Extended local convergence analysis of optimal eighth order method for solving equations in Banach space

Abstract

A local convergence analysis is developed for an eight-order method to solve Banach space defined nonlinear equation under $\omega$-continuity. Earlier efforts require the existence of the ninth derivative to show the convergence on the finite Euclidean space $\mathbb R^k$. However, high order derivatives do not appear in the method. Moreover, no error estimates are available. Therefore, the previous efforts cannot assure the convergence if these derivatives do not exist although the method may converge. The present article addresses these problems. In particular, the new convergence conditions require only the existence of the first derivative appearing in the method. Moreover, error estimates become available. Furthermore, a region is determined containing only one solution of the equation. The novelty of the developed process allows its usage on other methods, since it is independent of the method. The numerical example complements the theory.