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

Jian Li, Xiaomeng Wang and Kalyanasundaram Madhu
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Jian Li: Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China
Xiaomeng Wang: Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China
Kalyanasundaram Madhu: Department of Mathematics, Saveetha Engineering College, Chennai 602105, India

Mathematics, 2019, vol. 7, issue 11, 1-15

Abstract: 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.

Keywords: Kung–Traub conjecture; multipoint iterations; nonlinear equation; optimal order; basins of attraction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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