On two new families of iterative methods for solving nonlinear equations with optimal order

In this paper, two new families of eighth-order iterative methods for solving nonlinear equations is presented. These methods are developed by combining a class of optimal two-point methods and a modified Newton’s method in the third step. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to 1:682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus the new families of eighth-order methods agrees with the conjecture of Kung-Traub for the case n = 4. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.