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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 new file mode 100644 index 000000000..c975afa59 --- /dev/null +++ b/gcc-4.4.3/libstdc++-v3/include/tr1/hypergeometric.tcc @@ -0,0 +1,774 @@ +// 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 |