1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006, 2007, 2008
4 // Free Software Foundation, Inc.
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
9 // Free Software Foundation; either version 2, or (at your option)
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.
17 // You should have received a copy of the GNU General Public License along
18 // with this library; see the file COPYING. If not, write to the Free
19 // Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
22 // As a special exception, you may use this file as part of a free software
23 // library without restriction. Specifically, if other files instantiate
24 // templates or use macros or inline functions from this file, or you compile
25 // this file and link it with other files to produce an executable, this
26 // file does not by itself cause the resulting executable to be covered by
27 // the GNU General Public License. This exception does not however
28 // invalidate any other reasons why the executable file might be covered by
29 // the GNU General Public License.
31 /** @file tr1/bessel_function.tcc
32 * This is an internal header file, included by other library headers.
33 * You should not attempt to use it directly.
37 // ISO C++ 14882 TR1: 5.2 Special functions
40 // Written by Edward Smith-Rowland.
43 // (1) Handbook of Mathematical Functions,
44 // ed. Milton Abramowitz and Irene A. Stegun,
45 // Dover Publications,
46 // Section 9, pp. 355-434, Section 10 pp. 435-478
47 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
48 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
49 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
50 // 2nd ed, pp. 240-245
52 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
53 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
55 #include "special_function_util.h"
62 // [5.2] Special functions
64 // Implementation-space details.
69 * @brief Compute the gamma functions required by the Temme series
70 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
72 * \Gamma_1 = \frac{1}{2\mu}
73 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
77 * \Gamma_2 = \frac{1}{2}
78 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
80 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
81 * is the nearest integer to @f$ \nu @f$.
82 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
83 * are returned as well.
85 * The accuracy requirements on this are exquisite.
87 * @param __mu The input parameter of the gamma functions.
88 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$
89 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$
90 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$
91 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$
93 template <typename _Tp>
95 __gamma_temme(const _Tp __mu,
96 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
98 #if _GLIBCXX_USE_C99_MATH_TR1
99 __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu);
100 __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu);
102 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
103 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
106 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
107 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
109 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
111 __gam2 = (__gammi + __gampl) / (_Tp(2));
118 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
119 * @f$ N_\nu(x) @f$ functions and their first derivatives
120 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively.
121 * These four functions are computed together for numerical
124 * @param __nu The order of the Bessel functions.
125 * @param __x The argument of the Bessel functions.
126 * @param __Jnu The output Bessel function of the first kind.
127 * @param __Nnu The output Neumann function (Bessel function of the second kind).
128 * @param __Jpnu The output derivative of the Bessel function of the first kind.
129 * @param __Npnu The output derivative of the Neumann function.
131 template <typename _Tp>
133 __bessel_jn(const _Tp __nu, const _Tp __x,
134 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
143 else if (__nu == _Tp(1))
153 __Nnu = -std::numeric_limits<_Tp>::infinity();
154 __Npnu = std::numeric_limits<_Tp>::infinity();
158 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
159 // When the multiplier is N i.e.
160 // fp_min = N * min()
161 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
162 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
163 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
164 const int __max_iter = 15000;
165 const _Tp __x_min = _Tp(2);
167 const int __nl = (__x < __x_min
168 ? static_cast<int>(__nu + _Tp(0.5L))
169 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
171 const _Tp __mu = __nu - __nl;
172 const _Tp __mu2 = __mu * __mu;
173 const _Tp __xi = _Tp(1) / __x;
174 const _Tp __xi2 = _Tp(2) * __xi;
175 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi();
177 _Tp __h = __nu * __xi;
180 _Tp __b = __xi2 * __nu;
184 for (__i = 1; __i <= __max_iter; ++__i)
188 if (std::abs(__d) < __fp_min)
190 __c = __b - _Tp(1) / __c;
191 if (std::abs(__c) < __fp_min)
194 const _Tp __del = __c * __d;
198 if (std::abs(__del - _Tp(1)) < __eps)
201 if (__i > __max_iter)
202 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; "
203 "try asymptotic expansion."));
204 _Tp __Jnul = __isign * __fp_min;
205 _Tp __Jpnul = __h * __Jnul;
206 _Tp __Jnul1 = __Jnul;
207 _Tp __Jpnu1 = __Jpnul;
208 _Tp __fact = __nu * __xi;
209 for ( int __l = __nl; __l >= 1; --__l )
211 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul;
213 __Jpnul = __fact * __Jnutemp - __Jnul;
216 if (__Jnul == _Tp(0))
218 _Tp __f= __Jpnul / __Jnul;
219 _Tp __Nmu, __Nnu1, __Npmu, __Jmu;
222 const _Tp __x2 = __x / _Tp(2);
223 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
224 _Tp __fact = (std::abs(__pimu) < __eps
225 ? _Tp(1) : __pimu / std::sin(__pimu));
226 _Tp __d = -std::log(__x2);
227 _Tp __e = __mu * __d;
228 _Tp __fact2 = (std::abs(__e) < __eps
229 ? _Tp(1) : std::sinh(__e) / __e);
230 _Tp __gam1, __gam2, __gampl, __gammi;
231 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
232 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi())
233 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
235 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
236 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
237 const _Tp __pimu2 = __pimu / _Tp(2);
238 _Tp __fact3 = (std::abs(__pimu2) < __eps
239 ? _Tp(1) : std::sin(__pimu2) / __pimu2 );
240 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
243 _Tp __sum = __ff + __r * __q;
245 for (__i = 1; __i <= __max_iter; ++__i)
247 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
248 __c *= __d / _Tp(__i);
249 __p /= _Tp(__i) - __mu;
250 __q /= _Tp(__i) + __mu;
251 const _Tp __del = __c * (__ff + __r * __q);
253 const _Tp __del1 = __c * __p - __i * __del;
255 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) )
258 if ( __i > __max_iter )
259 std::__throw_runtime_error(__N("Bessel y series failed to converge "
262 __Nnu1 = -__sum1 * __xi2;
263 __Npmu = __mu * __xi * __Nmu - __Nnu1;
264 __Jmu = __w / (__Npmu - __f * __Nmu);
268 _Tp __a = _Tp(0.25L) - __mu2;
270 _Tp __p = -__xi / _Tp(2);
271 _Tp __br = _Tp(2) * __x;
273 _Tp __fact = __a * __xi / (__p * __p + __q * __q);
274 _Tp __cr = __br + __q * __fact;
275 _Tp __ci = __bi + __p * __fact;
276 _Tp __den = __br * __br + __bi * __bi;
277 _Tp __dr = __br / __den;
278 _Tp __di = -__bi / __den;
279 _Tp __dlr = __cr * __dr - __ci * __di;
280 _Tp __dli = __cr * __di + __ci * __dr;
281 _Tp __temp = __p * __dlr - __q * __dli;
282 __q = __p * __dli + __q * __dlr;
285 for (__i = 2; __i <= __max_iter; ++__i)
287 __a += _Tp(2 * (__i - 1));
289 __dr = __a * __dr + __br;
290 __di = __a * __di + __bi;
291 if (std::abs(__dr) + std::abs(__di) < __fp_min)
293 __fact = __a / (__cr * __cr + __ci * __ci);
294 __cr = __br + __cr * __fact;
295 __ci = __bi - __ci * __fact;
296 if (std::abs(__cr) + std::abs(__ci) < __fp_min)
298 __den = __dr * __dr + __di * __di;
301 __dlr = __cr * __dr - __ci * __di;
302 __dli = __cr * __di + __ci * __dr;
303 __temp = __p * __dlr - __q * __dli;
304 __q = __p * __dli + __q * __dlr;
306 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps)
309 if (__i > __max_iter)
310 std::__throw_runtime_error(__N("Lentz's method failed "
312 const _Tp __gam = (__p - __f) / __q;
313 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
314 #if _GLIBCXX_USE_C99_MATH_TR1
315 __Jmu = std::tr1::copysign(__Jmu, __Jnul);
317 if (__Jmu * __Jnul < _Tp(0))
320 __Nmu = __gam * __Jmu;
321 __Npmu = (__p + __q / __gam) * __Nmu;
322 __Nnu1 = __mu * __xi * __Nmu - __Npmu;
324 __fact = __Jmu / __Jnul;
325 __Jnu = __fact * __Jnul1;
326 __Jpnu = __fact * __Jpnu1;
327 for (__i = 1; __i <= __nl; ++__i)
329 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
334 __Npnu = __nu * __xi * __Nmu - __Nnu1;
341 * @brief This routine computes the asymptotic cylindrical Bessel
342 * and Neumann functions of order nu: \f$ J_{\nu} \f$,
346 * (1) Handbook of Mathematical Functions,
347 * ed. Milton Abramowitz and Irene A. Stegun,
348 * Dover Publications,
349 * Section 9 p. 364, Equations 9.2.5-9.2.10
351 * @param __nu The order of the Bessel functions.
352 * @param __x The argument of the Bessel functions.
353 * @param __Jnu The output Bessel function of the first kind.
354 * @param __Nnu The output Neumann function (Bessel function of the second kind).
356 template <typename _Tp>
358 __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x,
359 _Tp & __Jnu, _Tp & __Nnu)
361 const _Tp __coef = std::sqrt(_Tp(2)
362 / (__numeric_constants<_Tp>::__pi() * __x));
363 const _Tp __mu = _Tp(4) * __nu * __nu;
364 const _Tp __mum1 = __mu - _Tp(1);
365 const _Tp __mum9 = __mu - _Tp(9);
366 const _Tp __mum25 = __mu - _Tp(25);
367 const _Tp __mum49 = __mu - _Tp(49);
368 const _Tp __xx = _Tp(64) * __x * __x;
369 const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
370 * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
371 const _Tp __Q = __mum1 / (_Tp(8) * __x)
372 * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
374 const _Tp __chi = __x - (__nu + _Tp(0.5L))
375 * __numeric_constants<_Tp>::__pi_2();
376 const _Tp __c = std::cos(__chi);
377 const _Tp __s = std::sin(__chi);
379 __Jnu = __coef * (__c * __P - __s * __Q);
380 __Nnu = __coef * (__s * __P + __c * __Q);
387 * @brief This routine returns the cylindrical Bessel functions
388 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
389 * by series expansion.
391 * The modified cylindrical Bessel function is:
393 * Z_{\nu}(x) = \sum_{k=0}^{\infty}
394 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
396 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for
397 * \f$ Z = I \f$ or \f$ J \f$ respectively.
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
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.
413 template <typename _Tp>
415 __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn,
416 const unsigned int __max_iter)
419 const _Tp __x2 = __x / _Tp(2);
420 _Tp __fact = __nu * std::log(__x2);
421 #if _GLIBCXX_USE_C99_MATH_TR1
422 __fact -= std::tr1::lgamma(__nu + _Tp(1));
424 __fact -= __log_gamma(__nu + _Tp(1));
426 __fact = std::exp(__fact);
427 const _Tp __xx4 = __sgn * __x2 * __x2;
431 for (unsigned int __i = 1; __i < __max_iter; ++__i)
433 __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i)));
435 if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon())
439 return __fact * __Jn;
444 * @brief Return the Bessel function of order \f$ \nu \f$:
445 * \f$ J_{\nu}(x) \f$.
447 * The cylindrical Bessel function is:
449 * J_{\nu}(x) = \sum_{k=0}^{\infty}
450 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
453 * @param __nu The order of the Bessel function.
454 * @param __x The argument of the Bessel function.
455 * @return The output Bessel function.
457 template<typename _Tp>
459 __cyl_bessel_j(const _Tp __nu, const _Tp __x)
461 if (__nu < _Tp(0) || __x < _Tp(0))
462 std::__throw_domain_error(__N("Bad argument "
463 "in __cyl_bessel_j."));
464 else if (__isnan(__nu) || __isnan(__x))
465 return std::numeric_limits<_Tp>::quiet_NaN();
466 else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
467 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
468 else if (__x > _Tp(1000))
471 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
476 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
477 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
484 * @brief Return the Neumann function of order \f$ \nu \f$:
485 * \f$ N_{\nu}(x) \f$.
487 * The Neumann function is defined by:
489 * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
492 * where for integral \f$ \nu = n \f$ a limit is taken:
493 * \f$ lim_{\nu \to n} \f$.
495 * @param __nu The order of the Neumann function.
496 * @param __x The argument of the Neumann function.
497 * @return The output Neumann function.
499 template<typename _Tp>
501 __cyl_neumann_n(const _Tp __nu, const _Tp __x)
503 if (__nu < _Tp(0) || __x < _Tp(0))
504 std::__throw_domain_error(__N("Bad argument "
505 "in __cyl_neumann_n."));
506 else if (__isnan(__nu) || __isnan(__x))
507 return std::numeric_limits<_Tp>::quiet_NaN();
508 else if (__x > _Tp(1000))
511 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
516 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
517 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
524 * @brief Compute the spherical Bessel @f$ j_n(x) @f$
525 * and Neumann @f$ n_n(x) @f$ functions and their first
526 * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$
529 * @param __n The order of the spherical Bessel function.
530 * @param __x The argument of the spherical Bessel function.
531 * @param __j_n The output spherical Bessel function.
532 * @param __n_n The output spherical Neumann function.
533 * @param __jp_n The output derivative of the spherical Bessel function.
534 * @param __np_n The output derivative of the spherical Neumann function.
536 template <typename _Tp>
538 __sph_bessel_jn(const unsigned int __n, const _Tp __x,
539 _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n)
541 const _Tp __nu = _Tp(__n) + _Tp(0.5L);
543 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
544 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
546 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
549 __j_n = __factor * __J_nu;
550 __n_n = __factor * __N_nu;
551 __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x);
552 __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x);
559 * @brief Return the spherical Bessel function
560 * @f$ j_n(x) @f$ of order n.
562 * The spherical Bessel function is defined by:
564 * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
567 * @param __n The order of the spherical Bessel function.
568 * @param __x The argument of the spherical Bessel function.
569 * @return The output spherical Bessel function.
571 template <typename _Tp>
573 __sph_bessel(const unsigned int __n, const _Tp __x)
576 std::__throw_domain_error(__N("Bad argument "
577 "in __sph_bessel."));
578 else if (__isnan(__x))
579 return std::numeric_limits<_Tp>::quiet_NaN();
580 else if (__x == _Tp(0))
589 _Tp __j_n, __n_n, __jp_n, __np_n;
590 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
597 * @brief Return the spherical Neumann function
600 * The spherical Neumann function is defined by:
602 * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
605 * @param __n The order of the spherical Neumann function.
606 * @param __x The argument of the spherical Neumann function.
607 * @return The output spherical Neumann function.
609 template <typename _Tp>
611 __sph_neumann(const unsigned int __n, const _Tp __x)
614 std::__throw_domain_error(__N("Bad argument "
615 "in __sph_neumann."));
616 else if (__isnan(__x))
617 return std::numeric_limits<_Tp>::quiet_NaN();
618 else if (__x == _Tp(0))
619 return -std::numeric_limits<_Tp>::infinity();
622 _Tp __j_n, __n_n, __jp_n, __np_n;
623 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
628 } // namespace std::tr1::__detail
632 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC