| // Special functions -*- C++ -*- |
| |
| // Copyright (C) 2006-2021 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 (unsigned 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 |
