// Special functions -*- C++ -*-
// Copyright (C) 2006-2013 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/gamma.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. Milton Abramowitz and Irene A. Stegun,
// Dover Publications,
// Section 6, pp. 253-266
// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
// 2nd ed, pp. 213-216
// (4) Gamma, Exploring Euler's Constant, Julian Havil,
// Princeton, 2003.
#ifndef _GLIBCXX_TR1_GAMMA_TCC
#define _GLIBCXX_TR1_GAMMA_TCC 1
#include "special_function_util.h"
namespace std _GLIBCXX_VISIBILITY(default)
{
namespace tr1
{
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @brief This returns Bernoulli numbers from a table or by summation
* for larger values.
*
* Recursion is unstable.
*
* @param __n the order n of the Bernoulli number.
* @return The Bernoulli number of order n.
*/
template <typename _Tp>
_Tp
__bernoulli_series(unsigned int __n)
{
static const _Tp __num[28] = {
_Tp(1UL), -_Tp(1UL) / _Tp(2UL),
_Tp(1UL) / _Tp(6UL), _Tp(0UL),
-_Tp(1UL) / _Tp(30UL), _Tp(0UL),
_Tp(1UL) / _Tp(42UL), _Tp(0UL),
-_Tp(1UL) / _Tp(30UL), _Tp(0UL),
_Tp(5UL) / _Tp(66UL), _Tp(0UL),
-_Tp(691UL) / _Tp(2730UL), _Tp(0UL),
_Tp(7UL) / _Tp(6UL), _Tp(0UL),
-_Tp(3617UL) / _Tp(510UL), _Tp(0UL),
_Tp(43867UL) / _Tp(798UL), _Tp(0UL),
-_Tp(174611) / _Tp(330UL), _Tp(0UL),
_Tp(854513UL) / _Tp(138UL), _Tp(0UL),
-_Tp(236364091UL) / _Tp(2730UL), _Tp(0UL),
_Tp(8553103UL) / _Tp(6UL), _Tp(0UL)
};
if (__n == 0)
return _Tp(1);
if (__n == 1)
return -_Tp(1) / _Tp(2);
// Take care of the rest of the odd ones.
if (__n % 2 == 1)
return _Tp(0);
// Take care of some small evens that are painful for the series.
if (__n < 28)
return __num[__n];
_Tp __fact = _Tp(1);
if ((__n / 2) % 2 == 0)
__fact *= _Tp(-1);
for (unsigned int __k = 1; __k <= __n; ++__k)
__fact *= __k / (_Tp(2) * __numeric_constants<_Tp>::__pi());
__fact *= _Tp(2);
_Tp __sum = _Tp(0);
for (unsigned int __i = 1; __i < 1000; ++__i)
{
_Tp __term = std::pow(_Tp(__i), -_Tp(__n));
if (__term < std::numeric_limits<_Tp>::epsilon())
break;
__sum += __term;
}
return __fact * __sum;
}
/**
* @brief This returns Bernoulli number \f$B_n\f$.
*
* @param __n the order n of the Bernoulli number.
* @return The Bernoulli number of order n.
*/
template<typename _Tp>
inline _Tp
__bernoulli(int __n)
{ return __bernoulli_series<_Tp>(__n); }
/**
* @brief Return \f$log(\Gamma(x))\f$ by asymptotic expansion
* with Bernoulli number coefficients. This is like
* Sterling's approximation.
*
* @param __x The argument of the log of the gamma function.
* @return The logarithm of the gamma function.
*/
template<typename _Tp>
_Tp
__log_gamma_bernoulli(_Tp __x)
{
_Tp __lg = (__x - _Tp(0.5L)) * std::log(__x) - __x
+ _Tp(0.5L) * std::log(_Tp(2)
* __numeric_constants<_Tp>::__pi());
const _Tp __xx = __x * __x;
_Tp __help = _Tp(1) / __x;
for ( unsigned int __i = 1; __i < 20; ++__i )
{
const _Tp __2i = _Tp(2 * __i);
__help /= __2i * (__2i - _Tp(1)) * __xx;
__lg += __bernoulli<_Tp>(2 * __i) * __help;
}
return __lg;
}
/**
* @brief Return \f$log(\Gamma(x))\f$ by the Lanczos method.
* This method dominates all others on the positive axis I think.
*
* @param __x The argument of the log of the gamma function.
* @return The logarithm of the gamma function.
*/
template<typename _Tp>
_Tp
__log_gamma_lanczos(_Tp __x)
{
const _Tp __xm1 = __x - _Tp(1);
static const _Tp __lanczos_cheb_7[9] = {
_Tp( 0.99999999999980993227684700473478L),
_Tp( 676.520368121885098567009190444019L),
_Tp(-1259.13921672240287047156078755283L),
_Tp( 771.3234287776530788486528258894L),
_Tp(-176.61502916214059906584551354L),
_Tp( 12.507343278686904814458936853L),
_Tp(-0.13857109526572011689554707L),
_Tp( 9.984369578019570859563e-6L),
_Tp( 1.50563273514931155834e-7L)
};
static const _Tp __LOGROOT2PI
= _Tp(0.9189385332046727417803297364056176L);
_Tp __sum = __lanczos_cheb_7[0];
for(unsigned int __k = 1; __k < 9; ++__k)
__sum += __lanczos_cheb_7[__k] / (__xm1 + __k);
const _Tp __term1 = (__xm1 + _Tp(0.5L))
* std::log((__xm1 + _Tp(7.5L))
/ __numeric_constants<_Tp>::__euler());
const _Tp __term2 = __LOGROOT2PI + std::log(__sum);
const _Tp __result = __term1 + (__term2 - _Tp(7));
return __result;
}
/**
* @brief Return \f$ log(|\Gamma(x)|) \f$.
* This will return values even for \f$ x < 0 \f$.
* To recover the sign of \f$ \Gamma(x) \f$ for
* any argument use @a __log_gamma_sign.
*
* @param __x The argument of the log of the gamma function.
* @return The logarithm of the gamma function.
*/
template<typename _Tp>
_Tp
__log_gamma(_Tp __x)
{
if (__x > _Tp(0.5L))
return __log_gamma_lanczos(__x);
else
{
const _Tp __sin_fact
= std::abs(std::sin(__numeric_constants<_Tp>::__pi() * __x));
if (__sin_fact == _Tp(0))
std::__throw_domain_error(__N("Argument is nonpositive integer "
"in __log_gamma"));
return __numeric_constants<_Tp>::__lnpi()
- std::log(__sin_fact)
- __log_gamma_lanczos(_Tp(1) - __x);
}
}
/**
* @brief Return the sign of \f$ \Gamma(x) \f$.
* At nonpositive integers zero is returned.
*
* @param __x The argument of the gamma function.
* @return The sign of the gamma function.
*/
template<typename _Tp>
_Tp
__log_gamma_sign(_Tp __x)
{
if (__x > _Tp(0))
return _Tp(1);
else
{
const _Tp __sin_fact
= std::sin(__numeric_constants<_Tp>::__pi() * __x);
if (__sin_fact > _Tp(0))
return (1);
else if (__sin_fact < _Tp(0))
return -_Tp(1);
else
return _Tp(0);
}
}
/**
* @brief Return the logarithm of the binomial coefficient.
* The binomial coefficient is given by:
* @f[
* \left( \right) = \frac{n!}{(n-k)! k!}
* @f]
*
* @param __n The first argument of the binomial coefficient.
* @param __k The second argument of the binomial coefficient.
* @return The binomial coefficient.
*/
template<typename _Tp>
_Tp
__log_bincoef(unsigned int __n, unsigned int __k)
{
// Max e exponent before overflow.
static const _Tp __max_bincoeff
= std::numeric_limits<_Tp>::max_exponent10
* std::log(_Tp(10)) - _Tp(1);
#if _GLIBCXX_USE_C99_MATH_TR1
_Tp __coeff = std::tr1::lgamma(_Tp(1 + __n))
- std::tr1::lgamma(_Tp(1 + __k))
- std::tr1::lgamma(_Tp(1 + __n - __k));
#else
_Tp __coeff = __log_gamma(_Tp(1 + __n))
- __log_gamma(_Tp(1 + __k))
- __log_gamma(_Tp(1 + __n - __k));
#endif
}
/**
* @brief Return the binomial coefficient.
* The binomial coefficient is given by:
* @f[
* \left( \right) = \frac{n!}{(n-k)! k!}
* @f]
*
* @param __n The first argument of the binomial coefficient.
* @param __k The second argument of the binomial coefficient.
* @return The binomial coefficient.
*/
template<typename _Tp>
_Tp
__bincoef(unsigned int __n, unsigned int __k)
{
// Max e exponent before overflow.
static const _Tp __max_bincoeff
= std::numeric_limits<_Tp>::max_exponent10
* std::log(_Tp(10)) - _Tp(1);
const _Tp __log_coeff = __log_bincoef<_Tp>(__n, __k);
if (__log_coeff > __max_bincoeff)
return std::numeric_limits<_Tp>::quiet_NaN();
else
return std::exp(__log_coeff);
}
/**
* @brief Return \f$ \Gamma(x) \f$.
*
* @param __x The argument of the gamma function.
* @return The gamma function.
*/
template<typename _Tp>
inline _Tp
__gamma(_Tp __x)
{ return std::exp(__log_gamma(__x)); }
/**
* @brief Return the digamma function by series expansion.
* The digamma or @f$ \psi(x) @f$ function is defined by
* @f[
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
* @f]
*
* The series is given by:
* @f[
* \psi(x) = -\gamma_E - \frac{1}{x}
* \sum_{k=1}^{\infty} \frac{x}{k(x + k)}
* @f]
*/
template<typename _Tp>
_Tp
__psi_series(_Tp __x)
{
_Tp __sum = -__numeric_constants<_Tp>::__gamma_e() - _Tp(1) / __x;
const unsigned int __max_iter = 100000;
for (unsigned int __k = 1; __k < __max_iter; ++__k)
{
const _Tp __term = __x / (__k * (__k + __x));
__sum += __term;
if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
break;
}
return __sum;
}
/**
* @brief Return the digamma function for large argument.
* The digamma or @f$ \psi(x) @f$ function is defined by
* @f[
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
* @f]
*
* The asymptotic series is given by:
* @f[
* \psi(x) = \ln(x) - \frac{1}{2x}
* - \sum_{n=1}^{\infty} \frac{B_{2n}}{2 n x^{2n}}
* @f]
*/
template<typename _Tp>
_Tp
__psi_asymp(_Tp __x)
{
_Tp __sum = std::log(__x) - _Tp(0.5L) / __x;
const _Tp __xx = __x * __x;
_Tp __xp = __xx;
const unsigned int __max_iter = 100;
for (unsigned int __k = 1; __k < __max_iter; ++__k)
{
const _Tp __term = __bernoulli<_Tp>(2 * __k) / (2 * __k * __xp);
__sum -= __term;
if (std::abs(__term / __sum) < std::numeric_limits<_Tp>::epsilon())
break;
__xp *= __xx;
}
return __sum;
}
/**
* @brief Return the digamma function.
* The digamma or @f$ \psi(x) @f$ function is defined by
* @f[
* \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}
* @f]
* For negative argument the reflection formula is used:
* @f[
* \psi(x) = \psi(1-x) - \pi \cot(\pi x)
* @f]
*/
template<typename _Tp>
_Tp
__psi(_Tp __x)
{
const int __n = static_cast<int>(__x + 0.5L);
const _Tp __eps = _Tp(4) * std::numeric_limits<_Tp>::epsilon();
if (__n <= 0 && std::abs(__x - _Tp(__n)) < __eps)
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__x < _Tp(0))
{
const _Tp __pi = __numeric_constants<_Tp>::__pi();
return __psi(_Tp(1) - __x)
- __pi * std::cos(__pi * __x) / std::sin(__pi * __x);
}
else if (__x > _Tp(100))
return __psi_asymp(__x);
else
return __psi_series(__x);
}
/**
* @brief Return the polygamma function @f$ \psi^{(n)}(x) @f$.
*
* The polygamma function is related to the Hurwitz zeta function:
* @f[
* \psi^{(n)}(x) = (-1)^{n+1} m! \zeta(m+1,x)
* @f]
*/
template<typename _Tp>
_Tp
__psi(unsigned int __n, _Tp __x)
{
if (__x <= _Tp(0))
std::__throw_domain_error(__N("Argument out of range "
"in __psi"));
else if (__n == 0)
return __psi(__x);
else
{
const _Tp __hzeta = __hurwitz_zeta(_Tp(__n + 1), __x);
#if _GLIBCXX_USE_C99_MATH_TR1
const _Tp __ln_nfact = std::tr1::lgamma(_Tp(__n + 1));
#else
const _Tp __ln_nfact = __log_gamma(_Tp(__n + 1));
#endif
_Tp __result = std::exp(__ln_nfact) * __hzeta;
if (__n % 2 == 1)
__result = -__result;
return __result;
}
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std::tr1::__detail
}
}
#endif // _GLIBCXX_TR1_GAMMA_TCC