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On Unimodular Matrices of Difference Operators

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Computer Algebra in Scientific Computing (CASC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11077))

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Abstract

We consider matrices \(L \in \mathrm{Mat} _n(K[\sigma , \sigma ^{-1}])\) of scalar difference operators, where K is a difference field of characteristic 0 with an automorphism \(\sigma \). We discuss approaches to compute the dimension of the space of those solutions of the system of equations \(L(y)=0\) that belong to an adequate extension of K. On the base of one of those approaches, we propose a new algorithm for computing \(L^{-1}\in \mathrm{Mat} _n(K[\sigma , \sigma ^{-1}])\) whenever it exists. We investigate the worst-case complexity of the new algorithm, counting both arithmetic operations in K and shifts of elements of K. This complexity turns out to be smaller than in the earlier proposed algorithms for inverting matrices of difference operators.

Some experiments with our implementation in Maple of the algorithm are reported.

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Notes

  1. 1.

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Acknowledgments

The authors are thankful to anonymous referees for useful comments. Supported in part by the Russian Foundation for Basic Research, project No. 16-01-00174.

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Authors and Affiliations

  1. Dorodnicyn Computing Centre, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Vavilova, 40, Moscow, 119333, Russia

    S. A. Abramov & D. E. Khmelnov

Authors
  1. S. A. Abramov
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  2. D. E. Khmelnov
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Corresponding author

Correspondence to S. A. Abramov .

Editor information

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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Abramov, S.A., Khmelnov, D.E. (2018). On Unimodular Matrices of Difference Operators. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_2

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  • DOI: https://doi.org/10.1007/978-3-319-99639-4_2

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  • Online ISBN: 978-3-319-99639-4

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