| 1 | // Special functions -*- C++ -*-
|
|---|
| 2 |
|
|---|
| 3 | // Copyright (C) 2006-2021 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 <tr1/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$.
|
|---|
| 79 | */
|
|---|
| 80 | template<typename _Tp>
|
|---|
| 81 | _Tp
|
|---|
| 82 | __poly_legendre_p(unsigned int __l, _Tp __x)
|
|---|
| 83 | {
|
|---|
| 84 |
|
|---|
| 85 | if (__isnan(__x))
|
|---|
| 86 | return std::numeric_limits<_Tp>::quiet_NaN();
|
|---|
| 87 | else if (__x == +_Tp(1))
|
|---|
| 88 | return +_Tp(1);
|
|---|
| 89 | else if (__x == -_Tp(1))
|
|---|
| 90 | return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
|
|---|
| 91 | else
|
|---|
| 92 | {
|
|---|
| 93 | _Tp __p_lm2 = _Tp(1);
|
|---|
| 94 | if (__l == 0)
|
|---|
| 95 | return __p_lm2;
|
|---|
| 96 |
|
|---|
| 97 | _Tp __p_lm1 = __x;
|
|---|
| 98 | if (__l == 1)
|
|---|
| 99 | return __p_lm1;
|
|---|
| 100 |
|
|---|
| 101 | _Tp __p_l = 0;
|
|---|
| 102 | for (unsigned int __ll = 2; __ll <= __l; ++__ll)
|
|---|
| 103 | {
|
|---|
| 104 | // This arrangement is supposed to be better for roundoff
|
|---|
| 105 | // protection, Arfken, 2nd Ed, Eq 12.17a.
|
|---|
| 106 | __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
|
|---|
| 107 | - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
|
|---|
| 108 | __p_lm2 = __p_lm1;
|
|---|
| 109 | __p_lm1 = __p_l;
|
|---|
| 110 | }
|
|---|
| 111 |
|
|---|
| 112 | return __p_l;
|
|---|
| 113 | }
|
|---|
| 114 | }
|
|---|
| 115 |
|
|---|
| 116 |
|
|---|
| 117 | /**
|
|---|
| 118 | * @brief Return the associated Legendre function by recursion
|
|---|
| 119 | * on @f$ l @f$.
|
|---|
| 120 | *
|
|---|
| 121 | * The associated Legendre function is derived from the Legendre function
|
|---|
| 122 | * @f$ P_l(x) @f$ by the Rodrigues formula:
|
|---|
| 123 | * @f[
|
|---|
| 124 | * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
|
|---|
| 125 | * @f]
|
|---|
| 126 | * @note @f$ P_l^m(x) = 0 @f$ if @f$ m > l @f$.
|
|---|
| 127 | *
|
|---|
| 128 | * @param l The degree of the associated Legendre function.
|
|---|
| 129 | * @f$ l >= 0 @f$.
|
|---|
| 130 | * @param m The order of the associated Legendre function.
|
|---|
| 131 | * @param x The argument of the associated Legendre function.
|
|---|
| 132 | * @f$ |x| <= 1 @f$.
|
|---|
| 133 | * @param phase The phase of the associated Legendre function.
|
|---|
| 134 | * Use -1 for the Condon-Shortley phase convention.
|
|---|
| 135 | */
|
|---|
| 136 | template<typename _Tp>
|
|---|
| 137 | _Tp
|
|---|
| 138 | __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
|
|---|
| 139 | _Tp __phase = _Tp(+1))
|
|---|
| 140 | {
|
|---|
| 141 |
|
|---|
| 142 | if (__m > __l)
|
|---|
| 143 | return _Tp(0);
|
|---|
| 144 | else if (__isnan(__x))
|
|---|
| 145 | return std::numeric_limits<_Tp>::quiet_NaN();
|
|---|
| 146 | else if (__m == 0)
|
|---|
| 147 | return __poly_legendre_p(__l, __x);
|
|---|
| 148 | else
|
|---|
| 149 | {
|
|---|
| 150 | _Tp __p_mm = _Tp(1);
|
|---|
| 151 | if (__m > 0)
|
|---|
| 152 | {
|
|---|
| 153 | // Two square roots seem more accurate more of the time
|
|---|
| 154 | // than just one.
|
|---|
| 155 | _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
|
|---|
| 156 | _Tp __fact = _Tp(1);
|
|---|
| 157 | for (unsigned int __i = 1; __i <= __m; ++__i)
|
|---|
| 158 | {
|
|---|
| 159 | __p_mm *= __phase * __fact * __root;
|
|---|
| 160 | __fact += _Tp(2);
|
|---|
| 161 | }
|
|---|
| 162 | }
|
|---|
| 163 | if (__l == __m)
|
|---|
| 164 | return __p_mm;
|
|---|
| 165 |
|
|---|
| 166 | _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
|
|---|
| 167 | if (__l == __m + 1)
|
|---|
| 168 | return __p_mp1m;
|
|---|
| 169 |
|
|---|
| 170 | _Tp __p_lm2m = __p_mm;
|
|---|
| 171 | _Tp __P_lm1m = __p_mp1m;
|
|---|
| 172 | _Tp __p_lm = _Tp(0);
|
|---|
| 173 | for (unsigned int __j = __m + 2; __j <= __l; ++__j)
|
|---|
| 174 | {
|
|---|
| 175 | __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
|
|---|
| 176 | - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
|
|---|
| 177 | __p_lm2m = __P_lm1m;
|
|---|
| 178 | __P_lm1m = __p_lm;
|
|---|
| 179 | }
|
|---|
| 180 |
|
|---|
| 181 | return __p_lm;
|
|---|
| 182 | }
|
|---|
| 183 | }
|
|---|
| 184 |
|
|---|
| 185 |
|
|---|
| 186 | /**
|
|---|
| 187 | * @brief Return the spherical associated Legendre function.
|
|---|
| 188 | *
|
|---|
| 189 | * The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
|
|---|
| 190 | * and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
|
|---|
| 191 | * @f[
|
|---|
| 192 | * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
|
|---|
| 193 | * \frac{(l-m)!}{(l+m)!}]
|
|---|
| 194 | * P_l^m(\cos\theta) \exp^{im\phi}
|
|---|
| 195 | * @f]
|
|---|
| 196 | * is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
|
|---|
| 197 | * associated Legendre function.
|
|---|
| 198 | *
|
|---|
| 199 | * This function differs from the associated Legendre function by
|
|---|
| 200 | * argument (@f$x = \cos(\theta)@f$) and by a normalization factor
|
|---|
| 201 | * but this factor is rather large for large @f$ l @f$ and @f$ m @f$
|
|---|
| 202 | * and so this function is stable for larger differences of @f$ l @f$
|
|---|
| 203 | * and @f$ m @f$.
|
|---|
| 204 | * @note Unlike the case for __assoc_legendre_p the Condon-Shortley
|
|---|
| 205 | * phase factor @f$ (-1)^m @f$ is present here.
|
|---|
| 206 | * @note @f$ Y_l^m(\theta) = 0 @f$ if @f$ m > l @f$.
|
|---|
| 207 | *
|
|---|
| 208 | * @param l The degree of the spherical associated Legendre function.
|
|---|
| 209 | * @f$ l >= 0 @f$.
|
|---|
| 210 | * @param m The order of the spherical associated Legendre function.
|
|---|
| 211 | * @param theta The radian angle argument of the spherical associated
|
|---|
| 212 | * Legendre function.
|
|---|
| 213 | */
|
|---|
| 214 | template <typename _Tp>
|
|---|
| 215 | _Tp
|
|---|
| 216 | __sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
|
|---|
| 217 | {
|
|---|
| 218 | if (__isnan(__theta))
|
|---|
| 219 | return std::numeric_limits<_Tp>::quiet_NaN();
|
|---|
| 220 |
|
|---|
| 221 | const _Tp __x = std::cos(__theta);
|
|---|
| 222 |
|
|---|
| 223 | if (__m > __l)
|
|---|
| 224 | return _Tp(0);
|
|---|
| 225 | else if (__m == 0)
|
|---|
| 226 | {
|
|---|
| 227 | _Tp __P = __poly_legendre_p(__l, __x);
|
|---|
| 228 | _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
|
|---|
| 229 | / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
|
|---|
| 230 | __P *= __fact;
|
|---|
| 231 | return __P;
|
|---|
| 232 | }
|
|---|
| 233 | else if (__x == _Tp(1) || __x == -_Tp(1))
|
|---|
| 234 | {
|
|---|
| 235 | // m > 0 here
|
|---|
| 236 | return _Tp(0);
|
|---|
| 237 | }
|
|---|
| 238 | else
|
|---|
| 239 | {
|
|---|
| 240 | // m > 0 and |x| < 1 here
|
|---|
| 241 |
|
|---|
| 242 | // Starting value for recursion.
|
|---|
| 243 | // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
|
|---|
| 244 | // (-1)^m (1-x^2)^(m/2) / pi^(1/4)
|
|---|
| 245 | const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
|
|---|
| 246 | const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
|
|---|
| 247 | #if _GLIBCXX_USE_C99_MATH_TR1
|
|---|
| 248 | const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
|
|---|
| 249 | #else
|
|---|
| 250 | const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
|
|---|
| 251 | #endif
|
|---|
| 252 | // Gamma(m+1/2) / Gamma(m)
|
|---|
| 253 | #if _GLIBCXX_USE_C99_MATH_TR1
|
|---|
| 254 | const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
|
|---|
| 255 | - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
|
|---|
| 256 | #else
|
|---|
| 257 | const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
|
|---|
| 258 | - __log_gamma(_Tp(__m));
|
|---|
| 259 | #endif
|
|---|
| 260 | const _Tp __lnpre_val =
|
|---|
| 261 | -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
|
|---|
| 262 | + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
|
|---|
| 263 | const _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
|
|---|
| 264 | / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
|
|---|
| 265 | _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
|
|---|
| 266 | _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
|
|---|
| 267 |
|
|---|
| 268 | if (__l == __m)
|
|---|
| 269 | return __y_mm;
|
|---|
| 270 | else if (__l == __m + 1)
|
|---|
| 271 | return __y_mp1m;
|
|---|
| 272 | else
|
|---|
| 273 | {
|
|---|
| 274 | _Tp __y_lm = _Tp(0);
|
|---|
| 275 |
|
|---|
| 276 | // Compute Y_l^m, l > m+1, upward recursion on l.
|
|---|
| 277 | for (unsigned int __ll = __m + 2; __ll <= __l; ++__ll)
|
|---|
| 278 | {
|
|---|
| 279 | const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
|
|---|
| 280 | const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
|
|---|
| 281 | const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
|
|---|
| 282 | * _Tp(2 * __ll - 1));
|
|---|
| 283 | const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
|
|---|
| 284 | / _Tp(2 * __ll - 3));
|
|---|
| 285 | __y_lm = (__x * __y_mp1m * __fact1
|
|---|
| 286 | - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
|
|---|
| 287 | __y_mm = __y_mp1m;
|
|---|
| 288 | __y_mp1m = __y_lm;
|
|---|
| 289 | }
|
|---|
| 290 |
|
|---|
| 291 | return __y_lm;
|
|---|
| 292 | }
|
|---|
| 293 | }
|
|---|
| 294 | }
|
|---|
| 295 | } // namespace __detail
|
|---|
| 296 | #undef _GLIBCXX_MATH_NS
|
|---|
| 297 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
|
|---|
| 298 | } // namespace tr1
|
|---|
| 299 | #endif
|
|---|
| 300 |
|
|---|
| 301 | _GLIBCXX_END_NAMESPACE_VERSION
|
|---|
| 302 | }
|
|---|
| 303 |
|
|---|
| 304 | #endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
|
|---|
