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

Abstract

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 require three 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 without memory based on n evaluations could achieve optimal convergence order 2n−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.

A New Class of Optimal Eighth Order Method with Two Weight Functions for Solving Nonlinear Equation

“…Per iteration, the present methods require four function evaluations and therefore have the efficiency index equal to 1.682. This method has less number of weight functions which is a significant contribution when compared to a similar method found in [8]. Some of the proposed methods were also compared for their performance and efficiency with various other iteration methods of the same order and it was found that the new class of method produces better numerical results.…”

“…Thus the optimal order for three evaluations per iteration would be four, four evaluations per iteration would be eight and so on. Recently, some fourth and eighth order optimal iterative methods have been developed using weight functions (see [1,4,5,7,8,10,11,12,13,14] and references therein). A more extensive list of references as well as a survey on the progress made in the class of multi-point methods is found in the recent book by Petkovic et al [10].…”

“…A more extensive list of references as well as a survey on the progress made in the class of multi-point methods is found in the recent book by Petkovic et al [10]. In [8], a three step optimal eighth-order Ostrowski-type family of iterative method by using three weight functions is given. In this paper, we have presented a new class of optimal three-step eighth order method with two weight functions.…”

An Efficient Class of Multipoint Root-Solvers With andWithout Memory for Nonlinear Equations

“…Next, based on this new class of methods, we present an efficient class of multipoint methods with memory by suitable variation of a free parameter in each iterative step in Section 3. Numerical examples show better performance of our method than the Lotfi and Eftekhari's method (2), Kung and Traub's method (3), Thukral's method (4), and Eftekhari's method (5) in Section 4. Section 5 is a short conclusion.…”

“…Cordero et al in [1] extended the approaches of Zheng et al to provide a novel class of iterative methods with memory that use self-accelerating parameters calculated by Newton's interpolation with divided differences. In [5], a new family of optimal eighth-order Ostrowski-type iterative methods by using the method of weight functions has been derived by Lotfi and Eftekhari, which is written as:…”