A Newton-type midpoint method with high efficiency index

In this paper we present a Newton-type two-step method to approximate a solution of a nonlinear equation in Banach spaces. We establish a semilocal convergence theorem under Newton-Kantorovich-type conditions, both the convergence order and the efficiency index of the developed method are found. The iterative procedure presented here is a simple modification of the Newton-Kantorovich method, however, with the same number of function and derivative evaluations at each iteration, it is improved in two important aspects: Firstly, the convergence order is increased from 2 for the Newton-Kantorovich method to 1 + root 2 approximate to 2.414 for the new method. Secondly, the efficiency index (convergence order per function evaluation) is improved from root 2 approximate to 1.414 to (1 + root 2)(1/2) approximate to 1.553. We illustrate some of our results with a numerical example. (C) 2020 Elsevier Inc. All rights reserved.