A subdivision method for computing nearest gcd with certification

Authors:
Guillaume Chèze;André Galligo;Bernard Mourrain;Jean-Claude Yakoubsohn
Affiliations:
Institut Mathématique de Toulouse, équipe MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France;Laboratoire J.A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France and GALAAD, INRIA Méditerranée, 2004 route des Luciole ...;GALAAD, INRIA Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France;Institut Mathématique de Toulouse, équipe MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

A new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The exclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smale's @a-theorems to certify the existence and unicity of a solution in a cell. Under the condition of simple roots for the distance minimization problem, we analyze the complexity of the algorithm in terms of a condition number, which is the inverse of the distance to the set of degenerate systems. We report on some experimentation on representative examples to illustrate the behavior of the algorithm.