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// Special functions -*- C++ -*-

// Copyright (C) 2006, 2007, 2008, 2009, 2010

// Free Software Foundation, Inc.

//

// This file is part of the GNU ISO C++ Library.  This library is free

// software; you can redistribute it and/or modify it under the

// terms of the GNU General Public License as published by the

// Free Software Foundation; either version 3, or (at your option)

// any later version.

//

// This library is distributed in the hope that it will be useful,

// but WITHOUT ANY WARRANTY; without even the implied warranty of

// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the

// GNU General Public License for more details.

//

// Under Section 7 of GPL version 3, you are granted additional

// permissions described in the GCC Runtime Library Exception, version

// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and

// a copy of the GCC Runtime Library Exception along with this program;

// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see

// .

/** @file tr1/modified_bessel_func.tcc

 *  This is an internal header file, included by other library headers.

 *  Do not attempt to use it directly. @headername{tr1/cmath}

 */

//

// ISO C++ 14882 TR1: 5.2  Special functions

//

// Written by Edward Smith-Rowland.

//

// References:

//   (1) Handbook of Mathematical Functions,

//       Ed. Milton Abramowitz and Irene A. Stegun,

//       Dover Publications,

//       Section 9, pp. 355-434, Section 10 pp. 435-478

//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl

//   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,

//       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),

//       2nd ed, pp. 246-249.

#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC

#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1

#include "special_function_util.h"

namespace std _GLIBCXX_VISIBILITY(default)

{

namespace tr1

{

  // [5.2] Special functions

  // Implementation-space details.

  namespace __detail

  {

  _GLIBCXX_BEGIN_NAMESPACE_VERSION

    /**

     *   @brief  Compute the modified Bessel functions @f$ I_\nu(x) @f$ and

     *           @f$ K_\nu(x) @f$ and their first derivatives

     *           @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively.

     *           These four functions are computed together for numerical

     *           stability.

     *

     *   @param  __nu  The order of the Bessel functions.

     *   @param  __x   The argument of the Bessel functions.

     *   @param  __Inu  The output regular modified Bessel function.

     *   @param  __Knu  The output irregular modified Bessel function.

     *   @param  __Ipnu  The output derivative of the regular

     *                   modified Bessel function.

     *   @param  __Kpnu  The output derivative of the irregular

     *                   modified Bessel function.

     */

    template 

    void

    __bessel_ik(const _Tp __nu, const _Tp __x,

                _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu)

    {

      if (__x == _Tp(0))

        {

          if (__nu == _Tp(0))

            {

              __Inu = _Tp(1);

              __Ipnu = _Tp(0);

            }

          else if (__nu == _Tp(1))

            {

              __Inu = _Tp(0);

              __Ipnu = _Tp(0.5L);

            }

          else

            {

              __Inu = _Tp(0);

              __Ipnu = _Tp(0);

            }

          __Knu = std::numeric_limits<_Tp>::infinity();

          __Kpnu = -std::numeric_limits<_Tp>::infinity();

          return;

        }

      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();

      const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon();

      const int __max_iter = 15000;

      const _Tp __x_min = _Tp(2);

      const int __nl = static_cast(__nu + _Tp(0.5L));

      const _Tp __mu = __nu - __nl;

      const _Tp __mu2 = __mu * __mu;

      const _Tp __xi = _Tp(1) / __x;

      const _Tp __xi2 = _Tp(2) * __xi;

      _Tp __h = __nu * __xi;

      if ( __h < __fp_min )

        __h = __fp_min;

      _Tp __b = __xi2 * __nu;

      _Tp __d = _Tp(0);

      _Tp __c = __h;

      int __i;

      for ( __i = 1; __i <= __max_iter; ++__i )

        {

          __b += __xi2;

          __d = _Tp(1) / (__b + __d);

          __c = __b + _Tp(1) / __c;

          const _Tp __del = __c * __d;

          __h *= __del;

          if (std::abs(__del - _Tp(1)) < __eps)

            break;

        }

      if (__i > __max_iter)

        std::__throw_runtime_error(__N("Argument x too large "

                                       "in __bessel_jn; "

                                       "try asymptotic expansion."));

      _Tp __Inul = __fp_min;

      _Tp __Ipnul = __h * __Inul;

      _Tp __Inul1 = __Inul;

      _Tp __Ipnu1 = __Ipnul;

      _Tp __fact = __nu * __xi;

      for (int __l = __nl; __l >= 1; --__l)

        {

          const _Tp __Inutemp = __fact * __Inul + __Ipnul;

          __fact -= __xi;

          __Ipnul = __fact * __Inutemp + __Inul;

          __Inul = __Inutemp;

        }

      _Tp __f = __Ipnul / __Inul;

      _Tp __Kmu, __Knu1;

      if (__x < __x_min)

        {

          const _Tp __x2 = __x / _Tp(2);

          const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;

          const _Tp __fact = (std::abs(__pimu) < __eps

                            ? _Tp(1) : __pimu / std::sin(__pimu));

          _Tp __d = -std::log(__x2);

          _Tp __e = __mu * __d;

          const _Tp __fact2 = (std::abs(__e) < __eps

                            ? _Tp(1) : std::sinh(__e) / __e);

          _Tp __gam1, __gam2, __gampl, __gammi;

          __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);

          _Tp __ff = __fact

                   * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);

          _Tp __sum = __ff;

          __e = std::exp(__e);

          _Tp __p = __e / (_Tp(2) * __gampl);

          _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi);

          _Tp __c = _Tp(1);

          __d = __x2 * __x2;

          _Tp __sum1 = __p;

          int __i;

          for (__i = 1; __i <= __max_iter; ++__i)

            {

              __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);

              __c *= __d / __i;

              __p /= __i - __mu;

              __q /= __i + __mu;

              const _Tp __del = __c * __ff;

              __sum += __del;

              const _Tp __del1 = __c * (__p - __i * __ff);

              __sum1 += __del1;

              if (std::abs(__del) < __eps * std::abs(__sum))

                break;

            }

          if (__i > __max_iter)

            std::__throw_runtime_error(__N("Bessel k series failed to converge "

                                           "in __bessel_jn."));

          __Kmu = __sum;

          __Knu1 = __sum1 * __xi2;

        }

      else

        {

          _Tp __b = _Tp(2) * (_Tp(1) + __x);

          _Tp __d = _Tp(1) / __b;

          _Tp __delh = __d;

          _Tp __h = __delh;

          _Tp __q1 = _Tp(0);

          _Tp __q2 = _Tp(1);

          _Tp __a1 = _Tp(0.25L) - __mu2;

          _Tp __q = __c = __a1;

          _Tp __a = -__a1;

          _Tp __s = _Tp(1) + __q * __delh;

          int __i;

          for (__i = 2; __i <= __max_iter; ++__i)

            {

              __a -= 2 * (__i - 1);

              __c = -__a * __c / __i;

              const _Tp __qnew = (__q1 - __b * __q2) / __a;

              __q1 = __q2;

              __q2 = __qnew;

              __q += __c * __qnew;

              __b += _Tp(2);

              __d = _Tp(1) / (__b + __a * __d);

              __delh = (__b * __d - _Tp(1)) * __delh;

              __h += __delh;

              const _Tp __dels = __q * __delh;

              __s += __dels;

              if ( std::abs(__dels / __s) < __eps )

                break;

            }

          if (__i > __max_iter)

            std::__throw_runtime_error(__N("Steed's method failed "

                                           "in __bessel_jn."));

          __h = __a1 * __h;

          __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x))

                * std::exp(-__x) / __s;

          __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi;

        }

      _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1;

      _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu);

      __Inu = __Inumu * __Inul1 / __Inul;

      __Ipnu = __Inumu * __Ipnu1 / __Inul;

      for ( __i = 1; __i <= __nl; ++__i )

        {

          const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu;

          __Kmu = __Knu1;

          __Knu1 = __Knutemp;

        }

      __Knu = __Kmu;

      __Kpnu = __nu * __xi * __Kmu - __Knu1;

      return;

    }

    /**

     *   @brief  Return the regular modified Bessel function of order

     *           \f$ \nu \f$: \f$ I_{\nu}(x) \f$.

     *

     *   The regular modified cylindrical Bessel function is:

     *   @f[

     *    I_{\nu}(x) = \sum_{k=0}^{\infty}

     *              \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}

     *   @f]

     *

     *   @param  __nu  The order of the regular modified Bessel function.

     *   @param  __x   The argument of the regular modified Bessel function.

     *   @return  The output regular modified Bessel function.

     */

    template

    _Tp

    __cyl_bessel_i(const _Tp __nu, const _Tp __x)

    {

      if (__nu < _Tp(0) || __x < _Tp(0))

        std::__throw_domain_error(__N("Bad argument "

                                      "in __cyl_bessel_i."));

      else if (__isnan(__nu) || __isnan(__x))

        return std::numeric_limits<_Tp>::quiet_NaN();

      else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))

        return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200);

      else

        {

          _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;

          __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);

          return __I_nu;

        }

    }

    /**

     *   @brief  Return the irregular modified Bessel function

     *           \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$.

     *

     *   The irregular modified Bessel function is defined by:

     *   @f[

     *      K_{\nu}(x) = \frac{\pi}{2}

     *                   \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}

     *   @f]

     *   where for integral \f$ \nu = n \f$ a limit is taken:

     *   \f$ lim_{\nu \to n} \f$.

     *

     *   @param  __nu  The order of the irregular modified Bessel function.

     *   @param  __x   The argument of the irregular modified Bessel function.

     *   @return  The output irregular modified Bessel function.

     */

    template

    _Tp

    __cyl_bessel_k(const _Tp __nu, const _Tp __x)

    {

      if (__nu < _Tp(0) || __x < _Tp(0))

        std::__throw_domain_error(__N("Bad argument "

                                      "in __cyl_bessel_k."));

      else if (__isnan(__nu) || __isnan(__x))

        return std::numeric_limits<_Tp>::quiet_NaN();

      else

        {

          _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu;

          __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);

          return __K_nu;

        }

    }

    /**

     *   @brief  Compute the spherical modified Bessel functions

     *           @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first

     *           derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$

     *           respectively.

     *

     *   @param  __n  The order of the modified spherical Bessel function.

     *   @param  __x  The argument of the modified spherical Bessel function.

     *   @param  __i_n  The output regular modified spherical Bessel function.

     *   @param  __k_n  The output irregular modified spherical

     *                  Bessel function.

     *   @param  __ip_n  The output derivative of the regular modified

     *                   spherical Bessel function.

     *   @param  __kp_n  The output derivative of the irregular modified

     *                   spherical Bessel function.

     */

    template 

    void

    __sph_bessel_ik(const unsigned int __n, const _Tp __x,

                    _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n)

    {

      const _Tp __nu = _Tp(__n) + _Tp(0.5L);

      _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;

      __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu);

      const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()

                         / std::sqrt(__x);

      __i_n = __factor * __I_nu;

      __k_n = __factor * __K_nu;

      __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x);

      __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x);

      return;

    }

    /**

     *   @brief  Compute the Airy functions

     *           @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first

     *           derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$

     *           respectively.

     *

     *   @param  __n  The order of the Airy functions.

     *   @param  __x  The argument of the Airy functions.

     *   @param  __i_n  The output Airy function.

     *   @param  __k_n  The output Airy function.

     *   @param  __ip_n  The output derivative of the Airy function.

     *   @param  __kp_n  The output derivative of the Airy function.

     */

    template 

    void

    __airy(const _Tp __x,

           _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip)

    {

      const _Tp __absx = std::abs(__x);

      const _Tp __rootx = std::sqrt(__absx);

      const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3);

      if (__isnan(__x))

        return std::numeric_limits<_Tp>::quiet_NaN();

      else if (__x > _Tp(0))

        {

          _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu;

          __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);

          __Ai = __rootx * __K_nu

               / (__numeric_constants<_Tp>::__sqrt3()

                * __numeric_constants<_Tp>::__pi());

          __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi()

                 + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3());

          __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu);

          __Aip = -__x * __K_nu

                / (__numeric_constants<_Tp>::__sqrt3()

                 * __numeric_constants<_Tp>::__pi());

          __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi()

                      + _Tp(2) * __I_nu

                      / __numeric_constants<_Tp>::__sqrt3());

        }

      else if (__x < _Tp(0))

        {

          _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu;

          __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);

          __Ai = __rootx * (__J_nu

                    - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);

          __Bi = -__rootx * (__N_nu

                    + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2);

          __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu);

          __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3()

                          + __J_nu) / _Tp(2);

          __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3()

                          - __N_nu) / _Tp(2);

        }

      else

        {

          //  Reference:

          //    Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions.

          //  The number is Ai(0) = 3^{-2/3}/\Gamma(2/3).

          __Ai = _Tp(0.35502805388781723926L);

          __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3();

          //  Reference:

          //    Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions.

          //  The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3).

          __Aip = -_Tp(0.25881940379280679840L);

          __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3();

        }

      return;

    }

  _GLIBCXX_END_NAMESPACE_VERSION

  } // namespace std::tr1::__detail

}

}

#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC