# Over-constrained Weierstrass iteration and the nearest consistent system

1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method allows us to find the nearest system which has at least $k$ common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of ourmethod to the optimization problem formulated by Karmarkar and Lakshmanfor the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give a simple iterative method to compute the optimum. The arithmetic complexity of the iteration is detailed.
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https://hal.inria.fr/inria-00070779
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 9:36:28 PM
Last modification on : Thursday, January 11, 2018 - 4:02:51 PM
Document(s) archivé(s) le : Sunday, April 4, 2010 - 9:53:52 PM

### Identifiers

• HAL Id : inria-00070779, version 1

### Citation

Olivier Ruatta, Mark Sciabica, Agnes Szanto. Over-constrained Weierstrass iteration and the nearest consistent system. RR-5215, INRIA. 2004, pp.17. ⟨inria-00070779⟩

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