| [1166] | 1 | // Special functions -*- C++ -*-
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| 2 |
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| 3 | // Copyright (C) 2006-2021 Free Software Foundation, Inc.
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| 4 | //
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| 5 | // This file is part of the GNU ISO C++ Library. This library is free
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| 6 | // software; you can redistribute it and/or modify it under the
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| 7 | // terms of the GNU General Public License as published by the
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| 8 | // Free Software Foundation; either version 3, or (at your option)
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| 9 | // any later version.
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| 10 | //
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| 11 | // This library is distributed in the hope that it will be useful,
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| 12 | // but WITHOUT ANY WARRANTY; without even the implied warranty of
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| 13 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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| 14 | // GNU General Public License for more details.
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| 15 | //
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| 16 | // Under Section 7 of GPL version 3, you are granted additional
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| 17 | // permissions described in the GCC Runtime Library Exception, version
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| 18 | // 3.1, as published by the Free Software Foundation.
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| 19 |
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| 20 | // You should have received a copy of the GNU General Public License and
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| 21 | // a copy of the GCC Runtime Library Exception along with this program;
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| 22 | // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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| 23 | // <http://www.gnu.org/licenses/>.
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| 24 |
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| 25 | /** @file tr1/legendre_function.tcc
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| 26 | * This is an internal header file, included by other library headers.
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| 27 | * Do not attempt to use it directly. @headername{tr1/cmath}
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| 28 | */
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| 29 |
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| 30 | //
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| 31 | // ISO C++ 14882 TR1: 5.2 Special functions
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| 32 | //
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| 33 |
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| 34 | // Written by Edward Smith-Rowland based on:
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| 35 | // (1) Handbook of Mathematical Functions,
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| 36 | // ed. Milton Abramowitz and Irene A. Stegun,
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| 37 | // Dover Publications,
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| 38 | // Section 8, pp. 331-341
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| 39 | // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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| 40 | // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
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| 41 | // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
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| 42 | // 2nd ed, pp. 252-254
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| 43 |
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| 44 | #ifndef _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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| 45 | #define _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC 1
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| 46 |
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| 47 | #include <tr1/special_function_util.h>
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| 48 |
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| 49 | namespace std _GLIBCXX_VISIBILITY(default)
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| 50 | {
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| 51 | _GLIBCXX_BEGIN_NAMESPACE_VERSION
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| 52 |
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| 53 | #if _GLIBCXX_USE_STD_SPEC_FUNCS
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| 54 | # define _GLIBCXX_MATH_NS ::std
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| 55 | #elif defined(_GLIBCXX_TR1_CMATH)
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| 56 | namespace tr1
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| 57 | {
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| 58 | # define _GLIBCXX_MATH_NS ::std::tr1
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| 59 | #else
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| 60 | # error do not include this header directly, use <cmath> or <tr1/cmath>
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| 61 | #endif
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| 62 | // [5.2] Special functions
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| 63 |
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| 64 | // Implementation-space details.
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| 65 | namespace __detail
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| 66 | {
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| 67 | /**
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| 68 | * @brief Return the Legendre polynomial by recursion on degree
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| 69 | * @f$ l @f$.
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| 70 | *
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| 71 | * The Legendre function of @f$ l @f$ and @f$ x @f$,
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| 72 | * @f$ P_l(x) @f$, is defined by:
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| 73 | * @f[
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| 74 | * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}
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| 75 | * @f]
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| 76 | *
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| 77 | * @param l The degree of the Legendre polynomial. @f$l >= 0@f$.
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| 78 | * @param x The argument of the Legendre polynomial. @f$|x| <= 1@f$.
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| 79 | */
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| 80 | template<typename _Tp>
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| 81 | _Tp
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| 82 | __poly_legendre_p(unsigned int __l, _Tp __x)
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| 83 | {
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| 84 |
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| 85 | if (__isnan(__x))
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| 86 | return std::numeric_limits<_Tp>::quiet_NaN();
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| 87 | else if (__x == +_Tp(1))
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| 88 | return +_Tp(1);
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| 89 | else if (__x == -_Tp(1))
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| 90 | return (__l % 2 == 1 ? -_Tp(1) : +_Tp(1));
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| 91 | else
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| 92 | {
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| 93 | _Tp __p_lm2 = _Tp(1);
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| 94 | if (__l == 0)
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| 95 | return __p_lm2;
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| 96 |
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| 97 | _Tp __p_lm1 = __x;
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| 98 | if (__l == 1)
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| 99 | return __p_lm1;
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| 100 |
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| 101 | _Tp __p_l = 0;
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| 102 | for (unsigned int __ll = 2; __ll <= __l; ++__ll)
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| 103 | {
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| 104 | // This arrangement is supposed to be better for roundoff
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| 105 | // protection, Arfken, 2nd Ed, Eq 12.17a.
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| 106 | __p_l = _Tp(2) * __x * __p_lm1 - __p_lm2
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| 107 | - (__x * __p_lm1 - __p_lm2) / _Tp(__ll);
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| 108 | __p_lm2 = __p_lm1;
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| 109 | __p_lm1 = __p_l;
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| 110 | }
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| 111 |
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| 112 | return __p_l;
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| 113 | }
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| 114 | }
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| 115 |
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| 116 |
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| 117 | /**
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| 118 | * @brief Return the associated Legendre function by recursion
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| 119 | * on @f$ l @f$.
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| 120 | *
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| 121 | * The associated Legendre function is derived from the Legendre function
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| 122 | * @f$ P_l(x) @f$ by the Rodrigues formula:
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| 123 | * @f[
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| 124 | * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
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| 125 | * @f]
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| 126 | * @note @f$ P_l^m(x) = 0 @f$ if @f$ m > l @f$.
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| 127 | *
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| 128 | * @param l The degree of the associated Legendre function.
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| 129 | * @f$ l >= 0 @f$.
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| 130 | * @param m The order of the associated Legendre function.
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| 131 | * @param x The argument of the associated Legendre function.
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| 132 | * @f$ |x| <= 1 @f$.
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| 133 | * @param phase The phase of the associated Legendre function.
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| 134 | * Use -1 for the Condon-Shortley phase convention.
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| 135 | */
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| 136 | template<typename _Tp>
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| 137 | _Tp
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| 138 | __assoc_legendre_p(unsigned int __l, unsigned int __m, _Tp __x,
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| 139 | _Tp __phase = _Tp(+1))
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| 140 | {
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| 141 |
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| 142 | if (__m > __l)
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| 143 | return _Tp(0);
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| 144 | else if (__isnan(__x))
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| 145 | return std::numeric_limits<_Tp>::quiet_NaN();
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| 146 | else if (__m == 0)
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| 147 | return __poly_legendre_p(__l, __x);
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| 148 | else
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| 149 | {
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| 150 | _Tp __p_mm = _Tp(1);
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| 151 | if (__m > 0)
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| 152 | {
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| 153 | // Two square roots seem more accurate more of the time
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| 154 | // than just one.
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| 155 | _Tp __root = std::sqrt(_Tp(1) - __x) * std::sqrt(_Tp(1) + __x);
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| 156 | _Tp __fact = _Tp(1);
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| 157 | for (unsigned int __i = 1; __i <= __m; ++__i)
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| 158 | {
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| 159 | __p_mm *= __phase * __fact * __root;
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| 160 | __fact += _Tp(2);
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| 161 | }
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| 162 | }
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| 163 | if (__l == __m)
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| 164 | return __p_mm;
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| 165 |
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| 166 | _Tp __p_mp1m = _Tp(2 * __m + 1) * __x * __p_mm;
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| 167 | if (__l == __m + 1)
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| 168 | return __p_mp1m;
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| 169 |
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| 170 | _Tp __p_lm2m = __p_mm;
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| 171 | _Tp __P_lm1m = __p_mp1m;
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| 172 | _Tp __p_lm = _Tp(0);
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| 173 | for (unsigned int __j = __m + 2; __j <= __l; ++__j)
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| 174 | {
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| 175 | __p_lm = (_Tp(2 * __j - 1) * __x * __P_lm1m
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| 176 | - _Tp(__j + __m - 1) * __p_lm2m) / _Tp(__j - __m);
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| 177 | __p_lm2m = __P_lm1m;
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| 178 | __P_lm1m = __p_lm;
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| 179 | }
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| 180 |
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| 181 | return __p_lm;
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| 182 | }
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| 183 | }
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| 184 |
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| 185 |
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| 186 | /**
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| 187 | * @brief Return the spherical associated Legendre function.
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| 188 | *
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| 189 | * The spherical associated Legendre function of @f$ l @f$, @f$ m @f$,
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| 190 | * and @f$ \theta @f$ is defined as @f$ Y_l^m(\theta,0) @f$ where
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| 191 | * @f[
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| 192 | * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}
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| 193 | * \frac{(l-m)!}{(l+m)!}]
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| 194 | * P_l^m(\cos\theta) \exp^{im\phi}
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| 195 | * @f]
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| 196 | * is the spherical harmonic function and @f$ P_l^m(x) @f$ is the
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| 197 | * associated Legendre function.
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| 198 | *
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| 199 | * This function differs from the associated Legendre function by
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| 200 | * argument (@f$x = \cos(\theta)@f$) and by a normalization factor
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| 201 | * but this factor is rather large for large @f$ l @f$ and @f$ m @f$
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| 202 | * and so this function is stable for larger differences of @f$ l @f$
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| 203 | * and @f$ m @f$.
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| 204 | * @note Unlike the case for __assoc_legendre_p the Condon-Shortley
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| 205 | * phase factor @f$ (-1)^m @f$ is present here.
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| 206 | * @note @f$ Y_l^m(\theta) = 0 @f$ if @f$ m > l @f$.
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| 207 | *
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| 208 | * @param l The degree of the spherical associated Legendre function.
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| 209 | * @f$ l >= 0 @f$.
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| 210 | * @param m The order of the spherical associated Legendre function.
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| 211 | * @param theta The radian angle argument of the spherical associated
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| 212 | * Legendre function.
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| 213 | */
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| 214 | template <typename _Tp>
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| 215 | _Tp
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| 216 | __sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
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| 217 | {
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| 218 | if (__isnan(__theta))
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| 219 | return std::numeric_limits<_Tp>::quiet_NaN();
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| 220 |
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| 221 | const _Tp __x = std::cos(__theta);
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| 222 |
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| 223 | if (__m > __l)
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| 224 | return _Tp(0);
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| 225 | else if (__m == 0)
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| 226 | {
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| 227 | _Tp __P = __poly_legendre_p(__l, __x);
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| 228 | _Tp __fact = std::sqrt(_Tp(2 * __l + 1)
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| 229 | / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
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| 230 | __P *= __fact;
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| 231 | return __P;
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| 232 | }
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| 233 | else if (__x == _Tp(1) || __x == -_Tp(1))
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| 234 | {
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| 235 | // m > 0 here
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| 236 | return _Tp(0);
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| 237 | }
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| 238 | else
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| 239 | {
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| 240 | // m > 0 and |x| < 1 here
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| 241 |
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| 242 | // Starting value for recursion.
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| 243 | // Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) )
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| 244 | // (-1)^m (1-x^2)^(m/2) / pi^(1/4)
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| 245 | const _Tp __sgn = ( __m % 2 == 1 ? -_Tp(1) : _Tp(1));
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| 246 | const _Tp __y_mp1m_factor = __x * std::sqrt(_Tp(2 * __m + 3));
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| 247 | #if _GLIBCXX_USE_C99_MATH_TR1
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| 248 | const _Tp __lncirc = _GLIBCXX_MATH_NS::log1p(-__x * __x);
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| 249 | #else
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| 250 | const _Tp __lncirc = std::log(_Tp(1) - __x * __x);
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| 251 | #endif
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| 252 | // Gamma(m+1/2) / Gamma(m)
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| 253 | #if _GLIBCXX_USE_C99_MATH_TR1
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| 254 | const _Tp __lnpoch = _GLIBCXX_MATH_NS::lgamma(_Tp(__m + _Tp(0.5L)))
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| 255 | - _GLIBCXX_MATH_NS::lgamma(_Tp(__m));
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| 256 | #else
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| 257 | const _Tp __lnpoch = __log_gamma(_Tp(__m + _Tp(0.5L)))
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| 258 | - __log_gamma(_Tp(__m));
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| 259 | #endif
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| 260 | const _Tp __lnpre_val =
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| 261 | -_Tp(0.25L) * __numeric_constants<_Tp>::__lnpi()
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| 262 | + _Tp(0.5L) * (__lnpoch + __m * __lncirc);
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| 263 | const _Tp __sr = std::sqrt((_Tp(2) + _Tp(1) / __m)
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| 264 | / (_Tp(4) * __numeric_constants<_Tp>::__pi()));
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| 265 | _Tp __y_mm = __sgn * __sr * std::exp(__lnpre_val);
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| 266 | _Tp __y_mp1m = __y_mp1m_factor * __y_mm;
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| 267 |
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| 268 | if (__l == __m)
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| 269 | return __y_mm;
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| 270 | else if (__l == __m + 1)
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| 271 | return __y_mp1m;
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| 272 | else
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| 273 | {
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| 274 | _Tp __y_lm = _Tp(0);
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| 275 |
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| 276 | // Compute Y_l^m, l > m+1, upward recursion on l.
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| 277 | for (unsigned int __ll = __m + 2; __ll <= __l; ++__ll)
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| 278 | {
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| 279 | const _Tp __rat1 = _Tp(__ll - __m) / _Tp(__ll + __m);
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| 280 | const _Tp __rat2 = _Tp(__ll - __m - 1) / _Tp(__ll + __m - 1);
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| 281 | const _Tp __fact1 = std::sqrt(__rat1 * _Tp(2 * __ll + 1)
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| 282 | * _Tp(2 * __ll - 1));
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| 283 | const _Tp __fact2 = std::sqrt(__rat1 * __rat2 * _Tp(2 * __ll + 1)
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| 284 | / _Tp(2 * __ll - 3));
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| 285 | __y_lm = (__x * __y_mp1m * __fact1
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| 286 | - (__ll + __m - 1) * __y_mm * __fact2) / _Tp(__ll - __m);
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| 287 | __y_mm = __y_mp1m;
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| 288 | __y_mp1m = __y_lm;
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| 289 | }
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| 290 |
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| 291 | return __y_lm;
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| 292 | }
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| 293 | }
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| 294 | }
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| 295 | } // namespace __detail
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| 296 | #undef _GLIBCXX_MATH_NS
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| 297 | #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
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| 298 | } // namespace tr1
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| 299 | #endif
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| 300 |
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| 301 | _GLIBCXX_END_NAMESPACE_VERSION
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| 302 | }
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| 303 |
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| 304 | #endif // _GLIBCXX_TR1_LEGENDRE_FUNCTION_TCC
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