Ball Convergence for Two Optimal Eighth-Order Methods Using Only the First Derivative

Abstract

We present a local convergence analysis of two optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation defined on the real line or the complex plane. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative 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 of the solution with results 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.