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+2≈2.414 for the new method. Secondly, the efficiency index (convergence order per function evaluation) is improved from 2≈1.414 to (1+2)1/2≈1.553. We illustrate some of our results with a numerical example.