Preconditioned Conjugate Gradient Algorithms for Homotopy CurveTracking

Irani, Kashmira M. and Ribbens, Calvin J. and Walker, Homer F. and Watson, Layne T. and Kamat, Manohar P. (1989) Preconditioned Conjugate Gradient Algorithms for Homotopy CurveTracking. Technical Report TR-89-41, Computer Science, Virginia Polytechnic Institute and State University.

Abstract

These are alogorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking 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 alogorithms 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. Here variants of the conjugate gradient algorithms are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems.