2007-03-16 Benjamin Kosnik <bkoz@redhat.com>

	
	* testsuite/lib/dg-options.exp (dg-require-c-std): New.
	* testsuite/lib/libstdc++.exp (check_v3_target_c_std): New. Check to
	see if _GLIBCXX_USE_C99_MATH is active. 
	* testsuite/tr1/5_numerical_facilities/special_functions/
	08_cyl_bessel_i/check_nan.cc: Use dg-require-c-std.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	17_hyperg/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	13_ellint_2/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	01_assoc_laguerre/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	02_assoc_legendre/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	22_sph_legendre/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	10_cyl_bessel_k/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	09_cyl_bessel_j/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	21_sph_bessel/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	07_conf_hyperg/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	16_hermite/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	12_ellint_1/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	05_comp_ellint_2/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	18_laguerre/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	14_ellint_3/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	04_comp_ellint_1/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	06_comp_ellint_3/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	19_legendre/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	03_beta/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	11_cyl_neumann/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	15_expint/check_nan.cc: Same.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	23_sph_neumann/check_nan.cc: Same.

	* include/tr1/poly_laguerre.tcc (__poly_laguerre): Don't check if
	unsigned int can be a NaN.

2007-03-16  Edward M. Smith-Rowland  <3dw4rd@verizon.net>

	* docs/html/ext/tr1.html : Marked tr1 math special functions done.
	* docs/html/faq/index.html : Ditto.
	* include/Makefile.in : Added new special function header files.
	* include/Makefile.am : Ditto.	
	* include/tr1/common.h : Added type promotion routines for three
	and four args.

	* include/tr1/special_function_util.h: New.
	* include/tr1/bessel_function.tcc: New.
	* include/tr1/beta_function.tcc: New.
	* include/tr1/ell_integral.tcc: New.
	* include/tr1/exp_integral.tcc: New.
	* include/tr1/gamma.tcc: New.
	* include/tr1/hypergeometric.tcc: New.
	* include/tr1/legendre_function.tcc: New.
	* include/tr1/modified_bessel_func.tcc: New.
	* include/tr1/poly_hermite.tcc: New.
	* include/tr1/poly_laguerre.tcc: New.
	* include/tr1/riemann_zeta.tcc: New.
	* include/tr1/cmath : Included tr1 math special functions.
	* include/tr1/math.h : Ditto.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	testcase.h: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	01_assoc_laguerre/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	01_assoc_laguerre/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	01_assoc_laguerre/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	01_assoc_laguerre/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	02_assoc_legendre/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	02_assoc_legendre/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	02_assoc_legendre/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	02_assoc_legendre/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/03_beta/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/03_beta/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/03_beta/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/03_beta/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	04_comp_ellint_1/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	04_comp_ellint_1/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	04_comp_ellint_1/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	04_comp_ellint_1/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	05_comp_ellint_2/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	05_comp_ellint_2/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	05_comp_ellint_2/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	05_comp_ellint_2/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	06_comp_ellint_3/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	06_comp_ellint_3/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	06_comp_ellint_3/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	06_comp_ellint_3/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	07_conf_hyperg/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	07_conf_hyperg/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	07_conf_hyperg/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	07_conf_hyperg/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	08_cyl_bessel_i/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	08_cyl_bessel_i/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	08_cyl_bessel_i/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	08_cyl_bessel_i/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	09_cyl_bessel_j/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	09_cyl_bessel_j/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	09_cyl_bessel_j/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	09_cyl_bessel_j/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	10_cyl_bessel_k/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	10_cyl_bessel_k/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	10_cyl_bessel_k/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	10_cyl_bessel_k/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	11_cyl_neumann/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	11_cyl_neumann/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	11_cyl_neumann/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	11_cyl_neumann/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/12_ellint_1/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/12_ellint_1/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/12_ellint_1/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/12_ellint_1/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/13_ellint_2/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/13_ellint_2/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/13_ellint_2/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/13_ellint_2/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/14_ellint_3/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/14_ellint_3/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/14_ellint_3/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/14_ellint_3/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/15_expint/
	check_value_neg.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/15_expint/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/15_expint/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/15_expint/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/15_expint/
	check_value_pos.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/16_hermite/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/16_hermite/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/16_hermite/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/17_hyperg/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/17_hyperg/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/17_hyperg/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/17_hyperg/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/18_laguerre/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/18_laguerre/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/18_laguerre/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/18_laguerre/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/19_legendre/
	compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/19_legendre/
	check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/19_legendre/
	check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/19_legendre/
	compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/check_value_neg.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	20_riemann_zeta/check_value_pos.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	21_sph_bessel/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	21_sph_bessel/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	21_sph_bessel/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	21_sph_bessel/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	22_sph_legendre/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	22_sph_legendre/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	22_sph_legendre/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	22_sph_legendre/compile_2.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	23_sph_neumann/compile.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	23_sph_neumann/check_nan.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	23_sph_neumann/check_value.cc: New.
	* testsuite/tr1/5_numerical_facilities/special_functions/
	23_sph_neumann/ compile_2.cc: New.
	


git-svn-id: svn+ssh://gcc.gnu.org/svn/gcc/trunk@122986 138bc75d-0d04-0410-961f-82ee72b054a4

1 files changed, 788 insertions, 0 deletions

diff --git a/libstdc++-v3/include/tr1/hypergeometric.tcc b/libstdc++-v3/include/tr1/hypergeometric.tcc
new file mode 100644
index 00000000000..d54e6fecde3
--- /dev/null
+++ b/libstdc++-v3/include/tr1/hypergeometric.tcc
@@ -0,0 +1,788 @@
+// Special functions -*- C++ -*-
+
+// Copyright (C) 2006-2007
+// 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 2, 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.
+//
+// You should have received a copy of the GNU General Public License along
+// with this library; see the file COPYING.  If not, write to the Free
+// Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
+// USA.
+//
+// As a special exception, you may use this file as part of a free software
+// library without restriction.  Specifically, if other files instantiate
+// templates or use macros or inline functions from this file, or you compile
+// this file and link it with other files to produce an executable, this
+// file does not by itself cause the resulting executable to be covered by
+// the GNU General Public License.  This exception does not however
+// invalidate any other reasons why the executable file might be covered by
+// the GNU General Public License.
+
+/** @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 _TR1_HYPERGEOMETRIC_TCC
+#define _TR1_HYPERGEOMETRIC_TCC 1
+
+namespace std
+{
+_GLIBCXX_BEGIN_NAMESPACE(_GLIBCXX_TR1)
+
+  // [5.2] Special functions
+
+  /**
+   * @ingroup tr1_math_spec_func
+   * @{
+   */
+
+  //
+  // Implementation-space details.
+  //
+  namespace __detail
+  {
+
+    /**
+     *   @brief This routine returns the confluent hypereometric 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 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$.
+     * 
+     *   @todo  Handle b == nonpositive integer blowup - return NaN.
+     *
+     *   @param  __a  The "numerator" parameter.
+     *   @param  __c  The "denominator" parameter.
+     *   @param  __x  The argument of the confluent hypergeometric function.
+     *   @return  The confluent hypergeometric function.
+     */
+    template<typename _Tp>
+    inline _Tp
+    __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
+    {
+#if _GLIBCXX_USE_C99_MATH_TR1
+      const _Tp __c_nint = std::_GLIBCXX_TR1::nearbyint(__c);
+#else
+      const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
+#endif
+      if (__isnan(__a) || __isnan(__c) || __isnan(__x))
+        return std::numeric_limits<_Tp>::quiet_NaN();
+      else if (__c_nint == __c && __c_nint <= 0)
+        return std::numeric_limits<_Tp>::infinity();
+      else if (__a == _Tp(0))
+        return _Tp(1);
+      else if (__c == __a)
+        return std::exp(__x);
+      else if (__x < _Tp(0))
+        return __conf_hyperg_luke(__a, __c, __x);
+      else
+        return __conf_hyperg_series(__a, __c, __x);
+    }
+
+
+    /**
+     *   @brief Return the hypogeometric function @f$ _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::_GLIBCXX_TR1::nearbyint(__a);
+      const _Tp __b_nint = std::_GLIBCXX_TR1::nearbyint(__b);
+      const _Tp __c_nint = std::_GLIBCXX_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
+
+  /* @} */ // group tr1_math_spec_func
+
+_GLIBCXX_END_NAMESPACE
+}
+
+#endif // _TR1_HYPERGEOMETRIC_TCC