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// Special functions -*- C++ -*-
// Copyright (C) 2006-2019 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_hermite.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 22 pp. 773-802
#ifndef _GLIBCXX_TR1_POLY_HERMITE_TCC
#define _GLIBCXX_TR1_POLY_HERMITE_TCC 1
namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
#if _GLIBCXX_USE_STD_SPEC_FUNCS
#elif defined(_GLIBCXX_TR1_CMATH)
namespace 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 Hermite polynomial
* of order n: \f$ H_n(x) \f$ by recursion on n.
*
* The Hermite polynomial is defined by:
* @f[
* H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
* @f]
*
* @param __n The order of the Hermite polynomial.
* @param __x The argument of the Hermite polynomial.
* @return The value of the Hermite polynomial of order n
* and argument x.
*/
template<typename _Tp>
_Tp
__poly_hermite_recursion(unsigned int __n, _Tp __x)
{
// Compute H_0.
_Tp __H_0 = 1;
if (__n == 0)
return __H_0;
// Compute H_1.
_Tp __H_1 = 2 * __x;
if (__n == 1)
return __H_1;
// Compute H_n.
_Tp __H_n, __H_nm1, __H_nm2;
unsigned int __i;
for (__H_nm2 = __H_0, __H_nm1 = __H_1, __i = 2; __i <= __n; ++__i)
{
__H_n = 2 * (__x * __H_nm1 - (__i - 1) * __H_nm2);
__H_nm2 = __H_nm1;
__H_nm1 = __H_n;
}
return __H_n;
}
/**
* @brief This routine returns the Hermite polynomial
* of order n: \f$ H_n(x) \f$.
*
* The Hermite polynomial is defined by:
* @f[
* H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
* @f]
*
* @param __n The order of the Hermite polynomial.
* @param __x The argument of the Hermite polynomial.
* @return The value of the Hermite polynomial of order n
* and argument x.
*/
template<typename _Tp>
inline _Tp
__poly_hermite(unsigned int __n, _Tp __x)
{
if (__isnan(__x))
return std::numeric_limits<_Tp>::quiet_NaN();
else
return __poly_hermite_recursion(__n, __x);
}
} // namespace __detail
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
} // namespace tr1
#endif
_GLIBCXX_END_NAMESPACE_VERSION
}
#endif // _GLIBCXX_TR1_POLY_HERMITE_TCC
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