Adaptive multiprecision path tracking

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.