Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

¹ Institute of Mathematics, National University of Mongolia, Mongolia
² Joint Institute for Nuclear Research, Dubna, 141980 Moscow region, Russia
³ School of Applied Sciences, Mongolian University of Science and Technology, Mongolia

^* e-mail: tzhanlav@yahoo.com
^** e-mail: chuka@jinr.ru
^*** e-mail: ulzii@jinr.ru

Published online: 14 February 2018

Abstract

In this paper we propose a generating function method for constructing new two and three-point iterations with p (p = 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions for p-th (p = 4, 8) order convergence of the proposed iterations are given in terms of parameters τ_n and α_n. We also propose some generating functions for τ_n and α_n. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.