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An ODE-based method for computing the approximate greatest common divisor of polynomials

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Abstract

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

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Acknowledgements

Nicola Guglielmi thanks INdAM GNCS for financial support. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement number 258581 “Structured low-rank approximation: Theory, algorithms, and applications” and fund for Scientific Research (FWO-Vlaanderen), FWO projects G028015N ”Decoupling multivariate polynomials in nonlinear system identification” and G090117N ”Block-oriented nonlinear identification using Volterra series”; and the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

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Authors and Affiliations

  1. Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, 67100, L’ Aquila, Italy

    Antonio Fazzi & Nicola Guglielmi

  2. Università degli Studi di L’ Aquila, Via Vetoio 1 - Loc. Coppito, 67100, L’ Aquila, Italy

    Nicola Guglielmi

  3. Department ELEC, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050, Brussels, Belgium

    Ivan Markovsky

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  1. Antonio Fazzi
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  2. Nicola Guglielmi
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  3. Ivan Markovsky
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Correspondence to Antonio Fazzi.

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Fazzi, A., Guglielmi, N. & Markovsky, I. An ODE-based method for computing the approximate greatest common divisor of polynomials. Numer Algor 81, 719–740 (2019). https://doi.org/10.1007/s11075-018-0569-0

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  • DOI: https://doi.org/10.1007/s11075-018-0569-0

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