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The GPGCD Algorithm with the Bézout Matrix

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

For a given pair of univariate polynomials with real coefficients and a given degree, we propose a modification of the GPGCD algorithm, presented in our previous research, for calculating approximate greatest common divisor (GCD). In the proposed algorithm, the Bézout matrix is used in transferring the approximate GCD problem to a constrained minimization problem, whereas, in the original GPGCD algorithm, the Sylvester subresultant matrix is used. Experiments show that, in the case that the degree of the approximate GCD is large, the proposed algorithm computes more accurate approximate GCDs than those computed by the original algorithm. They also show that the computing time of the proposed algorithm is smaller than that of the SNTLS algorithm, which also uses the Bézout matrix, with a smaller amount of perturbations of the given polynomials and a higher stability.

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Notes

  1. 1.

    The coefficients are generated with the Mersenne Twister algorithm [13] by built-in function Generate with RandomTools:-MersenneTwister in Maple, which approximates a uniform distribution on \([-10, 10]\).

  2. 2.

    Remainders are calculated with built-in function SNAP:-Remainder.

  3. 3.

    We have excluded test polynomials from Group 6 to Group 10, because, for those polynomials, the computing time for the SNTLS algorithm is too long (over 100 min with tests in Group 6 whose degree of input polynomials is 60).

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Authors and Affiliations

  1. Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan

    Boming Chi

  2. Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan

    Akira Terui

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  1. Boming Chi
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  2. Akira Terui
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Correspondence to Akira Terui .

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Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Chi, B., Terui, A. (2020). The GPGCD Algorithm with the Bézout Matrix. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_10

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  • DOI: https://doi.org/10.1007/978-3-030-60026-6_10

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  • Online ISBN: 978-3-030-60026-6

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