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Diffstat (limited to 'gcc-4.4.3/libstdc++-v3/include/tr1/riemann_zeta.tcc')
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diff --git a/gcc-4.4.3/libstdc++-v3/include/tr1/riemann_zeta.tcc b/gcc-4.4.3/libstdc++-v3/include/tr1/riemann_zeta.tcc deleted file mode 100644 index c646c2e40..000000000 --- a/gcc-4.4.3/libstdc++-v3/include/tr1/riemann_zeta.tcc +++ /dev/null @@ -1,435 +0,0 @@ -// Special functions -*- C++ -*- - -// Copyright (C) 2006, 2007, 2008, 2009 -// Free Software Foundation, Inc. -// -// This file is part of the GNU ISO C++ Library. This library is free -// software; you can redistribute it and/or modify it under the -// terms of the GNU General Public License as published by the -// Free Software Foundation; either version 3, or (at your option) -// any later version. -// -// This library is distributed in the hope that it will be useful, -// but WITHOUT ANY WARRANTY; without even the implied warranty of -// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -// GNU General Public License for more details. -// -// Under Section 7 of GPL version 3, you are granted additional -// permissions described in the GCC Runtime Library Exception, version -// 3.1, as published by the Free Software Foundation. - -// You should have received a copy of the GNU General Public License and -// a copy of the GCC Runtime Library Exception along with this program; -// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see -// <http://www.gnu.org/licenses/>. - -/** @file tr1/riemann_zeta.tcc - * This is an internal header file, included by other library headers. - * You should not attempt to use it directly. - */ - -// -// ISO C++ 14882 TR1: 5.2 Special functions -// - -// Written by Edward Smith-Rowland based on: -// (1) Handbook of Mathematical Functions, -// Ed. by Milton Abramowitz and Irene A. Stegun, -// Dover Publications, New-York, Section 5, pp. 807-808. -// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl -// (3) Gamma, Exploring Euler's Constant, Julian Havil, -// Princeton, 2003. - -#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC -#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1 - -#include "special_function_util.h" - -namespace std -{ -namespace tr1 -{ - - // [5.2] Special functions - - // Implementation-space details. - namespace __detail - { - - /** - * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ - * by summation for s > 1. - * - * The Riemann zeta function is defined by: - * \f[ - * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 - * \f] - * For s < 1 use the reflection formula: - * \f[ - * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) - * \f] - */ - template<typename _Tp> - _Tp - __riemann_zeta_sum(const _Tp __s) - { - // A user shouldn't get to this. - if (__s < _Tp(1)) - std::__throw_domain_error(__N("Bad argument in zeta sum.")); - - const unsigned int max_iter = 10000; - _Tp __zeta = _Tp(0); - for (unsigned int __k = 1; __k < max_iter; ++__k) - { - _Tp __term = std::pow(static_cast<_Tp>(__k), -__s); - if (__term < std::numeric_limits<_Tp>::epsilon()) - { - break; - } - __zeta += __term; - } - - return __zeta; - } - - - /** - * @brief Evaluate the Riemann zeta function @f$ \zeta(s) @f$ - * by an alternate series for s > 0. - * - * The Riemann zeta function is defined by: - * \f[ - * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 - * \f] - * For s < 1 use the reflection formula: - * \f[ - * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) - * \f] - */ - template<typename _Tp> - _Tp - __riemann_zeta_alt(const _Tp __s) - { - _Tp __sgn = _Tp(1); - _Tp __zeta = _Tp(0); - for (unsigned int __i = 1; __i < 10000000; ++__i) - { - _Tp __term = __sgn / std::pow(__i, __s); - if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) - break; - __zeta += __term; - __sgn *= _Tp(-1); - } - __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); - - return __zeta; - } - - - /** - * @brief Evaluate the Riemann zeta function by series for all s != 1. - * Convergence is great until largish negative numbers. - * Then the convergence of the > 0 sum gets better. - * - * The series is: - * \f[ - * \zeta(s) = \frac{1}{1-2^{1-s}} - * \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} - * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} - * \f] - * Havil 2003, p. 206. - * - * The Riemann zeta function is defined by: - * \f[ - * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 - * \f] - * For s < 1 use the reflection formula: - * \f[ - * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) - * \f] - */ - template<typename _Tp> - _Tp - __riemann_zeta_glob(const _Tp __s) - { - _Tp __zeta = _Tp(0); - - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - // Max e exponent before overflow. - const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 - * std::log(_Tp(10)) - _Tp(1); - - // This series works until the binomial coefficient blows up - // so use reflection. - if (__s < _Tp(0)) - { -#if _GLIBCXX_USE_C99_MATH_TR1 - if (std::tr1::fmod(__s,_Tp(2)) == _Tp(0)) - return _Tp(0); - else -#endif - { - _Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s); - __zeta *= std::pow(_Tp(2) - * __numeric_constants<_Tp>::__pi(), __s) - * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) -#if _GLIBCXX_USE_C99_MATH_TR1 - * std::exp(std::tr1::lgamma(_Tp(1) - __s)) -#else - * std::exp(__log_gamma(_Tp(1) - __s)) -#endif - / __numeric_constants<_Tp>::__pi(); - return __zeta; - } - } - - _Tp __num = _Tp(0.5L); - const unsigned int __maxit = 10000; - for (unsigned int __i = 0; __i < __maxit; ++__i) - { - bool __punt = false; - _Tp __sgn = _Tp(1); - _Tp __term = _Tp(0); - for (unsigned int __j = 0; __j <= __i; ++__j) - { -#if _GLIBCXX_USE_C99_MATH_TR1 - _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) - - std::tr1::lgamma(_Tp(1 + __j)) - - std::tr1::lgamma(_Tp(1 + __i - __j)); -#else - _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) - - __log_gamma(_Tp(1 + __j)) - - __log_gamma(_Tp(1 + __i - __j)); -#endif - if (__bincoeff > __max_bincoeff) - { - // This only gets hit for x << 0. - __punt = true; - break; - } - __bincoeff = std::exp(__bincoeff); - __term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s); - __sgn *= _Tp(-1); - } - if (__punt) - break; - __term *= __num; - __zeta += __term; - if (std::abs(__term/__zeta) < __eps) - break; - __num *= _Tp(0.5L); - } - - __zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); - - return __zeta; - } - - - /** - * @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ - * using the product over prime factors. - * \f[ - * \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} - * \f] - * where @f$ {p_i} @f$ are the prime numbers. - * - * The Riemann zeta function is defined by: - * \f[ - * \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 - * \f] - * For s < 1 use the reflection formula: - * \f[ - * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) - * \f] - */ - template<typename _Tp> - _Tp - __riemann_zeta_product(const _Tp __s) - { - static const _Tp __prime[] = { - _Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19), - _Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47), - _Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79), - _Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109) - }; - static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp); - - _Tp __zeta = _Tp(1); - for (unsigned int __i = 0; __i < __num_primes; ++__i) - { - const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s); - __zeta *= __fact; - if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon()) - break; - } - - __zeta = _Tp(1) / __zeta; - - return __zeta; - } - - - /** - * @brief Return the Riemann zeta function @f$ \zeta(s) @f$. - * - * The Riemann zeta function is defined by: - * \f[ - * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 - * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) - * \Gamma (1 - s) \zeta (1 - s) for s < 1 - * \f] - * For s < 1 use the reflection formula: - * \f[ - * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) - * \f] - */ - template<typename _Tp> - _Tp - __riemann_zeta(const _Tp __s) - { - if (__isnan(__s)) - return std::numeric_limits<_Tp>::quiet_NaN(); - else if (__s == _Tp(1)) - return std::numeric_limits<_Tp>::infinity(); - else if (__s < -_Tp(19)) - { - _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); - __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) - * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) -#if _GLIBCXX_USE_C99_MATH_TR1 - * std::exp(std::tr1::lgamma(_Tp(1) - __s)) -#else - * std::exp(__log_gamma(_Tp(1) - __s)) -#endif - / __numeric_constants<_Tp>::__pi(); - return __zeta; - } - else if (__s < _Tp(20)) - { - // Global double sum or McLaurin? - bool __glob = true; - if (__glob) - return __riemann_zeta_glob(__s); - else - { - if (__s > _Tp(1)) - return __riemann_zeta_sum(__s); - else - { - _Tp __zeta = std::pow(_Tp(2) - * __numeric_constants<_Tp>::__pi(), __s) - * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) -#if _GLIBCXX_USE_C99_MATH_TR1 - * std::tr1::tgamma(_Tp(1) - __s) -#else - * std::exp(__log_gamma(_Tp(1) - __s)) -#endif - * __riemann_zeta_sum(_Tp(1) - __s); - return __zeta; - } - } - } - else - return __riemann_zeta_product(__s); - } - - - /** - * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ - * for all s != 1 and x > -1. - * - * The Hurwitz zeta function is defined by: - * @f[ - * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} - * @f] - * The Riemann zeta function is a special case: - * @f[ - * \zeta(s) = \zeta(1,s) - * @f] - * - * This functions uses the double sum that converges for s != 1 - * and x > -1: - * @f[ - * \zeta(x,s) = \frac{1}{s-1} - * \sum_{n=0}^{\infty} \frac{1}{n + 1} - * \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} - * @f] - */ - template<typename _Tp> - _Tp - __hurwitz_zeta_glob(const _Tp __a, const _Tp __s) - { - _Tp __zeta = _Tp(0); - - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - // Max e exponent before overflow. - const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 - * std::log(_Tp(10)) - _Tp(1); - - const unsigned int __maxit = 10000; - for (unsigned int __i = 0; __i < __maxit; ++__i) - { - bool __punt = false; - _Tp __sgn = _Tp(1); - _Tp __term = _Tp(0); - for (unsigned int __j = 0; __j <= __i; ++__j) - { -#if _GLIBCXX_USE_C99_MATH_TR1 - _Tp __bincoeff = std::tr1::lgamma(_Tp(1 + __i)) - - std::tr1::lgamma(_Tp(1 + __j)) - - std::tr1::lgamma(_Tp(1 + __i - __j)); -#else - _Tp __bincoeff = __log_gamma(_Tp(1 + __i)) - - __log_gamma(_Tp(1 + __j)) - - __log_gamma(_Tp(1 + __i - __j)); -#endif - if (__bincoeff > __max_bincoeff) - { - // This only gets hit for x << 0. - __punt = true; - break; - } - __bincoeff = std::exp(__bincoeff); - __term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s); - __sgn *= _Tp(-1); - } - if (__punt) - break; - __term /= _Tp(__i + 1); - if (std::abs(__term / __zeta) < __eps) - break; - __zeta += __term; - } - - __zeta /= __s - _Tp(1); - - return __zeta; - } - - - /** - * @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ - * for all s != 1 and x > -1. - * - * The Hurwitz zeta function is defined by: - * @f[ - * \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} - * @f] - * The Riemann zeta function is a special case: - * @f[ - * \zeta(s) = \zeta(1,s) - * @f] - */ - template<typename _Tp> - inline _Tp - __hurwitz_zeta(const _Tp __a, const _Tp __s) - { - return __hurwitz_zeta_glob(__a, __s); - } - - } // namespace std::tr1::__detail -} -} - -#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC |