A priori bounds for certified Krawczyk homotopy tracking

TL;DR

This paper provides the first complexity analysis for Krawczyk-based certified homotopy tracking, offering explicit stepsize bounds and iteration estimates that improve efficiency and reduce computational overhead.

Contribution

It introduces explicit a priori stepsize bounds and iteration bounds for Krawczyk homotopy tracking, enhancing its efficiency and reliability.

Findings

Fewer iterations needed compared to previous methods.

Validated bounds through experiments with a proof-of-concept implementation.

Reduced interval arithmetic overhead.

Abstract

We establish the first complexity analysis for Krawczyk-based certified homotopy tracking. It consists of explicit a priori stepsize bounds ensuring the success of the Krawczyk test, and an iteration count bound proportional to the weighted length of the solution path. Our a priori bounds reduce the overhead of interval arithmetic, resulting in fewer iterations than previous methods. Experiments using a proof-of-concept implementation validate the results.