1// Special functions -*- C++ -*- 
2 
3// Copyright (C) 2006-2019 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_NaN = std::numeric_limits<_Tp>::quiet_NaN(); 
381 const _Tp _S_inf = std::numeric_limits<_Tp>::infinity(); 
382 
383 if (__isnan(__x)) 
384 __Bip = __Aip = __Bi = __Ai = std::numeric_limits<_Tp>::quiet_NaN(); 
385 else if (__z == _S_inf) 
386 { 
387 __Aip = __Ai = _Tp(0); 
388 __Bip = __Bi = _S_inf; 
389 } 
390 else if (__z == -_S_inf) 
391 __Bip = __Aip = __Bi = __Ai = _Tp(0); 
392 else if (__x > _Tp(0)) 
393 { 
394 _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; 
395 
396 __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 
397 __Ai = __rootx * __K_nu 
398 / (__numeric_constants<_Tp>::__sqrt3() 
399 * __numeric_constants<_Tp>::__pi()); 
400 __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() 
401 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); 
402 
403 __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); 
404 __Aip = -__x * __K_nu 
405 / (__numeric_constants<_Tp>::__sqrt3() 
406 * __numeric_constants<_Tp>::__pi()); 
407 __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() 
408 + _Tp(2) * __I_nu 
409 / __numeric_constants<_Tp>::__sqrt3()); 
410 } 
411 else if (__x < _Tp(0)) 
412 { 
413 _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; 
414 
415 __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 
416 __Ai = __rootx * (__J_nu 
417 - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 
418 __Bi = -__rootx * (__N_nu 
419 + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); 
420 
421 __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); 
422 __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() 
423 + __J_nu) / _Tp(2); 
424 __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() 
425 - __N_nu) / _Tp(2); 
426 } 
427 else 
428 { 
429 // Reference: 
430 // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. 
431 // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). 
432 __Ai = _Tp(0.35502805388781723926L); 
433 __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); 
434 
435 // Reference: 
436 // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. 
437 // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). 
438 __Aip = -_Tp(0.25881940379280679840L); 
439 __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); 
440 } 
441 
442 return; 
443 } 
444 } // namespace __detail 
445#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) 
446} // namespace tr1 
447#endif 
448 
449_GLIBCXX_END_NAMESPACE_VERSION 
450} 
451 
452#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 
453