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b34f60ac | 1 | // Special functions -*- C++ -*- |
2 | ||
f1717362 | 3 | // Copyright (C) 2006-2016 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/modified_bessel_func.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. | |
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 | ||
c17b0a1c | 46 | #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |
47 | #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 | |
b34f60ac | 48 | |
49 | #include "special_function_util.h" | |
50 | ||
2948dd21 | 51 | namespace std _GLIBCXX_VISIBILITY(default) |
b34f60ac | 52 | { |
c17b0a1c | 53 | namespace tr1 |
54 | { | |
b34f60ac | 55 | // [5.2] Special functions |
56 | ||
b34f60ac | 57 | // Implementation-space details. |
b34f60ac | 58 | namespace __detail |
59 | { | |
2948dd21 | 60 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
b34f60ac | 61 | |
62 | /** | |
63 | * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and | |
64 | * @f$ K_\nu(x) @f$ and their first derivatives | |
65 | * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. | |
66 | * These four functions are computed together for numerical | |
67 | * stability. | |
68 | * | |
69 | * @param __nu The order of the Bessel functions. | |
70 | * @param __x The argument of the Bessel functions. | |
71 | * @param __Inu The output regular modified Bessel function. | |
72 | * @param __Knu The output irregular modified Bessel function. | |
73 | * @param __Ipnu The output derivative of the regular | |
74 | * modified Bessel function. | |
75 | * @param __Kpnu The output derivative of the irregular | |
76 | * modified Bessel function. | |
77 | */ | |
78 | template <typename _Tp> | |
79 | void | |
cd7f5f45 | 80 | __bessel_ik(_Tp __nu, _Tp __x, |
b34f60ac | 81 | _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) |
82 | { | |
83 | if (__x == _Tp(0)) | |
84 | { | |
85 | if (__nu == _Tp(0)) | |
86 | { | |
87 | __Inu = _Tp(1); | |
88 | __Ipnu = _Tp(0); | |
89 | } | |
90 | else if (__nu == _Tp(1)) | |
91 | { | |
92 | __Inu = _Tp(0); | |
93 | __Ipnu = _Tp(0.5L); | |
94 | } | |
95 | else | |
96 | { | |
97 | __Inu = _Tp(0); | |
98 | __Ipnu = _Tp(0); | |
99 | } | |
100 | __Knu = std::numeric_limits<_Tp>::infinity(); | |
101 | __Kpnu = -std::numeric_limits<_Tp>::infinity(); | |
102 | return; | |
103 | } | |
104 | ||
105 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
106 | const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); | |
107 | const int __max_iter = 15000; | |
108 | const _Tp __x_min = _Tp(2); | |
109 | ||
110 | const int __nl = static_cast<int>(__nu + _Tp(0.5L)); | |
111 | ||
112 | const _Tp __mu = __nu - __nl; | |
113 | const _Tp __mu2 = __mu * __mu; | |
114 | const _Tp __xi = _Tp(1) / __x; | |
115 | const _Tp __xi2 = _Tp(2) * __xi; | |
116 | _Tp __h = __nu * __xi; | |
117 | if ( __h < __fp_min ) | |
118 | __h = __fp_min; | |
119 | _Tp __b = __xi2 * __nu; | |
120 | _Tp __d = _Tp(0); | |
121 | _Tp __c = __h; | |
122 | int __i; | |
123 | for ( __i = 1; __i <= __max_iter; ++__i ) | |
124 | { | |
125 | __b += __xi2; | |
126 | __d = _Tp(1) / (__b + __d); | |
127 | __c = __b + _Tp(1) / __c; | |
128 | const _Tp __del = __c * __d; | |
129 | __h *= __del; | |
130 | if (std::abs(__del - _Tp(1)) < __eps) | |
131 | break; | |
132 | } | |
133 | if (__i > __max_iter) | |
134 | std::__throw_runtime_error(__N("Argument x too large " | |
cd7f5f45 | 135 | "in __bessel_ik; " |
b34f60ac | 136 | "try asymptotic expansion.")); |
137 | _Tp __Inul = __fp_min; | |
138 | _Tp __Ipnul = __h * __Inul; | |
139 | _Tp __Inul1 = __Inul; | |
140 | _Tp __Ipnu1 = __Ipnul; | |
141 | _Tp __fact = __nu * __xi; | |
142 | for (int __l = __nl; __l >= 1; --__l) | |
143 | { | |
144 | const _Tp __Inutemp = __fact * __Inul + __Ipnul; | |
145 | __fact -= __xi; | |
146 | __Ipnul = __fact * __Inutemp + __Inul; | |
147 | __Inul = __Inutemp; | |
148 | } | |
149 | _Tp __f = __Ipnul / __Inul; | |
150 | _Tp __Kmu, __Knu1; | |
151 | if (__x < __x_min) | |
152 | { | |
153 | const _Tp __x2 = __x / _Tp(2); | |
154 | const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; | |
155 | const _Tp __fact = (std::abs(__pimu) < __eps | |
156 | ? _Tp(1) : __pimu / std::sin(__pimu)); | |
157 | _Tp __d = -std::log(__x2); | |
158 | _Tp __e = __mu * __d; | |
159 | const _Tp __fact2 = (std::abs(__e) < __eps | |
160 | ? _Tp(1) : std::sinh(__e) / __e); | |
161 | _Tp __gam1, __gam2, __gampl, __gammi; | |
162 | __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); | |
163 | _Tp __ff = __fact | |
164 | * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); | |
165 | _Tp __sum = __ff; | |
166 | __e = std::exp(__e); | |
167 | _Tp __p = __e / (_Tp(2) * __gampl); | |
168 | _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); | |
169 | _Tp __c = _Tp(1); | |
170 | __d = __x2 * __x2; | |
171 | _Tp __sum1 = __p; | |
172 | int __i; | |
173 | for (__i = 1; __i <= __max_iter; ++__i) | |
174 | { | |
175 | __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); | |
176 | __c *= __d / __i; | |
177 | __p /= __i - __mu; | |
178 | __q /= __i + __mu; | |
179 | const _Tp __del = __c * __ff; | |
180 | __sum += __del; | |
181 | const _Tp __del1 = __c * (__p - __i * __ff); | |
182 | __sum1 += __del1; | |
183 | if (std::abs(__del) < __eps * std::abs(__sum)) | |
184 | break; | |
185 | } | |
186 | if (__i > __max_iter) | |
187 | std::__throw_runtime_error(__N("Bessel k series failed to converge " | |
cd7f5f45 | 188 | "in __bessel_ik.")); |
b34f60ac | 189 | __Kmu = __sum; |
190 | __Knu1 = __sum1 * __xi2; | |
191 | } | |
192 | else | |
193 | { | |
194 | _Tp __b = _Tp(2) * (_Tp(1) + __x); | |
195 | _Tp __d = _Tp(1) / __b; | |
196 | _Tp __delh = __d; | |
197 | _Tp __h = __delh; | |
198 | _Tp __q1 = _Tp(0); | |
199 | _Tp __q2 = _Tp(1); | |
200 | _Tp __a1 = _Tp(0.25L) - __mu2; | |
201 | _Tp __q = __c = __a1; | |
202 | _Tp __a = -__a1; | |
203 | _Tp __s = _Tp(1) + __q * __delh; | |
204 | int __i; | |
205 | for (__i = 2; __i <= __max_iter; ++__i) | |
206 | { | |
207 | __a -= 2 * (__i - 1); | |
208 | __c = -__a * __c / __i; | |
209 | const _Tp __qnew = (__q1 - __b * __q2) / __a; | |
210 | __q1 = __q2; | |
211 | __q2 = __qnew; | |
212 | __q += __c * __qnew; | |
213 | __b += _Tp(2); | |
214 | __d = _Tp(1) / (__b + __a * __d); | |
215 | __delh = (__b * __d - _Tp(1)) * __delh; | |
216 | __h += __delh; | |
217 | const _Tp __dels = __q * __delh; | |
218 | __s += __dels; | |
219 | if ( std::abs(__dels / __s) < __eps ) | |
220 | break; | |
221 | } | |
222 | if (__i > __max_iter) | |
223 | std::__throw_runtime_error(__N("Steed's method failed " | |
cd7f5f45 | 224 | "in __bessel_ik.")); |
b34f60ac | 225 | __h = __a1 * __h; |
226 | __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) | |
227 | * std::exp(-__x) / __s; | |
228 | __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; | |
229 | } | |
230 | ||
231 | _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; | |
232 | _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); | |
233 | __Inu = __Inumu * __Inul1 / __Inul; | |
234 | __Ipnu = __Inumu * __Ipnu1 / __Inul; | |
235 | for ( __i = 1; __i <= __nl; ++__i ) | |
236 | { | |
237 | const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; | |
238 | __Kmu = __Knu1; | |
239 | __Knu1 = __Knutemp; | |
240 | } | |
241 | __Knu = __Kmu; | |
242 | __Kpnu = __nu * __xi * __Kmu - __Knu1; | |
243 | ||
244 | return; | |
245 | } | |
246 | ||
247 | ||
248 | /** | |
249 | * @brief Return the regular modified Bessel function of order | |
250 | * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. | |
251 | * | |
252 | * The regular modified cylindrical Bessel function is: | |
253 | * @f[ | |
254 | * I_{\nu}(x) = \sum_{k=0}^{\infty} | |
255 | * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
256 | * @f] | |
257 | * | |
258 | * @param __nu The order of the regular modified Bessel function. | |
259 | * @param __x The argument of the regular modified Bessel function. | |
260 | * @return The output regular modified Bessel function. | |
261 | */ | |
262 | template<typename _Tp> | |
263 | _Tp | |
cd7f5f45 | 264 | __cyl_bessel_i(_Tp __nu, _Tp __x) |
b34f60ac | 265 | { |
266 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
267 | std::__throw_domain_error(__N("Bad argument " | |
268 | "in __cyl_bessel_i.")); | |
269 | else if (__isnan(__nu) || __isnan(__x)) | |
270 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
271 | else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) | |
272 | return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); | |
273 | else | |
274 | { | |
275 | _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; | |
276 | __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); | |
277 | return __I_nu; | |
278 | } | |
279 | } | |
280 | ||
281 | ||
282 | /** | |
283 | * @brief Return the irregular modified Bessel function | |
284 | * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. | |
285 | * | |
286 | * The irregular modified Bessel function is defined by: | |
287 | * @f[ | |
288 | * K_{\nu}(x) = \frac{\pi}{2} | |
289 | * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} | |
290 | * @f] | |
291 | * where for integral \f$ \nu = n \f$ a limit is taken: | |
292 | * \f$ lim_{\nu \to n} \f$. | |
293 | * | |
294 | * @param __nu The order of the irregular modified Bessel function. | |
295 | * @param __x The argument of the irregular modified Bessel function. | |
296 | * @return The output irregular modified Bessel function. | |
297 | */ | |
298 | template<typename _Tp> | |
299 | _Tp | |
cd7f5f45 | 300 | __cyl_bessel_k(_Tp __nu, _Tp __x) |
b34f60ac | 301 | { |
302 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
303 | std::__throw_domain_error(__N("Bad argument " | |
304 | "in __cyl_bessel_k.")); | |
305 | else if (__isnan(__nu) || __isnan(__x)) | |
306 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
307 | else | |
308 | { | |
309 | _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; | |
310 | __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); | |
311 | return __K_nu; | |
312 | } | |
313 | } | |
314 | ||
315 | ||
316 | /** | |
317 | * @brief Compute the spherical modified Bessel functions | |
318 | * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first | |
319 | * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ | |
320 | * respectively. | |
321 | * | |
322 | * @param __n The order of the modified spherical Bessel function. | |
323 | * @param __x The argument of the modified spherical Bessel function. | |
324 | * @param __i_n The output regular modified spherical Bessel function. | |
325 | * @param __k_n The output irregular modified spherical | |
326 | * Bessel function. | |
327 | * @param __ip_n The output derivative of the regular modified | |
328 | * spherical Bessel function. | |
329 | * @param __kp_n The output derivative of the irregular modified | |
330 | * spherical Bessel function. | |
331 | */ | |
332 | template <typename _Tp> | |
333 | void | |
cd7f5f45 | 334 | __sph_bessel_ik(unsigned int __n, _Tp __x, |
b34f60ac | 335 | _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) |
336 | { | |
337 | const _Tp __nu = _Tp(__n) + _Tp(0.5L); | |
338 | ||
339 | _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; | |
340 | __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); | |
341 | ||
342 | const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() | |
343 | / std::sqrt(__x); | |
344 | ||
345 | __i_n = __factor * __I_nu; | |
346 | __k_n = __factor * __K_nu; | |
347 | __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); | |
348 | __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); | |
349 | ||
350 | return; | |
351 | } | |
352 | ||
353 | ||
354 | /** | |
355 | * @brief Compute the Airy functions | |
356 | * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first | |
357 | * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ | |
358 | * respectively. | |
359 | * | |
b34f60ac | 360 | * @param __x The argument of the Airy functions. |
bf9c2a93 | 361 | * @param __Ai The output Airy function of the first kind. |
362 | * @param __Bi The output Airy function of the second kind. | |
363 | * @param __Aip The output derivative of the Airy function | |
364 | * of the first kind. | |
365 | * @param __Bip The output derivative of the Airy function | |
366 | * of the second kind. | |
b34f60ac | 367 | */ |
368 | template <typename _Tp> | |
369 | void | |
cd7f5f45 | 370 | __airy(_Tp __x, _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) |
b34f60ac | 371 | { |
372 | const _Tp __absx = std::abs(__x); | |
373 | const _Tp __rootx = std::sqrt(__absx); | |
374 | const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); | |
375 | ||
bf9c2a93 | 376 | if (__x > _Tp(0)) |
b34f60ac | 377 | { |
378 | _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; | |
379 | ||
380 | __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); | |
381 | __Ai = __rootx * __K_nu | |
382 | / (__numeric_constants<_Tp>::__sqrt3() | |
383 | * __numeric_constants<_Tp>::__pi()); | |
384 | __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() | |
385 | + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); | |
386 | ||
387 | __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); | |
388 | __Aip = -__x * __K_nu | |
389 | / (__numeric_constants<_Tp>::__sqrt3() | |
390 | * __numeric_constants<_Tp>::__pi()); | |
391 | __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() | |
392 | + _Tp(2) * __I_nu | |
393 | / __numeric_constants<_Tp>::__sqrt3()); | |
394 | } | |
395 | else if (__x < _Tp(0)) | |
396 | { | |
397 | _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; | |
398 | ||
399 | __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
400 | __Ai = __rootx * (__J_nu | |
401 | - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); | |
402 | __Bi = -__rootx * (__N_nu | |
403 | + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); | |
404 | ||
405 | __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
406 | __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() | |
407 | + __J_nu) / _Tp(2); | |
408 | __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() | |
409 | - __N_nu) / _Tp(2); | |
410 | } | |
411 | else | |
412 | { | |
413 | // Reference: | |
414 | // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. | |
415 | // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). | |
416 | __Ai = _Tp(0.35502805388781723926L); | |
417 | __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); | |
418 | ||
419 | // Reference: | |
420 | // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. | |
421 | // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). | |
422 | __Aip = -_Tp(0.25881940379280679840L); | |
423 | __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); | |
424 | } | |
425 | ||
426 | return; | |
427 | } | |
428 | ||
2948dd21 | 429 | _GLIBCXX_END_NAMESPACE_VERSION |
b34f60ac | 430 | } // namespace std::tr1::__detail |
c17b0a1c | 431 | } |
b34f60ac | 432 | } |
433 | ||
c17b0a1c | 434 | #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |