Direct secant updates of matrix factorizations

Abstract

This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newton-like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and q-superlinearly convergent, and some encouraging numerical results are presented.