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306 lines
10 KiB
C++
306 lines
10 KiB
C++
// Special functions -*- C++ -*-
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// Copyright (C) 2006, 2007, 2008, 2009
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// Free Software Foundation, Inc.
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//
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// This file is part of the GNU ISO C++ Library. This library is free
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// software; you can redistribute it and/or modify it under the
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// terms of the GNU General Public License as published by the
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// Free Software Foundation; either version 3, or (at your option)
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// any later version.
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//
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// This library is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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//
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// Under Section 7 of GPL version 3, you are granted additional
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// permissions described in the GCC Runtime Library Exception, version
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// 3.1, as published by the Free Software Foundation.
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// You should have received a copy of the GNU General Public License and
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// a copy of the GCC Runtime Library Exception along with this program;
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// <http://www.gnu.org/licenses/>.
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/** @file tr1/legendre_function.tcc
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* This is an internal header file, included by other library headers.
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* You should not attempt to use it directly.
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*/
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//
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// ISO C++ 14882 TR1: 5.2 Special functions
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//
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// Written by Edward Smith-Rowland based on:
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// (1) Handbook of Mathematical Functions,
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// ed. Milton Abramowitz and Irene A. Stegun,
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// Dover Publications,
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// Section 8, pp. 331-341
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
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// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
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// 2nd ed, pp. 252-254
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#ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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#define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
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#include "special_function_util.h"
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namespace std
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{
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namespace tr1
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{
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// [5.2] Special functions
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// Implementation-space details.
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namespace __detail
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{
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/**
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* @brief Return the Legendre polynomial by recursion on order
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* @f$ l @f$.
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*
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* The Legendre function of @f$ l @f$ and @f$ x @f$,
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* @f$ P_l(x) @f$, is defined by:
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* @f[
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* P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
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* @f]
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*
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* @param l The order of the Legendre polynomial. @f$l >= 0@f$.
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* @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
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*/
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template<typename _Tp>
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_Tp
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__poly_legendre_p(const unsigned int __l, const _Tp __x)
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{
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if ((__x < _Tp(-1)) || (__x > _Tp(+1)))
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std::__throw_domain_error(__N("Argument out of range"
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" in __poly_legendre_p."));
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else if (__isnan(__x))
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return std::numeric_limits<_Tp>::quiet_NaN();
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else if (__x == +_Tp(1))
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return +_Tp(1);
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else if (__x == -_Tp(1))
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return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
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else
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{
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_Tp __p_lm2 = _Tp(1);
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if (__l == 0)
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return __p_lm2;
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_Tp __p_lm1 = __x;
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if (__l == 1)
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return __p_lm1;
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_Tp __p_l = 0;
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for (unsigned int __ll = 2; __ll <= __l; ++__ll)
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{
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// This arrangement is supposed to be better for roundoff
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// protection, Arfken, 2nd Ed, Eq 12.17a.
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__p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
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- (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
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__p_lm2 = __p_lm1;
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__p_lm1 = __p_l;
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}
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return __p_l;
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}
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}
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/**
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* @brief Return the associated Legendre function by recursion
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* on @f$ l @f$.
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*
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* The associated Legendre function is derived from the Legendre function
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* @f$ P_l(x) @f$ by the Rodrigues formula:
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* @f[
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* P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
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* @f]
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*
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* @param l The order of the associated Legendre function.
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* @f$ l >= 0 @f$.
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* @param m The order of the associated Legendre function.
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* @f$ m <= l @f$.
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* @param x The argument of the associated Legendre function.
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* @f$ |x| <= 1 @f$.
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*/
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template<typename _Tp>
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_Tp
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__assoc_legendre_p(const unsigned int __l, const unsigned int __m,
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const _Tp __x)
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{
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if (__x < _Tp(-1) || __x > _Tp(+1))
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std::__throw_domain_error(__N("Argument out of range"
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" in __assoc_legendre_p."));
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else if (__m > __l)
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std::__throw_domain_error(__N("Degree out of range"
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" in __assoc_legendre_p."));
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else if (__isnan(__x))
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return std::numeric_limits<_Tp>::quiet_NaN();
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else if (__m == 0)
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return __poly_legendre_p(__l, __x);
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else
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{
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_Tp __p_mm = _Tp(1);
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if (__m > 0)
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{
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// Two square roots seem more accurate more of the time
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// than just one.
