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[/
Copyright 2019, Nick Thompson
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]
[section:jacobi Jacobi Polynomials]
[h4 Synopsis]
``
#include <boost/math/special_functions/jacobi.hpp>
``
namespace boost{ namespace math{
template<typename Real>
Real jacobi(unsigned n, Real alpha, Real beta, Real x);
template<typename Real>
Real jacobi_derivative(unsigned n, Real alpha, Real beta, Real x, unsigned k);
template<typename Real>
Real jacobi_prime(unsigned n, Real alpha, Real beta, Real x);
template<typename Real>
Real jacobi_double_prime(unsigned n, Real alpha, Real beta, Real x);
}} // namespaces
Jacobi polynomials are a family of orthogonal polynomials.
A basic usage is as follows:
using boost::math::jacobi;
double x = 0.5;
double alpha = 0.3;
double beta = 7.2;
unsigned n = 3;
double y = jacobi(n, alpha, beta, x);
All derivatives of the Jacobi polynomials are available.
The /k/-th derivative of the /n/-th Gegenbauer polynomial is given by
using boost::math::jacobi_derivative;
double x = 0.5;
double alpha = 0.3;
double beta = 7.2;
unsigned n = 3;
double y = jacobi_derivative(n, alpha, beta, x, k);
For consistency with the rest of the library, `jacobi_prime` is provided which simply returns `jacobi_derivative(n, lambda, x,1)`.
[$../graphs/jacobi.svg]
[h3 Implementation]
The implementation uses the 3-term recurrence for the Jacobi polynomials, rising.
[endsect]
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