| // Special functions -*- C++ -*- |
| |
| // Copyright (C) 2006, 2007, 2008, 2009, 2010 |
| // 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) |
| { |
| namespace tr1 |
| { |
| // [5.2] Special functions |
| |
| // Implementation-space details. |
| namespace __detail |
| { |
| _GLIBCXX_BEGIN_NAMESPACE_VERSION |
| |
| /** |
| * @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$, and argument x. |
| * |
| * This is from the GNU Scientific Library. |
| */ |
| template<typename _Tpa, typename _Tp> |
| _Tp |
| __poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1, |
| const _Tp __x) |
| { |
| const _Tp __a = -_Tp(__n); |
| const _Tp __b = _Tp(__alpha1) + _Tp(1); |
| const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; |
| const _Tp __cos2th = __x / __eta; |
| const _Tp __sin2th = _Tp(1) - __cos2th; |
| const _Tp __th = std::acos(std::sqrt(__cos2th)); |
| const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() |
| * __numeric_constants<_Tp>::__pi_2() |
| * __eta * __eta * __cos2th * __sin2th; |
| |
| #if _GLIBCXX_USE_C99_MATH_TR1 |
| const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b); |
| const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1)); |
| #else |
| const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); |
| const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); |
| #endif |
| |
| _Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) |
| * std::log(_Tp(0.25L) * __x * __eta); |
| _Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); |
| _Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x |
| + __pre_term1 - __pre_term2; |
| _Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); |
| _Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta |
| * (_Tp(2) * __th |
| - std::sin(_Tp(2) * __th)) |
| + __numeric_constants<_Tp>::__pi_4()); |
| _Tp __ser = __ser_term1 + __ser_term2; |
| |
| return std::exp(__lnpre) * __ser; |
| } |
| |
| |
| /** |
| * @brief Evaluate the polynomial based on the confluent hypergeometric |
| * function in a safe way, with no restriction on the arguments. |
| * |
| * The associated Laguerre function is defined by |
| * @f[ |
| * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
| * _1F_1(-n; \alpha + 1; x) |
| * @f] |
| * where @f$ (\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(const unsigned int __n, const _Tpa __alpha1, |
| const _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$, and argument x. |
| */ |
| template<typename _Tpa, typename _Tp> |
| _Tp |
| __poly_laguerre_recursion(const unsigned int __n, |
| const _Tpa __alpha1, const _Tp __x) |
| { |
| // Compute l_0. |
| _Tp __l_0 = _Tp(1); |
| if (__n == 0) |
| return __l_0; |
| |
| // Compute l_1^alpha. |
| _Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); |
| if (__n == 1) |
| return __l_1; |
| |
| // Compute l_n^alpha by recursion on n. |
| _Tp __l_n2 = __l_0; |
| _Tp __l_n1 = __l_1; |
| _Tp __l_n = _Tp(0); |
| for (unsigned int __nn = 2; __nn <= __n; ++__nn) |
| { |
| __l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) |
| * __l_n1 / _Tp(__nn) |
| - (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); |
| __l_n2 = __l_n1; |
| __l_n1 = __l_n; |
| } |
| |
| return __l_n; |
| } |
| |
| |
| /** |
| * @brief This routine returns the associated Laguerre polynomial |
| * of order n, degree @f$ \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$, and argument x. |
| */ |
| template<typename _Tpa, typename _Tp> |
| inline _Tp |
| __poly_laguerre(const unsigned int __n, const _Tpa __alpha1, |
| const _Tp __x) |
| { |
| if (__x < _Tp(0)) |
| std::__throw_domain_error(__N("Negative argument " |
| "in __poly_laguerre.")); |
| // Return NaN on NaN input. |
| else if (__isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else if (__n == 0) |
| return _Tp(1); |
| else if (__n == 1) |
| return _Tp(1) + _Tp(__alpha1) - __x; |
| else if (__x == _Tp(0)) |
| { |
| _Tp __prod = _Tp(__alpha1) + _Tp(1); |
| for (unsigned int __k = 2; __k <= __n; ++__k) |
| __prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); |
| return __prod; |
| } |
| else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) |
| && __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) |
| return __poly_laguerre_large_n(__n, __alpha1, __x); |
| else if (_Tp(__alpha1) >= _Tp(0) |
| || (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) |
| return __poly_laguerre_recursion(__n, __alpha1, __x); |
| else |
| return __poly_laguerre_hyperg(__n, __alpha1, __x); |
| } |
| |
| |
| /** |
| * @brief This routine returns the associated Laguerre polynomial |
| * of order n, degree m: @f$ 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(const unsigned int __n, const unsigned int __m, |
| const _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(const unsigned int __n, const _Tp __x) |
| { |
| return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); |
| } |
| |
| _GLIBCXX_END_NAMESPACE_VERSION |
| } // namespace std::tr1::__detail |
| } |
| } |
| |
| #endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC |
