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Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

Nearest common root of polynomials, approximate greatest common divisor and the structured... In this paper the following problem is considered: given two coprime polynomials, find the smallest perturbation in the magnitude of their coefficients such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

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Publisher
Oxford University Press
Copyright
The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Subject
Articles
ISSN
0265-0754
eISSN
1471-6887
DOI
10.1093/imamci/dns032
Publisher site
See Article on Publisher Site

Abstract

In this paper the following problem is considered: given two coprime polynomials, find the smallest perturbation in the magnitude of their coefficients such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints.

Journal

IMA Journal of Mathematical Control and InformationOxford University Press

Published: Dec 11, 2013

Keywords: approximate common root of polynomials approximate GCD Sylvester resultant matrix structured singular value distance to singularity structured approximations

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