We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
23/2019
Adaptive Step Size Control for Polynomial Homotopy Continuation Methods
In this paper we develop an adaptive step size control for the numerical tracking of implicitly defined paths in the context of polynomial homotopy continuation methods. We focus on the case where the paths are tracked using a predictor-corrector scheme with only a prescribed maximal number of allowed correction steps. The adaptive step size control changes the step size based on computational estimates of local geometric information, in particular a local Lipschitz constant and the local error of the used predictor method, as well as its order. The developed adaptive step size control is implemented in the software package HomotopyContinuation.jl and its efficiency over the currently commonly used adaptive step size control is demonstrated on several examples.
