// Special functions -*- C++ -*-
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// Special functions -*- C++ -*-
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// Copyright (C) 2006, 2007, 2008, 2009
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// Copyright (C) 2006, 2007, 2008, 2009
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// Free Software Foundation, Inc.
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// Free Software Foundation, Inc.
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//
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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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// 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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// 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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// 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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// Free Software Foundation; either version 3, or (at your option)
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// any later version.
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// any later version.
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//
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//
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// This library is distributed in the hope that it will be useful,
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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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// 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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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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// GNU General Public License for more details.
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//
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//
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// Under Section 7 of GPL version 3, you are granted additional
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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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// 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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// 3.1, as published by the Free Software Foundation.
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// You should have received a copy of the GNU General Public License and
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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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// 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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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// .
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// .
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/** @file tr1/poly_laguerre.tcc
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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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* This is an internal header file, included by other library headers.
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* You should not attempt to use it directly.
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* You should not attempt to use it directly.
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*/
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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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// 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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// Written by Edward Smith-Rowland based on:
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// (1) Handbook of Mathematical Functions,
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// (1) Handbook of Mathematical Functions,
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// Ed. Milton Abramowitz and Irene A. Stegun,
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// Ed. Milton Abramowitz and Irene A. Stegun,
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// Dover Publications,
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// Dover Publications,
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// Section 13, pp. 509-510, Section 22 pp. 773-802
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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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// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
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#ifndef _GLIBCXX_TR1_POLY_LAGUERRE_TCC
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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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#define _GLIBCXX_TR1_POLY_LAGUERRE_TCC 1
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namespace std
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namespace std
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{
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{
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namespace tr1
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namespace tr1
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{
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{
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// [5.2] Special functions
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// [5.2] Special functions
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// Implementation-space details.
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// Implementation-space details.
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namespace __detail
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namespace __detail
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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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* @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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* 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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* Abramowitz & Stegun, 13.5.21
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*
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*
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* @param __n The order of the Laguerre function.
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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 __alpha The degree of the Laguerre function.
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* @param __x The argument 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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* @return The value of the Laguerre function of order n,
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* degree @f$ \alpha @f$, and argument x.
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* degree @f$ \alpha @f$, and argument x.
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*
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*
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* This is from the GNU Scientific Library.
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* This is from the GNU Scientific Library.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1,
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__poly_laguerre_large_n(const unsigned __n, const _Tpa __alpha1,
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const _Tp __x)
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const _Tp __x)
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{
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{
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const _Tp __a = -_Tp(__n);
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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 __b = _Tp(__alpha1) + _Tp(1);
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const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
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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 __cos2th = __x / __eta;
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const _Tp __sin2th = _Tp(1) - __cos2th;
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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 __th = std::acos(std::sqrt(__cos2th));
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const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
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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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* __numeric_constants<_Tp>::__pi_2()
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* __eta * __eta * __cos2th * __sin2th;
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* __eta * __eta * __cos2th * __sin2th;
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#if _GLIBCXX_USE_C99_MATH_TR1
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#if _GLIBCXX_USE_C99_MATH_TR1
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const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
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const _Tp __lg_b = std::tr1::lgamma(_Tp(__n) + __b);
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const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
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const _Tp __lnfact = std::tr1::lgamma(_Tp(__n + 1));
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#else
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#else
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const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
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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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const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
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#endif
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#endif
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_Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
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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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* 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 __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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_Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
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+ __pre_term1 - __pre_term2;
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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_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 __ser_term2 = std::sin(_Tp(0.25L) * __eta
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* (_Tp(2) * __th
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* (_Tp(2) * __th
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- std::sin(_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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+ __numeric_constants<_Tp>::__pi_4());
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_Tp __ser = __ser_term1 + __ser_term2;
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_Tp __ser = __ser_term1 + __ser_term2;
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return std::exp(__lnpre) * __ser;
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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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* @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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* function in a safe way, with no restriction on the arguments.
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*
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*
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* The associated Laguerre function is defined by
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* The associated Laguerre function is defined by
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* @f[
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* @f[
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
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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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* _1F_1(-n; \alpha + 1; x)
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* @f]
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* @f]
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
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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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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
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*
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*
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* This function assumes x != 0.
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* This function assumes x != 0.
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*
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*
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* This is from the GNU Scientific Library.
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* This is from the GNU Scientific Library.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__poly_laguerre_hyperg(const unsigned int __n, const _Tpa __alpha1,
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__poly_laguerre_hyperg(const unsigned int __n, const _Tpa __alpha1,
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const _Tp __x)
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const _Tp __x)
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{
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{
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const _Tp __b = _Tp(__alpha1) + _Tp(1);
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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 __mx = -__x;
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const _Tp __tc_sgn = (__x < _Tp(0) ? _Tp(1)
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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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: ((__n % 2 == 1) ? -_Tp(1) : _Tp(1)));
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// Get |x|^n/n!
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// Get |x|^n/n!
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_Tp __tc = _Tp(1);
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_Tp __tc = _Tp(1);
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const _Tp __ax = std::abs(__x);
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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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for (unsigned int __k = 1; __k <= __n; ++__k)
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__tc *= (__ax / __k);
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__tc *= (__ax / __k);
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_Tp __term = __tc * __tc_sgn;
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_Tp __term = __tc * __tc_sgn;
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_Tp __sum = __term;
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_Tp __sum = __term;
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for (int __k = int(__n) - 1; __k >= 0; --__k)
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for (int __k = int(__n) - 1; __k >= 0; --__k)
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{
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{
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__term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k))
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__term *= ((__b + _Tp(__k)) / _Tp(int(__n) - __k))
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* _Tp(__k + 1) / __mx;
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* _Tp(__k + 1) / __mx;
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__sum += __term;
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__sum += __term;
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}
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}
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return __sum;
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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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* @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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* 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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* by recursion.
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*
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*
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* The associated Laguerre function is defined by
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* The associated Laguerre function is defined by
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* @f[
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* @f[
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
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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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* _1F_1(-n; \alpha + 1; x)
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* @f]
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* @f]
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
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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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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
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*
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*
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* The associated Laguerre polynomial is defined for integral
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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$ \alpha = m @f$ by:
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* @f[
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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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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
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* @f]
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* @f]
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* where the Laguerre polynomial is defined by:
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* where the Laguerre polynomial is defined by:
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* @f[
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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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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
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* @f]
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* @f]
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*
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*
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* @param __n The order of the Laguerre function.
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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 __alpha The degree of the Laguerre function.
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* @param __x The argument 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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* @return The value of the Laguerre function of order n,
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* degree @f$ \alpha @f$, and argument x.
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* degree @f$ \alpha @f$, and argument x.
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*/
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*/
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template
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template
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_Tp
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_Tp
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__poly_laguerre_recursion(const unsigned int __n,
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__poly_laguerre_recursion(const unsigned int __n,
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const _Tpa __alpha1, const _Tp __x)
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const _Tpa __alpha1, const _Tp __x)
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{
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{
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// Compute l_0.
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// Compute l_0.
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_Tp __l_0 = _Tp(1);
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_Tp __l_0 = _Tp(1);
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if (__n == 0)
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if (__n == 0)
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return __l_0;
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return __l_0;
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// Compute l_1^alpha.
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// Compute l_1^alpha.
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_Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
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_Tp __l_1 = -__x + _Tp(1) + _Tp(__alpha1);
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if (__n == 1)
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if (__n == 1)
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return __l_1;
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return __l_1;
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// Compute l_n^alpha by recursion on n.
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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_n2 = __l_0;
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_Tp __l_n1 = __l_1;
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_Tp __l_n1 = __l_1;
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_Tp __l_n = _Tp(0);
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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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for (unsigned int __nn = 2; __nn <= __n; ++__nn)
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{
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{
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__l_n = (_Tp(2 * __nn - 1) + _Tp(__alpha1) - __x)
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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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* __l_n1 / _Tp(__nn)
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- (_Tp(__nn - 1) + _Tp(__alpha1)) * __l_n2 / _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_n2 = __l_n1;
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__l_n1 = __l_n;
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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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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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* @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$.
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* of order n, degree @f$ \alpha @f$: @f$ L_n^alpha(x) @f$.
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*
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*
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* The associated Laguerre function is defined by
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* The associated Laguerre function is defined by
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* @f[
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* @f[
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
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* _1F_1(-n; \alpha + 1; x)
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* _1F_1(-n; \alpha + 1; x)
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* @f]
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* @f]
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
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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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* @f$ _1F_1(a; c; x) @f$ is the confluent hypergeometric function.
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*
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*
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* The associated Laguerre polynomial is defined for integral
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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$ \alpha = m @f$ by:
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* @f[
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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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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
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* @f]
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* @f]
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* where the Laguerre polynomial is defined by:
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* where the Laguerre polynomial is defined by:
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* @f[
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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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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
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* @f]
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* @f]
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*
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*
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* @param __n The order of the Laguerre function.
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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 __alpha The degree of the Laguerre function.
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* @param __x The argument 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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* @return The value of the Laguerre function of order n,
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* degree @f$ \alpha @f$, and argument x.
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* degree @f$ \alpha @f$, and argument x.
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*/
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*/
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template
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template
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inline _Tp
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inline _Tp
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__poly_laguerre(const unsigned int __n, const _Tpa __alpha1,
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__poly_laguerre(const unsigned int __n, const _Tpa __alpha1,
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const _Tp __x)
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const _Tp __x)
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{
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{
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if (__x < _Tp(0))
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if (__x < _Tp(0))
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std::__throw_domain_error(__N("Negative argument "
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std::__throw_domain_error(__N("Negative argument "
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"in __poly_laguerre."));
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"in __poly_laguerre."));
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// Return NaN on NaN input.
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// Return NaN on NaN input.
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else if (__isnan(__x))
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else if (__isnan(__x))
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return std::numeric_limits<_Tp>::quiet_NaN();
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return std::numeric_limits<_Tp>::quiet_NaN();
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else if (__n == 0)
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else if (__n == 0)
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return _Tp(1);
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return _Tp(1);
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else if (__n == 1)
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else if (__n == 1)
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return _Tp(1) + _Tp(__alpha1) - __x;
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return _Tp(1) + _Tp(__alpha1) - __x;
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else if (__x == _Tp(0))
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else if (__x == _Tp(0))
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{
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{
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_Tp __prod = _Tp(__alpha1) + _Tp(1);
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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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for (unsigned int __k = 2; __k <= __n; ++__k)
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__prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
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__prod *= (_Tp(__alpha1) + _Tp(__k)) / _Tp(__k);
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return __prod;
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return __prod;
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}
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}
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else if (__n > 10000000 && _Tp(__alpha1) > -_Tp(1)
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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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&& __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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return __poly_laguerre_large_n(__n, __alpha1, __x);
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else if (_Tp(__alpha1) >= _Tp(0)
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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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|| (__x > _Tp(0) && _Tp(__alpha1) < -_Tp(__n + 1)))
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return __poly_laguerre_recursion(__n, __alpha1, __x);
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return __poly_laguerre_recursion(__n, __alpha1, __x);
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else
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else
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return __poly_laguerre_hyperg(__n, __alpha1, __x);
|
return __poly_laguerre_hyperg(__n, __alpha1, __x);
|
}
|
}
|
|
|
|
|
/**
|
/**
|
* @brief This routine returns the associated Laguerre polynomial
|
* @brief This routine returns the associated Laguerre polynomial
|
* of order n, degree m: @f$ L_n^m(x) @f$.
|
* of order n, degree m: @f$ L_n^m(x) @f$.
|
*
|
*
|
* The associated Laguerre polynomial is defined for integral
|
* The associated Laguerre polynomial is defined for integral
|
* @f$ \alpha = m @f$ by:
|
* @f$ \alpha = m @f$ by:
|
* @f[
|
* @f[
|
* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
|
* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
|
* @f]
|
* @f]
|
* where the Laguerre polynomial is defined by:
|
* where the Laguerre polynomial is defined by:
|
* @f[
|
* @f[
|
* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
|
* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
|
* @f]
|
* @f]
|
*
|
*
|
* @param __n The order of the Laguerre polynomial.
|
* @param __n The order of the Laguerre polynomial.
|
* @param __m The degree of the Laguerre polynomial.
|
* @param __m The degree of the Laguerre polynomial.
|
* @param __x The argument of the Laguerre polynomial.
|
* @param __x The argument of the Laguerre polynomial.
|
* @return The value of the associated Laguerre polynomial of order n,
|
* @return The value of the associated Laguerre polynomial of order n,
|
* degree m, and argument x.
|
* degree m, and argument x.
|
*/
|
*/
|
template
|
template
|
inline _Tp
|
inline _Tp
|
__assoc_laguerre(const unsigned int __n, const unsigned int __m,
|
__assoc_laguerre(const unsigned int __n, const unsigned int __m,
|
const _Tp __x)
|
const _Tp __x)
|
{
|
{
|
return __poly_laguerre(__n, __m, __x);
|
return __poly_laguerre(__n, __m, __x);
|
}
|
}
|
|
|
|
|
/**
|
/**
|
* @brief This routine returns the Laguerre polynomial
|
* @brief This routine returns the Laguerre polynomial
|
* of order n: @f$ L_n(x) @f$.
|
* of order n: @f$ L_n(x) @f$.
|
*
|
*
|
* The Laguerre polynomial is defined by:
|
* The Laguerre polynomial is defined by:
|
* @f[
|
* @f[
|
* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
|
* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
|
* @f]
|
* @f]
|
*
|
*
|
* @param __n The order of the Laguerre polynomial.
|
* @param __n The order of the Laguerre polynomial.
|
* @param __x The argument of the Laguerre polynomial.
|
* @param __x The argument of the Laguerre polynomial.
|
* @return The value of the Laguerre polynomial of order n
|
* @return The value of the Laguerre polynomial of order n
|
* and argument x.
|
* and argument x.
|
*/
|
*/
|
template
|
template
|
inline _Tp
|
inline _Tp
|
__laguerre(const unsigned int __n, const _Tp __x)
|
__laguerre(const unsigned int __n, const _Tp __x)
|
{
|
{
|
return __poly_laguerre(__n, 0, __x);
|
return __poly_laguerre(__n, 0, __x);
|
}
|
}
|
|
|
} // namespace std::tr1::__detail
|
} // namespace std::tr1::__detail
|
}
|
}
|
}
|
}
|
|
|
#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC
|
#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC
|
|
|
