// (C) Copyright Nick Thompson 2019.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_JACOBI_HPP
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#define BOOST_MATH_SPECIAL_JACOBI_HPP
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#include <limits>
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#include <stdexcept>
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namespace boost { namespace math {
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template<typename Real>
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Real jacobi(unsigned n, Real alpha, Real beta, Real x)
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{
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static_assert(!std::is_integral<Real>::value, "Jacobi polynomials do not work with integer arguments.");
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if (n == 0) {
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return Real(1);
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}
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Real y0 = 1;
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Real y1 = (alpha+1) + (alpha+beta+2)*(x-1)/Real(2);
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Real yk = y1;
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Real k = 2;
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Real k_max = n*(1+std::numeric_limits<Real>::epsilon());
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while(k < k_max)
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{
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// Hoping for lots of common subexpression elimination by the compiler:
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Real denom = 2*k*(k+alpha+beta)*(2*k+alpha+beta-2);
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Real gamma1 = (2*k+alpha+beta-1)*( (2*k+alpha+beta)*(2*k+alpha+beta-2)*x + alpha*alpha -beta*beta);
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Real gamma0 = -2*(k+alpha-1)*(k+beta-1)*(2*k+alpha+beta);
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yk = (gamma1*y1 + gamma0*y0)/denom;
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y0 = y1;
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y1 = yk;
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k += 1;
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}
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return yk;
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}
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template<typename Real>
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Real jacobi_derivative(unsigned n, Real alpha, Real beta, Real x, unsigned k)
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{
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if (k > n) {
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return Real(0);
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}
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Real scale = 1;
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for(unsigned j = 1; j <= k; ++j) {
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scale *= (alpha + beta + n + j)/2;
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}
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return scale*jacobi<Real>(n-k, alpha + k, beta+k, x);
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}
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template<typename Real>
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Real jacobi_prime(unsigned n, Real alpha, Real beta, Real x)
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{
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return jacobi_derivative<Real>(n, alpha, beta, x, 1);
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}
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template<typename Real>
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Real jacobi_double_prime(unsigned n, Real alpha, Real beta, Real x)
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{
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return jacobi_derivative<Real>(n, alpha, beta, x, 2);
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}
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}}
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#endif
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