modified_bessel_func.tcc

Go to the documentation of this file.
00001 // Special functions -*- C++ -*-
00002 
00003 // Copyright (C) 2006, 2007, 2008, 2009
00004 // Free Software Foundation, Inc.
00005 //
00006 // This file is part of the GNU ISO C++ Library.  This library is free
00007 // software; you can redistribute it and/or modify it under the
00008 // terms of the GNU General Public License as published by the
00009 // Free Software Foundation; either version 3, or (at your option)
00010 // any later version.
00011 //
00012 // This library is distributed in the hope that it will be useful,
00013 // but WITHOUT ANY WARRANTY; without even the implied warranty of
00014 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
00015 // GNU General Public License for more details.
00016 //
00017 // Under Section 7 of GPL version 3, you are granted additional
00018 // permissions described in the GCC Runtime Library Exception, version
00019 // 3.1, as published by the Free Software Foundation.
00020 
00021 // You should have received a copy of the GNU General Public License and
00022 // a copy of the GCC Runtime Library Exception along with this program;
00023 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
00024 // <http://www.gnu.org/licenses/>.
00025 
00026 /** @file tr1/modified_bessel_func.tcc
00027  *  This is an internal header file, included by other library headers.
00028  *  You should not attempt to use it directly.
00029  */
00030 
00031 //
00032 // ISO C++ 14882 TR1: 5.2  Special functions
00033 //
00034 
00035 // Written by Edward Smith-Rowland.
00036 //
00037 // References:
00038 //   (1) Handbook of Mathematical Functions,
00039 //       Ed. Milton Abramowitz and Irene A. Stegun,
00040 //       Dover Publications,
00041 //       Section 9, pp. 355-434, Section 10 pp. 435-478
00042 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
00043 //   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
00044 //       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
00045 //       2nd ed, pp. 246-249.
00046 
00047 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
00048 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
00049 
00050 #include "special_function_util.h"
00051 
00052 namespace std
00053 {
00054 namespace tr1
00055 {
00056 
00057   // [5.2] Special functions
00058 
00059   // Implementation-space details.
00060   namespace __detail
00061   {
00062 
00063     /**
00064      *   @brief  Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
00065      *           @f$ K_\nu(x) @f$ and their first derivatives
00066      *           @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
00067      *           These four functions are computed together for numerical
00068      *           stability.
00069      *
00070      *   @param  __nu  The order of the Bessel functions.
00071      *   @param  __x   The argument of the Bessel functions.
00072      *   @param  __Inu  The output regular modified Bessel function.
00073      *   @param  __Knu  The output irregular modified Bessel function.
00074      *   @param  __Ipnu  The output derivative of the regular
00075      *                   modified Bessel function.
00076      *   @param  __Kpnu  The output derivative of the irregular
00077      *                   modified Bessel function.
00078      */
00079     template <typename _Tp>
00080     void
00081     __bessel_ik(const _Tp __nu, const _Tp __x,
00082                 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
00083     {
00084       if (__x == _Tp(0))
00085         {
00086           if (__nu == _Tp(0))
00087             {
00088               __Inu = _Tp(1);
00089               __Ipnu = _Tp(0);
00090             }
00091           else if (__nu == _Tp(1))
00092             {
00093               __Inu = _Tp(0);
00094               __Ipnu = _Tp(0.5L);
00095             }
00096           else
00097             {
00098               __Inu = _Tp(0);
00099               __Ipnu = _Tp(0);
00100             }
00101           __Knu = std::numeric_limits<_Tp>::infinity();
00102           __Kpnu = -std::numeric_limits<_Tp>::infinity();
00103           return;
00104         }
00105 
00106       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
00107       const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
00108       const int __max_iter = 15000;
00109       const _Tp __x_min = _Tp(2);
00110 
00111       const int __nl = static_cast<int>(__nu + _Tp(0.5L));
00112 
00113       const _Tp __mu = __nu - __nl;
00114       const _Tp __mu2 = __mu * __mu;
00115       const _Tp __xi = _Tp(1) / __x;
00116       const _Tp __xi2 = _Tp(2) * __xi;
00117       _Tp __h = __nu * __xi;
00118       if ( __h < __fp_min )
00119         __h = __fp_min;
00120       _Tp __b = __xi2 * __nu;
00121       _Tp __d = _Tp(0);
00122       _Tp __c = __h;
00123       int __i;
00124       for ( __i = 1; __i <= __max_iter; ++__i )
00125         {
00126           __b += __xi2;
00127           __d = _Tp(1) / (__b + __d);
00128           __c = __b + _Tp(1) / __c;
00129           const _Tp __del = __c * __d;
00130           __h *= __del;
00131           if (std::abs(__del - _Tp(1)) < __eps)
00132             break;
00133         }
00134       if (__i > __max_iter)
00135         std::__throw_runtime_error(__N("Argument x too large "
00136                                        "in __bessel_jn; "
00137                                        "try asymptotic expansion."));
00138       _Tp __Inul = __fp_min;
00139       _Tp __Ipnul = __h * __Inul;
00140       _Tp __Inul1 = __Inul;
00141       _Tp __Ipnu1 = __Ipnul;
00142       _Tp __fact = __nu * __xi;
00143       for (int __l = __nl; __l >= 1; --__l)
00144         {
00145           const _Tp __Inutemp = __fact * __Inul + __Ipnul;
00146           __fact -= __xi;
00147           __Ipnul = __fact * __Inutemp + __Inul;
00148           __Inul = __Inutemp;
00149         }
00150       _Tp __f = __Ipnul / __Inul;
00151       _Tp __Kmu, __Knu1;
00152       if (__x < __x_min)
00153         {
00154           const _Tp __x2 = __x / _Tp(2);
00155           const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
00156           const _Tp __fact = (std::abs(__pimu) < __eps
00157                             ? _Tp(1) : __pimu / std::sin(__pimu));
00158           _Tp __d = -std::log(__x2);
00159           _Tp __e = __mu * __d;
00160           const _Tp __fact2 = (std::abs(__e) < __eps
00161                             ? _Tp(1) : std::sinh(__e) / __e);
00162           _Tp __gam1, __gam2, __gampl, __gammi;
00163           __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
00164           _Tp __ff = __fact
00165                    * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
00166           _Tp __sum = __ff;
00167           __e = std::exp(__e);
00168           _Tp __p = __e / (_Tp(2) * __gampl);
00169           _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
00170           _Tp __c = _Tp(1);
00171           __d = __x2 * __x2;
00172           _Tp __sum1 = __p;
00173           int __i;
00174           for (__i = 1; __i <= __max_iter; ++__i)
00175             {
00176               __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
00177               __c *= __d / __i;
00178               __p /= __i - __mu;
00179               __q /= __i + __mu;
00180               const _Tp __del = __c * __ff;
00181               __sum += __del; 
00182               const _Tp __del1 = __c * (__p - __i * __ff);
00183               __sum1 += __del1;
00184               if (std::abs(__del) < __eps * std::abs(__sum))
00185                 break;
00186             }
00187           if (__i > __max_iter)
00188             std::__throw_runtime_error(__N("Bessel k series failed to converge "
00189                                            "in __bessel_jn."));
00190           __Kmu = __sum;
00191           __Knu1 = __sum1 * __xi2;
00192         }
00193       else
00194         {
00195           _Tp __b = _Tp(2) * (_Tp(1) + __x);
00196           _Tp __d = _Tp(1) / __b;
00197           _Tp __delh = __d;
00198           _Tp __h = __delh;
00199           _Tp __q1 = _Tp(0);
00200           _Tp __q2 = _Tp(1);
00201           _Tp __a1 = _Tp(0.25L) - __mu2;
00202           _Tp __q = __c = __a1;
00203           _Tp __a = -__a1;
00204           _Tp __s = _Tp(1) + __q * __delh;
00205           int __i;
00206           for (__i = 2; __i <= __max_iter; ++__i)
00207             {
00208               __a -= 2 * (__i - 1);
00209               __c = -__a * __c / __i;
00210               const _Tp __qnew = (__q1 - __b * __q2) / __a;
00211               __q1 = __q2;
00212               __q2 = __qnew;
00213               __q += __c * __qnew;
00214               __b += _Tp(2);
00215               __d = _Tp(1) / (__b + __a * __d);
00216               __delh = (__b * __d - _Tp(1)) * __delh;
00217               __h += __delh;
00218               const _Tp __dels = __q * __delh;
00219               __s += __dels;
00220               if ( std::abs(__dels / __s) < __eps )
00221                 break;
00222             }
00223           if (__i > __max_iter)
00224             std::__throw_runtime_error(__N("Steed's method failed "
00225                                            "in __bessel_jn."));
00226           __h = __a1 * __h;
00227           __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
00228                 * std::exp(-__x) / __s;
00229           __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
00230         }
00231 
00232       _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
00233       _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
00234       __Inu = __Inumu * __Inul1 / __Inul;
00235       __Ipnu = __Inumu * __Ipnu1 / __Inul;
00236       for ( __i = 1; __i <= __nl; ++__i )
00237         {
00238           const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
00239           __Kmu = __Knu1;
00240           __Knu1 = __Knutemp;
00241         }
00242       __Knu = __Kmu;
00243       __Kpnu = __nu * __xi * __Kmu - __Knu1;
00244   
00245       return;
00246     }
00247 
00248 
00249     /**
00250      *   @brief  Return the regular modified Bessel function of order
00251      *           \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
00252      *
00253      *   The regular modified cylindrical Bessel function is:
00254      *   @f[
00255      *    I_{\nu}(x) = \sum_{k=0}^{\infty}
00256      *              \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
00257      *   @f]
00258      *
00259      *   @param  __nu  The order of the regular modified Bessel function.
00260      *   @param  __x   The argument of the regular modified Bessel function.
00261      *   @return  The output regular modified Bessel function.
00262      */
00263     template<typename _Tp>
00264     _Tp
00265     __cyl_bessel_i(const _Tp __nu, const _Tp __x)
00266     {
00267       if (__nu < _Tp(0) || __x < _Tp(0))
00268         std::__throw_domain_error(__N("Bad argument "
00269                                       "in __cyl_bessel_i."));
00270       else if (__isnan(__nu) || __isnan(__x))
00271         return std::numeric_limits<_Tp>::quiet_NaN();
00272       else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
00273         return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
00274       else
00275         {
00276           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
00277           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00278           return __I_nu;
00279         }
00280     }
00281 
00282 
00283     /**
00284      *   @brief  Return the irregular modified Bessel function
00285      *           \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
00286      *
00287      *   The irregular modified Bessel function is defined by:
00288      *   @f[
00289      *      K_{\nu}(x) = \frac{\pi}{2}
00290      *                   \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
00291      *   @f]
00292      *   where for integral \f$ \nu = n \f$ a limit is taken:
00293      *   \f$ lim_{\nu \to n} \f$.
00294      *
00295      *   @param  __nu  The order of the irregular modified Bessel function.
00296      *   @param  __x   The argument of the irregular modified Bessel function.
00297      *   @return  The output irregular modified Bessel function.
00298      */
00299     template<typename _Tp>
00300     _Tp
00301     __cyl_bessel_k(const _Tp __nu, const _Tp __x)
00302     {
00303       if (__nu < _Tp(0) || __x < _Tp(0))
00304         std::__throw_domain_error(__N("Bad argument "
00305                                       "in __cyl_bessel_k."));
00306       else if (__isnan(__nu) || __isnan(__x))
00307         return std::numeric_limits<_Tp>::quiet_NaN();
00308       else
00309         {
00310           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
00311           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00312           return __K_nu;
00313         }
00314     }
00315 
00316 
00317     /**
00318      *   @brief  Compute the spherical modified Bessel functions
00319      *           @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
00320      *           derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
00321      *           respectively.
00322      *
00323      *   @param  __n  The order of the modified spherical Bessel function.
00324      *   @param  __x  The argument of the modified spherical Bessel function.
00325      *   @param  __i_n  The output regular modified spherical Bessel function.
00326      *   @param  __k_n  The output irregular modified spherical
00327      *                  Bessel function.
00328      *   @param  __ip_n  The output derivative of the regular modified
00329      *                   spherical Bessel function.
00330      *   @param  __kp_n  The output derivative of the irregular modified
00331      *                   spherical Bessel function.
00332      */
00333     template <typename _Tp>
00334     void
00335     __sph_bessel_ik(const unsigned int __n, const _Tp __x,
00336                     _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
00337     {
00338       const _Tp __nu = _Tp(__n) + _Tp(0.5L);
00339 
00340       _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
00341       __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00342 
00343       const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
00344                          / std::sqrt(__x);
00345 
00346       __i_n = __factor * __I_nu;
00347       __k_n = __factor * __K_nu;
00348       __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
00349       __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
00350 
00351       return;
00352     }
00353 
00354 
00355     /**
00356      *   @brief  Compute the Airy functions
00357      *           @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
00358      *           derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
00359      *           respectively.
00360      *
00361      *   @param  __n  The order of the Airy functions.
00362      *   @param  __x  The argument of the Airy functions.
00363      *   @param  __i_n  The output Airy function.
00364      *   @param  __k_n  The output Airy function.
00365      *   @param  __ip_n  The output derivative of the Airy function.
00366      *   @param  __kp_n  The output derivative of the Airy function.
00367      */
00368     template <typename _Tp>
00369     void
00370     __airy(const _Tp __x,
00371            _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
00372     {
00373       const _Tp __absx = std::abs(__x);
00374       const _Tp __rootx = std::sqrt(__absx);
00375       const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
00376 
00377       if (__isnan(__x))
00378         return std::numeric_limits<_Tp>::quiet_NaN();
00379       else if (__x > _Tp(0))
00380         {
00381           _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
00382 
00383           __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00384           __Ai = __rootx * __K_nu
00385                / (__numeric_constants<_Tp>::__sqrt3()
00386                 * __numeric_constants<_Tp>::__pi());
00387           __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
00388                  + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
00389 
00390           __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00391           __Aip = -__x * __K_nu
00392                 / (__numeric_constants<_Tp>::__sqrt3()
00393                  * __numeric_constants<_Tp>::__pi());
00394           __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
00395                       + _Tp(2) * __I_nu
00396                       / __numeric_constants<_Tp>::__sqrt3());
00397         }
00398       else if (__x < _Tp(0))
00399         {
00400           _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
00401 
00402           __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
00403           __Ai = __rootx * (__J_nu
00404                     - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
00405           __Bi = -__rootx * (__N_nu
00406                     + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
00407 
00408           __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
00409           __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
00410                           + __J_nu) / _Tp(2);
00411           __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
00412                           - __N_nu) / _Tp(2);
00413         }
00414       else
00415         {
00416           //  Reference:
00417           //    Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
00418           //  The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
00419           __Ai = _Tp(0.35502805388781723926L);
00420           __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
00421 
00422           //  Reference:
00423           //    Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
00424           //  The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
00425           __Aip = -_Tp(0.25881940379280679840L);
00426           __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
00427         }
00428 
00429       return;
00430     }
00431 
00432   } // namespace std::tr1::__detail
00433 }
00434 }
00435 
00436 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC

Generated on Thu Jul 23 21:16:11 2009 for libstdc++ by  doxygen 1.5.8