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1 | // Special functions -*- C++ -*- |
2 | ||
748086b7 | 3 | // Copyright (C) 2006, 2007, 2008, 2009 |
7c62b943 BK |
4 | // Free Software Foundation, Inc. |
5 | // | |
6 | // This file is part of the GNU ISO C++ Library. This library is free | |
7 | // software; you can redistribute it and/or modify it under the | |
8 | // terms of the GNU General Public License as published by the | |
748086b7 | 9 | // Free Software Foundation; either version 3, or (at your option) |
7c62b943 BK |
10 | // any later version. |
11 | // | |
12 | // This library is distributed in the hope that it will be useful, | |
13 | // but WITHOUT ANY WARRANTY; without even the implied warranty of | |
14 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
15 | // GNU General Public License for more details. | |
16 | // | |
748086b7 JJ |
17 | // Under Section 7 of GPL version 3, you are granted additional |
18 | // permissions described in the GCC Runtime Library Exception, version | |
19 | // 3.1, as published by the Free Software Foundation. | |
20 | ||
21 | // You should have received a copy of the GNU General Public License and | |
22 | // a copy of the GCC Runtime Library Exception along with this program; | |
23 | // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see | |
24 | // <http://www.gnu.org/licenses/>. | |
7c62b943 BK |
25 | |
26 | /** @file tr1/bessel_function.tcc | |
27 | * This is an internal header file, included by other library headers. | |
28 | * You should not attempt to use it directly. | |
29 | */ | |
30 | ||
31 | // | |
32 | // ISO C++ 14882 TR1: 5.2 Special functions | |
33 | // | |
34 | ||
35 | // Written by Edward Smith-Rowland. | |
36 | // | |
37 | // References: | |
38 | // (1) Handbook of Mathematical Functions, | |
39 | // ed. Milton Abramowitz and Irene A. Stegun, | |
40 | // Dover Publications, | |
41 | // Section 9, pp. 355-434, Section 10 pp. 435-478 | |
42 | // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl | |
43 | // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, | |
44 | // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), | |
45 | // 2nd ed, pp. 240-245 | |
46 | ||
e133ace8 PC |
47 | #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
48 | #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 | |
7c62b943 BK |
49 | |
50 | #include "special_function_util.h" | |
51 | ||
52 | namespace std | |
53 | { | |
e133ace8 PC |
54 | namespace tr1 |
55 | { | |
7c62b943 BK |
56 | |
57 | // [5.2] Special functions | |
58 | ||
7c62b943 | 59 | // Implementation-space details. |
7c62b943 BK |
60 | namespace __detail |
61 | { | |
62 | ||
63 | /** | |
64 | * @brief Compute the gamma functions required by the Temme series | |
65 | * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. | |
66 | * @f[ | |
67 | * \Gamma_1 = \frac{1}{2\mu} | |
68 | * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] | |
69 | * @f] | |
70 | * and | |
71 | * @f[ | |
72 | * \Gamma_2 = \frac{1}{2} | |
73 | * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] | |
74 | * @f] | |
75 | * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. | |
76 | * is the nearest integer to @f$ \nu @f$. | |
77 | * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ | |
78 | * are returned as well. | |
79 | * | |
80 | * The accuracy requirements on this are exquisite. | |
81 | * | |
82 | * @param __mu The input parameter of the gamma functions. | |
83 | * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ | |
84 | * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ | |
85 | * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ | |
86 | * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ | |
87 | */ | |
88 | template <typename _Tp> | |
89 | void | |
90 | __gamma_temme(const _Tp __mu, | |
91 | _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) | |
92 | { | |
93 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
e133ace8 PC |
94 | __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu); |
95 | __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu); | |
7c62b943 BK |
96 | #else |
97 | __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); | |
98 | __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); | |
99 | #endif | |
100 | ||
101 | if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) | |
102 | __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); | |
103 | else | |
104 | __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); | |
105 | ||
106 | __gam2 = (__gammi + __gampl) / (_Tp(2)); | |
107 | ||
108 | return; | |
109 | } | |
110 | ||
111 | ||
112 | /** | |
113 | * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann | |
114 | * @f$ N_\nu(x) @f$ functions and their first derivatives | |
115 | * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. | |
116 | * These four functions are computed together for numerical | |
117 | * stability. | |
118 | * | |
119 | * @param __nu The order of the Bessel functions. | |
120 | * @param __x The argument of the Bessel functions. | |
121 | * @param __Jnu The output Bessel function of the first kind. | |
28dac70a | 122 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
7c62b943 BK |
123 | * @param __Jpnu The output derivative of the Bessel function of the first kind. |
124 | * @param __Npnu The output derivative of the Neumann function. | |
125 | */ | |
126 | template <typename _Tp> | |
127 | void | |
128 | __bessel_jn(const _Tp __nu, const _Tp __x, | |
129 | _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) | |
130 | { | |
131 | if (__x == _Tp(0)) | |
132 | { | |
133 | if (__nu == _Tp(0)) | |
134 | { | |
135 | __Jnu = _Tp(1); | |
136 | __Jpnu = _Tp(0); | |
137 | } | |
138 | else if (__nu == _Tp(1)) | |
139 | { | |
140 | __Jnu = _Tp(0); | |
141 | __Jpnu = _Tp(0.5L); | |
142 | } | |
143 | else | |
144 | { | |
145 | __Jnu = _Tp(0); | |
146 | __Jpnu = _Tp(0); | |
147 | } | |
148 | __Nnu = -std::numeric_limits<_Tp>::infinity(); | |
149 | __Npnu = std::numeric_limits<_Tp>::infinity(); | |
150 | return; | |
151 | } | |
152 | ||
153 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
154 | // When the multiplier is N i.e. | |
155 | // fp_min = N * min() | |
156 | // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! | |
157 | //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); | |
158 | const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); | |
159 | const int __max_iter = 15000; | |
160 | const _Tp __x_min = _Tp(2); | |
161 | ||
162 | const int __nl = (__x < __x_min | |
163 | ? static_cast<int>(__nu + _Tp(0.5L)) | |
164 | : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); | |
165 | ||
166 | const _Tp __mu = __nu - __nl; | |
167 | const _Tp __mu2 = __mu * __mu; | |
168 | const _Tp __xi = _Tp(1) / __x; | |
169 | const _Tp __xi2 = _Tp(2) * __xi; | |
170 | _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); | |
171 | int __isign = 1; | |
172 | _Tp __h = __nu * __xi; | |
173 | if (__h < __fp_min) | |
174 | __h = __fp_min; | |
175 | _Tp __b = __xi2 * __nu; | |
176 | _Tp __d = _Tp(0); | |
177 | _Tp __c = __h; | |
178 | int __i; | |
179 | for (__i = 1; __i <= __max_iter; ++__i) | |
180 | { | |
181 | __b += __xi2; | |
182 | __d = __b - __d; | |
183 | if (std::abs(__d) < __fp_min) | |
184 | __d = __fp_min; | |
185 | __c = __b - _Tp(1) / __c; | |
186 | if (std::abs(__c) < __fp_min) | |
187 | __c = __fp_min; | |
188 | __d = _Tp(1) / __d; | |
189 | const _Tp __del = __c * __d; | |
190 | __h *= __del; | |
191 | if (__d < _Tp(0)) | |
192 | __isign = -__isign; | |
193 | if (std::abs(__del - _Tp(1)) < __eps) | |
194 | break; | |
195 | } | |
196 | if (__i > __max_iter) | |
197 | std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " | |
198 | "try asymptotic expansion.")); | |
199 | _Tp __Jnul = __isign * __fp_min; | |
200 | _Tp __Jpnul = __h * __Jnul; | |
201 | _Tp __Jnul1 = __Jnul; | |
202 | _Tp __Jpnu1 = __Jpnul; | |
