2012-03-27 23:13:14 +00:00
|
|
|
// Special functions -*- C++ -*-
|
|
|
|
|
2019-06-02 15:48:37 +00:00
|
|
|
// Copyright (C) 2006-2019 Free Software Foundation, Inc.
|
2012-03-27 23:13:14 +00:00
|
|
|
//
|
|
|
|
// 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/beta_function.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_BETA_FUNCTION_TCC
|
|
|
|
#define _GLIBCXX_TR1_BETA_FUNCTION_TCC 1
|
|
|
|
|
|
|
|
namespace std _GLIBCXX_VISIBILITY(default)
|
|
|
|
{
|
2018-12-28 15:30:48 +00:00
|
|
|
_GLIBCXX_BEGIN_NAMESPACE_VERSION
|
|
|
|
|
2017-10-07 00:16:47 +00:00
|
|
|
#if _GLIBCXX_USE_STD_SPEC_FUNCS
|
2017-04-10 11:32:00 +00:00
|
|
|
# define _GLIBCXX_MATH_NS ::std
|
|
|
|
#elif defined(_GLIBCXX_TR1_CMATH)
|
2012-03-27 23:13:14 +00:00
|
|
|
namespace tr1
|
|
|
|
{
|
2017-04-10 11:32:00 +00:00
|
|
|
# define _GLIBCXX_MATH_NS ::std::tr1
|
|
|
|
#else
|
|
|
|
# error do not include this header directly, use <cmath> or <tr1/cmath>
|
|
|
|
#endif
|
2012-03-27 23:13:14 +00:00
|
|
|
// [5.2] Special functions
|
|
|
|
|
|
|
|
// Implementation-space details.
|
|
|
|
namespace __detail
|
|
|
|
{
|
|
|
|
/**
|
|
|
|
* @brief Return the beta function: \f$B(x,y)\f$.
|
|
|
|
*
|
|
|
|
* The beta function is defined by
|
|
|
|
* @f[
|
|
|
|
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
|
|
|
|
* @f]
|
|
|
|
*
|
|
|
|
* @param __x The first argument of the beta function.
|
|
|
|
* @param __y The second argument of the beta function.
|
|
|
|
* @return The beta function.
|
|
|
|
*/
|
|
|
|
template<typename _Tp>
|
|
|
|
_Tp
|
|
|
|
__beta_gamma(_Tp __x, _Tp __y)
|
|
|
|
{
|
|
|
|
|
|
|
|
_Tp __bet;
|
|
|
|
#if _GLIBCXX_USE_C99_MATH_TR1
|
|
|
|
if (__x > __y)
|
|
|
|
{
|
2017-04-10 11:32:00 +00:00
|
|
|
__bet = _GLIBCXX_MATH_NS::tgamma(__x)
|
|
|
|
/ _GLIBCXX_MATH_NS::tgamma(__x + __y);
|
|
|
|
__bet *= _GLIBCXX_MATH_NS::tgamma(__y);
|
2012-03-27 23:13:14 +00:00
|
|
|
}
|
|
|
|
else
|
|
|
|
{
|
2017-04-10 11:32:00 +00:00
|
|
|
__bet = _GLIBCXX_MATH_NS::tgamma(__y)
|
|
|
|
/ _GLIBCXX_MATH_NS::tgamma(__x + __y);
|
|
|
|
__bet *= _GLIBCXX_MATH_NS::tgamma(__x);
|
2012-03-27 23:13:14 +00:00
|
|
|
}
|
|
|
|
#else
|
|
|
|
if (__x > __y)
|
|
|
|
{
|
|
|
|
__bet = __gamma(__x) / __gamma(__x + __y);
|
|
|
|
__bet *= __gamma(__y);
|
|
|
|
}
|
|
|
|
else
|
|
|
|
{
|
|
|
|
__bet = __gamma(__y) / __gamma(__x + __y);
|
|
|
|
__bet *= __gamma(__x);
|
|
|
|
}
|
|
|
|
#endif
|
|
|
|
|
|
|
|
return __bet;
|
|
|
|
}
|
|
|
|
|
|
|
|
/**
|
|
|
|
* @brief Return the beta function \f$B(x,y)\f$ using
|
|
|
|
* the log gamma functions.
|
|
|
|
*
|
|
|
|
* The beta function is defined by
|
|
|
|
* @f[
|
|
|
|
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
|
|
|
|
* @f]
|
|
|
|
*
|
|
|
|
* @param __x The first argument of the beta function.
|
|
|
|
* @param __y The second argument of the beta function.
|
|
|
|
* @return The beta function.
|
|
|
|
*/
|
|
|
|
template<typename _Tp>
|
|
|
|
_Tp
|
|
|
|
__beta_lgamma(_Tp __x, _Tp __y)
|
|
|
|
{
|
|
|
|
#if _GLIBCXX_USE_C99_MATH_TR1
|
2017-04-10 11:32:00 +00:00
|
|
|
_Tp __bet = _GLIBCXX_MATH_NS::lgamma(__x)
|
|
|
|
+ _GLIBCXX_MATH_NS::lgamma(__y)
|
|
|
|
- _GLIBCXX_MATH_NS::lgamma(__x + __y);
|
2012-03-27 23:13:14 +00:00
|
|
|
#else
|
|
|
|
_Tp __bet = __log_gamma(__x)
|
|
|
|
+ __log_gamma(__y)
|
|
|
|
- __log_gamma(__x + __y);
|
|
|
|
#endif
|
|
|
|
__bet = std::exp(__bet);
|
|
|
|
return __bet;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
* @brief Return the beta function \f$B(x,y)\f$ using
|
|
|
|
* the product form.
|
|
|
|
*
|
|
|
|
* The beta function is defined by
|
|
|
|
* @f[
|
|
|
|
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
|
|
|
|
* @f]
|
|
|
|
*
|
|
|
|
* @param __x The first argument of the beta function.
|
|
|
|
* @param __y The second argument of the beta function.
|
|
|
|
* @return The beta function.
|
|
|
|
*/
|
|
|
|
template<typename _Tp>
|
|
|
|
_Tp
|
|
|
|
__beta_product(_Tp __x, _Tp __y)
|
|
|
|
{
|
|
|
|
|
|
|
|
_Tp __bet = (__x + __y) / (__x * __y);
|
|
|
|
|
|
|
|
unsigned int __max_iter = 1000000;
|
|
|
|
for (unsigned int __k = 1; __k < __max_iter; ++__k)
|
|
|
|
{
|
|
|
|
_Tp __term = (_Tp(1) + (__x + __y) / __k)
|
|
|
|
/ ((_Tp(1) + __x / __k) * (_Tp(1) + __y / __k));
|
|
|
|
__bet *= __term;
|
|
|
|
}
|
|
|
|
|
|
|
|
return __bet;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
* @brief Return the beta function \f$ B(x,y) \f$.
|
|
|
|
*
|
|
|
|
* The beta function is defined by
|
|
|
|
* @f[
|
|
|
|
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
|
|
|
|
* @f]
|
|
|
|
*
|
|
|
|
* @param __x The first argument of the beta function.
|
|
|
|
* @param __y The second argument of the beta function.
|
|
|
|
* @return The beta function.
|
|
|
|
*/
|
|
|
|
template<typename _Tp>
|
|
|
|
inline _Tp
|
|
|
|
__beta(_Tp __x, _Tp __y)
|
|
|
|
{
|
|
|
|
if (__isnan(__x) || __isnan(__y))
|
|
|
|
return std::numeric_limits<_Tp>::quiet_NaN();
|
|
|
|
else
|
|
|
|
return __beta_lgamma(__x, __y);
|
|
|
|
}
|
2017-04-10 11:32:00 +00:00
|
|
|
} // namespace __detail
|
|
|
|
#undef _GLIBCXX_MATH_NS
|
2017-10-07 00:16:47 +00:00
|
|
|
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
|
2017-04-10 11:32:00 +00:00
|
|
|
} // namespace tr1
|
|
|
|
#endif
|
2018-12-28 15:30:48 +00:00
|
|
|
|
|
|
|
_GLIBCXX_END_NAMESPACE_VERSION
|
2012-03-27 23:13:14 +00:00
|
|
|
}
|
|
|
|
|
2015-08-28 15:33:40 +00:00
|
|
|
#endif // _GLIBCXX_TR1_BETA_FUNCTION_TCC
|