High-dimensional homotopy curve tracking on a shared-memory multiprocessor

Abstract

Results are reported for a series of experiments involving numerical curve tracking on a shared-memory parallel computer. Several algorithms exist for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, that is, with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then the tracking of some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. A parallel version of HOMPACK is implemented on a shared-memory parallel computer with various levels and degrees of parallelism (e.g., linear algebra, function, and Jacobian matrix evaluation), and a detailed study is presented for each of these levels with respect to the speedup in execution time obtained with the parallelism, the time spent implementing the parallel code, and the extra memory allocated by the parallel algorithm.