203 | _Tp __fact = __nu * __xi; | |
204 | for ( int __l = __nl; __l >= 1; --__l ) | |
205 | { | |
206 | const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; | |
207 | __fact -= __xi; | |
208 | __Jpnul = __fact * __Jnutemp - __Jnul; | |
209 | __Jnul = __Jnutemp; | |
210 | } | |
211 | if (__Jnul == _Tp(0)) | |
212 | __Jnul = __eps; | |
213 | _Tp __f= __Jpnul / __Jnul; | |
214 | _Tp __Nmu, __Nnu1, __Npmu, __Jmu; | |
215 | if (__x < __x_min) | |
216 | { | |
217 | const _Tp __x2 = __x / _Tp(2); | |
218 | const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; | |
219 | _Tp __fact = (std::abs(__pimu) < __eps | |
220 | ? _Tp(1) : __pimu / std::sin(__pimu)); | |
221 | _Tp __d = -std::log(__x2); | |
222 | _Tp __e = __mu * __d; | |
223 | _Tp __fact2 = (std::abs(__e) < __eps | |
224 | ? _Tp(1) : std::sinh(__e) / __e); | |
225 | _Tp __gam1, __gam2, __gampl, __gammi; | |
226 | __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); | |
227 | _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) | |
228 | * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); | |
229 | __e = std::exp(__e); | |
230 | _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); | |
231 | _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); | |
232 | const _Tp __pimu2 = __pimu / _Tp(2); | |
233 | _Tp __fact3 = (std::abs(__pimu2) < __eps | |
234 | ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); | |
235 | _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; | |
236 | _Tp __c = _Tp(1); | |
237 | __d = -__x2 * __x2; | |
238 | _Tp __sum = __ff + __r * __q; | |
239 | _Tp __sum1 = __p; | |
240 | for (__i = 1; __i <= __max_iter; ++__i) | |
241 | { | |
242 | __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); | |
243 | __c *= __d / _Tp(__i); | |
244 | __p /= _Tp(__i) - __mu; | |
245 | __q /= _Tp(__i) + __mu; | |
246 | const _Tp __del = __c * (__ff + __r * __q); | |
247 | __sum += __del; | |
248 | const _Tp __del1 = __c * __p - __i * __del; | |
249 | __sum1 += __del1; | |
250 | if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) | |
251 | break; | |
252 | } | |
253 | if ( __i > __max_iter ) | |
254 | std::__throw_runtime_error(__N("Bessel y series failed to converge " | |
255 | "in __bessel_jn.")); | |
256 | __Nmu = -__sum; | |
257 | __Nnu1 = -__sum1 * __xi2; | |
258 | __Npmu = __mu * __xi * __Nmu - __Nnu1; | |
259 | __Jmu = __w / (__Npmu - __f * __Nmu); | |
260 | } | |
261 | else | |
262 | { | |
263 | _Tp __a = _Tp(0.25L) - __mu2; | |
264 | _Tp __q = _Tp(1); | |
265 | _Tp __p = -__xi / _Tp(2); | |
266 | _Tp __br = _Tp(2) * __x; | |
267 | _Tp __bi = _Tp(2); | |
268 | _Tp __fact = __a * __xi / (__p * __p + __q * __q); | |
269 | _Tp __cr = __br + __q * __fact; | |
270 | _Tp __ci = __bi + __p * __fact; | |
271 | _Tp __den = __br * __br + __bi * __bi; | |
272 | _Tp __dr = __br / __den; | |
273 | _Tp __di = -__bi / __den; | |
274 | _Tp __dlr = __cr * __dr - __ci * __di; | |
275 | _Tp __dli = __cr * __di + __ci * __dr; | |
276 | _Tp __temp = __p * __dlr - __q * __dli; | |
277 | __q = __p * __dli + __q * __dlr; | |
278 | __p = __temp; | |
279 | int __i; | |
280 | for (__i = 2; __i <= __max_iter; ++__i) | |
281 | { | |
282 | __a += _Tp(2 * (__i - 1)); | |
283 | __bi += _Tp(2); | |
284 | __dr = __a * __dr + __br; | |
285 | __di = __a * __di + __bi; | |
286 | if (std::abs(__dr) + std::abs(__di) < __fp_min) | |
287 | __dr = __fp_min; | |
288 | __fact = __a / (__cr * __cr + __ci * __ci); | |
289 | __cr = __br + __cr * __fact; | |
290 | __ci = __bi - __ci * __fact; | |
291 | if (std::abs(__cr) + std::abs(__ci) < __fp_min) | |
292 | __cr = __fp_min; | |
293 | __den = __dr * __dr + __di * __di; | |
294 | __dr /= __den; | |
295 | __di /= -__den; | |
296 | __dlr = __cr * __dr - __ci * __di; | |
297 | __dli = __cr * __di + __ci * __dr; | |
298 | __temp = __p * __dlr - __q * __dli; | |
299 | __q = __p * __dli + __q * __dlr; | |
300 | __p = __temp; | |
301 | if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) | |
302 | break; | |
303 | } | |
304 | if (__i > __max_iter) | |
305 | std::__throw_runtime_error(__N("Lentz's method failed " | |
306 | "in __bessel_jn.")); | |
307 | const _Tp __gam = (__p - __f) / __q; | |
308 | __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); | |
309 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
e133ace8 | 310 | __Jmu = std::tr1::copysign(__Jmu, __Jnul); |
7c62b943 BK |
311 | #else |
312 | if (__Jmu * __Jnul < _Tp(0)) | |
313 | __Jmu = -__Jmu; | |
314 | #endif | |
315 | __Nmu = __gam * __Jmu; | |
316 | __Npmu = (__p + __q / __gam) * __Nmu; | |
317 | __Nnu1 = __mu * __xi * __Nmu - __Npmu; | |
318 | } | |
319 | __fact = __Jmu / __Jnul; | |
320 | __Jnu = __fact * __Jnul1; | |
321 | __Jpnu = __fact * __Jpnu1; | |
322 | for (__i = 1; __i <= __nl; ++__i) | |
323 | { | |
324 | const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; | |
325 | __Nmu = __Nnu1; | |
326 | __Nnu1 = __Nnutemp; | |
327 | } | |
328 | __Nnu = __Nmu; | |
329 | __Npnu = __nu * __xi * __Nmu - __Nnu1; | |
330 | ||
331 | return; | |
332 | } | |
333 | ||
334 | ||
335 | /** | |
28dac70a | 336 | * @brief This routine computes the asymptotic cylindrical Bessel |
7c62b943 BK |
337 | * and Neumann functions of order nu: \f$ J_{\nu} \f$, |
338 | * \f$ N_{\nu} \f$. | |
339 | * | |
340 | * References: | |
341 | * (1) Handbook of Mathematical Functions, | |
342 | * ed. Milton Abramowitz and Irene A. Stegun, | |
343 | * Dover Publications, | |
344 | * Section 9 p. 364, Equations 9.2.5-9.2.10 | |
345 | * | |
346 | * @param __nu The order of the Bessel functions. | |
347 | * @param __x The argument of the Bessel functions. | |
348 | * @param __Jnu The output Bessel function of the first kind. | |
28dac70a | 349 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
7c62b943 BK |
350 | */ |
351 | template <typename _Tp> | |
352 | void | |
353 | __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x, | |
354 | _Tp & __Jnu, _Tp & __Nnu) | |
355 | { | |
356 | const _Tp __coef = std::sqrt(_Tp(2) | |
357 | / (__numeric_constants<_Tp>::__pi() * __x)); | |
358 | const _Tp __mu = _Tp(4) * __nu * __nu; | |
359 | const _Tp __mum1 = __mu - _Tp(1); | |
360 | const _Tp __mum9 = __mu - _Tp(9); | |
361 | const _Tp __mum25 = __mu - _Tp(25); | |
362 | const _Tp __mum49 = __mu - _Tp(49); | |
363 | const _Tp __xx = _Tp(64) * __x * __x; | |
364 | const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) | |
365 | * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); | |
366 | const _Tp __Q = __mum1 / (_Tp(8) * __x) | |
367 | * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); | |
368 | ||
369 | const _Tp __chi = __x - (__nu + _Tp(0.5L)) | |
370 | * __numeric_constants<_Tp>::__pi_2(); | |
371 | const _Tp __c = std::cos(__chi); | |
372 | const _Tp __s = std::sin(__chi); | |
373 | ||
374 | __Jnu = __coef * (__c * __P - __s * __Q); | |
375 | __Nnu = __coef * (__s * __P + __c * __Q); | |
376 | ||
377 | return; | |
378 | } | |
379 | ||
380 | ||
381 | /** | |
382 | * @brief This routine returns the cylindrical Bessel functions | |
383 | * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ | |
384 | * by series expansion. | |
385 | * | |
386 | * The modified cylindrical Bessel function is: | |
387 | * @f[ | |
388 | * Z_{\nu}(x) = \sum_{k=0}^{\infty} | |
389 | * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
390 | * @f] | |
391 | * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for | |
28dac70a | 392 | * \f$ Z = I \f$ or \f$ J \f$ respectively. |
7c62b943 BK |
393 | * |
394 | * See Abramowitz & Stegun, 9.1.10 | |
395 | * Abramowitz & Stegun, 9.6.7 | |
396 | * (1) Handbook of Mathematical Functions, | |
397 | * ed. Milton Abramowitz and Irene A. Stegun, | |
398 | * Dover Publications, | |
399 | * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 | |
400 | * | |
401 | * @param __nu The order of the Bessel function. | |
402 | * @param __x The argument of the Bessel function. | |
403 | * @param __sgn The sign of the alternate terms | |
404 | * -1 for the Bessel function of the first kind. | |
405 | * +1 for the modified Bessel function of the first kind. | |
406 | * @return The output Bessel function. | |
407 | */ | |
408 | template <typename _Tp> | |
409 | _Tp | |
410 | __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn, | |
411 | const unsigned int __max_iter) | |
412 | { | |
413 | ||
414 | const _Tp __x2 = __x / _Tp(2); | |
415 | _Tp __fact = __nu * std::log(__x2); | |
416 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
e133ace8 | 417 | __fact -= std::tr1::lgamma(__nu + _Tp(1)); |
7c62b943 BK |
418 | #else |
419 | __fact -= __log_gamma(__nu + _Tp(1)); | |
420 | #endif | |
421 | __fact = std::exp(__fact); | |
422 | const _Tp __xx4 = __sgn * __x2 * __x2; | |
423 | _Tp __Jn = _Tp(1); | |
424 | _Tp __term = _Tp(1); | |
425 | ||
426 | for (unsigned int __i = 1; __i < __max_iter; ++__i) | |
427 | { | |
428 | __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); | |
429 | __Jn += __term; | |
430 | if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) | |
431 | break; | |
432 | } | |
433 | ||
434 | return __fact * __Jn; | |
435 | } | |
436 | ||
437 | ||
438 | /** | |
439 | * @brief Return the Bessel function of order \f$ \nu \f$: | |
440 | * \f$ J_{\nu}(x) \f$. | |
441 | * | |
442 | * The cylindrical Bessel function is: | |
443 | * @f[ | |
444 | * J_{\nu}(x) = \sum_{k=0}^{\infty} | |
445 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
446 | * @f] | |
447 | * | |
448 | * @param __nu The order of the Bessel function. | |
449 | * @param __x The argument of the Bessel function. | |
450 | * @return The output Bessel function. | |
451 | */ | |
452 | template<typename _Tp> | |
453 | _Tp | |
454 | __cyl_bessel_j(const _Tp __nu, const _Tp __x) | |
455 | { | |
456 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
457 | std::__throw_domain_error(__N("Bad argument " | |
458 | "in __cyl_bessel_j.")); | |
459 | else if (__isnan(__nu) || __isnan(__x)) | |
460 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
461 | else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) | |
462 | return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); | |
463 | else if (__x > _Tp(1000)) | |
464 | { | |
465 | _Tp __J_nu, __N_nu; | |
466 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
467 | return __J_nu; | |
468 | } | |
469 | else | |
470 | { | |
471 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
472 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
473 | return __J_nu; | |
474 | } | |
475 | } | |
476 | ||
477 | ||
478 | /** | |
28dac70a | 479 | * @brief Return the Neumann function of order \f$ \nu \f$: |
7c62b943 BK |
480 | * \f$ N_{\nu}(x) \f$. |
481 | * | |
482 | * The Neumann function is defined by: | |
483 | * @f[ | |
484 | * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} | |
485 | * {\sin \nu\pi} | |
486 | * @f] | |
487 | * where for integral \f$ \nu = n \f$ a limit is taken: | |
488 | * \f$ lim_{\nu \to n} \f$. | |
489 | * | |
490 | * @param __nu The order of the Neumann function. | |
491 | * @param __x The argument of the Neumann function. | |
492 | * @return The output Neumann function. | |
493 | */ | |
494 | template<typename _Tp> | |
495 | _Tp | |
496 | __cyl_neumann_n(const _Tp __nu, const _Tp __x) | |
497 | { | |
498 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
499 | std::__throw_domain_error(__N("Bad argument " | |
500 | "in __cyl_neumann_n.")); | |
501 | else if (__isnan(__nu) || __isnan(__x)) | |
502 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
503 | else if (__x > _Tp(1000)) | |
504 | { | |
505 | _Tp __J_nu, __N_nu; | |
506 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
507 | return __N_nu; | |
508 | } | |
509 | else | |
510 | { | |
511 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
512 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
513 | return __N_nu; | |
514 | } | |
515 | } | |
516 | ||
517 | ||
518 | /** | |
519 | * @brief Compute the spherical Bessel @f$ j_n(x) @f$ | |
520 | * and Neumann @f$ n_n(x) @f$ functions and their first | |
521 | * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ | |
522 | * respectively. | |
523 | * | |
524 | * @param __n The order of the spherical Bessel function. | |
525 | * @param __x The argument of the spherical Bessel function. | |
526 | * @param __j_n The output spherical Bessel function. | |
527 | * @param __n_n The output spherical Neumann function. | |
528 | * @param __jp_n The output derivative of the spherical Bessel function. | |
529 | * @param __np_n The output derivative of the spherical Neumann function. | |
530 | */ | |
531 | template <typename _Tp> | |
532 | void | |
533 | __sph_bessel_jn(const unsigned int __n, const _Tp __x, | |
534 | _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) | |
535 | { | |
536 | const _Tp __nu = _Tp(__n) + _Tp(0.5L); | |
537 | ||
538 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
539 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
540 | ||
541 | const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() | |
542 | / std::sqrt(__x); | |
543 | ||
544 | __j_n = __factor * __J_nu; | |
545 | __n_n = __factor * __N_nu; | |
546 | __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); | |
547 | __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); | |
548 | ||
549 | return; | |
550 | } | |
551 | ||
552 | ||
553 | /** | |
554 | * @brief Return the spherical Bessel function | |
555 | * @f$ j_n(x) @f$ of order n. | |
556 | * | |
557 | * The spherical Bessel function is defined by: | |
558 | * @f[ | |
559 | * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) | |
560 | * @f] | |
561 | * | |
562 | * @param __n The order of the spherical Bessel function. | |
563 | * @param __x The argument of the spherical Bessel function. | |
564 | * @return The output spherical Bessel function. | |
565 | */ | |
566 | template <typename _Tp> | |
567 | _Tp | |
568 | __sph_bessel(const unsigned int __n, const _Tp __x) | |
569 | { | |
570 | if (__x < _Tp(0)) | |
571 | std::__throw_domain_error(__N("Bad argument " | |
572 | "in __sph_bessel.")); | |
573 | else if (__isnan(__x)) | |
574 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
575 | else if (__x == _Tp(0)) | |
576 | { | |
577 | if (__n == 0) | |
578 | return _Tp(1); | |
579 | else | |
580 | return _Tp(0); | |
581 | } | |
582 | else | |
583 | { | |
584 | _Tp __j_n, __n_n, __jp_n, __np_n; | |
585 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
586 | return __j_n; | |
587 | } | |
588 | } | |
589 | ||
590 | ||
591 | /** | |
592 | * @brief Return the spherical Neumann function | |
593 | * @f$ n_n(x) @f$. | |
594 | * | |
595 | * The spherical Neumann function is defined by: | |
596 | * @f[ | |
597 | * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) | |
598 | * @f] | |
599 | * | |
600 | * @param __n The order of the spherical Neumann function. | |
601 | * @param __x The argument of the spherical Neumann function. | |
602 | * @return The output spherical Neumann function. | |
603 | */ | |
604 | template <typename _Tp> | |
605 | _Tp | |
606 | __sph_neumann(const unsigned int __n, const _Tp __x) | |
607 | { | |
608 | if (__x < _Tp(0)) | |
609 | std::__throw_domain_error(__N("Bad argument " | |
610 | "in __sph_neumann.")); | |
611 | else if (__isnan(__x)) | |
612 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
613 | else if (__x == _Tp(0)) | |
614 | return -std::numeric_limits<_Tp>::infinity(); | |
615 | else | |
616 | { | |
617 | _Tp __j_n, __n_n, __jp_n, __np_n; | |
618 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
619 | return __n_n; | |
620 | } | |
621 | } | |
622 | ||
623 | } // namespace std::tr1::__detail | |
e133ace8 | 624 | } |
7c62b943 BK |
625 | } |
626 | ||
e133ace8 | 627 | #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |