Comptes rendus de l’Acad´ emie bulgare des Sciences MATHEMATIQUES M´ ethodes de calculs num´ erique A NEW SEMILOCAL CONVERGENCE THEOREM FOR THE WEIERSTRASS METHOD FROM DATA AT ONE POINT P. D. Proinov (Submitted by Academician B. Boyanov on November 30, 2005 ) Abstract In this paper we present a new semilocal convergence theorem from data at one point for the Weierstrass iterative method for the simultaneous computation of polynomial zeros. The main result generalizes and improves all previous ones in this area. Key words: polynomial zeros, simultaneous methods, Weierstrass method, con- vergence theorems, point estimation 2000 Mathematics Subject Classification: 65H05 1. Introduction. Let f be a monic polynomial of degree n ≥ 2 with simple complex zeros. We consider the roots of f as a point in C n . Namely, a point ξ in C n with distinct coordinates is said to be a root-vector of f if each of its coordinates is a zero of f . Starting from an initial point z 0 in C n with distinct coordinates we build in C n the Weierstrass iterative sequence [ 1 ] (1) z k+1 = z k − W (z k ), k =0, 1, 2,..., where the operator W in C n is defined by W (z )=(W 1 (z ),...,W n (z )) with W i (z )= f (z i )  j =i (z i − z j ) (i =1, 2, ··· ,n). It is well-known that under some initial conditions the Weierstrass sequence (1) is well-defined and tends to a root-vector of f . Iteration formula (1) defines the famous Weierstrass method (known also as the Durand-Dochev-Kerner-Preˇ si´ c method) for finding all the zeros of f simultaneously. In 1962, Dochev [ 2,3 ] proved the first local convergence theorem for the Weier- strass method. Since 1980 a number of authors [ 4–15 ] have obtained semilocal conver- gence theorems for the Weierstrass method from data at one point (point estimation). In this note we present a new semilocal convergence theorem for the Weierstrass method which improves and generalizes all these results. 131