// 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