]>
Commit | Line | Data |
---|---|---|
7c62b943 BK |
1 | // Special functions -*- C++ -*- |
2 | ||
aa118a03 | 3 | // Copyright (C) 2006-2014 Free Software Foundation, Inc. |
7c62b943 BK |
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 | |
748086b7 | 8 | // Free Software Foundation; either version 3, or (at your option) |
7c62b943 BK |
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 | // | |
748086b7 JJ |
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/>. | |
7c62b943 BK |
24 | |
25 | /** @file tr1/poly_laguerre.tcc | |
26 | * This is an internal header file, included by other library headers. | |
f910786b | 27 | * Do not attempt to use it directly. @headername{tr1/cmath} |
7c62b943 BK |
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 13, pp. 509-510, Section 22 pp. 773-802 | |
39 | // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl | |
40 | ||
e133ace8 PC |
41 | #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC |
42 | #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 | |
7c62b943 | 43 | |
12ffa228 | 44 | namespace std _GLIBCXX_VISIBILITY(default) |
7c62b943 | 45 | { |
e133ace8 PC |
46 | namespace tr1 |
47 | { | |
7c62b943 BK |
48 | // [5.2] Special functions |
49 | ||
7c62b943 | 50 | // Implementation-space details. |
7c62b943 BK |
51 | namespace __detail |
52 | { | |
12ffa228 | 53 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
7c62b943 BK |
54 | |
55 | /** | |
56 | * @brief This routine returns the associated Laguerre polynomial | |
57 | * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. | |
58 | * Abramowitz & Stegun, 13.5.21 | |
59 | * | |
60 | * @param __n The order of the Laguerre function. | |
61 | * @param __alpha The degree of the Laguerre function. | |
62 | * @param __x The argument of the Laguerre function. | |
63 | * @return The value of the Laguerre function of order n, | |
64 | * degree @f$ \alpha @f$, and argument x. | |
65 | * | |
66 | * This is from the GNU Scientific Library. | |
67 | */ | |
68 | template<typename _Tpa, typename _Tp> | |
69 | _Tp | |
be59c932 | 70 | __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) |
7c62b943 BK |
71 | { |
72 | const _Tp __a = -_Tp(__n); | |
f070285a | 73 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |
7c62b943 BK |
74 | const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; |
75 | const _Tp __cos2th = __x / __eta; | |
76 | const _Tp __sin2th = _Tp(1) - __cos2th; | |
77 | const _Tp __th = std::acos(std::sqrt(__cos2th)); | |
78 | const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() | |
79 | * __numeric_constants<_Tp>::__pi_2() | |
80 | * __eta * __eta * __cos2th * __sin2th; | |
81 | ||
82 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
e133ace8 PC |
83 | const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b); |
84 | const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1)); | |
7c62b943 BK |
85 | #else |
86 | const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); | |
87 | const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); | |
88 | #endif | |
89 | ||
90 | _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) | |
91 | * std::log(_Tp(0.25L) * __x * __eta); | |
92 | _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); | |
93 | _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x | |
94 | + __pre_term1 - __pre_term2; | |
95 | _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); | |
96 | _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta | |
97 | * (_Tp(2) * __th | |
98 | - std::sin(_Tp(2) * __th)) | |
99 | + __numeric_constants<_Tp>::__pi_4()); | |
100 | _Tp __ser = __ser_term1 + __ser_term2; | |
101 | ||
102 | return std::exp(__lnpre) * __ser; | |
103 | } | |
104 | ||
105 | ||
106 | /** | |
107 | * @brief Evaluate the polynomial based on the confluent hypergeometric | |
108 | * function in a safe way, with no restriction on the arguments. | |
109 | * | |
110 | * The associated Laguerre function is defined by | |
111 | * @f[ | |
112 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
113 | * _1F_1(-n; \alpha + 1; x) | |
114 | * @f] | |
115 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
116 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
117 | * | |
118 | * This function assumes x != 0. | |
119 | * | |
120 | * This is from the GNU Scientific Library. | |
121 | */ | |
122 | template<typename _Tpa, typename _Tp> | |
123 | _Tp | |
be59c932 | 124 | __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) |
7c62b943 | 125 | { |
f070285a | 126 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |
7c62b943 BK |
127 | const _Tp __mx = -__x; |
128 | const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) | |
129 | : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); | |
130 | // Get |x|^n/n! | |
131 | _Tp __tc = _Tp(1); | |
132 | const _Tp __ax = std::abs(__x); | |
133 | for (unsigned int __k = 1; __k <= __n; ++__k) | |
134 | __tc *= (__ax / __k); | |
135 | ||
136 | _Tp __term = __tc * __tc_sgn; | |
137 | _Tp __sum = __term; | |
138 | for (int __k = int(__n) - 1; __k >= 0; --__k) | |
139 | { | |
140 | __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) | |
141 | * _Tp(__k + 1) / __mx; | |
142 | __sum += __term; | |
143 | } | |
144 | ||
145 | return __sum; | |
146 | } | |
147 | ||
148 | ||
149 | /** | |
150 | * @brief This routine returns the associated Laguerre polynomial | |
151 | * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ | |
152 | * by recursion. | |
153 | * | |
154 | * The associated Laguerre function is defined by | |
155 | * @f[ | |
156 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
157 | * _1F_1(-n; \alpha + 1; x) | |
158 | * @f] | |
159 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
160 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
161 | * | |
162 | * The associated Laguerre polynomial is defined for integral | |
163 | * @f$ \alpha = m @f$ by: | |
164 | * @f[ | |
165 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
166 | * @f] | |
167 | * where the Laguerre polynomial is defined by: | |
168 | * @f[ | |
169 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
170 | * @f] | |
171 | * | |
172 | * @param __n The order of the Laguerre function. | |
173 | * @param __alpha The degree of the Laguerre function. | |
174 | * @param __x The argument of the Laguerre function. | |
175 | * @return The value of the Laguerre function of order n, | |
176 | * degree @f$ \alpha @f$, and argument x. | |
177 | */ | |
178 | template<typename _Tpa, typename _Tp> | |
179 | _Tp | |
be59c932 | 180 | __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) |
7c62b943 BK |
181 | { |
182 | // Compute l_0. | |
183 | _Tp __l_0 = _Tp(1); | |
184 | if (__n == 0) | |
185 | return __l_0; | |
186 | ||
187 | // Compute l_1^alpha. | |
f070285a | 188 | _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); |
7c62b943 BK |
189 | if (__n == 1) |
190 | return __l_1; | |
191 | ||
192 | // Compute l_n^alpha by recursion on n. | |
193 | _Tp __l_n2 = __l_0; | |
194 | _Tp __l_n1 = __l_1; | |
195 | _Tp __l_n = _Tp(0); | |
196 | for (unsigned int __nn = 2; __nn <= __n; ++__nn) | |
197 | { | |
f070285a | 198 | __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) |
7c62b943 | 199 | * __l_n1 / _Tp(__nn) |
f070285a | 200 | - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); |
7c62b943 BK |
201 | __l_n2 = __l_n1; |
202 | __l_n1 = __l_n; | |
203 | } | |
204 | ||
205 | return __l_n; | |
206 | } | |
207 | ||
208 | ||
209 | /** | |
210 | * @brief This routine returns the associated Laguerre polynomial | |
211 | * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. | |
212 | * | |
213 | * The associated Laguerre function is defined by | |
214 | * @f[ | |
215 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
216 | * _1F_1(-n; \alpha + 1; x) | |
217 | * @f] | |
218 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
219 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
220 | * | |
221 | * The associated Laguerre polynomial is defined for integral | |
222 | * @f$ \alpha = m @f$ by: | |
223 | * @f[ | |
224 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
225 | * @f] | |
226 | * where the Laguerre polynomial is defined by: | |
227 | * @f[ | |
228 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
229 | * @f] | |
230 | * | |
231 | * @param __n The order of the Laguerre function. | |
232 | * @param __alpha The degree of the Laguerre function. | |
233 | * @param __x The argument of the Laguerre function. | |
234 | * @return The value of the Laguerre function of order n, | |
235 | * degree @f$ \alpha @f$, and argument x. | |
236 | */ | |
237 | template<typename _Tpa, typename _Tp> | |
be59c932 ESR |
238 | _Tp |
239 | __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) | |
7c62b943 BK |
240 | { |
241 | if (__x < _Tp(0)) | |
242 | std::__throw_domain_error(__N("Negative argument " | |
243 | "in __poly_laguerre.")); | |
244 | // Return NaN on NaN input. | |
245 | else if (__isnan(__x)) | |
246 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
247 | else if (__n == 0) | |
248 | return _Tp(1); | |
249 | else if (__n == 1) | |
f070285a | 250 | return _Tp(1) + _Tp(__alpha1) - __x; |
7c62b943 BK |
251 | else if (__x == _Tp(0)) |
252 | { | |
f070285a | 253 | _Tp __prod = _Tp(__alpha1) + _Tp(1); |
7c62b943 | 254 | for (unsigned int __k = 2; __k <= __n; ++__k) |
f070285a | 255 | __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); |
7c62b943 BK |
256 | return __prod; |
257 | } | |
f070285a RH |
258 | else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) |
259 | && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) | |
260 | return __poly_laguerre_large_n(__n, __alpha1, __x); | |
df848e82 | 261 | else if (_Tp(__alpha1) >= _Tp(0) |
f070285a RH |
262 | || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) |
263 | return __poly_laguerre_recursion(__n, __alpha1, __x); | |
7c62b943 | 264 | else |
f070285a | 265 | return __poly_laguerre_hyperg(__n, __alpha1, __x); |
7c62b943 BK |
266 | } |
267 | ||
268 | ||
269 | /** | |
270 | * @brief This routine returns the associated Laguerre polynomial | |
6165bbdd | 271 | * of order n, degree m: @f$ L_n^m(x) @f$. |
7c62b943 BK |
272 | * |
273 | * The associated Laguerre polynomial is defined for integral | |
274 | * @f$ \alpha = m @f$ by: | |
275 | * @f[ | |
276 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
277 | * @f] | |
278 | * where the Laguerre polynomial is defined by: | |
279 | * @f[ | |
280 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
281 | * @f] | |
282 | * | |
283 | * @param __n The order of the Laguerre polynomial. | |
284 | * @param __m The degree of the Laguerre polynomial. | |
285 | * @param __x The argument of the Laguerre polynomial. | |
286 | * @return The value of the associated Laguerre polynomial of order n, | |
287 | * degree m, and argument x. | |
288 | */ | |
289 | template<typename _Tp> | |
290 | inline _Tp | |
be59c932 ESR |
291 | __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |
292 | { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } | |
7c62b943 BK |
293 | |
294 | ||
295 | /** | |
6165bbdd | 296 | * @brief This routine returns the Laguerre polynomial |
7c62b943 BK |
297 | * of order n: @f$ L_n(x) @f$. |
298 | * | |
299 | * The Laguerre polynomial is defined by: | |
300 | * @f[ | |
301 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
302 | * @f] | |
303 | * | |
304 | * @param __n The order of the Laguerre polynomial. | |
305 | * @param __x The argument of the Laguerre polynomial. | |
306 | * @return The value of the Laguerre polynomial of order n | |
307 | * and argument x. | |
308 | */ | |
309 | template<typename _Tp> | |
310 | inline _Tp | |
be59c932 ESR |
311 | __laguerre(unsigned int __n, _Tp __x) |
312 | { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } | |
7c62b943 | 313 | |
12ffa228 | 314 | _GLIBCXX_END_NAMESPACE_VERSION |
7c62b943 | 315 | } // namespace std::tr1::__detail |
e133ace8 | 316 | } |
7c62b943 BK |
317 | } |
318 | ||
e133ace8 | 319 | #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC |