A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods requirethree evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration withoutmemory based on n evaluations could achieve optimal convergence order2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods