A new fifth-order iterative method for solving non-linear equations using weight function technique and the basins of attraction

Volume 28, Issue 3, pp 281--293 http://dx.doi.org/10.22436/jmcs.028.03.06

Publication Date: June 26, 2022 Submission Date: July 28, 2021 Revision Date: September 25, 2021 Accteptance Date: May 20, 2022

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Authors

M. Q. Khirallah - Department of Mathematics and Computer Science, Faculty of Science, Ibb University, Yemen. - Department of Mathematics, Faculty of Science and Arts, Najran University, Najran 1988, Saudi Arabia. A. M. Alkhomsan - Department of Mathematics, Faculty of Science and Arts, Najran University, Najran 1988, Saudi Arabia.

Abstract

In this paper, new iterative method is presented of fifth-order for solving non-linear equations \(f\left(x\right)=0\) a devoid of the second derivative which requires two derivative functions and evaluations for each step, using both weight functions and synthesis techniques together. This method improves Newton's method and thus the efficiency index has been improved from \(1.414\) to \(1.495\). The convergence analysis for the new method is discussed. We provide some numerical examples that illustrate the performance of our proposed method by comparing them with numerical methods of fifth-order also the complex dynamics and basins of attraction is discussed, comparing it with several methods of the same order, thus comparisons show that new method gives the best results.

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Keywords

• Nonlinear equations
• basins of attraction efficiency index
• iterative methods
• complex dynamics

MSC

•  41A25
•  65H05
•  65K05

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