472 lines
14 KiB
C++
472 lines
14 KiB
C++
// Special functions -*- C++ -*-
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// Copyright (C) 2006, 2007, 2008, 2009
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// Free Software Foundation, Inc.
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//
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// This file is part of the GNU ISO C++ Library. This library is free
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// software; you can redistribute it and/or modify it under the
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// terms of the GNU General Public License as published by the
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// Free Software Foundation; either version 3, or (at your option)
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// any later version.
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//
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// This library is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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//
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// Under Section 7 of GPL version 3, you are granted additional
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// permissions described in the GCC Runtime Library Exception, version
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// 3.1, as published by the Free Software Foundation.
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// You should have received a copy of the GNU General Public License and
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// a copy of the GCC Runtime Library Exception along with this program;
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// <http://www.gnu.org/licenses/>.
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/** @file tr1/gamma.tcc
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* This is an internal header file, included by other library headers.
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* You should not attempt to use it directly.
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*/
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//
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// ISO C++ 14882 TR1: 5.2 Special functions
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//
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// Written by Edward Smith-Rowland based on:
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// (1) Handbook of Mathematical Functions,
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// ed. Milton Abramowitz and Irene A. Stegun,
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// Dover Publications,
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// Section 6, pp. 253-266
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
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// 2nd ed, pp. 213-216
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// (4) Gamma, Exploring Euler's Constant, Julian Havil,
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// Princeton, 2003.
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#ifndef _TR1_GAMMA_TCC
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#define _TR1_GAMMA_TCC 1
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#include "special_function_util.h"
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namespace std
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{
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namespace tr1
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{
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// Implementation-space details.
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namespace __detail
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{
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/**
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* @brief This returns Bernoulli numbers from a table or by summation
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* for larger values.
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*
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* Recursion is unstable.
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*
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* @param __n the order n of the Bernoulli number.
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* @return The Bernoulli number of order n.
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*/
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template <typename _Tp>
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_Tp __bernoulli_series(unsigned int __n)
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{
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static const _Tp __num[28] = {
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_Tp(1UL), -_Tp(1UL) / _Tp(2UL),
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_Tp(1UL) / _Tp(6UL), _Tp(0UL),
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-_Tp(1UL) / _Tp(30UL), _Tp(0UL),
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_Tp(1UL) / _Tp(42UL), _Tp(0UL),
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-_Tp(1UL) / _Tp(30UL), _Tp(0UL),
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_Tp(5UL) / _Tp(66UL), _Tp(0UL),
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-_Tp(691UL) / _Tp(2730UL), _Tp(0UL),
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_Tp(7UL) / _Tp(6UL), _Tp(0UL),
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-_Tp(3617UL) / _Tp(510UL), _Tp(0UL),
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_Tp(43867UL) / _Tp(798UL), _Tp(0UL),
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-_Tp(174611) / _Tp(330UL), _Tp(0UL),
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_Tp(854513UL) / _Tp(138UL), _Tp(0UL),
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-_Tp(236364091UL) / _Tp(2730UL), _Tp(0UL),
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_Tp(8553103UL) / _Tp(6UL), _Tp(0UL)
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};
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if (__n == 0)
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return _Tp(1);
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if (__n == 1)
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return -_Tp(1) / _Tp(2);
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// Take care of the rest of the odd ones.
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if (__n % 2 == 1)
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return _Tp(0);
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// Take care of some small evens that are painful for the series.
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if (__n < 28)
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return __num[__n];
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_Tp __fact = _Tp(1);
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if ((__n / 2) % 2 == 0)
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__fact *= _Tp(-1);
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for (unsigned int __k = 1; __k <= __n; ++__k)
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__fact *= __k / (_Tp(2) * __numeric_constants<_Tp>::__pi());
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__fact *= _Tp(2);
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_Tp __sum = _Tp(0);
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for (unsigned int __i = 1; __i < 1000; ++__i)
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{
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_Tp __term = std::pow(_Tp(__i), -_Tp(__n));
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if (__term < std::numeric_limits<_Tp>::epsilon())
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break;
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__sum += __term;
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}
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return __fact * __sum;
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}
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/**
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* @brief This returns Bernoulli number \f$B_n\f$.
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*
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* @param __n the order n of the Bernoulli number.
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* @return The Bernoulli number of order n.
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*/
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template<typename _Tp>
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inline _Tp
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__bernoulli(const int __n)
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{
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return __bernoulli_series<_Tp>(__n);
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}
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/**
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* @brief Return \f$log(\Gamma(x))\f$ by asymptotic expansion
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* with Bernoulli number coefficients. This is like
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* Sterling's approximation.
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*
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* @param __x The argument of the log of the gamma function.
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* @return The logarithm of the gamma function.
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*/
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template<typename _Tp>
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_Tp
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__log_gamma_bernoulli(const _Tp __x)
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{
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_Tp __lg = (__x - _Tp(0.5L)) * std::log(__x) - __x
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+ _Tp(0.5L) * std::log(_Tp(2)
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* __numeric_constants<_Tp>::__pi());
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const _Tp __xx = __x * __x;
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_Tp __help = _Tp(1) / __x;
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for ( unsigned int __i = 1; __i < 20; ++__i )
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{
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const _Tp __2i = _Tp(2 * __i);
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__help /= __2i * (__2i - _Tp(1)) * __xx;
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__lg += __bernoulli<_Tp>(2 * __i) * __help;
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}
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return __lg;
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}
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/**
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* @brief Return \f$log(\Gamma(x))\f$ by the Lanczos method.
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* This method dominates all others on the positive axis I think.
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*
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* @param __x The argument of the log of the gamma function.
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* @return The logarithm of the gamma function.
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*/
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template<typename _Tp>
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_Tp
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__log_gamma_lanczos(const _Tp __x)
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{
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const _Tp __xm1 = __x - _Tp(1);
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static const _Tp __lanczos_cheb_7[9] = {
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_Tp( 0.99999999999980993227684700473478L),
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_Tp( 676.520368121885098567009190444019L),
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_Tp(-1259.13921672240287047156078755283L),
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_Tp( 771.3234287776530788486528258894L),
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_Tp(-176.61502916214059906584551354L),
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_Tp( 12.507343278686904814458936853L),
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_Tp(-0.13857109526572011689554707L),
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_Tp( 9.984369578019570859563e-6L),
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_Tp( 1.50563273514931155834e-7L)
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};
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static const _Tp __LOGROOT2PI
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= _Tp(0.9189385332046727417803297364056176L);
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_Tp __sum = __lanczos_cheb_7[0];
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for(unsigned int __k = 1; __k < 9; ++__k)
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__sum += __lanczos_cheb_7[__k] / (__xm1 + __k);
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const _Tp __term1 = (__xm1 + _Tp(0.5L))
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* std::log((__xm1 + _Tp(7.5L))
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/ __numeric_constants<_Tp>::__euler());
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const _Tp __term2 = __LOGROOT2PI + std::log(__sum);
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const _Tp __result = __term1 + (__term2 - _Tp(7));
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return __result;
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}
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/**
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* @brief Return \f$ log(|\Gamma(x)|) \f$.
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* This will return values even for \f$ x < 0 \f$.
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* To recover the sign of \f$ \Gamma(x) \f$ for
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* any argument use @a __log_gamma_sign.
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*
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* @param __x The argument of the log of the gamma function.
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* @return The logarithm of the gamma function.
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*/
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template<typename _Tp>
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_Tp
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__log_gamma(const _Tp __x)
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{
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if (__x > _Tp(0.5L))
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return __log_gamma_lanczos(__x);
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else
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{
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const _Tp __sin_fact
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= std::abs(std::sin(__numeric_constants<_Tp>::__pi() * __x));
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if (__sin_fact == _Tp(0))
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std::__throw_domain_error(__N("Argument is nonpositive integer "
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"in __log_gamma"));
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return __numeric_constants<_Tp>::__lnpi()
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- std::log(__sin_fact)
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- __log_gamma_lanczos(_Tp(1) - __x);
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}
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}
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/**
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* @brief Return the sign of \f$ \Gamma(x) \f$.
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* At nonpositive integers zero is returned.
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*
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* @param __x The argument of the gamma function.
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* @return The sign of the gamma function.
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*/
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template<typename _Tp>
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_Tp
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__log_gamma_sign(const _Tp __x)
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{
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if (__x > _Tp(0))
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return _Tp(1);
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else
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{
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const _Tp __sin_fact
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= std::sin(__numeric_constants<_Tp>::__pi() * __x);
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if (__sin_fact > _Tp(0))
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return (1);
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else if (__sin_fact < _Tp(0))
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return -_Tp(1);
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else
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return _Tp(0);
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}
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}
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/**
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* @brief Return the logarithm of the binomial coefficient.
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* The binomial coefficient is given by:
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* @f[
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* \left( \right) = \frac{n!}{(n-k)! k!}
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* @f]
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*
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* @param __n The first argument of the binomial coefficient.
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* @param __k The second argument of the binomial coefficient.
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* @return The binomial coefficient.
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*/
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template<typename _Tp>
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_Tp
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__log_bincoef(const unsigned int __n, const unsigned int __k)
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{
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// Max e exponent before overflow.
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static const _Tp __max_bincoeff
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= std::numeric_limits<_Tp>::max_exponent10
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* std::log(_Tp(10)) - _Tp(1);
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#if _GLIBCXX_USE_C99_MATH_TR1
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_Tp __coeff = std::tr1::lgamma(_Tp(1 + __n))
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- std::tr1::lgamma(_Tp(1 + __k))
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- std::tr1::lgamma(_Tp(1 + __n - __k));
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#else
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_Tp __coeff = __log_gamma(_Tp(1 + __n))
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- __log_gamma(_Tp(1 + __k))
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- __log_gamma(_Tp(1 + __n - __k));
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#endif
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}
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/**
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* @brief Return the binomial coefficient.
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* The binomial coefficient is given by:
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* @f[
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* \left( \right) = \frac{n!}{(n-k)! k!}
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* @f]
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*
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* @param __n The first argument of the binomial coefficient.
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* @param __k The second argument of the binomial coefficient.
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* @return The binomial coefficient.
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*/
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template<typename _Tp>
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_Tp
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__bincoef(const unsigned int __n, const unsigned int __k)
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{
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// Max e exponent before overflow.
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static const _Tp __max_bincoeff
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= std::numeric_limits<_Tp>::max_exponent10
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* std::log(_Tp(10)) - _Tp(1);
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const _Tp __log_coeff = __log_bincoef<_Tp>(__n, __k);
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if (__log_coeff > __max_bincoeff)
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return std::numeric_limits<_Tp>::quiet_NaN();
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else
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return std::exp(__log_coeff);
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}
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/**
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* @brief Return \f$ \Gamma(x) \f$.
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*
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* @param __x The argument of the gamma function.
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* @return The gamma function.
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*/
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template<typename _Tp>
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inline _Tp
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__gamma(const _Tp __x)
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{
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return std::exp(__log_gamma(__x));
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}
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/**
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* @brief Return the digamma function by series expansion.
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* The digamma or @f$ \psi(x) @f$ function is defined by
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* @f[
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* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
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* @f]
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*
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* The series is given by:
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* @f[
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* \psi(x) = -\gamma_E - \frac{1}{x}
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* \sum_{k=1}^{\infty} \frac{x}{k(x + k)}
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* @f]
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*/
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template<typename _Tp>
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_Tp
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__psi_series(const _Tp __x)
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{
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_Tp __sum = -__numeric_constants<_Tp>::__gamma_e() - _Tp(1) / __x;
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const unsigned int __max_iter = 100000;
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for (unsigned int __k = 1; __k < __max_iter; ++__k)
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{
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const _Tp __term = __x / (__k * (__k + __x));
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__sum += __term;
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if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
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break;
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}
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return __sum;
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}
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/**
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* @brief Return the digamma function for large argument.
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* The digamma or @f$ \psi(x) @f$ function is defined by
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* @f[
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* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
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* @f]
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*
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* The asymptotic series is given by:
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* @f[
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* \psi(x) = \ln(x) - \frac{1}{2x}
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* - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}}
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* @f]
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*/
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template<typename _Tp>
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_Tp
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__psi_asymp(const _Tp __x)
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{
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_Tp __sum = std::log(__x) - _Tp(0.5L) / __x;
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const _Tp __xx = __x * __x;
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_Tp __xp = __xx;
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const unsigned int __max_iter = 100;
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for (unsigned int __k = 1; __k < __max_iter; ++__k)
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{
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const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp);
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__sum -= __term;
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if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
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break;
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__xp *= __xx;
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}
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return __sum;
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}
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/**
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* @brief Return the digamma function.
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* The digamma or @f$ \psi(x) @f$ function is defined by
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* @f[
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* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
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* @f]
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* For negative argument the reflection formula is used:
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* @f[
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* \psi(x) = \psi(1-x) - \pi \cot(\pi x)
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* @f]
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*/
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template<typename _Tp>
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_Tp
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__psi(const _Tp __x)
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{
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const int __n = static_cast<int>(__x + 0.5L);
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const _Tp __eps = _Tp(4) * std::numeric_limits<_Tp>::epsilon();
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if (__n <= 0 && std::abs(__x - _Tp(__n)) < __eps)
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return std::numeric_limits<_Tp>::quiet_NaN();
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else if (__x < _Tp(0))
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{
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const _Tp __pi = __numeric_constants<_Tp>::__pi();
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return __psi(_Tp(1) - __x)
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- __pi * std::cos(__pi * __x) / std::sin(__pi * __x);
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}
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else if (__x > _Tp(100))
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return __psi_asymp(__x);
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else
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return __psi_series(__x);
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}
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/**
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* @brief Return the polygamma function @f$ \psi^{(n)}(x) @f$.
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*
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* The polygamma function is related to the Hurwitz zeta function:
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* @f[
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* \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x)
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* @f]
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*/
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template<typename _Tp>
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_Tp
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__psi(const unsigned int __n, const _Tp __x)
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{
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if (__x <= _Tp(0))
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std::__throw_domain_error(__N("Argument out of range "
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"in __psi"));
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else if (__n == 0)
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return __psi(__x);
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else
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{
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const _Tp __hzeta = __hurwitz_zeta(_Tp(__n + 1), __x);
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#if _GLIBCXX_USE_C99_MATH_TR1
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const _Tp __ln_nfact = std::tr1::lgamma(_Tp(__n + 1));
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#else
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const _Tp __ln_nfact = __log_gamma(_Tp(__n + 1));
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#endif
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_Tp __result = std::exp(__ln_nfact) * __hzeta;
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if (__n % 2 == 1)
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__result = -__result;
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return __result;
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}
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}
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} // namespace std::tr1::__detail
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}
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}
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#endif // _TR1_GAMMA_TCC
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