Local and semilocal analysis of a class of fourth order methods under common set of assumptions

Abstract

This study presents an efficient class of fourth-order iterative methods introduced by Ali Zein (2024) in a more abstract Banach space setting. The Convergence Order of this class is proved by bypassing the Taylor expansion. We use the mean value theorem and relax the differentiability assumptions of the involved function. At the outset, we provide a semilocal analysis, and then, using the results and the same set of assumptions, we study the local convergence. This approach has the advantage that we do not need to use any assumptions on the unknown solution to study the local convergence. This technique can be used to extend the applicability of other methods along the same lines. Examples from both the chemical and the physical sciences are studied to analyze the performance of the class. The dynamics of the class are also studied.