Computers in mathematical research: the study of three-point root-finding methods
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Computers in mathematical research: the study of three-point root-finding methods Ivan Petkovi´c1 · -Dord¯e Herceg2 Received: 21 May 2019 / Accepted: 12 August 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract This paper is motivated by results of extensive comparative study of three-point methods for solving nonlinear equations presented in the paper (C. Chun, B. Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algor. 74, 1169–1201, 2017). In the first part, we show that the best ranking method, constructed by Sharma and Arora in 2016, is actually a special case of the family of three-point methods presented in the paper (J. Dˇzuni´c, M. S. Petkovi´c, L. D. Petkovi´c, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217, 7612–7619, 2011). Since the mentioned Chun-Neta’s comparative study was carried out by testing algebraic polynomials of low degree (not higher than 6), in the second part we have extended comparative study to higher-order polynomials, polynomials with random coefficients, and polynomials with clusters of zeros. The performed experiments confirmed results of Chun and Neta. As in the cited paper of Chun and Neta, we have also applied the quality study based on computer visualization plotting the basins of attraction. Keywords Comparison of root finding · CPU time empirical model · Solving polynomial equations · Computer graphics · Convergence behavior Mathematics Subject Classification (2010) 65H05 · 65D18 · 68W40 · 33F05
Ivan Petkovi´c
[email protected] -Dord¯e Herceg [email protected] 1
Faculty of Electronic Engineering, Department of Computer Science, University of Niˇs, 18000 Niˇs, Serbia
2
Faculty of Science, Department of Mathematics and Informatics, University of Novi Sad, 21000 Novi Sad, Serbia
Numerical Algorithms
1 Introduction Multipoint iterative methods for solving nonlinear equations are of great practical importance, since they overcome theoretical limits related to the order of convergence and computational efficiency of any one-point method (such as Newton’s, Halley’s, Laguerre’s, and so on). The so-called optimal n-point methods are of highest practical interest since they reach the order of convergence 2n requiring only n+1 function evaluations. For this reason, only optimal methods will be considered in this paper. Multipoint methods were studied for the first time by Ostrowski [1] and Traub [2] and later in the papers [3–5] and [6]. An extensive and systematic review of multipoint methods was given in the monograph Multipoint Methods for Solving Nonlinear Equations [7] and the survey paper [8], and references cited there. In this paper, we will restrict our attention to three-point methods, motivated by results presented by Sharma and Arora [9] in 2016 and by Chun and Neta [10] in 2017. In the latter paper [10], the authors have performed an extensive comparative study of three-point methods an
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