1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
|
// 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
|
