A Local Convergence Theory for Combined Inexact-Newton/Finite-Difference Projection Methods

Finite-difference versions of some recently developed Krylov subspace projection methods are presented and analysed in the context of solving systems of nonlinear equations using Inexact-Newton Methods. The specific projection methods considered are Arnoldi’s Method, the Generalized Minimum Residual Method (GMRES), and the Generalized Conjugate Residual Method (GCR). A local convergence theory is given for the combined Inexact-Newton/Finite-Difference Projection Methods, and the main tool used in the convergence proofs relates the results of the finite-difference method to those obtained by applying the regular projection method to a perturbed problem. In addition, several results are given which show that, under certain mild restrictions, the approximate Newton steps computed by any of the above projection methods are descent directions for the full nonlinear problem.