// Special functions -*- C++ -*- // Copyright (C) 2006-2020 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. * Do not attempt to use it directly. @headername{tr1/cmath} */ // // 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 _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION #if _GLIBCXX_USE_STD_SPEC_FUNCS # define _GLIBCXX_MATH_NS ::std #elif defined(_GLIBCXX_TR1_CMATH) namespace tr1 { # define _GLIBCXX_MATH_NS ::std::tr1 #else # error do not include this header directly, use <cmath> or <tr1/cmath> #endif // [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(_Tp __a, _Tp __c, _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(_Tp __a, _Tp __c, _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$. * * @todo Handle b == nonpositive integer blowup - return NaN. * * @param __a The @a numerator parameter. * @param __c The @a denominator parameter. * @param __x The argument of the confluent hypergeometric function. * @return The confluent hypergeometric function. */ template<typename _Tp> _Tp __conf_hyperg(_Tp __a, _Tp __c, _Tp __x) { #if _GLIBCXX_USE_C99_MATH_TR1 const _Tp __c_nint = _GLIBCXX_MATH_NS::nearbyint(__c); #else const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L)); #endif if (__isnan(__a) || __isnan(__c) || __isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__c_nint == __c && __c_nint <= 0) return std::numeric_limits<_Tp>::infinity(); else if (__a == _Tp(0)) return _Tp(1); else if (__c == __a) return std::exp(__x); else if (__x < _Tp(0)) return __conf_hyperg_luke(__a, __c, __x); else return __conf_hyperg_series(__a, __c, __x); } /** * @brief Return the hypogeometric function @f$ _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 @a numerator parameter. * @param __a The second @a numerator parameter. * @param __c The @a denominator parameter. * @param __x The argument of the confluent hypergeometric function. * @return The confluent hypergeometric function. */ template<typename _Tp> _Tp __hyperg_series(_Tp __a, _Tp __b, _Tp __c, _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(_Tp __a, _Tp __b, _Tp __c, _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(_Tp __a, _Tp __b, _Tp __c, _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 @a numerator parameter. * @param __a The second @a numerator parameter. * @param __c The @a denominator parameter. * @param __x The argument of the confluent hypergeometric function. * @return The confluent hypergeometric function. */ template<typename _Tp> _Tp __hyperg(_Tp __a, _Tp __b, _Tp __c, _Tp __x) { #if _GLIBCXX_USE_C99_MATH_TR1 const _Tp __a_nint = _GLIBCXX_MATH_NS::nearbyint(__a); const _Tp __b_nint = _GLIBCXX_MATH_NS::nearbyint(__b); const _Tp __c_nint = _GLIBCXX_MATH_NS::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 __detail #undef _GLIBCXX_MATH_NS #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) } // namespace tr1 #endif _GLIBCXX_END_NAMESPACE_VERSION } #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC