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

   1 // Special functions -*- C++ -*-
   2
   3 // Copyright (C) 2006, 2007, 2008, 2009, 2010, 2011
   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/modified_bessel_func.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. 246-249.
  46
  47 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
  48 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
  49
  50 #include "special_function_util.h"
  51
  52 namespace std _GLIBCXX_VISIBILITY(default)
  53 {
  54 namespace tr1
  55 {
  56   // [5.2] Special functions
  57
  58   // Implementation-space details.
  59   namespace __detail
  60   {
  61   _GLIBCXX_BEGIN_NAMESPACE_VERSION
  62
  63     /**
  64      *   @brief  Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
  65      *           @f$ K_\nu(x) @f$ and their first derivatives
  66      *           @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
  67      *           These four functions are computed together for numerical
  68      *           stability.
  69      *
  70      *   @param  __nu  The order of the Bessel functions.
  71      *   @param  __x   The argument of the Bessel functions.
  72      *   @param  __Inu  The output regular modified Bessel function.
  73      *   @param  __Knu  The output irregular modified Bessel function.
  74      *   @param  __Ipnu  The output derivative of the regular
  75      *                   modified Bessel function.
  76      *   @param  __Kpnu  The output derivative of the irregular
  77      *                   modified Bessel function.
  78      */
  79     template <typename _Tp>
  80     void
  81     __bessel_ik(const _Tp __nu, const _Tp __x,
  82                 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
  83     {
  84       if (__x == _Tp(0))
  85         {
  86           if (__nu == _Tp(0))
  87             {
  88               __Inu = _Tp(1);
  89               __Ipnu = _Tp(0);
  90             }
  91           else if (__nu == _Tp(1))
  92             {
  93               __Inu = _Tp(0);
  94               __Ipnu = _Tp(0.5L);
  95             }
  96           else
  97             {
  98               __Inu = _Tp(0);
  99               __Ipnu = _Tp(0);
 100             }
 101           __Knu = std::numeric_limits<_Tp>::infinity();
 102           __Kpnu = -std::numeric_limits<_Tp>::infinity();
 103           return;
 104         }
 105
 106       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
 107       const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
 108       const int __max_iter = 15000;
 109       const _Tp __x_min = _Tp(2);
 110
 111       const int __nl = static_cast<int>(__nu + _Tp(0.5L));
 112
 113       const _Tp __mu = __nu - __nl;
 114       const _Tp __mu2 = __mu * __mu;
 115       const _Tp __xi = _Tp(1) / __x;
 116       const _Tp __xi2 = _Tp(2) * __xi;
 117       _Tp __h = __nu * __xi;
 118       if ( __h < __fp_min )
 119         __h = __fp_min;
 120       _Tp __b = __xi2 * __nu;
 121       _Tp __d = _Tp(0);
 122       _Tp __c = __h;
 123       int __i;
 124       for ( __i = 1; __i <= __max_iter; ++__i )
 125         {
 126           __b += __xi2;
 127           __d = _Tp(1) / (__b + __d);
 128           __c = __b + _Tp(1) / __c;
 129           const _Tp __del = __c * __d;
 130           __h *= __del;
 131           if (std::abs(__del - _Tp(1)) < __eps)
 132             break;
 133         }
 134       if (__i > __max_iter)
 135         std::__throw_runtime_error(__N("Argument x too large "
 136                                        "in __bessel_jn; "
 137                                        "try asymptotic expansion."));
 138       _Tp __Inul = __fp_min;
 139       _Tp __Ipnul = __h * __Inul;
 140       _Tp __Inul1 = __Inul;
 141       _Tp __Ipnu1 = __Ipnul;
 142       _Tp __fact = __nu * __xi;
 143       for (int __l = __nl; __l >= 1; --__l)
 144         {
 145           const _Tp __Inutemp = __fact * __Inul + __Ipnul;
 146           __fact -= __xi;
 147           __Ipnul = __fact * __Inutemp + __Inul;
 148           __Inul = __Inutemp;
 149         }
 150       _Tp __f = __Ipnul / __Inul;
 151       _Tp __Kmu, __Knu1;
 152       if (__x < __x_min)
 153         {
 154           const _Tp __x2 = __x / _Tp(2);
 155           const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
 156           const _Tp __fact = (std::abs(__pimu) < __eps
 157                             ? _Tp(1) : __pimu / std::sin(__pimu));
 158           _Tp __d = -std::log(__x2);
 159           _Tp __e = __mu * __d;
 160           const _Tp __fact2 = (std::abs(__e) < __eps
 161                             ? _Tp(1) : std::sinh(__e) / __e);
 162           _Tp __gam1, __gam2, __gampl, __gammi;
 163           __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
 164           _Tp __ff = __fact
 165                    * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
 166           _Tp __sum = __ff;
 167           __e = std::exp(__e);
 168           _Tp __p = __e / (_Tp(2) * __gampl);
 169           _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
 170           _Tp __c = _Tp(1);
 171           __d = __x2 * __x2;
 172           _Tp __sum1 = __p;
 173           int __i;
 174           for (__i = 1; __i <= __max_iter; ++__i)
 175             {
 176               __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
 177               __c *= __d / __i;
 178               __p /= __i - __mu;
 179               __q /= __i + __mu;
 180               const _Tp __del = __c * __ff;
 181               __sum += __del;
 182               const _Tp __del1 = __c * (__p - __i * __ff);
 183               __sum1 += __del1;
 184               if (std::abs(__del) < __eps * std::abs(__sum))
 185                 break;
 186             }
 187           if (__i > __max_iter)
 188             std::__throw_runtime_error(__N("Bessel k series failed to converge "
 189                                            "in __bessel_jn."));
 190           __Kmu = __sum;
 191           __Knu1 = __sum1 * __xi2;
 192         }
 193       else
 194         {
 195           _Tp __b = _Tp(2) * (_Tp(1) + __x);
 196           _Tp __d = _Tp(1) / __b;
 197           _Tp __delh = __d;
 198           _Tp __h = __delh;
 199           _Tp __q1 = _Tp(0);
 200           _Tp __q2 = _Tp(1);
 201           _Tp __a1 = _Tp(0.25L) - __mu2;
 202           _Tp __q = __c = __a1;
 203           _Tp __a = -__a1;
 204           _Tp __s = _Tp(1) + __q * __delh;
 205           int __i;
 206           for (__i = 2; __i <= __max_iter; ++__i)
 207             {
 208               __a -= 2 * (__i - 1);
 209               __c = -__a * __c / __i;
 210               const _Tp __qnew = (__q1 - __b * __q2) / __a;
 211               __q1 = __q2;
 212               __q2 = __qnew;
 213               __q += __c * __qnew;
 214               __b += _Tp(2);
 215               __d = _Tp(1) / (__b + __a * __d);
 216               __delh = (__b * __d - _Tp(1)) * __delh;
 217               __h += __delh;
 218               const _Tp __dels = __q * __delh;
 219               __s += __dels;
 220               if ( std::abs(__dels / __s) < __eps )
 221                 break;
 222             }
 223           if (__i > __max_iter)
 224             std::__throw_runtime_error(__N("Steed's method failed "
 225                                            "in __bessel_jn."));
 226           __h = __a1 * __h;
 227           __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
 228                 * std::exp(-__x) / __s;
 229           __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
 230         }
 231
 232       _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
 233       _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
 234       __Inu = __Inumu * __Inul1 / __Inul;
 235       __Ipnu = __Inumu * __Ipnu1 / __Inul;
 236       for ( __i = 1; __i <= __nl; ++__i )
 237         {
 238           const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
 239           __Kmu = __Knu1;
 240           __Knu1 = __Knutemp;
 241         }
 242       __Knu = __Kmu;
 243       __Kpnu = __nu * __xi * __Kmu - __Knu1;
 244
 245       return;
 246     }
 247
 248
 249     /**
 250      *   @brief  Return the regular modified Bessel function of order
 251      *           \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
 252      *
 253      *   The regular modified cylindrical Bessel function is:
 254      *   @f[
 255      *    I_{\nu}(x) = \sum_{k=0}^{\infty}
 256      *              \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
 257      *   @f]
 258      *
 259      *   @param  __nu  The order of the regular modified Bessel function.
 260      *   @param  __x   The argument of the regular modified Bessel function.
 261      *   @return  The output regular modified Bessel function.
 262      */
 263     template<typename _Tp>
 264     _Tp
 265     __cyl_bessel_i(const _Tp __nu, const _Tp __x)
 266     {
 267       if (__nu < _Tp(0) || __x < _Tp(0))
 268         std::__throw_domain_error(__N("Bad argument "
 269                                       "in __cyl_bessel_i."));
 270       else if (__isnan(__nu) || __isnan(__x))
 271         return std::numeric_limits<_Tp>::quiet_NaN();
 272       else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
 273         return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
 274       else
 275         {
 276           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
 277           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
 278           return __I_nu;
 279         }
 280     }
 281
 282
 283     /**
 284      *   @brief  Return the irregular modified Bessel function
 285      *           \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
 286      *
 287      *   The irregular modified Bessel function is defined by:
 288      *   @f[
 289      *      K_{\nu}(x) = \frac{\pi}{2}
 290      *                   \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
 291      *   @f]
 292      *   where for integral \f$ \nu = n \f$ a limit is taken:
 293      *   \f$ lim_{\nu \to n} \f$.
 294      *
 295      *   @param  __nu  The order of the irregular modified Bessel function.
 296      *   @param  __x   The argument of the irregular modified Bessel function.
 297      *   @return  The output irregular modified Bessel function.
 298      */
 299     template<typename _Tp>
 300     _Tp
 301     __cyl_bessel_k(const _Tp __nu, const _Tp __x)
 302     {
 303       if (__nu < _Tp(0) || __x < _Tp(0))
 304         std::__throw_domain_error(__N("Bad argument "
 305                                       "in __cyl_bessel_k."));
 306       else if (__isnan(__nu) || __isnan(__x))
 307         return std::numeric_limits<_Tp>::quiet_NaN();
 308       else
 309         {
 310           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
 311           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
 312           return __K_nu;
 313         }
 314     }
 315
 316
 317     /**
 318      *   @brief  Compute the spherical modified Bessel functions
 319      *           @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
 320      *           derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
 321      *           respectively.
 322      *
 323      *   @param  __n  The order of the modified spherical Bessel function.
 324      *   @param  __x  The argument of the modified spherical Bessel function.
 325      *   @param  __i_n  The output regular modified spherical Bessel function.
 326      *   @param  __k_n  The output irregular modified spherical
 327      *                  Bessel function.
 328      *   @param  __ip_n  The output derivative of the regular modified
 329      *                   spherical Bessel function.
 330      *   @param  __kp_n  The output derivative of the irregular modified
 331      *                   spherical Bessel function.
 332      */
 333     template <typename _Tp>
 334     void
 335     __sph_bessel_ik(const unsigned int __n, const _Tp __x,
 336                     _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
 337     {
 338       const _Tp __nu = _Tp(__n) + _Tp(0.5L);
 339
 340       _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
 341       __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
 342
 343       const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
 344                          / std::sqrt(__x);
 345
 346       __i_n = __factor * __I_nu;
 347       __k_n = __factor * __K_nu;
 348       __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
 349       __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
 350
 351       return;
 352     }
 353
 354
 355     /**
 356      *   @brief  Compute the Airy functions
 357      *           @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
 358      *           derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
 359      *           respectively.
 360      *
 361      *   @param  __n  The order of the Airy functions.
 362      *   @param  __x  The argument of the Airy functions.
 363      *   @param  __i_n  The output Airy function.
 364      *   @param  __k_n  The output Airy function.
 365      *   @param  __ip_n  The output derivative of the Airy function.
 366      *   @param  __kp_n  The output derivative of the Airy function.
 367      */
 368     template <typename _Tp>
 369     void
 370     __airy(const _Tp __x,
 371            _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
 372     {
 373       const _Tp __absx = std::abs(__x);
 374       const _Tp __rootx = std::sqrt(__absx);
 375       const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
 376
 377       if (__isnan(__x))
 378         return std::numeric_limits<_Tp>::quiet_NaN();
 379       else if (__x > _Tp(0))
 380         {
 381           _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
 382
 383           __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
 384           __Ai = __rootx * __K_nu
 385                / (__numeric_constants<_Tp>::__sqrt3()
 386                 * __numeric_constants<_Tp>::__pi());
 387           __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
 388                  + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
 389
 390           __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
 391           __Aip = -__x * __K_nu
 392                 / (__numeric_constants<_Tp>::__sqrt3()
 393                  * __numeric_constants<_Tp>::__pi());
 394           __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
 395                       + _Tp(2) * __I_nu
 396                       / __numeric_constants<_Tp>::__sqrt3());
 397         }
 398       else if (__x < _Tp(0))
 399         {
 400           _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
 401
 402           __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
 403           __Ai = __rootx * (__J_nu
 404                     - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
 405           __Bi = -__rootx * (__N_nu
 406                     + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
 407
 408           __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
 409           __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
 410                           + __J_nu) / _Tp(2);
 411           __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
 412                           - __N_nu) / _Tp(2);
 413         }
 414       else
 415         {
 416           //  Reference:
 417           //    Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
 418           //  The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
 419           __Ai = _Tp(0.35502805388781723926L);
 420           __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
 421
 422           //  Reference:
 423           //    Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
 424           //  The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
 425           __Aip = -_Tp(0.25881940379280679840L);
 426           __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
 427         }
 428
 429       return;
 430     }
 431
 432   _GLIBCXX_END_NAMESPACE_VERSION
 433   } // namespace std::tr1::__detail
 434 }
 435 }
 436
 437 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC