305 lines
10 KiB
C++
305 lines
10 KiB
C++
// Special functions -*- C++ -*-
|
|
|
|
// Copyright (C) 2006-2020 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.
|
|
* Do not attempt to use it directly. @headername{tr1/cmath}
|
|
*/
|
|
|
|
//
|
|
// 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 <tr1/special_function_util.h>
|
|
|
|
namespace std _GLIBCXX_VISIBILITY(default)
|
|
{
|
|
_GLIBCXX_BEGIN_NAMESPACE_VERSION
|
|
|
|
#if _GLIBCXX_USE_STD_SPEC_FUNCS
|
|
# define _GLIBCXX_MATH_NS ::std
|
|
#elif defined(_GLIBCXX_TR1_CMATH)
|
|
namespace tr1
|
|
{
|
|
# define _GLIBCXX_MATH_NS ::std::tr1
|
|
#else
|
|
# error do not include this header directly, use <cmath> or <tr1/cmath>
|
|
#endif
|
|
// [5.2] Special functions
|
|
|
|
// Implementation-space details.
|
|
namespace __detail
|
|
{
|
|
/**
|
|
* @brief Return the Legendre polynomial by recursion on degree
|
|
* @f$ l @f$.
|
|
*
|
|
* The Legendre function of @f$ l @f$ and @f$ x @f$,
|
|
* @f$ P_l(x) @f$, is defined by:
|
|
* @f[
|
|
* P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
|
|
* @f]
|
|
*
|
|
* @param l The degree of the Legendre polynomial. @f$l >= 0@f$.
|
|
* @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
|
|
*/
|
|
template<typename _Tp>
|
|
_Tp
|
|
__poly_legendre_p(unsigned int __l, _Tp __x)
|
|
{
|
|
|
|
if (__isnan(__x))
|
|
return std::numeric_limits<_Tp>::quiet_NaN();
|
|
else if (__x == +_Tp(1))
|
|
return +_Tp(1);
|
|
else if (__x == -_Tp(1))
|
|
return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
|
|
else
|
|
{
|
|
_Tp __p_lm2 = _Tp(1);
|
|
if (__l == 0)
|
|
return __p_lm2;
|
|
|
|
_Tp __p_lm1 = __x;
|
|
if (__l == 1)
|
|
return __p_lm1;
|
|
|
|
_Tp __p_l = 0;
|
|
for (unsigned int __ll = 2; __ll <= __l; ++__ll)
|
|
{
|
|
// This arrangement is supposed to be better for roundoff
|
|
// protection, Arfken, 2nd Ed, Eq 12.17a.
|
|
__p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
|
|
- (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
|
|
__p_lm2 = __p_lm1;
|
|
__p_lm1 = __p_l;
|
|
}
|
|
|
|
return __p_l;
|
|
}
|
|
}
|
|
|
|
|
|
/**
|
|
* @brief Return the associated Legendre function by recursion
|
|
* on @f$ 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]
|
|
* @note @f$ P_l^m(x) = 0 @f$ if @f$ m > l @f$.
|
|
*
|
|
* @param l The degree of the associated Legendre function.
|
|
* @f$ l >= 0 @f$.
|
|
* @param m The order of the associated Legendre function.
|
|
* @param x The argument of the associated Legendre function.
|
|
* @f$ |x| <= 1 @f$.
|
|
* @param phase The phase of the associated Legendre function.
|
|
* Use -1 for the Condon-Shortley phase convention.
|
|
*/
|
|
template<typename _Tp>
|
|
_Tp
|
|
__assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
|
|
_Tp __phase = _Tp(+1))
|
|
{
|
|
|
|
if (__m > __l)
|
|
return _Tp(0);
|
|
else if (__isnan(__x))
|
|
return std::numeric_limits<_Tp>::quiet_NaN();
|
|
else if (__m == 0)
|
|
return __poly_legendre_p(__l, __x);
|
|
else
|
|
{
|
|
_Tp __p_mm = _Tp(1);
|
|
if (__m > 0)
|
|
{
|
|
// Two square roots seem more accurate more of the time
|
|
// than just one.
|
|
_Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
|
|
_Tp __fact = _Tp(1);
|
|
for (unsigned int __i = 1; __i <= __m; ++__i)
|
|
{
|
|
__p_mm *= __phase * __fact * __root;
|
|
__fact += _Tp(2);
|
|
}
|
|
}
|
|
if (__l == __m)
|
|
return __p_mm;
|
|
|
|
_Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
|
|
if (__l == __m + 1)
|
|
return __p_mp1m;
|
|
|
|
_Tp __p_lm2m = __p_mm;
|
|
_Tp __P_lm1m = __p_mp1m;
|
|
_Tp __p_lm = _Tp(0);
|
|
for (unsigned int __j = __m + 2; __j <= __l; ++__j)
|
|
{
|
|
__p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
|
|
- _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
|
|
__p_lm2m = __P_lm1m;
|
|
__P_lm1m = __p_lm;
|
|
}
|
|
|
|
return __p_lm;
|
|
}
|
|
}
|
|
|
|
|
|
/**
|
|
* @brief Return the spherical associated Legendre function.
|
|
*
|
|
* The spherical associated Legendre function of @f$ 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$.
|
|
* @note Unlike the case for __assoc_legendre_p the Condon-Shortley
|
|
* phase factor @f$ (-1)^m @f$ is present here.
|
|
* @note @f$ Y_l^m(\theta) = 0 @f$ if @f$ m > l @f$.
|
|
*
|
|
* @param l The degree of the spherical associated Legendre function.
|
|
* @f$ l >= 0 @f$.
|
|
* @param m The order of the spherical associated Legendre function.
|
|
* @param theta The radian angle argument of the spherical associated
|
|
* Legendre function.
|
|
*/
|
|
template <typename _Tp>
|
|
_Tp
|
|
__sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
|
|
{
|
|
if (__isnan(__theta))
|
|
return std::numeric_limits<_Tp>::quiet_NaN();
|
|
|
|
const _Tp __x = std::cos(__theta);
|
|
|
|
if (__m > __l)
|
|
return _Tp(0);
|
|
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 = _GLIBCXX_MATH_NS::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 = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
|
|
- _GLIBCXX_MATH_NS::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);
|
|
const _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 __detail
|
|
#undef _GLIBCXX_MATH_NS
|
|
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
|
|
} // namespace tr1
|
|
#endif
|
|
|
|
_GLIBCXX_END_NAMESPACE_VERSION
|
|
}
|
|
|
|
#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
|