// Special functions -*- C++ -*-
// Copyright (C) 2006-2017 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)
{
#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
{
_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(_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); }
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace __detail
#undef _GLIBCXX_MATH_NS
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
} // namespace tr1
#endif
}
#endif // _GLIBCXX_TR1_RIEMANN_ZETA_TCC