Stepsize control for adaptive multiprecision path tracking

When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. When precision and stepsize are adapted separately, the performance of the path tracker can be suboptimal and, in certain circumstances, failure may occur. This paper presents a strategy for adjusting precision and stepsize together to eliminate path failures while reducing 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. The objective of this paper is to describe a heurstic for adjusting precision and stepsize together to reduce the computational cost of tracking the path while maintaining high reliability. Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80528 (bates@math.colostate.edu, 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 (jhauenst@nd.edu, 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 (sommese@nd.edu, 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 (Charles.W.Wampler@gm.com, www.nd.edu/∼cwample1) This author was supported by NSF grant DMS-0410047 and NSF DMS-0712910.