Preconditioned Iterative Methods for Homotopy Curve Tracking

Abstract

Homotopy algorithms are a class of methods for solving systems of nonlinear equations that are globally convergent with probability one. All homotopy algorithms are based on the construction of an appropriate homotopy map and then the tracking of a curve in the zero set of this homotopy map. The fundamental linear algebra step in these algorithms is the computation of the kernel of the homotopy Jacobian matrix. Problems with large, sparse Jacobian matrices are considered. The curve-tracking algorithms used here require the solution of a series of very special systems. In particular, each $(n + 1) \times (n + 1)$ system is in general nonsymmetric but has a leading symmetric indefinite $n \times n$ submatrix (typical of large structural mechanics problems, for example). Furthermore, the last row of each system may by chosen (almost) arbitrarily. The authors seek to take advantage of these special properties. The iterative methods studied here include Craig’s variant of the conjugate gradient algorithm and the SYMMLQ algorithm for symmetric indefinite problems. The effectiveness of various preconditioning strategies in this context are also investigated, and several choices for the last row of the systems to be solved are explored.