| // 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 |
