A New One Parameter Family of Iterative Methods With Eighth-Order of Convergence for Solving Nonlinear Equations

Abstract

In this paper, a new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations is presented and analyzed. This new family of iterative methods is obtained by composing an iterative method proposed by Chun [3] with Newton's method and approximating the first-appeared derivative in the last step by a combination of already evaluated function values. The proposed family is optimal since its efficiency index is 8 1/4 ≈ 1.6818. The convergence analysis of the new family is studied in this paper. Several numerical examples are presented to illustrate the efficiency and accuracy of the family.

Ball comparison for three optimal eight order methods under weak conditions

“…The approach in this paper establishes the local convergence result under hypotheses only on the first derivative and give a larger convergence ball than the earlier studies, under weaker hypotheses. Notice that in earlier studies [19,23] the convergence is shown under hypotheses on the eighth derivative. The same technique can be used to other methods.…”

“…where F : D ⊆ S → T is a Fréchet-differentiable operator defined on a convex set D, where S, T are subsets of R or C. Equation of the form (1.1) is used to study problems in Computational Sciences and other disciplines [3,7,14,16,20]. Newton-like iterative methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] are famous for approximating a solution of the equation (1.1).…”

“…In this paper, we study the local convergence analysis of the methods defined for each n = 0, 1, 2, • • • by Siyyam et al [19]…”

“…f (x) = 6x ln x 2 + 20x 3 + 12x 2 + 10x and f (x) = 6 ln x 2 + 60x 2 − 24x + 22. Notice that f (x) is unbounded on D. Hence, the results in [19,23], cannot apply to show the convergence of method (1.2) (see also the numerical examples).…”

Three-Step Iterative Method for Solving Nonlinear Equations

“…J. Yun [10] proposed a three-step iterative method, which is a significant improvement of the method proposed by Noor and Noor [8]. Siyyam [9] derived and analyzed a new fourth-step iterative method with fifth order of convergence. Al-Subaihi and Alqarni [3] developed optimal three-step methods with eighth order of convergence.…”