Extended Traub–Woźniakowski convergence and complexity of Newton iteration in Banach space

Abstract

An optimal convergence condition for Newton iteration is presented which is at least as weak as the one obtained by Traub and Woźniakowski leading also to an at least as precise complexity. The novelty of the paper is the introduction of a restricted convergence domain. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local convergence as well as the complexity of Newton iteration. Numerical examples further justify the theoretical results.