// Special functions -*- C++ -*-
// Copyright (C) 2006-2013 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)
{
namespace tr1
{
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
/**
* @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$.
*
* @todo Handle b == nonpositive integer blowup - return NaN.
*
* @param __a The @a numerator parameter.
* @param __c The @a denominator parameter.
* @param __x The argument of the confluent hypergeometric function.
* @return The confluent hypergeometric function.
*/
template<typename _Tp>
_Tp
__conf_hyperg(_Tp __a, _Tp __c, _Tp __x)
{
#if _GLIBCXX_USE_C99_MATH_TR1
const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
if (__isnan(__a) || __isnan(__c) || __isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else if (__c_nint == __c && __c_nint <= 0)
return std::numeric_limits<_Tp>::infinity();
else if (__a == _Tp(0))
return _Tp(1);
else if (__c == __a)
return std::exp(__x);
else if (__x < _Tp(0))
return __conf_hyperg_luke(__a, __c, __x);
else
return __conf_hyperg_series(__a, __c, __x);
}
/**
* @brief Return the hypogeometric function @f$ _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 @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 = 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);
}
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace std::tr1::__detail
}
}
#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC