Ball convergence of a novel Newton-Traub composition for solving equations

We present a local convergence analysis of a Newton-Traub composition method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The method was shown to be of convergence order five, if defined on the $ m- $dimensional Euclidean space using Taylor’s expansion and hypotheses reaching up to the fifth derivative (Hueso, Martinez, & Tervel, 2015; Sharma, 2014). We expand the applicability of this method using contractive techniques and hypotheses only on the first Fréchet-derivative of the operator involved. Moreover, we provide computable radius of convergence and error estimates on the distances involved not given in the earlier studies (Hueso et al., 2015; Sharma, 2014). Numerical examples illustrating the theoretical results are also presented in this study