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/modified_bessel_func.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.
35//
36// References:
37// (1) Handbook of Mathematical Functions,
38// Ed. Milton Abramowitz and Irene A. Stegun,
39// Dover Publications,
40// Section 9, pp. 355-434, Section 10 pp. 435-478
41// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
42// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
43// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
44// 2nd ed, pp. 246-249.
45
46#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
47#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
48
49#include <tr1/special_function_util.h>
50
51namespace std _GLIBCXX_VISIBILITY(default)
52{
53_GLIBCXX_BEGIN_NAMESPACE_VERSION
54
55#if _GLIBCXX_USE_STD_SPEC_FUNCS
56#elif defined(_GLIBCXX_TR1_CMATH)
57namespace tr1
58{
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 Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
69 * @f$ K_\nu(x) @f$ and their first derivatives
70 * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
71 * These four functions are computed together for numerical
72 * stability.
73 *
74 * @param __nu The order of the Bessel functions.
75 * @param __x The argument of the Bessel functions.
76 * @param __Inu The output regular modified Bessel function.
77 * @param __Knu The output irregular modified Bessel function.
78 * @param __Ipnu The output derivative of the regular
79 * modified Bessel function.
80 * @param __Kpnu The output derivative of the irregular
81 * modified Bessel function.
82 */
83 template <typename _Tp>
84 void
85 __bessel_ik(_Tp __nu, _Tp __x,
86 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
87 {
88 if (__x == _Tp(0))
89 {
90 if (__nu == _Tp(0))
91 {
92 __Inu = _Tp(1);
93 __Ipnu = _Tp(0);
94 }
95 else if (__nu == _Tp(1))
96 {
97 __Inu = _Tp(0);
98 __Ipnu = _Tp(0.5L);
99 }
100 else
101 {
102 __Inu = _Tp(0);
103 __Ipnu = _Tp(0);
104 }
105 __Knu = std::numeric_limits<_Tp>::infinity();
106 __Kpnu = -std::numeric_limits<_Tp>::infinity();
107 return;
108 }
109
110 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
111 const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
112 const int __max_iter = 15000;
113 const _Tp __x_min = _Tp(2);
114
115 const int __nl = static_cast<int>(__nu + _Tp(0.5L));
116
117 const _Tp __mu = __nu - __nl;
118 const _Tp __mu2 = __mu * __mu;
119 const _Tp __xi = _Tp(1) / __x;
120 const _Tp __xi2 = _Tp(2) * __xi;
121 _Tp __h = __nu * __xi;
122 if ( __h < __fp_min )
123 __h = __fp_min;
124 _Tp __b = __xi2 * __nu;
125 _Tp __d = _Tp(0);
126 _Tp __c = __h;
127 int __i;
128 for ( __i = 1; __i <= __max_iter; ++__i )
129 {
130 __b += __xi2;
131 __d = _Tp(1) / (__b + __d);
132 __c = __b + _Tp(1) / __c;
133 const _Tp __del = __c * __d;
134 __h *= __del;
135 if (std::abs(__del - _Tp(1)) < __eps)
136 break;
137 }
138 if (__i > __max_iter)
139 std::__throw_runtime_error(__N("Argument x too large "
140 "in __bessel_ik; "
141 "try asymptotic expansion."));
142 _Tp __Inul = __fp_min;
143 _Tp __Ipnul = __h * __Inul;
144 _Tp __Inul1 = __Inul;
145 _Tp __Ipnu1 = __Ipnul;
146 _Tp __fact = __nu * __xi;
147 for (int __l = __nl; __l >= 1; --__l)
148 {
149 const _Tp __Inutemp = __fact * __Inul + __Ipnul;
150 __fact -= __xi;
151 __Ipnul = __fact * __Inutemp + __Inul;
152 __Inul = __Inutemp;
153 }
154 _Tp __f = __Ipnul / __Inul;
155 _Tp __Kmu, __Knu1;
156 if (__x < __x_min)
157 {
158 const _Tp __x2 = __x / _Tp(2);
159 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
160 const _Tp __fact = (std::abs(__pimu) < __eps
161 ? _Tp(1) : __pimu / std::sin(__pimu));
162 _Tp __d = -std::log(__x2);
163 _Tp __e = __mu * __d;
164 const _Tp __fact2 = (std::abs(__e) < __eps
165 ? _Tp(1) : std::sinh(__e) / __e);
166 _Tp __gam1, __gam2, __gampl, __gammi;
167 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
168 _Tp __ff = __fact
169 * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
170 _Tp __sum = __ff;
171 __e = std::exp(__e);
172 _Tp __p = __e / (_Tp(2) * __gampl);
173 _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
174 _Tp __c = _Tp(1);
175 __d = __x2 * __x2;
176 _Tp __sum1 = __p;
177 int __i;
178 for (__i = 1; __i <= __max_iter; ++__i)
179 {
180 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
181 __c *= __d / __i;
182 __p /= __i - __mu;
183 __q /= __i + __mu;
184 const _Tp __del = __c * __ff;
185 __sum += __del;
186 const _Tp __del1 = __c * (__p - __i * __ff);
187 __sum1 += __del1;
188 if (std::abs(__del) < __eps * std::abs(__sum))
189 break;
190 }
191 if (__i > __max_iter)
192 std::__throw_runtime_error(__N("Bessel k series failed to converge "
193 "in __bessel_ik."));
194 __Kmu = __sum;
195 __Knu1 = __sum1 * __xi2;
196 }
197 else
198 {
199 _Tp __b = _Tp(2) * (_Tp(1) + __x);
200 _Tp __d = _Tp(1) / __b;
201 _Tp __delh = __d;
202 _Tp __h = __delh;
203 _Tp __q1 = _Tp(0);
204 _Tp __q2 = _Tp(1);
205 _Tp __a1 = _Tp(0.25L) - __mu2;
206 _Tp __q = __c = __a1;
207 _Tp __a = -__a1;
208 _Tp __s = _Tp(1) + __q * __delh;
209 int __i;
210 for (__i = 2; __i <= __max_iter; ++__i)
211 {
212 __a -= 2 * (__i - 1);
213 __c = -__a * __c / __i;
214 const _Tp __qnew = (__q1 - __b * __q2) / __a;
215 __q1 = __q2;
216 __q2 = __qnew;
217 __q += __c * __qnew;
218 __b += _Tp(2);
219 __d = _Tp(1) / (__b + __a * __d);
220 __delh = (__b * __d - _Tp(1)) * __delh;
221 __h += __delh;
222 const _Tp __dels = __q * __delh;
223 __s += __dels;
224 if ( std::abs(__dels / __s) < __eps )
225 break;
226 }
227 if (__i > __max_iter)
228 std::__throw_runtime_error(__N("Steed's method failed "
229 "in __bessel_ik."));
230 __h = __a1 * __h;
231 __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
232 * std::exp(-__x) / __s;
233 __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
234 }
235
236 _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
237 _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
238 __Inu = __Inumu * __Inul1 / __Inul;
239 __Ipnu = __Inumu * __Ipnu1 / __Inul;
240 for ( __i = 1; __i <= __nl; ++__i )
241 {
242 const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
243 __Kmu = __Knu1;
244 __Knu1 = __Knutemp;
245 }
246 __Knu = __Kmu;
247 __Kpnu = __nu * __xi * __Kmu - __Knu1;
248
249 return;
250 }
251
252
253 /**
254 * @brief Return the regular modified Bessel function of order
255 * \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
256 *
257 * The regular modified cylindrical Bessel function is:
258 * @f[
259 * I_{\nu}(x) = \sum_{k=0}^{\infty}
260 * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
261 * @f]
262 *
263 * @param __nu The order of the regular modified Bessel function.
264 * @param __x The argument of the regular modified Bessel function.
265 * @return The output regular modified Bessel function.
266 */
267 template<typename _Tp>
268 _Tp
269 __cyl_bessel_i(_Tp __nu, _Tp __x)
270 {
271 if (__nu < _Tp(0) || __x < _Tp(0))
272 std::__throw_domain_error(__N("Bad argument "
273 "in __cyl_bessel_i."));
274 else if (__isnan(__nu) || __isnan(__x))
275 return std::numeric_limits<_Tp>::quiet_NaN();
276 else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
277 return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
278 else
279 {
280 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
281 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
282 return __I_nu;
283 }
284 }
285
286
287 /**
288 * @brief Return the irregular modified Bessel function
289 * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
290 *
291 * The irregular modified Bessel function is defined by:
292 * @f[
293 * K_{\nu}(x) = \frac{\pi}{2}
294 * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
295 * @f]
296 * where for integral \f$ \nu = n \f$ a limit is taken:
297 * \f$ lim_{\nu \to n} \f$.
298 *
299 * @param __nu The order of the irregular modified Bessel function.
300 * @param __x The argument of the irregular modified Bessel function.
301 * @return The output irregular modified Bessel function.
302 */
303 template<typename _Tp>
304 _Tp
305 __cyl_bessel_k(_Tp __nu, _Tp __x)
306 {
307 if (__nu < _Tp(0) || __x < _Tp(0))
308 std::__throw_domain_error(__N("Bad argument "
309 "in __cyl_bessel_k."));
310 else if (__isnan(__nu) || __isnan(__x))
311 return std::numeric_limits<_Tp>::quiet_NaN();
312 else
313 {
314 _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
315 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
316 return __K_nu;
317 }
318 }
319
320
321 /**
322 * @brief Compute the spherical modified Bessel functions
323 * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
324 * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
325 * respectively.
326 *
327 * @param __n The order of the modified spherical Bessel function.
328 * @param __x The argument of the modified spherical Bessel function.
329 * @param __i_n The output regular modified spherical Bessel function.
330 * @param __k_n The output irregular modified spherical
331 * Bessel function.
332 * @param __ip_n The output derivative of the regular modified
333 * spherical Bessel function.
334 * @param __kp_n The output derivative of the irregular modified
335 * spherical Bessel function.
336 */
337 template <typename _Tp>
338 void
339 __sph_bessel_ik(unsigned int __n, _Tp __x,
340 _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
341 {
342 const _Tp __nu = _Tp(__n) + _Tp(0.5L);
343
344 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
345 __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
346
347 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
348 / std::sqrt(__x);
349
350 __i_n = __factor * __I_nu;
351 __k_n = __factor * __K_nu;
352 __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
353 __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
354
355 return;
356 }
357
358
359 /**
360 * @brief Compute the Airy functions
361 * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
362 * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
363 * respectively.
364 *
365 * @param __x The argument of the Airy functions.
366 * @param __Ai The output Airy function of the first kind.
367 * @param __Bi The output Airy function of the second kind.
368 * @param __Aip The output derivative of the Airy function
369 * of the first kind.
370 * @param __Bip The output derivative of the Airy function
371 * of the second kind.
372 */
373 template <typename _Tp>
374 void
375 __airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
376 {
377 const _Tp __absx = std::abs(__x);
378 const _Tp __rootx = std::sqrt(__absx);
379 const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
380 const _Tp _S_inf = std::numeric_limits<_Tp>::infinity();
381
382 if (__isnan(__x))
383 __Bip = __Aip = __Bi = __Ai = std::numeric_limits<_Tp>::quiet_NaN();
384 else if (__z == _S_inf)
385 {
386 __Aip = __Ai = _Tp(0);
387 __Bip = __Bi = _S_inf;
388 }
389 else if (__z == -_S_inf)
390 __Bip = __Aip = __Bi = __Ai = _Tp(0);
391 else if (__x > _Tp(0))
392 {
393 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
394
395 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
396 __Ai = __rootx * __K_nu
397 / (__numeric_constants<_Tp>::__sqrt3()
398 * __numeric_constants<_Tp>::__pi());
399 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
400 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
401
402 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
403 __Aip = -__x * __K_nu
404 / (__numeric_constants<_Tp>::__sqrt3()
405 * __numeric_constants<_Tp>::__pi());
406 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
407 + _Tp(2) * __I_nu
408 / __numeric_constants<_Tp>::__sqrt3());
409 }
410 else if (__x < _Tp(0))
411 {
412 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
413
414 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
415 __Ai = __rootx * (__J_nu
416 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
417 __Bi = -__rootx * (__N_nu
418 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
419
420 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
421 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
422 + __J_nu) / _Tp(2);
423 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
424 - __N_nu) / _Tp(2);
425 }
426 else
427 {
428 // Reference:
429 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
430 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
431 __Ai = _Tp(0.35502805388781723926L);
432 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
433
434 // Reference:
435 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
436 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
437 __Aip = -_Tp(0.25881940379280679840L);
438 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
439 }
440
441 return;
442 }
443 } // namespace __detail
444#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
445} // namespace tr1
446#endif
447
448_GLIBCXX_END_NAMESPACE_VERSION
449}
450
451#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
452