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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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