An analysis of a family of Maheshwari-based optimal eighth order methods

Keywords: Iterative methods Order of convergence Basin of attraction Extraneous fixed points Weight functions a b s t r a c t In this paper we analyze an optimal eighth-order family of methods based on Mahesh-wari's fourth order method. This family of methods uses a weight function. We analyze the family using the information on the extraneous fixed points. Two measures of closeness of an extraneous points set to the imaginary axis are considered and applied to the members of the family to find its best performer. The results are compared to a modified version of Wang–Liu method. ''Calculating zeros of a scalar function f ranks among the most significant problems in the theory and practice not only of applied mathematics, but also of many branches of engineering sciences, physics, computer science, finance, to mention only some fields'' [1]. For example, to minimize a function FðxÞ one has to find the points where the derivative vanishes, i.e. F 0 ðxÞ ¼ 0. There are many algorithms for the solution of nonlinear equations, see e.g. Traub [2], Neta [3] and the recent book by Petkovicét al. [1]. The methods can be classified as one step and multistep. One step methods are of the form