Zeros of analytic functions a b s t r a c tLet T : D ⊂ X → X be an iteration function in a complete metric space X . In this paper we present some new general complete convergence theorems for the Picard iteration x n+1 = Tx n with order of convergence at least r ≥ 1. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions of T and a convergence function of T . We study the convergence of the Picard iteration associated to T with respect to a function of initial conditions E: D → X . The initial conditions in our convergence results utilize only information at the starting point x 0 . More precisely, the initial conditions are given in the form E(x 0 ) ∈ J, where J is an interval on R + containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the famous semilocal α-theorem of Smale for analytic functions.

a b s t r a c tGeneral local convergence theorems with order of convergence r ≥ 1 are provided for iterative processes of the type x n+1 = Tx n , where T : D ⊂ X → X is an iteration function in a metric space X . The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Dočev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695-701; J.F. Traub, H. Woźniakowski, Convergence and complexity of Newton iteration for operator equations, J.

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In this paper, we prove some general convergence theorems for the Picard iteration in cone metric spaces over a solid vector space. As an application, we provide a detailed convergence analysis of the Weierstrass iterative method for computing all zeros of a polynomial simultaneously. These results improve and generalize existing ones in the literature.Keywords: iterative methods, cone metric space, convergence analysis, error estimates, Weierstrass method, polynomial zeros 2000 MSC: 65J15, 54H25, 65H04, 12Y05where T : D ⊂ X → X is an iteration function in a cone metric space (X, d) over a solid vector space (Y, ). Cone metric spaces have a long history (see Collatz [3], Zabrejko [43], Janković, Kadelburg and Radenović [10], Proinov [29] and references therein). For an overview of the theory of cone metric spaces over a solid vector space, we refer the reader to [29] and [31, Section 2].In the second part of the paper, we study the convergence of the famous Weierstrass method [39] for computing all zeros of a polynomial simultaneously. This method was introduced and studied for the first time by Weierstrass in 1891. In 1960-1966, the method was rediscovered by Durand [6] (in Email address: proinov@uni-plovdiv.bg (Petko D. Proinov)

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