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Diffstat (limited to 'gcc-4.4.3/libstdc++-v3/include/tr1/hypergeometric.tcc')
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diff --git a/gcc-4.4.3/libstdc++-v3/include/tr1/hypergeometric.tcc b/gcc-4.4.3/libstdc++-v3/include/tr1/hypergeometric.tcc deleted file mode 100644 index c975afa59..000000000 --- a/gcc-4.4.3/libstdc++-v3/include/tr1/hypergeometric.tcc +++ /dev/null @@ -1,774 +0,0 @@ -// Special functions -*- C++ -*- - -// Copyright (C) 2006, 2007, 2008, 2009 -// 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/hypergeometric.tcc - * This is an internal header file, included by other library headers. - * You should not attempt to use it directly. - */ - -// -// ISO C++ 14882 TR1: 5.2 Special functions -// - -// Written by Edward Smith-Rowland based: -// (1) Handbook of Mathematical Functions, -// ed. Milton Abramowitz and Irene A. Stegun, -// Dover Publications, -// Section 6, pp. 555-566 -// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl - -#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC -#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1 - -namespace std -{ -namespace tr1 -{ - - // [5.2] Special functions - - // Implementation-space details. - namespace __detail - { - - /** - * @brief This routine returns the confluent hypergeometric function - * by series expansion. - * - * @f[ - * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)} - * \sum_{n=0}^{\infty} - * \frac{\Gamma(a+n)}{\Gamma(c+n)} - * \frac{x^n}{n!} - * @f] - * - * If a and b are integers and a < 0 and either b > 0 or b < a then the - * series is a polynomial with a finite number of terms. If b is an integer - * and b <= 0 the confluent hypergeometric function is undefined. - * - * @param __a The "numerator" parameter. - * @param __c The "denominator" parameter. - * @param __x The argument of the confluent hypergeometric function. - * @return The confluent hypergeometric function. - */ - template<typename _Tp> - _Tp - __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x) - { - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - - _Tp __term = _Tp(1); - _Tp __Fac = _Tp(1); - const unsigned int __max_iter = 100000; - unsigned int __i; - for (__i = 0; __i < __max_iter; ++__i) - { - __term *= (__a + _Tp(__i)) * __x - / ((__c + _Tp(__i)) * _Tp(1 + __i)); - if (std::abs(__term) < __eps) - { - break; - } - __Fac += __term; - } - if (__i == __max_iter) - std::__throw_runtime_error(__N("Series failed to converge " - "in __conf_hyperg_series.")); - - return __Fac; - } - - - /** - * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ - * by an iterative procedure described in - * Luke, Algorithms for the Computation of Mathematical Functions. - * - * Like the case of the 2F1 rational approximations, these are - * probably guaranteed to converge for x < 0, barring gross - * numerical instability in the pre-asymptotic regime. - */ - template<typename _Tp> - _Tp - __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin) - { - const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); - const int __nmax = 20000; - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - const _Tp __x = -__xin; - const _Tp __x3 = __x * __x * __x; - const _Tp __t0 = __a / __c; - const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c); - const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1))); - _Tp __F = _Tp(1); - _Tp __prec; - - _Tp __Bnm3 = _Tp(1); - _Tp __Bnm2 = _Tp(1) + __t1 * __x; - _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); - - _Tp __Anm3 = _Tp(1); - _Tp __Anm2 = __Bnm2 - __t0 * __x; - _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x - + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; - - int __n = 3; - while(1) - { - _Tp __npam1 = _Tp(__n - 1) + __a; - _Tp __npcm1 = _Tp(__n - 1) + __c; - _Tp __npam2 = _Tp(__n - 2) + __a; - _Tp __npcm2 = _Tp(__n - 2) + __c; - _Tp __tnm1 = _Tp(2 * __n - 1); - _Tp __tnm3 = _Tp(2 * __n - 3); - _Tp __tnm5 = _Tp(2 * __n - 5); - _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1); - _Tp __F2 = (_Tp(__n) + __a) * __npam1 - / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); - _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a) - / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 - * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); - _Tp __E = -__npam1 * (_Tp(__n - 1) - __c) - / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); - - _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 - + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; - _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 - + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; - _Tp __r = __An / __Bn; - - __prec = std::abs((__F - __r) / __F); - __F = __r; - - if (__prec < __eps || __n > __nmax) - break; - - if (std::abs(__An) > __big || std::abs(__Bn) > __big) - { - __An /= __big; - __Bn /= __big; - __Anm1 /= __big; - __Bnm1 /= __big; - __Anm2 /= __big; - __Bnm2 /= __big; - __Anm3 /= __big; - __Bnm3 /= __big; - } - else if (std::abs(__An) < _Tp(1) / __big - || std::abs(__Bn) < _Tp(1) / __big) - { - __An *= __big; - __Bn *= __big; - __Anm1 *= __big; - __Bnm1 *= __big; - __Anm2 *= __big; - __Bnm2 *= __big; - __Anm3 *= __big; - __Bnm3 *= __big; - } - - ++__n; - __Bnm3 = __Bnm2; - __Bnm2 = __Bnm1; - __Bnm1 = __Bn; - __Anm3 = __Anm2; - __Anm2 = __Anm1; - __Anm1 = __An; - } - - if (__n >= __nmax) - std::__throw_runtime_error(__N("Iteration failed to converge " - "in __conf_hyperg_luke.")); - - return __F; - } - - - /** - * @brief Return the confluent hypogeometric function - * @f$ _1F_1(a;c;x) @f$. - * - * @todo Handle b == nonpositive integer blowup - return NaN. - * - * @param __a The "numerator" parameter. - * @param __c The "denominator" parameter. - * @param __x The argument of the confluent hypergeometric function. - * @return The confluent hypergeometric function. - */ - template<typename _Tp> - inline _Tp - __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x) - { -#if _GLIBCXX_USE_C99_MATH_TR1 - const _Tp __c_nint = std::tr1::nearbyint(__c); -#else - const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); -#endif - if (__isnan(__a) || __isnan(__c) || __isnan(__x)) - return std::numeric_limits<_Tp>::quiet_NaN(); - else if (__c_nint == __c && __c_nint <= 0) - return std::numeric_limits<_Tp>::infinity(); - else if (__a == _Tp(0)) - return _Tp(1); - else if (__c == __a) - return std::exp(__x); - else if (__x < _Tp(0)) - return __conf_hyperg_luke(__a, __c, __x); - else - return __conf_hyperg_series(__a, __c, __x); - } - - - /** - * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ - * by series expansion. - * - * The hypogeometric function is defined by - * @f[ - * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} - * \sum_{n=0}^{\infty} - * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} - * \frac{x^n}{n!} - * @f] - * - * This works and it's pretty fast. - * - * @param __a The first "numerator" parameter. - * @param __a The second "numerator" parameter. - * @param __c The "denominator" parameter. - * @param __x The argument of the confluent hypergeometric function. - * @return The confluent hypergeometric function. - */ - template<typename _Tp> - _Tp - __hyperg_series(const _Tp __a, const _Tp __b, - const _Tp __c, const _Tp __x) - { - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - - _Tp __term = _Tp(1); - _Tp __Fabc = _Tp(1); - const unsigned int __max_iter = 100000; - unsigned int __i; - for (__i = 0; __i < __max_iter; ++__i) - { - __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x - / ((__c + _Tp(__i)) * _Tp(1 + __i)); - if (std::abs(__term) < __eps) - { - break; - } - __Fabc += __term; - } - if (__i == __max_iter) - std::__throw_runtime_error(__N("Series failed to converge " - "in __hyperg_series.")); - - return __Fabc; - } - - - /** - * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ - * by an iterative procedure described in - * Luke, Algorithms for the Computation of Mathematical Functions. - */ - template<typename _Tp> - _Tp - __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c, - const _Tp __xin) - { - const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L)); - const int __nmax = 20000; - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - const _Tp __x = -__xin; - const _Tp __x3 = __x * __x * __x; - const _Tp __t0 = __a * __b / __c; - const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c); - const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2)) - / (_Tp(2) * (__c + _Tp(1))); - - _Tp __F = _Tp(1); - - _Tp __Bnm3 = _Tp(1); - _Tp __Bnm2 = _Tp(1) + __t1 * __x; - _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x); - - _Tp __Anm3 = _Tp(1); - _Tp __Anm2 = __Bnm2 - __t0 * __x; - _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x - + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x; - - int __n = 3; - while (1) - { - const _Tp __npam1 = _Tp(__n - 1) + __a; - const _Tp __npbm1 = _Tp(__n - 1) + __b; - const _Tp __npcm1 = _Tp(__n - 1) + __c; - const _Tp __npam2 = _Tp(__n - 2) + __a; - const _Tp __npbm2 = _Tp(__n - 2) + __b; - const _Tp __npcm2 = _Tp(__n - 2) + __c; - const _Tp __tnm1 = _Tp(2 * __n - 1); - const _Tp __tnm3 = _Tp(2 * __n - 3); - const _Tp __tnm5 = _Tp(2 * __n - 5); - const _Tp __n2 = __n * __n; - const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n - + _Tp(2) - __a * __b - _Tp(2) * (__a + __b)) - / (_Tp(2) * __tnm3 * __npcm1); - const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n - + _Tp(2) - __a * __b) * __npam1 * __npbm1 - / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1); - const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1 - * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b)) - / (_Tp(8) * __tnm3 * __tnm3 * __tnm5 - * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1); - const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c) - / (_Tp(2) * __tnm3 * __npcm2 * __npcm1); - - _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1 - + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3; - _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1 - + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3; - const _Tp __r = __An / __Bn; - - const _Tp __prec = std::abs((__F - __r) / __F); - __F = __r; - - if (__prec < __eps || __n > __nmax) - break; - - if (std::abs(__An) > __big || std::abs(__Bn) > __big) - { - __An /= __big; - __Bn /= __big; - __Anm1 /= __big; - __Bnm1 /= __big; - __Anm2 /= __big; - __Bnm2 /= __big; - __Anm3 /= __big; - __Bnm3 /= __big; - } - else if (std::abs(__An) < _Tp(1) / __big - || std::abs(__Bn) < _Tp(1) / __big) - { - __An *= __big; - __Bn *= __big; - __Anm1 *= __big; - __Bnm1 *= __big; - __Anm2 *= __big; - __Bnm2 *= __big; - __Anm3 *= __big; - __Bnm3 *= __big; - } - - ++__n; - __Bnm3 = __Bnm2; - __Bnm2 = __Bnm1; - __Bnm1 = __Bn; - __Anm3 = __Anm2; - __Anm2 = __Anm1; - __Anm1 = __An; - } - - if (__n >= __nmax) - std::__throw_runtime_error(__N("Iteration failed to converge " - "in __hyperg_luke.")); - - return __F; - } - - - /** - * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection - * formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral - * and formula 15.3.11 for d = c - a - b integral. - * This assumes a, b, c != negative integer. - * - * The hypogeometric function is defined by - * @f[ - * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} - * \sum_{n=0}^{\infty} - * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} - * \frac{x^n}{n!} - * @f] - * - * The reflection formula for nonintegral @f$ d = c - a - b @f$ is: - * @f[ - * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)} - * _2F_1(a,b;1-d;1-x) - * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)} - * _2F_1(c-a,c-b;1+d;1-x) - * @f] - * - * The reflection formula for integral @f$ m = c - a - b @f$ is: - * @f[ - * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)} - * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k} - * - - * @f] - */ - template<typename _Tp> - _Tp - __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c, - const _Tp __x) - { - const _Tp __d = __c - __a - __b; - const int __intd = std::floor(__d + _Tp(0.5L)); - const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); - const _Tp __toler = _Tp(1000) * __eps; - const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max()); - const bool __d_integer = (std::abs(__d - __intd) < __toler); - - if (__d_integer) - { - const _Tp __ln_omx = std::log(_Tp(1) - __x); - const _Tp __ad = std::abs(__d); - _Tp __F1, __F2; - - _Tp __d1, __d2; - if (__d >= _Tp(0)) - { - __d1 = __d; - __d2 = _Tp(0); - } - else - { - __d1 = _Tp(0); - __d2 = __d; - } - - const _Tp __lng_c = __log_gamma(__c); - - // Evaluate F1. - if (__ad < __eps) - { - // d = c - a - b = 0. - __F1 = _Tp(0); - } - else - { - - bool __ok_d1 = true; - _Tp __lng_ad, __lng_ad1, __lng_bd1; - __try - { - __lng_ad = __log_gamma(__ad); - __lng_ad1 = __log_gamma(__a + __d1); - __lng_bd1 = __log_gamma(__b + __d1); - } - __catch(...) - { - __ok_d1 = false; - } - - if (__ok_d1) - { - /* Gamma functions in the denominator are ok. - * Proceed with evaluation. - */ - _Tp __sum1 = _Tp(1); - _Tp __term = _Tp(1); - _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx - - __lng_ad1 - __lng_bd1; - - /* Do F1 sum. - */ - for (int __i = 1; __i < __ad; ++__i) - { - const int __j = __i - 1; - __term *= (__a + __d2 + __j) * (__b + __d2 + __j) - / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x); - __sum1 += __term; - } - - if (__ln_pre1 > __log_max) - std::__throw_runtime_error(__N("Overflow of gamma functions " - "in __hyperg_luke.")); - else - __F1 = std::exp(__ln_pre1) * __sum1; - } - else - { - // Gamma functions in the denominator were not ok. - // So the F1 term is zero. - __F1 = _Tp(0); - } - } // end F1 evaluation - - // Evaluate F2. - bool __ok_d2 = true; - _Tp __lng_ad2, __lng_bd2; - __try - { - __lng_ad2 = __log_gamma(__a + __d2); - __lng_bd2 = __log_gamma(__b + __d2); - } - __catch(...) - { - __ok_d2 = false; - } - - if (__ok_d2) - { - // Gamma functions in the denominator are ok. - // Proceed with evaluation. - const int __maxiter = 2000; - const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e(); - const _Tp __psi_1pd = __psi(_Tp(1) + __ad); - const _Tp __psi_apd1 = __psi(__a + __d1); - const _Tp __psi_bpd1 = __psi(__b + __d1); - - _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1 - - __psi_bpd1 - __ln_omx; - _Tp __fact = _Tp(1); - _Tp __sum2 = __psi_term; - _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx - - __lng_ad2 - __lng_bd2; - - // Do F2 sum. - int __j; - for (__j = 1; __j < __maxiter; ++__j) - { - // Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5 - const _Tp __term1 = _Tp(1) / _Tp(__j) - + _Tp(1) / (__ad + __j); - const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1)) - + _Tp(1) / (__b + __d1 + _Tp(__j - 1)); - __psi_term += __term1 - __term2; - __fact *= (__a + __d1 + _Tp(__j - 1)) - * (__b + __d1 + _Tp(__j - 1)) - / ((__ad + __j) * __j) * (_Tp(1) - __x); - const _Tp __delta = __fact * __psi_term; - __sum2 += __delta; - if (std::abs(__delta) < __eps * std::abs(__sum2)) - break; - } - if (__j == __maxiter) - std::__throw_runtime_error(__N("Sum F2 failed to converge " - "in __hyperg_reflect")); - - if (__sum2 == _Tp(0)) - __F2 = _Tp(0); - else - __F2 = std::exp(__ln_pre2) * __sum2; - } - else - { - // Gamma functions in the denominator not ok. - // So the F2 term is zero. - __F2 = _Tp(0); - } // end F2 evaluation - - const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1)); - const _Tp __F = __F1 + __sgn_2 * __F2; - - return __F; - } - else - { - // d = c - a - b not an integer. - - // These gamma functions appear in the denominator, so we - // catch their harmless domain errors and set the terms to zero. - bool __ok1 = true; - _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0); - _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0); - __try - { - __sgn_g1ca = __log_gamma_sign(__c - __a); - __ln_g1ca = __log_gamma(__c - __a); - __sgn_g1cb = __log_gamma_sign(__c - __b); - __ln_g1cb = __log_gamma(__c - __b); - } - __catch(...) - { - __ok1 = false; - } - - bool __ok2 = true; - _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0); - _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0); - __try - { - __sgn_g2a = __log_gamma_sign(__a); - __ln_g2a = __log_gamma(__a); - __sgn_g2b = __log_gamma_sign(__b); - __ln_g2b = __log_gamma(__b); - } - __catch(...) - { - __ok2 = false; - } - - const _Tp __sgn_gc = __log_gamma_sign(__c); - const _Tp __ln_gc = __log_gamma(__c); - const _Tp __sgn_gd = __log_gamma_sign(__d); - const _Tp __ln_gd = __log_gamma(__d); - const _Tp __sgn_gmd = __log_gamma_sign(-__d); - const _Tp __ln_gmd = __log_gamma(-__d); - - const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb; - const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b; - - _Tp __pre1, __pre2; - if (__ok1 && __ok2) - { - _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; - _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b - + __d * std::log(_Tp(1) - __x); - if (__ln_pre1 < __log_max && __ln_pre2 < __log_max) - { - __pre1 = std::exp(__ln_pre1); - __pre2 = std::exp(__ln_pre2); - __pre1 *= __sgn1; - __pre2 *= __sgn2; - } - else - { - std::__throw_runtime_error(__N("Overflow of gamma functions " - "in __hyperg_reflect")); - } - } - else if (__ok1 && !__ok2) - { - _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb; - if (__ln_pre1 < __log_max) - { - __pre1 = std::exp(__ln_pre1); - __pre1 *= __sgn1; - __pre2 = _Tp(0); - } - else - { - std::__throw_runtime_error(__N("Overflow of gamma functions " - "in __hyperg_reflect")); - } - } - else if (!__ok1 && __ok2) - { - _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b - + __d * std::log(_Tp(1) - __x); - if (__ln_pre2 < __log_max) - { - __pre1 = _Tp(0); - __pre2 = std::exp(__ln_pre2); - __pre2 *= __sgn2; - } - else - { - std::__throw_runtime_error(__N("Overflow of gamma functions " - "in __hyperg_reflect")); - } - } - else - { - __pre1 = _Tp(0); - __pre2 = _Tp(0); - std::__throw_runtime_error(__N("Underflow of gamma functions " - "in __hyperg_reflect")); - } - - const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d, - _Tp(1) - __x); - const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d, - _Tp(1) - __x); - - const _Tp __F = __pre1 * __F1 + __pre2 * __F2; - - return __F; - } - } - - - /** - * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$. - * - * The hypogeometric function is defined by - * @f[ - * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} - * \sum_{n=0}^{\infty} - * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} - * \frac{x^n}{n!} - * @f] - * - * @param __a The first "numerator" parameter. - * @param __a The second "numerator" parameter. - * @param __c The "denominator" parameter. - * @param __x The argument of the confluent hypergeometric function. - * @return The confluent hypergeometric function. - */ - template<typename _Tp> - inline _Tp - __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x) - { -#if _GLIBCXX_USE_C99_MATH_TR1 - const _Tp __a_nint = std::tr1::nearbyint(__a); - const _Tp __b_nint = std::tr1::nearbyint(__b); - const _Tp __c_nint = std::tr1::nearbyint(__c); -#else - const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L)); - const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L)); - const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); -#endif - const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon(); - if (std::abs(__x) >= _Tp(1)) - std::__throw_domain_error(__N("Argument outside unit circle " - "in __hyperg.")); - else if (__isnan(__a) || __isnan(__b) - || __isnan(__c) || __isnan(__x)) - return std::numeric_limits<_Tp>::quiet_NaN(); - else if (__c_nint == __c && __c_nint <= _Tp(0)) - return std::numeric_limits<_Tp>::infinity(); - else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler) - return std::pow(_Tp(1) - __x, __c - __a - __b); - else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0) - && __x >= _Tp(0) && __x < _Tp(0.995L)) - return __hyperg_series(__a, __b, __c, __x); - else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10)) - { - // For integer a and b the hypergeometric function is a finite polynomial. - if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler) - return __hyperg_series(__a_nint, __b, __c, __x); - else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler) - return __hyperg_series(__a, __b_nint, __c, __x); - else if (__x < -_Tp(0.25L)) - return __hyperg_luke(__a, __b, __c, __x); - else if (__x < _Tp(0.5L)) - return __hyperg_series(__a, __b, __c, __x); - else - if (std::abs(__c) > _Tp(10)) - return __hyperg_series(__a, __b, __c, __x); - else - return __hyperg_reflect(__a, __b, __c, __x); - } - else - return __hyperg_luke(__a, __b, __c, __x); - } - - } // namespace std::tr1::__detail -} -} - -#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC |