1// Special functions -*- C++ -*-
2
3// Copyright (C) 2006-2022 Free Software Foundation, Inc.
4//
5// This file is part of the GNU ISO C++ Library. This library is free
6// software; you can redistribute it and/or modify it under the
7// terms of the GNU General Public License as published by the
8// Free Software Foundation; either version 3, or (at your option)
9// any later version.
10//
11// This library is distributed in the hope that it will be useful,
12// but WITHOUT ANY WARRANTY; without even the implied warranty of
13// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14// GNU General Public License for more details.
15//
16// Under Section 7 of GPL version 3, you are granted additional
17// permissions described in the GCC Runtime Library Exception, version
18// 3.1, as published by the Free Software Foundation.
19
20// You should have received a copy of the GNU General Public License and
21// a copy of the GCC Runtime Library Exception along with this program;
22// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
23// <http://www.gnu.org/licenses/>.
24
25/** @file tr1/legendre_function.tcc
26 * This is an internal header file, included by other library headers.
27 * Do not attempt to use it directly. @headername{tr1/cmath}
28 */
29
30//
31// ISO C++ 14882 TR1: 5.2 Special functions
32//
33
34// Written by Edward Smith-Rowland based on:
35// (1) Handbook of Mathematical Functions,
36// ed. Milton Abramowitz and Irene A. Stegun,
37// Dover Publications,
38// Section 8, pp. 331-341
39// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
40// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
41// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
42// 2nd ed, pp. 252-254
43
44#ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
45#define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
46
47#include <tr1/special_function_util.h>
48
49namespace std _GLIBCXX_VISIBILITY(default)
50{
51_GLIBCXX_BEGIN_NAMESPACE_VERSION
52
53#if _GLIBCXX_USE_STD_SPEC_FUNCS
54# define _GLIBCXX_MATH_NS ::std
55#elif defined(_GLIBCXX_TR1_CMATH)
56namespace tr1
57{
58# define _GLIBCXX_MATH_NS ::std::tr1
59#else
60# error do not include this header directly, use <cmath> or <tr1/cmath>
61#endif
62 // [5.2] Special functions
63
64 // Implementation-space details.
65 namespace __detail
66 {
67 /**
68 * @brief Return the Legendre polynomial by recursion on degree
69 * @f$ l @f$.
70 *
71 * The Legendre function of @f$ l @f$ and @f$ x @f$,
72 * @f$ P_l(x) @f$, is defined by:
73 * @f[
74 * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
75 * @f]
76 *
77 * @param l The degree of the Legendre polynomial. @f$l >= 0@f$.
78 * @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
79 */
80 template<typename _Tp>
81 _Tp
82 __poly_legendre_p(unsigned int __l, _Tp __x)
83 {
84
85 if (__isnan(__x))
86 return std::numeric_limits<_Tp>::quiet_NaN();
87 else if (__x == +_Tp(1))
88 return +_Tp(1);
89 else if (__x == -_Tp(1))
90 return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
91 else
92 {
93 _Tp __p_lm2 = _Tp(1);
94 if (__l == 0)
95 return __p_lm2;
96
97 _Tp __p_lm1 = __x;
98 if (__l == 1)
99 return __p_lm1;
100
101 _Tp __p_l = 0;
102 for (unsigned int __ll = 2; __ll <= __l; ++__ll)
103 {
104 // This arrangement is supposed to be better for roundoff
105 // protection, Arfken, 2nd Ed, Eq 12.17a.
106 __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
107 - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
108 __p_lm2 = __p_lm1;
109 __p_lm1 = __p_l;
110 }
111
112 return __p_l;
113 }
114 }
115
116
117 /**
118 * @brief Return the associated Legendre function by recursion
119 * on @f$ l @f$.
120 *
121 * The associated Legendre function is derived from the Legendre function
122 * @f$ P_l(x) @f$ by the Rodrigues formula:
123 * @f[
124 * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
125 * @f]
126 * @note @f$ P_l^m(x) = 0 @f$ if @f$ m > l @f$.
127 *
128 * @param l The degree of the associated Legendre function.
129 * @f$ l >= 0 @f$.
130 * @param m The order of the associated Legendre function.
131 * @param x The argument of the associated Legendre function.
132 * @f$ |x| <= 1 @f$.
133 * @param phase The phase of the associated Legendre function.
134 * Use -1 for the Condon-Shortley phase convention.
135 */
136 template<typename _Tp>
137 _Tp
138 __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
139 _Tp __phase = _Tp(+1))
140 {
141
142 if (__m > __l)
143 return _Tp(0);
144 else if (__isnan(__x))
145 return std::numeric_limits<_Tp>::quiet_NaN();
146 else if (__m == 0)
147 return __poly_legendre_p(__l, __x);
148 else
149 {
150 _Tp __p_mm = _Tp(1);
151 if (__m > 0)
152 {
153 // Two square roots seem more accurate more of the time
154 // than just one.
155 _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
156 _Tp __fact = _Tp(1);
157 for (unsigned int __i = 1; __i <= __m; ++__i)
158 {
159 __p_mm *= __phase * __fact * __root;
160 __fact += _Tp(2);
161 }
162 }
163 if (__l == __m)
164 return __p_mm;
165
166 _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
167 if (__l == __m + 1)
168 return __p_mp1m;
169
170 _Tp __p_lm2m = __p_mm;
171 _Tp __P_lm1m = __p_mp1m;
172 _Tp __p_lm = _Tp(0);
173 for (unsigned int __j = __m + 2; __j <= __l; ++__j)
174 {
175 __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
176 - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
177 __p_lm2m = __P_lm1m;
178 __P_lm1m = __p_lm;
179 }
180
181 return __p_lm;
182 }
183 }
184
185
186 /**
187 * @brief Return the spherical associated Legendre function.
188 *
189 * The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
190 * and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
191 * @f[
192 * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
193 * \frac{(l-m)!}{(l+m)!}]
194 * P_l^m(\cos\theta) \exp^{im\phi}
195 * @f]
196 * is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
197 * associated Legendre function.
198 *
199 * This function differs from the associated Legendre function by
200 * argument (@f$x = \cos(\theta)@f$) and by a normalization factor
201 * but this factor is rather large for large @f$ l @f$ and @f$ m @f$
202 * and so this function is stable for larger differences of @f$ l @f$
203 * and @f$ m @f$.
204 * @note Unlike the case for __assoc_legendre_p the Condon-Shortley
205 * phase factor @f$ (-1)^m @f$ is present here.
206 * @note @f$ Y_l^m(\theta) = 0 @f$ if @f$ m > l @f$.
207 *
208 * @param l The degree of the spherical associated Legendre function.
209 * @f$ l >= 0 @f$.
210 * @param m The order of the spherical associated Legendre function.
211 * @param theta The radian angle argument of the spherical associated
212 * Legendre function.
213 */
214 template <typename _Tp>
215 _Tp
216 __sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
217 {
218 if (__isnan(__theta))
219 return std::numeric_limits<_Tp>::quiet_NaN();
220
221 const _Tp __x = std::cos(__theta);
222
223 if (__m > __l)
224 return _Tp(0);
225 else if (__m == 0)
226 {
227 _Tp __P = __poly_legendre_p(__l, __x);
228 _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
229 / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
230 __P *= __fact;
231 return __P;
232 }
233 else if (__x == _Tp(1) || __x == -_Tp(1))
234 {
235 // m > 0 here
236 return _Tp(0);
237 }
238 else
239 {
240 // m > 0 and |x| < 1 here
241
242 // Starting value for recursion.
243 // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
244 // (-1)^m (1-x^2)^(m/2) / pi^(1/4)
245 const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
246 const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
247#if _GLIBCXX_USE_C99_MATH_TR1
248 const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
249#else
250 const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
251#endif
252 // Gamma(m+1/2) / Gamma(m)
253#if _GLIBCXX_USE_C99_MATH_TR1
254 const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
255 - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
256#else
257 const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
258 - __log_gamma(_Tp(__m));
259#endif
260 const _Tp __lnpre_val =
261 -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
262 + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
263 const _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
264 / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
265 _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
266 _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
267
268 if (__l == __m)
269 return __y_mm;
270 else if (__l == __m + 1)
271 return __y_mp1m;
272 else
273 {
274 _Tp __y_lm = _Tp(0);
275
276 // Compute Y_l^m, l > m+1, upward recursion on l.
277 for (unsigned int __ll = __m + 2; __ll <= __l; ++__ll)
278 {
279 const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
280 const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
281 const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
282 * _Tp(2 * __ll - 1));
283 const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
284 / _Tp(2 * __ll - 3));
285 __y_lm = (__x * __y_mp1m * __fact1
286 - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
287 __y_mm = __y_mp1m;
288 __y_mp1m = __y_lm;
289 }
290
291 return __y_lm;
292 }
293 }
294 }
295 } // namespace __detail
296#undef _GLIBCXX_MATH_NS
297#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
298} // namespace tr1
299#endif
300
301_GLIBCXX_END_NAMESPACE_VERSION
302}
303
304#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
305