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

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 τn and αn. We also propose some generating functions for τn and αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.