mirror of
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271 lines
12 KiB
Plaintext
271 lines
12 KiB
Plaintext
[library Math Toolkit
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[quickbook 1.3]
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[copyright 2006 John Maddock and Paul A. Bristow]
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[purpose Various Special Functions, Statistical Distributions, and Numerical Tools]
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[license
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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[@http://www.boost.org/LICENSE_1_0.txt])
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]
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[authors [Maddock, John], [Bristow, Paul A.]]
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[category math]
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[/last-revision $Date$]
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]
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[include HTML4_symbols.qbk]
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[def __effects [*Effects: ]]
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[def __formula [*Formula: ]]
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[def __exm1 '''<code>e<superscript>x</superscript> - 1</code>''']
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[def __ex '''<code>e<superscript>x</superscript></code>''']
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[def __te '''2ɛ'''] [/small Latin letter open e]
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[/@http://www.oreilly.com/catalog/docbook/chapter/book/iso-grk1.html Greek DocBook]
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[def __alpha '''α''']
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[def __lambda '''λ'''] [/lower case]
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[def __Lambda '''Λ'''] [/upper case]
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[def __gamma '''γ''']
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[def __Gamma '''Γ''']
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[def __sigma '''σ''']
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[def __Sigma '''Σ''']
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[/@http://www.oreilly.com/catalog/docbook/chapter/book/iso-tech.html technical]
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[def __ceilR '''⌉''']
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[def __ceilL '''ऄ''']
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[def __floorR '''⌋''']
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[def __floorL '''⌊''']
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[def __infin '''∞''']
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[def __integral '''∫''']
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[def __aacute '''á''']
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[def __eacute '''é''']
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[def __quarter '''¼''']
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[def __nearequal '''≊''']
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[def __quarter '''¼''']
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[def __space '''​''']
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[def __NTL_RR [@http://shoup.net/ntl/doc/RR.txt NTL::RR]]
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[def __NTL_quad_float [@http://shoup.net/ntl/doc/quad_float.txt NTL::quad_float]]
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[def __godfrey [link godfrey Godfrey]]
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[def __pugh [link pugh Pugh]]
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[def __caution This is not an official Boost library, it is a library under
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construction, the code is fully functional and robust, but
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interfaces, library structure, and function and distribution names
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may be changed without notice.]
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[def __domain_error [link domain_error domain_error]]
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[def __pole_error [link pole_error pole_error]]
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[def __overflow_error [link overflow_error overflow_error]]
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[def __underflow_error [link underflow_error underflow_error]]
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[def __denorm_error [link denorm_error denorm_error]]
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[def __logic_error [link logic_error logic_error]]
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[def __checked_narrowing_cast [link checked_narrowing_cast checked_narrowing_cast]]
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[/ The following macros expand to links to the various special functions
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and use the function's name as the link text]
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[def __lanczos [link math_toolkit.backgrounders.lanczos Lanczos approximation]]
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[def __zero_error [link zero_error effectively zero error]]
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[def __relative_error [link math_toolkit.backgrounders.relative_error relative zero error]]
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[def __lgamma [link math_toolkit.special.lgamma lgamma]]
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[def __digamma [link math_toolkit.special.digamma digamma]]
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[def __tgamma_ratio [link math_toolkit.special.gamma_ratios tgamma_ratio]]
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[def __tgamma_delta_ratio [link math_toolkit.special.gamma_ratios tgamma_delta_ratio]]
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[def __factorial [link math_toolkit.special.factorials factorial]]
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[def __unchecked_factorial [link math_toolkit.special.factorials unchecked_factorial]]
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[def __max_factorial [link math_toolkit.special.factorials max_factorial]]
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[def __tgamma [link math_toolkit.special.tgamma tgamma]]
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[def __tgamma1pm1 [link math_toolkit.special.tgamma tgamma1pm1]]
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[def __tgamma_lower [link math_toolkit.special.igamma tgamma_lower]]
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[def __gamma_P [link math_toolkit.special.igamma gamma_P]]
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[def __gamma_Q [link math_toolkit.special.igamma gamma_Q]]
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[def __gamma_Q_inv [link math_toolkit.special.igamma_inv gamma_Q_inv]]
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[def __gamma_P_inv [link math_toolkit.special.igamma_inv gamma_P_inv]]
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[def __gamma_Q_inva [link math_toolkit.special.igamma_inv gamma_Q_inva]]
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[def __gamma_P_inva [link math_toolkit.special.igamma_inv gamma_P_inva]]
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[def __erf [link math_toolkit.special.error_function erf]]
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[def __erfc [link math_toolkit.special.error_function erfc]]
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[def __erf_inv [link math_toolkit.special.error_inv erf_inv]]
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[def __erfc_inv [link math_toolkit.special.error_inv erfc_inv]]
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[def __beta [link math_toolkit.special.beta_function beta]]
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[def __beta3 [link math_toolkit.special.ibeta_function beta]]
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[def __betac [link math_toolkit.special.ibeta_function betac]]
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[def __ibeta [link math_toolkit.special.ibeta_function ibeta]]
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[def __ibetac [link math_toolkit.special.ibeta_function ibetac]]
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[def __ibeta_inv [link math_toolkit.special.ibeta_inv_function ibeta_inv]]
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[def __ibetac_inv [link math_toolkit.special.ibeta_inv_function ibetac_inv]]
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[def __ibeta_inva [link math_toolkit.special.ibeta_inv_function ibeta_inva]]
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[def __ibetac_inva [link math_toolkit.special.ibeta_inv_function ibetac_inva]]
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[def __ibeta_invb [link math_toolkit.special.ibeta_inv_function ibeta_invb]]
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[def __ibetac_invb [link math_toolkit.special.ibeta_inv_function ibetac_invb]]
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[def __gamma_P_derivative [link math_toolkit.special.beta_gamma_derivatives gamma_P_derivative]]
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[def __ibeta_derivative [link math_toolkit.special.beta_gamma_derivatives ibeta_derivative]]
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[def __fpclassify [link math_toolkit.special.fpclass fpclassify]]
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[def __isfinite [link math_toolkit.special.fpclass isfinite]]
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[def __isnan [link math_toolkit.special.fpclass isnan]]
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[def __isinf [link math_toolkit.special.fpclass isinf]]
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[def __isnormal [link math_toolkit.special.fpclass isnormal]]
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[def __expm1 [link math_toolkit.special.powers expm1]]
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[def __log1p [link math_toolkit.special.powers log1p]]
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[def __cbrt [link math_toolkit.special.powers cbrt]]
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[def __sqrt1pm1 [link math_toolkit.special.powers sqrt1pm1]]
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[def __powm1 [link math_toolkit.special.powers powm1]]
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[/ distribution non-members]
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[def __cdf [link math.dist.cdf Cumulative Distribution Function]]
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[def __pdf [link math.dist.pdf Probability Density Function]]
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[def __ccdf [link math.dist.ccdf Complement of the Cumulative Distribution Function]]
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[def __quantile [link math.dist.quantile Quantile]]
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[def __quantile_c [link math.dist.quantile_c Quantile from the complement of the probability]]
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[def __mean [link math.dist.mean mean]]
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[def __mode [link math.dist.mode mode]]
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[def __skewness [link math.dist.skewness skewness]]
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[def __kurtosis [link math.dist.kurtosis kurtosis]]
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[def __kurtosis_excess [link math.dist.kurtosis_excess kurtosis_excess]]
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[def __variance [link math.dist.variance variance]]
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[def __sd [link math.dist.sd standard deviation]]
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[def __hazard [link math.dist.hazard Hazard Function]]
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[def __chf [link math.dist.chf Cumulative Hazard Function]]
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[/ distribution def names end in distrib to avoid clashes]
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[def __beta_distrib [link math_toolkit.dist.dist_ref.dists.beta_dist Beta Distribution]]
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[def __binomial_distrib [link math_toolkit.dist.dist_ref.dists.binomial_dist Binomial Distribution]]
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[def __negative_binomial_distrib [link math_toolkit.dist.dist_ref.dists.negative_binomial_dist Negative Binomial Distribution]]
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[def __chi_squared_distrib [link math_toolkit.dist.dist_ref.dists.chi_squared_dist Chi Squared Distribution]]
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[def __normal_distrib [link math_toolkit.dist.dist_ref.dists.normal_dist Normal Distribution]]
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[def __F_distrib [link math_toolkit.dist.dist_ref.dists.f_dist Fisher F Distribution]]
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[def __students_t_distrib [link math_toolkit.dist.dist_ref.dists.students_t_dist Students t Distribution]]
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[def __usual_accessors __cdf, __pdf, __quantile, __hazard,
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__chf, __mean, __mode, __variance, __sd, __skewness, __kurtosis and __kurtosis_excess]
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[/ Some composite templates]
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[template super[x]'''<superscript>'''[x]'''</superscript>''']
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[template sub[x]'''<subscript>'''[x]'''</subscript>''']
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[template floor[x]'''⌊'''[x]'''⌋''']
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[template floorlr[x][lfloor][x][rfloor]]
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[template ceil[x] '''⌈'''[x]'''⌉''']
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[section:intro About the Math Toolkit] [/was Introduction]
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This library is work in progress, it is intended to fulfil several needs:
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# To provide a reasonably comprehensive set of
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[@http://en.wikipedia.org/wiki/List_of_probability_distributions statistical distributions],
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upon which higher level statistical tests can be built.
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The initial focus is on the central
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[@http://en.wikipedia.org/wiki/Univariate univariate ]
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[@http://mathworld.wolfram.com/StatisticalDistribution.html distributions].\n
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Both [@http://mathworld.wolfram.com/ContinuousDistribution.html continuous]
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(like [@http://mathworld.wolfram.com/NormalDistribution.html normal]
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& [@http://mathworld.wolfram.com/F-Distribution.html Fisher])
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and [@http://mathworld.wolfram.com/DiscreteDistribution.html discrete]
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(like [@http://mathworld.wolfram.com/BinomialDistribution.html binomial]
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& [@http://mathworld.wolfram.com/PoissonDistribution.html Poisson]) distributions are provided.
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# To provide a small number of high quality
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[@http://en.wikipedia.org/wiki/Special_functions special functions],
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initially these will be concentrated on functions used in statistical applications.
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The functions currently implemented are the gamma, beta & erf functions
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along with the incomplete gamma and beta functions (four variants
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of each) and all the possible inverses of these. All the implementations
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are fully generic and support the use of arbitrary "real-number" types,
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although they are optimised for use with types with known-about
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[@http://en.wikipedia.org/wiki/Significand significand (or mantissa)]
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sizes: typically `float`, `double` or `long double`.
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Use of these functions with interval arithmetic (Boost.Interval)
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isn't quite supported yet, but it's something that needs to be investigated.
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# To provide at least some of the tools required to implement
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mathematical special functions, hopefully the presence of
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these will encourage other authors to contribute more special
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function implementations in the future. Currently implemented
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are helpers for the evaluation of infinite series, continued
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fractions and rational approximations. There are also
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a fairly comprehensive set of root finding algorithms:
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both with and without derivative support. There are also classes
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for the manipulation of polynomials, for testing a special function
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against tabulated test data, and for the rapid generation of test
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data and/or data for output to an external graphing application.
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# To provide the most common
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[@http://en.wikipedia.org/wiki/List_of_probability_distributions probability distributions]
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used for statistics as C++ templated objects, fully generic,
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to support the use of arbitrary "real-number" types, so that they can be used not only with
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built-in floating-point types but also User-Defined Types (UDT) conforming to the
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real_concept archetype.
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These distributions are implemented using the special functions above.
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There are several examples of using the distributions for
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some typical applications in statistics.
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[endsect][/section:intro Introduction]
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[include structure.qbk] [/getting about]
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[section:dist Statistical Distributions and Functions]
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[include dist_tutorial.qbk]
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[include dist_reference.qbk]
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[endsect][/section:dist Statistical Distributions and Functions]
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[section:special Special Functions]
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[include tgamma.qbk]
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[include lgamma.qbk]
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[include digamma.qbk]
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[include gamma_ratios.qbk]
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[include factorials.qbk]
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[include igamma.qbk]
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[include igamma_inv.qbk]
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[include erf.qbk]
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[include erf_inv.qbk]
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[include beta.qbk]
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[include ibeta.qbk]
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[include ibeta_inv.qbk]
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[include beta_gamma_derivatives.qbk]
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[include fpclassify.qbk]
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[include powers.qbk]
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[include error_handling.qbk]
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[endsect][/section:special Special Functions]
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[section:toolkit Toolkit]
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[include series.qbk]
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[include fraction.qbk]
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[include rational.qbk]
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[include roots.qbk]
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[include roots_without_derivatives.qbk]
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[include minima.qbk]
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[section:internals Tools for Internal Testing and Development Use (Experimental)]
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[include internals_overview.qbk]
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[include polynomial.qbk]
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[include minimax.qbk]
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[include relative_error.qbk]
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[include test_data.qbk]
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[endsect][/section: internal tools]
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[endsect][/section:toolkit Toolkit]
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[section:Using_UDT Use with User Defined Floating-Point Types]
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[include concepts.qbk]
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[endsect][/section:Using_UDT Use with User Defined Floating-Point Types]
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[section:backgrounders Backgrounders]
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[include implementation.qbk]
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[include error.qbk][/relative error NOT handling]
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[include lanczos.qbk]
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[include remez.qbk]
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[endsect][/section:backgrounds Backgrounders]
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[section:status Status and Roadmap]
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[include roadmap.qbk]
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[include compilers.qbk]
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[include issues.qbk]
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[/include test_HTML4_symbols.qbk]
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[endsect][/section:status Status and Roadmap]
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[/ math.qbk
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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