330 lines
11 KiB
C++
330 lines
11 KiB
C++
// Special functions -*- C++ -*-
|
|
|
|
// Copyright (C) 2006-2017 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)
|
|
{
|
|
#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
|
|
{
|
|
_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$, and argument x.
|
|
*
|
|
* This is from the GNU Scientific Library.
|
|
*/
|
|
template<typename _Tpa, typename _Tp>
|
|
_Tp
|
|
__poly_laguerre_large_n(unsigned __n, _Tpa __alpha1, _Tp __x)
|
|
{
|
|
const _Tp __a = -_Tp(__n);
|
|
const _Tp __b = _Tp(__alpha1) + _Tp(1);
|
|
const _Tp __eta = _Tp(2) * __b - _Tp(4) * __a;
|
|
const _Tp __cos2th = __x / __eta;
|
|
const _Tp __sin2th = _Tp(1) - __cos2th;
|
|
const _Tp __th = std::acos(std::sqrt(__cos2th));
|
|
const _Tp __pre_h = __numeric_constants<_Tp>::__pi_2()
|
|
* __numeric_constants<_Tp>::__pi_2()
|
|
* __eta * __eta * __cos2th * __sin2th;
|
|
|
|
#if _GLIBCXX_USE_C99_MATH_TR1
|
|
const _Tp __lg_b = _GLIBCXX_MATH_NS::lgamma(_Tp(__n) + __b);
|
|
const _Tp __lnfact = _GLIBCXX_MATH_NS::lgamma(_Tp(__n + 1));
|
|
#else
|
|
const _Tp __lg_b = __log_gamma(_Tp(__n) + __b);
|
|
const _Tp __lnfact = __log_gamma(_Tp(__n + 1));
|
|
#endif
|
|
|
|
_Tp __pre_term1 = _Tp(0.5L) * (_Tp(1) - __b)
|
|
* std::log(_Tp(0.25L) * __x * __eta);
|
|
_Tp __pre_term2 = _Tp(0.25L) * std::log(__pre_h);
|
|
_Tp __lnpre = __lg_b - __lnfact + _Tp(0.5L) * __x
|
|
+ __pre_term1 - __pre_term2;
|
|
_Tp __ser_term1 = std::sin(__a * __numeric_constants<_Tp>::__pi());
|
|
_Tp __ser_term2 = std::sin(_Tp(0.25L) * __eta
|
|
* (_Tp(2) * __th
|
|
- std::sin(_Tp(2) * __th))
|
|
+ __numeric_constants<_Tp>::__pi_4());
|
|
_Tp __ser = __ser_term1 + __ser_term2;
|
|
|
|
return std::exp(__lnpre) * __ser;
|
|
}
|
|
|
|
|
|
/**
|
|
* @brief Evaluate the polynomial based on the confluent hypergeometric
|
|
* function in a safe way, with no restriction on the arguments.
|
|
*
|
|
* The associated Laguerre function is defined by
|
|
* @f[
|
|
* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
|
|
* _1F_1(-n; \alpha + 1; x)
|
|
* @f]
|
|
* where @f$ (\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); }
|
|
|
|
_GLIBCXX_END_NAMESPACE_VERSION
|
|
} // namespace __detail
|
|
#undef _GLIBCXX_MATH_NS
|
|
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
|
|
} // namespace tr1
|
|
#endif
|
|
}
|
|
|
|
#endif // _GLIBCXX_TR1_POLY_LAGUERRE_TCC
|