1// Special functions -*- C++ -*- 
2 
3// Copyright (C) 2006-2019 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 
55namespace 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) 
62namespace 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(0, 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