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// Special functions -*- C++ -*- |
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|
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// Copyright (C) 2006-2021 Free Software Foundation, Inc. |
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// |
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// This file is part of the GNU ISO C++ Library. This library is free |
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// software; you can redistribute it and/or modify it under the |
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// terms of the GNU General Public License as published by the |
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// Free Software Foundation; either version 3, or (at your option) |
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// any later version. |
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// |
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// This library is distributed in the hope that it will be useful, |
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// but WITHOUT ANY WARRANTY; without even the implied warranty of |
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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// GNU General Public License for more details. |
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// |
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// Under Section 7 of GPL version 3, you are granted additional |
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// permissions described in the GCC Runtime Library Exception, version |
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// 3.1, as published by the Free Software Foundation. |
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|
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// You should have received a copy of the GNU General Public License and |
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// a copy of the GCC Runtime Library Exception along with this program; |
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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// <http://www.gnu.org/licenses/>. |
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|
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/** @file tr1/poly_laguerre.tcc |
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* This is an internal header file, included by other library headers. |
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* Do not attempt to use it directly. @headername{tr1/cmath} |
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*/ |
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|
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// |
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// ISO C++ 14882 TR1: 5.2 Special functions |
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// |
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|
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// Written by Edward Smith-Rowland based on: |
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// (1) Handbook of Mathematical Functions, |
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// Ed. Milton Abramowitz and Irene A. Stegun, |
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// Dover Publications, |
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// Section 13, pp. 509-510, Section 22 pp. 773-802 |
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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
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|
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#ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC |
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#define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1 |
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|
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namespace std _GLIBCXX_VISIBILITY(default) |
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{ |
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_GLIBCXX_BEGIN_NAMESPACE_VERSION |
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|
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#if _GLIBCXX_USE_STD_SPEC_FUNCS |
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# define _GLIBCXX_MATH_NS ::std |
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#elif defined(_GLIBCXX_TR1_CMATH) |
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namespace tr1 |
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{ |
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# define _GLIBCXX_MATH_NS ::std::tr1 |
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#else |
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# error do not include this header directly, use <cmath> or <tr1/cmath> |
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#endif |
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// [5.2] Special functions |
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|
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// Implementation-space details. |
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namespace __detail |
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{ |
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/** |
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* @brief This routine returns the associated Laguerre polynomial |
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* of order @f$ n @f$, degree @f$ \alpha @f$ for large n. |
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* Abramowitz & Stegun, 13.5.21 |
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* |
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* @param __n The order of the Laguerre function. |
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* @param __alpha The degree of the Laguerre function. |
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* @param __x The argument of the Laguerre function. |
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* @return The value of the Laguerre function of order n, |
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* degree @f$ \alpha @f$, and argument x. |
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* |
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* This is from the GNU Scientific Library. |
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*/ |
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template<typename _Tpa, typename _Tp> |
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_Tp |
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__poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x) |
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{ |
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const _Tp __a = -_Tp(__n); |
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const _Tp __b = _Tp(__alpha1) + _Tp(1); |
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const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a; |
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const _Tp __cos2th = __x / __eta; |
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const _Tp __sin2th = _Tp(1) - __cos2th; |
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const _Tp __th = std::acos(std::sqrt(__cos2th)); |
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const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2() |
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* __numeric_constants<_Tp>::__pi_2() |
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* __eta * __eta * __cos2th * __sin2th; |
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|
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#if _GLIBCXX_USE_C99_MATH_TR1 |
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const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b); |
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const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1)); |
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#else |
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const _Tp __lg_b = __log_gamma(_Tp(__n) + __b); |
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const _Tp __lnfact = __log_gamma(_Tp(__n + 1)); |
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#endif |
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|
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_Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b) |
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* std::log(_Tp(0.25L) * __x * __eta); |
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_Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h); |
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_Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x |
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+ __pre_term1 - __pre_term2; |
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_Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi()); |
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_Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta |
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* (_Tp(2) * __th |
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- std::sin(_Tp(2) * __th)) |
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+ __numeric_constants<_Tp>::__pi_4()); |
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_Tp __ser = __ser_term1 + __ser_term2; |
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|
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return std::exp(__lnpre) * __ser; |
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} |
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|
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|
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/** |
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* @brief Evaluate the polynomial based on the confluent hypergeometric |
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* function in a safe way, with no restriction on the arguments. |
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* |
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* The associated Laguerre function is defined by |
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* @f[ |
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
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* _1F_1(-n; \alpha + 1; x) |
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* @f] |
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |
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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |
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* |
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* This function assumes x != 0. |
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* |
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* This is from the GNU Scientific Library. |
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*/ |
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template<typename _Tpa, typename _Tp> |
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_Tp |
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__poly_laguerre_hyperg(unsigned int __n, _Tpa __alpha1, _Tp __x) |
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{ |
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const _Tp __b = _Tp(__alpha1) + _Tp(1); |
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const _Tp __mx = -__x; |
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const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1) |
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: ((__n % 2 == 1) ? -_Tp(1) : _Tp(1))); |
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// Get |x|^n/n! |
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_Tp __tc = _Tp(1); |
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const _Tp __ax = std::abs(__x); |
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for (unsigned int __k = 1; __k <= __n; ++__k) |
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__tc *= (__ax / __k); |
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|
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_Tp __term = __tc * __tc_sgn; |
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_Tp __sum = __term; |
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for (int __k = int(__n) - 1; __k >= 0; --__k) |
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{ |
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__term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k)) |
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* _Tp(__k + 1) / __mx; |
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__sum += __term; |
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} |
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|
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return __sum; |
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} |
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|
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|
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/** |
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* @brief This routine returns the associated Laguerre polynomial |
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* of order @f$ n @f$, degree @f$ \alpha @f$: @f$ L_n^\alpha(x) @f$ |
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* by recursion. |
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* |
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* The associated Laguerre function is defined by |
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* @f[ |
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
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* _1F_1(-n; \alpha + 1; x) |
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* @f] |
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |
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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |
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* |
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* The associated Laguerre polynomial is defined for integral |
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* @f$ \alpha = m @f$ by: |
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* @f[ |
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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |
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* @f] |
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* where the Laguerre polynomial is defined by: |
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* @f[ |
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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
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* @f] |
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* |
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* @param __n The order of the Laguerre function. |
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* @param __alpha The degree of the Laguerre function. |
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* @param __x The argument of the Laguerre function. |
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* @return The value of the Laguerre function of order n, |
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* degree @f$ \alpha @f$, and argument x. |
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*/ |
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template<typename _Tpa, typename _Tp> |
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_Tp |
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__poly_laguerre_recursion(unsigned int __n, _Tpa __alpha1, _Tp __x) |
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{ |
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// Compute l_0. |
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_Tp __l_0 = _Tp(1); |
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if (__n == 0) |
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return __l_0; |
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|
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// Compute l_1^alpha. |
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_Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1); |
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if (__n == 1) |
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return __l_1; |
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|
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// Compute l_n^alpha by recursion on n. |
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_Tp __l_n2 = __l_0; |
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_Tp __l_n1 = __l_1; |
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_Tp __l_n = _Tp(0); |
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for (unsigned int __nn = 2; __nn <= __n; ++__nn) |
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{ |
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__l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x) |
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* __l_n1 / _Tp(__nn) |
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- (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _Tp(__nn); |
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__l_n2 = __l_n1; |
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__l_n1 = __l_n; |
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} |
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|
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return __l_n; |
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} |
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|
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|
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/** |
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* @brief This routine returns the associated Laguerre polynomial |
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* of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$. |
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* |
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* The associated Laguerre function is defined by |
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* @f[ |
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
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* _1F_1(-n; \alpha + 1; x) |
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* @f] |
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |
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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function. |
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* |
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* The associated Laguerre polynomial is defined for integral |
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* @f$ \alpha = m @f$ by: |
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* @f[ |
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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |
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* @f] |
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* where the Laguerre polynomial is defined by: |
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* @f[ |
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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
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* @f] |
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* |
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* @param __n The order of the Laguerre function. |
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* @param __alpha The degree of the Laguerre function. |
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* @param __x The argument of the Laguerre function. |
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* @return The value of the Laguerre function of order n, |
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* degree @f$ \alpha @f$, and argument x. |
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*/ |
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template<typename _Tpa, typename _Tp> |
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_Tp |
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__poly_laguerre(unsigned int __n, _Tpa __alpha1, _Tp __x) |
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{ |
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if (__x < _Tp(0)) |
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std::__throw_domain_error(__N("Negative argument " |
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"in __poly_laguerre.")); |
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// Return NaN on NaN input. |
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else if (__isnan(__x)) |
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return std::numeric_limits<_Tp>::quiet_NaN(); |
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else if (__n == 0) |
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return _Tp(1); |
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else if (__n == 1) |
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return _Tp(1) + _Tp(__alpha1) - __x; |
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else if (__x == _Tp(0)) |
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{ |
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_Tp __prod = _Tp(__alpha1) + _Tp(1); |
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for (unsigned int __k = 2; __k <= __n; ++__k) |
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__prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k); |
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return __prod; |
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} |
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else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1) |
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&& __x < _Tp(2) * (_Tp(__alpha1) + _Tp(1)) + _Tp(4 * __n)) |
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return __poly_laguerre_large_n(__n, __alpha1, __x); |
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else if (_Tp(__alpha1) >= _Tp(0) |
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|| (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1))) |
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return __poly_laguerre_recursion(__n, __alpha1, __x); |
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else |
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return __poly_laguerre_hyperg(__n, __alpha1, __x); |
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} |
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|
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|
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/** |
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* @brief This routine returns the associated Laguerre polynomial |
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* of order n, degree m: @f$ L_n^m(x) @f$. |
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* |
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* The associated Laguerre polynomial is defined for integral |
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* @f$ \alpha = m @f$ by: |
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* @f[ |
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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |
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* @f] |
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* where the Laguerre polynomial is defined by: |
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* @f[ |
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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
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* @f] |
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* |
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* @param __n The order of the Laguerre polynomial. |
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* @param __m The degree of the Laguerre polynomial. |
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* @param __x The argument of the Laguerre polynomial. |
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* @return The value of the associated Laguerre polynomial of order n, |
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* degree m, and argument x. |
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*/ |
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template<typename _Tp> |
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inline _Tp |
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__assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |
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{ return __poly_laguerre<unsigned int, _Tp>(__n, __m, __x); } |
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|
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|
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/** |
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* @brief This routine returns the Laguerre polynomial |
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* of order n: @f$ L_n(x) @f$. |
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* |
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* The Laguerre polynomial is defined by: |
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* @f[ |
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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
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* @f] |
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* |
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* @param __n The order of the Laguerre polynomial. |
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* @param __x The argument of the Laguerre polynomial. |
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* @return The value of the Laguerre polynomial of order n |
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* and argument x. |
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*/ |
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template<typename _Tp> |
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inline _Tp |
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__laguerre(unsigned int __n, _Tp __x) |
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{ return __poly_laguerre<unsigned int, _Tp>(__n, 0, __x); } |
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} // namespace __detail |
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#undef _GLIBCXX_MATH_NS |
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#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH) |
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} // namespace tr1 |
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#endif |
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|
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_GLIBCXX_END_NAMESPACE_VERSION |
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} |
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|
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#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC |
