FM / tr1

/usr/include/c++/4.4.4/tr1/

/usr/include/c++/4.4.4/tr1

Name	Size	Mode	Actions
array	1802	0644	edit dl rm
bessel_function.tcc	22103	0644	edit dl rm
beta_function.tcc	5510	0644	edit dl rm
ccomplex	1256	0644	edit dl rm
cctype	1670	0644	edit dl rm
cfenv	1729	0644	edit dl rm
cfloat	1387	0644	edit dl rm
cinttypes	2006	0644	edit dl rm
climits	1467	0644	edit dl rm
cmath	16362	0644	edit dl rm
complex	2689	0644	edit dl rm
complex.h	1262	0644	edit dl rm
cstdarg	1253	0644	edit dl rm
cstdbool	1351	0644	edit dl rm
cstdint	2355	0644	edit dl rm
cstdio	1767	0644	edit dl rm
cstdlib	1775	0644	edit dl rm
ctgmath	1255	0644	edit dl rm
ctime	1241	0644	edit dl rm
ctype.h	1210	0644	edit dl rm
cwchar	1773	0644	edit dl rm
cwctype	1781	0644	edit dl rm
ell_integral.tcc	27553	0644	edit dl rm
exp_integral.tcc	15728	0644	edit dl rm
fenv.h	1205	0644	edit dl rm
float.h	1210	0644	edit dl rm
functional	2034	0644	edit dl rm
functional_hash.h	1912	0644	edit dl rm
gamma.tcc	14296	0644	edit dl rm
hashtable.h	1866	0644	edit dl rm
hypergeometric.tcc	27804	0644	edit dl rm
inttypes.h	1274	0644	edit dl rm
legendre_function.tcc	10561	0644	edit dl rm
limits.h	1215	0644	edit dl rm
math.h	4560	0644	edit dl rm
memory	2265	0644	edit dl rm
modified_bessel_func.tcc	15686	0644	edit dl rm
poly_hermite.tcc	3628	0644	edit dl rm
poly_laguerre.tcc	11481	0644	edit dl rm
random	2045	0644	edit dl rm
regex	1984	0644	edit dl rm
riemann_zeta.tcc	13633	0644	edit dl rm
shared_ptr.h	29180	0644	edit dl rm
special_function_util.h	4771	0644	edit dl rm
stdarg.h	1215	0644	edit dl rm
stdbool.h	1220	0644	edit dl rm
stdint.h	1215	0644	edit dl rm
stdio.h	1210	0644	edit dl rm
stdlib.h	1487	0644	edit dl rm
tgmath.h	1262	0644	edit dl rm
tuple	12049	0644	edit dl rm
type_traits	7236	0644	edit dl rm
unordered_map	2069	0644	edit dl rm
unordered_set	2069	0644	edit dl rm
utility	1858	0644	edit dl rm
wchar.h	1256	0644	edit dl rm
wctype.h	1262	0644	edit dl rm

Edit: /usr/include/c++/4.4.4/tr1/legendre_function.tcc (10561B)

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

// Copyright (C) 2006, 2007, 2008, 2009
// 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/legendre_function.tcc
 *  This is an internal header file, included by other library headers.
 *  You should not attempt to use it directly.
 */

//
// 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
{
namespace tr1
{

// [5.2] Special functions

// Implementation-space details.
  namespace __detail
  {

/**
     *   @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<typename _Tp>
    _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<typename _Tp>
    _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 <typename _Tp>
    _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;
            }
        }
    }

} // namespace std::tr1::__detail
}
}

#endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC

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