English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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., 82 (1977), 427–437] and Gargantini [I. Gargantini, SIAM J. Numer. Anal., 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, 10 (1967), 107–108]. We provide two types of local convergence results for the Gargantini-FarmerLoizou method. The first main result improves the results of [N. V. Kyurkchiev, A. Andreev, V. Popov, Ann. Univ. Sofia Fac. Math. Mech., 78 (1984), 178–185] and [A. I. Iliev, C. R. Acad. Bulg. Sci., 49 (1996), 23–26] for this method. Both main results of the paper generalize the results of Proinov [P. D. Proinov, Calcolo, 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.

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