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1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006-2023 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.
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/>.
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 */
30 /* __cyl_bessel_jn_asymp adapted from GNU GSL version 2.4 specfunc/bessel_j.c
31 * Copyright (C) 1996-2003 Gerard Jungman
32 */
34 //
35 // ISO C++ 14882 TR1: 5.2 Special functions
36 //
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
50 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
51 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
53 #include <tr1/special_function_util.h>
55 namespace std _GLIBCXX_VISIBILITY(default)
56 {
57 _GLIBCXX_BEGIN_NAMESPACE_VERSION
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
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
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);
116 __gam2 = (__gammi + __gampl) / (_Tp(2));
118 return;
119 }
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 }
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);
172 const int __nl = (__x < __x_min
173 ? static_cast<int>(__nu + _Tp(0.5L))
174 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
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;
341 return;
342 }
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$.
349 *
350 * References:
351 * (1) Handbook of Mathematical Functions,
352 * ed. Milton Abramowitz and Irene A. Stegun,
353 * Dover Publications,
354 * Section 9 p. 364, Equations 9.2.5-9.2.10
355 *
356 * @param __nu The order of the Bessel functions.
357 * @param __x The argument of the Bessel functions.
358 * @param __Jnu The output Bessel function of the first kind.
359 * @param __Nnu The output Neumann function (Bessel function of the second kind).
360 */
361 template <typename _Tp>
362 void
363 __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu)
364 {
365 const _Tp __mu = _Tp(4) * __nu * __nu;
366 const _Tp __8x = _Tp(8) * __x;
368 _Tp __P = _Tp(0);
369 _Tp __Q = _Tp(0);
371 _Tp __k = _Tp(0);
372 _Tp __term = _Tp(1);
374 int __epsP = 0;
375 int __epsQ = 0;
377 _Tp __eps = std::numeric_limits<_Tp>::epsilon();
379 do
380 {
381 __term *= (__k == 0
382 ? _Tp(1)
383 : -(__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x));
385 __epsP = std::abs(__term) < __eps * std::abs(__P);
386 __P += __term;
388 __k++;
390 __term *= (__mu - (2 * __k - 1) * (2 * __k - 1)) / (__k * __8x);
391 __epsQ = std::abs(__term) < __eps * std::abs(__Q);
392 __Q += __term;
394 if (__epsP && __epsQ && __k > (__nu / 2.))
395 break;
397 __k++;
398 }
399 while (__k < 1000);
401 const _Tp __chi = __x - (__nu + _Tp(0.5L))
402 * __numeric_constants<_Tp>::__pi_2();
404 const _Tp __c = std::cos(__chi);
405 const _Tp __s = std::sin(__chi);
407 const _Tp __coef = std::sqrt(_Tp(2)
408 / (__numeric_constants<_Tp>::__pi() * __x));
410 __Jnu = __coef * (__c * __P - __s * __Q);
411 __Nnu = __coef * (__s * __P + __c * __Q);
413 return;
414 }
417 /**
418 * @brief This routine returns the cylindrical Bessel functions
419 * of order \f$ \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);
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);
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 }
472 return __fact * __Jn;
473 }
476 /**
477 * @brief Return the Bessel function of order \f$ \nu \f$:
478 * \f$ J_{\nu}(x) \f$.
479 *
480 * The cylindrical Bessel function is:
481 * @f[
482 * J_{\nu}(x) = \sum_{k=0}^{\infty}
483 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
484 * @f]
485 *
486 * @param __nu The order of the Bessel function.
487 * @param __x The argument of the Bessel function.
488 * @return The output Bessel function.
489 */
490 template<typename _Tp>
491 _Tp
492 __cyl_bessel_j(_Tp __nu, _Tp __x)
493 {
494 if (__nu < _Tp(0) || __x < _Tp(0))
495 std::__throw_domain_error(__N("Bad argument "
496 "in __cyl_bessel_j."));
497 else if (__isnan(__nu) || __isnan(__x))
498 return std::numeric_limits<_Tp>::quiet_NaN();
499 else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
500 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
501 else if (__x > _Tp(1000))
502 {
503 _Tp __J_nu, __N_nu;
504 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
505 return __J_nu;
506 }
507 else
508 {
509 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
510 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
511 return __J_nu;
512 }
513 }
516 /**
517 * @brief Return the Neumann function of order \f$ \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$.
527 *
528 * @param __nu The order of the Neumann function.
529 * @param __x The argument of the Neumann function.
530 * @return The output Neumann function.
531 */
532 template<typename _Tp>
533 _Tp
534 __cyl_neumann_n(_Tp __nu, _Tp __x)
535 {
536 if (__nu < _Tp(0) || __x < _Tp(0))
537 std::__throw_domain_error(__N("Bad argument "
538 "in __cyl_neumann_n."));
539 else if (__isnan(__nu) || __isnan(__x))
540 return std::numeric_limits<_Tp>::quiet_NaN();
541 else if (__x > _Tp(1000))
542 {
543 _Tp __J_nu, __N_nu;
544 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
545 return __N_nu;
546 }
547 else
548 {
549 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
550 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
551 return __N_nu;
552 }
553 }
556 /**
557 * @brief Compute the spherical Bessel @f$ 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);
576 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
577 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
579 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
580 / std::sqrt(__x);
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);
587 return;
588 }
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 }
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
666 _GLIBCXX_END_NAMESPACE_VERSION
667 }
669 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC