libstdc++-v3/include/tr1/legendre_function.tcc

   1 // Special functions -*- C++ -*-
   2
   3 // Copyright (C) 2006-2019 Free Software Foundation, Inc.
   4 //
   5 // This file is part of the GNU ISO C++ Library.  This library is free
   6 // software; you can redistribute it and/or modify it under the
   7 // terms of the GNU General Public License as published by the
   8 // Free Software Foundation; either version 3, or (at your option)
   9 // any later version.
  10 //
  11 // This library is distributed in the hope that it will be useful,
  12 // but WITHOUT ANY WARRANTY; without even the implied warranty of
  13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14 // GNU General Public License for more details.
  15 //
  16 // Under Section 7 of GPL version 3, you are granted additional
  17 // permissions described in the GCC Runtime Library Exception, version
  18 // 3.1, as published by the Free Software Foundation.
  19
  20 // You should have received a copy of the GNU General Public License and
  21 // a copy of the GCC Runtime Library Exception along with this program;
  22 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
  23 // <http://www.gnu.org/licenses/>.
  24
  25 /** @file tr1/legendre_function.tcc
  26  *  This is an internal header file, included by other library headers.
  27  *  Do not attempt to use it directly. @headername{tr1/cmath}
  28  */
  29
  30 //
  31 // ISO C++ 14882 TR1: 5.2  Special functions
  32 //
  33
  34 // Written by Edward Smith-Rowland based on:
  35 //   (1) Handbook of Mathematical Functions,
  36 //       ed. Milton Abramowitz and Irene A. Stegun,
  37 //       Dover Publications,
  38 //       Section 8, pp. 331-341
  39 //   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
  40 //   (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
  41 //       W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
  42 //       2nd ed, pp. 252-254
  43
  44 #ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
  45 #define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
  46
  47 #include "special_function_util.h"
  48
  49 namespace std _GLIBCXX_VISIBILITY(default)
  50 {
  51 _GLIBCXX_BEGIN_NAMESPACE_VERSION
  52
  53 #if _GLIBCXX_USE_STD_SPEC_FUNCS
  54 # define _GLIBCXX_MATH_NS ::std
  55 #elif defined(_GLIBCXX_TR1_CMATH)
  56 namespace tr1
  57 {
  58 # define _GLIBCXX_MATH_NS ::std::tr1
  59 #else
  60 # error do not include this header directly, use <cmath> or <tr1/cmath>
  61 #endif
  62   // [5.2] Special functions
  63
  64   // Implementation-space details.
  65   namespace __detail
  66   {
  67     /**
  68      *   @brief  Return the Legendre polynomial by recursion on degree
  69      *           @f$ l @f$.
  70      *
  71      *   The Legendre function of @f$ l @f$ and @f$ x @f$,
  72      *   @f$ P_l(x) @f$, is defined by:
  73      *   @f[
  74      *     P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
  75      *   @f]
  76      *
  77      *   @param  l  The degree of the Legendre polynomial.  @f$l >= 0@f$.
  78      *   @param  x  The argument of the Legendre polynomial.  @f$|x| <= 1@f$.
  79      */
  80     template<typename _Tp>
  81     _Tp
  82     __poly_legendre_p(unsigned int __l, _Tp __x)
  83     {
  84
  85       if ((__x < _Tp(-1)) || (__x > _Tp(+1)))
  86         std::__throw_domain_error(__N("Argument out of range"
  87                                       " in __poly_legendre_p."));
  88       else if (__isnan(__x))
  89         return std::numeric_limits<_Tp>::quiet_NaN();
  90       else if (__x == +_Tp(1))
  91         return +_Tp(1);
  92       else if (__x == -_Tp(1))
  93         return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
  94       else
  95         {
  96           _Tp __p_lm2 = _Tp(1);
  97           if (__l == 0)
  98             return __p_lm2;
  99
 100           _Tp __p_lm1 = __x;
 101           if (__l == 1)
 102             return __p_lm1;
 103
 104           _Tp __p_l = 0;
 105           for (unsigned int __ll = 2; __ll <= __l; ++__ll)
 106             {
 107               //  This arrangement is supposed to be better for roundoff
 108               //  protection, Arfken, 2nd Ed, Eq 12.17a.
 109               __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
 110                     - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
 111               __p_lm2 = __p_lm1;
 112               __p_lm1 = __p_l;
 113             }
 114
 115           return __p_l;
 116         }
 117     }
 118
 119
 120     /**
 121      *   @brief  Return the associated Legendre function by recursion
 122      *           on @f$ l @f$.
 123      *
 124      *   The associated Legendre function is derived from the Legendre function
 125      *   @f$ P_l(x) @f$ by the Rodrigues formula:
 126      *   @f[
 127      *     P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
 128      *   @f]
 129      *
 130      *   @param  l  The degree of the associated Legendre function.
 131      *              @f$ l >= 0 @f$.
 132      *   @param  m  The order of the associated Legendre function.
 133      *              @f$ m <= l @f$.
 134      *   @param  x  The argument of the associated Legendre function.
 135      *              @f$ |x| <= 1 @f$.
 136      *   @param  phase  The phase of the associated Legendre function.
 137      *                  Use -1 for the Condon-Shortley phase convention.
 138      */
 139     template<typename _Tp>
 140     _Tp
 141     __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
 142                        _Tp __phase = _Tp(+1))
 143     {
 144
 145       if (__x < _Tp(-1) || __x > _Tp(+1))
 146         std::__throw_domain_error(__N("Argument out of range"
 147                                       " in __assoc_legendre_p."));
 148       else if (__m > __l)
 149         std::__throw_domain_error(__N("Degree out of range"
 150                                       " in __assoc_legendre_p."));
 151       else if (__isnan(__x))
 152         return std::numeric_limits<_Tp>::quiet_NaN();
 153       else if (__m == 0)
 154         return __poly_legendre_p(__l, __x);
 155       else
 156         {
 157           _Tp __p_mm = _Tp(1);
 158           if (__m > 0)
 159             {
 160               //  Two square roots seem more accurate more of the time
 161               //  than just one.
 162               _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
 163               _Tp __fact = _Tp(1);
 164               for (unsigned int __i = 1; __i <= __m; ++__i)
 165                 {
 166                   __p_mm *= __phase * __fact * __root;
 167                   __fact += _Tp(2);
 168                 }
 169             }
 170           if (__l == __m)
 171             return __p_mm;
 172
 173           _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
 174           if (__l == __m + 1)
 175             return __p_mp1m;
 176
 177           _Tp __p_lm2m = __p_mm;
 178           _Tp __P_lm1m = __p_mp1m;
 179           _Tp __p_lm = _Tp(0);
 180           for (unsigned int __j = __m + 2; __j <= __l; ++__j)
 181             {
 182               __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
 183                       - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
 184               __p_lm2m = __P_lm1m;
 185               __P_lm1m = __p_lm;
 186             }
 187
 188           return __p_lm;
 189         }
 190     }
 191
 192
 193     /**
 194      *   @brief  Return the spherical associated Legendre function.
 195      *
 196      *   The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
 197      *   and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
 198      *   @f[
 199      *      Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
 200      *                                  \frac{(l-m)!}{(l+m)!}]
 201      *                     P_l^m(\cos\theta) \exp^{im\phi}
 202      *   @f]
 203      *   is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
 204      *   associated Legendre function.
 205      *
 206      *   This function differs from the associated Legendre function by
 207      *   argument (@f$x = \cos(\theta)@f$) and by a normalization factor
 208      *   but this factor is rather large for large @f$ l @f$ and @f$ m @f$
 209      *   and so this function is stable for larger differences of @f$ l @f$
 210      *   and @f$ m @f$.
 211      *   @note Unlike the case for __assoc_legendre_p the Condon-Shortley
 212      *   phase factor @f$ (-1)^m @f$ is present here.
 213      *
 214      *   @param  l  The degree of the spherical associated Legendre function.
 215      *              @f$ l >= 0 @f$.
 216      *   @param  m  The order of the spherical associated Legendre function.
 217      *              @f$ m <= l @f$.
 218      *   @param  theta  The radian angle argument of the spherical associated
 219      *                  Legendre function.
 220      */
 221     template <typename _Tp>
 222     _Tp
 223     __sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
 224     {
 225       if (__isnan(__theta))
 226         return std::numeric_limits<_Tp>::quiet_NaN();
 227
 228       const _Tp __x = std::cos(__theta);
 229
 230       if (__l < __m)
 231         {
 232           std::__throw_domain_error(__N("Bad argument "
 233                                         "in __sph_legendre."));
 234         }
 235       else if (__m == 0)
 236         {
 237           _Tp __P = __poly_legendre_p(__l, __x);
 238           _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
 239                      / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
 240           __P *= __fact;
 241           return __P;
 242         }
 243       else if (__x == _Tp(1) || __x == -_Tp(1))
 244         {
 245           //  m > 0 here
 246           return _Tp(0);
 247         }
 248       else
 249         {
 250           // m > 0 and |x| < 1 here
 251
 252           // Starting value for recursion.
 253           // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
 254           //             (-1)^m (1-x^2)^(m/2) / pi^(1/4)
 255           const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
 256           const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
 257 #if _GLIBCXX_USE_C99_MATH_TR1
 258           const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
 259 #else
 260           const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
 261 #endif
 262           //  Gamma(m+1/2) / Gamma(m)
 263 #if _GLIBCXX_USE_C99_MATH_TR1
 264           const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
 265                              - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
 266 #else
 267           const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
 268                              - __log_gamma(_Tp(__m));
 269 #endif
 270           const _Tp __lnpre_val =
 271                     -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
 272                     + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
 273           const _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
 274                          / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
 275           _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
 276           _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
 277
 278           if (__l == __m)
 279             return __y_mm;
 280           else if (__l == __m + 1)
 281             return __y_mp1m;
 282           else
 283             {
 284               _Tp __y_lm = _Tp(0);
 285
 286               // Compute Y_l^m, l > m+1, upward recursion on l.
 287               for (unsigned int __ll = __m + 2; __ll <= __l; ++__ll)
 288                 {
 289                   const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
 290                   const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
 291                   const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
 292                                                        * _Tp(2 * __ll - 1));
 293                   const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
 294                                                                 / _Tp(2 * __ll - 3));
 295                   __y_lm = (__x * __y_mp1m * __fact1
 296                          - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
 297                   __y_mm = __y_mp1m;
 298                   __y_mp1m = __y_lm;
 299                 }
 300
 301               return __y_lm;
 302             }
 303         }
 304     }
 305   } // namespace __detail
 306 #undef _GLIBCXX_MATH_NS
 307 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
 308 } // namespace tr1
 309 #endif
 310
 311 _GLIBCXX_END_NAMESPACE_VERSION
 312 }
 313
 314 #endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC