Efficient Three-point Iterative Methods for Solving Nonlinear Equations

Abstract

In this paper, we present a new family of three-point Newton type iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Numerical examples are demonstrated to confirm theoretical results. Introduction Solving nonlinear equations is one of the most important problems in scientific computation. Since 1960’s, many iterative methods have been proposed for solving nonlinear equations, see [1-5] and the references therein. Specifically, Wang and Zhang in [1] developed the following iterative method 2 1 ( ) , ( ) ( ) ( ) (1 2 ), 2 ( ) ( ) n n n n n n n n n n n n f x y x f x f x f y x y s s f x f x λ τ λ +  = −  ′ +    = − + + ′  +  (1) where , R λ τ ∈ and ( ) / ( ). n n n s f y f x = This paper is organized as follows. In Section 2, based on Wu’s method, we derive a new family of Newton type iterative methods without memory for solving nonlinear equations. We prove that the order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Numerical examples are given in Section 4 to confirm theoretical results. The New Methods without Memory Now, we consider the following iteration scheme