Three-step iterative methods with optimal eighth-order convergence

Abstract

In this paper, based on Ostrowski's method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub's conjecture. Numerical comparisons are made to show the performance of the new family.

A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

“…In [9] Kung and Traub conjectured that the iterative method which requires n + 1 function evaluations per iteration can reach at most 2 n convergence order in general. The methods that satisfy KungTraub conjecture are known as optimal methods (see [10], [11], [12], [13], [14], [15], [16], [14], [18], [19]). This research presents a new family of such optimal eighth order methods.…”

An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

“…It has been about half a century since Traub [1] performed in the 1960s the extensive analyses on qualitative as well as quantitative viewpoints of iterative methods locating numerical roots for nonlinear equations. A number of authors [2][3][4][5][6][7] have developed high-order multipoint methods to solve a given nonlinear equation in the form of f (x) = 0. In 2011, Džunić et al [5] extensively investigated a family of optimal three-point methods for solving nonlinear equations using two parametric functions.…”

“…They showed that many new methods are just special cases or reformulation of known methods. A numerical scheme is said to be optimal according to Kung-Traub's conjecture [9] that any multipoint method [8] without memory can attain its convergence order of at most 2 r−1 for r functional evaluations with r ∈ N. For the purpose of comparison, we employ several existing eighth-order methods in [3,4,7], being respectively presented by (1.1), (1.5), and (1.6).…”

Comparative study of eighth-order methods for finding simple roots of nonlinear equations

“…Sharma and Sharma [38] have used α = 1. (xvii) Cordero et al's optimal family of methods [39] y n = x n − u n ,…”

“…Remark: Cordero et al [39] have used β 1 = β 3 = 0 and β 2 = 1. (xviii) Chun and Lee's optimal family of methods [40] y n = x n − u n ,…”