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A third-order Newton-type method to solve systems of nonlinear equations.pdf

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A third-order Newton-type method to solve systems of nonlinear equations

Applied Mathematics and Computation 187 (2007) 630–635 /locate/amcA third-order Newton-type method to solve systems of nonlinear equations M.T. Darvishi *, A. Barati Department of Mathematics, Razi University, Kermanshah 67149, IranAbstract In this paper, we present a third-order Newton-type method to solve systems of nonlinear equations. In the first we present theoretical preliminaries of the method. Secondly, we solve some systems of nonlinear equations. All test problems show the third-order convergence of our method. 2006 Elsevier Inc. All rights reserved. Keywords: Systems of nonlinear equations; Newton-type method; Third-order convergence1. Introduction In recent papers [4,5] a new family of third-order convergence methods have been obtained, by using an integral interpolation of Newton’s method to solve equation f(x) = 0. There has been another approach based on the Adomian decomposition method on developing iterative method to solve the equation f(x) = 0 (see [3]). Recently, there has been some progress on Newton-type methods with cubic convergence that do not require the computation of second derivatives [6]. Chun [2] by improving Newton–Raphson method presented a new iterative method to solve nonlinear equations. His work is based on modification of the Abbasbandy’s proposal [1] on improving the order of accuracy of Newton–Raphson method. Frontini and Sormani [4] pre- sented a third-order iterative method for solving systems of nonlinear equations. In this paper, we construct a high order iterative method based on Adomian decomposition method to solve the systems of nonlinear equations. This paper is organized as follows: In the next section we introduce an iterative method based on Adomian decomposition method to solve f(x) = 0, this method introduced by Chun [2]. In Section 3 we extend the Chun’s method to solve systems of nonlinear equations. In that section we state and prove a theorem that shows the cubic convergence of the method. Numerical result