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Expanding the applicability of an eighth-order method in banach space under weak conditions

Abstract:

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