Triangular Decomposition Methods for Solving Reducible Nonlinear Systems of Equations

Abstract

This paper generalizes to the nonlinear case a standard way to solve general sparse systems of linear equations. In particular, Duff [J. Inst. Math. Appl., 19 (1977), pp. 339–342] has suggested that row and column interchanges be applied to permute the coefficient matrix of a linear system into block lower triangular forma The linear system then can be solved by using the associated block Gauss–Seidel or forward block substitution scheme. This is the approach taken in the Harwell MA28 routine. If more than one matrix with the same sparsity pattern is to be factored, then the reordering need not be redone. In extending this approach to nonlinear problems, it is necessary to assume as in the linear case that component equations can be evaluated separately from equations in other blocks. The algorithm for doing the reordering is very fast, and if the equations can be put into block lower triangular form with each block size much less than the dimension of the system, then a large savings is in order because only the diagonal blocks need to be factored. In the nonlinear variants considered here, there are additional advantages. Not only are just the diagonal blocks of the Jacobian matrix computed and factored, but the off diagonal partial derivatives need not even exist. Numerical tests and analytic results affirm the intuition that these variants are superior locally to Newton’s method. Current work is concerned with globalizing these methods and with variants especially suited to parallel implementations.