Foundations of Computational Mathematics

, Volume 5, Issue 3, pp 257–311 | Cite as

On Location and Approximation of Clusters of Zeros of Analytic Functions

  • M. Giusti
  • G. Lecerf
  • B. Salvy
  • J.-C. Yakoubsohn


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 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 Schroder'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.

Theory Cluster approximation Cluster location Cluster of zeros Newton's operator Pellet's criterion Rouche's theorem Schroder's operator Zeros of analytic functions 


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Copyright information

© Society for the Foundations of Computational Mathematics 2005

Authors and Affiliations

  1. 1.Laboratoire STIX, Ecole polytechnique, 91128 PalaiseauFrance
  2. 2.Laboratoire de Mathematiques, Universite de Versailles Saint-Quentin-en-Yvelines, 45 avenue des Etats-Unis, 78035 Versailles France
  3. 3.Projet ALGO, INRIA Rocquencourt, 78153 Le ChesnayFrance
  4. 4.Laboratoire MIP, Bureau 131, Universite Paul Sabatier, 118 route de Narbonne, 31062 ToulouseFrance

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