Certified evaluations of H\"older continuous functions at roots of polynomials

TL;DR

This paper introduces a new method for obtaining certified estimates of H"older continuous functions evaluated at polynomial roots, extending existing techniques beyond analytic functions.

Contribution

It develops and analyzes an alternative approach for certified evaluation of H"older continuous functions at polynomial roots, which was previously challenging.

Findings

Method is effective for H"older continuous functions

Implementation in Maple shows efficiency

Extends certified evaluation techniques beyond analytic functions

Abstract

Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an objective function at critical points. For analytic evaluation functions, Newton's method naturally applies to yield certified estimates. These estimates no longer apply, however, for H\"older continuous functions, which are a generalization of Lipschitz continuous functions where continuous derivatives need not exist. This work develops and analyzes an alternative approach for certified estimates of evaluating locally H\"older continuous functions at roots of polynomials. An implementation of the method in Maple demonstrates efficacy and efficiency.