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

   1 // Special functions -*- C++ -*-
   2
   3 // Copyright (C) 2006-2017 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 order
  69      *           @f$ l @f$.
  70      *
  71      *   The Legendre function of @f$ l @f$ and @f$ x @f$,
  72      *   @f$ P_l(x) @f$, is defined by:
  73      *   @f[
  74      *     P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
  75      *   @f]
  76      *
  77      *   @param  l  The order of the Legendre polynomial.  @f$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 order 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      */
 137     template<typename _Tp>
 138     _Tp
 139     __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x)
 140     {
 141
 142       if (__x < _Tp(-1) || __x > _Tp(+1))
 143         std::__throw_domain_error(__N("Argument out of range"
 144                                       " in __assoc_legendre_p."));
 145       else if (__m > __l)
 146         std::__throw_domain_error(__N("Degree out of range"
 147                                       " in __assoc_legendre_p."));
 148       else if (__isnan(__x))
 149         return std::numeric_limits<_Tp>::quiet_NaN();
 150       else if (__m == 0)
 151         return __poly_legendre_p(__l, __x);
 152       else
 153         {
 154           _Tp __p_mm = _Tp(1);
 155           if (__m > 0)
 156             {
 157               //  Two square roots seem more accurate more of the time
 158               //  than just one.
 159               _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
 160               _Tp __fact = _Tp(1);
 161               for (unsigned int __i = 1; __i <= __m; ++__i)
 162                 {
 163                   __p_mm *= -__fact * __root;
 164                   __fact += _Tp(2);
 165                 }
 166             }
 167           if (__l == __m)
 168             return __p_mm;
 169
 170           _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
 171           if (__l == __m + 1)
 172             return __p_mp1m;
 173
 174           _Tp __p_lm2m = __p_mm;
 175           _Tp __P_lm1m = __p_mp1m;
 176           _Tp __p_lm = _Tp(0);
 177           for (unsigned int __j = __m + 2; __j <= __l; ++__j)
 178             {
 179               __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
 180                       - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
 181               __p_lm2m = __P_lm1m;
 182               __P_lm1m = __p_lm;
 183             }
 184
 185           return __p_lm;
 186         }
 187     }
 188
 189
 190     /**
 191      *   @brief  Return the spherical associated Legendre function.
 192      *
 193      *   The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
 194      *   and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
 195      *   @f[
 196      *      Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
 197      *                                  \frac{(l-m)!}{(l+m)!}]
 198      *                     P_l^m(\cos\theta) \exp^{im\phi}
 199      *   @f]
 200      *   is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
 201      *   associated Legendre function.
 202      *
 203      *   This function differs from the associated Legendre function by
 204      *   argument (@f$x = \cos(\theta)@f$) and by a normalization factor
 205      *   but this factor is rather large for large @f$ l @f$ and @f$ m @f$
 206      *   and so this function is stable for larger differences of @f$ l @f$
 207      *   and @f$ m @f$.
 208      *
 209      *   @param  l  The order of the spherical associated Legendre function.
 210      *              @f$ l >= 0 @f$.
 211      *   @param  m  The order of the spherical associated Legendre function.
 212      *              @f$ m <= l @f$.
 213      *   @param  theta  The radian angle argument of the spherical associated
 214      *                  Legendre function.
 215      */
 216     template <typename _Tp>
 217     _Tp
 218     __sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
 219     {
 220       if (__isnan(__theta))
 221         return std::numeric_limits<_Tp>::quiet_NaN();
 222
 223       const _Tp __x = std::cos(__theta);
 224
 225       if (__l < __m)
 226         {
 227           std::__throw_domain_error(__N("Bad argument "
 228                                         "in __sph_legendre."));
 229         }
 230       else if (__m == 0)
 231         {
 232           _Tp __P = __poly_legendre_p(__l, __x);
 233           _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
 234                      / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
 235           __P *= __fact;
 236           return __P;
 237         }
 238       else if (__x == _Tp(1) || __x == -_Tp(1))
 239         {
 240           //  m > 0 here
 241           return _Tp(0);
 242         }
 243       else
 244         {
 245           // m > 0 and |x| < 1 here
 246
 247           // Starting value for recursion.
 248           // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
 249           //             (-1)^m (1-x^2)^(m/2) / pi^(1/4)
 250           const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
 251           const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
 252 #if _GLIBCXX_USE_C99_MATH_TR1
 253           const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
 254 #else
 255           const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
 256 #endif
 257           //  Gamma(m+1/2) / Gamma(m)
 258 #if _GLIBCXX_USE_C99_MATH_TR1
 259           const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
 260                              - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
 261 #else
 262           const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
 263                              - __log_gamma(_Tp(__m));
 264 #endif
 265           const _Tp __lnpre_val =
 266                     -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
 267                     + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
 268           _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
 269                    / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
 270           _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
 271           _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
 272
 273           if (__l == __m)
 274             {
 275               return __y_mm;
 276             }
 277           else if (__l == __m + 1)
 278             {
 279               return __y_mp1m;
 280             }
 281           else
 282             {
 283               _Tp __y_lm = _Tp(0);
 284
 285               // Compute Y_l^m, l > m+1, upward recursion on l.
 286               for ( int __ll = __m + 2; __ll <= __l; ++__ll)
 287                 {
 288                   const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
 289                   const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
 290                   const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
 291                                                        * _Tp(2 * __ll - 1));
 292                   const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
 293                                                                 / _Tp(2 * __ll - 3));
 294                   __y_lm = (__x * __y_mp1m * __fact1
 295                          - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
 296                   __y_mm = __y_mp1m;
 297                   __y_mp1m = __y_lm;
 298                 }
 299
 300               return __y_lm;
 301             }
 302         }
 303     }
 304   } // namespace __detail
 305 #undef _GLIBCXX_MATH_NS
 306 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
 307 } // namespace tr1
 308 #endif
 309
 310 _GLIBCXX_END_NAMESPACE_VERSION
 311 }
 312
 313 #endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC