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Nearest common root of a set of polynomials: A structured singular value approach

Limantseva, O., Halikias, G. ORCID: 0000-0003-1260-1383 and Karcanias, N. ORCID: 0000-0002-1889-6314 (2020). Nearest common root of a set of polynomials: A structured singular value approach. Linear Algebra and its Applications, 584, pp. 233-256. doi: 10.1016/j.laa.2019.09.005


The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations.

Publication Type: Article
Additional Information: © Elsevier 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Publisher Keywords: Structured singular value; Sylvester resultant matrix; Approximate GCD; Distance to singularity; Almost common root
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Engineering > Electrical & Electronic Engineering
Date Deposited: 07 Oct 2019 08:46
Text - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.

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