An Approach for Certifying Homotopy Continuation Paths: Univariate Case

Abstract

Homotopy continuation is a well-known method in numerical root-finding. Recently, certified algorithms for homotopy continuation based on Smale's alpha-theory have been developed. This approach enforces very strong requirements at each step, leading to small step sizes. In this paper, we propose an approach that is independent of alpha-theory. It is based on the weaker notion of well-isolated approximations to the roots. We apply it to univariate polynomials and provide experimental evidence of its feasibility.