On a Three-Step Efficient Fourth-Order Method for Computing the Numerical Solution of System of Nonlinear Equations and

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RESEARCH PAPER

On a Three-Step Efficient Fourth-Order Method for Computing the Numerical Solution of System of Nonlinear Equations and Its Applications Anuradha Singh1

Received: 18 August 2017 / Revised: 22 April 2019 / Accepted: 17 May 2019 Ó The National Academy of Sciences, India 2019

Abstract In this paper, a new fourth-order iterative scheme for finding the zeros of systems of nonlinear equations has been built and analyzed. Theoretical proof has been given to confirm the convergence order of the new method. The effectiveness of the proposed method is shown by the comparison of traditional as well as flops-like efficiency index with recent existing same order schemes. Numerical examples confirm that the new iterative method is efficient and gives tough competition to some existing fourth-order methods. We have also discussed the application of our proposed method for finding numerical solution of nonlinear ODE and PDE. Keywords Nonlinear equation  Nonlinear system  Order of convergence  Efficiency index Mathematics Subject Classification 65H10  41A25  65N22

1 Introduction In many applications, the system of equations to be solved arises in the attempt to find a minimizer or a critical point of a related nonlinear functional. We consider the following system of nonlinear equations FðxÞ ¼ 0;

& Anuradha Singh [email protected] 1

Department of Basic Science and Engineering, Indian Institute of Information Technology, Nagpur, Maharashtra 440006, India

ð1Þ

has a zero (root) n 2 D where FðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; . . .; fn ðxÞt and x ¼ ½x1 ; x2 ; . . .; xn t . The most celebrated Newton’s method for several variables is defined by xðkþ1Þ ¼ xðkÞ  Jk1 FðxðkÞ Þ; k ¼ 0; 1; 2; . . .;

ð2Þ

where Jk1 is the Jacobian matrix of the functions f1 ; f2 ; . . .; fn evaluated at xðkÞ . Ortega et al. [1] have discussed the theoretical proof of Newton’s method and some of its variations. Inspired from Newton’s method, many researchers have developed different higher-order methods for finding a solution of Eq. (1). In recent days, some works have been done to develop higher-order iterative methods for solving nonlinear systems [2–4]. Noor and Waseem [5] and Darvishi and Barati [6] have developed third-order variants of Newton’s method. In particular, Darvishi and Barati [7] implemented a fourth-order iterative scheme which needed two functions and three derivative evaluations. Cordero et al. [8] and Abad et al. [9] have presented fourth- and fifth-order nonlinear solvers each of which requires two functions and two derivative evaluations. The evaluation of derivative increases the cost and time of the system. The main aim of this paper is to build the higher-order iterative method in several variables in order to minimize the number of function evaluations (mainly derivatives) and matrix inversion. The efficiency index of iterative schemes increases by minimizing the number of evaluations. Various higher-order methods for solving nonlinear systems require the computation of two or more Jacobian matrices; for a