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1 | // Special functions -*- C++ -*- |
2 | ||
cbe34bb5 | 3 | // Copyright (C) 2006-2017 Free Software Foundation, Inc. |
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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 | |
748086b7 | 8 | // Free Software Foundation; either version 3, or (at your option) |
7c62b943 BK |
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 | // | |
748086b7 JJ |
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/>. | |
7c62b943 BK |
24 | |
25 | /** @file tr1/bessel_function.tcc | |
26 | * This is an internal header file, included by other library headers. | |
f910786b | 27 | * Do not attempt to use it directly. @headername{tr1/cmath} |
7c62b943 BK |
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. 240-245 | |
45 | ||
e133ace8 PC |
46 | #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |
47 | #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1 | |
7c62b943 BK |
48 | |
49 | #include "special_function_util.h" | |
50 | ||
12ffa228 | 51 | namespace std _GLIBCXX_VISIBILITY(default) |
7c62b943 | 52 | { |
f8571e51 | 53 | #if _GLIBCXX_USE_STD_SPEC_FUNCS |
2be75957 ESR |
54 | # define _GLIBCXX_MATH_NS ::std |
55 | #elif defined(_GLIBCXX_TR1_CMATH) | |
e133ace8 PC |
56 | namespace tr1 |
57 | { | |
2be75957 ESR |
58 | # define _GLIBCXX_MATH_NS ::std::tr1 |
59 | #else | |
60 | # error do not include this header directly, use <cmath> or <tr1/cmath> | |
61 | #endif | |
7c62b943 BK |
62 | // [5.2] Special functions |
63 | ||
7c62b943 | 64 | // Implementation-space details. |
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65 | namespace __detail |
66 | { | |
12ffa228 | 67 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
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68 | |
69 | /** | |
70 | * @brief Compute the gamma functions required by the Temme series | |
71 | * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$. | |
72 | * @f[ | |
73 | * \Gamma_1 = \frac{1}{2\mu} | |
74 | * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}] | |
75 | * @f] | |
76 | * and | |
77 | * @f[ | |
78 | * \Gamma_2 = \frac{1}{2} | |
79 | * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}] | |
80 | * @f] | |
81 | * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$. | |
82 | * is the nearest integer to @f$ \nu @f$. | |
83 | * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$ | |
84 | * are returned as well. | |
85 | * | |
86 | * The accuracy requirements on this are exquisite. | |
87 | * | |
88 | * @param __mu The input parameter of the gamma functions. | |
89 | * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$ | |
90 | * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$ | |
91 | * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$ | |
92 | * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$ | |
93 | */ | |
94 | template <typename _Tp> | |
95 | void | |
be59c932 ESR |
96 | __gamma_temme(_Tp __mu, |
97 | _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi) | |
7c62b943 BK |
98 | { |
99 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
2be75957 ESR |
100 | __gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu); |
101 | __gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu); | |
7c62b943 BK |
102 | #else |
103 | __gampl = _Tp(1) / __gamma(_Tp(1) + __mu); | |
104 | __gammi = _Tp(1) / __gamma(_Tp(1) - __mu); | |
105 | #endif | |
106 | ||
107 | if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon()) | |
108 | __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e()); | |
109 | else | |
110 | __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu); | |
111 | ||
112 | __gam2 = (__gammi + __gampl) / (_Tp(2)); | |
113 | ||
114 | return; | |
115 | } | |
116 | ||
117 | ||
118 | /** | |
119 | * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann | |
120 | * @f$ N_\nu(x) @f$ functions and their first derivatives | |
121 | * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively. | |
122 | * These four functions are computed together for numerical | |
123 | * stability. | |
124 | * | |
125 | * @param __nu The order of the Bessel functions. | |
126 | * @param __x The argument of the Bessel functions. | |
127 | * @param __Jnu The output Bessel function of the first kind. | |
28dac70a | 128 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
7c62b943 BK |
129 | * @param __Jpnu The output derivative of the Bessel function of the first kind. |
130 | * @param __Npnu The output derivative of the Neumann function. | |
131 | */ | |
132 | template <typename _Tp> | |
133 | void | |
be59c932 | 134 | __bessel_jn(_Tp __nu, _Tp __x, |
7c62b943 BK |
135 | _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu) |
136 | { | |
137 | if (__x == _Tp(0)) | |
138 | { | |
139 | if (__nu == _Tp(0)) | |
140 | { | |
141 | __Jnu = _Tp(1); | |
142 | __Jpnu = _Tp(0); | |
143 | } | |
144 | else if (__nu == _Tp(1)) | |
145 | { | |
146 | __Jnu = _Tp(0); | |
147 | __Jpnu = _Tp(0.5L); | |
148 | } | |
149 | else | |
150 | { | |
151 | __Jnu = _Tp(0); | |
152 | __Jpnu = _Tp(0); | |
153 | } | |
154 | __Nnu = -std::numeric_limits<_Tp>::infinity(); | |
155 | __Npnu = std::numeric_limits<_Tp>::infinity(); | |
156 | return; | |
157 | } | |
158 | ||
159 | const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); | |
160 | // When the multiplier is N i.e. | |
161 | // fp_min = N * min() | |
162 | // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)! | |
163 | //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min(); | |
164 | const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min()); | |
165 | const int __max_iter = 15000; | |
166 | const _Tp __x_min = _Tp(2); | |
167 | ||
168 | const int __nl = (__x < __x_min | |
169 | ? static_cast<int>(__nu + _Tp(0.5L)) | |
170 | : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L)))); | |
171 | ||
172 | const _Tp __mu = __nu - __nl; | |
173 | const _Tp __mu2 = __mu * __mu; | |
174 | const _Tp __xi = _Tp(1) / __x; | |
175 | const _Tp __xi2 = _Tp(2) * __xi; | |
176 | _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi(); | |
177 | int __isign = 1; | |
178 | _Tp __h = __nu * __xi; | |
179 | if (__h < __fp_min) | |
180 | __h = __fp_min; | |
181 | _Tp __b = __xi2 * __nu; | |
182 | _Tp __d = _Tp(0); | |
183 | _Tp __c = __h; | |
184 | int __i; | |
185 | for (__i = 1; __i <= __max_iter; ++__i) | |
186 | { | |
187 | __b += __xi2; | |
188 | __d = __b - __d; | |
189 | if (std::abs(__d) < __fp_min) | |
190 | __d = __fp_min; | |
191 | __c = __b - _Tp(1) / __c; | |
192 | if (std::abs(__c) < __fp_min) | |
193 | __c = __fp_min; | |
194 | __d = _Tp(1) / __d; | |
195 | const _Tp __del = __c * __d; | |
196 | __h *= __del; | |
197 | if (__d < _Tp(0)) | |
198 | __isign = -__isign; | |
199 | if (std::abs(__del - _Tp(1)) < __eps) | |
200 | break; | |
201 | } | |
202 | if (__i > __max_iter) | |
203 | std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; " | |
204 | "try asymptotic expansion.")); | |
205 | _Tp __Jnul = __isign * __fp_min; | |
206 | _Tp __Jpnul = __h * __Jnul; | |
207 | _Tp __Jnul1 = __Jnul; | |
208 | _Tp __Jpnu1 = __Jpnul; | |
209 | _Tp __fact = __nu * __xi; | |
210 | for ( int __l = __nl; __l >= 1; --__l ) | |
211 | { | |
212 | const _Tp __Jnutemp = __fact * __Jnul + __Jpnul; | |
213 | __fact -= __xi; | |
214 | __Jpnul = __fact * __Jnutemp - __Jnul; | |
215 | __Jnul = __Jnutemp; | |
216 | } | |
217 | if (__Jnul == _Tp(0)) | |
218 | __Jnul = __eps; | |
219 | _Tp __f= __Jpnul / __Jnul; | |
220 | _Tp __Nmu, __Nnu1, __Npmu, __Jmu; | |
221 | if (__x < __x_min) | |
222 | { | |
223 | const _Tp __x2 = __x / _Tp(2); | |
224 | const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; | |
225 | _Tp __fact = (std::abs(__pimu) < __eps | |
226 | ? _Tp(1) : __pimu / std::sin(__pimu)); | |
227 | _Tp __d = -std::log(__x2); | |
228 | _Tp __e = __mu * __d; | |
229 | _Tp __fact2 = (std::abs(__e) < __eps | |
230 | ? _Tp(1) : std::sinh(__e) / __e); | |
231 | _Tp __gam1, __gam2, __gampl, __gammi; | |
232 | __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); | |
233 | _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi()) | |
234 | * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); | |
235 | __e = std::exp(__e); | |
236 | _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl); | |
237 | _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi); | |
238 | const _Tp __pimu2 = __pimu / _Tp(2); | |
239 | _Tp __fact3 = (std::abs(__pimu2) < __eps | |
240 | ? _Tp(1) : std::sin(__pimu2) / __pimu2 ); | |
241 | _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3; | |
242 | _Tp __c = _Tp(1); | |
243 | __d = -__x2 * __x2; | |
244 | _Tp __sum = __ff + __r * __q; | |
245 | _Tp __sum1 = __p; | |
246 | for (__i = 1; __i <= __max_iter; ++__i) | |
247 | { | |
248 | __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); | |
249 | __c *= __d / _Tp(__i); | |
250 | __p /= _Tp(__i) - __mu; | |
251 | __q /= _Tp(__i) + __mu; | |
252 | const _Tp __del = __c * (__ff + __r * __q); | |
253 | __sum += __del; | |
254 | const _Tp __del1 = __c * __p - __i * __del; | |
255 | __sum1 += __del1; | |
256 | if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) ) | |
257 | break; | |
258 | } | |
259 | if ( __i > __max_iter ) | |
260 | std::__throw_runtime_error(__N("Bessel y series failed to converge " | |
261 | "in __bessel_jn.")); | |
262 | __Nmu = -__sum; | |
263 | __Nnu1 = -__sum1 * __xi2; | |
264 | __Npmu = __mu * __xi * __Nmu - __Nnu1; | |
265 | __Jmu = __w / (__Npmu - __f * __Nmu); | |
266 | } | |
267 | else | |
268 | { | |
269 | _Tp __a = _Tp(0.25L) - __mu2; | |
270 | _Tp __q = _Tp(1); | |
271 | _Tp __p = -__xi / _Tp(2); | |
272 | _Tp __br = _Tp(2) * __x; | |
273 | _Tp __bi = _Tp(2); | |
274 | _Tp __fact = __a * __xi / (__p * __p + __q * __q); | |
275 | _Tp __cr = __br + __q * __fact; | |
276 | _Tp __ci = __bi + __p * __fact; | |
277 | _Tp __den = __br * __br + __bi * __bi; | |
278 | _Tp __dr = __br / __den; | |
279 | _Tp __di = -__bi / __den; | |
280 | _Tp __dlr = __cr * __dr - __ci * __di; | |
281 | _Tp __dli = __cr * __di + __ci * __dr; | |
282 | _Tp __temp = __p * __dlr - __q * __dli; | |
283 | __q = __p * __dli + __q * __dlr; | |
284 | __p = __temp; | |
285 | int __i; | |
286 | for (__i = 2; __i <= __max_iter; ++__i) | |
287 | { | |
288 | __a += _Tp(2 * (__i - 1)); | |
289 | __bi += _Tp(2); | |
290 | __dr = __a * __dr + __br; | |
291 | __di = __a * __di + __bi; | |
292 | if (std::abs(__dr) + std::abs(__di) < __fp_min) | |
293 | __dr = __fp_min; | |
294 | __fact = __a / (__cr * __cr + __ci * __ci); | |
295 | __cr = __br + __cr * __fact; | |
296 | __ci = __bi - __ci * __fact; | |
297 | if (std::abs(__cr) + std::abs(__ci) < __fp_min) | |
298 | __cr = __fp_min; | |
299 | __den = __dr * __dr + __di * __di; | |
300 | __dr /= __den; | |
301 | __di /= -__den; | |
302 | __dlr = __cr * __dr - __ci * __di; | |
303 | __dli = __cr * __di + __ci * __dr; | |
304 | __temp = __p * __dlr - __q * __dli; | |
305 | __q = __p * __dli + __q * __dlr; | |
306 | __p = __temp; | |
307 | if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps) | |
308 | break; | |
309 | } | |
310 | if (__i > __max_iter) | |
311 | std::__throw_runtime_error(__N("Lentz's method failed " | |
312 | "in __bessel_jn.")); | |
313 | const _Tp __gam = (__p - __f) / __q; | |
314 | __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q)); | |
315 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
2be75957 | 316 | __Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul); |
7c62b943 BK |
317 | #else |
318 | if (__Jmu * __Jnul < _Tp(0)) | |
319 | __Jmu = -__Jmu; | |
320 | #endif | |
321 | __Nmu = __gam * __Jmu; | |
322 | __Npmu = (__p + __q / __gam) * __Nmu; | |
323 | __Nnu1 = __mu * __xi * __Nmu - __Npmu; | |
324 | } | |
325 | __fact = __Jmu / __Jnul; | |
326 | __Jnu = __fact * __Jnul1; | |
327 | __Jpnu = __fact * __Jpnu1; | |
328 | for (__i = 1; __i <= __nl; ++__i) | |
329 | { | |
330 | const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu; | |
331 | __Nmu = __Nnu1; | |
332 | __Nnu1 = __Nnutemp; | |
333 | } | |
334 | __Nnu = __Nmu; | |
335 | __Npnu = __nu * __xi * __Nmu - __Nnu1; | |
336 | ||
337 | return; | |
338 | } | |
339 | ||
340 | ||
341 | /** | |
28dac70a | 342 | * @brief This routine computes the asymptotic cylindrical Bessel |
7c62b943 BK |
343 | * and Neumann functions of order nu: \f$ J_{\nu} \f$, |
344 | * \f$ N_{\nu} \f$. | |
345 | * | |
346 | * References: | |
347 | * (1) Handbook of Mathematical Functions, | |
348 | * ed. Milton Abramowitz and Irene A. Stegun, | |
349 | * Dover Publications, | |
350 | * Section 9 p. 364, Equations 9.2.5-9.2.10 | |
351 | * | |
352 | * @param __nu The order of the Bessel functions. | |
353 | * @param __x The argument of the Bessel functions. | |
354 | * @param __Jnu The output Bessel function of the first kind. | |
28dac70a | 355 | * @param __Nnu The output Neumann function (Bessel function of the second kind). |
7c62b943 BK |
356 | */ |
357 | template <typename _Tp> | |
358 | void | |
be59c932 | 359 | __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu) |
7c62b943 | 360 | { |
7c62b943 BK |
361 | const _Tp __mu = _Tp(4) * __nu * __nu; |
362 | const _Tp __mum1 = __mu - _Tp(1); | |
363 | const _Tp __mum9 = __mu - _Tp(9); | |
364 | const _Tp __mum25 = __mu - _Tp(25); | |
365 | const _Tp __mum49 = __mu - _Tp(49); | |
366 | const _Tp __xx = _Tp(64) * __x * __x; | |
367 | const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx) | |
368 | * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx)); | |
369 | const _Tp __Q = __mum1 / (_Tp(8) * __x) | |
370 | * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx)); | |
371 | ||
372 | const _Tp __chi = __x - (__nu + _Tp(0.5L)) | |
373 | * __numeric_constants<_Tp>::__pi_2(); | |
374 | const _Tp __c = std::cos(__chi); | |
375 | const _Tp __s = std::sin(__chi); | |
376 | ||
be59c932 ESR |
377 | const _Tp __coef = std::sqrt(_Tp(2) |
378 | / (__numeric_constants<_Tp>::__pi() * __x)); | |
7c62b943 BK |
379 | __Jnu = __coef * (__c * __P - __s * __Q); |
380 | __Nnu = __coef * (__s * __P + __c * __Q); | |
381 | ||
382 | return; | |
383 | } | |
384 | ||
385 | ||
386 | /** | |
387 | * @brief This routine returns the cylindrical Bessel functions | |
388 | * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$ | |
389 | * by series expansion. | |
390 | * | |
391 | * The modified cylindrical Bessel function is: | |
392 | * @f[ | |
393 | * Z_{\nu}(x) = \sum_{k=0}^{\infty} | |
394 | * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
395 | * @f] | |
396 | * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for | |
28dac70a | 397 | * \f$ Z = I \f$ or \f$ J \f$ respectively. |
7c62b943 BK |
398 | * |
399 | * See Abramowitz & Stegun, 9.1.10 | |
400 | * Abramowitz & Stegun, 9.6.7 | |
401 | * (1) Handbook of Mathematical Functions, | |
402 | * ed. Milton Abramowitz and Irene A. Stegun, | |
403 | * Dover Publications, | |
404 | * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375 | |
405 | * | |
406 | * @param __nu The order of the Bessel function. | |
407 | * @param __x The argument of the Bessel function. | |
408 | * @param __sgn The sign of the alternate terms | |
409 | * -1 for the Bessel function of the first kind. | |
410 | * +1 for the modified Bessel function of the first kind. | |
411 | * @return The output Bessel function. | |
412 | */ | |
413 | template <typename _Tp> | |
414 | _Tp | |
be59c932 ESR |
415 | __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn, |
416 | unsigned int __max_iter) | |
7c62b943 | 417 | { |
be59c932 | 418 | if (__x == _Tp(0)) |
0112ed60 FD |
419 | return __nu == _Tp(0) ? _Tp(1) : _Tp(0); |
420 | ||
7c62b943 BK |
421 | const _Tp __x2 = __x / _Tp(2); |
422 | _Tp __fact = __nu * std::log(__x2); | |
423 | #if _GLIBCXX_USE_C99_MATH_TR1 | |
2be75957 | 424 | __fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1)); |
7c62b943 BK |
425 | #else |
426 | __fact -= __log_gamma(__nu + _Tp(1)); | |
427 | #endif | |
428 | __fact = std::exp(__fact); | |
429 | const _Tp __xx4 = __sgn * __x2 * __x2; | |
430 | _Tp __Jn = _Tp(1); | |
431 | _Tp __term = _Tp(1); | |
432 | ||
433 | for (unsigned int __i = 1; __i < __max_iter; ++__i) | |
434 | { | |
435 | __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i))); | |
436 | __Jn += __term; | |
437 | if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon()) | |
438 | break; | |
439 | } | |
440 | ||
441 | return __fact * __Jn; | |
442 | } | |
443 | ||
444 | ||
445 | /** | |
446 | * @brief Return the Bessel function of order \f$ \nu \f$: | |
447 | * \f$ J_{\nu}(x) \f$. | |
448 | * | |
449 | * The cylindrical Bessel function is: | |
450 | * @f[ | |
451 | * J_{\nu}(x) = \sum_{k=0}^{\infty} | |
452 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
453 | * @f] | |
454 | * | |
455 | * @param __nu The order of the Bessel function. | |
456 | * @param __x The argument of the Bessel function. | |
457 | * @return The output Bessel function. | |
458 | */ | |
459 | template<typename _Tp> | |
460 | _Tp | |
be59c932 | 461 | __cyl_bessel_j(_Tp __nu, _Tp __x) |
7c62b943 BK |
462 | { |
463 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
464 | std::__throw_domain_error(__N("Bad argument " | |
465 | "in __cyl_bessel_j.")); | |
466 | else if (__isnan(__nu) || __isnan(__x)) | |
467 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
468 | else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) | |
469 | return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200); | |
470 | else if (__x > _Tp(1000)) | |
471 | { | |
472 | _Tp __J_nu, __N_nu; | |
473 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
474 | return __J_nu; | |
475 | } | |
476 | else | |
477 | { | |
478 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
479 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
480 | return __J_nu; | |
481 | } | |
482 | } | |
483 | ||
484 | ||
485 | /** | |
28dac70a | 486 | * @brief Return the Neumann function of order \f$ \nu \f$: |
7c62b943 BK |
487 | * \f$ N_{\nu}(x) \f$. |
488 | * | |
489 | * The Neumann function is defined by: | |
490 | * @f[ | |
491 | * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} | |
492 | * {\sin \nu\pi} | |
493 | * @f] | |
494 | * where for integral \f$ \nu = n \f$ a limit is taken: | |
495 | * \f$ lim_{\nu \to n} \f$. | |
496 | * | |
497 | * @param __nu The order of the Neumann function. | |
498 | * @param __x The argument of the Neumann function. | |
499 | * @return The output Neumann function. | |
500 | */ | |
501 | template<typename _Tp> | |
502 | _Tp | |
be59c932 | 503 | __cyl_neumann_n(_Tp __nu, _Tp __x) |
7c62b943 BK |
504 | { |
505 | if (__nu < _Tp(0) || __x < _Tp(0)) | |
506 | std::__throw_domain_error(__N("Bad argument " | |
507 | "in __cyl_neumann_n.")); | |
508 | else if (__isnan(__nu) || __isnan(__x)) | |
509 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
510 | else if (__x > _Tp(1000)) | |
511 | { | |
512 | _Tp __J_nu, __N_nu; | |
513 | __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu); | |
514 | return __N_nu; | |
515 | } | |
516 | else | |
517 | { | |
518 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
519 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
520 | return __N_nu; | |
521 | } | |
522 | } | |
523 | ||
524 | ||
525 | /** | |
526 | * @brief Compute the spherical Bessel @f$ j_n(x) @f$ | |
527 | * and Neumann @f$ n_n(x) @f$ functions and their first | |
528 | * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$ | |
529 | * respectively. | |
530 | * | |
531 | * @param __n The order of the spherical Bessel function. | |
532 | * @param __x The argument of the spherical Bessel function. | |
533 | * @param __j_n The output spherical Bessel function. | |
534 | * @param __n_n The output spherical Neumann function. | |
12ffa228 BK |
535 | * @param __jp_n The output derivative of the spherical Bessel function. |
536 | * @param __np_n The output derivative of the spherical Neumann function. | |
7c62b943 BK |
537 | */ |
538 | template <typename _Tp> | |
539 | void | |
be59c932 | 540 | __sph_bessel_jn(unsigned int __n, _Tp __x, |
7c62b943 BK |
541 | _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n) |
542 | { | |
543 | const _Tp __nu = _Tp(__n) + _Tp(0.5L); | |
544 | ||
545 | _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu; | |
546 | __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu); | |
547 | ||
548 | const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() | |
549 | / std::sqrt(__x); | |
550 | ||
551 | __j_n = __factor * __J_nu; | |
552 | __n_n = __factor * __N_nu; | |
553 | __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x); | |
554 | __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x); | |
555 | ||
556 | return; | |
557 | } | |
558 | ||
559 | ||
560 | /** | |
561 | * @brief Return the spherical Bessel function | |
562 | * @f$ j_n(x) @f$ of order n. | |
563 | * | |
564 | * The spherical Bessel function is defined by: | |
565 | * @f[ | |
566 | * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) | |
567 | * @f] | |
568 | * | |
569 | * @param __n The order of the spherical Bessel function. | |
570 | * @param __x The argument of the spherical Bessel function. | |
571 | * @return The output spherical Bessel function. | |
572 | */ | |
573 | template <typename _Tp> | |
574 | _Tp | |
be59c932 | 575 | __sph_bessel(unsigned int __n, _Tp __x) |
7c62b943 BK |
576 | { |
577 | if (__x < _Tp(0)) | |
578 | std::__throw_domain_error(__N("Bad argument " | |
579 | "in __sph_bessel.")); | |
580 | else if (__isnan(__x)) | |
581 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
582 | else if (__x == _Tp(0)) | |
583 | { | |
584 | if (__n == 0) | |
585 | return _Tp(1); | |
586 | else | |
587 | return _Tp(0); | |
588 | } | |
589 | else | |
590 | { | |
591 | _Tp __j_n, __n_n, __jp_n, __np_n; | |
592 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
593 | return __j_n; | |
594 | } | |
595 | } | |
596 | ||
597 | ||
598 | /** | |
599 | * @brief Return the spherical Neumann function | |
600 | * @f$ n_n(x) @f$. | |
601 | * | |
602 | * The spherical Neumann function is defined by: | |
603 | * @f[ | |
604 | * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) | |
605 | * @f] | |
606 | * | |
607 | * @param __n The order of the spherical Neumann function. | |
608 | * @param __x The argument of the spherical Neumann function. | |
609 | * @return The output spherical Neumann function. | |
610 | */ | |
611 | template <typename _Tp> | |
612 | _Tp | |
be59c932 | 613 | __sph_neumann(unsigned int __n, _Tp __x) |
7c62b943 BK |
614 | { |
615 | if (__x < _Tp(0)) | |
616 | std::__throw_domain_error(__N("Bad argument " | |
617 | "in __sph_neumann.")); | |
618 | else if (__isnan(__x)) | |
619 | return std::numeric_limits<_Tp>::quiet_NaN(); | |
620 | else if (__x == _Tp(0)) | |
621 | return -std::numeric_limits<_Tp>::infinity(); | |
622 | else | |
623 | { | |
624 | _Tp __j_n, __n_n, __jp_n, __np_n; | |
625 | __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n); | |
626 | return __n_n; | |
627 | } | |
628 | } | |
629 | ||
12ffa228 | 630 | _GLIBCXX_END_NAMESPACE_VERSION |
2be75957 ESR |
631 | } // namespace __detail |
632 | #undef _GLIBCXX_MATH_NS | |
f8571e51 | 633 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
2be75957 ESR |
634 | } // namespace tr1 |
635 | #endif | |
7c62b943 BK |
636 | } |
637 | ||
e133ace8 | 638 | #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC |