The Laurent-Horner method for validated evaluation of Chebyshev expansions

TL;DR

This paper introduces a two-step algorithm that efficiently encloses Chebyshev expansions by transforming them into Laurent basis and applying interval Horner method, especially effective for high-degree polynomials or boundary evaluations.

Contribution

It presents a novel, simple, and linear-cost algorithm for validated Chebyshev expansion evaluation, outperforming eigenvalue-based methods in certain scenarios.

Findings

Algorithm is efficient for high-degree polynomials.

Outperforms existing eigenvalue-based methods near domain boundaries.

Cost is linear in polynomial degree.

Abstract

We develop a simple two-step algorithm for enclosing Chebyshev expansions whose cost is linear in terms of the polynomial degree. The algorithm first transforms the expansion from Chebyshev to the Laurent basis and then applies the interval Horner method. It outperforms the existing eigenvalue-based methods if the degree is high or the evaluation point is close to the boundaries of the domain.