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b34f60ac | 1 | // Special functions -*- C++ -*- |
2 | ||
fbd26352 | 3 | // Copyright (C) 2006-2019 Free Software Foundation, Inc. |
b34f60ac | 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 | |
6bc9506f | 8 | // Free Software Foundation; either version 3, or (at your option) |
b34f60ac | 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 | // | |
6bc9506f | 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/>. | |
b34f60ac | 24 | |
25 | /** @file tr1/poly_laguerre.tcc | |
26 | * This is an internal header file, included by other library headers. | |
5846aeac | 27 | * Do not attempt to use it directly. @headername{tr1/cmath} |
b34f60ac | 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 | ||
c17b0a1c | 41 | #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC |
42 | #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 | |
b34f60ac | 43 | |
2948dd21 | 44 | namespace std _GLIBCXX_VISIBILITY(default) |
b34f60ac | 45 | { |
ae6a4ce9 | 46 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
47 | ||
dc043b87 | 48 | #if _GLIBCXX_USE_STD_SPEC_FUNCS |
ca5b12a4 | 49 | # define _GLIBCXX_MATH_NS ::std |
50 | #elif defined(_GLIBCXX_TR1_CMATH) | |
c17b0a1c | 51 | namespace tr1 |
52 | { | |
ca5b12a4 | 53 | # define _GLIBCXX_MATH_NS ::std::tr1 |
54 | #else | |
55 | # error do not include this header directly, use <cmath> or <tr1/cmath> | |
56 | #endif | |
b34f60ac | 57 | // [5.2] Special functions |
58 | ||
b34f60ac | 59 | // Implementation-space details. |
b34f60ac | 60 | namespace __detail |
61 | { | |
b34f60ac | 62 | /** |
63 | * @brief This routine returns the associated Laguerre polynomial | |
64 | * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. | |
65 | * Abramowitz & Stegun, 13.5.21 | |
66 | * | |
67 | * @param __n The order of the Laguerre function. | |
68 | * @param __alpha The degree of the Laguerre function. | |
69 | * @param __x The argument of the Laguerre function. | |
70 | * @return The value of the Laguerre function of order n, | |
71 | * degree @f$ \alpha @f$, and argument x. | |
72 | * | |
73 | * This is from the GNU Scientific Library. | |
74 | */ | |
75 | template<typename _Tpa, typename _Tp> | |
76 | _Tp | |
cd7f5f45 | 77 | __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) |
b34f60ac | 78 | { |
79 | const _Tp __a = -_Tp(__n); | |
41456429 | 80 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |
b34f60ac | 81 | const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; |
82 | const _Tp __cos2th = __x / __eta; | |
83 | const _Tp __sin2th = _Tp(1) - __cos2th; | |
84 | const _Tp __th = std::acos(std::sqrt(__cos2th)); | |
85 | const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() | |
86 | * __numeric_constants<_Tp>::__pi_2() | |
87 | * __eta * __eta * __cos2th * __sin2th; | |
88 | ||
89 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
ca5b12a4 | 90 | const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b); |
91 | const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); | |
b34f60ac | 92 | #else |
93 | const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); | |
94 | const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); | |
95 | #endif | |
96 | ||
97 | _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) | |
98 | * std::log(_Tp(0.25L) * __x * __eta); | |
99 | _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); | |
100 | _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x | |
101 | + __pre_term1 - __pre_term2; | |
102 | _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); | |
103 | _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta | |
104 | * (_Tp(2) * __th | |
105 | - std::sin(_Tp(2) * __th)) | |
106 | + __numeric_constants<_Tp>::__pi_4()); | |
107 | _Tp __ser = __ser_term1 + __ser_term2; | |
108 | ||
109 | return std::exp(__lnpre) * __ser; | |
110 | } | |
111 | ||
112 | ||
113 | /** | |
114 | * @brief Evaluate the polynomial based on the confluent hypergeometric | |
115 | * function in a safe way, with no restriction on the arguments. | |
116 | * | |
117 | * The associated Laguerre function is defined by | |
118 | * @f[ | |
119 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
120 | * _1F_1(-n; \alpha + 1; x) | |
121 | * @f] | |
122 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
123 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
124 | * | |
125 | * This function assumes x != 0. | |
126 | * | |
127 | * This is from the GNU Scientific Library. | |
128 | */ | |
129 | template<typename _Tpa, typename _Tp> | |
130 | _Tp | |
cd7f5f45 | 131 | __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) |
b34f60ac | 132 | { |
41456429 | 133 | const _Tp __b = _Tp(__alpha1) + _Tp(1); |
b34f60ac | 134 | const _Tp __mx = -__x; |
135 | const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) | |
136 | : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); | |
137 | // Get |x|^n/n! | |
138 | _Tp __tc = _Tp(1); | |
139 | const _Tp __ax = std::abs(__x); | |
140 | for (unsigned int __k = 1; __k <= __n; ++__k) | |
141 | __tc *= (__ax / __k); | |
142 | ||
143 | _Tp __term = __tc * __tc_sgn; | |
144 | _Tp __sum = __term; | |
145 | for (int __k = int(__n) - 1; __k >= 0; --__k) | |
146 | { | |
147 | __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) | |
148 | * _Tp(__k + 1) / __mx; | |
149 | __sum += __term; | |
150 | } | |
151 | ||
152 | return __sum; | |
153 | } | |
154 | ||
155 | ||
156 | /** | |
157 | * @brief This routine returns the associated Laguerre polynomial | |
158 | * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ | |
159 | * by recursion. | |
160 | * | |
161 | * The associated Laguerre function is defined by | |
162 | * @f[ | |
163 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
164 | * _1F_1(-n; \alpha + 1; x) | |
165 | * @f] | |
166 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
167 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
168 | * | |
169 | * The associated Laguerre polynomial is defined for integral | |
170 | * @f$ \alpha = m @f$ by: | |
171 | * @f[ | |
172 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
173 | * @f] | |
174 | * where the Laguerre polynomial is defined by: | |
175 | * @f[ | |
176 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
177 | * @f] | |
178 | * | |
179 | * @param __n The order of the Laguerre function. | |
180 | * @param __alpha The degree of the Laguerre function. | |
181 | * @param __x The argument of the Laguerre function. | |
182 | * @return The value of the Laguerre function of order n, | |
183 | * degree @f$ \alpha @f$, and argument x. | |
184 | */ | |
185 | template<typename _Tpa, typename _Tp> | |
186 | _Tp | |
cd7f5f45 | 187 | __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) |
b34f60ac | 188 | { |
189 | // Compute l_0. | |
190 | _Tp __l_0 = _Tp(1); | |
191 | if (__n == 0) | |
192 | return __l_0; | |
193 | ||
194 | // Compute l_1^alpha. | |
41456429 | 195 | _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); |
b34f60ac | 196 | if (__n == 1) |
197 | return __l_1; | |
198 | ||
199 | // Compute l_n^alpha by recursion on n. | |
200 | _Tp __l_n2 = __l_0; | |
201 | _Tp __l_n1 = __l_1; | |
202 | _Tp __l_n = _Tp(0); | |
203 | for (unsigned int __nn = 2; __nn <= __n; ++__nn) | |
204 | { | |
41456429 | 205 | __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) |
b34f60ac | 206 | * __l_n1 / _Tp(__nn) |
41456429 | 207 | - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); |
b34f60ac | 208 | __l_n2 = __l_n1; |
209 | __l_n1 = __l_n; | |
210 | } | |
211 | ||
212 | return __l_n; | |
213 | } | |
214 | ||
215 | ||
216 | /** | |
217 | * @brief This routine returns the associated Laguerre polynomial | |
218 | * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. | |
219 | * | |
220 | * The associated Laguerre function is defined by | |
221 | * @f[ | |
222 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} | |
223 | * _1F_1(-n; \alpha + 1; x) | |
224 | * @f] | |
225 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and | |
226 | * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. | |
227 | * | |
228 | * The associated Laguerre polynomial is defined for integral | |
229 | * @f$ \alpha = m @f$ by: | |
230 | * @f[ | |
231 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
232 | * @f] | |
233 | * where the Laguerre polynomial is defined by: | |
234 | * @f[ | |
235 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
236 | * @f] | |
237 | * | |
238 | * @param __n The order of the Laguerre function. | |
239 | * @param __alpha The degree of the Laguerre function. | |
240 | * @param __x The argument of the Laguerre function. | |
241 | * @return The value of the Laguerre function of order n, | |
242 | * degree @f$ \alpha @f$, and argument x. | |
243 | */ | |
244 | template<typename _Tpa, typename _Tp> | |
cd7f5f45 | 245 | _Tp |
246 | __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) | |
b34f60ac | 247 | { |
248 | if (__x < _Tp(0)) | |
249 | std::__throw_domain_error(__N("Negative argument " | |
250 | "in __poly_laguerre.")); | |
251 | // Return NaN on NaN input. | |
252 | else if (__isnan(__x)) | |
253 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
254 | else if (__n == 0) | |
255 | return _Tp(1); | |
256 | else if (__n == 1) | |
41456429 | 257 | return _Tp(1) + _Tp(__alpha1) - __x; |
b34f60ac | 258 | else if (__x == _Tp(0)) |
259 | { | |
41456429 | 260 | _Tp __prod = _Tp(__alpha1) + _Tp(1); |
b34f60ac | 261 | for (unsigned int __k = 2; __k <= __n; ++__k) |
41456429 | 262 | __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); |
b34f60ac | 263 | return __prod; |
264 | } | |
41456429 | 265 | else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) |
266 | && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) | |
267 | return __poly_laguerre_large_n(__n, __alpha1, __x); | |
cbf772d6 | 268 | else if (_Tp(__alpha1) >= _Tp(0) |
41456429 | 269 | || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) |
270 | return __poly_laguerre_recursion(__n, __alpha1, __x); | |
b34f60ac | 271 | else |
41456429 | 272 | return __poly_laguerre_hyperg(__n, __alpha1, __x); |
b34f60ac | 273 | } |
274 | ||
275 | ||
276 | /** | |
277 | * @brief This routine returns the associated Laguerre polynomial | |
048ff85f | 278 | * of order n, degree m: @f$ L_n^m(x) @f$. |
b34f60ac | 279 | * |
280 | * The associated Laguerre polynomial is defined for integral | |
281 | * @f$ \alpha = m @f$ by: | |
282 | * @f[ | |
283 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) | |
284 | * @f] | |
285 | * where the Laguerre polynomial is defined by: | |
286 | * @f[ | |
287 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
288 | * @f] | |
289 | * | |
290 | * @param __n The order of the Laguerre polynomial. | |
291 | * @param __m The degree of the Laguerre polynomial. | |
292 | * @param __x The argument of the Laguerre polynomial. | |
293 | * @return The value of the associated Laguerre polynomial of order n, | |
294 | * degree m, and argument x. | |
295 | */ | |
296 | template<typename _Tp> | |
297 | inline _Tp | |
cd7f5f45 | 298 | __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |
299 | { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } | |
b34f60ac | 300 | |
301 | ||
302 | /** | |
048ff85f | 303 | * @brief This routine returns the Laguerre polynomial |
b34f60ac | 304 | * of order n: @f$ L_n(x) @f$. |
305 | * | |
306 | * The Laguerre polynomial is defined by: | |
307 | * @f[ | |
308 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
309 | * @f] | |
310 | * | |
311 | * @param __n The order of the Laguerre polynomial. | |
312 | * @param __x The argument of the Laguerre polynomial. | |
313 | * @return The value of the Laguerre polynomial of order n | |
314 | * and argument x. | |
315 | */ | |
316 | template<typename _Tp> | |
317 | inline _Tp | |
cd7f5f45 | 318 | __laguerre(unsigned int __n, _Tp __x) |
319 | { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } | |
ca5b12a4 | 320 | } // namespace __detail |
321 | #undef _GLIBCXX_MATH_NS | |
dc043b87 | 322 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
ca5b12a4 | 323 | } // namespace tr1 |
324 | #endif | |
ae6a4ce9 | 325 | |
326 | _GLIBCXX_END_NAMESPACE_VERSION | |
b34f60ac | 327 | } |
328 | ||
c17b0a1c | 329 | #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC |