A Quasi-Newton Method Employing Direct Secant Updates of Matrix Factorizations

Abstract

A quasi-Newton algorithm is presented which employs an initial factorization of the approximate Jacobian matrix. The method then updates the upper and lower triangular factors directly at each step. Iterates are generated using forward and backward substitution employing the updated factorizations. The method is shown to be locally and q-superlinearly convergent, and a numerical algorithm is presented.

Some secant approximations for Rosenbrock W-methods

“…Now, because of the sparsity assumptions, it would be a nonsense to consider the application of formula (5) for the computation of (I − h m+1 γ W m ) −1 , and it is more suitable to use a sparse LU factorization of (I − h m+1 γ W m ). Actually, there exists a formula for updating the LU factorization [7] but it is based on the updating of U and L −1 . Since L −1 generally looses the original bandwidth, this approach does not seem reliable.…”

“…Since we are mainly interested in these kind of problems the methods have been implemented in this manner. For the good Broyden's update, since we also need the matrix W m to perform the update of (I − h m+1 γ W m ) −1 we use the recursion (similar to (38)) arising from (7). In this way, with respect to a standard ROW-method these two update approaches are able to work with only one factorization (unless we need to recompute the Jacobian) and the cost of the computation of the Jacobian is substituted by the cost for the computation of the update vectors.…”

On the relation between two local convergence theories of least-change secant update methods

“…In the decade of the 80's some new methods appeared which preserve the structure of the true Jacobian in a way not covered by the Dennis-Walker theory. We have mainly in mind the family of Partitioned Quasi-Newton methods [20,21,22,23,35], the family of superlinear methods with direct secant updates of matrix factorizations [25,5,27], and the Secant Finite Difference method of Dennis and Li [11].…”

Local Convergence Theory of Inexact Newton Methods Based on Structured Least Change Updates

“…Their method is not locally convergent unless implemented with a convenient restart procedure. However, using their idea, Johnson and Austria [24], and Chadee [5] introduced superlinearly convergent direct secant update methods, and Martinez [31] introduced a family of quasi-Newton methods with direct secant updates of matrix factorizations with superlinear convergence properties.…”