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,M // Special functions -*- C++ -*-
// Copyright (C) 2006-2013 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
// .
/** @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
_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 = 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;