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

Proinov, Petko D. ¹

¹ Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 9, p. 1045-1055.

Résumé

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.

Détail

DOI : 10.22436/jnsa.011.09.03

Classification : 65H05, 47H09, 47H17, 12Y05
Keywords: Iterative methods, simultaneous methods, Ehrlich method, multiple polynomial zeros, Gargantini-Farmer-Loizou method, local convergence, error estimates

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