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

TL;DR

This paper introduces a fast, stable method for transforming Chebyshev expansion coefficients to Jacobi expansion coefficients using Hahn's asymptotic formula and stabilized recurrence, improving numerical stability near endpoints.

Contribution

It develops a new algorithm combining Hahn's asymptotic formula and stable recurrence for efficient Chebyshev--Jacobi transforms, extending Reinsch's modification for better endpoint stability.

Findings

The method is fast and simple to implement.

It achieves numerical stability across the entire interval.

The approach outperforms existing algorithms in stability near endpoints.

Abstract

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 a 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.