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

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-Farmer-Loizou 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.