-    ++n;
-  }
-);
-
-

-

-

- Define the floating-point type for the generated file, either built-in - double, - float, or long double, - or a user defined type like cpp_dec_float_50. -

-

-

-
std::string fp_type = "double";
-
-std::cout << "Generating an `std::array` or `boost::array` for floating-point type: "
-  << fp_type << ". " << std::endl;
-
-

-

-

- By default, output would only show the standard 6 decimal digits, so set - precision to show enough significant digits for the chosen floating-point - type. For cpp_dec_float_50 - is 50. (50 decimal digits should be ample for most applications). -

-

-

-
std::streamsize precision = std::numeric_limits<cpp_dec_float_50>::digits10;
-
-//  std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
-std::cout << precision << " decimal digits precision. " << std::endl;
-
-

-

-

- Of course, one could also choose less, for example, 36 would be sufficient - for the most precise current long - double implementations using 128-bit. - In general, it should be a couple of decimal digits more (guard digits) - than std::numeric_limits<RealType>::max_digits10 for the target system floating-point - type. If the implementation does not provide max_digits10, - the the Kahan formula std::numeric_limits<RealType>::digits - * 3010/10000 + - 2 can be used instead. -

-

- The compiler will read these values as decimal digits strings and use the - nearest representation for the floating-point type. -

-

- Now output all the sine table, to a file of your chosen name. -

-

-

-
  const char sines_name[] = "sines.hpp";  // In same directory as .exe
-
-  std::ofstream fout(sines_name, std::ios_base::out);  // Creates if no file exists,
-  // & uses default overwrite/ ios::replace.
-  if (fout.is_open() == false)
-  {  // failed to open OK!
-    std::cout << "Open file " << sines_name << " failed!" << std::endl;
-    return EXIT_FAILURE;
-  }
-  else
-  {
-    std::cout << "Open file " << sines_name << " for output OK." << std::endl;
-    fout << prolog << "// Table of " << sin_values.size() << " values with "
-      << precision << " decimal digits precision,\n"
-      "// generated by program fft_sines_table.cpp.\n" << std::endl;
-
-    fout <<
-"#ifdef BOOST_NO_CXX11_HDR_ARRAY""\n"
- "  static const boost::array<double, " << size << "> sines =\n"
-"#else""\n"
-"  static const std::array<double, " << size << "> sines =\n"
-"#endif""\n"
-    "{{\n"; // 2nd { needed for some GCC compiler versions.
-    fout.precision(precision);
-
-    for (unsigned int i = 0U; ;)
-    {
-      fout << "  " << sin_values[i];
-      if (i == sin_values.size()-1)
-      { // next is last value.
-        fout << "\n}};\n"; // 2nd } needed for some GCC compiler versions.
-        break;
-      }
-      else
-      {
-        fout << ",\n";
-        i++;
-      }
-    }
-
-    fout.close();
-    std::cout << "Close file " << sines_name << " for output OK." << std::endl;
-
-  }
-
-

-

-

- The output file generated can be seen at ..\sines.hpp -

-

- The table output is: -

-

-

-
The printed table is:
-
-  1
-  0.70710678118654752440084436210484903928483593768847
-  0.38268343236508977172845998403039886676134456248563
-  0.19509032201612826784828486847702224092769161775195
-  0.098017140329560601994195563888641845861136673167501
-  0.049067674327418014254954976942682658314745363025753
-  0.024541228522912288031734529459282925065466119239451
-  0.012271538285719926079408261951003212140372319591769
-  0.0061358846491544753596402345903725809170578863173913
-  0.003067956762965976270145365490919842518944610213452
-  0.0015339801862847656123036971502640790799548645752374
-  0.00076699031874270452693856835794857664314091945206328
-  0.00038349518757139558907246168118138126339502603496474
-  0.00019174759731070330743990956198900093346887403385916
-  9.5873799095977345870517210976476351187065612851145e-05
-  4.7936899603066884549003990494658872746866687685767e-05
-  2.3968449808418218729186577165021820094761474895673e-05
-  1.1984224905069706421521561596988984804731977538387e-05
-  5.9921124526424278428797118088908617299871778780951e-06
-  2.9960562263346607504548128083570598118251878683408e-06
-  1.4980281131690112288542788461553611206917585861527e-06
-  7.4901405658471572113049856673065563715595930217207e-07
-  3.7450702829238412390316917908463317739740476297248e-07
-  1.8725351414619534486882457659356361712045272098287e-07
-  9.3626757073098082799067286680885620193236507169473e-08
-  4.681337853654909269511551813854009695950362701667e-08
-  2.3406689268274552759505493419034844037886207223779e-08
-  1.1703344634137277181246213503238103798093456639976e-08
-  5.8516723170686386908097901008341396943900085051757e-09
-  2.9258361585343193579282304690689559020175857150074e-09
-  1.4629180792671596805295321618659637103742615227834e-09
-*/
-
-

-

-

- The output can be copied as text and readily integrated into a given source - code. Alternatively, the output can be written to a text or even be used - within a self-written automatic code generator as this example. -

-

- A computer algebra system can be used to verify the results obtained from - Boost.Math and Boost.Multiprecision. For example, the Wolfram - Mathematica computer algebra system can obtain a similar table - with the command: -

-
Table[N[Sin[Pi / (2^n)], 50], {n, 1, 31, 1}]
-
-

- The Wolfram Alpha computational - knowledge engine can also be used to generate this table. The same command - can be pasted into the compute box. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/use_ntl.html b/doc/html/math_toolkit/using_udt/high_precision/use_ntl.html deleted file mode 100644 index 9881e6d02..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/use_ntl.html +++ /dev/null @@ -1,67 +0,0 @@ - - - -Using NTL Library - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- NTL::RR (an arbitrarily-fixed - precision floating-point number type), can be used via the bindings in - boost/math/bindings/rr.hpp. - For details, see NTL: A Library for - doing Number Theory by Victor Shoup. -

-

- New projects are recommended to use Boost.Multiprecision - instead. -

-

- 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. -

-

- For those functions that are based upon the Lanczos - approximation, the bindings defines a series of approximations with - up to 61 terms and accuracy up to approximately 3e-113. This therefore - sets the upper limit for accuracy to the majority of functions defined - this library when used with NTL::RR. -

-

- There is a concept checking test program for NTL support here. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/using_test.html b/doc/html/math_toolkit/using_udt/high_precision/using_test.html deleted file mode 100644 index fae6ea8e3..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/using_test.html +++ /dev/null @@ -1,131 +0,0 @@ - - - -Using without expression templates for Boost.Test and others - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- As noted in the Boost.Multiprecision - documentation, certain program constructs will not compile when using expression - templates. One example that many users may encounter is Boost.Test (1.54 - and earlier) when using macro BOOST_CHECK_CLOSE and BOOST_CHECK_CLOSE_FRACTION. -

-

- If, for example, you wish to use any multiprecision type like cpp_dec_float_50 in place of double to give more precision, you will - need to override the default boost::multiprecision::et_on - with boost::multiprecision::et_off. -

-

-

-
#include <boost/multiprecision/cpp_dec_float.hpp>
-
-

-

-

- To define a 50 decimal digit type using cpp_dec_float, - you must pass two template parameters to boost::multiprecision::number. -

-

- It may be more legible to use a two-staged type definition such as this: -

-

-

-
typedef boost::multiprecision::cpp_dec_float<50> mp_backend;
-typedef boost::multiprecision::number<mp_backend, boost::multiprecision::et_off> cpp_dec_float_50_noet;
-
-

-

-

- Here, we first define mp_backend - as cpp_dec_float with 50 - digits. The second step passes this backend to boost::multiprecision::number - with boost::multiprecision::et_off, an enumerated type. -

-
typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<50>, boost::multiprecision::et_off>
-cpp_dec_float_50_noet;
-
-

- You can reduce typing with a using - directive using namespace - boost::multiprecision; - if desired, as shown below. -

-

-

-
using namespace boost::multiprecision;
-
-typedef number<cpp_dec_float<50>, et_off> cpp_dec_float_50_noet;
-
-

-

-

- Now cpp_dec_float_50_noet - can be used as a direct replacement for built-in types like double etc. -

-

-

-
BOOST_AUTO_TEST_CASE(cpp_float_test_check_close)
-{
-
-  std::cout.precision(std::numeric_limits<cpp_dec_float_50_noet>::digits10); // All significant digits.
-  std::cout << std::showpoint << std::endl; // Show trailing zeros.
-
-  cpp_dec_float_50_noet a ("1.");
-  cpp_dec_float_50_noet b ("1.");
-  b += std::numeric_limits<cpp_dec_float_50_noet>::epsilon(); // Increment least significant decimal digit.
-
-  cpp_dec_float_50_noet eps = std::numeric_limits<cpp_dec_float_50_noet>::epsilon();
-
-  std::cout <<"a = " << a << ",\nb = " << b << ",\neps = " << eps << std::endl;
-
-  BOOST_CHECK_CLOSE(a, b, eps * 100); // Expected to pass (because tolerance is as percent).
-  BOOST_CHECK_CLOSE_FRACTION(a, b, eps); // Expected to pass (because tolerance is as fraction).
-
-

-

-

- Using cpp_dec_float_50 - with the default expression template use switched on, the compiler error - message for `BOOST_CHECK_CLOSE_FRACTION(a, b, eps); would be: -

-

-

-
// failure floating_point_comparison.hpp(59): error C2440: 'static_cast' :
-// cannot convert from 'int' to 'boost::multiprecision::detail::expression<tag,Arg1,Arg2,Arg3,Arg4>'
-
-

-

-

- A full example code is at test_cpp_float_close_fraction.cpp -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/using_udt/high_precision/why_high_precision.html b/doc/html/math_toolkit/using_udt/high_precision/why_high_precision.html deleted file mode 100644 index 93adb9043..000000000 --- a/doc/html/math_toolkit/using_udt/high_precision/why_high_precision.html +++ /dev/null @@ -1,127 +0,0 @@ - - - -Why use a high-precision library rather than built-in floating-point types? - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- For nearly all applications, the built-in floating-point types, double (and long - double if this offers higher precision - than double) offer enough - precision, typically a dozen decimal digits. -

-

- Some reasons why one would want to use a higher precision: -

-
    -
  • - A much more precise result (many more digits) is just a requirement. -
    • -
  • - The range of the computed value exceeds the range of the type: factorials - are the textbook example. -
    • -
  • - Using double is (or may be) too inaccurate. -
    • -
  • - Using long double (or may be) is too inaccurate. -
    • -
  • - Using an extended precision type implemented in software as double-double - (Darwin) - is sometimes unpredictably inaccurate. -
    • -
  • - Loss of precision or inaccuracy caused by extreme arguments or cancellation - error. -
    • -
  • - An accuracy as good as possible for a chosen built-in floating-point - type is required. -
    • -
  • - As a reference value, for example, to determine the inaccuracy of a - value computed with a built-in floating point type, (perhaps even using - some quick'n'dirty algorithm). The accuracy of many functions and distributions - in Boost.Math has been measured in this way from tables of very high - precision (up to 1000 decimal digits). -
    • -
-

- Many functions and distributions have differences from exact values that - are only a few least significant bits - computation noise. Others, often - those for which analytical solutions are not available, require approximations - and iteration: these may lose several decimal digits of precision. -

-

- Much larger loss of precision can occur for boundary - or corner cases, - often caused by cancellation - errors. -

-

- (Some of the worst and most common examples of cancellation - error or loss of significance can be avoided by using complements: - see why complements?). -

-

- If you require a value which is as accurate as can be represented in the - floating-point type, and is thus the closest representable value and has - an error less than 1/2 a least - significant bit or ulp - it may be useful to use a higher-precision type, for example, cpp_dec_float_50, to generate this value. - Conversion of this value to a built-in floating-point type ('float', double or long - double) will not cause any further - loss of precision. A decimal digit string will also be 'read' precisely - by the compiler into a built-in floating-point type to the nearest representable - value. -

-
- - - - - -
[Note]Note

- In contrast, reading a value from an std::istream - into a built-in floating-point type is not guaranteed - by the C++ Standard to give the nearest representable value. -

-

- William Kahan coined the term Table-Maker's - Dilemma for the problem of correctly rounding functions. Using - a much higher precision (50 or 100 decimal digits) is a practical way of - generating (almost always) correctly rounded values. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils.html b/doc/html/math_toolkit/utils.html deleted file mode 100644 index 538d03e47..000000000 --- a/doc/html/math_toolkit/utils.html +++ /dev/null @@ -1,48 +0,0 @@ - - - -Floating Point Utilities - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Rounding Truncation and - Integer Conversion
-
Floating-Point Classification: - Infinities and NaNs
-
Sign Manipulation - Functions
-
Facets for Floating-Point - Infinities and NaNs
-
Floating-Point Representation - Distance (ULP), and Finding Adjacent Floating-Point Values
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets.html b/doc/html/math_toolkit/utils/fp_facets.html deleted file mode 100644 index aca8c9044..000000000 --- a/doc/html/math_toolkit/utils/fp_facets.html +++ /dev/null @@ -1,82 +0,0 @@ - - - -Facets for Floating-Point Infinities and NaNs - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Introduction
-
Reference
-
Examples
-
Portability
-
Design Rationale
-
-
- - Synopsis -
-
namespace boost{ namespace math
-{
-  // Values for flags. 
-  const int legacy;
-  const int signed_zero;
-  const int trap_infinity;
-  const int trap_nan;
-
-  template<
-      class CharType,
-      class OutputIterator = std::ostreambuf_iterator<CharType>
-  >
-  class nonfinite_num_put : public std::num_put<CharType, OutputIterator>
-  {
-  public:
-      explicit nonfinite_num_put(int flags = 0);
-  };
-
-  template<
-      class CharType,
-      class InputIterator = std::istreambuf_iterator<CharType>
-  >
-  class nonfinite_num_get : public std::num_get<CharType, InputIterator>
-  {
-  public:
-      explicit nonfinite_num_get(int flags = 0);  // legacy, sign_zero ...
-  };
-}} // namespace boost namespace math
-
-

- To use these facets -

-
#include <boost\math\special_functions\nonfinite_num_facets.hpp>
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets/examples.html b/doc/html/math_toolkit/utils/fp_facets/examples.html deleted file mode 100644 index d93a218ae..000000000 --- a/doc/html/math_toolkit/utils/fp_facets/examples.html +++ /dev/null @@ -1,261 +0,0 @@ - - - -Examples - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Simple - example with std::stringstreams -
-

-

-
locale old_locale;
-locale tmp_locale(old_locale, new nonfinite_num_put<char>);
-locale new_locale(tmp_locale, new nonfinite_num_get<char>);
-
-

-

-

-

-
stringstream ss;
-ss.imbue(new_locale);
-double inf = numeric_limits<double>::infinity();
-ss << inf; // Write out.
-assert(ss.str() == "inf");
-double r;
-ss >> r; // Read back in.
-assert(inf == r); // Confirms that the double values really are identical.
-
-cout << "infinity output was " << ss.str() << endl;
-cout << "infinity input was " << r << endl;
-// But the string representation of r displayed will be the native type
-// because, when it was constructed, cout had NOT been imbued
-// with the new locale containing the nonfinite_numput facet.
-// So the cout output will be "1.#INF on MS platforms
-// and may be "inf" or other string representation on other platforms.
-
-

-

-
- - Use - with lexical_cast -
-
- - - - - -
[Note]Note

- From Boost 1.48, lexical_cast no longer uses stringstreams internally, - and is now able to handle infinities and NaNs natively on most platforms. -

-

- Without using a new locale that contains the nonfinite facets, previous - versions of lexical_cast - using stringstream were not portable (and often failed) if nonfinite values - are found. -

-

-

-
locale old_locale;
-locale tmp_locale(old_locale, new nonfinite_num_put<char>);
-locale new_locale(tmp_locale, new nonfinite_num_get<char>);
-
-

-

-

- Although other examples imbue individual streams with the new locale, for - the streams constructed inside lexical_cast, it was necesary to assign - to a global locale. -

-
locale::global(new_locale);
-
-

- lexical_cast then works - as expected, even with infinity and NaNs. -

-
double x = boost::lexical_cast<double>("inf");
-assert(x == std::numeric:limits<double>::infinity());
-
-string s = boost::lexical_cast<string>(numeric_limits<double>::infinity());
-assert(s == "inf");
-
-
- - - - - -
[Warning]Warning

- If you use stringstream inside your functions, you may still need to - use a global locale to handle nonfinites correctly. Or you need to imbue - your stringstream with suitable get and put facets. -

-
- - - - - -
[Warning]Warning

- You should be aware that the C++ specification does not explicitly require - that input from decimal digits strings converts with rounding to the - nearest representable floating-point binary value. (In contrast, decimal - digits read by the compiler, for example by an assignment like double d - = 1.234567890123456789, - are guaranteed to assign the nearest representable value to double d). - This implies that, no matter how many decimal digits you provide, there - is a potential uncertainty of 1 least significant bit in the resulting - binary value. -

-

- See for - more information on nearest representable and rounding. -

-

- Most iostream libraries do in fact achieve the desirable nearest - representable floating-point binary value for all values of - input. However one popular STL library does not quite achieve this for - 64-bit doubles. See Decimal - digit string input to double may be 1 bit wrong for the bizarre - full details. -

-

- If you are expecting to 'round-trip' lexical_cast - or serialization, for example - archiving and loading, and want to be absolutely - certain that you will always get an exactly identical double value binary - pattern, you should use the suggested 'workaround' below that - is believed to work on all platforms. -

-

- You should output using all potentially significant decimal digits, by - setting stream precision to std::numeric_limits<double>::max_digits10, - (or for the appropriate floating-point type, if not double) and crucially, - require scientific - format, not fixed - or automatic (default), for example: -

-
double output_value = any value;
-std::stringstream s;
-s << setprecison(std::numeric_limits<double>::max_digits10) << scientific << output_value;
-s >> input_value;
-
-
- - Use - with serialization archives -
-

- It is vital that the same locale is used when an archive is saved and when - it is loaded. Otherwise, loading the archive may fail. By default, archives - are saved and loaded with a classic C locale with a boost::archive::codecvt_null - facet added. Normally you do not have to worry about that. -

-

- The constructors for the archive classes, as a side-effect, imbue the stream - with such a locale. However, if you want to use the facets nonfinite_num_put and nonfinite_num_get - with archives, then you have to manage the locale manually. That is done - by calling the archive constructor with the flag boost::archive::no_codecvt, - thereby ensuring that the archive constructor will not - imbue the stream with a new locale. -

-

- The following code shows how to use nonfinite_num_put - with a text_oarchive. -

-
locale default_locale(locale::classic(), new boost::archive::codecvt_null<char>);
-locale my_locale(default_locale, new nonfinite_num_put<char>);
-
-ofstream ofs("test.txt");
-ofs.imbue(my_locale);
-
-boost::archive::text_oarchive oa(ofs, no_codecvt);
-
-double x = numeric_limits<double>::infinity();
-oa & x;
-
-

- The same method works with nonfinite_num_get - and text_iarchive. -

-

- If you use the nonfinite_num_put - with trap_infinity and/or - trap_nan flag with a serialization - archive, then you must set the exception mask of the stream. Serialization - archives do not check the stream state. -

-
- - Other examples -
-

- nonfinite_facet_simple.cpp - give some more simple demonstrations of the difference between using classic - C locale and constructing a C99 infinty and NaN compliant locale for input - and output. -

-

- See nonfinite_facet_sstream.cpp - for this example of use with std::stringstreams. -

-

- For an example of how to enforce the MSVC 'legacy' "1.#INF" and - "1.#QNAN" representations of infinity and NaNs, for input and - output, see nonfinite_legacy.cpp. -

-

- Treatment of signaling NaN is demonstrated at ../../../example/nonfinite_signaling_NaN.cpp -

-

- Example ../../../example/nonfinite_loopback_ok.cpp - shows loopback works OK. -

-

- Example ../../../example/nonfinite_num_facet.cpp - shows output and re-input of various finite and nonfinite values. -

-

- A simple example of trapping nonfinite output is at nonfinite_num_facet_trap.cpp. -

-

- A very basic example of using Boost.Archive is at ../../../example/nonfinite_serialization_archives.cpp. -

-

- A full demonstration of serialization by Francois Mauger is at ../../../example/nonfinite_num_facet_serialization.cpp -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets/intro.html b/doc/html/math_toolkit/utils/fp_facets/intro.html deleted file mode 100644 index efb31ed0c..000000000 --- a/doc/html/math_toolkit/utils/fp_facets/intro.html +++ /dev/null @@ -1,382 +0,0 @@ - - - -Introduction - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - The - Problem -
-

- The C++98 standard does not specify how infinity and - NaN are represented in text streams. As a result, - different platforms use different string representations. This can cause - undefined behavior when text files are moved between different platforms. - Some platforms cannot even input parse their own output! So 'route-tripping' - or loopback of output to input is not possible. For instance, the following - test fails with MSVC: -

-
stringstream ss;
-double inf = numeric_limits<double>::infinity();
-double r;
-ss << inf; // Write out.
-ss >> r; // Read back in.
-
-cout << "infinity output was " << inf << endl; // 1.#INF
-cout << "infinity input was " << r << endl; // 1
-
-assert(inf == y); // Fails!
-
-
- - The - Solution -
-

- The facets nonfinite_num_put - and nonfinite_num_get format - and parse all floating-point numbers, including infinity - and NaN, in a consistent - and portable manner. -

-

- The following test succeeds with MSVC. -

-

-

-
locale old_locale;
-locale tmp_locale(old_locale, new nonfinite_num_put<char>);
-locale new_locale(tmp_locale, new nonfinite_num_get<char>);
-
-

-

-
- - - - - -
[Tip]Tip
-

- To add two facets, nonfinite_num_put - and nonfinite_num_get, - you may have to add one at a time, using a temporary locale. -

-

- Or you can create a new locale in one step -

-

- std::locale new_locale(std::locale(std::locale(std::locale(), new boost::math::nonfinite_num_put<char>), new boost::math::nonfinite_num_get<char>)); -

-

- and, for example, use it to imbue an input and output stringstream. -

-
-
- - - - - -
[Tip]Tip

- To just change an input or output stream, you can concisely write cout.imbue (std::locale(std::locale(), - new boost::math::nonfinite_num_put<char>)); or cin.imbue (std::locale(std::locale(), new boost::math::nonfinite_num_get<char>)); -

-

-

-
stringstream ss;
-ss.imbue(new_locale);
-double inf = numeric_limits<double>::infinity();
-ss << inf; // Write out.
-assert(ss.str() == "inf");
-double r;
-ss >> r; // Read back in.
-assert(inf == r); // Confirms that the double values really are identical.
-
-cout << "infinity output was " << ss.str() << endl;
-cout << "infinity input was " << r << endl;
-// But the string representation of r displayed will be the native type
-// because, when it was constructed, cout had NOT been imbued
-// with the new locale containing the nonfinite_numput facet.
-// So the cout output will be "1.#INF on MS platforms
-// and may be "inf" or other string representation on other platforms.
-
-

-

-
- - C++0X - standard for output of infinity and NaN -
-

- C++0X - (final) draft standard does not explicitly specify the representation - (and input) of nonfinite values, leaving it implementation-defined. So - without some specific action, input and output of nonfinite values is not - portable. -

-
- - C99 - standard for output of infinity and NaN -
-

- The C99 - standard does specify how infinity - and NaN are formatted by printf and similar output functions, and parsed - by scanf and similar input functions. -

-

- The following string representations are used: -

-
-

Table 51. C99 Representation of Infinity and NaN

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- number -

- 		-

- string -

-
-

- Positive infinity -

- 		-

- "inf" or "infinity" -

-
-

- Positive NaN -

- 		-

- "nan" or "nan(...)" -

-
-

- Negative infinity -

- 		-

- "-inf" or "-infinity" -

-
-

- Negative NaN -

- 		-

- "-nan" or "-nan(...)" -

-
-
-

- So following C99 provides a sensible 'standard' way of handling input and - output of nonfinites in C++, and this implementation follows most of these - formats. -

-
- - Signaling - NaNs -
-

- A particular type of NaN is the signaling NaN. The usual mechanism of signaling - is by raising a floating-point exception. Signaling NaNs are defined by - IEEE - 754-2008. -

-

- Floating-point values with layout s111 1111 1axx - xxxx xxxx xxxx xxxx xxxx where s is the sign, x - is the payload, and bit a determines the type of NaN. -

-

- If bit a = 1, it is a quiet NaN. -

-

- If bit a is zero and the payload x - is nonzero, then it is a signaling NaN. -

-

- Although there has been theoretical interest in the ability of a signaling - NaN to raise an exception, for example to prevent use of an uninitialised - variable, in practice there appears to be no useful application of signaling - NaNs for most current processors. C++0X - 18.3.2.2 still specifies a (implementation-defined) representation - for signaling NaN, and static constexpr bool - has_signaling_NaN a method of - checking if a floating-point type has a representation for signaling NaN. -

-

- But in practice, most platforms treat signaling NaNs in the same as quiet - NaNs. So, for example, they are represented by "nan" on output - in C99 - format, and output as 1.#QNAN - by Microsoft compilers. -

-
- - - - - -
[Note]Note
-

- The C99 standard does not distinguish between the quiet NaN and signaling - NaN values. A quiet NaN propagates through almost every arithmetic operation - without raising a floating-point exception; a signaling NaN generally - raises a floating-point exception when occurring as an arithmetic operand. -

-

- C99 specification does not define the behavior of signaling NaNs. NaNs - created by IEC 60559 operations are always quiet. Therefore this implementation - follows C99, and treats the signaling NaN bit as just a part of the NaN - payload field. So this implementation does not distinguish between the - two classes of NaN. -

-
-
- - - - - -
[Note]Note
-

- An implementation may give zero and non-numeric values (such as infinities - and NaNs) a sign or may leave them unsigned. Wherever such values are - unsigned, any requirement in the C99 Standard to retrieve the sign shall - produce an unspecified sign, and any requirement to set the sign shall - be ignored. -

-

- This might apply to user-defined types, but in practice built-in floating-point - types float, double and long - double have well-behaved signs. -

-
-

- The numbers can be of type float, - double and long - double. An optional + sign can be - used with positive numbers (controlled by ios manipulator showpos). The function printf and similar C++ functions use - standard formatting flags to put all lower or all upper case (controlled - by std::ios manipulator uppercase - and lowercase). -

-

- The function scanf and - similar input functions are case-insensitive. -

-

- The dots in nan(...) - stand for an arbitrary string. The meaning of that string is implementation - dependent. It can be used to convey extra information about the NaN, from - the 'payload'. A particular value of the payload might be used to indicate - a missing value, for example. -

-

- This library uses the string representations specified by the C99 standard. -

-

- An example of an implementation that optionally includes the NaN payload - information is at AIX - NaN fprintf. That implementation specifies for Binary Floating - Point NANs: -

-
    -
  • - A NaN ordinal sequence is a left-parenthesis character '(', followed - by a digit sequence representing an integer n, where 1 <= n <= - INT_MAX-1, followed by a right-parenthesis character ')'. -
    • -
  • - The integer value, n, is determined by the fraction bits of the NaN - argument value as follows: -
    • -
  • - For a signalling NaN value, NaN fraction bits are reversed (left to - right) to produce bits (right to left) of an even integer value, 2*n. - Then formatted output functions produce a (signalling) NaN ordinal - sequence corresponding to the integer value n. -
    • -
  • - For a quiet NaN value, NaN fraction bits are reversed (left to right) - to produce bits (right to left) of an odd integer value, 2*n-1. Then - formatted output functions produce a (quiet) NaN ordinal sequence corresponding - to the integer value n. -
    • -
-
- - - - - -
[Warning]Warning

- This implementation does not (yet) provide output of, or access to, the - NaN payload. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets/portability.html b/doc/html/math_toolkit/utils/fp_facets/portability.html deleted file mode 100644 index 2c52660b8..000000000 --- a/doc/html/math_toolkit/utils/fp_facets/portability.html +++ /dev/null @@ -1,41 +0,0 @@ - - - -Portability - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- This library uses the floating-point number classification and sign-bit - from Boost.Math library, and should work on all platforms where that library - works. See the portability information for that library. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets/rationale.html b/doc/html/math_toolkit/utils/fp_facets/rationale.html deleted file mode 100644 index 331a9df87..000000000 --- a/doc/html/math_toolkit/utils/fp_facets/rationale.html +++ /dev/null @@ -1,58 +0,0 @@ - - - -Design Rationale - - - - - - - - -
-PrevUpHomeNext -
-
-
-
    -
  • - The flags are implemented as a const data member of the facet. Facets - are reference counted, and locales can share facets. Therefore changing - the flags of a facet would have effects that are hard to predict. An - alternative design would be to implement the flags using std::ios_base::xalloc and std::ios_base::iword. - Then one could safely modify the flags, and one could define manipulators - that do so. However, for that to work with dynamically linked libraries, - a .cpp - file would have to be added to the library. It was judged be more desirable - to have a headers only library, than to have mutable flags and manipulators. -
    • -
  • - The facet nonfinite_num_put - throws an exception when the trap_infinity - or trap_nan flag is - set and an attempt is made to format infinity or NaN. It would be better - if the facet set the fail bit of the stream. However, facets derived - from std::num_put do not have access to the - stream state. -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fp_facets/reference.html b/doc/html/math_toolkit/utils/fp_facets/reference.html deleted file mode 100644 index 05b4df4d8..000000000 --- a/doc/html/math_toolkit/utils/fp_facets/reference.html +++ /dev/null @@ -1,482 +0,0 @@ - - - -Reference - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - The - Facet nonfinite_num_put -
-
template<
-  class CharType, class OutputIterator = std::ostreambuf_iterator<CharType>
-        >
-class nonfinite_num_put;
-
-

- The class nonfinite_num_put<CharType, OutputIterator> is derived from std::num_put<CharType, OutputIterator>. Thus it is a facet that formats numbers. - The first template argument is the character type of the formatted strings, - usually char or wchar_t. The second template argument is - the type of iterator used to write the strings. It is required to be an - output iterator. Usually the default std::ostreambuf_iterator - is used. The public interface of the class consists of a single constructor - only: -

-
nonfinite_num_put(int flags = 0);
-
-

- The flags argument (effectively optional because a default of no_flags is provided) is discussed below. - The class template nonfinite_num_put - is defined in the header boost/math/nonfinite_num_facets.hpp - and lives in the namespace boost::math. -

-

- Unlike the C++ Standard facet std::num_put, - the facet nonfinite_num_put - formats infinity and NaN in a consistent and portable manner. - It uses the following string representations: -

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- Number -

- 		-

- String -

-
-

- Positive infinity -

- 		-

- inf -

-
-

- Positive NaN -

- 		-

- nan -

-
-

- Negative infinity -

- 		-

- -inf -

-
-

- Negative NaN -

- 		-

- -nan -

-
-

- The numbers can be of type float, - double and long - double. The strings can be in all - lower case or all upper case. An optional + sign can be used with positive - numbers. This can be controlled with the uppercase, - lowercase, showpos and noshowpos - manipulators. Formatting of integers, boolean values and finite floating-point - numbers is simply delegated to the normal std::num_put. -

-
- - Facet - nonfinite_num_get -
-
template<class CharType, class InputIterator = std::istreambuf_iterator<CharType> > class nonfinite_num_get;
-
-

- The class nonfinite_num_get<CharType, InputIterator> is derived from std::num_get<CharType, IntputIterator>. Thus it is a facet that parses strings - that represent numbers. The first template argument is the character type - of the strings, usually char - or wchar_t. The second template - argument is the type of iterator used to read the strings. It is required - to be an input iterator. Usually the default is used. The public interface - of the class consists of a single constructor only: -

-
nonfinite_num_get(int flags = 0);
-
-

- The flags argument is discussed below. The class - template nonfinite_num_get - is defined in the header boost/math/nonfinite_num_facets.hpp - and lives in the namespace boost::math. -

-

- Unlike the facet std::num_get, the facet nonfinite_num_get - parses strings that represent infinity - and NaN in a consistent - and portable manner. It recognizes precisely the string representations - specified by the C99 standard: -

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- Number -

- 		-

- String -

-
-

- Positive infinity -

- 		-

- inf, infinity -

-
-

- Positive NaN -

- 		-

- nan, nan(...) -

-
-

- Negative infinity -

- 		-

- -inf, -infinity -

-
-

- Negative NaN -

- 		-

- -nan, -nan(...) -

-
-

- The numbers can be of type float, - double and long - double. The facet is case-insensitive. - An optional + sign can be used with positive numbers. The dots in nan(...) - stand for an arbitrary string usually containing the NaN payload. - Parsing of strings that represent integers, boolean values and finite floating-point - numbers is delegated to std::num_get. -

-

- When the facet parses a string that represents infinity - on a platform that lacks infinity, then the fail bit of the stream is set. -

-

- When the facet parses a string that represents NaN - on a platform that lacks NaN, then the fail bit of the stream is set. -

-
- - Flags -
-

- The constructors for nonfinite_num_put - and nonfinite_num_get take - an optional bit flags argument. There are four different bit flags: -

-
    -
  • - legacy -
    • -
  • - signed_zero -
    • -
  • - trap_infinity -
    • -
  • - trap_nan -
    • -
-

- The flags can be combined with the OR operator|. -

-

- The flags are defined in the header boost/math/nonfinite_num_facets.hpp - and live in the namespace boost::math. -

-
- - legacy -
-

- The legacy flag has no effect with the output facet nonfinite_num_put. -

-

- If the legacy flag is used with the nonfinite_num_get - input facet, then the facet will recognize all the following string representations - of infinity and NaN: -

-
---- - - - - - - - - - - - - - - - - - - - - - - -
-

- Number -

- 		-

- String -

-
-

- Positive infinity -

- 		-

- inf, infinity, one#inf -

-
-

- Positive NaN -

- 		-

- nan, nan(...), nanq, nans, qnan, snan, one#ind, one#qnan, one#snan -

-
-

- Negative infinity -

- 		-

- -inf, -infinity, -one#inf -

-
-

- Negative NaN -

- 		-

- -nan, -nan(...), -nanq, -nans, -qnan, -snan, -one#ind, - one#qnan, - -one#snan -

-
-
    -
  • - The numbers can be of type float, - double and long double. -
    • -
  • - The facet is case-insensitive. -
    • -
  • - An optional + sign can - be used with the positive values. -
    • -
  • - The dots in nan(...) - stand for an arbitrary string. -
    • -
  • - one stands for any - string that std::num_get parses as the number 1, typically "1.#INF", "1.QNAN" - but also "000001.#INF"... -
    • -
-

- The list includes a number of non-standard string representations of infinity - and NaN that are used by various existing implementations of the C++ standard - library, and also string representations used by other programming languages. -

-
- - signed_zero -
-

- If the signed_zero flag - is used with nonfinite_num_put, - then the facet will always distinguish between positive and negative zero. - It will format positive zero as "0" or "+0" and negative - zero as "-0". The string representation of positive zero can - be controlled with the showpos - and noshowpos manipulators. -

-

- The signed_zero flag - has no effect with the input facet nonfinite_num_get. - The input facet nonfinite_num_get - always parses "0" and "+0" as positive zero and "-0" - as negative zero, as do most implementations of std::num_get. -

-
- - - - - -
[Note]Note

- If the signed_zero flag - is not set (the default), then a negative zero value will be displayed - on output in whatever way the platform normally handles it. For most - platforms, this it will format positive zero as "0" or "+0" - and negative zero as "-0". But setting the signed_zero - flag may be more portable. -

-
- - - - - -
[Tip]Tip

- A negative zero value can be portably produced using the changesign function - (changesign)(static_cast<ValType>(0)) where ValType - is float, double or long - double, or a User-Defined floating-point - type (UDT) provided that this UDT has a sign and that the changesign - function is implemented. -

-
- - trap_infinity -
-

- If the trap_infinity flag - is used with nonfinite_num_put, - then the facet will throw an exception of type std::ios_base::failure - when an attempt is made to format positive or negative infinity. If the - facet is called from a stream insertion operator, then the stream will - catch that exception and set either its fail - bit or its bad - bit. Which bit is set is platform - dependent. -

-

- If the trap_infinity flag - is used with nonfinite_num_get, - then the facet will set the fail - bit of the stream when an attempt - is made to parse a string that represents positive or negative infinity. -

-

- (See Design Rationale below for a discussion of this inconsistency.) -

-
- - trap_nan -
-

- Same as trap_infinity, - but positive and negative NaN are trapped instead. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/fpclass.html b/doc/html/math_toolkit/utils/fpclass.html deleted file mode 100644 index c2e75681d..000000000 --- a/doc/html/math_toolkit/utils/fpclass.html +++ /dev/null @@ -1,243 +0,0 @@ - - - -Floating-Point Classification: Infinities and NaNs - - - - - - - - -
-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); // Neither infinity nor NaN.
-
-template <class T>
-bool isinf(T t); // Infinity (+ or -).
-
-template <class T>
-bool isnan(T t); // NaN.
-
-template <class T>
-bool isnormal(T t); // isfinite and not denormalised.
-
-#include <boost\math\special_functions\fpclassify.hpp>
-
-

- to use these functions. -

-
- - 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 explicit namespace qualification.
-(boost::math::isnan)(z);
-//
-// This will cause a compiler error if isnan is a native macro:
-boost::math::isnan(z);
-// So always use instead:
-(boost::math::isnan)(z);
-//
-// You can also add a using statement,
-// globally to a .cpp file, or to a local function in a .hpp file.
-using boost::math::isnan;
-// so you can write the shorter and less cluttered
-(isnan)(z)
-// But, as above, if isnan is a native macro, this causes a compiler error,
-// because the macro always 'gets' the name first, unless enclosed in () brackets.
-
-

- 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). -

-
- - Floating-point - format -
-

- If you wish to find details of the floating-point format for any particular - processor, there is a program -

-

- inspect_fp.cpp -

-

- by Johan Rade which can be used to print out the processor type, endianness, - and detailed bit layout of a selection of floating-point values, including - infinity and NaNs. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float.html b/doc/html/math_toolkit/utils/next_float.html deleted file mode 100644 index b25c73093..000000000 --- a/doc/html/math_toolkit/utils/next_float.html +++ /dev/null @@ -1,74 +0,0 @@ - - - -Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Finding the - Next Representable Value in a Specific Direction (nextafter)
-
Finding the - Next Greater Representable Value (float_next)
-
Finding - the Next Smaller Representable Value (float_prior)
-
Calculating - the Representation Distance Between Two Floating Point Values (ULP) float_distance
-
Advancing - a Floating Point Value by a Specific Representation Distance (ULP) float_advance
-
-

- Unit of - Least Precision or Unit in the Last Place is the gap between two - different, but as close as possible, floating-point numbers. -

-

- Most decimal values, for example 0.1, cannot be exactly represented as floating-point - values, but will be stored as the closest - representable floating-point. -

-

- Functions are provided for finding adjacent greater and lesser floating-point - values, and estimating the number of gaps between any two floating-point - values. -

-

- The floating-point type FPT must have has a fixed number of bits in the representation. - The number of bits may set at runtime, but must be the same for all numbers. - For example, NTL::quad_float - type (fixed 128-bit representation) or NTL::RR - type (arbitrary but fixed decimal digits, default 150) but not - a type that extends the representation to provide an exact representation - for any number, for example XRC - eXact Real in C. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float/float_advance.html b/doc/html/math_toolkit/utils/next_float/float_advance.html deleted file mode 100644 index 29f22e26a..000000000 --- a/doc/html/math_toolkit/utils/next_float/float_advance.html +++ /dev/null @@ -1,67 +0,0 @@ - - - -Advancing a Floating Point Value by a Specific Representation Distance (ULP) float_advance - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Function float_advance advances a floating point number by a specified - number of ULP. -

-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/next.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class FPT>
-FPT float_advance(FPT val, int distance);
-
-}} // namespaces
-
-
- - Description - - float_advance -
-

- Returns a floating point number r such that float_distance(val, r) == - distance. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float/float_distance.html b/doc/html/math_toolkit/utils/next_float/float_distance.html deleted file mode 100644 index 7aa704017..000000000 --- a/doc/html/math_toolkit/utils/next_float/float_distance.html +++ /dev/null @@ -1,101 +0,0 @@ - - - -Calculating the Representation Distance Between Two Floating Point Values (ULP) float_distance - - - - - - - - -
-PrevUpHomeNext -
-
-
-

- Function float_distance finds the number of gaps/bits/ULP between any two - floating-point values. If the significands of floating-point numbers are - viewed as integers, then their difference is the number of ULP/gaps/bits - different. -

-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/next.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class FPT>
-FPT float_distance(FPT a, FPT b);
-
-}} // namespaces
-
-
- - Description - - float_distance -
-

- Returns the distance between a and b: - the result is always a signed integer value (stored in floating-point type - FPT) representing the number of distinct representations between a - and b. -

-

- Note that -

-
    -
  • - float_distance(a, a) - always returns 0. -
    • -
  • - float_distance(float_next(a), a) - always returns -1. -
    • -
  • - float_distance(float_prior(a), a) - always returns 1. -
    • -
-

- The function float_distance - is equivalent to calculating the number of ULP (Units in the Last Place) - between a and b except that it - returns a signed value indicating whether a - > b - or not. -

-

- If the distance is too great then it may not be able to be represented - as an exact integer by type FPT, but in practice this is unlikely to be - a issue. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float/float_next.html b/doc/html/math_toolkit/utils/next_float/float_next.html deleted file mode 100644 index 91d30a692..000000000 --- a/doc/html/math_toolkit/utils/next_float/float_next.html +++ /dev/null @@ -1,69 +0,0 @@ - - - -Finding the Next Greater Representable Value (float_next) - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/next.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class FPT>
-FPT float_next(FPT val);
-
-}} // namespaces
-
-
- - Description - - float_next -
-

- Returns the next representable value which is greater than x. - If x is non-finite then returns the result of a domain_error. If there is no such value greater - than x then returns an overflow_error. -

-

- Has the same effect as -

-
nextafter(val, (std::numeric_limits<FPT>::max)());
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float/float_prior.html b/doc/html/math_toolkit/utils/next_float/float_prior.html deleted file mode 100644 index 7a9b959e7..000000000 --- a/doc/html/math_toolkit/utils/next_float/float_prior.html +++ /dev/null @@ -1,69 +0,0 @@ - - - -Finding the Next Smaller Representable Value (float_prior) - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/next.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class FPT>
-FPT float_prior(FPT val);
-
-}} // namespaces
-
-
- - Description - - float_prior -
-

- Returns the next representable value which is less than x. - If x is non-finite then returns the result of a domain_error. If there is no such value less - than x then returns an overflow_error. -

-

- Has the same effect as -

-
nextafter(val, -(std::numeric_limits<FPT>::max)());  // Note most negative value -max.
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/next_float/nextafter.html b/doc/html/math_toolkit/utils/next_float/nextafter.html deleted file mode 100644 index 4529cff0b..000000000 --- a/doc/html/math_toolkit/utils/next_float/nextafter.html +++ /dev/null @@ -1,123 +0,0 @@ - - - -Finding the Next Representable Value in a Specific Direction (nextafter) - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/next.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template <class FPT>
-FPT nextafter(FPT val, FPT direction);
-
-}} // namespaces
-
-
- - Description - - nextafter -
-

- This is an implementation of the nextafter - function included in the C99 standard. (It is also effectively an implementation - of the C99 'nexttoward' legacy function which differs only having a long - double direction, and can generally serve in its place if required). -

-
- - - - - -
[Note]Note

- The C99 functions must use suffixes f and l to distinguish float and - long double versions. C++ uses the template mechanism instead. -

-

- Returns the next representable value after x in the - direction of y. If x - == y - then returns x. If x is non-finite - then returns the result of a domain_error. - If there is no such value in the direction of y then - returns an overflow_error. -

-
- - - - - -
[Warning]Warning

- The template parameter FTP must be a floating-point type. An integer - type, for example, will produce an unhelpful error message. -

-
- - - - - -
[Tip]Tip

- Nearly always, you just want the next or prior representable value, so - instead use float_next - or float_prior below. -

-
- - Examples - - nextafter -
-

- The two representations using a 32-bit float either side of unity are: -

-
The nearest (exact) representation of 1.F is      1.00000000
-nextafter(1.F, 999) is                            1.00000012
-nextafter(1/f, -999) is                           0.99999994
-
-The nearest (not exact) representation of 0.1F is 0.100000001
-nextafter(0.1F, 10) is                            0.100000009
-nextafter(0.1F, 10) is                            0.099999994
-
-

-

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/rounding.html b/doc/html/math_toolkit/utils/rounding.html deleted file mode 100644 index 33763b93c..000000000 --- a/doc/html/math_toolkit/utils/rounding.html +++ /dev/null @@ -1,43 +0,0 @@ - - - -Rounding Truncation and Integer Conversion - - - - - - - - -
-PrevUpHomeNext -
-
-
-
-
Rounding Functions
-
Truncation Functions
-
Integer and Fractional - Part Splitting (modf)
-
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/rounding/modf.html b/doc/html/math_toolkit/utils/rounding/modf.html deleted file mode 100644 index 0d18934fd..000000000 --- a/doc/html/math_toolkit/utils/rounding/modf.html +++ /dev/null @@ -1,77 +0,0 @@ - - - -Integer and Fractional Part Splitting (modf) - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/special_functions/modf.hpp>
-

-

-
template <class T>
-T modf(const T& v, T* ipart);
-
-template <class T, class Policy>
-T modf(const T& v, T* ipart, const Policy&);
-
-template <class T>
-T modf(const T& v, int* ipart);
-
-template <class T, class Policy>
-T modf(const T& v, int* ipart, const Policy&);
-
-template <class T>
-T modf(const T& v, long* ipart);
-
-template <class T, class Policy>
-T modf(const T& v, long* ipart, const Policy&);
-
-template <class T>
-T modf(const T& v, long long* ipart);
-
-template <class T, class Policy>
-T modf(const T& v, long long* ipart, const Policy&);
-
-

- The modf functions store - the integer part of v in *ipart and return the fractional part - of v. The sign of the integer and fractional parts - are the same as the sign of v. -

-

- If the argument v is either non-finite or else outside - the range of the result type, then returns the result of rounding_error: - by default this throws an instance of boost::math::rounding_error. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/rounding/round.html b/doc/html/math_toolkit/utils/rounding/round.html deleted file mode 100644 index f6bf4bc3e..000000000 --- a/doc/html/math_toolkit/utils/rounding/round.html +++ /dev/null @@ -1,77 +0,0 @@ - - - -Rounding Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/special_functions/round.hpp>
-

-

-
template <class T>
-T round(const T& v);
-
-template <class T, class Policy>
-T round(const T& v, const Policy&);
-
-template <class T>
-int iround(const T& v);
-
-template <class T, class Policy>
-int iround(const T& v, const Policy&);
-
-template <class T>
-long lround(const T& v);
-
-template <class T, class Policy>
-long lround(const T& v, const Policy&);
-
-template <class T>
-long long llround(const T& v);
-
-template <class T, class Policy>
-long long llround(const T& v, const Policy&);
-
-

- These functions return the closest integer to the argument v. -

-

- Halfway cases are rounded away from zero, regardless of the current rounding - direction. -

-

- If the argument v is either non-finite or else outside - the range of the result type, then returns the result of rounding_error: - by default this throws an instance of boost::math::rounding_error. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/rounding/trunc.html b/doc/html/math_toolkit/utils/rounding/trunc.html deleted file mode 100644 index 0b9d4f323..000000000 --- a/doc/html/math_toolkit/utils/rounding/trunc.html +++ /dev/null @@ -1,79 +0,0 @@ - - - -Truncation Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-

-

-
#include <boost/math/special_functions/trunc.hpp>
-

-

-
template <class T>
-T trunc(const T& v);
-
-template <class T, class Policy>
-T trunc(const T& v, const Policy&);
-
-template <class T>
-int itrunc(const T& v);
-
-template <class T, class Policy>
-int itrunc(const T& v, const Policy&);
-
-template <class T>
-long ltrunc(const T& v);
-
-template <class T, class Policy>
-long ltrunc(const T& v, const Policy&);
-
-template <class T>
-long long lltrunc(const T& v);
-
-template <class T, class Policy>
-long long lltrunc(const T& v, const Policy&);
-
-

- The trunc functions round their argument to the integer value, nearest - to but no larger in magnitude than the argument. -

-

- For example itrunc(3.7) would - return 3 and ltrunc(-4.6) would - return -4. -

-

- If the argument v is either non-finite or else outside - the range of the result type, then returns the result of rounding_error: - by default this throws an instance of boost::math::rounding_error. -

-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html/math_toolkit/utils/sign_functions.html b/doc/html/math_toolkit/utils/sign_functions.html deleted file mode 100644 index fd20576ec..000000000 --- a/doc/html/math_toolkit/utils/sign_functions.html +++ /dev/null @@ -1,246 +0,0 @@ - - - -Sign Manipulation Functions - - - - - - - - -
-PrevUpHomeNext -
-
-
-
- - Synopsis -
-

-

-
#include <boost/math/special_functions/sign.hpp>
-
-

-

-
namespace boost{ namespace math{
-
-template<class T>
-int signbit(T x);
-
-template <class T>
-int sign (const T& z);
-
-template <class T, class U>
-T copysign (const T& x, const U& y);
-
-template <class T>
-calculated-result-type changesign (const T& z);
-
-}} // namespaces
-
-
- - Description -
-
template<class T>
-int signbit(T x);
-
-

- Returns a non-zero value if the sign bit is set in variable x, - otherwise 0. -

-
- - - - - -
[Important]Important

- The return value from this function is zero or not-zero - and not zero or one. -

-
template <class T>
-int sign (const T& z);
-
-

- Returns 1 if x - > 0, - -1 - if x < 0, and 0 - if x is zero. -

-
template <class T, class U>
-calculated-result-type copysign (const T& x, const U& y);
-
-

- Sets the sign of x to be the same as the sign of y. -

-

- See C99 - 7.12.11.1 The copysign functions for more detail. -

-
template <class T>
-T changesign (const T& z);
-
-

- Returns a floating point number with a binary representation where the signbit - is the opposite of the sign bit in x, and where the - other bits are the same as in x. -

-

- This function is widely available, but not specified in any standards. -

-

- Rationale: Not specified by TR1, but changesign(x) - is both easier to read and more efficient than -

-
copysign(x, signbit(x) ? 1.0 : -1.0);
-
-

- For finite values, this function has the same effect as simple negation, - the assignment z = -z, but for nonfinite values, infinities - and NaNs, the changesign(x) function - may be the only portable way to ensure that the sign bit is changed. -

-
- - Sign - bits -
-

- One of the bits in the binary representation of a floating-point number gives - the sign, and the remaining bits give the absolute value. That bit is known - as the sign bit. The sign bit is set = 1 for negative numbers, and is not - set = 0 for positive numbers. (This is true for all binary representations - of floating point numbers that are used by modern microprocessors.) -

-

- C++ - TR1 specifies copysign - functions and function templates for accessing the sign bit. -

-

- For user-defined types (UDT), the sign may be stored in some other way. They - may also not provide infinity or NaNs. To use these functions with a UDT, - it may be necessary to explicitly specialize then for UDT type T. -

-
- - Examples -
-
signbit(3.5) is zero (or false)
-signbit(-7.1) is 1 (or true)
-copysign(4.2, 7.9) is 4.2
-copysign(3.5 -1.4) is -3.5
-copysign(-4.2, 1.0) is 4.2
-copysign(-8.6, -3.3) is -8.6
-changesign(6.9) is -6.9
-changesign(-1.8) is 1.8
-
-
- - Portability -
-

- The library supports the following binary floating-point formats: -

-
    -
  • - IEEE 754 single precision -
    • -
  • - IEEE 754 double precision -
    • -
  • - IEEE 754 extended double precision with 15 exponent bits -
    • -
  • - Intel extended double precision -
    • -
  • - PowerPC extended double precision -
    • -
  • - Motorola 68K extended double precision -
    • -
-

- The library does not support the VAX floating-point formats. (These are available - on VMS, but the default on VMS is the IEEE 754 floating-point format.) -

-

- The main portability issues are: -

-
    -
  • - Unsupported floating point formats -
    • -
  • - The library depends on the header boost/detail/endian.hpp -
    • -
  • - Code such as #if defined(__ia64) || defined(__ia64__) - || defined(_M_IA64) is used to determine the processor type. -
    • -
-

- The library has passed all tests on the following platforms: -

-
    -
  • - Win32 / MSVC 7.1 / 10.0 / x86 -
    • -
  • - Win32 / Intel C++ 7.1, 8.1, 9.1 / x86 -
    • -
  • - Mac OS X / GCC 3.3, 4.0 / ppc -
    • -
  • - Linux / Intel C++ 9.1 / x86, ia64 -
    • -
  • - Linux / GCC 3.3 / x86, x64, ia64, ppc, hppa, mips, m68k -
    • -
  • - Linux / GCC 3.4 / x64 -
    • -
  • - HP-UX / aCC, GCC 4.1 / ia64 -
    • -
  • - HP-UX / aCC / hppa -
    • -
  • - Tru64 / Compaq C++ 7.1 / alpha -
    • -
  • - VMS / HP C++ 7.1 / alpha (in IEEE floating point mode) -
    • -
  • - VMS / HP C++ 7.2 / ia64 (in IEEE floating point mode) -
    • -
-
- - - -
Copyright © 2006-2010, 2012, 2013 John Maddock, Paul A. Bristow, - Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs - van den Berg, Benjamin Sobotta and Christopher Kormanyos

- Distributed under the 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/html4_symbols.qbk b/doc/html4_symbols.qbk index c3bd27784..d44d3a158 100644 --- a/doc/html4_symbols.qbk +++ b/doc/html4_symbols.qbk @@ -11,7 +11,7 @@ [/Unicode Latin extended http://www.unicode.org/charts/U0080.pdf] [/ Also some miscellaneous math characters added to this list - see the end.] -[/ For others see also math_symbols.qbk] +[/ For others see also math_toolkit.symbols.qbk] [/ To use, enclose the template name in square brackets, for example: [pi]] diff --git a/doc/internals/minimax.qbk b/doc/internals/minimax.qbk index a0fc0cde1..61a14f43d 100644 --- a/doc/internals/minimax.qbk +++ b/doc/internals/minimax.qbk @@ -13,7 +13,7 @@ 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 +you should first read the [link math_toolkit.remez brief background article explaining the principles behind the Remez algorithm]. @@ -35,7 +35,7 @@ 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.high_precision.use_ntl patched NTL library to compile]. +a [link math_toolkit.high_precision.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 diff --git a/doc/internals/rational.qbk b/doc/internals/rational.qbk index 61c82392e..eafa07a92 100644 --- a/doc/internals/rational.qbk +++ b/doc/internals/rational.qbk @@ -159,10 +159,10 @@ 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 +[link math_toolkit.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 +[link math_toolkit.perf_test_app performance test application] before changing the default settings. [endsect][/section:rational Polynomial and Rational Function Evaluation] diff --git a/doc/internals/roots.qbk b/doc/internals/roots.qbk index c2a8a5892..5a88dea46 100644 --- a/doc/internals/roots.qbk +++ b/doc/internals/roots.qbk @@ -34,10 +34,10 @@ These functions all perform iterative root finding using derivatives: * `newton_raphson_iterate` performs second order -[link newton Newton-Raphson iteration], +[link math_toolkit.internals1.roots.newton Newton-Raphson iteration], * `halley_iterate` and`schroeder_iterate` perform third order -[link halley Halley] and [link schroeder Schroeder] iteration. +[link math_toolkit.internals1.roots.halley Halley] and [link math_toolkit.internals1.roots.schroeder Schroeder] iteration. The functions all take the same parameters: @@ -109,8 +109,7 @@ be able to compute the ratio of derivatives used by these methods more efficient than computing the derivatives themselves. As ever, algebraic simplification can be a big win. -[#newton] -[h4 Newton Raphson Method] +[h4:newton Newton Raphson Method] Given an initial guess x0 the subsequent values are computed using: [equation roots1] @@ -119,8 +118,7 @@ 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] +[h4:halley Halley's Method] Given an initial guess x0 the subsequent values are computed using: @@ -134,8 +132,7 @@ 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] +[h4:schroeder Schroeder's Method] Given an initial guess x0 the subsequent values are computed using: diff --git a/doc/internals/test_data.qbk b/doc/internals/test_data.qbk index b7a17ed58..b329a6f73 100644 --- a/doc/internals/test_data.qbk +++ b/doc/internals/test_data.qbk @@ -389,7 +389,6 @@ 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: diff --git a/doc/math.qbk b/doc/math.qbk index 6ee12d4ed..20c6e7e55 100644 --- a/doc/math.qbk +++ b/doc/math.qbk @@ -1,16 +1,21 @@ -[article Math Toolkit - [quickbook 1.5] - [copyright 2006, 2007, 2008, 2009, 2010, 2012, 2013 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg, Benjamin Sobotta and Christopher Kormanyos] +[book Math Toolkit + [quickbook 1.6] + [copyright 2006, 2007, 2008, 2009, 2010, 2012, 2013 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg, Daryle Walker, Benjamin Sobotta and Christopher Kormanyos] [/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 [Bristow, Paul A.], [Holin, Hubert], [Kormanyos, Christopher], [Lalande, Bruno], [Maddock, John], [Råde, Johan], [Sobotta, Benjamin], [Sewani, Gautam], [van den Berg, Thijs], [Zhang, Xiaogang] ] + [authors [Bristow, Paul A.], [Holin, Hubert], [Kormanyos, Christopher], [Lalande, Bruno], [Maddock, John], [Råde, Johan], [Sobotta, Benjamin], [Sewani, Gautam], [van den Berg, Thijs], [Walker, Daryle], [Zhang, Xiaogang] ] [/last-revision $Date$] ] +[template mathpart[id title] +[block ''''''[title]'''''']] +[template endmathpart[] +[block '''''']] + [/ \u00E5 = Latin small letter A with ring above doesn't seem to work in authors with Quickbook 1.5] [template equation[name] ''' @@ -31,14 +36,14 @@ '''] -[include ../../../tools/auto_index/include/auto_index_helpers.qbk] +[import ../../../tools/auto_index/include/auto_index_helpers.qbk] [/ Must be first included file!] -[include html4_symbols.qbk] +[import html4_symbols.qbk] -[include overview/common_overviews.qbk] [/ overviews that appear in more than one place!] -[include overview/tr1.qbk] [/tr1 docs also appear in more than one place!] -[include overview/roadmap.qbk] [/ for history] +[import overview/common_overviews.qbk] [/ overviews that appear in more than one place!] +[import overview/tr1.qbk] [/tr1 docs also appear in more than one place!] +[import overview/roadmap.qbk] [/ for history] [def __effects [*Effects: ]] [def __formula [*Formula: ]] @@ -65,238 +70,238 @@ [template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]] [template C99[] [@http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1256.pdf C99 Standard ISO/IEC 9899:1999]] -[template jm_rationals[] [link math_toolkit.backgrounders.implementation.rational_approximations_used devised by JM]] +[template jm_rationals[] [link math_toolkit.implementation0.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 __rounding_error [link rounding_error rounding_error]] -[def __indeterminate_result_error [link indeterminate_result_error indeterminate_result_error]] -[def __checked_narrowing_cast [link checked_narrowing_cast checked_narrowing_cast]] +[def __domain_error [link math_toolkit.error_handling.domain_error domain_error]] +[def __pole_error [link math_toolkit.error_handling.pole_error pole_error]] +[def __overflow_error [link math_toolkit.error_handling.overflow_error overflow_error]] +[def __underflow_error [link math_toolkit.error_handling.underflow_error underflow_error]] +[def __denorm_error [link math_toolkit.error_handling.denorm_error denorm_error]] +[def __evaluation_error [link math_toolkit.error_handling.evaluation_error evaluation_error]] +[def __rounding_error [link math_toolkit.error_handling.rounding_error rounding_error]] +[def __indeterminate_result_error [link math_toolkit.error_handling.indeterminate_result_error indeterminate_result_error]] +[def __checked_narrowing_cast [link math_toolkit.error_handling.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]]] +[def __arg_pomotion_rules [link math_toolkit.result_type ['result type calculation rules]]] +[def __sf_result [link math_toolkit.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]] -[def __root_finding_with_derivatives [link math_toolkit.toolkit.internals1.roots root-finding with derivatives]] +[def __lanczos [link math_toolkit.lanczos Lanczos approximation]] +[def __zero_error [link math_toolkit.relative_error.zero_error effectively zero error]] +[def __relative_error [link math_toolkit.relative_error relative zero error]] +[def __root_finding_with_derivatives [link math_toolkit.internals1.roots root-finding with derivatives]] [/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]] +[def __lgamma [link math_toolkit.sf_gamma.lgamma lgamma]] +[def __digamma [link math_toolkit.sf_gamma.digamma digamma]] +[def __tgamma_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_ratio]] +[def __tgamma_delta_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_delta_ratio]] +[def __tgamma [link math_toolkit.sf_gamma.tgamma tgamma]] +[def __tgamma1pm1 [link math_toolkit.sf_gamma.tgamma tgamma1pm1]] +[def __tgamma_lower [link math_toolkit.sf_gamma.igamma tgamma_lower]] +[def __gamma_p [link math_toolkit.sf_gamma.igamma gamma_p]] +[def __gamma_q [link math_toolkit.sf_gamma.igamma gamma_q]] +[def __gamma_q_inv [link math_toolkit.sf_gamma.igamma_inv gamma_q_inv]] +[def __gamma_p_inv [link math_toolkit.sf_gamma.igamma_inv gamma_p_inv]] +[def __gamma_q_inva [link math_toolkit.sf_gamma.igamma_inv gamma_q_inva]] +[def __gamma_p_inva [link math_toolkit.sf_gamma.igamma_inv gamma_p_inva]] +[def __gamma_p_derivative [link math_toolkit.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]] +[def __factorial [link math_toolkit.factorials.sf_factorial factorial]] +[def __unchecked_factorial [link math_toolkit.factorials.sf_factorial unchecked_factorial]] +[def __max_factorial [link math_toolkit.factorials.sf_factorial max_factorial]] +[def __double_factorial [link math_toolkit.factorials.sf_double_factorial double_factorial]] +[def __rising_factorial [link math_toolkit.factorials.sf_rising_factorial rising_factorial]] +[def __falling_factorial [link math_toolkit.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]] +[def __erf [link math_toolkit.sf_erf.error_function erf]] +[def __erfc [link math_toolkit.sf_erf.error_function erfc]] +[def __erf_inv [link math_toolkit.sf_erf.error_inv erf_inv]] +[def __erfc_inv [link math_toolkit.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]] +[def __beta [link math_toolkit.sf_beta.beta_function beta]] +[def __beta3 [link math_toolkit.sf_beta.ibeta_function beta]] +[def __betac [link math_toolkit.sf_beta.ibeta_function betac]] +[def __ibeta [link math_toolkit.sf_beta.ibeta_function ibeta]] +[def __ibetac [link math_toolkit.sf_beta.ibeta_function ibetac]] +[def __ibeta_inv [link math_toolkit.sf_beta.ibeta_inv_function ibeta_inv]] +[def __ibetac_inv [link math_toolkit.sf_beta.ibeta_inv_function ibetac_inv]] +[def __ibeta_inva [link math_toolkit.sf_beta.ibeta_inv_function ibeta_inva]] +[def __ibetac_inva [link math_toolkit.sf_beta.ibeta_inv_function ibetac_inva]] +[def __ibeta_invb [link math_toolkit.sf_beta.ibeta_inv_function ibeta_invb]] +[def __ibetac_invb [link math_toolkit.sf_beta.ibeta_inv_function ibetac_invb]] +[def __ibeta_derivative [link math_toolkit.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]] +[def __ellint_rj [link math_toolkit.ellint.ellint_carlson ellint_rj]] +[def __ellint_rf [link math_toolkit.ellint.ellint_carlson ellint_rf]] +[def __ellint_rc [link math_toolkit.ellint.ellint_carlson ellint_rc]] +[def __ellint_rd [link math_toolkit.ellint.ellint_carlson ellint_rd]] +[def __ellint_1 [link math_toolkit.ellint.ellint_1 ellint_1]] +[def __ellint_2 [link math_toolkit.ellint.ellint_2 ellint_2]] +[def __ellint_3 [link math_toolkit.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]] +[def __cyl_bessel_j [link math_toolkit.bessel.bessel cyl_bessel_j]] +[def __cyl_neumann [link math_toolkit.bessel.bessel cyl_neumann]] +[def __cyl_bessel_i [link math_toolkit.bessel.mbessel cyl_bessel_i]] +[def __cyl_bessel_k [link math_toolkit.bessel.mbessel cyl_bessel_k]] +[def __sph_bessel [link math_toolkit.bessel.sph_bessel sph_bessel]] +[def __sph_neumann [link math_toolkit.bessel.sph_bessel sph_neumann]] [/Hankel Functions] -[def __cyl_hankel_1 [link math_toolkit.special.hankel.cyl_hankel cyl_hankel_1]] -[def __cyl_hankel_2 [link math_toolkit.special.hankel.cyl_hankel cyl_hankel_2]] -[def __sph_hankel_1 [link math_toolkit.special.hankel.sph_hankel sph_hankel_1]] -[def __sph_hankel_2 [link math_toolkit.special.hankel.sph_hankel sph_hankel_2]] +[def __cyl_hankel_1 [link math_toolkit.hankel.cyl_hankel cyl_hankel_1]] +[def __cyl_hankel_2 [link math_toolkit.hankel.cyl_hankel cyl_hankel_2]] +[def __sph_hankel_1 [link math_toolkit.hankel.sph_hankel sph_hankel_1]] +[def __sph_hankel_2 [link math_toolkit.hankel.sph_hankel sph_hankel_2]] [/Airy Functions] -[def __airy_ai [link math_toolkit.special.airy.ai airy_ai]] -[def __airy_bi [link math_toolkit.special.airy.bi airy_bi]] -[def __airy_bi_prime [link math_toolkit.special.airy.bip airy_bi_prime]] -[def __airy_ai_prime [link math_toolkit.special.airy.aip airy_ai_prime]] +[def __airy_ai [link math_toolkit.airy.ai airy_ai]] +[def __airy_bi [link math_toolkit.airy.bi airy_bi]] +[def __airy_bi_prime [link math_toolkit.airy.bip airy_bi_prime]] +[def __airy_ai_prime [link math_toolkit.airy.aip airy_ai_prime]] [/Jacobi Elliptic Functions] -[def __jacobi_elliptic [link math_toolkit.special.jacobi.jacobi_elliptic jacobi_elliptic]] -[def __jacobi_cd [link math_toolkit.special.jacobi.jacobi_cd jacobi_cd]] -[def __jacobi_cn [link math_toolkit.special.jacobi.jacobi_cn jacobi_cn]] -[def __jacobi_cs [link math_toolkit.special.jacobi.jacobi_cs jacobi_cs]] -[def __jacobi_dc [link math_toolkit.special.jacobi.jacobi_dc jacobi_dc]] -[def __jacobi_dn [link math_toolkit.special.jacobi.jacobi_dn jacobi_dn]] -[def __jacobi_ds [link math_toolkit.special.jacobi.jacobi_ds jacobi_ds]] -[def __jacobi_nd [link math_toolkit.special.jacobi.jacobi_nd jacobi_nd]] -[def __jacobi_nc [link math_toolkit.special.jacobi.jacobi_nc jacobi_nc]] -[def __jacobi_ns [link math_toolkit.special.jacobi.jacobi_ns jacobi_ns]] -[def __jacobi_sd [link math_toolkit.special.jacobi.jacobi_sd jacobi_sd]] -[def __jacobi_sn [link math_toolkit.special.jacobi.jacobi_sn jacobi_sn]] -[def __jacobi_sc [link math_toolkit.special.jacobi.jacobi_sc jacobi_sc]] +[def __jacobi_elliptic [link math_toolkit.jacobi.jacobi_elliptic jacobi_elliptic]] +[def __jacobi_cd [link math_toolkit.jacobi.jacobi_cd jacobi_cd]] +[def __jacobi_cn [link math_toolkit.jacobi.jacobi_cn jacobi_cn]] +[def __jacobi_cs [link math_toolkit.jacobi.jacobi_cs jacobi_cs]] +[def __jacobi_dc [link math_toolkit.jacobi.jacobi_dc jacobi_dc]] +[def __jacobi_dn [link math_toolkit.jacobi.jacobi_dn jacobi_dn]] +[def __jacobi_ds [link math_toolkit.jacobi.jacobi_ds jacobi_ds]] +[def __jacobi_nd [link math_toolkit.jacobi.jacobi_nd jacobi_nd]] +[def __jacobi_nc [link math_toolkit.jacobi.jacobi_nc jacobi_nc]] +[def __jacobi_ns [link math_toolkit.jacobi.jacobi_ns jacobi_ns]] +[def __jacobi_sd [link math_toolkit.jacobi.jacobi_sd jacobi_sd]] +[def __jacobi_sn [link math_toolkit.jacobi.jacobi_sn jacobi_sn]] +[def __jacobi_sc [link math_toolkit.jacobi.jacobi_sc jacobi_sc]] [/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]] +[def __sinc_pi [link math_toolkit.sinc.sinc_pi sinc_pi]] +[def __sinhc_pi [link math_toolkit.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]] +[def __acosh [link math_toolkit.inv_hyper.acosh acosh]] +[def __asinh [link math_toolkit.inv_hyper.asinh asinh]] +[def __atanh [link math_toolkit.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]] -[def __fp_facets [link math_toolkit.utils.fp_facets nonfinite fp_facets]] +[def __fpclassify [link math_toolkit.fpclass fpclassify]] +[def __isfinite [link math_toolkit.fpclass isfinite]] +[def __isnan [link math_toolkit.fpclass isnan]] +[def __isinf [link math_toolkit.fpclass isinf]] +[def __isnormal [link math_toolkit.fpclass isnormal]] +[def __fp_facets [link math_toolkit.fp_facets nonfinite fp_facets]] [/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]] -[def __pow [link math_toolkit.special.powers.ct_pow pow]] +[def __expm1 [link math_toolkit.powers.expm1 expm1]] +[def __log1p [link math_toolkit.powers.log1p log1p]] +[def __cbrt [link math_toolkit.powers.cbrt cbrt]] +[def __sqrt1pm1 [link math_toolkit.powers.sqrt1pm1 sqrt1pm1]] +[def __powm1 [link math_toolkit.powers.powm1 powm1]] +[def __hypot [link math_toolkit.powers.hypot hypot]] +[def __pow [link math_toolkit.powers.ct_pow pow]] [/zeta] -[def __zeta [link math_toolkit.special.zetas.zeta zeta]] +[def __zeta [link math_toolkit.zetas.zeta zeta]] [/expint] -[def __expint_i [link math_toolkit.special.expint.expint_i zeta]] -[def __expint_n [link math_toolkit.special.expint.expint_n zeta]] +[def __expint_i [link math_toolkit.expint.expint_i zeta]] +[def __expint_n [link math_toolkit.expint.expint_n zeta]] [/rounding] -[def __round [link math_toolkit.utils.rounding.round round]] -[def __lround [link math_toolkit.utils.rounding.round round]] -[def __llround [link math_toolkit.utils.rounding.round round]] -[def __trunc [link math_toolkit.utils.rounding.trunc trunc]] -[def __modf [link math_toolkit.utils.rounding.modf modf]] +[def __round [link math_toolkit.rounding.round round]] +[def __lround [link math_toolkit.rounding.round round]] +[def __llround [link math_toolkit.rounding.round round]] +[def __trunc [link math_toolkit.rounding.trunc trunc]] +[def __modf [link math_toolkit.rounding.modf modf]] [/polynomials] -[def __laguerre [link math_toolkit.special.sf_poly.laguerre laguerre]] -[def __legendre [link math_toolkit.special.sf_poly.legendre legendre_p]] -[def __hermite [link math_toolkit.special.sf_poly.hermite hermite]] +[def __laguerre [link math_toolkit.sf_poly.laguerre laguerre]] +[def __legendre [link math_toolkit.sf_poly.legendre legendre_p]] +[def __hermite [link math_toolkit.sf_poly.hermite hermite]] [/Misc] -[def __expint [link math_toolkit.special.expint.expint_i expint]] -[def __spherical_harmonic [link math_toolkit.special.sf_poly.sph_harm spherical_harmonic]] -[def __owens_t [link math_toolkit.special.owens_t Owens T]] +[def __expint [link math_toolkit.expint.expint_i expint]] +[def __spherical_harmonic [link math_toolkit.sf_poly.sph_harm spherical_harmonic]] +[def __owens_t [link math_toolkit.owens_t Owens T]] [/tools] -[def __tuple [link math_toolkit.toolkit.internals1.tuples boost::math::tuple]] +[def __tuple [link math_toolkit.internals1.tuples boost::math::tuple]] [/ 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]] +[def __cdf [link math_toolkit.dist_ref.nmp.cdf Cumulative Distribution Function]] +[def __pdf [link math_toolkit.dist_ref.nmp.pdf Probability Density Function]] +[def __ccdf [link math_toolkit.dist_ref.nmp.ccdf Complement of the Cumulative Distribution Function]] +[def __quantile [link math_toolkit.dist_ref.nmp.quantile Quantile]] +[def __quantile_c [link math_toolkit.dist_ref.nmp.quantile_c Quantile from the complement of the probability]] +[def __mean [link math_toolkit.dist_ref.nmp.mean mean]] +[def __median [link math_toolkit.dist_ref.nmp.median median]] +[def __mode [link math_toolkit.dist_ref.nmp.mode mode]] +[def __skewness [link math_toolkit.dist_ref.nmp.skewness skewness]] +[def __kurtosis [link math_toolkit.dist_ref.nmp.kurtosis kurtosis]] +[def __kurtosis_excess [link math_toolkit.dist_ref.nmp.kurtosis_excess kurtosis_excess]] +[def __variance [link math_toolkit.dist_ref.nmp.variance variance]] +[def __sd [link math_toolkit.dist_ref.nmp.sd standard deviation]] +[def __hazard [link math_toolkit.dist_ref.nmp.hazard Hazard Function]] +[def __chf [link math_toolkit.dist_ref.nmp.chf Cumulative Hazard Function]] +[def __range [link math_toolkit.dist_ref.nmp.range range]] +[def __support [link math_toolkit.dist_ref.nmp.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 __extreme_distrib [link math_toolkit.dist.dist_ref.dists.extreme_dist Extreme Distributions]] -[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 __geometric_distrib [link math_toolkit.dist.dist_ref.dists.geometric_dist Geometric Distribution]] -[def __hypergeometric_distrib [link math_toolkit.dist.dist_ref.dists.hypergeometric_dist hypergeometric Distribution]] -[def __inverse_gamma_distrib [link math_toolkit.dist.dist_ref.dists.inverse_gamma_dist Inverse Gamma Distribution]] -[def __inverse_gaussian_distrib [link math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist Inverse Gaussian Distribution]] -[def __inverse_chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist Inverse chi squared Distribution]] -[def __laplace_distrib [link math_toolkit.dist.dist_ref.dists.laplace_dist Laplace Distribution]] -[def __logistic_distrib [link math_toolkit.dist.dist_ref.dists.logistic_dist Logistic 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 __non_central_chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist Noncentral Chi Squared Distribution]] -[def __non_central_beta_distrib [link math_toolkit.dist.dist_ref.dists.nc_beta_dist Noncentral Beta Distribution]] -[def __non_central_F_distrib [link math_toolkit.dist.dist_ref.dists.nc_f_dist Noncentral F Distribution]] -[def __non_central_T_distrib [link math_toolkit.dist.dist_ref.dists.nc_t_dist Noncentral T Distribution]] -[def __non_central_t_distrib [link math_toolkit.dist.dist_ref.dists.nc_t_dist noncentral T 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 __pareto_distrib [link math_toolkit.dist.dist_ref.dists.pareto_dist Pareto Distribution]] -[def __students_t_distrib [link math_toolkit.dist.dist_ref.dists.students_t_dist Students t Distribution]] -[def __skew_normal_distrib [link math_toolkit.dist.dist_ref.dists.skew_normal_dist Skew Normal Distribution]] -[def __weibull_distrib [link math_toolkit.dist.dist_ref.dists.weibull_dist Weibull Distribution]] +[def __beta_distrib [link math_toolkit.dist_ref.dists.beta_dist Beta Distribution]] +[def __binomial_distrib [link math_toolkit.dist_ref.dists.binomial_dist Binomial Distribution]] +[def __cauchy_distrib [link math_toolkit.dist_ref.dists.cauchy_dist Cauchy Distribution]] +[def __chi_squared_distrib [link math_toolkit.dist_ref.dists.chi_squared_dist Chi Squared Distribution]] +[def __extreme_distrib [link math_toolkit.dist_ref.dists.extreme_dist Extreme Distributions]] +[def __exp_distrib [link math_toolkit.dist_ref.dists.exp_dist Exponential Distribution]] +[def __F_distrib [link math_toolkit.dist_ref.dists.f_dist Fisher F Distribution]] +[def __gamma_distrib [link math_toolkit.dist_ref.dists.gamma_dist Gamma Distribution]] +[def __geometric_distrib [link math_toolkit.dist_ref.dists.geometric_dist Geometric Distribution]] +[def __hypergeometric_distrib [link math_toolkit.dist_ref.dists.hypergeometric_dist hypergeometric Distribution]] +[def __inverse_gamma_distrib [link math_toolkit.dist_ref.dists.inverse_gamma_dist Inverse Gamma Distribution]] +[def __inverse_gaussian_distrib [link math_toolkit.dist_ref.dists.inverse_gaussian_dist Inverse Gaussian Distribution]] +[def __inverse_chi_squared_distrib [link math_toolkit.dist_ref.dists.inverse_chi_squared_dist Inverse chi squared Distribution]] +[def __laplace_distrib [link math_toolkit.dist_ref.dists.laplace_dist Laplace Distribution]] +[def __logistic_distrib [link math_toolkit.dist_ref.dists.logistic_dist Logistic Distribution]] +[def __lognormal_distrib [link math_toolkit.dist_ref.dists.lognormal_dist Log-normal Distribution]] +[def __negative_binomial_distrib [link math_toolkit.dist_ref.dists.negative_binomial_dist Negative Binomial Distribution]] +[def __non_central_chi_squared_distrib [link math_toolkit.dist_ref.dists.nc_chi_squared_dist Noncentral Chi Squared Distribution]] +[def __non_central_beta_distrib [link math_toolkit.dist_ref.dists.nc_beta_dist Noncentral Beta Distribution]] +[def __non_central_F_distrib [link math_toolkit.dist_ref.dists.nc_f_dist Noncentral F Distribution]] +[def __non_central_T_distrib [link math_toolkit.dist_ref.dists.nc_t_dist Noncentral T Distribution]] +[def __non_central_t_distrib [link math_toolkit.dist_ref.dists.nc_t_dist noncentral T distribution]] +[def __normal_distrib [link math_toolkit.dist_ref.dists.normal_dist Normal Distribution]] +[def __poisson_distrib [link math_toolkit.dist_ref.dists.poisson_dist Poisson Distribution]] +[def __pareto_distrib [link math_toolkit.dist_ref.dists.pareto_dist Pareto Distribution]] +[def __students_t_distrib [link math_toolkit.dist_ref.dists.students_t_dist Students t Distribution]] +[def __skew_normal_distrib [link math_toolkit.dist_ref.dists.skew_normal_dist Skew Normal Distribution]] +[def __weibull_distrib [link math_toolkit.dist_ref.dists.weibull_dist 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 to.] -[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 __promotion_policy [link math_toolkit.policy.pol_ref.internal_promotion internal promotion policy]] -[def __precision_policy [link math_toolkit.policy.pol_ref.precision_pol precision policy]] -[def __policy_macros [link math_toolkit.policy.pol_ref.policy_defaults Using Macros to Change the Policy Defaults]] +[def __Policy [link policy Policy]] [/ Used in distribution template specifications] +[def __policy_section [link policy Policies]] [/ Used in text to refer to.] +[def __policy_class [link math_toolkit.pol_ref.pol_ref_ref policies::policy<>]] +[def __math_undefined [link math_toolkit.pol_ref.assert_undefined mathematically undefined function]] +[def __policy_ref [link math_toolkit.pol_ref policy reference]] +[def __math_discrete [link math_toolkit.pol_ref.discrete_quant_ref discrete functions]] +[def __error_policy [link math_toolkit.pol_ref.error_handling_policies error handling policies]] +[def __changing_policy_defaults [link math_toolkit.pol_ref.policy_defaults changing policies defaults]] +[def __user_error_handling [link math_toolkit.pol_tutorial.user_def_err_pol user error handling]] +[def __promotion_policy [link math_toolkit.pol_ref.internal_promotion internal promotion policy]] +[def __precision_policy [link math_toolkit.pol_ref.precision_pol precision policy]] +[def __policy_macros [link math_toolkit.pol_ref.policy_defaults Using Macros to Change the Policy Defaults]] [def __random_variate [@http://en.wikipedia.org/wiki/Random_variate random variate]] [def __random_variable [@http://en.wikipedia.org/wiki/Random_variable random variable]] -[def __probability_distribution [@[@http://en.wikipedia.org/wiki/Probability_distribution probability_distribution]] +[def __probability_distribution [@http://en.wikipedia.org/wiki/Probability_distribution probability_distribution]] [def __usual_accessors __cdf, __pdf, __quantile, __hazard, __chf, __mean, __median, __mode, __variance, __sd, __skewness, @@ -324,9 +329,9 @@ and use the function's name as the link text.] [def __TOMS748 [@http://portal.acm.org/citation.cfm?id=210111 TOMS Algorithm 748: enclosing zeros of continuous functions]] [def __TOMS910 [@http://portal.acm.org/citation.cfm?id=1916469 TOMS Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations]] [def __why_complements [link why_complements why complements?]] -[def __complements [link complements complements]] -[def __performance [link math_toolkit.perf performance]] -[def __building [link math_toolkit.main_overview.building building libraries]] +[def __complements [link math_toolkit.stat_tut.overview.complements complements]] +[def __performance [link perf performance]] +[def __building [link math_toolkit.building building libraries]] [def __e_float [@http://calgo.acm.org/910.zip e_float (TOMS Algorithm 910)]] [def __Abramowitz_Stegun M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964)] [def _DMLF [@http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]] @@ -344,7 +349,7 @@ and use the function's name as the link text.] 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].] +[link policy policy documentation for more details].] [template discrete_quantile_warning[NAME] [caution @@ -364,23 +369,25 @@ 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 +[link math_toolkit.pol_overview Policies]. It is strongly recommended that you read the tutorial -[link math_toolkit.policy.pol_tutorial.understand_dis_quant +[link math_toolkit.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] +[link math_toolkit.pol_ref.discrete_quant_ref reference docs] describe how to change the rounding policy for these distributions. ] [/ caution] ] [/ template discrete_quantile_warning] +[/ This manual is also available in [@http://sourceforge.net/projects/boost/files/boost-docs/ printer friendly PDF format], and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. +] -[section:main_overview Overview] +[mathpart overview Overview] [include overview/overview.qbk] [include overview/structure.qbk] [/getting about, directory and file structure.] [include overview/result_type_calc.qbk] @@ -409,14 +416,14 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [include overview/faq.qbk] [include overview/contact_info.qbk] -[endsect] [/section:main_overview Overview] +[endmathpart] [/section:main_overview Overview] -[section:dist Statistical Distributions and Functions] +[mathpart dist Statistical Distributions and Functions] [include distributions/dist_tutorial.qbk] [include distributions/dist_reference.qbk] [/includes all individual distribution.qbk files] -[endsect] [/section:dist Statistical Distributions and Functions] +[endmathpart] [/section:dist Statistical Distributions and Functions] -[section:special Special Functions] +[mathpart special Special Functions] [section:sf_gamma Gamma Functions] [include sf/tgamma.qbk] @@ -482,23 +489,23 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [include sf/owens_t.qbk] -[endsect] [/section:special Special Functions] +[endmathpart] [/section:special Special Functions] -[section:utils Floating Point Utilities] +[mathpart utils Floating Point Utilities] [include fp_utilities/rounding_func.qbk] [include fp_utilities/fpclassify.qbk] [include fp_utilities/sign.qbk] [include fp_utilities/fp_facets.qbk] [include fp_utilities/float_next.qbk] -[endsect] +[endmathpart] -[section:extern_c TR1 and C99 external "C" Functions] +[mathpart extern_c TR1 and C99 external "C" Functions] [section:tr1 C99 and TR1 C Functions Overview] [tr1_overview] [endsect] [include tr1/c99_ref.qbk] [include tr1/tr1_ref.qbk] -[endsect] +[endmathpart] [include constants/constants.qbk] @@ -507,7 +514,7 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [include octonion/math-octonion.qbk] [include gcd/math-gcd.qbk] -[section:toolkit Internals and Internal Details] +[mathpart toolkit Internals and Internal Details] [include internals/internals_overview.qbk] @@ -528,25 +535,25 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [include internals/test_data.qbk] [endsect] [/section:internals2 Testing and Development] -[endsect] [/section:toolkit Toolkit] +[endmathpart] [/section:toolkit Toolkit] -[section:using_udt Use with User-Defined Floating-Point Types] +[mathpart using_udt Use with User-Defined Floating-Point Types] [include concepts/concepts.qbk] -[endsect] [/section:using_udt Use with User Defined Floating-Point Types] +[endmathpart] [/section:using_udt Use with User Defined Floating-Point Types] [include policies/policy.qbk] [include performance/performance.qbk] -[section:backgrounders Backgrounders] +[mathpart backgrounders Backgrounders] [include background/implementation.qbk] [include background/error.qbk] [/relative error NOT handling] [include background/lanczos.qbk] [include background/remez.qbk] [include background/references.qbk] -[endsect] [/section:backgrounders Backgrounders] +[endmathpart] [/section:backgrounders Backgrounders] -[section:status Library Status] +[mathpart status Library Status] [section:history1 History and What's New] [history] [endsect] @@ -555,9 +562,9 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [/include test_HTML4_symbols.qbk] [/include test_Latin1_symbols.qbk] -[endsect] [/section:status Status and Roadmap] +[endmathpart] [/section:status Status and Roadmap] -[section:indexes Indexes] +[mathpart indexes Indexes] [named_index function_name Function Index] [named_index class_name Class Index] @@ -565,7 +572,7 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [named_index macro_name Macro Index] [index] -[endsect] +[endmathpart] [/ math.qbk Copyright 2008, 2010, 2012 John Maddock and Paul A. Bristow. diff --git a/doc/octonion/math-octonion.qbk b/doc/octonion/math-octonion.qbk index 817f73226..955306677 100644 --- a/doc/octonion/math-octonion.qbk +++ b/doc/octonion/math-octonion.qbk @@ -11,7 +11,7 @@ [def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]] [def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]] -[section:octonions Octonions] +[mathpart octonions Octonions] [section:overview Overview] @@ -55,96 +55,96 @@ The interface and implementation are both supplied by the header file namespace boost{ namespace math{ - template class ``[link math_toolkit.octonions.octonion octonion]``; - template<> class ``[link math_toolkit.octonions.oct_specialization octonion]``; - template<> class ``[link math_toolkit.octonion_double octonion]``; - template<> class ``[link math_toolkit.octonion_long_double octonion]``; + template class ``[link math_toolkit.octonion octonion]``; + template<> class ``[link math_toolkit.oct_specialization octonion]``; + template<> class ``[link math_octonion_double octonion]``; + template<> class ``[link math_octonion_long_double octonion]``; // operators - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, T const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_addition_operators operator +]`` (octonion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (T const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (octonion const & lhs, T const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_addition_operators operator +]`` (octonion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, T const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (T const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (octonion const & lhs, T const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_subtraction_operators operator -]`` (octonion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, T const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (T const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (octonion const & lhs, T const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_multiplication_operators operator *]`` (octonion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (T const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, T const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::std::complex const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.binary_division_operators operator /]`` (octonion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (T const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (octonion const & lhs, T const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (::std::complex const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (octonion const & lhs, ::std::complex const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template octonion ``[link math_toolkit.non_mem0.binary_division_operators operator /]`` (octonion const & lhs, octonion const & rhs); - template octonion ``[link math_toolkit.octonions.non_mem.unary_plus_and_minus_operators operator +]`` (octonion const & o); - template octonion ``[link math_toolkit.octonions.non_mem.unary_plus_and_minus_operators operator -]`` (octonion const & o); + template octonion ``[link math_toolkit.non_mem0.unary_plus_and_minus_operators operator +]`` (octonion const & o); + template octonion ``[link math_toolkit.non_mem0.unary_plus_and_minus_operators operator -]`` (octonion const & o); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, T const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (::std::complex const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::std::complex const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (T const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (octonion const & lhs, T const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (::std::complex const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (octonion const & lhs, ::std::complex const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_equality_operators operator ==]`` (octonion const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, T const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (::std::complex const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::std::complex const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion const & lhs, octonion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); - template bool ``[link math_toolkit.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (T const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (octonion const & lhs, T const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (::std::complex const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (octonion const & lhs, ::std::complex const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (::boost::math::quaternion const & lhs, octonion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); + template bool ``[link math_toolkit.non_mem0.binary_inequality_operators operator !=]`` (octonion const & lhs, octonion const & rhs); template - ::std::basic_istream & ``[link math_toolkit.octonions.non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, octonion & o); + ::std::basic_istream & ``[link math_toolkit.non_mem0.stream_extractor operator >>]`` (::std::basic_istream & is, octonion & o); template - ::std::basic_ostream & ``[link math_toolkit.octonions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, octonion const & o); + ::std::basic_ostream & ``[link math_toolkit.non_mem0.stream_inserter operator <<]`` (::std::basic_ostream & os, octonion const & o); // values - template T ``[link math_toolkit.octonions.oct_value_ops.real_and_unreal real]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_value_ops.real_and_unreal unreal]``(octonion const & o); + template T ``[link math_toolkit.oct_value_ops.real_and_unreal real]``(octonion const & o); + template octonion ``[link math_toolkit.oct_value_ops.real_and_unreal unreal]``(octonion const & o); - template T ``[link math_toolkit.octonions.oct_value_ops.sup sup]``(octonion const & o); - template T ``[link math_toolkit.octonions.oct_value_ops.l1 l1]``(octonionconst & o); - template T ``[link math_toolkit.octonions.oct_value_ops.abs abs]``(octonion const & o); - template T ``[link math_toolkit.octonions.oct_value_ops.norm norm]``(octonionconst & o); - template octonion ``[link math_toolkit.octonions.oct_value_ops.conj conj]``(octonion const & o); + template T ``[link math_toolkit.oct_value_ops.sup sup]``(octonion const & o); + template T ``[link math_toolkit.oct_value_ops.l1 l1]``(octonionconst & o); + template T ``[link math_toolkit.oct_value_ops.abs abs]``(octonion const & o); + template T ``[link math_toolkit.oct_value_ops.norm norm]``(octonionconst & o); + template octonion ``[link math_toolkit.oct_value_ops.conj conj]``(octonion const & o); - template octonion ``[link math_toolkit.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 math_toolkit.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 math_toolkit.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); + template octonion ``[link math_toolkit.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 math_toolkit.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 math_toolkit.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 math_toolkit.octonions.oct_trans.exp exp]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.cos cos]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.sin sin]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.tan tan]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.cosh cosh]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.sinh sinh]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.tanh tanh]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.exp exp]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.cos cos]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.sin sin]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.tan tan]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.cosh cosh]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.sinh sinh]``(octonion const & o); + template octonion ``[link math_toolkit.oct_trans.tanh tanh]``(octonion const & o); - template octonion ``[link math_toolkit.octonions.oct_trans.pow pow]``(octonion const & o, int n); + template octonion ``[link math_toolkit.oct_trans.pow pow]``(octonion const & o, int n); } } // namespaces @@ -160,62 +160,62 @@ The interface and implementation are both supplied by the header file public: typedef T value_type; - explicit ``[link math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit ``[link math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); template - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); - T ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts real]``() const; - octonion ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts unreal]``() const; + T ``[link math_toolkit.mem_fun0.real_and_unreal_parts real]``() const; + octonion ``[link math_toolkit.mem_fun0.real_and_unreal_parts unreal]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_1]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_2]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_3]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_4]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_5]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_6]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_7]``() const; - T ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_8]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_1]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_2]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_3]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_4]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_5]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_6]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_7]``() const; + T ``[link math_toolkit.mem_fun0.individual_real_components R_component_8]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_1]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_2]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_3]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_2]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_3]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_4]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_2]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_2]``() const; - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); template - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (T const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (T const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (T const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (T const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (T const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (T const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (T const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (T const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (T const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (T const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (octonion const & rhs); }; } } // namespaces @@ -232,65 +232,65 @@ The interface and implementation are both supplied by the header file public: typedef float value_type; - explicit ``[link math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); - float ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts real]``() const; - octonion ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts unreal]``() const; + float ``[link math_toolkit.mem_fun0.real_and_unreal_parts real]``() const; + octonion ``[link math_toolkit.mem_fun0.real_and_unreal_parts unreal]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_1]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_2]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_3]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_4]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_5]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_6]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_7]``() const; - float ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_8]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_1]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_2]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_3]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_4]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_5]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_6]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_7]``() const; + float ``[link math_toolkit.mem_fun0.individual_real_components R_component_8]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_1]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_2]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_3]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_2]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_3]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_4]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_2]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_2]``() const; - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); template - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (float const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (float const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (float const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (float const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (float const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (float const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (float const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (float const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (float const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (float const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (octonion const & rhs); }; -[#math_toolkit.octonion_double] +[#math_octonion_double] template<> class octonion @@ -298,65 +298,65 @@ The interface and implementation are both supplied by the header file public: typedef double value_type; - explicit ``[link math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); - double ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts real]``() const; - octonion ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts unreal]``() const; + double ``[link math_toolkit.mem_fun0.real_and_unreal_parts real]``() const; + octonion ``[link math_toolkit.mem_fun0.real_and_unreal_parts unreal]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_1]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_2]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_3]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_4]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_5]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_6]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_7]``() const; - double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_8]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_1]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_2]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_3]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_4]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_5]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_6]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_7]``() const; + double ``[link math_toolkit.mem_fun0.individual_real_components R_component_8]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_1]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_2]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_3]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_2]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_3]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_4]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_2]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_2]``() const; - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); template - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (double const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (double const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (octonion const & rhs); }; -[#math_toolkit.octonion_long_double] +[#math_octonion_long_double] template<> class octonion @@ -364,62 +364,62 @@ The interface and implementation are both supplied by the header file public: typedef long double value_type; - explicit ``[link math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.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 math_toolkit.octonions.mem_fun.constructors octonion]``( ::boost::math::quaternion const & q0, ::boost::math::quaternion const & z1 = ::boost::math::quaternion()); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); - explicit ``[link math_toolkit.octonions.mem_fun.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.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 math_toolkit.mem_fun0.constructors octonion]``( ::boost::math::quaternion const & q0, ::boost::math::quaternion const & z1 = ::boost::math::quaternion()); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); + explicit ``[link math_toolkit.mem_fun0.constructors octonion]``(octonion const & a_recopier); - long double ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts real]``() const; - octonion ``[link math_toolkit.octonions.mem_fun.real_and_unreal_parts unreal]``() const; + long double ``[link math_toolkit.mem_fun0.real_and_unreal_parts real]``() const; + octonion ``[link math_toolkit.mem_fun0.real_and_unreal_parts unreal]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_1]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_2]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_3]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_4]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_5]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_6]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_7]``() const; - long double ``[link math_toolkit.octonions.mem_fun.individual_real_components R_component_8]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_1]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_2]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_3]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_4]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_5]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_6]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_7]``() const; + long double ``[link math_toolkit.mem_fun0.individual_real_components R_component_8]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_1]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_2]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_3]``() const; - ::std::complex ``[link math_toolkit.octonions.mem_fun.individual_complex_components C_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_2]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_3]``() const; + ::std::complex ``[link math_toolkit.mem_fun0.individual_complex_components C_component_4]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_1]``() const; - ::boost::math::quaternion ``[link math_toolkit.octonions.mem_fun.individual_quaternion_components H_component_2]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_1]``() const; + ::boost::math::quaternion ``[link math_toolkit.mem_fun0.individual_quaternion_components H_component_2]``() const; - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); template - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (long double const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (octonion const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (long double const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::std::complex const & a_affecter); + octonion & ``[link math_toolkit.mem_fun0.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (long double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (long double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator +=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator +=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (long double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (long double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator -=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator -=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (long double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (long double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator *=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator *=]`` (octonion const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (long double const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (long double const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::std::complex const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template - octonion & ``[link math_toolkit.octonions.mem_fun.other_member_operators operator /=]`` (octonion const & rhs); + octonion & ``[link math_toolkit.mem_fun0.other_member_operators operator /=]`` (octonion const & rhs); }; } } // namespaces @@ -979,4 +979,4 @@ History section. Thank you to all who contributed to the discussion about this l [endsect] -[endsect] +[endmathpart] diff --git a/doc/overview/building.qbk b/doc/overview/building.qbk index f2c30f161..05618d36a 100644 --- a/doc/overview/building.qbk +++ b/doc/overview/building.qbk @@ -10,7 +10,7 @@ For most simple uses, including a header (or few) is best for compile time and program size. Refer to -[link math_toolkit.extern_c C99 and C++ TR1 C-style Functions] +[link extern_c C99 and C++ TR1 C-style Functions] for pros and cons of using the TR1 components as opposed to the header only ones. diff --git a/doc/overview/common_overviews.qbk b/doc/overview/common_overviews.qbk index 0fba60765..3d0483d35 100644 --- a/doc/overview/common_overviews.qbk +++ b/doc/overview/common_overviews.qbk @@ -4,11 +4,11 @@ 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]. +available in the [link math_toolkit.pol_tutorial policy tutorial] +and the [link math_toolkit.pol_ref policy reference]. Generally speaking, unless you find that the -[link math_toolkit.policy.pol_tutorial.policy_tut_defaults +[link math_toolkit.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. @@ -22,13 +22,13 @@ 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], +* [link math_toolkit.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] +* How [link math_toolkit.pol_ref.internal_promotion accuracy is controlled by internal promotion] to use more precise types. +* What working [link math_toolkit.pol_ref.precision_pol precision] should be used to calculate results. +* What to do when a [link math_toolkit.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], +* Whether [link math_toolkit.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 @@ -36,16 +36,16 @@ Using policies you can control: You can control policies: -* Using [link math_toolkit.policy.pol_ref.policy_defaults macros] to +* Using [link math_toolkit.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 +* At your chosen [link math_toolkit.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 +* In an ad-hoc manner [link math_toolkit.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 +[link math_toolkit.pol_tutorial.ad_hoc_dist_policies statistical distribution]. ] @@ -70,12 +70,12 @@ 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 +and "debug" builds can easily be a [link math_toolkit.getting_best factor of 20]. +# Pick your compiler carefully: [link math_toolkit.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 +The [link 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. diff --git a/doc/overview/config_macros.qbk b/doc/overview/config_macros.qbk index 8cecdb0b6..22c933bd5 100644 --- a/doc/overview/config_macros.qbk +++ b/doc/overview/config_macros.qbk @@ -49,10 +49,10 @@ Designed primarily for internal use and development.]] [table Boost.Math Tuning [[ Macros for Tuning performance options for specific compilers] [Notes]] -[[`BOOST_MATH_POLY_METHOD`] [See the [link math_toolkit.perf.tuning performance tuning section].]] -[[`BOOST_MATH_RATIONAL_METHOD`] [See the [link math_toolkit.perf.tuning performance tuning section].]] -[[`BOOST_MATH_MAX_POLY_ORDER`] [See the [link math_toolkit.perf.tuning performance tuning section].]] -[[`BOOST_MATH_INT_TABLE_TYPE`] [See the [link math_toolkit.perf.tuning performance tuning section].]] +[[`BOOST_MATH_POLY_METHOD`] [See the [link math_toolkit.tuning performance tuning section].]] +[[`BOOST_MATH_RATIONAL_METHOD`] [See the [link math_toolkit.tuning performance tuning section].]] +[[`BOOST_MATH_MAX_POLY_ORDER`] [See the [link math_toolkit.tuning performance tuning section].]] +[[`BOOST_MATH_INT_TABLE_TYPE`] [See the [link math_toolkit.tuning performance tuning section].]] [[`BOOST_MATH_INT_VALUE_SUFFIX`] [Helper macro for appending the correct suffix to integer constants which may actually be stored as reals depending on the value of BOOST_MATH_INT_TABLE_TYPE.]] ] [/table] diff --git a/doc/overview/error_handling.qbk b/doc/overview/error_handling.qbk index b3001063e..4ad2147f5 100644 --- a/doc/overview/error_handling.qbk +++ b/doc/overview/error_handling.qbk @@ -53,7 +53,7 @@ The available actions are: 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].]] + [link math_toolkit.pol_tutorial.user_def_err_pol user-supplied error handler].]] ] The following tables show all the permutations of errors and actions, @@ -65,7 +65,7 @@ with the *default action for each error shown in bold*: [[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 + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -75,7 +75,7 @@ with the *default action for each error shown in bold*: [[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 + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -85,7 +85,7 @@ with the *default action for each error shown in bold*: [[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 + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -95,7 +95,7 @@ with the *default action for each error shown in bold*: [[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 + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -105,7 +105,7 @@ with the *default action for each error shown in bold*: [[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 + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -117,7 +117,7 @@ with the *default action for each error shown in bold*: [[ignore_error][[*Returns the largest representable value of the target integer type (or the most negative value if the argument to the function was less than zero).]]] [[user_error][Returns the result of `boost::math::policies::user_rounding_error`: - [link math_toolkit.policy.pol_tutorial.user_def_err_pol + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -127,7 +127,7 @@ with the *default action for each error shown in bold*: [[errno_on_error][Sets `::errno` to `EDOM` and returns the closest approximation found.]] [[ignore_error][Returns the closest approximation found.]] [[user_error][Returns the result of `boost::math::policies::user_evaluation_error`: - [link math_toolkit.policy.pol_tutorial.user_def_err_pol + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -137,7 +137,7 @@ with the *default action for each error shown in bold*: [[errno_on_error][Sets `::errno` to `EDOM` and returns the same value as `ignore_error`.]] [[ignore_error][[*Returns a default result that depends on the function where the error occurred.]]] [[user_error][Returns the result of `boost::math::policies::user_indeterminate_result_error`: - [link math_toolkit.policy.pol_tutorial.user_def_err_pol + [link math_toolkit.pol_tutorial.user_def_err_pol this function must be defined by the user].]] ] @@ -162,21 +162,21 @@ only the first detected will be reported in the throw message. [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 +[link math_toolkit.pol_ref.policy_defaults +change the policy defaults]. See also the [link policy policy section]. An example is at the Policy tutorial in -[link math_toolkit.policy.pol_tutorial.changing_policy_defaults +[link math_toolkit.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]. +is [link math_toolkit.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] +[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 @@ -224,7 +224,7 @@ 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] +[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 @@ -251,7 +251,7 @@ 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] +[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: @@ -270,7 +270,7 @@ 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] +[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: @@ -287,7 +287,7 @@ 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] +[heading:denorm_error Denormalisation Errors] If the result of a special function is a denormalised value /z/ then the function returns the result of: @@ -304,7 +304,7 @@ 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] +[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: @@ -329,7 +329,7 @@ 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 [#indeterminate_result_error]Indeterminate Result Errors] +[heading:indeterminate_result_error Indeterminate Result Errors] When the result of a special function is indeterminate for the value that was passed to it, then the function returns the result of: @@ -350,7 +350,7 @@ undefined or indeterminate, but there is none the less a convention for what the result should be: for example the C99 standard specifies that the result of 0[super 0] is 1, even though the result is actually mathematically indeterminate. -[heading [#rounding_error]Rounding Errors] +[heading:rounding_error Rounding Errors] When one of the rounding functions __round, __trunc or __modf is called with an argument that has no integer representation, or @@ -376,7 +376,7 @@ 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] +[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 @@ -389,8 +389,8 @@ the result to the narrower result type. The function: 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]. +listed above on [link math_toolkit.error_handling.overflow_error overflow], +[link math_toolkit.error_handling.underflow_error underflow] or [link math_toolkit.error_handling.denorm_error denormalisation]. [endsect][/section:error_handling Error Handling] diff --git a/doc/overview/overview.qbk b/doc/overview/overview.qbk index a52dceb08..aff683415 100644 --- a/doc/overview/overview.qbk +++ b/doc/overview/overview.qbk @@ -5,29 +5,29 @@ This library is divided into three interconnected parts: [h4 Statistical Distributions] Provides a reasonably comprehensive set of -[link math_toolkit.dist statistical distributions], +[link 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]) +(like [link math_toolkit.dist_ref.dists.normal_dist normal] +& [link math_toolkit.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]) +(like [link math_toolkit.dist_ref.dists.binomial_dist binomial] +& [link math_toolkit.dist_ref.dists.poisson_dist Poisson]) distributions are provided. -A [link math_toolkit.dist.stat_tut comprehensive tutorial is provided], +A [link math_toolkit.stat_tut comprehensive tutorial is provided], along with a series of -[link math_toolkit.dist.stat_tut.weg worked examples] illustrating +[link math_toolkit.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], +[link special special functions], initially these were concentrated on functions used in statistical applications along with those in the [tr1]. @@ -49,7 +49,7 @@ sizes: typically `float`, `double` or `long double`. [h4 Implementation Toolkit] -Provides [link math_toolkit.toolkit many of the tools] required to implement +Provides [link 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 @@ -57,27 +57,27 @@ 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 +[link math_toolkit.internals1.series_evaluation evaluation of infinite series], -[link math_toolkit.toolkit.internals1.cf continued -fractions] and [link math_toolkit.toolkit.internals1.rational +[link math_toolkit.internals1.cf continued +fractions] and [link math_toolkit.internals1.rational rational approximations]. There is a fairly comprehensive set of root finding and -[link math_toolkit.toolkit.internals1.minima function minimisation +[link math_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. +[link math_toolkit.internals1.roots with] and +[link math_toolkit.internals1.roots2 without] derivative support. -A [link math_toolkit.toolkit.internals2.minimax +A [link math_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 +[link math_toolkit.internals2.polynomials manipulation of polynomials], for +[link math_toolkit.internals2.error_test testing a special function against tabulated test data], and for -the [link math_toolkit.toolkit.internals2.test_data +the [link math_toolkit.internals2.test_data rapid generation of test data] and/or data for output to an external graphing application. diff --git a/doc/overview/result_type_calc.qbk b/doc/overview/result_type_calc.qbk index c16250c5b..ff3b607fd 100644 --- a/doc/overview/result_type_calc.qbk +++ b/doc/overview/result_type_calc.qbk @@ -68,7 +68,7 @@ And for user-defined types, all of the following return an `NTL::RR` result: 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.high_precision.use_ntl patched NTL library]. +a [link math_toolkit.high_precision.use_ntl patched NTL library]. These rules are chosen to be compatible with the behaviour of ['ISO/IEC 9899:1999 Programming languages - C] diff --git a/doc/overview/roadmap.qbk b/doc/overview/roadmap.qbk index 7b37d9da8..a27423928 100644 --- a/doc/overview/roadmap.qbk +++ b/doc/overview/roadmap.qbk @@ -39,7 +39,7 @@ and [@https://svn.boost.org/trac/boost/ticket/7416 #7416], [@https://svn.boost.o [h4 Boost-1.52] * Corrected moments for small degrees of freedom [@https://svn.boost.org/trac/boost/ticket/7177 #7177] (reported by Thomas Mang). -* Added [link math_toolkit.special.airy Airy functions] and [link math_toolkit.special.jacobi Jacobi Elliptic functions]. +* Added [link math_toolkit.airy Airy functions] and [link math_toolkit.jacobi Jacobi Elliptic functions]. * Corrected failure to detect bad parameters in many distributions [@https://svn.boost.org/trac/boost/ticket/6934 #6934] (reported by Florian Schoppmann) by adding a function check_out_of_range to test many possible bad parameters. diff --git a/doc/overview/tr1.qbk b/doc/overview/tr1.qbk index 9f1ef8b41..3a5d83993 100644 --- a/doc/overview/tr1.qbk +++ b/doc/overview/tr1.qbk @@ -93,13 +93,13 @@ no templates being instantiated any more), and also speed up debugging runtimes - since the externally compiled libraries can be compiler optimised, rather than built using full settings - the difference in performance between -[link math_toolkit.perf.getting_best +[link math_toolkit.getting_best release and debug builds can be as much as 20 times], so for complex applications this can be a big win. [h4 Supported C99 Functions] -See also the [link math_toolkit.extern_c.c99 +See also the [link math_toolkit.c99 quick reference guide for these functions]. namespace boost{ namespace math{ namespace tr1{ extern "C"{ @@ -191,7 +191,7 @@ quick reference guide for these functions]. [h4 Supported TR1 Functions] -See also the [link math_toolkit.extern_c.tr1 +See also the [link math_toolkit.tr1 quick reference guide for these functions]. namespace boost{ namespace math{ namespace tr1{ extern "C"{ diff --git a/doc/performance/performance.qbk b/doc/performance/performance.qbk index f434068c8..3607f00f0 100644 --- a/doc/performance/performance.qbk +++ b/doc/performance/performance.qbk @@ -1,7 +1,7 @@ [template perf[name value] [value]] [template para[text] ''''''[text]''''''] -[section:perf Performance] +[mathpart perf Performance] [section:perf_over Performance Overview] [performance_overview] @@ -202,18 +202,18 @@ suited to highly vectorisable processors in the future. ] 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` +[link 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 +[link math_toolkit.pol_ref.policy_defaults defining the macro `BOOST_MATH_PROMOTE_DOUBLE_POLICY`] to `false`, or -[link math_toolkit.policy.pol_ref.internal_promotion +[link math_toolkit.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]. +[link math_toolkit.pol_tutorial policy tutorial]. [table Performance Comparison with and Without Internal Promotion to long double [[Function] @@ -432,7 +432,7 @@ To run this application yourself, build the all the .cpp files in 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]. +[link math_toolkit.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: @@ -441,7 +441,7 @@ __policy_section you want to test set as the default. [endsect] -[endsect] +[endmathpart] [/ Copyright 2006 John Maddock and Paul A. Bristow. @@ -451,15 +451,3 @@ __policy_section you want to test set as the default. ] - - - - - - - - - - - - diff --git a/doc/policies/policy.qbk b/doc/policies/policy.qbk index ae6aafcb4..78e0a2267 100644 --- a/doc/policies/policy.qbk +++ b/doc/policies/policy.qbk @@ -1,4 +1,6 @@ -[section:policy Policies] +[def foo______ bar] + +[mathpart policy Policies] [section:pol_overview Policy Overview] [policy_overview] @@ -123,7 +125,7 @@ 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 +the [link math_toolkit.pol_tutorial.user_def_err_pol tutorial here]. [h4 Kinds of Error Raised] @@ -884,9 +886,9 @@ actually use. [endsect][/section:pol_ref_ref Policy Class Reference] [endsect][/section:pol_ref Policy Reference] -[endsect][/section:policy Policies] +[endmathpart][/section:policy Policies] -[/ policy.qbk +[/ qbk Copyright 2007, 2010 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 diff --git a/doc/policies/policy_tutorial.qbk b/doc/policies/policy_tutorial.qbk index f2b231cd8..4d5c3343a 100644 --- a/doc/policies/policy_tutorial.qbk +++ b/doc/policies/policy_tutorial.qbk @@ -65,12 +65,12 @@ if these work OK for you then you can stop reading now!] 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 + [link math_toolkit.pol_overview Policies]. It is strongly recommended that you read the tutorial - [link math_toolkit.policy.pol_tutorial.understand_dis_quant + [link math_toolkit.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] + [link math_toolkit.pol_ref.discrete_quant_ref reference docs] describe how to change the rounding policy for these distributions. ]] diff --git a/doc/quaternion/math-quaternion.qbk b/doc/quaternion/math-quaternion.qbk index 6f886665a..8f89056ac 100644 --- a/doc/quaternion/math-quaternion.qbk +++ b/doc/quaternion/math-quaternion.qbk @@ -9,7 +9,7 @@ [def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] [def __not_equal ['[^xy '''≠''' yx]]] -[section:quaternions Quaternions] +[mathpart quaternions Quaternions] [section:overview Overview] @@ -61,82 +61,82 @@ The interface and implementation are both supplied by the header file namespace boost{ namespace math{ - template class ``[link math_toolkit.quaternions.quat quaternion]``; - template<> class ``[link math_toolkit.quaternions.spec quaternion]``; - template<> class ``[link math_toolkit.quaternion_double quaternion]``; - template<> class ``[link math_toolkit.quaternion_long_double quaternion]``; + template class ``[link math_toolkit.quat quaternion]``; + template<> class ``[link math_toolkit.spec quaternion]``; + template<> class ``[link math_quaternion_double quaternion]``; + template<> class ``[link math_quaternion_long_double quaternion]``; // operators - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_addition_operators operator +]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_addition_operators operator +]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_division_operators operator /]`` (T const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, T const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, ::std::complex const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_division_operators operator /]`` (T const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, T const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, quaternion const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, ::std::complex const & rhs); + template quaternion ``[link math_toolkit.non_mem.binary_division_operators operator /]`` (quaternion const & lhs, quaternion const & rhs); - template quaternion ``[link math_toolkit.quaternions.non_mem.unary_plus operator +]`` (quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.non_mem.unary_minus operator -]`` (quaternion const & q); + template quaternion ``[link math_toolkit.non_mem.unary_plus operator +]`` (quaternion const & q); + template quaternion ``[link math_toolkit.non_mem.unary_minus operator -]`` (quaternion const & q); - template bool ``[link math_toolkit.quaternions.non_mem.equality_operators operator ==]`` (T const & lhs, quaternion const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, T const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.equality_operators operator ==]`` (::std::complex const & lhs, quaternion const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, ::std::complex const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.equality_operators operator ==]`` (quaternion const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.equality_operators operator ==]`` (T const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.equality_operators operator ==]`` (quaternion const & lhs, T const & rhs); + template bool ``[link math_toolkit.non_mem.equality_operators operator ==]`` (::std::complex const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.equality_operators operator ==]`` (quaternion const & lhs, ::std::complex const & rhs); + template bool ``[link math_toolkit.non_mem.equality_operators operator ==]`` (quaternion const & lhs, quaternion const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.inequality_operators operator !=]`` (T const & lhs, quaternion const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, T const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.inequality_operators operator !=]`` (::std::complex const & lhs, quaternion const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, ::std::complex const & rhs); - template bool ``[link math_toolkit.quaternions.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.inequality_operators operator !=]`` (T const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, T const & rhs); + template bool ``[link math_toolkit.non_mem.inequality_operators operator !=]`` (::std::complex const & lhs, quaternion const & rhs); + template bool ``[link math_toolkit.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, ::std::complex const & rhs); + template bool ``[link math_toolkit.non_mem.inequality_operators operator !=]`` (quaternion const & lhs, quaternion const & rhs); template - ::std::basic_istream& ``[link math_toolkit.quaternions.non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, quaternion & q); + ::std::basic_istream& ``[link math_toolkit.non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, quaternion & q); template - ::std::basic_ostream& operator ``[link math_toolkit.quaternions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, quaternion const & q); + ::std::basic_ostream& operator ``[link math_toolkit.non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, quaternion const & q); // values - template T ``[link math_toolkit.quaternions.value_op.real_and_unreal real]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.value_op.real_and_unreal unreal]``(quaternion const & q); + template T ``[link math_toolkit.value_op.real_and_unreal real]``(quaternion const & q); + template quaternion ``[link math_toolkit.value_op.real_and_unreal unreal]``(quaternion const & q); - template T ``[link math_toolkit.quaternions.value_op.sup sup]``(quaternion const & q); - template T ``[link math_toolkit.quaternions.value_op.l1 l1]``(quaternion const & q); - template T ``[link math_toolkit.quaternions.value_op.abs abs]``(quaternion const & q); - template T ``[link math_toolkit.quaternions.value_op.norm norm]``(quaternionconst & q); - template quaternion ``[link math_toolkit.quaternions.value_op.conj conj]``(quaternion const & q); + template T ``[link math_toolkit.value_op.sup sup]``(quaternion const & q); + template T ``[link math_toolkit.value_op.l1 l1]``(quaternion const & q); + template T ``[link math_toolkit.value_op.abs abs]``(quaternion const & q); + template T ``[link math_toolkit.value_op.norm norm]``(quaternionconst & q); + template quaternion ``[link math_toolkit.value_op.conj conj]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.creation_spherical spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); - template quaternion ``[link math_toolkit.quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); - template quaternion ``[link math_toolkit.quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); - template quaternion ``[link math_toolkit.quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); - template quaternion ``[link math_toolkit.quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); + template quaternion ``[link math_quaternions.creation_spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); + template quaternion ``[link math_quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); + template quaternion ``[link math_quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); + template quaternion ``[link math_quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); + template quaternion ``[link math_quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); // transcendentals - template quaternion ``[link math_toolkit.quaternions.trans.exp exp]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.cos cos]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.sin sin]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.tan tan]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.cosh cosh]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.sinh sinh]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.tanh tanh]``(quaternion const & q); - template quaternion ``[link math_toolkit.quaternions.trans.pow pow]``(quaternion const & q, int n); + template quaternion ``[link math_toolkit.trans.exp exp]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.cos cos]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.sin sin]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.tan tan]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.cosh cosh]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.sinh sinh]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.tanh tanh]``(quaternion const & q); + template quaternion ``[link math_toolkit.trans.pow pow]``(quaternion const & q, int n); } // namespace math } // namespace boost @@ -152,47 +152,47 @@ The interface and implementation are both supplied by the header file { public: - typedef T ``[link math_toolkit.quaternions.mem_typedef value_type]``; + typedef T ``[link math_toolkit.mem_typedef value_type]``; - explicit ``[link math_toolkit.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 math_toolkit.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link math_toolkit.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 math_toolkit.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); template - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - T ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts real]``() const; - quaternion ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; - T ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_1]``() const; - T ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_2]``() const; - T ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_3]``() const; - T ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_4]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_1]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + T ``[link math_toolkit.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link math_toolkit.mem_fun.real_and_unreal_parts unreal]``() const; + T ``[link math_toolkit.mem_fun.individual_real_components R_component_1]``() const; + T ``[link math_toolkit.mem_fun.individual_real_components R_component_2]``() const; + T ``[link math_toolkit.mem_fun.individual_real_components R_component_3]``() const; + T ``[link math_toolkit.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_2]``() const; - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(T const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(T const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(T const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(T const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(T const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(T const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(T const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(T const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(T const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(T const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; } // namespace math @@ -208,145 +208,145 @@ The interface and implementation are both supplied by the header file class quaternion { public: - typedef float ``[link math_toolkit.quaternions.mem_typedef value_type]``; + typedef float ``[link math_toolkit.mem_typedef value_type]``; - explicit ``[link math_toolkit.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 math_toolkit.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.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 math_toolkit.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - float ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts real]``() const; - quaternion ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; - float ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_1]``() const; - float ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_2]``() const; - float ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_3]``() const; - float ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_4]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_1]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + float ``[link math_toolkit.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link math_toolkit.mem_fun.real_and_unreal_parts unreal]``() const; + float ``[link math_toolkit.mem_fun.individual_real_components R_component_1]``() const; + float ``[link math_toolkit.mem_fun.individual_real_components R_component_2]``() const; + float ``[link math_toolkit.mem_fun.individual_real_components R_component_3]``() const; + float ``[link math_toolkit.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_2]``() const; - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(float const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(float const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(float const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(float const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(float const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(float const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(float const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(float const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(float const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(float const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; -[#math_toolkit.quaternion_double] +[#math_quaternion_double] template<> class quaternion { public: - typedef double ``[link math_toolkit.quaternions.mem_typedef value_type]``; + typedef double ``[link math_toolkit.mem_typedef value_type]``; - explicit ``[link math_toolkit.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 math_toolkit.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.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 math_toolkit.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - double ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts real]``() const; - quaternion ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; - double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_1]``() const; - double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_2]``() const; - double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_3]``() const; - double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_4]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_1]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + double ``[link math_toolkit.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link math_toolkit.mem_fun.real_and_unreal_parts unreal]``() const; + double ``[link math_toolkit.mem_fun.individual_real_components R_component_1]``() const; + double ``[link math_toolkit.mem_fun.individual_real_components R_component_2]``() const; + double ``[link math_toolkit.mem_fun.individual_real_components R_component_3]``() const; + double ``[link math_toolkit.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_2]``() const; - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(double const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(double const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; -[#math_toolkit.quaternion_long_double] +[#math_quaternion_long_double] template<> class quaternion { public: - typedef long double ``[link math_toolkit.quaternions.mem_typedef value_type]``; + typedef long double ``[link math_toolkit.mem_typedef value_type]``; - explicit ``[link math_toolkit.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 math_toolkit.quaternions.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - explicit ``[link math_toolkit.quaternions.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.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 math_toolkit.mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); + explicit ``[link math_toolkit.mem_fun.constructors quaternion]``(quaternion const & a_recopier); - long double ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts real]``() const; - quaternion ``[link math_toolkit.quaternions.mem_fun.real_and_unreal_parts unreal]``() const; - long double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_1]``() const; - long double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_2]``() const; - long double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_3]``() const; - long double ``[link math_toolkit.quaternions.mem_fun.individual_real_components R_component_4]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_1]``() const; - ::std::complex ``[link math_toolkit.quaternions.mem_fun.individual_complex__components C_component_2]``() const; + long double ``[link math_toolkit.mem_fun.real_and_unreal_parts real]``() const; + quaternion ``[link math_toolkit.mem_fun.real_and_unreal_parts unreal]``() const; + long double ``[link math_toolkit.mem_fun.individual_real_components R_component_1]``() const; + long double ``[link math_toolkit.mem_fun.individual_real_components R_component_2]``() const; + long double ``[link math_toolkit.mem_fun.individual_real_components R_component_3]``() const; + long double ``[link math_toolkit.mem_fun.individual_real_components R_component_4]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_1]``() const; + ::std::complex ``[link math_toolkit.mem_fun.individual_complex_components C_component_2]``() const; - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(long double const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(long double const & a_affecter); + quaternion& ``[link math_toolkit.mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(long double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(long double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.addition_operators operator += ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.addition_operators operator += ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(long double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(long double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(long double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(long double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(long double const & rhs); - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(long double const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template - quaternion& ``[link math_toolkit.quaternions.mem_fun.division_operators operator /= ]``(quaternion const & rhs); + quaternion& ``[link math_toolkit.mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; } // namespace math @@ -709,7 +709,7 @@ Euclidian norm (and quadratic functionals). For this version of 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. -[#math_toolkit.quaternions.creation_spherical] `spherical` is a simple transposition of `polar`, it takes as inputs +[#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 @@ -718,23 +718,23 @@ 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. -[#math_toolkit.quaternions.creation_cylindrical] `cylindrical` is likewise a simple transposition of the usual +[#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. -[#math_toolkit.quaternions.creation_multipolar] `multipolar` is yet another simple generalization of polar coordinates. +[#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. -[#math_toolkit.quaternions.creation_cylindrospherical] `cylindrospherical` is specific to quaternions. It is often interesting to +[#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. -[#math_toolkit.quaternions.creation_semipolar] `semipolar` is another generator which is specific to quaternions. +[#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 @@ -843,7 +843,7 @@ 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 math_toolkit_TEST_VERBOSE, you will get +If you define the symbol 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 @@ -903,4 +903,4 @@ Thank you to all who contributed to the discution about this library. implementation to an efficient one). [endsect] -[endsect] +[endmathpart] diff --git a/doc/sf/ellint_introduction.qbk b/doc/sf/ellint_introduction.qbk index c93dd5fec..12a375a5f 100644 --- a/doc/sf/ellint_introduction.qbk +++ b/doc/sf/ellint_introduction.qbk @@ -25,7 +25,7 @@ As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic inte All variables are real numbers unless otherwise noted. -[h4 [#ellint_def]Definition] +[h4 Definition] [equation ellint1] @@ -138,13 +138,13 @@ where ['x] is nonnegative and ['y] is nonzero. ['R[sub D](x, y, z) = R[sub J](x, y, z, z)]] -[h4 [#ellint_theorem]Duplication Theorem] +[h4 Duplication Theorem] Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that [equation ellint13] -[h4 [#ellint_formula]Carlson's Formulas] +[h4 Carlson's Formulas] The Legendre form and Carlson form of elliptic integrals are related by equations: @@ -165,7 +165,7 @@ third kind. Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] [[link provides a unified method for all three kinds of elliptic integrals with satisfactory precisions. -[h4 [#ellint_refs]References] +[h4 References] Special mention goes to: diff --git a/doc/sf/erf.qbk b/doc/sf/erf.qbk index d5f120462..1b276a98a 100644 --- a/doc/sf/erf.qbk +++ b/doc/sf/erf.qbk @@ -180,7 +180,7 @@ 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: +For `z > 0.5` we observe that over a small interval \[a, b) then: erfc(z) * exp(z*z) * z ~ c diff --git a/doc/sf/erf_inv.qbk b/doc/sf/erf_inv.qbk index 79d3ddca5..e4f0494cf 100644 --- a/doc/sf/erf_inv.qbk +++ b/doc/sf/erf_inv.qbk @@ -60,7 +60,7 @@ value x such that: 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]. +[link math_toolkit.sf_erf.error_function forward error functions]. [h4 Testing] diff --git a/doc/sf/inv_hyper.qbk b/doc/sf/inv_hyper.qbk index ae9b48a37..d5ea84417 100644 --- a/doc/sf/inv_hyper.qbk +++ b/doc/sf/inv_hyper.qbk @@ -84,7 +84,7 @@ and can be computed as [equation special_functions_blurb18]. ``__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 +[link math_toolkit.inv_hyper.inv_hyper_over the hyperbolic cosine function], at x. Values returned are positive. If x is in the range __form2 then returns the result of __domain_error. @@ -150,7 +150,7 @@ primary definition] is used: ``__sf_result`` asinh(const T x, const ``__Policy``&); Computes the reciprocal of -[link math_toolkit.special.inv_hyper.inv_hyper_over +[link math_toolkit.inv_hyper.inv_hyper_over the hyperbolic sine function]. The return type of this function is computed using the __arg_pomotion_rules: @@ -213,7 +213,7 @@ primary definition]: ``__sf_result`` atanh(const T x, const ``__Policy``&); Computes the reciprocal of -[link math_toolkit.special.inv_hyper.inv_hyper_over +[link math_toolkit.inv_hyper.inv_hyper_over the hyperbolic tangent function], at x. [optional_policy] diff --git a/doc/sf/math.qbk b/doc/sf/math.qbk deleted file mode 100644 index 9fa249b9b..000000000 --- a/doc/sf/math.qbk +++ /dev/null @@ -1,571 +0,0 @@ -[article Math Toolkit - [quickbook 1.5] - [copyright 2006, 2007, 2008, 2009, 2010, 2012, 2013 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg, Benjamin Sobotta and Christopher Kormanyos] - [/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], [Lalande, Bruno], [Råde, Johan], [Sewani, Gautam], [van den Berg, Thijs], [Sobotta, Benjamin], [Kormanyos, Christopher] ] - [/last-revision $Date$] -] - -[/ \u00E5 = Latin small letter A with ring above doesn't seem to work in authors with Quickbook 1.5] - -[template equation[name] ''' - - - - - - -'''] - -[template graph[name] ''' - - - - - - -'''] - -[include ../../../../tools/auto_index/include/auto_index_helpers.qbk] -[/ Must be first included file!] - -[include html4_symbols.qbk] -[/include math_symbols.qbk] - -[/include latin1_symbols.qbk] [/ just for testing] -[include common_overviews.qbk] [/ overviews that appear in more than one place!] -[include tr1.qbk] [/tr1 docs also appear in more than one place!] -[include roadmap.qbk] [/ for history] - -[def __effects [*Effects: ]] -[def __formula [*Formula: ]] -[def __exm1 '''ex - 1'''] [/e^x -1] -[def __ex '''ex'''] [/e^x] -[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 __spaces '''  '''] [/ two spaces - useful for an indent.] - -[def __caution This is now an official Boost library, but remains a library under - development, 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 C99[] [@http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1256.pdf C99 Standard ISO/IEC 9899:1999]] -[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 __rounding_error [link rounding_error rounding_error]] -[def __indeterminate_result_error [link indeterminate_result_error indeterminate_result_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]] -[def __root_finding_with_derivatives [link math_toolkit.toolkit.internals1.roots root-finding with derivatives]] - -[/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]] - -[/Hankel Functions] -[def __cyl_hankel_1 [link math_toolkit.special.hankel.cyl_hankel cyl_hankel_1]] -[def __cyl_hankel_2 [link math_toolkit.special.hankel.cyl_hankel cyl_hankel_2]] -[def __sph_hankel_1 [link math_toolkit.special.hankel.sph_hankel sph_hankel_1]] -[def __sph_hankel_2 [link math_toolkit.special.hankel.sph_hankel sph_hankel_2]] - -[/Airy Functions] -[def __airy_ai [link math_toolkit.special.airy.ai airy_ai]] -[def __airy_bi [link math_toolkit.special.airy.bi airy_bi]] -[def __airy_bi_prime [link math_toolkit.special.airy.bip airy_bi_prime]] -[def __airy_ai_prime [link math_toolkit.special.airy.aip airy_ai_prime]] - -[/Jacobi Elliptic Functions] -[def __jacobi_elliptic [link math_toolkit.special.jacobi.jacobi_elliptic jacobi_elliptic]] -[def __jacobi_cd [link math_toolkit.special.jacobi.jacobi_cd jacobi_cd]] -[def __jacobi_cn [link math_toolkit.special.jacobi.jacobi_cn jacobi_cn]] -[def __jacobi_cs [link math_toolkit.special.jacobi.jacobi_cs jacobi_cs]] -[def __jacobi_dc [link math_toolkit.special.jacobi.jacobi_dc jacobi_dc]] -[def __jacobi_dn [link math_toolkit.special.jacobi.jacobi_dn jacobi_dn]] -[def __jacobi_ds [link math_toolkit.special.jacobi.jacobi_ds jacobi_ds]] -[def __jacobi_nd [link math_toolkit.special.jacobi.jacobi_nd jacobi_nd]] -[def __jacobi_nc [link math_toolkit.special.jacobi.jacobi_nc jacobi_nc]] -[def __jacobi_ns [link math_toolkit.special.jacobi.jacobi_ns jacobi_ns]] -[def __jacobi_sd [link math_toolkit.special.jacobi.jacobi_sd jacobi_sd]] -[def __jacobi_sn [link math_toolkit.special.jacobi.jacobi_sn jacobi_sn]] -[def __jacobi_sc [link math_toolkit.special.jacobi.jacobi_sc jacobi_sc]] - -[/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]] -[def __fp_facets [link math_toolkit.utils.fp_facets nonfinite fp_facets]] - -[/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]] -[def __pow [link math_toolkit.special.powers.ct_pow pow]] - -[/zeta] -[def __zeta [link math_toolkit.special.zetas.zeta zeta]] - -[/expint] -[def __expint_i [link math_toolkit.special.expint.expint_i zeta]] -[def __expint_n [link math_toolkit.special.expint.expint_n zeta]] - -[/rounding] -[def __round [link math_toolkit.utils.rounding.round round]] -[def __lround [link math_toolkit.utils.rounding.round round]] -[def __llround [link math_toolkit.utils.rounding.round round]] -[def __trunc [link math_toolkit.utils.rounding.trunc trunc]] -[def __modf [link math_toolkit.utils.rounding.modf modf]] - -[/polynomials] -[def __laguerre [link math_toolkit.special.sf_poly.laguerre laguerre]] -[def __legendre [link math_toolkit.special.sf_poly.legendre legendre_p]] -[def __hermite [link math_toolkit.special.sf_poly.hermite hermite]] - -[/Misc] -[def __expint [link math_toolkit.special.expint.expint_i expint]] -[def __spherical_harmonic [link math_toolkit.special.sf_poly.sph_harm spherical_harmonic]] -[def __owens_t [link math_toolkit.special.owens_t Owens T]] - - -[/tools] -[def __tuple [link math_toolkit.toolkit.internals1.tuples boost::math::tuple]] - -[/ 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 __extreme_distrib [link math_toolkit.dist.dist_ref.dists.extreme_dist Extreme Distributions]] -[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 __geometric_distrib [link math_toolkit.dist.dist_ref.dists.geometric_dist Geometric Distribution]] -[def __hypergeometric_distrib [link math_toolkit.dist.dist_ref.dists.hypergeometric_dist hypergeometric Distribution]] -[def __inverse_gamma_distrib [link math_toolkit.dist.dist_ref.dists.inverse_gamma_dist Inverse Gamma Distribution]] -[def __inverse_gaussian_distrib [link math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist Inverse Gaussian Distribution]] -[def __inverse_chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist Inverse chi squared Distribution]] -[def __laplace_distrib [link math_toolkit.dist.dist_ref.dists.laplace_dist Laplace Distribution]] -[def __logistic_distrib [link math_toolkit.dist.dist_ref.dists.logistic_dist Logistic 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 __non_central_chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.nc_chi_squared_dist Noncentral Chi Squared Distribution]] -[def __non_central_beta_distrib [link math_toolkit.dist.dist_ref.dists.nc_beta_dist Noncentral Beta Distribution]] -[def __non_central_F_distrib [link math_toolkit.dist.dist_ref.dists.nc_f_dist Noncentral F Distribution]] -[def __non_central_T_distrib [link math_toolkit.dist.dist_ref.dists.nc_t_dist Noncentral T Distribution]] -[def __non_central_t_distrib [link math_toolkit.dist.dist_ref.dists.nc_t_dist noncentral T 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 __pareto_distrib [link math_toolkit.dist.dist_ref.dists.pareto_dist Pareto Distribution]] -[def __students_t_distrib [link math_toolkit.dist.dist_ref.dists.students_t_dist Students t Distribution]] -[def __skew_normal_distrib [link math_toolkit.dist.dist_ref.dists.skew_normal_dist Skew Normal Distribution]] -[def __weibull_distrib [link math_toolkit.dist.dist_ref.dists.weibull_dist 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 to.] -[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 __promotion_policy [link math_toolkit.policy.pol_ref.internal_promotion internal promotion policy]] -[def __precision_policy [link math_toolkit.policy.pol_ref.precision_pol precision policy]] -[def __policy_macros [link math_toolkit.policy.pol_ref.policy_defaults Using Macros to Change the Policy Defaults]] - -[def __random_variate [@http://en.wikipedia.org/wiki/Random_variate random variate]] -[def __random_variable [@http://en.wikipedia.org/wiki/Random_variable random variable]] -[def __probability_distribution [@[@http://en.wikipedia.org/wiki/Probability_distribution probability_distribution]] - -[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 __NTL [@http://www.shoup.net/ntl/ NTL A Library for doing Number Theory]] -[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 __MPFR [@http://www.mpfr.org/ GNU MPFR library]] -[def __GMP [@http://gmplib.org/ GNU Multiple Precision Arithmetic Library]] -[def __multiprecision [@http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/index.html Boost.Multiprecision]] -[def __cpp_dec_float [@http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html cpp_dec_float]] -[def __R [@http://www.r-project.org/ The R Project for Statistical Computing]] -[def __godfrey [link godfrey Godfrey]] -[def __pugh [link pugh Pugh]] -[def __NaN [@http://en.wikipedia.org/wiki/NaN NaN]] -[def __errno [@http://en.wikipedia.org/wiki/Errno `::errno`]] -[def __Mathworld [@http://mathworld.wolfram.com Wolfram MathWorld]] -[def __Mathematica [@http://www.wolfram.com/products/mathematica/index.html Wolfram Mathematica]] -[def __WolframAlpha [@http://www.wolframalpha.com/ Wolfram Alpha]] -[def __TOMS748 [@http://portal.acm.org/citation.cfm?id=210111 TOMS Algorithm 748: enclosing zeros of continuous functions]] -[def __TOMS910 [@http://portal.acm.org/citation.cfm?id=1916469 TOMS Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations]] -[def __why_complements [link why_complements why complements?]] -[def __complements [link complements complements]] -[def __performance [link math_toolkit.perf performance]] -[def __building [link math_toolkit.main_overview.building building libraries]] -[def __e_float [@http://calgo.acm.org/910.zip e_float (TOMS Algorithm 910)]] -[def __Abramowitz_Stegun M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964)] -[def _DMLF [@http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]] - -[/ 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] - -This manual is also available in -[@http://sourceforge.net/projects/boost/files/boost-docs/ -printer friendly PDF format], -and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. - -[section:main_overview Overview] -[include overview.qbk] -[include structure.qbk] [/getting about, directory and file structure.] -[include result_type_calc.qbk] -[include error_handling.qbk] - -[section:compilers_overview Compilers] -[compilers_overview] -[endsect] -[include config_macros.qbk] -[section:pol_overview Policies] -[policy_overview] -[endsect] - -[include thread_safety.qbk] - -[section:perf_over Performance] -[performance_overview] -[endsect] -[include building.qbk] -[section:history1 History and What's New] -[history] -[endsect] -[section:tr1 C99 and C++ TR1 C-style Functions] -[tr1_overview] -[endsect] -[include faq.qbk] -[include contact_info.qbk] - -[endsect] [/section:main_overview Overview] - -[section:dist Statistical Distributions and Functions] -[include dist_tutorial.qbk] -[include dist_reference.qbk] [/includes all individual distribution.qbk files] -[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] - -[/Hankel functions] -[include hankel.qbk] - -[/Airy Functions] -[include airy.qbk] - -[section:ellint Elliptic Integrals] -[include ellint_introduction.qbk] -[include ellint_carlson.qbk] -[include ellint_legendre.qbk] -[endsect] [/section:ellint Elliptic Integrals] - -[include jacobi_elliptic.qbk] - -[section:zetas Zeta Functions] -[include zeta.qbk] -[endsect] - -[include expint.qbk] - -[include powers.qbk] -[include sinc.qbk] -[include inv_hyper.qbk] - -[include owens_t.qbk] - -[endsect] [/section:special Special Functions] - -[section:utils Floating Point Utilities] -[include rounding_func.qbk] -[include fpclassify.qbk] -[include sign.qbk] -[include fp_facets.qbk] -[include float_next.qbk] -[endsect] - -[section:extern_c TR1 and C99 external "C" Functions] -[section:tr1 C99 and TR1 C Functions Overview] -[tr1_overview] -[endsect] -[include c99_ref.qbk] -[include tr1_ref.qbk] -[endsect] - -[include constants.qbk] - -[section:toolkit Tools and Internal Details] - -[include internals_overview.qbk] - -[section:internals1 Utilities & Tools] -[include series.qbk] -[include fraction.qbk] -[include rational.qbk] -[include roots.qbk] -[include roots_without_derivatives.qbk] -[include minima.qbk] -[include tuple.qbk] -[endsect] [/section:internals1 Utilities & Tools] - -[section:internals2 Testing and Development] -[include polynomial.qbk] -[include minimax.qbk] -[include relative_error.qbk] [/ uncertainty of floating-point calculations.] -[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:backgrounders Backgrounders] - -[section:status Library Status] -[section:history1 History and What's New] -[history] -[endsect] -[include issues.qbk] -[include credits.qbk] -[/include test_HTML4_symbols.qbk] -[/include test_Latin1_symbols.qbk] - -[endsect] [/section:status Status and Roadmap] - -[named_index function_name Function Index] -[named_index class_name Class Index] -[named_index typedef_name Typedef Index] -[named_index macro_name Macro Index] -[index] - -[/ math.qbk - Copyright 2008, 2010, 2012 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/pow.qbk b/doc/sf/pow.qbk index de2c94280..e4a622c62 100644 --- a/doc/sf/pow.qbk +++ b/doc/sf/pow.qbk @@ -73,7 +73,7 @@ this operation is a division by 0 (it equals to 1/0). * In case of null base and null exponent, an __indeterminate_result_error occurs since the result of this operation is indeterminate. Those errors follow the -[link math_toolkit.main_overview.error_handling +[link math_toolkit.error_handling general policies of error handling in Boost.Math]. The default overflow error policy is `throw_on_error`. A call like `pow<-2>(0)` diff --git a/doc/sf/sinc.qbk b/doc/sf/sinc.qbk index 3c9e12e8e..bd00557b1 100644 --- a/doc/sf/sinc.qbk +++ b/doc/sf/sinc.qbk @@ -55,7 +55,7 @@ and [@http://mathworld.wolfram.com/Octonion.html octonions]. U sinc_pi(const U x, const ``__Policy``&); Computes -[link math_toolkit.special.sinc.sinc_overview +[link math_toolkit.sinc.sinc_overview the Sinus Cardinal] of x: sinc_pi(x) = sin(x) / x @@ -89,7 +89,7 @@ to ensure accuracy. U sinhc_pi(const U x, const ``__Policy``&); Computes http://mathworld.wolfram.com/SinhcFunction.html -[link math_toolkit.special.sinc.sinc_overview +[link math_toolkit.sinc.sinc_overview the Hyperbolic Sinus Cardinal] of x: sinhc_pi(x) = sinh(x) / x diff --git a/example/binomial_quiz_example.cpp b/example/binomial_quiz_example.cpp index 2fde414b4..9b48a0a6f 100644 --- a/example/binomial_quiz_example.cpp +++ b/example/binomial_quiz_example.cpp @@ -345,13 +345,13 @@ Deciles 1 to 6 //[discrete_quantile_real /*` Quantiles values are controlled by the -[link math_toolkit.policy.pol_ref.discrete_quant_ref discrete quantile policy] +[link math_toolkit.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]. +[link math_toolkit.pol_tutorial.understand_dis_quant Understanding Discrete Quantile Policy]. We could control the policy for *all* distributions by diff --git a/example/fft_sines_table.cpp b/example/fft_sines_table.cpp index 397c02658..96dd28c4b 100644 --- a/example/fft_sines_table.cpp +++ b/example/fft_sines_table.cpp @@ -188,7 +188,7 @@ Now output all the sine table, to a file of your chosen name. std::cout << "Close file " << sines_name << " for output OK." << std::endl; } -//`The output file generated can be seen at [@..\sines.hpp] +//`The output file generated can be seen at [@..\\sines.hpp] //] [/fft_sines_table_example_1] return EXIT_SUCCESS; diff --git a/example/find_location_example.cpp b/example/find_location_example.cpp index 97c5302a5..789b270a0 100644 --- a/example/find_location_example.cpp +++ b/example/find_location_example.cpp @@ -133,7 +133,7 @@ although it is not required, as the various examples below show. l = find_location(z, p, sd, policy >()); /*` If we want to use a probability that is the -[link complements complement of our probability], +[link math_toolkit.stat_tut.overview.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. */ diff --git a/example/find_mean_and_sd_normal.cpp b/example/find_mean_and_sd_normal.cpp index 9a09c7a8c..2dd6246ce 100644 --- a/example/find_mean_and_sd_normal.cpp +++ b/example/find_mean_and_sd_normal.cpp @@ -117,7 +117,7 @@ cout << "Setting the packer to " << nominal_mean << " will mean that " /*` This calculation is generalized as the free function called -[link math_toolkit.dist.dist_ref.dist_algorithms find_location]. +[link math_toolkit.dist_ref.dist_algorithms find_location]. To use this we will need to */ @@ -205,7 +205,7 @@ 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] +See [link math_toolkit.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. @@ -262,7 +262,7 @@ cout <<"Fraction of packs >= " << minimum_weight << " with a mean of " << mean 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]. +[link math_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. diff --git a/example/find_root_example.cpp b/example/find_root_example.cpp index 65b2f3b97..d318ef81c 100644 --- a/example/find_root_example.cpp +++ b/example/find_root_example.cpp @@ -132,7 +132,7 @@ Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95 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]. +[link math_toolkit.internals1.roots2 Root Finding Without Derivatives]. But in this normal distribution case, we could be even smarter and make a direct calculation. */ diff --git a/example/geometric_examples.cpp b/example/geometric_examples.cpp index b10ff1f7b..df171e957 100644 --- a/example/geometric_examples.cpp +++ b/example/geometric_examples.cpp @@ -127,7 +127,7 @@ using the inverse or quantile. /*`Note that the value returned is not an integer: if you want an integer result you should use either floor, round or ceil functions, or use the policies mechanism. -See [link math_toolkit.policy.pol_tutorial.understand_dis_quant +See [link math_toolkit.pol_tutorial.understand_dis_quant Understanding Quantiles of Discrete Distributions] The geometric distribution is related to the negative binomial diff --git a/example/negative_binomial_example1.cpp b/example/negative_binomial_example1.cpp index 928811d09..69ae91864 100644 --- a/example/negative_binomial_example1.cpp +++ b/example/negative_binomial_example1.cpp @@ -137,7 +137,7 @@ 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)] +and obtain the [link math_toolkit.dist_ref.nmp.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. diff --git a/example/normal_misc_examples.cpp b/example/normal_misc_examples.cpp index 4503847dd..f109642ca 100644 --- a/example/normal_misc_examples.cpp +++ b/example/normal_misc_examples.cpp @@ -354,7 +354,7 @@ Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95 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]. +[link math_toolkit.internals1.roots2 Root Finding Without Derivatives]. But in this normal distribution case, we could be even smarter and make a direct calculation. */