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gcc/libstdc++-v3/include/tr1/riemann_zeta.tcc
Jonathan Wakely 42d9f14bab PR libstdc++/88066 Use <> for includes not "" 
These headers were missed in the previous commit for this bug.

There are also several "" includes in the profile mode headers, but
because they're deprecated I'm not fixing them.

	* include/backward/hash_map: Use <> for includes not "".
	* include/backward/hash_set: Likewise.
	* include/backward/strstream: Likewise.
	* include/tr1/bessel_function.tcc: Likewise.
	* include/tr1/exp_integral.tcc: Likewise.
	* include/tr1/legendre_function.tcc: Likewise.
	* include/tr1/modified_bessel_func.tcc: Likewise.
	* include/tr1/riemann_zeta.tcc: Likewise.

From-SVN: r269835
2019-03-21 12:07:10 +00:00

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// Special functions -*- C++ -*-
// Copyright (C) 2006-2019 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
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