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-// Special functions -*- C++ -*-
-
-// Copyright (C) 2006, 2007, 2008, 2009
-// Free Software Foundation, Inc.
-//
-// This file is part of the GNU ISO C++ Library. This library is free
-// software; you can redistribute it and/or modify it under the
-// terms of the GNU General Public License as published by the
-// Free Software Foundation; either version 3, or (at your option)
-// any later version.
-//
-// This library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU General Public License for more details.
-//
-// Under Section 7 of GPL version 3, you are granted additional
-// permissions described in the GCC Runtime Library Exception, version
-// 3.1, as published by the Free Software Foundation.
-
-// You should have received a copy of the GNU General Public License and
-// a copy of the GCC Runtime Library Exception along with this program;
-// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
-// <http://www.gnu.org/licenses/>.
-
-/** @file tr1/legendre_function.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 on:
-// (1) Handbook of Mathematical Functions,
-// ed. Milton Abramowitz and Irene A. Stegun,
-// Dover Publications,
-// Section 8, pp. 331-341
-// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
-// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
-// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
-// 2nd ed, pp. 252-254
-
-#ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
-#define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
-
-#include "special_function_util.h"
-
-namespace std
-{
-namespace tr1
-{
-
- // [5.2] Special functions
-
- // Implementation-space details.
- namespace __detail
- {
-
- /**
- * @brief Return the Legendre polynomial by recursion on order
- * @f$ l @f$.
- *
- * The Legendre function of @f$ l @f$ and @f$ x @f$,
- * @f$ P_l(x) @f$, is defined by:
- * @f[
- * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
- * @f]
- *
- * @param l The order of the Legendre polynomial. @f$l >= 0@f$.
- * @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
- */
- template<typename _Tp>
- _Tp
- __poly_legendre_p(const unsigned int __l, const _Tp __x)
- {
-
- if ((__x < _Tp(-1)) || (__x > _Tp(+1)))
- std::__throw_domain_error(__N("Argument out of range"
- " in __poly_legendre_p."));
- else if (__isnan(__x))
- return std::numeric_limits<_Tp>::quiet_NaN();
- else if (__x == +_Tp(1))
- return +_Tp(1);
- else if (__x == -_Tp(1))
- return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
- else
- {
- _Tp __p_lm2 = _Tp(1);
- if (__l == 0)
- return __p_lm2;
-
- _Tp __p_lm1 = __x;
- if (__l == 1)
- return __p_lm1;
-
- _Tp __p_l = 0;
- for (unsigned int __ll = 2; __ll <= __l; ++__ll)
- {
- // This arrangement is supposed to be better for roundoff
- // protection, Arfken, 2nd Ed, Eq 12.17a.
- __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
- - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
- __p_lm2 = __p_lm1;
- __p_lm1 = __p_l;
- }
-
- return __p_l;
- }
- }
-
-
- /**
- * @brief Return the associated Legendre function by recursion
- * on @f$ l @f$.
- *
- * The associated Legendre function is derived from the Legendre function
- * @f$ P_l(x) @f$ by the Rodrigues formula:
- * @f[
- * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
- * @f]
- *
- * @param l The order of the associated Legendre function.
- * @f$ l >= 0 @f$.
- * @param m The order of the associated Legendre function.
- * @f$ m <= l @f$.
- * @param x The argument of the associated Legendre function.
- * @f$ |x| <= 1 @f$.
- */
- template<typename _Tp>
- _Tp
- __assoc_legendre_p(const unsigned int __l, const unsigned int __m,
- const _Tp __x)
- {
-
- if (__x < _Tp(-1) || __x > _Tp(+1))
- std::__throw_domain_error(__N("Argument out of range"
- " in __assoc_legendre_p."));
- else if (__m > __l)
- std::__throw_domain_error(__N("Degree out of range"
- " in __assoc_legendre_p."));
- else if (__isnan(__x))
- return std::numeric_limits<_Tp>::quiet_NaN();
- else if (__m == 0)
- return __poly_legendre_p(__l, __x);
- else
- {
- _Tp __p_mm = _Tp(1);
- if (__m > 0)
- {
- // Two square roots seem more accurate more of the time
- // than just one.
- _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
- _Tp __fact = _Tp(1);
- for (unsigned int __i = 1; __i <= __m; ++__i)
- {
- __p_mm *= -__fact * __root;
- __fact += _Tp(2);
- }
- }
- if (__l == __m)
- return __p_mm;
-
- _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
- if (__l == __m + 1)
- return __p_mp1m;
-
- _Tp __p_lm2m = __p_mm;
- _Tp __P_lm1m = __p_mp1m;
- _Tp __p_lm = _Tp(0);
- for (unsigned int __j = __m + 2; __j <= __l; ++__j)
- {
- __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
- - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
- __p_lm2m = __P_lm1m;
- __P_lm1m = __p_lm;
- }
-
- return __p_lm;
- }
- }
-
-
- /**
- * @brief Return the spherical associated Legendre function.
- *
- * The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
- * and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
- * @f[
- * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
- * \frac{(l-m)!}{(l+m)!}]
- * P_l^m(\cos\theta) \exp^{im\phi}
- * @f]
- * is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
- * associated Legendre function.
- *
- * This function differs from the associated Legendre function by
- * argument (@f$x = \cos(\theta)@f$) and by a normalization factor
- * but this factor is rather large for large @f$ l @f$ and @f$ m @f$
- * and so this function is stable for larger differences of @f$ l @f$
- * and @f$ m @f$.
- *
- * @param l The order of the spherical associated Legendre function.
- * @f$ l >= 0 @f$.
- * @param m The order of the spherical associated Legendre function.
- * @f$ m <= l @f$.
- * @param theta The radian angle argument of the spherical associated
- * Legendre function.
- */
- template <typename _Tp>
- _Tp
- __sph_legendre(const unsigned int __l, const unsigned int __m,
- const _Tp __theta)
- {
- if (__isnan(__theta))
- return std::numeric_limits<_Tp>::quiet_NaN();
-
- const _Tp __x = std::cos(__theta);
-
- if (__l < __m)
- {
- std::__throw_domain_error(__N("Bad argument "
- "in __sph_legendre."));
- }
- else if (__m == 0)
- {
- _Tp __P = __poly_legendre_p(__l, __x);
- _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
- / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
- __P *= __fact;
- return __P;
- }
- else if (__x == _Tp(1) || __x == -_Tp(1))
- {
- // m > 0 here
- return _Tp(0);
- }
- else
- {
- // m > 0 and |x| < 1 here
-
- // Starting value for recursion.
- // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
- // (-1)^m (1-x^2)^(m/2) / pi^(1/4)
- const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
- const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
-#if _GLIBCXX_USE_C99_MATH_TR1
- const _Tp __lncirc = std::tr1::log1p(-__x * __x);
-#else
- const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
-#endif
- // Gamma(m+1/2) / Gamma(m)
-#if _GLIBCXX_USE_C99_MATH_TR1
- const _Tp __lnpoch = std::tr1::lgamma(_Tp(__m + _Tp(0.5L)))
- - std::tr1::lgamma(_Tp(__m));
-#else
- const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
- - __log_gamma(_Tp(__m));
-#endif
- const _Tp __lnpre_val =
- -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
- + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
- _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
- / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
- _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
- _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
-
- if (__l == __m)
- {
- return __y_mm;
- }
- else if (__l == __m + 1)
- {
- return __y_mp1m;
- }
- else
- {
- _Tp __y_lm = _Tp(0);
-
- // Compute Y_l^m, l > m+1, upward recursion on l.
- for ( int __ll = __m + 2; __ll <= __l; ++__ll)
- {
- const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
- const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
- const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
- * _Tp(2 * __ll - 1));
- const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
- / _Tp(2 * __ll - 3));
- __y_lm = (__x * __y_mp1m * __fact1
- - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
- __y_mm = __y_mp1m;
- __y_mp1m = __y_lm;
- }
-
- return __y_lm;
- }
- }
- }
-
- } // namespace std::tr1::__detail
-}
-}
-
-#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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