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-// Special functions -*- C++ -*-
-
-// Copyright (C) 2006, 2007, 2008, 2009
-// 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.
- * You should not attempt to use it directly.
- */
-
-//
-// 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
-{
-namespace tr1
-{
-
- // [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(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);
- }
-
- } // namespace std::tr1::__detail
-}
-}
-
-#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC