A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

TL;DR

This paper introduces a robust numerical path tracking algorithm for polynomial homotopy continuation that uses adaptive stepsize control and Padé techniques to prevent path jumping, often functioning effectively in double precision.

Contribution

The paper presents a novel path tracking algorithm that enhances robustness and efficiency in polynomial homotopy continuation, with adaptive stepsize and difficulty detection features.

Findings

Effective prevention of path jumping in numerical path tracking.

Comparable or improved performance in numerical examples.

Potential to operate reliably in double precision arithmetic.

Abstract

We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is `robust' in the sense that it is designed to prevent path jumping and in many cases, it can be used in (only) double precision arithmetic. It is based on an adaptive stepsize predictor that uses Pad\'e techniques to detect local difficulties for function approximation and danger for path jumping. We show the potential of the new path tracking algorithm through several numerical examples and compare with existing implementations.