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On location and approximation of clusters of zeros of analytic functions. (English) Zbl 1109.65046
Authors’ abstract: At the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newton’s method for multivariate analytic maps. In particular, their $$\alpha$$-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem.
In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schröder’s iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.

##### MSC:
 65H05 Numerical computation of solutions to single equations 30B10 Power series (including lacunary series) in one complex variable 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
MultRoot; na20
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