libstdc++-v3/include/tr1/bessel_function.tcc

   1 // Special functions -*- C++ -*-
   2
   3 // Copyright (C) 2006, 2007, 2008, 2009
   4 // Free Software Foundation, Inc.
   5 //
   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 3, or (at your option)
  10 // any later version.
  11 //
  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.
  16 //
  17 // Under Section 7 of GPL version 3, you are granted additional
  18 // permissions described in the GCC Runtime Library Exception, version
  19 // 3.1, as published by the Free Software Foundation.
  20
  21 // You should have received a copy of the GNU General Public License and
  22 // a copy of the GCC Runtime Library Exception along with this program;
  23 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
  24 // <http://www.gnu.org/licenses/>.
  25
  26 /** @file tr1/bessel_function.tcc
  27  *  This is an internal header file, included by other library headers.
  28  *  Do not attempt to use it directly. @headername{tr1/cmath}
  29  */
  30
  31 //
  32 // ISO C++ 14882 TR1: 5.2  Special functions
  33 //
  34
  35 // Written by Edward Smith-Rowland.
  36 //
  37 // References:
  38 //   (1) Handbook of Mathematical Functions,
  39 //       ed. Milton Abramowitz and Irene A. Stegun,
  40 //       Dover Publications,
  41 //       Section 9, pp. 355-434, Section 10 pp. 435-478
  42 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
  43 //   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
  44 //       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
  45 //       2nd ed, pp. 240-245
  46
  47 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
  48 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
  49
  50 #include "special_function_util.h"
  51
  52 namespace std
  53 {
  54 namespace tr1
  55 {
  56
  57   // [5.2] Special functions
  58
  59   // Implementation-space details.
  60   namespace __detail
  61   {
  62
  63     /**
  64      *   @brief Compute the gamma functions required by the Temme series
  65      *          expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
  66      *   @f[
  67      *     \Gamma_1 = \frac{1}{2\mu}
  68      *                [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
  69      *   @f]
  70      *   and
  71      *   @f[
  72      *     \Gamma_2 = \frac{1}{2}
  73      *                [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
  74      *   @f]
  75      *   where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
  76      *   is the nearest integer to @f$ \nu @f$.
  77      *   The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
  78      *   are returned as well.
  79      *
  80      *   The accuracy requirements on this are exquisite.
  81      *
  82      *   @param __mu     The input parameter of the gamma functions.
  83      *   @param __gam1   The output function \f$ \Gamma_1(\mu) \f$
  84      *   @param __gam2   The output function \f$ \Gamma_2(\mu) \f$
  85      *   @param __gampl  The output function \f$ \Gamma(1 + \mu) \f$
  86      *   @param __gammi  The output function \f$ \Gamma(1 - \mu) \f$
  87      */
  88     template <typename _Tp>
  89     void
  90     __gamma_temme(const _Tp __mu,
  91                    _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
  92     {
  93 #if _GLIBCXX_USE_C99_MATH_TR1
  94       __gampl = _Tp(1) / std::tr1::tgamma(_Tp(1) + __mu);
  95       __gammi = _Tp(1) / std::tr1::tgamma(_Tp(1) - __mu);
  96 #else
  97       __gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
  98       __gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
  99 #endif
 100
 101       if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
 102         __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
 103       else
 104         __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
 105
 106       __gam2 = (__gammi + __gampl) / (_Tp(2));
 107
 108       return;
 109     }
 110
 111
 112     /**
 113      *   @brief  Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
 114      *           @f$ N_\nu(x) @f$ functions and their first derivatives
 115      *           @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively.
 116      *           These four functions are computed together for numerical
 117      *           stability.
 118      *
 119      *   @param  __nu  The order of the Bessel functions.
 120      *   @param  __x   The argument of the Bessel functions.
 121      *   @param  __Jnu  The output Bessel function of the first kind.
 122      *   @param  __Nnu  The output Neumann function (Bessel function of the second kind).
 123      *   @param  __Jpnu  The output derivative of the Bessel function of the first kind.
 124      *   @param  __Npnu  The output derivative of the Neumann function.
 125      */
 126     template <typename _Tp>
 127     void
 128     __bessel_jn(const _Tp __nu, const _Tp __x,
 129                 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
 130     {
 131       if (__x == _Tp(0))
 132         {
 133           if (__nu == _Tp(0))
 134             {
 135               __Jnu = _Tp(1);
 136               __Jpnu = _Tp(0);
 137             }
 138           else if (__nu == _Tp(1))
 139             {
 140               __Jnu = _Tp(0);
 141               __Jpnu = _Tp(0.5L);
 142             }
 143           else
 144             {
 145               __Jnu = _Tp(0);
 146               __Jpnu = _Tp(0);
 147             }
 148           __Nnu = -std::numeric_limits<_Tp>::infinity();
 149           __Npnu = std::numeric_limits<_Tp>::infinity();
 150           return;
 151         }
 152
 153       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
 154       //  When the multiplier is N i.e.
 155       //  fp_min = N * min()
 156       //  Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
 157       //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
 158       const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
 159       const int __max_iter = 15000;
 160       const _Tp __x_min = _Tp(2);
 161
 162       const int __nl = (__x < __x_min
 163                     ? static_cast<int>(__nu + _Tp(0.5L))
 164                     : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
 165
 166       const _Tp __mu = __nu - __nl;
 167       const _Tp __mu2 = __mu * __mu;
 168       const _Tp __xi = _Tp(1) / __x;
 169       const _Tp __xi2 = _Tp(2) * __xi;
 170       _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi();
 171       int __isign = 1;
 172       _Tp __h = __nu * __xi;
 173       if (__h < __fp_min)
 174         __h = __fp_min;
 175       _Tp __b = __xi2 * __nu;
 176       _Tp __d = _Tp(0);
 177       _Tp __c = __h;
 178       int __i;
 179       for (__i = 1; __i <= __max_iter; ++__i)
 180         {
 181           __b += __xi2;
 182           __d = __b - __d;
 183           if (std::abs(__d) < __fp_min)
 184             __d = __fp_min;
 185           __c = __b - _Tp(1) / __c;
 186           if (std::abs(__c) < __fp_min)
 187             __c = __fp_min;
 188           __d = _Tp(1) / __d;
 189           const _Tp __del = __c * __d;
 190           __h *= __del;
 191           if (__d < _Tp(0))
 192             __isign = -__isign;
 193           if (std::abs(__del - _Tp(1)) < __eps)
 194             break;
 195         }
 196       if (__i > __max_iter)
 197         std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; "
 198                                        "try asymptotic expansion."));
 199       _Tp __Jnul = __isign * __fp_min;
 200       _Tp __Jpnul = __h * __Jnul;
 201       _Tp __Jnul1 = __Jnul;
 202       _Tp __Jpnu1 = __Jpnul;
 203       _Tp __fact = __nu * __xi;
 204       for ( int __l = __nl; __l >= 1; --__l )
 205         {
 206           const _Tp __Jnutemp = __fact * __Jnul + __Jpnul;
 207           __fact -= __xi;
 208           __Jpnul = __fact * __Jnutemp - __Jnul;
 209           __Jnul = __Jnutemp;
 210         }
 211       if (__Jnul == _Tp(0))
 212         __Jnul = __eps;
 213       _Tp __f= __Jpnul / __Jnul;
 214       _Tp __Nmu, __Nnu1, __Npmu, __Jmu;
 215       if (__x < __x_min)
 216         {
 217           const _Tp __x2 = __x / _Tp(2);
 218           const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
 219           _Tp __fact = (std::abs(__pimu) < __eps
 220                       ? _Tp(1) : __pimu / std::sin(__pimu));
 221           _Tp __d = -std::log(__x2);
 222           _Tp __e = __mu * __d;
 223           _Tp __fact2 = (std::abs(__e) < __eps
 224                        ? _Tp(1) : std::sinh(__e) / __e);
 225           _Tp __gam1, __gam2, __gampl, __gammi;
 226           __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
 227           _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi())
 228                    * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
 229           __e = std::exp(__e);
 230           _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
 231           _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
 232           const _Tp __pimu2 = __pimu / _Tp(2);
 233           _Tp __fact3 = (std::abs(__pimu2) < __eps
 234                        ? _Tp(1) : std::sin(__pimu2) / __pimu2 );
 235           _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
 236           _Tp __c = _Tp(1);
 237           __d = -__x2 * __x2;
 238           _Tp __sum = __ff + __r * __q;
 239           _Tp __sum1 = __p;
 240           for (__i = 1; __i <= __max_iter; ++__i)
 241             {
 242               __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
 243               __c *= __d / _Tp(__i);
 244               __p /= _Tp(__i) - __mu;
 245               __q /= _Tp(__i) + __mu;
 246               const _Tp __del = __c * (__ff + __r * __q);
 247               __sum += __del;
 248               const _Tp __del1 = __c * __p - __i * __del;
 249               __sum1 += __del1;
 250               if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) )
 251                 break;
 252             }
 253           if ( __i > __max_iter )
 254             std::__throw_runtime_error(__N("Bessel y series failed to converge "
 255                                            "in __bessel_jn."));
 256           __Nmu = -__sum;
 257           __Nnu1 = -__sum1 * __xi2;
 258           __Npmu = __mu * __xi * __Nmu - __Nnu1;
 259           __Jmu = __w / (__Npmu - __f * __Nmu);
 260         }
 261       else
 262         {
 263           _Tp __a = _Tp(0.25L) - __mu2;
 264           _Tp __q = _Tp(1);
 265           _Tp __p = -__xi / _Tp(2);
 266           _Tp __br = _Tp(2) * __x;
 267           _Tp __bi = _Tp(2);
 268           _Tp __fact = __a * __xi / (__p * __p + __q * __q);
 269           _Tp __cr = __br + __q * __fact;
 270           _Tp __ci = __bi + __p * __fact;
 271           _Tp __den = __br * __br + __bi * __bi;
 272           _Tp __dr = __br / __den;
 273           _Tp __di = -__bi / __den;
 274           _Tp __dlr = __cr * __dr - __ci * __di;
 275           _Tp __dli = __cr * __di + __ci * __dr;
 276           _Tp __temp = __p * __dlr - __q * __dli;
 277           __q = __p * __dli + __q * __dlr;
 278           __p = __temp;
 279           int __i;
 280           for (__i = 2; __i <= __max_iter; ++__i)
 281             {
 282               __a += _Tp(2 * (__i - 1));
 283               __bi += _Tp(2);
 284               __dr = __a * __dr + __br;
 285               __di = __a * __di + __bi;
 286               if (std::abs(__dr) + std::abs(__di) < __fp_min)
 287                 __dr = __fp_min;
 288               __fact = __a / (__cr * __cr + __ci * __ci);
 289               __cr = __br + __cr * __fact;
 290               __ci = __bi - __ci * __fact;
 291               if (std::abs(__cr) + std::abs(__ci) < __fp_min)
 292                 __cr = __fp_min;
 293               __den = __dr * __dr + __di * __di;
 294               __dr /= __den;
 295               __di /= -__den;
 296               __dlr = __cr * __dr - __ci * __di;
 297               __dli = __cr * __di + __ci * __dr;
 298               __temp = __p * __dlr - __q * __dli;
 299               __q = __p * __dli + __q * __dlr;
 300               __p = __temp;
 301               if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps)
 302                 break;
 303           }
 304           if (__i > __max_iter)
 305             std::__throw_runtime_error(__N("Lentz's method failed "
 306                                            "in __bessel_jn."));
 307           const _Tp __gam = (__p - __f) / __q;
 308           __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
 309 #if _GLIBCXX_USE_C99_MATH_TR1
 310           __Jmu = std::tr1::copysign(__Jmu, __Jnul);
 311 #else
 312           if (__Jmu * __Jnul < _Tp(0))
 313             __Jmu = -__Jmu;
 314 #endif
 315           __Nmu = __gam * __Jmu;
 316           __Npmu = (__p + __q / __gam) * __Nmu;
 317           __Nnu1 = __mu * __xi * __Nmu - __Npmu;
 318       }
 319       __fact = __Jmu / __Jnul;
 320       __Jnu = __fact * __Jnul1;
 321       __Jpnu = __fact * __Jpnu1;
 322       for (__i = 1; __i <= __nl; ++__i)
 323         {
 324           const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
 325           __Nmu = __Nnu1;
 326           __Nnu1 = __Nnutemp;
 327         }
 328       __Nnu = __Nmu;
 329       __Npnu = __nu * __xi * __Nmu - __Nnu1;
 330
 331       return;
 332     }
 333
 334
 335     /**
 336      *   @brief This routine computes the asymptotic cylindrical Bessel
 337      *          and Neumann functions of order nu: \f$ J_{\nu} \f$,
 338      *          \f$ N_{\nu} \f$.
 339      *
 340      *   References:
 341      *    (1) Handbook of Mathematical Functions,
 342      *        ed. Milton Abramowitz and Irene A. Stegun,
 343      *        Dover Publications,
 344      *        Section 9 p. 364, Equations 9.2.5-9.2.10
 345      *
 346      *   @param  __nu  The order of the Bessel functions.
 347      *   @param  __x   The argument of the Bessel functions.
 348      *   @param  __Jnu  The output Bessel function of the first kind.
 349      *   @param  __Nnu  The output Neumann function (Bessel function of the second kind).
 350      */
 351     template <typename _Tp>
 352     void
 353     __cyl_bessel_jn_asymp(const _Tp __nu, const _Tp __x,
 354                           _Tp & __Jnu, _Tp & __Nnu)
 355     {
 356       const _Tp __coef = std::sqrt(_Tp(2)
 357                              / (__numeric_constants<_Tp>::__pi() * __x));
 358       const _Tp __mu   = _Tp(4) * __nu * __nu;
 359       const _Tp __mum1 = __mu - _Tp(1);
 360       const _Tp __mum9 = __mu - _Tp(9);
 361       const _Tp __mum25 = __mu - _Tp(25);
 362       const _Tp __mum49 = __mu - _Tp(49);
 363       const _Tp __xx = _Tp(64) * __x * __x;
 364       const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
 365                     * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
 366       const _Tp __Q = __mum1 / (_Tp(8) * __x)
 367                     * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
 368
 369       const _Tp __chi = __x - (__nu + _Tp(0.5L))
 370                             * __numeric_constants<_Tp>::__pi_2();
 371       const _Tp __c = std::cos(__chi);
 372       const _Tp __s = std::sin(__chi);
 373
 374       __Jnu = __coef * (__c * __P - __s * __Q);
 375       __Nnu = __coef * (__s * __P + __c * __Q);
 376
 377       return;
 378     }
 379
 380
 381     /**
 382      *   @brief This routine returns the cylindrical Bessel functions
 383      *          of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
 384      *          by series expansion.
 385      *
 386      *   The modified cylindrical Bessel function is:
 387      *   @f[
 388      *    Z_{\nu}(x) = \sum_{k=0}^{\infty}
 389      *              \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
 390      *   @f]
 391      *   where \f$ \sigma = +1 \f$ or\f$  -1 \f$ for
 392      *   \f$ Z = I \f$ or \f$ J \f$ respectively.
 393      *
 394      *   See Abramowitz & Stegun, 9.1.10
 395      *       Abramowitz & Stegun, 9.6.7
 396      *    (1) Handbook of Mathematical Functions,
 397      *        ed. Milton Abramowitz and Irene A. Stegun,
 398      *        Dover Publications,
 399      *        Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375
 400      *
 401      *   @param  __nu  The order of the Bessel function.
 402      *   @param  __x   The argument of the Bessel function.
 403      *   @param  __sgn  The sign of the alternate terms
 404      *                  -1 for the Bessel function of the first kind.
 405      *                  +1 for the modified Bessel function of the first kind.
 406      *   @return  The output Bessel function.
 407      */
 408     template <typename _Tp>
 409     _Tp
 410     __cyl_bessel_ij_series(const _Tp __nu, const _Tp __x, const _Tp __sgn,
 411                            const unsigned int __max_iter)
 412     {
 413
 414       const _Tp __x2 = __x / _Tp(2);
 415       _Tp __fact = __nu * std::log(__x2);
 416 #if _GLIBCXX_USE_C99_MATH_TR1
 417       __fact -= std::tr1::lgamma(__nu + _Tp(1));
 418 #else
 419       __fact -= __log_gamma(__nu + _Tp(1));
 420 #endif
 421       __fact = std::exp(__fact);
 422       const _Tp __xx4 = __sgn * __x2 * __x2;
 423       _Tp __Jn = _Tp(1);
 424       _Tp __term = _Tp(1);
 425
 426       for (unsigned int __i = 1; __i < __max_iter; ++__i)
 427         {
 428           __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i)));
 429           __Jn += __term;
 430           if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon())
 431             break;
 432         }
 433
 434       return __fact * __Jn;
 435     }
 436
 437
 438     /**
 439      *   @brief  Return the Bessel function of order \f$ \nu \f$:
 440      *           \f$ J_{\nu}(x) \f$.
 441      *
 442      *   The cylindrical Bessel function is:
 443      *   @f[
 444      *    J_{\nu}(x) = \sum_{k=0}^{\infty}
 445      *              \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
 446      *   @f]
 447      *
 448      *   @param  __nu  The order of the Bessel function.
 449      *   @param  __x   The argument of the Bessel function.
 450      *   @return  The output Bessel function.
 451      */
 452     template<typename _Tp>
 453     _Tp
 454     __cyl_bessel_j(const _Tp __nu, const _Tp __x)
 455     {
 456       if (__nu < _Tp(0) || __x < _Tp(0))
 457         std::__throw_domain_error(__N("Bad argument "
 458                                       "in __cyl_bessel_j."));
 459       else if (__isnan(__nu) || __isnan(__x))
 460         return std::numeric_limits<_Tp>::quiet_NaN();
 461       else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
 462         return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
 463       else if (__x > _Tp(1000))
 464         {
 465           _Tp __J_nu, __N_nu;
 466           __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
 467           return __J_nu;
 468         }
 469       else
 470         {
 471           _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
 472           __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
 473           return __J_nu;
 474         }
 475     }
 476
 477
 478     /**
 479      *   @brief  Return the Neumann function of order \f$ \nu \f$:
 480      *           \f$ N_{\nu}(x) \f$.
 481      *
 482      *   The Neumann function is defined by:
 483      *   @f[
 484      *      N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
 485      *                        {\sin \nu\pi}
 486      *   @f]
 487      *   where for integral \f$ \nu = n \f$ a limit is taken:
 488      *   \f$ lim_{\nu \to n} \f$.
 489      *
 490      *   @param  __nu  The order of the Neumann function.
 491      *   @param  __x   The argument of the Neumann function.
 492      *   @return  The output Neumann function.
 493      */
 494     template<typename _Tp>
 495     _Tp
 496     __cyl_neumann_n(const _Tp __nu, const _Tp __x)
 497     {
 498       if (__nu < _Tp(0) || __x < _Tp(0))
 499         std::__throw_domain_error(__N("Bad argument "
 500                                       "in __cyl_neumann_n."));
 501       else if (__isnan(__nu) || __isnan(__x))
 502         return std::numeric_limits<_Tp>::quiet_NaN();
 503       else if (__x > _Tp(1000))
 504         {
 505           _Tp __J_nu, __N_nu;
 506           __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
 507           return __N_nu;
 508         }
 509       else
 510         {
 511           _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
 512           __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
 513           return __N_nu;
 514         }
 515     }
 516
 517
 518     /**
 519      *   @brief  Compute the spherical Bessel @f$ j_n(x) @f$
 520      *           and Neumann @f$ n_n(x) @f$ functions and their first
 521      *           derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$
 522      *           respectively.
 523      *
 524      *   @param  __n  The order of the spherical Bessel function.
 525      *   @param  __x  The argument of the spherical Bessel function.
 526      *   @param  __j_n  The output spherical Bessel function.
 527      *   @param  __n_n  The output spherical Neumann function.
 528      *   @param  __jp_n  The output derivative of the spherical Bessel function.
 529      *   @param  __np_n  The output derivative of the spherical Neumann function.
 530      */
 531     template <typename _Tp>
 532     void
 533     __sph_bessel_jn(const unsigned int __n, const _Tp __x,
 534                     _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n)
 535     {
 536       const _Tp __nu = _Tp(__n) + _Tp(0.5L);
 537
 538       _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
 539       __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
 540
 541       const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
 542                          / std::sqrt(__x);
 543
 544       __j_n = __factor * __J_nu;
 545       __n_n = __factor * __N_nu;
 546       __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x);
 547       __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x);
 548
 549       return;
 550     }
 551
 552
 553     /**
 554      *   @brief  Return the spherical Bessel function
 555      *           @f$ j_n(x) @f$ of order n.
 556      *
 557      *   The spherical Bessel function is defined by:
 558      *   @f[
 559      *    j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
 560      *   @f]
 561      *
 562      *   @param  __n  The order of the spherical Bessel function.
 563      *   @param  __x  The argument of the spherical Bessel function.
 564      *   @return  The output spherical Bessel function.
 565      */
 566     template <typename _Tp>
 567     _Tp
 568     __sph_bessel(const unsigned int __n, const _Tp __x)
 569     {
 570       if (__x < _Tp(0))
 571         std::__throw_domain_error(__N("Bad argument "
 572                                       "in __sph_bessel."));
 573       else if (__isnan(__x))
 574         return std::numeric_limits<_Tp>::quiet_NaN();
 575       else if (__x == _Tp(0))
 576         {
 577           if (__n == 0)
 578             return _Tp(1);
 579           else
 580             return _Tp(0);
 581         }
 582       else
 583         {
 584           _Tp __j_n, __n_n, __jp_n, __np_n;
 585           __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
 586           return __j_n;
 587         }
 588     }
 589
 590
 591     /**
 592      *   @brief  Return the spherical Neumann function
 593      *           @f$ n_n(x) @f$.
 594      *
 595      *   The spherical Neumann function is defined by:
 596      *   @f[
 597      *    n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
 598      *   @f]
 599      *
 600      *   @param  __n  The order of the spherical Neumann function.
 601      *   @param  __x  The argument of the spherical Neumann function.
 602      *   @return  The output spherical Neumann function.
 603      */
 604     template <typename _Tp>
 605     _Tp
 606     __sph_neumann(const unsigned int __n, const _Tp __x)
 607     {
 608       if (__x < _Tp(0))
 609         std::__throw_domain_error(__N("Bad argument "
 610                                       "in __sph_neumann."));
 611       else if (__isnan(__x))
 612         return std::numeric_limits<_Tp>::quiet_NaN();
 613       else if (__x == _Tp(0))
 614         return -std::numeric_limits<_Tp>::infinity();
 615       else
 616         {
 617           _Tp __j_n, __n_n, __jp_n, __np_n;
 618           __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
 619           return __n_n;
 620         }
 621     }
 622
 623   } // namespace std::tr1::__detail
 624 }
 625 }
 626
 627 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC