modified_bessel_func.tcc

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00001 // Special functions -*- C++ -*-
00002 
00003 // Copyright (C) 2006, 2007, 2008
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 2, 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 // You should have received a copy of the GNU General Public License along
00018 // with this library; see the file COPYING.  If not, write to the Free
00019 // Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301,
00020 // USA.
00021 //
00022 // As a special exception, you may use this file as part of a free software
00023 // library without restriction.  Specifically, if other files instantiate
00024 // templates or use macros or inline functions from this file, or you compile
00025 // this file and link it with other files to produce an executable, this
00026 // file does not by itself cause the resulting executable to be covered by
00027 // the GNU General Public License.  This exception does not however
00028 // invalidate any other reasons why the executable file might be covered by
00029 // the GNU General Public License.
00030 
00031 /** @file tr1/modified_bessel_func.tcc
00032  *  This is an internal header file, included by other library headers.
00033  *  You should not attempt to use it directly.
00034  */
00035 
00036 //
00037 // ISO C++ 14882 TR1: 5.2  Special functions
00038 //
00039 
00040 // Written by Edward Smith-Rowland.
00041 //
00042 // References:
00043 //   (1) Handbook of Mathematical Functions,
00044 //       Ed. Milton Abramowitz and Irene A. Stegun,
00045 //       Dover Publications,
00046 //       Section 9, pp. 355-434, Section 10 pp. 435-478
00047 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
00048 //   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
00049 //       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
00050 //       2nd ed, pp. 246-249.
00051 
00052 #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
00053 #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1
00054 
00055 #include "special_function_util.h"
00056 
00057 namespace std
00058 {
00059 namespace tr1
00060 {
00061 
00062   // [5.2] Special functions
00063 
00064   // Implementation-space details.
00065   namespace __detail
00066   {
00067 
00068     /**
00069      *   @brief  Compute the modified Bessel functions @f$ I_\nu(x) @f$ and
00070      *           @f$ K_\nu(x) @f$ and their first derivatives
00071      *           @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.
00072      *           These four functions are computed together for numerical
00073      *           stability.
00074      *
00075      *   @param  __nu  The order of the Bessel functions.
00076      *   @param  __x   The argument of the Bessel functions.
00077      *   @param  __Inu  The output regular modified Bessel function.
00078      *   @param  __Knu  The output irregular modified Bessel function.
00079      *   @param  __Ipnu  The output derivative of the regular
00080      *                   modified Bessel function.
00081      *   @param  __Kpnu  The output derivative of the irregular
00082      *                   modified Bessel function.
00083      */
00084     template <typename _Tp>
00085     void
00086     __bessel_ik(const _Tp __nu, const _Tp __x,
00087                 _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)
00088     {
00089       if (__x == _Tp(0))
00090         {
00091           if (__nu == _Tp(0))
00092             {
00093               __Inu = _Tp(1);
00094               __Ipnu = _Tp(0);
00095             }
00096           else if (__nu == _Tp(1))
00097             {
00098               __Inu = _Tp(0);
00099               __Ipnu = _Tp(0.5L);
00100             }
00101           else
00102             {
00103               __Inu = _Tp(0);
00104               __Ipnu = _Tp(0);
00105             }
00106           __Knu = std::numeric_limits<_Tp>::infinity();
00107           __Kpnu = -std::numeric_limits<_Tp>::infinity();
00108           return;
00109         }
00110 
00111       const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
00112       const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();
00113       const int __max_iter = 15000;
00114       const _Tp __x_min = _Tp(2);
00115 
00116       const int __nl = static_cast<int>(__nu + _Tp(0.5L));
00117 
00118       const _Tp __mu = __nu - __nl;
00119       const _Tp __mu2 = __mu * __mu;
00120       const _Tp __xi = _Tp(1) / __x;
00121       const _Tp __xi2 = _Tp(2) * __xi;
00122       _Tp __h = __nu * __xi;
00123       if ( __h < __fp_min )
00124         __h = __fp_min;
00125       _Tp __b = __xi2 * __nu;
00126       _Tp __d = _Tp(0);
00127       _Tp __c = __h;
00128       int __i;
00129       for ( __i = 1; __i <= __max_iter; ++__i )
00130         {
00131           __b += __xi2;
00132           __d = _Tp(1) / (__b + __d);
00133           __c = __b + _Tp(1) / __c;
00134           const _Tp __del = __c * __d;
00135           __h *= __del;
00136           if (std::abs(__del - _Tp(1)) < __eps)
00137             break;
00138         }
00139       if (__i > __max_iter)
00140         std::__throw_runtime_error(__N("Argument x too large "
00141                                        "in __bessel_jn; "
00142                                        "try asymptotic expansion."));
00143       _Tp __Inul = __fp_min;
00144       _Tp __Ipnul = __h * __Inul;
00145       _Tp __Inul1 = __Inul;
00146       _Tp __Ipnu1 = __Ipnul;
00147       _Tp __fact = __nu * __xi;
00148       for (int __l = __nl; __l >= 1; --__l)
00149         {
00150           const _Tp __Inutemp = __fact * __Inul + __Ipnul;
00151           __fact -= __xi;
00152           __Ipnul = __fact * __Inutemp + __Inul;
00153           __Inul = __Inutemp;
00154         }
00155       _Tp __f = __Ipnul / __Inul;
00156       _Tp __Kmu, __Knu1;
00157       if (__x < __x_min)
00158         {
00159           const _Tp __x2 = __x / _Tp(2);
00160           const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
00161           const _Tp __fact = (std::abs(__pimu) < __eps
00162                             ? _Tp(1) : __pimu / std::sin(__pimu));
00163           _Tp __d = -std::log(__x2);
00164           _Tp __e = __mu * __d;
00165           const _Tp __fact2 = (std::abs(__e) < __eps
00166                             ? _Tp(1) : std::sinh(__e) / __e);
00167           _Tp __gam1, __gam2, __gampl, __gammi;
00168           __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
00169           _Tp __ff = __fact
00170                    * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
00171           _Tp __sum = __ff;
00172           __e = std::exp(__e);
00173           _Tp __p = __e / (_Tp(2) * __gampl);
00174           _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);
00175           _Tp __c = _Tp(1);
00176           __d = __x2 * __x2;
00177           _Tp __sum1 = __p;
00178           int __i;
00179           for (__i = 1; __i <= __max_iter; ++__i)
00180             {
00181               __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
00182               __c *= __d / __i;
00183               __p /= __i - __mu;
00184               __q /= __i + __mu;
00185               const _Tp __del = __c * __ff;
00186               __sum += __del; 
00187               const _Tp __del1 = __c * (__p - __i * __ff);
00188               __sum1 += __del1;
00189               if (std::abs(__del) < __eps * std::abs(__sum))
00190                 break;
00191             }
00192           if (__i > __max_iter)
00193             std::__throw_runtime_error(__N("Bessel k series failed to converge "
00194                                            "in __bessel_jn."));
00195           __Kmu = __sum;
00196           __Knu1 = __sum1 * __xi2;
00197         }
00198       else
00199         {
00200           _Tp __b = _Tp(2) * (_Tp(1) + __x);
00201           _Tp __d = _Tp(1) / __b;
00202           _Tp __delh = __d;
00203           _Tp __h = __delh;
00204           _Tp __q1 = _Tp(0);
00205           _Tp __q2 = _Tp(1);
00206           _Tp __a1 = _Tp(0.25L) - __mu2;
00207           _Tp __q = __c = __a1;
00208           _Tp __a = -__a1;
00209           _Tp __s = _Tp(1) + __q * __delh;
00210           int __i;
00211           for (__i = 2; __i <= __max_iter; ++__i)
00212             {
00213               __a -= 2 * (__i - 1);
00214               __c = -__a * __c / __i;
00215               const _Tp __qnew = (__q1 - __b * __q2) / __a;
00216               __q1 = __q2;
00217               __q2 = __qnew;
00218               __q += __c * __qnew;
00219               __b += _Tp(2);
00220               __d = _Tp(1) / (__b + __a * __d);
00221               __delh = (__b * __d - _Tp(1)) * __delh;
00222               __h += __delh;
00223               const _Tp __dels = __q * __delh;
00224               __s += __dels;
00225               if ( std::abs(__dels / __s) < __eps )
00226                 break;
00227             }
00228           if (__i > __max_iter)
00229             std::__throw_runtime_error(__N("Steed's method failed "
00230                                            "in __bessel_jn."));
00231           __h = __a1 * __h;
00232           __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))
00233                 * std::exp(-__x) / __s;
00234           __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;
00235         }
00236 
00237       _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;
00238       _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);
00239       __Inu = __Inumu * __Inul1 / __Inul;
00240       __Ipnu = __Inumu * __Ipnu1 / __Inul;
00241       for ( __i = 1; __i <= __nl; ++__i )
00242         {
00243           const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;
00244           __Kmu = __Knu1;
00245           __Knu1 = __Knutemp;
00246         }
00247       __Knu = __Kmu;
00248       __Kpnu = __nu * __xi * __Kmu - __Knu1;
00249   
00250       return;
00251     }
00252 
00253 
00254     /**
00255      *   @brief  Return the regular modified Bessel function of order
00256      *           \f$ \nu \f$: \f$ I_{\nu}(x) \f$.
00257      *
00258      *   The regular modified cylindrical Bessel function is:
00259      *   @f[
00260      *    I_{\nu}(x) = \sum_{k=0}^{\infty}
00261      *              \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
00262      *   @f]
00263      *
00264      *   @param  __nu  The order of the regular modified Bessel function.
00265      *   @param  __x   The argument of the regular modified Bessel function.
00266      *   @return  The output regular modified Bessel function.
00267      */
00268     template<typename _Tp>
00269     _Tp
00270     __cyl_bessel_i(const _Tp __nu, const _Tp __x)
00271     {
00272       if (__nu < _Tp(0) || __x < _Tp(0))
00273         std::__throw_domain_error(__N("Bad argument "
00274                                       "in __cyl_bessel_i."));
00275       else if (__isnan(__nu) || __isnan(__x))
00276         return std::numeric_limits<_Tp>::quiet_NaN();
00277       else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
00278         return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);
00279       else
00280         {
00281           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
00282           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00283           return __I_nu;
00284         }
00285     }
00286 
00287 
00288     /**
00289      *   @brief  Return the irregular modified Bessel function
00290      *           \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.
00291      *
00292      *   The irregular modified Bessel function is defined by:
00293      *   @f[
00294      *      K_{\nu}(x) = \frac{\pi}{2}
00295      *                   \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}
00296      *   @f]
00297      *   where for integral \f$ \nu = n \f$ a limit is taken:
00298      *   \f$ lim_{\nu \to n} \f$.
00299      *
00300      *   @param  __nu  The order of the irregular modified Bessel function.
00301      *   @param  __x   The argument of the irregular modified Bessel function.
00302      *   @return  The output irregular modified Bessel function.
00303      */
00304     template<typename _Tp>
00305     _Tp
00306     __cyl_bessel_k(const _Tp __nu, const _Tp __x)
00307     {
00308       if (__nu < _Tp(0) || __x < _Tp(0))
00309         std::__throw_domain_error(__N("Bad argument "
00310                                       "in __cyl_bessel_k."));
00311       else if (__isnan(__nu) || __isnan(__x))
00312         return std::numeric_limits<_Tp>::quiet_NaN();
00313       else
00314         {
00315           _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;
00316           __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00317           return __K_nu;
00318         }
00319     }
00320 
00321 
00322     /**
00323      *   @brief  Compute the spherical modified Bessel functions
00324      *           @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first
00325      *           derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$
00326      *           respectively.
00327      *
00328      *   @param  __n  The order of the modified spherical Bessel function.
00329      *   @param  __x  The argument of the modified spherical Bessel function.
00330      *   @param  __i_n  The output regular modified spherical Bessel function.
00331      *   @param  __k_n  The output irregular modified spherical
00332      *                  Bessel function.
00333      *   @param  __ip_n  The output derivative of the regular modified
00334      *                   spherical Bessel function.
00335      *   @param  __kp_n  The output derivative of the irregular modified
00336      *                   spherical Bessel function.
00337      */
00338     template <typename _Tp>
00339     void
00340     __sph_bessel_ik(const unsigned int __n, const _Tp __x,
00341                     _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)
00342     {
00343       const _Tp __nu = _Tp(__n) + _Tp(0.5L);
00344 
00345       _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
00346       __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00347 
00348       const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
00349                          / std::sqrt(__x);
00350 
00351       __i_n = __factor * __I_nu;
00352       __k_n = __factor * __K_nu;
00353       __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);
00354       __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);
00355 
00356       return;
00357     }
00358 
00359 
00360     /**
00361      *   @brief  Compute the Airy functions
00362      *           @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first
00363      *           derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$
00364      *           respectively.
00365      *
00366      *   @param  __n  The order of the Airy functions.
00367      *   @param  __x  The argument of the Airy functions.
00368      *   @param  __i_n  The output Airy function.
00369      *   @param  __k_n  The output Airy function.
00370      *   @param  __ip_n  The output derivative of the Airy function.
00371      *   @param  __kp_n  The output derivative of the Airy function.
00372      */
00373     template <typename _Tp>
00374     void
00375     __airy(const _Tp __x,
00376            _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)
00377     {
00378       const _Tp __absx = std::abs(__x);
00379       const _Tp __rootx = std::sqrt(__absx);
00380       const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);
00381 
00382       if (__isnan(__x))
00383         return std::numeric_limits<_Tp>::quiet_NaN();
00384       else if (__x > _Tp(0))
00385         {
00386           _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;
00387 
00388           __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00389           __Ai = __rootx * __K_nu
00390                / (__numeric_constants<_Tp>::__sqrt3()
00391                 * __numeric_constants<_Tp>::__pi());
00392           __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()
00393                  + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());
00394 
00395           __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);
00396           __Aip = -__x * __K_nu
00397                 / (__numeric_constants<_Tp>::__sqrt3()
00398                  * __numeric_constants<_Tp>::__pi());
00399           __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()
00400                       + _Tp(2) * __I_nu
00401                       / __numeric_constants<_Tp>::__sqrt3());
00402         }
00403       else if (__x < _Tp(0))
00404         {
00405           _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;
00406 
00407           __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
00408           __Ai = __rootx * (__J_nu
00409                     - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
00410           __Bi = -__rootx * (__N_nu
00411                     + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);
00412 
00413           __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);
00414           __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()
00415                           + __J_nu) / _Tp(2);
00416           __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()
00417                           - __N_nu) / _Tp(2);
00418         }
00419       else
00420         {
00421           //  Reference:
00422           //    Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.
00423           //  The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).
00424           __Ai = _Tp(0.35502805388781723926L);
00425           __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();
00426 
00427           //  Reference:
00428           //    Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.
00429           //  The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).
00430           __Aip = -_Tp(0.25881940379280679840L);
00431           __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();
00432         }
00433 
00434       return;
00435     }
00436 
00437   } // namespace std::tr1::__detail
00438 }
00439 }
00440 
00441 #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC

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