// Special functions -*- C++ -*- // Copyright (C) 2006-2020 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 <tr1/special_function_util.h> namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION #if _GLIBCXX_USE_STD_SPEC_FUNCS # define _GLIBCXX_MATH_NS ::std #elif defined(_GLIBCXX_TR1_CMATH) namespace tr1 { # define _GLIBCXX_MATH_NS ::std::tr1 #else # error do not include this header directly, use <cmath> or <tr1/cmath> #endif // [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(_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 (_GLIBCXX_MATH_NS::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(_GLIBCXX_MATH_NS::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 = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) - _GLIBCXX_MATH_NS::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$. * * The Riemann zeta function is defined by: * \f[ * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 * \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) * \Gamma (1 - s) \zeta (1 - s) for s < 1 * \f] * For s < 1 use the reflection formula: * \f[ * \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) * \f] */ template<typename _Tp> _Tp __riemann_zeta(_Tp __s) { if (__isnan(__s)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__s == _Tp(1)) return std::numeric_limits<_Tp>::infinity(); else if (__s < -_Tp(19)) { _Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); __zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) #if _GLIBCXX_USE_C99_MATH_TR1 * std::exp(_GLIBCXX_MATH_NS::lgamma(_Tp(1) - __s)) #else * std::exp(__log_gamma(_Tp(1) - __s)) #endif / __numeric_constants<_Tp>::__pi(); return __zeta; } else if (__s < _Tp(20)) { // Global double sum or McLaurin? bool __glob = true; if (__glob) return __riemann_zeta_glob(__s); else { if (__s > _Tp(1)) return __riemann_zeta_sum(__s); else { _Tp __zeta = std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) * std::sin(__numeric_constants<_Tp>::__pi_2() * __s) #if _GLIBCXX_USE_C99_MATH_TR1 * _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __s) #else * std::exp(__log_gamma(_Tp(1) - __s)) #endif * __riemann_zeta_sum(_Tp(1) - __s); return __zeta; } } } else return __riemann_zeta_product(__s); } /** * @brief Return the Hurwitz zeta function @f$ \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 = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) - _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) - _GLIBCXX_MATH_NS::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); } } // namespace __detail #undef _GLIBCXX_MATH_NS #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) } // namespace tr1 #endif _GLIBCXX_END_NAMESPACE_VERSION } #endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC