444 lines
14 KiB
C++
444 lines
14 KiB
C++
// Special functions -*- C++ -*-
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// Copyright (C) 2006-2017 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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// 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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/** @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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// ISO C++ 14882 TR1: 5.2 Special functions
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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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#ifndef _GLIBCXX_TR1_RIEMANN_ZETA_TCC
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#define _GLIBCXX_TR1_RIEMANN_ZETA_TCC 1
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#include "special_function_util.h"
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namespace std _GLIBCXX_VISIBILITY(default)
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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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// Implementation-space details.
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namespace __detail
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{
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_GLIBCXX_BEGIN_NAMESPACE_VERSION
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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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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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return __zeta;
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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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return __zeta;
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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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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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// 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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_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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__zeta /= _Tp(1) - std::pow(_Tp(2), _Tp(1) - __s);
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return __zeta;
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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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_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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__zeta = _Tp(1) / __zeta;
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return __zeta;
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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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* @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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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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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);
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if (std::abs(__term / __zeta) < __eps)
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break;
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__zeta += __term;
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}
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__zeta /= __s - _Tp(1);
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return __zeta;
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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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template<typename _Tp>
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inline _Tp
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__hurwitz_zeta(_Tp __a, _Tp __s)
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{ return __hurwitz_zeta_glob(__a, __s); }
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_GLIBCXX_END_NAMESPACE_VERSION
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} // namespace __detail
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#undef _GLIBCXX_MATH_NS
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#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
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} // namespace tr1
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#endif
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}
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#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC
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