// Special functions -*- C++ -*-
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// Special functions -*- C++ -*-
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// Copyright (C) 2006, 2007, 2008, 2009
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// Copyright (C) 2006, 2007, 2008, 2009
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// Free Software Foundation, Inc.
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// Free Software Foundation, Inc.
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//
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//
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// This file is part of the GNU ISO C++ Library. This library is free
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// This file is part of the GNU ISO C++ Library. This library is free
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// software; you can redistribute it and/or modify it under the
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// software; you can redistribute it and/or modify it under the
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// terms of the GNU General Public License as published by the
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// terms of the GNU General Public License as published by the
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// Free Software Foundation; either version 3, or (at your option)
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// Free Software Foundation; either version 3, or (at your option)
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// any later version.
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// any later version.
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//
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//
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// This library is distributed in the hope that it will be useful,
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// This library is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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// GNU General Public License for more details.
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//
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//
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// Under Section 7 of GPL version 3, you are granted additional
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// Under Section 7 of GPL version 3, you are granted additional
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// permissions described in the GCC Runtime Library Exception, version
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// permissions described in the GCC Runtime Library Exception, version
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// 3.1, as published by the Free Software Foundation.
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// 3.1, as published by the Free Software Foundation.
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// You should have received a copy of the GNU General Public License and
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// You should have received a copy of the GNU General Public License and
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// a copy of the GCC Runtime Library Exception along with this program;
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// a copy of the GCC Runtime Library Exception along with this program;
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// .
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// .
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/** @file tr1/beta_function.tcc
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/** @file tr1/beta_function.tcc
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* This is an internal header file, included by other library headers.
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* This is an internal header file, included by other library headers.
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* You should not attempt to use it directly.
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* You should not attempt to use it directly.
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*/
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*/
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//
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//
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// ISO C++ 14882 TR1: 5.2 Special functions
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// ISO C++ 14882 TR1: 5.2 Special functions
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//
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//
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// Written by Edward Smith-Rowland based on:
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// Written by Edward Smith-Rowland based on:
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// (1) Handbook of Mathematical Functions,
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// (1) Handbook of Mathematical Functions,
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// ed. Milton Abramowitz and Irene A. Stegun,
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// ed. Milton Abramowitz and Irene A. Stegun,
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// Dover Publications,
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// Dover Publications,
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// Section 6, pp. 253-266
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// Section 6, pp. 253-266
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
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// 2nd ed, pp. 213-216
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// 2nd ed, pp. 213-216
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// (4) Gamma, Exploring Euler's Constant, Julian Havil,
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// (4) Gamma, Exploring Euler's Constant, Julian Havil,
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// Princeton, 2003.
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// Princeton, 2003.
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#ifndef _GLIBCXX_TR1_BETA_FUNCTION_TCC
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#ifndef _GLIBCXX_TR1_BETA_FUNCTION_TCC
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#define _GLIBCXX_TR1_BETA_FUNCTION_TCC 1
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#define _GLIBCXX_TR1_BETA_FUNCTION_TCC 1
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namespace std
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namespace std
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{
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{
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namespace tr1
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namespace tr1
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{
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{
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// [5.2] Special functions
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// [5.2] Special functions
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// Implementation-space details.
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// Implementation-space details.
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namespace __detail
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namespace __detail
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{
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{
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/**
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/**
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* @brief Return the beta function: \f$B(x,y)\f$.
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* @brief Return the beta function: \f$B(x,y)\f$.
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*
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*
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* The beta function is defined by
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* The beta function is defined by
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* @f[
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* @f[
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* @f]
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* @f]
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*
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*
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* @param __x The first argument of the beta function.
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* @param __x The first argument of the beta function.
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* @param __y The second argument of the beta function.
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* @param __y The second argument of the beta function.
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* @return The beta function.
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* @return The beta function.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__beta_gamma(_Tp __x, _Tp __y)
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__beta_gamma(_Tp __x, _Tp __y)
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{
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{
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_Tp __bet;
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_Tp __bet;
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#if _GLIBCXX_USE_C99_MATH_TR1
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#if _GLIBCXX_USE_C99_MATH_TR1
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if (__x > __y)
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if (__x > __y)
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{
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{
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__bet = std::tr1::tgamma(__x)
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__bet = std::tr1::tgamma(__x)
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/ std::tr1::tgamma(__x + __y);
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/ std::tr1::tgamma(__x + __y);
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__bet *= std::tr1::tgamma(__y);
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__bet *= std::tr1::tgamma(__y);
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}
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}
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else
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else
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{
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{
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__bet = std::tr1::tgamma(__y)
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__bet = std::tr1::tgamma(__y)
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/ std::tr1::tgamma(__x + __y);
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/ std::tr1::tgamma(__x + __y);
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__bet *= std::tr1::tgamma(__x);
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__bet *= std::tr1::tgamma(__x);
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}
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}
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#else
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#else
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if (__x > __y)
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if (__x > __y)
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{
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{
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__bet = __gamma(__x) / __gamma(__x + __y);
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__bet = __gamma(__x) / __gamma(__x + __y);
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__bet *= __gamma(__y);
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__bet *= __gamma(__y);
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}
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}
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else
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else
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{
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{
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__bet = __gamma(__y) / __gamma(__x + __y);
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__bet = __gamma(__y) / __gamma(__x + __y);
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__bet *= __gamma(__x);
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__bet *= __gamma(__x);
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}
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}
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#endif
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#endif
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return __bet;
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return __bet;
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}
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}
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/**
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/**
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* @brief Return the beta function \f$B(x,y)\f$ using
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* @brief Return the beta function \f$B(x,y)\f$ using
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* the log gamma functions.
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* the log gamma functions.
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*
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*
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* The beta function is defined by
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* The beta function is defined by
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* @f[
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* @f[
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* @f]
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* @f]
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*
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*
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* @param __x The first argument of the beta function.
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* @param __x The first argument of the beta function.
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* @param __y The second argument of the beta function.
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* @param __y The second argument of the beta function.
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* @return The beta function.
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* @return The beta function.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__beta_lgamma(_Tp __x, _Tp __y)
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__beta_lgamma(_Tp __x, _Tp __y)
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{
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{
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#if _GLIBCXX_USE_C99_MATH_TR1
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#if _GLIBCXX_USE_C99_MATH_TR1
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_Tp __bet = std::tr1::lgamma(__x)
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_Tp __bet = std::tr1::lgamma(__x)
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+ std::tr1::lgamma(__y)
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+ std::tr1::lgamma(__y)
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- std::tr1::lgamma(__x + __y);
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- std::tr1::lgamma(__x + __y);
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#else
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#else
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_Tp __bet = __log_gamma(__x)
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_Tp __bet = __log_gamma(__x)
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+ __log_gamma(__y)
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+ __log_gamma(__y)
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- __log_gamma(__x + __y);
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- __log_gamma(__x + __y);
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#endif
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#endif
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__bet = std::exp(__bet);
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__bet = std::exp(__bet);
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return __bet;
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return __bet;
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}
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}
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/**
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/**
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* @brief Return the beta function \f$B(x,y)\f$ using
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* @brief Return the beta function \f$B(x,y)\f$ using
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* the product form.
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* the product form.
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*
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*
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* The beta function is defined by
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* The beta function is defined by
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* @f[
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* @f[
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* @f]
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* @f]
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*
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*
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* @param __x The first argument of the beta function.
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* @param __x The first argument of the beta function.
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* @param __y The second argument of the beta function.
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* @param __y The second argument of the beta function.
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* @return The beta function.
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* @return The beta function.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__beta_product(_Tp __x, _Tp __y)
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__beta_product(_Tp __x, _Tp __y)
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{
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{
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_Tp __bet = (__x + __y) / (__x * __y);
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_Tp __bet = (__x + __y) / (__x * __y);
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unsigned int __max_iter = 1000000;
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unsigned int __max_iter = 1000000;
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for (unsigned int __k = 1; __k < __max_iter; ++__k)
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for (unsigned int __k = 1; __k < __max_iter; ++__k)
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{
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{
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_Tp __term = (_Tp(1) + (__x + __y) / __k)
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_Tp __term = (_Tp(1) + (__x + __y) / __k)
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/ ((_Tp(1) + __x / __k) * (_Tp(1) + __y / __k));
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/ ((_Tp(1) + __x / __k) * (_Tp(1) + __y / __k));
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__bet *= __term;
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__bet *= __term;
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}
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}
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return __bet;
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return __bet;
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}
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}
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/**
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/**
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* @brief Return the beta function \f$ B(x,y) \f$.
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* @brief Return the beta function \f$ B(x,y) \f$.
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*
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*
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* The beta function is defined by
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* The beta function is defined by
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* @f[
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* @f[
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
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* @f]
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* @f]
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*
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*
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* @param __x The first argument of the beta function.
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* @param __x The first argument of the beta function.
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* @param __y The second argument of the beta function.
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* @param __y The second argument of the beta function.
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* @return The beta function.
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* @return The beta function.
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*/
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*/
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template
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template
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inline _Tp
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inline _Tp
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__beta(_Tp __x, _Tp __y)
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__beta(_Tp __x, _Tp __y)
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{
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{
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if (__isnan(__x) || __isnan(__y))
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if (__isnan(__x) || __isnan(__y))
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return std::numeric_limits<_Tp>::quiet_NaN();
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return std::numeric_limits<_Tp>::quiet_NaN();
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else
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else
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return __beta_lgamma(__x, __y);
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return __beta_lgamma(__x, __y);
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}
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}
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} // namespace std::tr1::__detail
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} // namespace std::tr1::__detail
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}
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}
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}
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}
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#endif // __GLIBCXX_TR1_BETA_FUNCTION_TCC
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#endif // __GLIBCXX_TR1_BETA_FUNCTION_TCC
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