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Sparse Gaussian Elimination Modulo p: An Update

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Book cover Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

This paper considers elimination algorithms for sparse matrices over finite fields. We mostly focus on computing the rank, because it raises the same challenges as solving linear systems, while being slightly simpler.

We developed a new sparse elimination algorithm inspired by the Gilbert-Peierls sparse LU factorization, which is well-known in the numerical computation community. We benchmarked it against the usual right-looking sparse gaussian elimination and the Wiedemann algorithm using the Sparse Integer Matrix Collection of Jean-Guillaume Dumas.

We obtain large speedups (1000\(\times \) and more) on many cases. In particular, we are able to compute the rank of several large sparse matrices in seconds or minutes, compared to days with previous methods.

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Acknowledgement

Claire Delaplace was supported by the french ANR under the BRUTUS project. We thank the anonymous reviewers for their comments.

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Authors and Affiliations

  1. Univ. Lille, CNRS, Centrale Lille, UMR 9189 - CRIStAL - Centre de Recherche en Informatique Signal et Automatique de Lille, 59000, Lille, France

    Charles Bouillaguet & Claire Delaplace

  2. Université de Rennes-1/IRISA, Rennes, France

    Claire Delaplace

Authors
  1. Charles Bouillaguet
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  2. Claire Delaplace
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Correspondence to Charles Bouillaguet .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Bouillaguet, C., Delaplace, C. (2016). Sparse Gaussian Elimination Modulo p: An Update. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_8

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