--- /dev/null
+// 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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