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Open AccessArticle

Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

1
Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China
2
Department of Mathematics, Saveetha Engineering College, Chennai 602105, India
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Authors to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1052; https://doi.org/10.3390/math7111052
Received: 8 September 2019 / Revised: 12 October 2019 / Accepted: 16 October 2019 / Published: 4 November 2019
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane. View Full-Text
Keywords: Kung–Traub conjecture; multipoint iterations; nonlinear equation; optimal order; basins of attraction Kung–Traub conjecture; multipoint iterations; nonlinear equation; optimal order; basins of attraction
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Li, J.; Wang, X.; Madhu, K. Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction. Mathematics 2019, 7, 1052.

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