On Location and Approximation of Clusters of Zeros of Analytic Functions

  • Authors:

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

  • Affiliations:

  • Laboratoire STIX, Ecole polytechnique, 91128 Palaiseau, France;Laboratoire de Mathematiques, Universite de Versailles Saint-Quentin-en-Yvelines, 45 avenue des Etats-Unis, 78035 Versailles, France;Projet ALGO, INRIA Rocquencourt, 78153 Le Chesnay, France;Laboratoire MIP, Bureau 131, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

  • Venue:
  • Foundations of Computational Mathematics
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

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 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.