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.
We state precise results on the complexity of a classical bisection-exclusion method to locate zeros of univariate analytic functions contained in a square. The output of this algorithm is a list of squares containing all the zeros. It is also a robust method to locate clusters of zeros. We show that the global complexity depends on the following quantities: the size of the square, the desired precision, the number of clusters of zeros in the square, the distance between the clusters and the global behavior of the analytic function and its derivatives. We also prove that, closed to a cluster of zeros, the complexity depends only on the number of zeros inside the cluster. In particular, for a polynomial which has d simple roots separated by a distance greater than sep, we will prove the bisection-exclusion algorithm needs O(d 3 log(d/sep)) tests to isolate the d roots and the number of squares suspected to contain a zero is bounded by 4d. Moreover, always in the polynomial case, we will see the arithmetic complexity can be reduced to O(d 2 (log d) 2 log(d/sep)) using log d steps of the Graeffe iteration.
Abstract. Isolated multiple zeroes or clusters of zeroes of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeroes, i.e., where the map has corank zero. In this article we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeroes and a fast algorithm for approximating them, with quadratic convergence. In case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.
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