Open Access

ARTICLE

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

Obadah Said Solaiman¹, Samsul Ariffin Abdul Karim², Ishak Hashim^1,*

1 Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi Selangor, 43600, Malaysia
2 Department of Fundamental and Applied Sciences, Center for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Seri Iskandar, Perak DR, 32610, Malaysia

* Corresponding Author: Ishak Hashim. Email:

Computers, Materials & Continua 2021, 67(2), 1951-1962. https://doi.org/10.32604/cmc.2021.015344

Received 16 November 2020; Accepted 06 December 2020; Issue published 05 February 2021

Abstract

There are several ways that can be used to classify or compare iterative methods for nonlinear equations, for instance; order of convergence, informational efficiency, and efficiency index. In this work, we use another way, namely the basins of attraction of the method. The purpose of this study is to compare several iterative schemes for nonlinear equations. All the selected schemes are of the third-order of convergence and most of them have the same efficiency index. The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees. As a comparison, we determine the CPU time (in seconds) needed by each scheme to obtain the basins of attraction, besides, we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods. Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders, furthermore, they vary for iterative methods of the same order even if they have the same efficiency index. Consequently, this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

---

Keywords

---