Local convergence for multistep high order methods under weak conditions

We present a local convergence analysis for an eighth-order convergent method in order to find a solution of a nonlinear equation in a Banach space setting. In contrast to the earlier studies using hypotheses up to the seventh Fréchet derivative, we use only hypotheses on the first-order Fréchet derivative and Lipschitz constants. This way, we not only expand the applicability of these methods but also propose a computable radius of convergence for these methods. Finally, concrete numerical examples demonstrate that our results apply to nonlinear equations not covered before.