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Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines

1
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2
Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
3
ISEP-Institute of Engineering, Polytechnic of Porto Department of Electrical Engineering, 431 4294-015 Porto, Portugal
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(4), 586; https://doi.org/10.3390/sym11040586
Received: 27 March 2019 / Revised: 16 April 2019 / Accepted: 17 April 2019 / Published: 23 April 2019
(This article belongs to the Special Issue Symmetry in Complex Systems)
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PDF [265 KB, uploaded 23 April 2019]

Abstract

This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function Ω in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique. View Full-Text
Keywords: divided difference; radius of convergence; Kung–Traub method; local convergence; Lipschitz constant; Banach space divided difference; radius of convergence; Kung–Traub method; local convergence; Lipschitz constant; Banach space
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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MDPI and ACS Style

Behl, R.; Argyros, I.K.; Mallawi, F.O.; Tenreiro Machado, J.A. Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines. Symmetry 2019, 11, 586.

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