## On location and approximation of clusters of zeroes of analytic functions (2005)

Venue: | Found. Comput. Math |

Citations: | 14 - 4 self |

### BibTeX

@ARTICLE{Giusti05onlocation,

author = {M. Giusti and G. Lecerf and B. Salvy and J. -c. Yakoubsohn},

title = {On location and approximation of clusters of zeroes of analytic functions},

journal = {Found. Comput. Math},

year = {2005},

volume = {5},

pages = {257--311}

}

### OpenURL

### Abstract

Abstract. In the beginning of the eighties, 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 initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this article we focus on one complex variable functions. 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