1 | // Special functions -*- C++ -*- |
2 | |
3 | // Copyright (C) 2006-2021 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/bessel_function.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 | /* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c |
31 | * Copyright (C) 1996-2003 Gerard Jungman |
32 | */ |
33 | |
34 | // |
35 | // ISO C++ 14882 TR1: 5.2 Special functions |
36 | // |
37 | |
38 | // Written by Edward Smith-Rowland. |
39 | // |
40 | // References: |
41 | // (1) Handbook of Mathematical Functions, |
42 | // ed. Milton Abramowitz and Irene A. Stegun, |
43 | // Dover Publications, |
44 | // Section 9, pp. 355-434, Section 10 pp. 435-478 |
45 | // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
46 | // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, |
47 | // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), |
48 | // 2nd ed, pp. 240-245 |
49 | |
50 | #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
51 | #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 |
52 | |
53 | #include <tr1/special_function_util.h> |
54 | |
55 | namespace std _GLIBCXX_VISIBILITY(default) |
56 | { |
57 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
58 | |
59 | #if _GLIBCXX_USE_STD_SPEC_FUNCS |
60 | # define _GLIBCXX_MATH_NS ::std |
61 | #elif defined(_GLIBCXX_TR1_CMATH) |
62 | namespace tr1 |
63 | { |
64 | # define _GLIBCXX_MATH_NS ::std::tr1 |
65 | #else |
66 | # error do not include this header directly, use <cmath> or <tr1/cmath> |
67 | #endif |
68 | // [5.2] Special functions |
69 | |
70 | // Implementation-space details. |
71 | namespace __detail |
72 | { |
73 | /** |
74 | * @brief Compute the gamma functions required by the Temme series |
75 | * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. |
76 | * @f[ |
77 | * \Gamma_1 = \frac{1}{2\mu} |
78 | * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] |
79 | * @f] |
80 | * and |
81 | * @f[ |
82 | * \Gamma_2 = \frac{1}{2} |
83 | * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] |
84 | * @f] |
85 | * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. |
86 | * is the nearest integer to @f$ \nu @f$. |
87 | * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ |
88 | * are returned as well. |
89 | * |
90 | * The accuracy requirements on this are exquisite. |
91 | * |
92 | * @param __mu The input parameter of the gamma functions. |
93 | * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ |
94 | * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ |
95 | * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ |
96 | * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ |
97 | */ |
98 | template <typename _Tp> |
99 | void |
100 | __gamma_temme(_Tp __mu, |
101 | _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) |
102 | { |
103 | #if _GLIBCXX_USE_C99_MATH_TR1 |
104 | __gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu); |
105 | __gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu); |
106 | #else |
107 | __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); |
108 | __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); |
109 | #endif |
110 | |
111 | if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) |
112 | __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); |
113 | else |
114 | __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); |
115 | |
116 | __gam2 = (__gammi + __gampl) / (_Tp(2)); |
117 | |
118 | return; |
119 | } |
120 | |
121 | |
122 | /** |
123 | * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann |
124 | * @f$ N_\nu(x) @f$ functions and their first derivatives |
125 | * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. |
126 | * These four functions are computed together for numerical |
127 | * stability. |
128 | * |
129 | * @param __nu The order of the Bessel functions. |
130 | * @param __x The argument of the Bessel functions. |
131 | * @param __Jnu The output Bessel function of the first kind. |
132 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
133 | * @param __Jpnu The output derivative of the Bessel function of the first kind. |
134 | * @param __Npnu The output derivative of the Neumann function. |
135 | */ |
136 | template <typename _Tp> |
137 | void |
138 | __bessel_jn(_Tp __nu, _Tp __x, |
139 | _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) |
140 | { |
141 | if (__x == _Tp(0)) |
142 | { |
143 | if (__nu == _Tp(0)) |
144 | { |
145 | __Jnu = _Tp(1); |
146 | __Jpnu = _Tp(0); |
147 | } |
148 | else if (__nu == _Tp(1)) |
149 | { |
150 | __Jnu = _Tp(0); |
151 | __Jpnu = _Tp(0.5L); |
152 | } |
153 | else |
154 | { |
155 | __Jnu = _Tp(0); |
156 | __Jpnu = _Tp(0); |
157 | } |
158 | __Nnu = -std::numeric_limits<_Tp>::infinity(); |
159 | __Npnu = std::numeric_limits<_Tp>::infinity(); |
160 | return; |
161 | } |
162 | |
163 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
164 | // When the multiplier is N i.e. |
165 | // fp_min = N * min() |
166 | // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! |
167 | //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); |
168 | const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); |
169 | const int __max_iter = 15000; |
170 | const _Tp __x_min = _Tp(2); |
171 | |
172 | const int __nl = (__x < __x_min |
173 | ? static_cast<int>(__nu + _Tp(0.5L)) |
174 | : std::max(a: 0, b: static_cast<int>(__nu - __x + _Tp(1.5L)))); |
175 | |
176 | const _Tp __mu = __nu - __nl; |
177 | const _Tp __mu2 = __mu * __mu; |
178 | const _Tp __xi = _Tp(1) / __x; |
179 | const _Tp __xi2 = _Tp(2) * __xi; |
180 | _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); |
181 | int __isign = 1; |
182 | _Tp __h = __nu * __xi; |
183 | if (__h < __fp_min) |
184 | __h = __fp_min; |
185 | _Tp __b = __xi2 * __nu; |
186 | _Tp __d = _Tp(0); |
187 | _Tp __c = __h; |
188 | int __i; |
189 | for (__i = 1; __i <= __max_iter; ++__i) |
190 | { |
191 | __b += __xi2; |
192 | __d = __b - __d; |
193 | if (std::abs(__d) < __fp_min) |
194 | __d = __fp_min; |
195 | __c = __b - _Tp(1) / __c; |
196 | if (std::abs(__c) < __fp_min) |
197 | __c = __fp_min; |
198 | __d = _Tp(1) / __d; |
199 | const _Tp __del = __c * __d; |
200 | __h *= __del; |
201 | if (__d < _Tp(0)) |
202 | __isign = -__isign; |
203 | if (std::abs(__del - _Tp(1)) < __eps) |
204 | break; |
205 | } |
206 | if (__i > __max_iter) |
207 | std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " |
208 | "try asymptotic expansion." )); |
209 | _Tp __Jnul = __isign * __fp_min; |
210 | _Tp __Jpnul = __h * __Jnul; |
211 | _Tp __Jnul1 = __Jnul; |
212 | _Tp __Jpnu1 = __Jpnul; |
213 | _Tp __fact = __nu * __xi; |
214 | for ( int __l = __nl; __l >= 1; --__l ) |
215 | { |
216 | const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; |
217 | __fact -= __xi; |
218 | __Jpnul = __fact * __Jnutemp - __Jnul; |
219 | __Jnul = __Jnutemp; |
220 | } |
221 | if (__Jnul == _Tp(0)) |
222 | __Jnul = __eps; |
223 | _Tp __f= __Jpnul / __Jnul; |
224 | _Tp __Nmu, __Nnu1, __Npmu, __Jmu; |
225 | if (__x < __x_min) |
226 | { |
227 | const _Tp __x2 = __x / _Tp(2); |
228 | const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; |
229 | _Tp __fact = (std::abs(__pimu) < __eps |
230 | ? _Tp(1) : __pimu / std::sin(__pimu)); |
231 | _Tp __d = -std::log(__x2); |
232 | _Tp __e = __mu * __d; |
233 | _Tp __fact2 = (std::abs(__e) < __eps |
234 | ? _Tp(1) : std::sinh(__e) / __e); |
235 | _Tp __gam1, __gam2, __gampl, __gammi; |
236 | __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); |
237 | _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) |
238 | * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); |
239 | __e = std::exp(__e); |
240 | _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); |
241 | _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); |
242 | const _Tp __pimu2 = __pimu / _Tp(2); |
243 | _Tp __fact3 = (std::abs(__pimu2) < __eps |
244 | ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); |
245 | _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; |
246 | _Tp __c = _Tp(1); |
247 | __d = -__x2 * __x2; |
248 | _Tp __sum = __ff + __r * __q; |
249 | _Tp __sum1 = __p; |
250 | for (__i = 1; __i <= __max_iter; ++__i) |
251 | { |
252 | __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); |
253 | __c *= __d / _Tp(__i); |
254 | __p /= _Tp(__i) - __mu; |
255 | __q /= _Tp(__i) + __mu; |
256 | const _Tp __del = __c * (__ff + __r * __q); |
257 | __sum += __del; |
258 | const _Tp __del1 = __c * __p - __i * __del; |
259 | __sum1 += __del1; |
260 | if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) |
261 | break; |
262 | } |
263 | if ( __i > __max_iter ) |
264 | std::__throw_runtime_error(__N("Bessel y series failed to converge " |
265 | "in __bessel_jn." )); |
266 | __Nmu = -__sum; |
267 | __Nnu1 = -__sum1 * __xi2; |
268 | __Npmu = __mu * __xi * __Nmu - __Nnu1; |
269 | __Jmu = __w / (__Npmu - __f * __Nmu); |
270 | } |
271 | else |
272 | { |
273 | _Tp __a = _Tp(0.25L) - __mu2; |
274 | _Tp __q = _Tp(1); |
275 | _Tp __p = -__xi / _Tp(2); |
276 | _Tp __br = _Tp(2) * __x; |
277 | _Tp __bi = _Tp(2); |
278 | _Tp __fact = __a * __xi / (__p * __p + __q * __q); |
279 | _Tp __cr = __br + __q * __fact; |
280 | _Tp __ci = __bi + __p * __fact; |
281 | _Tp __den = __br * __br + __bi * __bi; |
282 | _Tp __dr = __br / __den; |
283 | _Tp __di = -__bi / __den; |
284 | _Tp __dlr = __cr * __dr - __ci * __di; |
285 | _Tp __dli = __cr * __di + __ci * __dr; |
286 | _Tp __temp = __p * __dlr - __q * __dli; |
287 | __q = __p * __dli + __q * __dlr; |
288 | __p = __temp; |
289 | int __i; |
290 | for (__i = 2; __i <= __max_iter; ++__i) |
291 | { |
292 | __a += _Tp(2 * (__i - 1)); |
293 | __bi += _Tp(2); |
294 | __dr = __a * __dr + __br; |
295 | __di = __a * __di + __bi; |
296 | if (std::abs(__dr) + std::abs(__di) < __fp_min) |
297 | __dr = __fp_min; |
298 | __fact = __a / (__cr * __cr + __ci * __ci); |
299 | __cr = __br + __cr * __fact; |
300 | __ci = __bi - __ci * __fact; |
301 | if (std::abs(__cr) + std::abs(__ci) < __fp_min) |
302 | __cr = __fp_min; |
303 | __den = __dr * __dr + __di * __di; |
304 | __dr /= __den; |
305 | __di /= -__den; |
306 | __dlr = __cr * __dr - __ci * __di; |
307 | __dli = __cr * __di + __ci * __dr; |
308 | __temp = __p * __dlr - __q * __dli; |
309 | __q = __p * __dli + __q * __dlr; |
310 | __p = __temp; |
311 | if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) |
312 | break; |
313 | } |
314 | if (__i > __max_iter) |
315 | std::__throw_runtime_error(__N("Lentz's method failed " |
316 | "in __bessel_jn." )); |
317 | const _Tp __gam = (__p - __f) / __q; |
318 | __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); |
319 | #if _GLIBCXX_USE_C99_MATH_TR1 |
320 | __Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul); |
321 | #else |
322 | if (__Jmu * __Jnul < _Tp(0)) |
323 | __Jmu = -__Jmu; |
324 | #endif |
325 | __Nmu = __gam * __Jmu; |
326 | __Npmu = (__p + __q / __gam) * __Nmu; |
327 | __Nnu1 = __mu * __xi * __Nmu - __Npmu; |
328 | } |
329 | __fact = __Jmu / __Jnul; |
330 | __Jnu = __fact * __Jnul1; |
331 | __Jpnu = __fact * __Jpnu1; |
332 | for (__i = 1; __i <= __nl; ++__i) |
333 | { |
334 | const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; |
335 | __Nmu = __Nnu1; |
336 | __Nnu1 = __Nnutemp; |
337 | } |
338 | __Nnu = __Nmu; |
339 | __Npnu = __nu * __xi * __Nmu - __Nnu1; |
340 | |
341 | return; |
342 | } |
343 | |
344 | |
345 | /** |
346 | * @brief This routine computes the asymptotic cylindrical Bessel |
347 | * and Neumann functions of order nu: \f$ J_{\nu} \f$, |
348 | * \f$ N_{\nu} \f$. |
349 | * |
350 | * References: |
351 | * (1) Handbook of Mathematical Functions, |
352 | * ed. Milton Abramowitz and Irene A. Stegun, |
353 | * Dover Publications, |
354 | * Section 9 p. 364, Equations 9.2.5-9.2.10 |
355 | * |
356 | * @param __nu The order of the Bessel functions. |
357 | * @param __x The argument of the Bessel functions. |
358 | * @param __Jnu The output Bessel function of the first kind. |
359 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
360 | */ |
361 | template <typename _Tp> |
362 | void |
363 | __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) |
364 | { |
365 | const _Tp __mu = _Tp(4) * __nu * __nu; |
366 | const _Tp __8x = _Tp(8) * __x; |
367 | |
368 | _Tp __P = _Tp(0); |
369 | _Tp __Q = _Tp(0); |
370 | |
371 | _Tp __k = _Tp(0); |
372 | _Tp __term = _Tp(1); |
373 | |
374 | int __epsP = 0; |
375 | int __epsQ = 0; |
376 | |
377 | _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
378 | |
379 | do |
380 | { |
381 | __term *= (__k == 0 |
382 | ? _Tp(1) |
383 | : -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x)); |
384 | |
385 | __epsP = std::abs(__term) < __eps * std::abs(__P); |
386 | __P += __term; |
387 | |
388 | __k++; |
389 | |
390 | __term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x); |
391 | __epsQ = std::abs(__term) < __eps * std::abs(__Q); |
392 | __Q += __term; |
393 | |
394 | if (__epsP && __epsQ && __k > (__nu / 2.)) |
395 | break; |
396 | |
397 | __k++; |
398 | } |
399 | while (__k < 1000); |
400 | |
401 | const _Tp __chi = __x - (__nu + _Tp(0.5L)) |
402 | * __numeric_constants<_Tp>::__pi_2(); |
403 | |
404 | const _Tp __c = std::cos(__chi); |
405 | const _Tp __s = std::sin(__chi); |
406 | |
407 | const _Tp __coef = std::sqrt(_Tp(2) |
408 | / (__numeric_constants<_Tp>::__pi() * __x)); |
409 | |
410 | __Jnu = __coef * (__c * __P - __s * __Q); |
411 | __Nnu = __coef * (__s * __P + __c * __Q); |
412 | |
413 | return; |
414 | } |
415 | |
416 | |
417 | /** |
418 | * @brief This routine returns the cylindrical Bessel functions |
419 | * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ |
420 | * by series expansion. |
421 | * |
422 | * The modified cylindrical Bessel function is: |
423 | * @f[ |
424 | * Z_{\nu}(x) = \sum_{k=0}^{\infty} |
425 | * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
426 | * @f] |
427 | * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for |
428 | * \f$ Z = I \f$ or \f$ J \f$ respectively. |
429 | * |
430 | * See Abramowitz & Stegun, 9.1.10 |
431 | * Abramowitz & Stegun, 9.6.7 |
432 | * (1) Handbook of Mathematical Functions, |
433 | * ed. Milton Abramowitz and Irene A. Stegun, |
434 | * Dover Publications, |
435 | * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 |
436 | * |
437 | * @param __nu The order of the Bessel function. |
438 | * @param __x The argument of the Bessel function. |
439 | * @param __sgn The sign of the alternate terms |
440 | * -1 for the Bessel function of the first kind. |
441 | * +1 for the modified Bessel function of the first kind. |
442 | * @return The output Bessel function. |
443 | */ |
444 | template <typename _Tp> |
445 | _Tp |
446 | __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, |
447 | unsigned int __max_iter) |
448 | { |
449 | if (__x == _Tp(0)) |
450 | return __nu == _Tp(0) ? _Tp(1) : _Tp(0); |
451 | |
452 | const _Tp __x2 = __x / _Tp(2); |
453 | _Tp __fact = __nu * std::log(__x2); |
454 | #if _GLIBCXX_USE_C99_MATH_TR1 |
455 | __fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1)); |
456 | #else |
457 | __fact -= __log_gamma(__nu + _Tp(1)); |
458 | #endif |
459 | __fact = std::exp(__fact); |
460 | const _Tp __xx4 = __sgn * __x2 * __x2; |
461 | _Tp __Jn = _Tp(1); |
462 | _Tp __term = _Tp(1); |
463 | |
464 | for (unsigned int __i = 1; __i < __max_iter; ++__i) |
465 | { |
466 | __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); |
467 | __Jn += __term; |
468 | if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) |
469 | break; |
470 | } |
471 | |
472 | return __fact * __Jn; |
473 | } |
474 | |
475 | |
476 | /** |
477 | * @brief Return the Bessel function of order \f$ \nu \f$: |
478 | * \f$ J_{\nu}(x) \f$. |
479 | * |
480 | * The cylindrical Bessel function is: |
481 | * @f[ |
482 | * J_{\nu}(x) = \sum_{k=0}^{\infty} |
483 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
484 | * @f] |
485 | * |
486 | * @param __nu The order of the Bessel function. |
487 | * @param __x The argument of the Bessel function. |
488 | * @return The output Bessel function. |
489 | */ |
490 | template<typename _Tp> |
491 | _Tp |
492 | __cyl_bessel_j(_Tp __nu, _Tp __x) |
493 | { |
494 | if (__nu < _Tp(0) || __x < _Tp(0)) |
495 | std::__throw_domain_error(__N("Bad argument " |
496 | "in __cyl_bessel_j." )); |
497 | else if (__isnan(__nu) || __isnan(__x)) |
498 | return std::numeric_limits<_Tp>::quiet_NaN(); |
499 | else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) |
500 | return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); |
501 | else if (__x > _Tp(1000)) |
502 | { |
503 | _Tp __J_nu, __N_nu; |
504 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
505 | return __J_nu; |
506 | } |
507 | else |
508 | { |
509 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
510 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
511 | return __J_nu; |
512 | } |
513 | } |
514 | |
515 | |
516 | /** |
517 | * @brief Return the Neumann function of order \f$ \nu \f$: |
518 | * \f$ N_{\nu}(x) \f$. |
519 | * |
520 | * The Neumann function is defined by: |
521 | * @f[ |
522 | * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} |
523 | * {\sin \nu\pi} |
524 | * @f] |
525 | * where for integral \f$ \nu = n \f$ a limit is taken: |
526 | * \f$ lim_{\nu \to n} \f$. |
527 | * |
528 | * @param __nu The order of the Neumann function. |
529 | * @param __x The argument of the Neumann function. |
530 | * @return The output Neumann function. |
531 | */ |
532 | template<typename _Tp> |
533 | _Tp |
534 | __cyl_neumann_n(_Tp __nu, _Tp __x) |
535 | { |
536 | if (__nu < _Tp(0) || __x < _Tp(0)) |
537 | std::__throw_domain_error(__N("Bad argument " |
538 | "in __cyl_neumann_n." )); |
539 | else if (__isnan(__nu) || __isnan(__x)) |
540 | return std::numeric_limits<_Tp>::quiet_NaN(); |
541 | else if (__x > _Tp(1000)) |
542 | { |
543 | _Tp __J_nu, __N_nu; |
544 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); |
545 | return __N_nu; |
546 | } |
547 | else |
548 | { |
549 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
550 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
551 | return __N_nu; |
552 | } |
553 | } |
554 | |
555 | |
556 | /** |
557 | * @brief Compute the spherical Bessel @f$ j_n(x) @f$ |
558 | * and Neumann @f$ n_n(x) @f$ functions and their first |
559 | * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ |
560 | * respectively. |
561 | * |
562 | * @param __n The order of the spherical Bessel function. |
563 | * @param __x The argument of the spherical Bessel function. |
564 | * @param __j_n The output spherical Bessel function. |
565 | * @param __n_n The output spherical Neumann function. |
566 | * @param __jp_n The output derivative of the spherical Bessel function. |
567 | * @param __np_n The output derivative of the spherical Neumann function. |
568 | */ |
569 | template <typename _Tp> |
570 | void |
571 | __sph_bessel_jn(unsigned int __n, _Tp __x, |
572 | _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) |
573 | { |
574 | const _Tp __nu = _Tp(__n) + _Tp(0.5L); |
575 | |
576 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; |
577 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
578 | |
579 | const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() |
580 | / std::sqrt(__x); |
581 | |
582 | __j_n = __factor * __J_nu; |
583 | __n_n = __factor * __N_nu; |
584 | __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); |
585 | __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); |
586 | |
587 | return; |
588 | } |
589 | |
590 | |
591 | /** |
592 | * @brief Return the spherical Bessel function |
593 | * @f$ j_n(x) @f$ of order n. |
594 | * |
595 | * The spherical Bessel function is defined by: |
596 | * @f[ |
597 | * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) |
598 | * @f] |
599 | * |
600 | * @param __n The order of the spherical Bessel function. |
601 | * @param __x The argument of the spherical Bessel function. |
602 | * @return The output spherical Bessel function. |
603 | */ |
604 | template <typename _Tp> |
605 | _Tp |
606 | __sph_bessel(unsigned int __n, _Tp __x) |
607 | { |
608 | if (__x < _Tp(0)) |
609 | std::__throw_domain_error(__N("Bad argument " |
610 | "in __sph_bessel." )); |
611 | else if (__isnan(__x)) |
612 | return std::numeric_limits<_Tp>::quiet_NaN(); |
613 | else if (__x == _Tp(0)) |
614 | { |
615 | if (__n == 0) |
616 | return _Tp(1); |
617 | else |
618 | return _Tp(0); |
619 | } |
620 | else |
621 | { |
622 | _Tp __j_n, __n_n, __jp_n, __np_n; |
623 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); |
624 | return __j_n; |
625 | } |
626 | } |
627 | |
628 | |
629 | /** |
630 | * @brief Return the spherical Neumann function |
631 | * @f$ n_n(x) @f$. |
632 | * |
633 | * The spherical Neumann function is defined by: |
634 | * @f[ |
635 | * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) |
636 | * @f] |
637 | * |
638 | * @param __n The order of the spherical Neumann function. |
639 | * @param __x The argument of the spherical Neumann function. |
640 | * @return The output spherical Neumann function. |
641 | */ |
642 | template <typename _Tp> |
643 | _Tp |
644 | __sph_neumann(unsigned int __n, _Tp __x) |
645 | { |
646 | if (__x < _Tp(0)) |
647 | std::__throw_domain_error(__N("Bad argument " |
648 | "in __sph_neumann." )); |
649 | else if (__isnan(__x)) |
650 | return std::numeric_limits<_Tp>::quiet_NaN(); |
651 | else if (__x == _Tp(0)) |
652 | return -std::numeric_limits<_Tp>::infinity(); |
653 | else |
654 | { |
655 | _Tp __j_n, __n_n, __jp_n, __np_n; |
656 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); |
657 | return __n_n; |
658 | } |
659 | } |
660 | } // namespace __detail |
661 | #undef _GLIBCXX_MATH_NS |
662 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
663 | } // namespace tr1 |
664 | #endif |
665 | |
666 | _GLIBCXX_END_NAMESPACE_VERSION |
667 | } |
668 | |
669 | #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
670 | |