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// Special functions -*- C++ -*- |
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|
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// Copyright (C) 2006-2021 Free Software Foundation, Inc. |
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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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// 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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// Free Software Foundation; either version 3, or (at your option) |
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// any later version. |
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// |
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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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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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// GNU General Public License for more details. |
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// |
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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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// 3.1, as published by the Free Software Foundation. |
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|
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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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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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// <http://www.gnu.org/licenses/>. |
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|
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/** @file tr1/hypergeometric.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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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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// |
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|
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// Written by Edward Smith-Rowland based: |
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// (1) Handbook of Mathematical Functions, |
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// ed. Milton Abramowitz and Irene A. Stegun, |
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// Dover Publications, |
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// Section 6, pp. 555-566 |
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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|
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#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC |
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#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1 |
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|
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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|
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#if _GLIBCXX_USE_STD_SPEC_FUNCS |
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# define _GLIBCXX_MATH_NS ::std |
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#elif defined(_GLIBCXX_TR1_CMATH) |
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namespace tr1 |
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{ |
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# define _GLIBCXX_MATH_NS ::std::tr1 |
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#else |
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# error do not include this header directly, use <cmath> or <tr1/cmath> |
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#endif |
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// [5.2] Special functions |
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|
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// Implementation-space details. |
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namespace __detail |
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{ |
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/** |
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* @brief This routine returns the confluent hypergeometric function |
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* by series expansion. |
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* |
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* @f[ |
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* _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} |
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* \sum_{n=0}^{\infty} |
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* \frac{\Gamma(a+n)}{\Gamma(c+n)} |
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* \frac{x^n}{n!} |
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* @f] |
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* |
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* If a and b are integers and a < 0 and either b > 0 or b < a |
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* then the series is a polynomial with a finite number of |
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* terms. If b is an integer and b <= 0 the confluent |
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* hypergeometric function is undefined. |
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* |
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* @param __a The "numerator" parameter. |
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* @param __c The "denominator" parameter. |
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* @param __x The argument of the confluent hypergeometric function. |
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* @return The confluent hypergeometric function. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__conf_hyperg_series(_Tp __a, _Tp __c, _Tp __x) |
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{ |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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|
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_Tp __term = _Tp(1); |
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_Tp __Fac = _Tp(1); |
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const unsigned int __max_iter = 100000; |
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unsigned int __i; |
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for (__i = 0; __i < __max_iter; ++__i) |
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{ |
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__term *= (__a + _Tp(__i)) * __x |
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/ ((__c + _Tp(__i)) * _Tp(1 + __i)); |
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if (std::abs(__term) < __eps) |
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{ |
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break; |
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} |
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__Fac += __term; |
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} |
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if (__i == __max_iter) |
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std::__throw_runtime_error(__N("Series failed to converge " |
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"in __conf_hyperg_series.")); |
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|
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return __Fac; |
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} |
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|
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|
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/** |
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* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ |
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* by an iterative procedure described in |
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* Luke, Algorithms for the Computation of Mathematical Functions. |
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* |
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* Like the case of the 2F1 rational approximations, these are |
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* probably guaranteed to converge for x < 0, barring gross |
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* numerical instability in the pre-asymptotic regime. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__conf_hyperg_luke(_Tp __a, _Tp __c, _Tp __xin) |
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{ |
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const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); |
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const int __nmax = 20000; |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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const _Tp __x = -__xin; |
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const _Tp __x3 = __x * __x * __x; |
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const _Tp __t0 = __a / __c; |
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const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); |
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const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); |
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_Tp __F = _Tp(1); |
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_Tp __prec; |
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|
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_Tp __Bnm3 = _Tp(1); |
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_Tp __Bnm2 = _Tp(1) + __t1 * __x; |
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_Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); |
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|
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_Tp __Anm3 = _Tp(1); |
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_Tp __Anm2 = __Bnm2 - __t0 * __x; |
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_Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x |
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+ __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; |
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|
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int __n = 3; |
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while(1) |
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{ |
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_Tp __npam1 = _Tp(__n - 1) + __a; |
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_Tp __npcm1 = _Tp(__n - 1) + __c; |
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_Tp __npam2 = _Tp(__n - 2) + __a; |
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_Tp __npcm2 = _Tp(__n - 2) + __c; |
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_Tp __tnm1 = _Tp(2 * __n - 1); |
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_Tp __tnm3 = _Tp(2 * __n - 3); |
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_Tp __tnm5 = _Tp(2 * __n - 5); |
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_Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); |
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_Tp __F2 = (_Tp(__n) + __a) * __npam1 |
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/ (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); |
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_Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) |
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/ (_Tp(8) * __tnm3 * __tnm3 * __tnm5 |
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* (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); |
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_Tp __E = -__npam1 * (_Tp(__n - 1) - __c) |
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/ (_Tp(2) * __tnm3 * __npcm2 * __npcm1); |
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|
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_Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 |
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+ (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; |
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_Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 |
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+ (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; |
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_Tp __r = __An / __Bn; |
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|
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__prec = std::abs((__F - __r) / __F); |
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__F = __r; |
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|
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if (__prec < __eps || __n > __nmax) |
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break; |
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|
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if (std::abs(__An) > __big || std::abs(__Bn) > __big) |
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{ |
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__An /= __big; |
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__Bn /= __big; |
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__Anm1 /= __big; |
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__Bnm1 /= __big; |
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__Anm2 /= __big; |
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__Bnm2 /= __big; |
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__Anm3 /= __big; |
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__Bnm3 /= __big; |
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} |
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else if (std::abs(__An) < _Tp(1) / __big |
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|| std::abs(__Bn) < _Tp(1) / __big) |
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{ |
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__An *= __big; |
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__Bn *= __big; |
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__Anm1 *= __big; |
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__Bnm1 *= __big; |
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__Anm2 *= __big; |
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__Bnm2 *= __big; |
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__Anm3 *= __big; |
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__Bnm3 *= __big; |
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} |
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|
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++__n; |
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__Bnm3 = __Bnm2; |
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__Bnm2 = __Bnm1; |
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__Bnm1 = __Bn; |
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__Anm3 = __Anm2; |
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__Anm2 = __Anm1; |
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__Anm1 = __An; |
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} |
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|
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if (__n >= __nmax) |
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std::__throw_runtime_error(__N("Iteration failed to converge " |
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"in __conf_hyperg_luke.")); |
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|
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return __F; |
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} |
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|
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|
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/** |
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* @brief Return the confluent hypogeometric function |
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* @f$ _1F_1(a;c;x) @f$. |
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* |
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* @todo Handle b == nonpositive integer blowup - return NaN. |
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* |
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* @param __a The @a numerator parameter. |
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* @param __c The @a denominator parameter. |
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* @param __x The argument of the confluent hypergeometric function. |
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* @return The confluent hypergeometric function. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__conf_hyperg(_Tp __a, _Tp __c, _Tp __x) |
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{ |
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c); |
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#else |
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const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); |
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#endif |
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if (__isnan(__a) || __isnan(__c) || __isnan(__x)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (__c_nint == __c && __c_nint <= 0) |
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return std::numeric_limits<_Tp>::infinity(); |
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else if (__a == _Tp(0)) |
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return _Tp(1); |
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else if (__c == __a) |
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return std::exp(__x); |
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else if (__x < _Tp(0)) |
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return __conf_hyperg_luke(__a, __c, __x); |
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else |
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return __conf_hyperg_series(__a, __c, __x); |
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} |
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|
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|
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/** |
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* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ |
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* by series expansion. |
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* |
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* The hypogeometric function is defined by |
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* @f[ |
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* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} |
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* \sum_{n=0}^{\infty} |
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* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} |
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* \frac{x^n}{n!} |
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* @f] |
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* |
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* This works and it's pretty fast. |
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* |
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* @param __a The first @a numerator parameter. |
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* @param __a The second @a numerator parameter. |
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* @param __c The @a denominator parameter. |
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* @param __x The argument of the confluent hypergeometric function. |
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* @return The confluent hypergeometric function. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__hyperg_series(_Tp __a, _Tp __b, _Tp __c, _Tp __x) |
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{ |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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|
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_Tp __term = _Tp(1); |
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_Tp __Fabc = _Tp(1); |
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const unsigned int __max_iter = 100000; |
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unsigned int __i; |
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for (__i = 0; __i < __max_iter; ++__i) |
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{ |
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__term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x |
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/ ((__c + _Tp(__i)) * _Tp(1 + __i)); |
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if (std::abs(__term) < __eps) |
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{ |
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break; |
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} |
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__Fabc += __term; |
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} |
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if (__i == __max_iter) |
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std::__throw_runtime_error(__N("Series failed to converge " |
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"in __hyperg_series.")); |
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|
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return __Fabc; |
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} |
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|
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|
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/** |
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* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ |
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* by an iterative procedure described in |
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* Luke, Algorithms for the Computation of Mathematical Functions. |
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*/ |
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template<typename _Tp> |
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_Tp |
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__hyperg_luke(_Tp __a, _Tp __b, _Tp __c, _Tp __xin) |
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{ |
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const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); |
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const int __nmax = 20000; |
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const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
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const _Tp __x = -__xin; |
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const _Tp __x3 = __x * __x * __x; |
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const _Tp __t0 = __a * __b / __c; |
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const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); |
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const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) |
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/ (_Tp(2) * (__c + _Tp(1))); |
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|
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_Tp __F = _Tp(1); |
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|
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_Tp __Bnm3 = _Tp(1); |
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_Tp __Bnm2 = _Tp(1) + __t1 * __x; |
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_Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); |
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|
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_Tp __Anm3 = _Tp(1); |
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_Tp __Anm2 = __Bnm2 - __t0 * __x; |
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_Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x |
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+ __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; |
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|
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int __n = 3; |
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while (1) |
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{ |
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const _Tp __npam1 = _Tp(__n - 1) + __a; |
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const _Tp __npbm1 = _Tp(__n - 1) + __b; |
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const _Tp __npcm1 = _Tp(__n - 1) + __c; |
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const _Tp __npam2 = _Tp(__n - 2) + __a; |
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const _Tp __npbm2 = _Tp(__n - 2) + __b; |
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const _Tp __npcm2 = _Tp(__n - 2) + __c; |
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const _Tp __tnm1 = _Tp(2 * __n - 1); |
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const _Tp __tnm3 = _Tp(2 * __n - 3); |
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const _Tp __tnm5 = _Tp(2 * __n - 5); |
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const _Tp __n2 = __n * __n; |
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const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n |
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+ _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) |
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/ (_Tp(2) * __tnm3 * __npcm1); |
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const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n |
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+ _Tp(2) - __a * __b) * __npam1 * __npbm1 |
| 347 |
/ (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); |
| 348 |
const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 |
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* (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) |
| 350 |
/ (_Tp(8) * __tnm3 * __tnm3 * __tnm5 |
| 351 |
* (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); |
| 352 |
const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) |
| 353 |
/ (_Tp(2) * __tnm3 * __npcm2 * __npcm1); |
| 354 |
|
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_Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 |
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+ (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; |
| 357 |
_Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 |
| 358 |
+ (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; |
| 359 |
const _Tp __r = __An / __Bn; |
| 360 |
|
| 361 |
const _Tp __prec = std::abs((__F - __r) / __F); |
| 362 |
__F = __r; |
| 363 |
|
| 364 |
if (__prec < __eps || __n > __nmax) |
| 365 |
break; |
| 366 |
|
| 367 |
if (std::abs(__An) > __big || std::abs(__Bn) > __big) |
| 368 |
{ |
| 369 |
__An /= __big; |
| 370 |
__Bn /= __big; |
| 371 |
__Anm1 /= __big; |
| 372 |
__Bnm1 /= __big; |
| 373 |
__Anm2 /= __big; |
| 374 |
__Bnm2 /= __big; |
| 375 |
__Anm3 /= __big; |
| 376 |
__Bnm3 /= __big; |
| 377 |
} |
| 378 |
else if (std::abs(__An) < _Tp(1) / __big |
| 379 |
|| std::abs(__Bn) < _Tp(1) / __big) |
| 380 |
{ |
| 381 |
__An *= __big; |
| 382 |
__Bn *= __big; |
| 383 |
__Anm1 *= __big; |
| 384 |
__Bnm1 *= __big; |
| 385 |
__Anm2 *= __big; |
| 386 |
__Bnm2 *= __big; |
| 387 |
__Anm3 *= __big; |
| 388 |
__Bnm3 *= __big; |
| 389 |
} |
| 390 |
|
| 391 |
++__n; |
| 392 |
__Bnm3 = __Bnm2; |
| 393 |
__Bnm2 = __Bnm1; |
| 394 |
__Bnm1 = __Bn; |
| 395 |
__Anm3 = __Anm2; |
| 396 |
__Anm2 = __Anm1; |
| 397 |
__Anm1 = __An; |
| 398 |
} |
| 399 |
|
| 400 |
if (__n >= __nmax) |
| 401 |
std::__throw_runtime_error(__N("Iteration failed to converge " |
| 402 |
"in __hyperg_luke.")); |
| 403 |
|
| 404 |
return __F; |
| 405 |
} |
| 406 |
|
| 407 |
|
| 408 |
/** |
| 409 |
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ |
| 410 |
* by the reflection formulae in Abramowitz & Stegun formula |
| 411 |
* 15.3.6 for d = c - a - b not integral and formula 15.3.11 for |
| 412 |
* d = c - a - b integral. This assumes a, b, c != negative |
| 413 |
* integer. |
| 414 |
* |
| 415 |
* The hypogeometric function is defined by |
| 416 |
* @f[ |
| 417 |
* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} |
| 418 |
* \sum_{n=0}^{\infty} |
| 419 |
* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} |
| 420 |
* \frac{x^n}{n!} |
| 421 |
* @f] |
| 422 |
* |
| 423 |
* The reflection formula for nonintegral @f$ d = c - a - b @f$ is: |
| 424 |
* @f[ |
| 425 |
* _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} |
| 426 |
* _2F_1(a,b;1-d;1-x) |
| 427 |
* + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} |
| 428 |
* _2F_1(c-a,c-b;1+d;1-x) |
| 429 |
* @f] |
| 430 |
* |
| 431 |
* The reflection formula for integral @f$ m = c - a - b @f$ is: |
| 432 |
* @f[ |
| 433 |
* _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} |
| 434 |
* \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} |
| 435 |
* - |
| 436 |
* @f] |
| 437 |
*/ |
| 438 |
template<typename _Tp> |
