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// Special functions -*- C++ -*- |
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|
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// Copyright (C) 2006-2021 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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|
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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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|
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/** @file tr1/riemann_zeta.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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*/ |
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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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|
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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. by Milton Abramowitz and Irene A. Stegun, |
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// Dover Publications, New-York, Section 5, pp. 807-808. |
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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// (3) Gamma, Exploring Euler's Constant, Julian Havil, |
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// Princeton, 2003. |
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|
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#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC |
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#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1 |
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|
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#include <tr1/special_function_util.h> |
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|
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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|
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#if _GLIBCXX_USE_STD_SPEC_FUNCS |
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# define _GLIBCXX_MATH_NS ::std |
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#elif defined(_GLIBCXX_TR1_CMATH) |
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namespace tr1 |
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{ |
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# define _GLIBCXX_MATH_NS ::std::tr1 |
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#else |
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# error do not include this header directly, use <cmath> or <tr1/cmath> |
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#endif |
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// [5.2] Special functions |
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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 Compute the Riemann zeta function @f$ \zeta(s) @f$ |
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* by summation for s > 1. |
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* |
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* The Riemann zeta function is defined by: |
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* \f[ |
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* \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 |
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* \f] |
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* For s < 1 use the reflection formula: |
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* \f[ |
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* \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) |
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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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__riemann_zeta_sum(_Tp __s) |
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{ |
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// A user shouldn't get to this. |
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if (__s < _Tp(1)) |
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std::__throw_domain_error(__N("Bad argument in zeta sum.")); |
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|
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const unsigned int max_iter = 10000; |
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_Tp __zeta = _Tp(0); |
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for (unsigned int __k = 1; __k < max_iter; ++__k) |
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{ |
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_Tp __term = std::pow(static_cast<_Tp>(__k), -__s); |
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if (__term < std::numeric_limits<_Tp>::epsilon()) |
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{ |
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break; |
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} |
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__zeta += __term; |
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} |
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|
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return __zeta; |
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} |
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|
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|
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/** |
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* @brief Evaluate the Riemann zeta function @f$ \zeta(s) @f$ |
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* by an alternate series for s > 0. |
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* |
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* The Riemann zeta function is defined by: |
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* \f[ |
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* \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 |
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* \f] |
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* For s < 1 use the reflection formula: |
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* \f[ |
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* \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) |
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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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__riemann_zeta_alt(_Tp __s) |
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{ |
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_Tp __sgn = _Tp(1); |
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_Tp __zeta = _Tp(0); |
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for (unsigned int __i = 1; __i < 10000000; ++__i) |
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{ |
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_Tp __term = __sgn / std::pow(__i, __s); |
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if (std::abs(__term) < std::numeric_limits<_Tp>::epsilon()) |
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break; |
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__zeta += __term; |
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__sgn *= _Tp(-1); |
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} |
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__zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); |
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|
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return __zeta; |
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} |
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|
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|
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/** |
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* @brief Evaluate the Riemann zeta function by series for all s != 1. |
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* Convergence is great until largish negative numbers. |
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* Then the convergence of the > 0 sum gets better. |
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* |
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* The series is: |
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* \f[ |
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* \zeta(s) = \frac{1}{1-2^{1-s}} |
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* \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} |
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* \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (k+1)^{-s} |
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* \f] |
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* Havil 2003, p. 206. |
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* |
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* The Riemann zeta function is defined by: |
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* \f[ |
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* \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 |
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* \f] |
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* For s < 1 use the reflection formula: |
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* \f[ |
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* \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) |
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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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__riemann_zeta_glob(_Tp __s) |
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{ |
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_Tp __zeta = _Tp(0); |
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|
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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// Max e exponent before overflow. |
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const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 |
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* std::log(_Tp(10)) - _Tp(1); |
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|
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// This series works until the binomial coefficient blows up |
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// so use reflection. |
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if (__s < _Tp(0)) |
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{ |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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if (_GLIBCXX_MATH_NS::fmod(__s,_Tp(2)) == _Tp(0)) |
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return _Tp(0); |
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else |
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#endif |
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{ |
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_Tp __zeta = __riemann_zeta_glob(_Tp(1) - __s); |
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__zeta *= std::pow(_Tp(2) |
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* __numeric_constants<_Tp>::__pi(), __s) |
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* std::sin(__numeric_constants<_Tp>::__pi_2() * __s) |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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* std::exp(_GLIBCXX_MATH_NS::lgamma(_Tp(1) - __s)) |
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#else |
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* std::exp(__log_gamma(_Tp(1) - __s)) |
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#endif |
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/ __numeric_constants<_Tp>::__pi(); |
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return __zeta; |
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} |
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} |
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|
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_Tp __num = _Tp(0.5L); |
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const unsigned int __maxit = 10000; |
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for (unsigned int __i = 0; __i < __maxit; ++__i) |
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{ |
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bool __punt = false; |
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_Tp __sgn = _Tp(1); |
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_Tp __term = _Tp(0); |
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for (unsigned int __j = 0; __j <= __i; ++__j) |
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{ |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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_Tp __bincoeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) |
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- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) |
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- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i - __j)); |
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#else |
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_Tp __bincoeff = __log_gamma(_Tp(1 + __i)) |
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- __log_gamma(_Tp(1 + __j)) |
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- __log_gamma(_Tp(1 + __i - __j)); |
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#endif |
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if (__bincoeff > __max_bincoeff) |
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{ |
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// This only gets hit for x << 0. |
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__punt = true; |
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break; |
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} |
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__bincoeff = std::exp(__bincoeff); |
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__term += __sgn * __bincoeff * std::pow(_Tp(1 + __j), -__s); |
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__sgn *= _Tp(-1); |
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} |
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if (__punt) |
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break; |
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__term *= __num; |
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__zeta += __term; |
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if (std::abs(__term/__zeta) < __eps) |
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break; |
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__num *= _Tp(0.5L); |
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} |
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|
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__zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s); |
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|
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return __zeta; |
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} |
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|
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|
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/** |
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* @brief Compute the Riemann zeta function @f$ \zeta(s) @f$ |
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* using the product over prime factors. |
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* \f[ |
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* \zeta(s) = \Pi_{i=1}^\infty \frac{1}{1 - p_i^{-s}} |
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* \f] |
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* where @f$ {p_i} @f$ are the prime numbers. |
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* |
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* The Riemann zeta function is defined by: |
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* \f[ |
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* \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} for s > 1 |
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* \f] |
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* For s < 1 use the reflection formula: |
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* \f[ |
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* \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) |
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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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__riemann_zeta_product(_Tp __s) |
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{ |
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static const _Tp __prime[] = { |
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_Tp(2), _Tp(3), _Tp(5), _Tp(7), _Tp(11), _Tp(13), _Tp(17), _Tp(19), |
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_Tp(23), _Tp(29), _Tp(31), _Tp(37), _Tp(41), _Tp(43), _Tp(47), |
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_Tp(53), _Tp(59), _Tp(61), _Tp(67), _Tp(71), _Tp(73), _Tp(79), |
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_Tp(83), _Tp(89), _Tp(97), _Tp(101), _Tp(103), _Tp(107), _Tp(109) |
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}; |
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static const unsigned int __num_primes = sizeof(__prime) / sizeof(_Tp); |
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|
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_Tp __zeta = _Tp(1); |
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for (unsigned int __i = 0; __i < __num_primes; ++__i) |
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{ |
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const _Tp __fact = _Tp(1) - std::pow(__prime[__i], -__s); |
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__zeta *= __fact; |
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if (_Tp(1) - __fact < std::numeric_limits<_Tp>::epsilon()) |
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break; |
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} |
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|
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__zeta = _Tp(1) / __zeta; |
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|
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return __zeta; |
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} |
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|
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|
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/** |
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* @brief Return the Riemann zeta function @f$ \zeta(s) @f$. |
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* |
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* The Riemann zeta function is defined by: |
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* \f[ |
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* \zeta(s) = \sum_{k=1}^{\infty} k^{-s} for s > 1 |
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* \frac{(2\pi)^s}{pi} sin(\frac{\pi s}{2}) |
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* \Gamma (1 - s) \zeta (1 - s) for s < 1 |
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* \f] |
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* For s < 1 use the reflection formula: |
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* \f[ |
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* \zeta(s) = 2^s \pi^{s-1} \Gamma(1-s) \zeta(1-s) |
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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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__riemann_zeta(_Tp __s) |
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{ |
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if (__isnan(__s)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (__s == _Tp(1)) |
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return std::numeric_limits<_Tp>::infinity(); |
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else if (__s < -_Tp(19)) |
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{ |
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_Tp __zeta = __riemann_zeta_product(_Tp(1) - __s); |
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__zeta *= std::pow(_Tp(2) * __numeric_constants<_Tp>::__pi(), __s) |
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* std::sin(__numeric_constants<_Tp>::__pi_2() * __s) |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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* std::exp(_GLIBCXX_MATH_NS::lgamma(_Tp(1) - __s)) |
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#else |
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* std::exp(__log_gamma(_Tp(1) - __s)) |
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#endif |
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/ __numeric_constants<_Tp>::__pi(); |
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return __zeta; |
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} |
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else if (__s < _Tp(20)) |
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{ |
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// Global double sum or McLaurin? |
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bool __glob = true; |
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if (__glob) |
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return __riemann_zeta_glob(__s); |
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else |
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{ |
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if (__s > _Tp(1)) |
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return __riemann_zeta_sum(__s); |
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else |
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{ |
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_Tp __zeta = std::pow(_Tp(2) |
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* __numeric_constants<_Tp>::__pi(), __s) |
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* std::sin(__numeric_constants<_Tp>::__pi_2() * __s) |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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* _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __s) |
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#else |
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* std::exp(__log_gamma(_Tp(1) - __s)) |
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#endif |
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* __riemann_zeta_sum(_Tp(1) - __s); |
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return __zeta; |
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} |
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} |
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} |
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else |
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return __riemann_zeta_product(__s); |
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} |
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|
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|
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/** |
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* @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ |
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* for all s != 1 and x > -1. |
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* |
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* The Hurwitz zeta function is defined by: |
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* @f[ |
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* \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} |
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* @f] |
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* The Riemann zeta function is a special case: |
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* @f[ |
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* \zeta(s) = \zeta(1,s) |
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* @f] |
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* |
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* This functions uses the double sum that converges for s != 1 |
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* and x > -1: |
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* @f[ |
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* \zeta(x,s) = \frac{1}{s-1} |
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* \sum_{n=0}^{\infty} \frac{1}{n + 1} |
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* \sum_{k=0}^{n} (-1)^k \frac{n!}{(n-k)!k!} (x+k)^{-s} |
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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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__hurwitz_zeta_glob(_Tp __a, _Tp __s) |
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{ |
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_Tp __zeta = _Tp(0); |
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|
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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// Max e exponent before overflow. |
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const _Tp __max_bincoeff = std::numeric_limits<_Tp>::max_exponent10 |
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* std::log(_Tp(10)) - _Tp(1); |
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|
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const unsigned int __maxit = 10000; |
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for (unsigned int __i = 0; __i < __maxit; ++__i) |
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{ |
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bool __punt = false; |
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_Tp __sgn = _Tp(1); |
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_Tp __term = _Tp(0); |
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for (unsigned int __j = 0; __j <= __i; ++__j) |
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{ |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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_Tp __bincoeff = _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i)) |
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- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __j)) |
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- _GLIBCXX_MATH_NS::lgamma(_Tp(1 + __i - __j)); |
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#else |
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_Tp __bincoeff = __log_gamma(_Tp(1 + __i)) |
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- __log_gamma(_Tp(1 + __j)) |
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- __log_gamma(_Tp(1 + __i - __j)); |
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#endif |
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if (__bincoeff > __max_bincoeff) |
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{ |
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// This only gets hit for x << 0. |
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__punt = true; |
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break; |
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} |
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__bincoeff = std::exp(__bincoeff); |
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__term += __sgn * __bincoeff * std::pow(_Tp(__a + __j), -__s); |
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__sgn *= _Tp(-1); |
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} |
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if (__punt) |
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break; |
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__term /= _Tp(__i + 1); |
| 406 |
if (std::abs(__term / __zeta) < __eps) |
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break; |
| 408 |
__zeta += __term; |
| 409 |
} |
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|
| 411 |
__zeta /= __s - _Tp(1); |
| 412 |
|
| 413 |
return __zeta; |
| 414 |
} |
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|
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|
| 417 |
/** |
| 418 |
* @brief Return the Hurwitz zeta function @f$ \zeta(x,s) @f$ |
| 419 |
* for all s != 1 and x > -1. |
| 420 |
* |
| 421 |
* The Hurwitz zeta function is defined by: |
| 422 |
* @f[ |
| 423 |
* \zeta(x,s) = \sum_{n=0}^{\infty} \frac{1}{(n + x)^s} |
| 424 |
* @f] |
| 425 |
* The Riemann zeta function is a special case: |
| 426 |
* @f[ |
| 427 |
* \zeta(s) = \zeta(1,s) |
| 428 |
* @f] |
| 429 |
*/ |
| 430 |
template<typename _Tp> |
| 431 |
inline _Tp |
| 432 |
__hurwitz_zeta(_Tp __a, _Tp __s) |
| 433 |
{ return __hurwitz_zeta_glob(__a, __s); } |
| 434 |
} // namespace __detail |
| 435 |
#undef _GLIBCXX_MATH_NS |
| 436 |
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
| 437 |
} // namespace tr1 |
| 438 |
#endif |
| 439 |
|
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_GLIBCXX_END_NAMESPACE_VERSION |
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} |
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|
| 443 |
#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC |
