Local convergence for multipoint methods using only the first derivative

Abstract

We present a local convergence analysis of two methods in order to approximate a locally unique solution of an equation in a Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative (or even higher order derivatives) although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness results for the solution based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.