Local and semi-local convergence analysis of a multi-step method

This paper is concerned with an m-step (m 2) method of order m + 1, which was first studied by Ahmad, Tohidi, and Carrasco in [Numer.Algorithms 71 (2016) 631-653] to solve nonlinear systems defined on the finite Euclidean space.The method depends on a single real parameter and uses one inverse per a complete step.But the local convergence was shown using Taylor series expansions and by assuming the existence of the seventh derivative; this assumption, however, does not pertain to the method itself.Other constraints include the lack of a priori error estimates and isolation of the solution results.The semi-local convergence has not been considered previously either.In this paper, both local and semi-local convergence analyses are performed using only the operators inherent to the method, namely the operator and its derivative.Moreover, the convergence analysis is carried out in the more general setting of Banach space, employing generalized continuity to control the derivative and to sharpen the error distances.In addition, isolation of the solution results are provided.The proposed technique is quite general and can be employed to extend the applicability of other iterative methods.