Stepsize control for adaptive multiprecision path tracking

Numerical difficulties encountered when following paths using methods such as homotopy continuation may be overcome by combining adaptive stepsize and adaptive multiprecision. In the paper Adaptive multiprecision path tracking [1], precision and stepsize are adapted separately. This can lead to suboptimal performance and even failure in certain circumstances. This paper presents a strategy for adjusting precision and stepsize together to eliminate path failures while minimizing the computational effort expended per unit advance along the path. This paper concerns path tracking algorithms for tracing out a one dimensional path defined implicitly by n equations in n + 1 unknowns. In particular, we consider such algorithms when multiprecision calculations are available, that is, when the precision of the computations can be changed during the computation. We treat a common type of path tracker that uses an Euler predictor to step ahead along the tangent to the path and a Newton corrector to bring the predicted point closer to the path. Our objective is to adjust precision and stepsize together to minimize the computational cost of tracking the path while maintaining high reliability. ∗Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80528 ([email protected], http://www.math.colostate.edu/∼bates). This author was supported by Colorado State University and the Institute for Mathematics and Its Applications (IMA). †Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 ([email protected], http://www.nd.edu/∼jhauenst). This author was supported by the Duncan Chair of the University of Notre Dame; the University of Notre Dame Center for Applied Mathematics; and NSF grants DMS-0410047 and NSF DMS-0712910. ‡Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 ([email protected], http://www.nd.edu/∼sommese). This author was supported by the Duncan Chair of the University of Notre Dame; and NSF grants DMS-0410047 and NSF DMS0712910. §General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090 ([email protected], www.nd.edu/∼cwample1) This author was supported by NSF grant DMS-0410047 and NSF DMS-0712910.