On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

  • 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;Project 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:
  • 2007

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Abstract

Isolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This paper is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the 1980s. This theory restricts to simple zeros, i.e., where the map has corank zero. In this paper 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 zeros and a fast algorithm for approximating them, with quadratic convergence. In the case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.