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author | Ben Cheng <bccheng@google.com> | 2014-03-25 22:37:19 -0700 |
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committer | Ben Cheng <bccheng@google.com> | 2014-03-25 22:37:19 -0700 |
commit | 1bc5aee63eb72b341f506ad058502cd0361f0d10 (patch) | |
tree | c607e8252f3405424ff15bc2d00aa38dadbb2518 /gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc | |
parent | 283a0bf58fcf333c58a2a92c3ebbc41fb9eb1fdb (diff) | |
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Initial checkin of GCC 4.9.0 from trunk (r208799).
Change-Id: I48a3c08bb98542aa215912a75f03c0890e497dba
Diffstat (limited to 'gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc')
-rw-r--r-- | gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc | 433 |
1 files changed, 433 insertions, 0 deletions
diff --git a/gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc b/gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc new file mode 100644 index 0000000..dbedfe0 --- /dev/null +++ b/gcc-4.9/libstdc++-v3/include/tr1/riemann_zeta.tcc @@ -0,0 +1,433 @@ +// Special functions -*- C++ -*- + +// Copyright (C) 2006-2014 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. + * Do not attempt to use it directly. @headername{tr1/cmath} + */ + +// +// 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 _GLIBCXX_VISIBILITY(default) +{ +namespace tr1 +{ + // [5.2] Special functions + + // Implementation-space details. + namespace __detail + { + _GLIBCXX_BEGIN_NAMESPACE_VERSION + + /** + * @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(_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(_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(_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(_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$. + * + * The Riemann zeta function is defined by: + * \f[ + * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 + * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) + * \Gamma (1 - s) \zeta (1 - s) for s < 1 + * \f] + * For s < 1 use the reflection formula: + * \f[ + * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) + * \f] + */ + template<typename _Tp> + _Tp + __riemann_zeta(_Tp __s) + { + if (__isnan(__s)) + return std::numeric_limits<_Tp>::quiet_NaN(); + else if (__s == _Tp(1)) + return std::numeric_limits<_Tp>::infinity(); + else if (__s < -_Tp(19)) + { + _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); + __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) + * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) +#if _GLIBCXX_USE_C99_MATH_TR1 + * std::exp(std::tr1::lgamma(_Tp(1) - __s)) +#else + * std::exp(__log_gamma(_Tp(1) - __s)) +#endif + / __numeric_constants<_Tp>::__pi(); + return __zeta; + } + else if (__s < _Tp(20)) + { + // Global double sum or McLaurin? + bool __glob = true; + if (__glob) + return __riemann_zeta_glob(__s); + else + { + if (__s > _Tp(1)) + return __riemann_zeta_sum(__s); + else + { + _Tp __zeta = std::pow(_Tp(2) + * __numeric_constants<_Tp>::__pi(), __s) + * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) +#if _GLIBCXX_USE_C99_MATH_TR1 + * std::tr1::tgamma(_Tp(1) - __s) +#else + * std::exp(__log_gamma(_Tp(1) - __s)) +#endif + * __riemann_zeta_sum(_Tp(1) - __s); + return __zeta; + } + } + } + else + return __riemann_zeta_product(__s); + } + + + /** + * @brief Return the Hurwitz zeta function @f$ \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(_Tp __a, _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(_Tp __a, _Tp __s) + { return __hurwitz_zeta_glob(__a, __s); } + + _GLIBCXX_END_NAMESPACE_VERSION + } // namespace std::tr1::__detail +} +} + +#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC |