legendre.hpp

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/** \file legendre.hpp
 * \brief Declares and defines Legendre polynomials.
 * \ingroup gen_math_files
 * \author Jared R. Males (jaredmales@gmail.com)
 *
 */
 
//***********************************************************************//
// Copyright 2021 Jared R. Males (jaredmales@gmail.com)
//
// This file is part of mxlib.
//
// mxlib 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 of the License, or
// (at your option) any later version.
//
// mxlib 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.
//
// You should have received a copy of the GNU General Public License
// along with mxlib.  If not, see <http://www.gnu.org/licenses/>.
//***********************************************************************//
 
#ifndef mx_legendre_hpp
#define mx_legendre_hpp
 
#include <type_traits>
 
#ifdef MX_INCLUDE_BOOST
#include <boost/math/special_functions/legendre.hpp>
#endif
 
namespace mx
{
namespace math
{
namespace func
{
 
/// Legendre Polynomials
/** See https://en.wikipedia.org/wiki/Legendre_polynomials
 * and https://www.boost.org/doc/libs/1_75_0/libs/math/doc/html/math_toolkit/sf_poly/legendre.html
 *
 * \returns the value of the n-th Legendre polynomial at x.
 *
 * \ingroup functions
 */
template <typename T>
T legendre_p( int n, ///< [in] the order of the Legendre polynomial, n>=0.
              T x    ///< [in] the argument, -1 <= x <= 1.
)
{
#ifdef MX_INCLUDE_BOOST
    return boost::math::legendre_p<T>( n, x );
#else
    static_assert(
        std::is_fundamental<T>::value || !std::is_fundamental<T>::value,
        "legendre_p<T> not specialized for type T, and MX_INCLUDE_BOOST is not defined, so I can't just use boost." );
    return 0;
#endif
}
 
template <>
float legendre_p<float>( int n, float x );
 
template <>
double legendre_p<double>( int n, double x );
 
template <>
long double legendre_p<long double>( int n, long double x );
 
#ifdef HASQUAD
template <>
__float128 legendre_p<__float128>( int n, __float128 x );
#endif
 
/// The orthonormal Legendre polynomials
/** A version of the Legendre polynomials which are orthonormal on the interval
 * -1 <= x <= 1.
 *
 * \returns the value of the n-th orthonormal Legendre polynomial at x.
 */
template <typename T>
T orthoNormalLegendre( int n, T x )
{
    return sqrt( ( static_cast<T>( 2 * n + 1 ) ) / static_cast<T>( 2 ) ) * legendre_p<T>( n, x );
}
 
} // namespace func
} // namespace math
} // namespace mx
 
#endif // mx_legendre_hpp