The Laurent-Horner method for validated evaluation of Chebyshev expansions

TL;DR

This paper tackles validated evaluation of Chebyshev expansions p(x)=\sum_{k=0}^n c_k T_k(x) by introducing the Laurent-Horner method, a two-step, linear-time approach. It first converts the problem to a Laurent polynomial via the inverse Joukowski map x=J(z)=(z+z^{-1})/2 and then applies interval Horner to obtain guaranteed enclosures. Unlike prior spectral methods that rely on eigenvector transformations, Laurent-Horner avoids V and V^{-1}, reducing wrapping effects near the domain endpoints x=±1. Extensive experiments on high-degree expansions (e.g., n=9150) show Laurent-Horner is faster and often yields the smallest enclosure radii among competing methods, validating its practicality for computer-assisted proofs and forward-error analysis in Chebyshev-based models.

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.

Figures (2)

• Figure 1: Radius of enclosures for a degree 9150 interval polynomial at 1000 intervals. Only 50 points are depicted to make the curves easier to distinguish.
• Figure 2: Average number of correct digits (left) and computing time (right) of different methods for bounding the range of a Chebyshev expansion of degree 9150 at 1000 points.