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/poly_laguerre.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 13, pp. 509-510, Section 22 pp. 773-802
39// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
40
41#ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC
42#define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1
43
44namespace std _GLIBCXX_VISIBILITY(default)
45{
46_GLIBCXX_BEGIN_NAMESPACE_VERSION
47
48#if _GLIBCXX_USE_STD_SPEC_FUNCS
49# define _GLIBCXX_MATH_NS ::std
50#elif defined(_GLIBCXX_TR1_CMATH)
51namespace tr1
52{
53# define _GLIBCXX_MATH_NS ::std::tr1
54#else
55# error do not include this header directly, use <cmath> or <tr1/cmath>
56#endif
57 // [5.2] Special functions
58
59 // Implementation-space details.
60 namespace __detail
61 {
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$, and argument x.
72 *
73 * This is from the GNU Scientific Library.
74 */
75 template<typename _Tpa, typename _Tp>
76 _Tp
77 __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x)
78 {
79 const _Tp __a = -_Tp(__n);
80 const _Tp __b = _Tp(__alpha1) + _Tp(1);
81 const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
82 const _Tp __cos2th = __x / __eta;
83 const _Tp __sin2th = _Tp(1) - __cos2th;
84 const _Tp __th = std::acos(std::sqrt(__cos2th));
85 const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
86 * __numeric_constants<_Tp>::__pi_2()
87 * __eta * __eta * __cos2th * __sin2th;
88
89#if _GLIBCXX_USE_C99_MATH_TR1
90 const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b);
91 const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1));
92#else
93 const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
94 const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
95#endif
96
97 _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
98 * std::log(_Tp(0.25L) * __x * __eta);
99 _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
100 _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
101 + __pre_term1 - __pre_term2;
102 _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
103 _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
104 * (_Tp(2) * __th
105 - std::sin(_Tp(2) * __th))
106 + __numeric_constants<_Tp>::__pi_4());
107 _Tp __ser = __ser_term1 + __ser_term2;
108
109 return std::exp(__lnpre) * __ser;
110 }
111
112
113 /**
114 * @brief Evaluate the polynomial based on the confluent hypergeometric
115 * function in a safe way, with no restriction on the arguments.
116 *
117 * The associated Laguerre function is defined by
118 * @f[
119 * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
120 * _1F_1(-n; \alpha + 1; x)
121 * @f]
122 * where @f$ (\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
131 __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x)
132 {
133 const _Tp __b = _Tp(__alpha1) + _Tp(1);
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$, and argument x.
184 */
185 template<typename _Tpa, typename _Tp>
186 _Tp
187 __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x)
188 {
189 // Compute l_0.
190 _Tp __l_0 = _Tp(1);
191 if (__n == 0)
192 return __l_0;
193
194 // Compute l_1^alpha.
195 _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
196 if (__n == 1)
197 return __l_1;
198
199 // Compute l_n^alpha by recursion on n.
200 _Tp __l_n2 = __l_0;
201 _Tp __l_n1 = __l_1;
202 _Tp __l_n = _Tp(0);
203 for (unsigned int __nn = 2; __nn <= __n; ++__nn)
204 {
205 __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
206 * __l_n1 / _Tp(__nn)
207 - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn);
208 __l_n2 = __l_n1;
209 __l_n1 = __l_n;
210 }
211
212 return __l_n;
213 }
214
215
216 /**
217 * @brief This routine returns the associated Laguerre polynomial
218 * of order n, degree @f$ \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$, and argument x.
243 */
244 template<typename _Tpa, typename _Tp>
245 _Tp
246 __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x)
247 {
248 if (__x < _Tp(0))
249 std::__throw_domain_error(__N("Negative argument "
250 "in __poly_laguerre."));
251 // Return NaN on NaN input.
252 else if (__isnan(__x))
253 return std::numeric_limits<_Tp>::quiet_NaN();
254 else if (__n == 0)
255 return _Tp(1);
256 else if (__n == 1)
257 return _Tp(1) + _Tp(__alpha1) - __x;
258 else if (__x == _Tp(0))
259 {
260 _Tp __prod = _Tp(__alpha1) + _Tp(1);
261 for (unsigned int __k = 2; __k <= __n; ++__k)
262 __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
263 return __prod;
264 }
265 else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
266 && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n))
267 return __poly_laguerre_large_n(__n, __alpha1, __x);
268 else if (_Tp(__alpha1) >= _Tp(0)
269 || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
270 return __poly_laguerre_recursion(__n, __alpha1, __x);
271 else
272 return __poly_laguerre_hyperg(__n, __alpha1, __x);
273 }
274
275
276 /**
277 * @brief This routine returns the associated Laguerre polynomial
278 * of order n, degree m: @f$ L_n^m(x) @f$.
279 *
280 * The associated Laguerre polynomial is defined for integral
281 * @f$ \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
298 __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)
299 { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); }
300
301
302 /**
303 * @brief This routine returns the Laguerre polynomial
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
318 __laguerre(unsigned int __n, _Tp __x)
319 { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); }
320 } // namespace __detail
321#undef _GLIBCXX_MATH_NS
322#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
323} // namespace tr1
324#endif
325
326_GLIBCXX_END_NAMESPACE_VERSION
327}
328
329#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC
330