an:05057741
Zbl 1109.65046
Giusti, M.; Lecerf, G.; Salvy, B.; Yakoubsohn, J.-C.
On location and approximation of clusters of zeros of analytic functions
EN
Found. Comput. Math. 5, No. 3, 257-311 (2005).
00123284
2005
j
65H05 30B10 30C15 65E05
\(\alpha\)-theory; Cluster approximations; Cluster location; Newton's operator; Pellet's criterion; Rouch??'s theorem; Schr??der's operator
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 Schr??der'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.
Luigi Gatteschi (Torino)