A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

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.