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Mathematics 2019, 7(4), 339; https://doi.org/10.3390/math7040339

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

1
Department of Mathematics, King Abdualziz University, Jeddah 21589, Saudi Arabia
2
Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
3
Fac. de Ciencias Económicas, Universidad Laica “Eloy Alfaro de Manabí”, Manabí 130214, Ecuador
4
Departamento de Matemática Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
*
Author to whom correspondence should be addressed.
Received: 15 January 2019 / Revised: 21 March 2019 / Accepted: 22 March 2019 / Published: 9 April 2019
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Abstract

The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results. View Full-Text
Keywords: nonlinear equations; multiple roots; Chebyshev–Halley-type; optimal iterative methods; efficiency index nonlinear equations; multiple roots; Chebyshev–Halley-type; optimal iterative methods; efficiency index
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Behl, R.; Martínez, E.; Cevallos, F.; Alarcón, D. A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics 2019, 7, 339.

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