On Location and Approximation of Clusters of Zeros of Analytic Functions

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 α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the Date 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.

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…For works that focus on classical exclusion/exclusion algorithms but without bit complexity bounds we refer the reader to [17,35,14]. For other approaches for approximating curves and surfaces we refer the reader to [11,4,3,10] and the references therein.…”

“…For other approaches for approximating curves and surfaces we refer the reader to [11,4,3,10] and the references therein. Recently, there is an extension to the case of analytic functions [17,38]. For the problem of isolating the roots of polynomials we refer the reader to [23,20,19,12,36,24,7,27,9] and the references therein.…”

On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

“…Then we exhibit and analyze a natural approximation algorithm and show that quadratic approximation is possible with an overhead cost that is only polynomial in the multiplicity: roughly speaking, the main feature of our algorithm is that it stops the iteration at a distance of the cluster which is proved to be about the diameter of this cluster. Our techniques naturally extend and rely on the results of [12] for location and approximation of clusters of zeroes of analytic functions in one variable.…”

“…which provides an efficient way of computing Taylor expansions of h. For the sake of completeness, we recall the location criterion of [12] for univariate maps, that we use in next sections. We also recall the relation between diameters of clusters and β m,l : briefly speaking, for a univariate function q, if α m,l (q; a) is sufficiently small then q (l) admits a cluster of m − l zeroes in a ball centered at a of radius about β m,l (q; a).…”

“…This means finding (a) a point; (b) an estimate for the radius of a ball centered at this point and containing zeroes; (c) the radius of a zero-free region beyond this ball. We generalize [12,Section 1], that restricts to univariate functions and [9], that deals with our present case of study but restricting to clusters containing 2 zeroes (that is m = 2 in our context).…”