Ball Convergence for Higher Order Methods Under Weak Conditions

Abstract

We present a local convergence analysis for higher order methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies, Taylor expansions and hypotheses on higher order Fréchet-derivatives are used. We expand the applicability of these methods using only hypotheses on the first Fréchet derivative. Moreover, we obtain a radius of convergence and computable error bounds using Lipschitz constants not given before. Numerical examples are also presented in this study.