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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/legendre_function.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 based on:

//   (1) Handbook of Mathematical Functions,

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

//       Dover Publications,

//       Section 8, pp. 331-341

//   (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. 252-254

#ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC

#define _GLIBCXX_TR1_LEGENDRE_FUNCTION_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  Return the Legendre polynomial by recursion on order

     *           @f$ l @f$.

     *

     *   The Legendre function of @f$ l @f$ and @f$ x @f$,

     *   @f$ P_l(x) @f$, is defined by:

     *   @f[

     *     P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}

     *   @f]

     *

     *   @param  l  The order of the Legendre polynomial.  @f$l >= 0@f$.

     *   @param  x  The argument of the Legendre polynomial.  @f$|x| <= 1@f$.

     */

    template

    _Tp

    __poly_legendre_p(const unsigned int __l, const _Tp __x)

    {

      if ((__x < _Tp(-1)) || (__x > _Tp(+1)))

        std::__throw_domain_error(__N("Argument out of range"

                                      " in __poly_legendre_p."));

      else if (__isnan(__x))

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

      else if (__x == +_Tp(1))

        return +_Tp(1);

      else if (__x == -_Tp(1))

        return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));

      else

        {

          _Tp __p_lm2 = _Tp(1);

          if (__l == 0)

            return __p_lm2;

          _Tp __p_lm1 = __x;

          if (__l == 1)

            return __p_lm1;

          _Tp __p_l = 0;

          for (unsigned int __ll = 2; __ll <= __l; ++__ll)

            {

              //  This arrangement is supposed to be better for roundoff

              //  protection, Arfken, 2nd Ed, Eq 12.17a.

              __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2

                    - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);

              __p_lm2 = __p_lm1;

              __p_lm1 = __p_l;

            }

          return __p_l;

        }

    }

    /**

     *   @brief  Return the associated Legendre function by recursion

     *           on @f$ l @f$.

     *

     *   The associated Legendre function is derived from the Legendre function

     *   @f$ P_l(x) @f$ by the Rodrigues formula:

     *   @f[

     *     P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)

     *   @f]

     *

     *   @param  l  The order of the associated Legendre function.

     *              @f$ l >= 0 @f$.

     *   @param  m  The order of the associated Legendre function.

     *              @f$ m <= l @f$.

     *   @param  x  The argument of the associated Legendre function.

     *              @f$ |x| <= 1 @f$.

     */

    template

    _Tp

    __assoc_legendre_p(const unsigned int __l, const unsigned int __m,

                       const _Tp __x)

    {

      if (__x < _Tp(-1) || __x > _Tp(+1))

        std::__throw_domain_error(__N("Argument out of range"

                                      " in __assoc_legendre_p."));

      else if (__m > __l)

        std::__throw_domain_error(__N("Degree out of range"

                                      " in __assoc_legendre_p."));

      else if (__isnan(__x))

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

      else if (__m == 0)

        return __poly_legendre_p(__l, __x);

      else

        {

          _Tp __p_mm = _Tp(1);

          if (__m > 0)

            {

              //  Two square roots seem more accurate more of the time

              //  than just one.

              _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);

              _Tp __fact = _Tp(1);

              for (unsigned int __i = 1; __i <= __m; ++__i)

                {

                  __p_mm *= -__fact * __root;

                  __fact += _Tp(2);

                }

            }

          if (__l == __m)

            return __p_mm;

          _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;

          if (__l == __m + 1)

            return __p_mp1m;

          _Tp __p_lm2m = __p_mm;

          _Tp __P_lm1m = __p_mp1m;

          _Tp __p_lm = _Tp(0);

          for (unsigned int __j = __m + 2; __j <= __l; ++__j)

            {

              __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m

                      - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);

              __p_lm2m = __P_lm1m;

              __P_lm1m = __p_lm;

            }

          return __p_lm;

        }

    }

    /**

     *   @brief  Return the spherical associated Legendre function.

     *

     *   The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,

     *   and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where

     *   @f[

     *      Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}

     *                                  \frac{(l-m)!}{(l+m)!}]

     *                     P_l^m(\cos\theta) \exp^{im\phi}

     *   @f]

     *   is the spherical harmonic function and @f$ P_l^m(x) @f$ is the

     *   associated Legendre function.

     *

     *   This function differs from the associated Legendre function by

     *   argument (@f$x = \cos(\theta)@f$) and by a normalization factor

     *   but this factor is rather large for large @f$ l @f$ and @f$ m @f$

     *   and so this function is stable for larger differences of @f$ l @f$

     *   and @f$ m @f$.

     *

     *   @param  l  The order of the spherical associated Legendre function.

     *              @f$ l >= 0 @f$.

     *   @param  m  The order of the spherical associated Legendre function.

     *              @f$ m <= l @f$.

     *   @param  theta  The radian angle argument of the spherical associated

     *                  Legendre function.

     */

    template 

    _Tp

    __sph_legendre(const unsigned int __l, const unsigned int __m,

                   const _Tp __theta)

    {

      if (__isnan(__theta))

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

      const _Tp __x = std::cos(__theta);

      if (__l < __m)

        {

          std::__throw_domain_error(__N("Bad argument "

                                        "in __sph_legendre."));

        }

      else if (__m == 0)

        {

          _Tp __P = __poly_legendre_p(__l, __x);

          _Tp __fact = std::sqrt(_Tp(2 * __l + 1)

                     / (_Tp(4) * __numeric_constants<_Tp>::__pi()));

          __P *= __fact;

          return __P;

        }

      else if (__x == _Tp(1) || __x == -_Tp(1))

        {

          //  m > 0 here

          return _Tp(0);

        }

      else

        {

          // m > 0 and |x| < 1 here

          // Starting value for recursion.

          // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )

          //             (-1)^m (1-x^2)^(m/2) / pi^(1/4)

          const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));

          const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));

#if _GLIBCXX_USE_C99_MATH_TR1

          const _Tp __lncirc = std::tr1::log1p(-__x * __x);

#else

          const _Tp __lncirc = std::log(_Tp(1) - __x * __x);

#endif

          //  Gamma(m+1/2) / Gamma(m)

#if _GLIBCXX_USE_C99_MATH_TR1

          const _Tp __lnpoch = std::tr1::lgamma(_Tp(__m + _Tp(0.5L)))

                             - std::tr1::lgamma(_Tp(__m));

#else

          const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))

                             - __log_gamma(_Tp(__m));

#endif

          const _Tp __lnpre_val =

                    -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()

                    + _Tp(0.5L) * (__lnpoch + __m * __lncirc);

          _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)

                   / (_Tp(4) * __numeric_constants<_Tp>::__pi()));

          _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);

          _Tp __y_mp1m = __y_mp1m_factor * __y_mm;

          if (__l == __m)

            {

              return __y_mm;

            }

          else if (__l == __m + 1)

            {

              return __y_mp1m;

            }

          else

            {

              _Tp __y_lm = _Tp(0);

              // Compute Y_l^m, l > m+1, upward recursion on l.

              for ( int __ll = __m + 2; __ll <= __l; ++__ll)

                {

                  const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);

                  const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);

                  const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)

                                                       * _Tp(2 * __ll - 1));

                  const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)

                                                                / _Tp(2 * __ll - 3));

                  __y_lm = (__x * __y_mp1m * __fact1

                         - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);

                  __y_mm = __y_mp1m;

                  __y_mp1m = __y_lm;

                }

              return __y_lm;

            }

        }

    }

  _GLIBCXX_END_NAMESPACE_VERSION

  } // namespace std::tr1::__detail

}

}

#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC