Certified homotopy tracking using the Krawczyk method

TL;DR

This paper introduces a new, general approach for certifying the correctness of approximate solution paths in polynomial systems using a parametric Krawczyk method, with theoretical guarantees and competitive experimental results.

Contribution

It presents a novel, broadly applicable certification method for homotopy continuation based on the Krawczyk method, including a new preconditioning strategy and theoretical analysis.

Findings

Method is applicable to general parameter homotopies.

Theoretical correctness and termination are established.

Experimental results show competitive performance.

Abstract

We revisit the problem of certifying the correctness of approximate solution paths computed by numerical homotopy continuation methods. We propose a conceptually simple approach based on a parametric variant of the Krawczyk method from interval arithmetic. Unlike most previous methods for certified path-tracking, our approach is applicable in the general setting of parameter homotopies commonly used to solve polynomial systems of equations. We also describe a novel preconditioning strategy and give theoretical correctness and termination results. Experiments using a preliminary implementation of the method indicate that our approach is competitive with specialized methods appearing previously in the literature, in spite of our more general setting.