Certified numerical homotopy tracking

TL;DR

This paper introduces a rigorous algorithm for homotopy path tracking in polynomial systems, leveraging alpha theory to ensure accuracy and practicality, with experimental validation in Macaulay2.

Contribution

It presents a new certified homotopy tracking algorithm based on alpha theory, with proven complexity and demonstrated effectiveness through computational experiments.

Findings

Algorithm successfully tracks homotopy paths with certification.

Experimental results confirm theoretical complexity bounds.

Supports conjecture on optimal initial pairs for homotopy continuation.

Abstract

Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with the implementation in the numerical algebraic geometry package of Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.