| 439 |
_Tp |
| 440 |
__hyperg_reflect(_Tp __a, _Tp __b, _Tp __c, _Tp __x) |
| 441 |
{ |
| 442 |
const _Tp __d = __c - __a - __b; |
| 443 |
const int __intd = std::floor(__d + _Tp(0.5L)); |
| 444 |
const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
| 445 |
const _Tp __toler = _Tp(1000) * __eps; |
| 446 |
const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); |
| 447 |
const bool __d_integer = (std::abs(__d - __intd) < __toler); |
| 448 |
|
| 449 |
if (__d_integer) |
| 450 |
{ |
| 451 |
const _Tp __ln_omx = std::log(_Tp(1) - __x); |
| 452 |
const _Tp __ad = std::abs(__d); |
| 453 |
_Tp __F1, __F2; |
| 454 |
|
| 455 |
_Tp __d1, __d2; |
| 456 |
if (__d >= _Tp(0)) |
| 457 |
{ |
| 458 |
__d1 = __d; |
| 459 |
__d2 = _Tp(0); |
| 460 |
} |
| 461 |
else |
| 462 |
{ |
| 463 |
__d1 = _Tp(0); |
| 464 |
__d2 = __d; |
| 465 |
} |
| 466 |
|
| 467 |
const _Tp __lng_c = __log_gamma(__c); |
| 468 |
|
| 469 |
// Evaluate F1. |
| 470 |
if (__ad < __eps) |
| 471 |
{ |
| 472 |
// d = c - a - b = 0. |
| 473 |
__F1 = _Tp(0); |
| 474 |
} |
| 475 |
else |
| 476 |
{ |
| 477 |
|
| 478 |
bool __ok_d1 = true; |
| 479 |
_Tp __lng_ad, __lng_ad1, __lng_bd1; |
| 480 |
__try |
| 481 |
{ |
| 482 |
__lng_ad = __log_gamma(__ad); |
| 483 |
__lng_ad1 = __log_gamma(__a + __d1); |
| 484 |
__lng_bd1 = __log_gamma(__b + __d1); |
| 485 |
} |
| 486 |
__catch(...) |
| 487 |
{ |
| 488 |
__ok_d1 = false; |
| 489 |
} |
| 490 |
|
| 491 |
if (__ok_d1) |
| 492 |
{ |
| 493 |
/* Gamma functions in the denominator are ok. |
| 494 |
* Proceed with evaluation. |
| 495 |
*/ |
| 496 |
_Tp __sum1 = _Tp(1); |
| 497 |
_Tp __term = _Tp(1); |
| 498 |
_Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx |
| 499 |
- __lng_ad1 - __lng_bd1; |
| 500 |
|
| 501 |
/* Do F1 sum. |
| 502 |
*/ |
| 503 |
for (int __i = 1; __i < __ad; ++__i) |
| 504 |
{ |
| 505 |
const int __j = __i - 1; |
| 506 |
__term *= (__a + __d2 + __j) * (__b + __d2 + __j) |
| 507 |
/ (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); |
| 508 |
__sum1 += __term; |
| 509 |
} |
| 510 |
|
| 511 |
if (__ln_pre1 > __log_max) |
| 512 |
std::__throw_runtime_error(__N("Overflow of gamma functions" |
| 513 |
" in __hyperg_luke.")); |
| 514 |
else |
| 515 |
__F1 = std::exp(__ln_pre1) * __sum1; |
| 516 |
} |
| 517 |
else |
| 518 |
{ |
| 519 |
// Gamma functions in the denominator were not ok. |
| 520 |
// So the F1 term is zero. |
| 521 |
__F1 = _Tp(0); |
| 522 |
} |
| 523 |
} // end F1 evaluation |
| 524 |
|
| 525 |
// Evaluate F2. |
| 526 |
bool __ok_d2 = true; |
| 527 |
_Tp __lng_ad2, __lng_bd2; |
| 528 |
__try |
| 529 |
{ |
| 530 |
__lng_ad2 = __log_gamma(__a + __d2); |
| 531 |
__lng_bd2 = __log_gamma(__b + __d2); |
| 532 |
} |
| 533 |
__catch(...) |
| 534 |
{ |
| 535 |
__ok_d2 = false; |
| 536 |
} |
| 537 |
|
| 538 |
if (__ok_d2) |
| 539 |
{ |
| 540 |
// Gamma functions in the denominator are ok. |
| 541 |
// Proceed with evaluation. |
| 542 |
const int __maxiter = 2000; |
| 543 |
const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); |
| 544 |
const _Tp __psi_1pd = __psi(_Tp(1) + __ad); |
| 545 |
const _Tp __psi_apd1 = __psi(__a + __d1); |
| 546 |
const _Tp __psi_bpd1 = __psi(__b + __d1); |
| 547 |
|
| 548 |
_Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 |
| 549 |
- __psi_bpd1 - __ln_omx; |
| 550 |
_Tp __fact = _Tp(1); |
| 551 |
_Tp __sum2 = __psi_term; |
| 552 |
_Tp __ln_pre2 = __lng_c + __d1 * __ln_omx |
| 553 |
- __lng_ad2 - __lng_bd2; |
| 554 |
|
| 555 |
// Do F2 sum. |
| 556 |
int __j; |
| 557 |
for (__j = 1; __j < __maxiter; ++__j) |
| 558 |
{ |
| 559 |
// Values for psi functions use recurrence; |
| 560 |
// Abramowitz & Stegun 6.3.5 |
| 561 |
const _Tp __term1 = _Tp(1) / _Tp(__j) |
| 562 |
+ _Tp(1) / (__ad + __j); |
| 563 |
const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) |
| 564 |
+ _Tp(1) / (__b + __d1 + _Tp(__j - 1)); |
| 565 |
__psi_term += __term1 - __term2; |
| 566 |
__fact *= (__a + __d1 + _Tp(__j - 1)) |
| 567 |
* (__b + __d1 + _Tp(__j - 1)) |
| 568 |
/ ((__ad + __j) * __j) * (_Tp(1) - __x); |
| 569 |
const _Tp __delta = __fact * __psi_term; |
| 570 |
__sum2 += __delta; |
| 571 |
if (std::abs(__delta) < __eps * std::abs(__sum2)) |
| 572 |
break; |
| 573 |
} |
| 574 |
if (__j == __maxiter) |
| 575 |
std::__throw_runtime_error(__N("Sum F2 failed to converge " |
| 576 |
"in __hyperg_reflect")); |
| 577 |
|
| 578 |
if (__sum2 == _Tp(0)) |
| 579 |
__F2 = _Tp(0); |
| 580 |
else |
| 581 |
__F2 = std::exp(__ln_pre2) * __sum2; |
| 582 |
} |
| 583 |
else |
| 584 |
{ |
| 585 |
// Gamma functions in the denominator not ok. |
| 586 |
// So the F2 term is zero. |
| 587 |
__F2 = _Tp(0); |
| 588 |
} // end F2 evaluation |
| 589 |
|
| 590 |
const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); |
| 591 |
const _Tp __F = __F1 + __sgn_2 * __F2; |
| 592 |
|
| 593 |
return __F; |
| 594 |
} |
| 595 |
else |
| 596 |
{ |
| 597 |
// d = c - a - b not an integer. |
| 598 |
|
| 599 |
// These gamma functions appear in the denominator, so we |
| 600 |
// catch their harmless domain errors and set the terms to zero. |
| 601 |
bool __ok1 = true; |
| 602 |
_Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); |
| 603 |
_Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); |
| 604 |
__try |
| 605 |
{ |
| 606 |
__sgn_g1ca = __log_gamma_sign(__c - __a); |
| 607 |
__ln_g1ca = __log_gamma(__c - __a); |
| 608 |
__sgn_g1cb = __log_gamma_sign(__c - __b); |
| 609 |
__ln_g1cb = __log_gamma(__c - __b); |
| 610 |
} |
| 611 |
__catch(...) |
| 612 |
{ |
| 613 |
__ok1 = false; |
| 614 |
} |
| 615 |
|
| 616 |
bool __ok2 = true; |
| 617 |
_Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); |
| 618 |
_Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); |
| 619 |
__try |
| 620 |
{ |
| 621 |
__sgn_g2a = __log_gamma_sign(__a); |
| 622 |
__ln_g2a = __log_gamma(__a); |
| 623 |
__sgn_g2b = __log_gamma_sign(__b); |
| 624 |
__ln_g2b = __log_gamma(__b); |
| 625 |
} |
| 626 |
__catch(...) |
| 627 |
{ |
| 628 |
__ok2 = false; |
| 629 |
} |
| 630 |
|
| 631 |
const _Tp __sgn_gc = __log_gamma_sign(__c); |
| 632 |
const _Tp __ln_gc = __log_gamma(__c); |
| 633 |
const _Tp __sgn_gd = __log_gamma_sign(__d); |
| 634 |
const _Tp __ln_gd = __log_gamma(__d); |
| 635 |
const _Tp __sgn_gmd = __log_gamma_sign(-__d); |
| 636 |
const _Tp __ln_gmd = __log_gamma(-__d); |
| 637 |
|
| 638 |
const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; |
| 639 |
const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; |
| 640 |
|
| 641 |
_Tp __pre1, __pre2; |
| 642 |
if (__ok1 && __ok2) |
| 643 |
{ |
| 644 |
_Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; |
| 645 |
_Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b |
| 646 |
+ __d * std::log(_Tp(1) - __x); |
| 647 |
if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) |
| 648 |
{ |
| 649 |
__pre1 = std::exp(__ln_pre1); |
| 650 |
__pre2 = std::exp(__ln_pre2); |
| 651 |
__pre1 *= __sgn1; |
| 652 |
__pre2 *= __sgn2; |
| 653 |
} |
| 654 |
else |
| 655 |
{ |
| 656 |
std::__throw_runtime_error(__N("Overflow of gamma functions " |
| 657 |
"in __hyperg_reflect")); |
| 658 |
} |
| 659 |
} |
| 660 |
else if (__ok1 && !__ok2) |
| 661 |
{ |
| 662 |
_Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; |
| 663 |
if (__ln_pre1 < __log_max) |
| 664 |
{ |
| 665 |
__pre1 = std::exp(__ln_pre1); |
| 666 |
__pre1 *= __sgn1; |
| 667 |
__pre2 = _Tp(0); |
| 668 |
} |
| 669 |
else |
| 670 |
{ |
| 671 |
std::__throw_runtime_error(__N("Overflow of gamma functions " |
| 672 |
"in __hyperg_reflect")); |
| 673 |
} |
| 674 |
} |
| 675 |
else if (!__ok1 && __ok2) |
| 676 |
{ |
| 677 |
_Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b |
| 678 |
+ __d * std::log(_Tp(1) - __x); |
| 679 |
if (__ln_pre2 < __log_max) |
| 680 |
{ |
| 681 |
__pre1 = _Tp(0); |
| 682 |
__pre2 = std::exp(__ln_pre2); |
| 683 |
__pre2 *= __sgn2; |
| 684 |
} |
| 685 |
else |
| 686 |
{ |
| 687 |
std::__throw_runtime_error(__N("Overflow of gamma functions " |
| 688 |
"in __hyperg_reflect")); |
| 689 |
} |
| 690 |
} |
| 691 |
else |
| 692 |
{ |
| 693 |
__pre1 = _Tp(0); |
| 694 |
__pre2 = _Tp(0); |
| 695 |
std::__throw_runtime_error(__N("Underflow of gamma functions " |
| 696 |
"in __hyperg_reflect")); |
| 697 |
} |
| 698 |
|
| 699 |
const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, |
| 700 |
_Tp(1) - __x); |
| 701 |
const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, |
| 702 |
_Tp(1) - __x); |
| 703 |
|
| 704 |
const _Tp __F = __pre1 * __F1 + __pre2 * __F2; |
| 705 |
|
| 706 |
return __F; |
| 707 |
} |
| 708 |
} |
| 709 |
|
| 710 |
|
| 711 |
/** |
| 712 |
* @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. |
| 713 |
* |
| 714 |
* The hypogeometric function is defined by |
| 715 |
* @f[ |
| 716 |
* _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} |
| 717 |
* \sum_{n=0}^{\infty} |
| 718 |
* \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} |
| 719 |
* \frac{x^n}{n!} |
| 720 |
* @f] |
| 721 |
* |
| 722 |
* @param __a The first @a numerator parameter. |
| 723 |
* @param __a The second @a numerator parameter. |
| 724 |
* @param __c The @a denominator parameter. |
| 725 |
* @param __x The argument of the confluent hypergeometric function. |
| 726 |
* @return The confluent hypergeometric function. |
| 727 |
*/ |
| 728 |
template<typename _Tp> |
| 729 |
_Tp |
| 730 |
__hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x) |
| 731 |
{ |
| 732 |
#if _GLIBCXX_USE_C99_MATH_TR1 |
| 733 |
const _Tp __a_nint = _GLIBCXX_MATH_NS::nearbyint(__a); |
| 734 |
const _Tp __b_nint = _GLIBCXX_MATH_NS::nearbyint(__b); |
| 735 |
const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c); |
| 736 |
#else |
| 737 |
const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); |
| 738 |
const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); |
| 739 |
const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); |
| 740 |
#endif |
| 741 |
const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); |
| 742 |
if (std::abs(__x) >= _Tp(1)) |
| 743 |
std::__throw_domain_error(__N("Argument outside unit circle " |
| 744 |
"in __hyperg.")); |
| 745 |
else if (__isnan(__a) || __isnan(__b) |
| 746 |
|| __isnan(__c) || __isnan(__x)) |
| 747 |
return std::numeric_limits<_Tp>::quiet_NaN(); |
| 748 |
else if (__c_nint == __c && __c_nint <= _Tp(0)) |
| 749 |
return std::numeric_limits<_Tp>::infinity(); |
| 750 |
else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) |
| 751 |
return std::pow(_Tp(1) - __x, __c - __a - __b); |
| 752 |
else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) |
| 753 |
&& __x >= _Tp(0) && __x < _Tp(0.995L)) |
| 754 |
return __hyperg_series(__a, __b, __c, __x); |
| 755 |
else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) |
| 756 |
{ |
| 757 |
// For integer a and b the hypergeometric function is a |
| 758 |
// finite polynomial. |
| 759 |
if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) |
| 760 |
return __hyperg_series(__a_nint, __b, __c, __x); |
| 761 |
else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) |
| 762 |
return __hyperg_series(__a, __b_nint, __c, __x); |
| 763 |
else if (__x < -_Tp(0.25L)) |
| 764 |
return __hyperg_luke(__a, __b, __c, __x); |
| 765 |
else if (__x < _Tp(0.5L)) |
| 766 |
return __hyperg_series(__a, __b, __c, __x); |
| 767 |
else |
| 768 |
if (std::abs(__c) > _Tp(10)) |
| 769 |
return __hyperg_series(__a, __b, __c, __x); |
| 770 |
else |
| 771 |
return __hyperg_reflect(__a, __b, __c, __x); |
| 772 |
} |
| 773 |
else |
| 774 |
return __hyperg_luke(__a, __b, __c, __x); |
| 775 |
} |
| 776 |
} // namespace __detail |
| 777 |
#undef _GLIBCXX_MATH_NS |
| 778 |
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
| 779 |
} // namespace tr1 |
| 780 |
#endif |
| 781 |
|
| 782 |
_GLIBCXX_END_NAMESPACE_VERSION |
| 783 |
} |
| 784 |
|
| 785 |
#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC |
