| // Special functions -*- C++ -*- |
| |
| // Copyright (C) 2006, 2007, 2008, 2009, 2010 |
| // 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(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); |
| } |
| |
| _GLIBCXX_END_NAMESPACE_VERSION |
| } // namespace std::tr1::__detail |
| } |
| } |
| |
| #endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC |
