Construction of optimal derivative-free techniques without memory
type: Item
instance of: scholarly article
title: Construction of optimal derivative-free techniques without memory
zbMATH Open document ID: 1268.65069
DOI: 10.1155/2012/497023
author: Fazlollah Soleymani
author: D. K. R. Babajee
author: S. Shateyi
author: Sandile Sydney Motsa
published in: Journal of Applied Mathematics
publication date: 2013-06-03T00:00:00Z
review text: Summary: The construction of iterative processes without memory, which are both optimal according to the Kung-Traub hypothesis (cf. [\textit{H. T. Kung} and \textit{J. F. Traub}, J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)]) and derivative-free, is considered in this paper. For this reason, techniques with four and five function evaluations per iteration, which reach to the optimal orders eight and sixteen, respectively, are discussed theoretically. These schemes can be viewed as the generalizations of the recent optimal derivative-free family of \textit{Q. Zheng} et al. [Appl. Math. Comput. 217, No. 23, 9592--9597 (2011; Zbl 1227.65044)]. This procedure also provides an \(n\)-step family using \(n + 1\) function evaluations per full cycle to possess the optimal order \(2^{n}\). The analytical proofs of the main contributions are given and numerical examples are included to confirm the outstanding convergence speed of the presented iterative methods using only few function evaluations. The second aim of this work will be furnished when a hybrid algorithm for capturing all the zeros in an interval has been proposed. The novel algorithm could deal with nonlinear functions having finitely many zeros in an interval.
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 6169925
zbMATH Keywords: nonlinear equations
zbMATH Keywords: iterative processes
zbMATH Keywords: Kung-Traub hypothesis
zbMATH Keywords: derivative-free
zbMATH Keywords: numerical examples
zbMATH Keywords: convergence
zbMATH Keywords: algorithm
Wikidata QID: Q58906003
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/497023
OpenAlex ID: W1972868758
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description: scientific article; zbMATH DE number 6169925