On the local convergence of Gargantini-Farmer-Loizou method for simultaneous approximation of multiple polynomial zeros

Volume 11, Issue 9, pp 1045--1055 http://dx.doi.org/10.22436/jnsa.011.09.03

Publication Date: June 19, 2018 Submission Date: April 15, 2018 Revision Date: May 04, 2018 Accteptance Date: May 12, 2018

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Authors

Petko D. Proinov - Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria.

Abstract

The paper deals with a well known iterative method for simultaneous computation of all zeros (of known multiplicities) of a polynomial with coefficients in a valued field. This method was independently introduced by Farmer and Loizou [M. R. Farmer, G. Loizou, Math. Proc. Cambridge Philos. Soc., \({\bf 82}\) (1977), 427--437] and Gargantini [I. Gargantini, SIAM J. Numer. Anal., \({\bf 15}\) (1978), 497--510]. If all zeros of the polynomial are simple, the method coincides with the famous Ehrlich's method [L. W. Ehrlich, Commun. ACM, \({\bf 10}\) (1967), 107--108]. We provide two types of local convergence results for the Gargantini-Farmer-Loizou method. The first main result improves the results of [N. V. Kyurkchiev, A. Andreev, V. Popov, Ann. Univ. Sofia Fac. Math. Mech., \({\bf 78}\) (1984), 178--185] and [A. I. Iliev, C. R. Acad. Bulg. Sci., \({\bf 49}\) (1996), 23--26] for this method. Both main results of the paper generalize the results of Proinov [P. D. Proinov, Calcolo, \({\bf 53}\) (2016), 413--426] for Ehrlich's method. The results in the present paper are obtained by applying a new approach for convergence analysis of Picard type iterative methods in finite-dimensional vector spaces.

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Keywords

• Iterative methods
• simultaneous methods
• Ehrlich method
• multiple polynomial zeros
• Gargantini-Farmer-Loizou method
• local convergence
• error estimates

MSC

•  65H05
•  47H09
•  47H17
•  12Y05

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