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Comparison of Some Iterative Methods for Finding Simple Root of Nonlinear Equations
Current Issue
Volume 5, 2018
Issue 6 (November)
Pages: 167-170   |   Vol. 5, No. 6, November 2018   |   Follow on         
Paper in PDF Downloads: 63   Since Oct. 30, 2018 Views: 1009   Since Oct. 30, 2018
Authors
[1]
Mehdi Salimi, Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran.
[2]
Davood Nezami Behrooz, Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran.
Abstract
With the advancement of digital computer, advanced computer arithmetic and symbolic computation, several scholars presented a good number of iterative methods without memory for approximating a simple root of nonlinear equations. A main tool for solving nonlinear problems is the approximation of simple roots x* of a nonlinear equation f(x*) equal to zero, with a scalar function f, from D in R to R which is defined on an open interval D. In this paper, some third and fourth order methods without memory to compute the approximate simple roots of nonlinear equations in one variable presented in recent years are compared. Per iteration, the methods need two evaluations of the function and one evaluation of its first derivation. The efficacy of the present methods is tested on a number of numerical examples. The efficiency index of the methods is equal to 1.442 for non-optimal methods and 1.587 for optimal methods. Consequently, this method possesses very high computational efficiency. The principal aim and motivation in constructing the iterative methods for solving nonlinear equations is to reach the highest attainable order of convergence with the minimum number of function and derivation evaluations for each iteration. The main purpose of present research is to adjust the third and fourth order methods, which are rediscovery of Traub's method [1], for solving nonlinear equations for simple zeros with the equal order of convergence and the efficiency index. In the simple words, the methods are tested on several test problems and compared together with the same convergence order methods.
Keywords
Simple Root, Two-point Iterative Method, Kung and Traub Conjecture, Optimal Order of Convergence, Computational Efficacy
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