Evaluation of Chebyshev polynomials on intervals and application to root finding

TL;DR

This paper analyzes the evaluation of Chebyshev polynomials on intervals, introduces a ball arithmetic-based variant of the Clenshaw algorithm that reduces interval width growth, and applies it to develop an efficient root-finding method.

Contribution

It presents a novel ball arithmetic-based Clenshaw algorithm variant that improves interval width control and demonstrates its application in root-finding.

Findings

Interval width grows quadratically with degree using the new method

The new algorithm is efficient for small interval widths

Application to root finding improves accuracy and efficiency

Abstract

In approximation theory, it is standard to approximate functions by polynomials expressed in the Chebyshev basis. Evaluating a polynomial $f$ of degree n given in the Chebyshev basis can be done in $O(n)$ arithmetic operations using the Clenshaw algorithm. Unfortunately, the evaluation of $f$ on an interval $I$ using the Clenshaw algorithm with interval arithmetic returns an interval of width exponential in $n$. We describe a variant of the Clenshaw algorithm based on ball arithmetic that returns an interval of width quadratic in $n$ for an interval of small enough width. As an application, our variant of the Clenshaw algorithm can be used to design an efficient root finding algorithm.