Stepsize control for 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. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy for adjusting precision and stepsize together to eliminate certain types of path failures that can occur when adapting the two quantities independently, while also 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. In fixed precision tracking, a trial-and-error approach to setting the stepsize is effective: shorten the step upon failure, and lengthen it upon repeated successes. If the level of precision is inadequate, the step may fail no matter how small the step is made, so the trial-and-error approach repeatedly shortens the stepsize until failure is declared due to lack of progress. 2000 Mathematics Subject Classification. Primary 65H10; Secondary 65H20, 65G50, 14Q99. Bates was supported by Colorado State University and the Institute for Mathematics and Its Applications (IMA). Hauenstein 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 DMS0712910. Sommese was supported by the Duncan Chair of the University of Notre Dame; and NSF grants DMS-0410047 and DMS-0712910. Wampler was supported by NSF grants DMS-0410047 and DMS-0712910.