// Special functions -*- C++ -*- // Copyright (C) 2006-2020 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/poly_laguerre.tcc * This is an internal header file, included by other library headers. * Do not attempt to use it directly. @headername{tr1/cmath} */ // // 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 13, pp. 509-510, Section 22 pp. 773-802 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl #ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC #define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION #if _GLIBCXX_USE_STD_SPEC_FUNCS # define _GLIBCXX_MATH_NS ::std #elif defined(_GLIBCXX_TR1_CMATH) namespace tr1 { # define _GLIBCXX_MATH_NS ::std::tr1 #else # error do not include this header directly, use <cmath> or <tr1/cmath> #endif // [5.2] Special functions // Implementation-space details. namespace __detail { /** * @brief This routine returns the associated Laguerre polynomial * of order @f$ n @f$, degree @f$ \alpha @f$ for large n. * Abramowitz & Stegun, 13.5.21 * * @param __n The order of the Laguerre function. * @param __alpha The degree of the Laguerre function. * @param __x The argument of the Laguerre function. * @return The value of the Laguerre function of order n, * degree @f$ \alpha @f$, and argument x. * * This is from the GNU Scientific Library. */ template<typename _Tpa, typename _Tp> _Tp __poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) { const _Tp __a = -_Tp(__n); const _Tp __b = _Tp(__alpha1) + _Tp(1); const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; const _Tp __cos2th = __x / __eta; const _Tp __sin2th = _Tp(1) - __cos2th; const _Tp __th = std::acos(std::sqrt(__cos2th)); const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() * __numeric_constants<_Tp>::__pi_2() * __eta * __eta * __cos2th * __sin2th; #if _GLIBCXX_USE_C99_MATH_TR1 const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b); const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); #else const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); #endif _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) * std::log(_Tp(0.25L) * __x * __eta); _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x + __pre_term1 - __pre_term2; _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta * (_Tp(2) * __th - std::sin(_Tp(2) * __th)) + __numeric_constants<_Tp>::__pi_4()); _Tp __ser = __ser_term1 + __ser_term2; return std::exp(__lnpre) * __ser; } /** * @brief Evaluate the polynomial based on the confluent hypergeometric * function in a safe way, with no restriction on the arguments. * * The associated Laguerre function is defined by * @f[ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} * _1F_1(-n; \alpha + 1; x) * @f] * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. * * This function assumes x != 0. * * This is from the GNU Scientific Library. */ template<typename _Tpa, typename _Tp> _Tp __poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) { const _Tp __b = _Tp(__alpha1) + _Tp(1); const _Tp __mx = -__x; const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) : ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); // Get |x|^n/n! _Tp __tc = _Tp(1); const _Tp __ax = std::abs(__x); for (unsigned int __k = 1; __k <= __n; ++__k) __tc *= (__ax / __k); _Tp __term = __tc * __tc_sgn; _Tp __sum = __term; for (int __k = int(__n) - 1; __k >= 0; --__k) { __term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) * _Tp(__k + 1) / __mx; __sum += __term; } return __sum; } /** * @brief This routine returns the associated Laguerre polynomial * of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ * by recursion. * * The associated Laguerre function is defined by * @f[ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} * _1F_1(-n; \alpha + 1; x) * @f] * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. * * The associated Laguerre polynomial is defined for integral * @f$ \alpha = m @f$ by: * @f[ * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) * @f] * where the Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * * @param __n The order of the Laguerre function. * @param __alpha The degree of the Laguerre function. * @param __x The argument of the Laguerre function. * @return The value of the Laguerre function of order n, * degree @f$ \alpha @f$, and argument x. */ template<typename _Tpa, typename _Tp> _Tp __poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) { // Compute l_0. _Tp __l_0 = _Tp(1); if (__n == 0) return __l_0; // Compute l_1^alpha. _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); if (__n == 1) return __l_1; // Compute l_n^alpha by recursion on n. _Tp __l_n2 = __l_0; _Tp __l_n1 = __l_1; _Tp __l_n = _Tp(0); for (unsigned int __nn = 2; __nn <= __n; ++__nn) { __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) * __l_n1 / _Tp(__nn) - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); __l_n2 = __l_n1; __l_n1 = __l_n; } return __l_n; } /** * @brief This routine returns the associated Laguerre polynomial * of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. * * The associated Laguerre function is defined by * @f[ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} * _1F_1(-n; \alpha + 1; x) * @f] * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and * @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. * * The associated Laguerre polynomial is defined for integral * @f$ \alpha = m @f$ by: * @f[ * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) * @f] * where the Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * * @param __n The order of the Laguerre function. * @param __alpha The degree of the Laguerre function. * @param __x The argument of the Laguerre function. * @return The value of the Laguerre function of order n, * degree @f$ \alpha @f$, and argument x. */ template<typename _Tpa, typename _Tp> _Tp __poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) { if (__x < _Tp(0)) std::__throw_domain_error(__N("Negative argument " "in __poly_laguerre.")); // Return NaN on NaN input. else if (__isnan(__x)) return std::numeric_limits<_Tp>::quiet_NaN(); else if (__n == 0) return _Tp(1); else if (__n == 1) return _Tp(1) + _Tp(__alpha1) - __x; else if (__x == _Tp(0)) { _Tp __prod = _Tp(__alpha1) + _Tp(1); for (unsigned int __k = 2; __k <= __n; ++__k) __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); return __prod; } else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) return __poly_laguerre_large_n(__n, __alpha1, __x); else if (_Tp(__alpha1) >= _Tp(0) || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) return __poly_laguerre_recursion(__n, __alpha1, __x); else return __poly_laguerre_hyperg(__n, __alpha1, __x); } /** * @brief This routine returns the associated Laguerre polynomial * of order n, degree m: @f$ L_n^m(x) @f$. * * The associated Laguerre polynomial is defined for integral * @f$ \alpha = m @f$ by: * @f[ * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) * @f] * where the Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * * @param __n The order of the Laguerre polynomial. * @param __m The degree of the Laguerre polynomial. * @param __x The argument of the Laguerre polynomial. * @return The value of the associated Laguerre polynomial of order n, * degree m, and argument x. */ template<typename _Tp> inline _Tp __assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) { return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } /** * @brief This routine returns the Laguerre polynomial * of order n: @f$ L_n(x) @f$. * * The Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * * @param __n The order of the Laguerre polynomial. * @param __x The argument of the Laguerre polynomial. * @return The value of the Laguerre polynomial of order n * and argument x. */ template<typename _Tp> inline _Tp __laguerre(unsigned int __n, _Tp __x) { return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } } // namespace __detail #undef _GLIBCXX_MATH_NS #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) } // namespace tr1 #endif _GLIBCXX_END_NAMESPACE_VERSION } #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC