On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev-Jacobi transform

摘要

We describe a fast, simple and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients, and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev-Lobatto points. This is achieved via decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw-Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.