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_Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
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_Tp __fact = _Tp(1);
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for (unsigned int __i = 1; __i <= __m; ++__i)
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{
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__p_mm *= -__fact * __root;
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__fact += _Tp(2);
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}
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}
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if (__l == __m)
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return __p_mm;
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_Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
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if (__l == __m + 1)
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return __p_mp1m;
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_Tp __p_lm2m = __p_mm;
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_Tp __P_lm1m = __p_mp1m;
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_Tp __p_lm = _Tp(0);
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for (unsigned int __j = __m + 2; __j <= __l; ++__j)
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{
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__p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
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- _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
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__p_lm2m = __P_lm1m;
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__P_lm1m = __p_lm;
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}
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return __p_lm;
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}
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}
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/**
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* @brief Return the spherical associated Legendre function.
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*
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* The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
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* and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
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* @f[
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* Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
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* \frac{(l-m)!}{(l+m)!}]
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* P_l^m(\cos\theta) \exp^{im\phi}
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* @f]
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* is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
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* associated Legendre function.
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*
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* This function differs from the associated Legendre function by
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* argument (@f$x = \cos(\theta)@f$) and by a normalization factor
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* but this factor is rather large for large @f$ l @f$ and @f$ m @f$
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* and so this function is stable for larger differences of @f$ l @f$
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* and @f$ m @f$.
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*
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* @param l The order of the spherical associated Legendre function.
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* @f$ l >= 0 @f$.
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* @param m The order of the spherical associated Legendre function.
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* @f$ m <= l @f$.
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* @param theta The radian angle argument of the spherical associated
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* Legendre function.
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*/
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template <typename _Tp>
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_Tp
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__sph_legendre(const unsigned int __l, const unsigned int __m,
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const _Tp __theta)
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{
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if (__isnan(__theta))
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return std::numeric_limits<_Tp>::quiet_NaN();
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const _Tp __x = std::cos(__theta);
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if (__l < __m)
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{
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std::__throw_domain_error(__N("Bad argument "
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"in __sph_legendre."));
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}
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else if (__m == 0)
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{
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_Tp __P = __poly_legendre_p(__l, __x);
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_Tp __fact = std::sqrt(_Tp(2 * __l + 1)
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/ (_Tp(4) * __numeric_constants<_Tp>::__pi()));
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__P *= __fact;
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return __P;
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}
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else if (__x == _Tp(1) || __x == -_Tp(1))
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{
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// m > 0 here
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return _Tp(0);
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}
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else
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{
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// m > 0 and |x| < 1 here
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// Starting value for recursion.
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// Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
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// (-1)^m (1-x^2)^(m/2) / pi^(1/4)
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const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
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const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
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#if _GLIBCXX_USE_C99_MATH_TR1
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const _Tp __lncirc = std::tr1::log1p(-__x * __x);
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#else
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const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
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#endif
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// Gamma(m+1/2) / Gamma(m)
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#if _GLIBCXX_USE_C99_MATH_TR1
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const _Tp __lnpoch = std::tr1::lgamma(_Tp(__m + _Tp(0.5L)))
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- std::tr1::lgamma(_Tp(__m));
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#else
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const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
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- __log_gamma(_Tp(__m));
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#endif
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const _Tp __lnpre_val =
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-_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
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+ _Tp(0.5L) * (__lnpoch + __m * __lncirc);
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_Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
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/ (_Tp(4) * __numeric_constants<_Tp>::__pi()));
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_Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
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_Tp __y_mp1m = __y_mp1m_factor * __y_mm;
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if (__l == __m)
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{
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return __y_mm;
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}
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else if (__l == __m + 1)
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{
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return __y_mp1m;
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}
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else
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{
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_Tp __y_lm = _Tp(0);
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// Compute Y_l^m, l > m+1, upward recursion on l.
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for ( int __ll = __m + 2; __ll <= __l; ++__ll)
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{
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const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
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const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
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const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
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* _Tp(2 * __ll - 1));
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const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
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/ _Tp(2 * __ll - 3));
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__y_lm = (__x * __y_mp1m * __fact1
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- (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
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__y_mm = __y_mp1m;
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__y_mp1m = __y_lm;
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}
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return __y_lm;
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}
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}
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}
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} // namespace std::tr1::__detail
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}
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}
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#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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