text: // Special functions -*- C++ -*-
  
  // Copyright (C) 2006-2017 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
  // <http://www.gnu.org/licenses/>.
  
  /** @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)
  {
  #if _GLIBCXX_USE_STD_SPEC_FUNCS
  #elif defined(_GLIBCXX_TR1_CMATH)
  namespace tr1
  {
  #else
  # error do not include this header directly, use <cmath> or <tr1/cmath>
  #endif
    // [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 <typename _Tp>
      void
      __bessel_ik(_Tp __nu, _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<int>(__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_ik; "
                                         "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_ik."));
            __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_ik."));
            __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<typename _Tp>
      _Tp
      __cyl_bessel_i(_Tp __nu, _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<typename _Tp>
      _Tp
      __cyl_bessel_k(_Tp __nu, _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 <typename _Tp>
      void
      __sph_bessel_ik(unsigned int __n, _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  __x  The argument of the Airy functions.
       *   @param  __Ai  The output Airy function of the first kind.
       *   @param  __Bi  The output Airy function of the second kind.
       *   @param  __Aip  The output derivative of the Airy function
       *                  of the first kind.
       *   @param  __Bip  The output derivative of the Airy function
       *                  of the second kind.
       */
      template <typename _Tp>
      void
      __airy(_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 (__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 __detail
  #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
  } // namespace tr1
  #endif
  }
  
  #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC
  