Skip to main content
SpringerLink
Log in

Inversion of circulant matrices over Zm

  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1443))

Included in the following conference series:

Abstract

In this paper we consider the problem of inverting an n × n circulant matrix with entries over Zm. We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Zm. We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of m and n, and their costs range - roughly - from n log n log log n to n log2 n log log n log m operations over Zm. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear Cellular Automata.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Massachussets, 1974.

    Google Scholar 

  2. D. Bini, G. M. Del Corso, G. Manzini, and L. Margara. Inversion of Circulant Matrices over Zm. Technical Report B4-97-14, Istituto di Matematica Computazionale, CNR, Pisa, Italy, 1997.

    Google Scholar 

  3. D. Bini and V. Y. Pan. Polynomial and Matrix Computations, Fundamental Algorithms, volume 1. Birkhäuser, 1994.

    Google Scholar 

  4. P. Chaudhuri, D. Chowdhury, S. Nandi, and S. Chattopadhyay. Additive Cellular Automata Theory and Applications, Vol. 1. IEEE Press, 1997.

    Google Scholar 

  5. P. Feinsilver. Circulants, inversion of circulants, and some related matrix algebras. Linear Algebra and Appl., 56:29–43, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Ito, N. Osato, and M. Nasu. Linear cellular automata over Zm. Journal of Computer and System Sciences, 27:125–140, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Manzini and L. Margara. Invertible linear cellular automata over Zm: Algorithmic and dynamical aspects. Journal of Computer and System Sciences. To appear. A preliminary version appeared in Proc. MFCS '97, LNCS n. 1295, Springer Verlag.

    Google Scholar 

  8. G. Manzini and L. Margara. A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Zm. In 24th International Colloquium on Automata Languages and Programming (ICALP '97). LNCS n. 1256, Springer Verlag, 1997.

    Google Scholar 

  9. O. Martin, A. Odlyzko, and S. Wolfran. Algebraic properties of cellular automata. Comm. Math. Phys., 93:219–258, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. M. Ostrowski. Recherches sur la méthode de graeffe et les zéros des polynomes et des series de laurent. Acta Math., 72:99–257, 1940.

    MATH  MathSciNet  Google Scholar 

  11. A. Schönhage and V. Strassen. Schnelle Multiplikation grosse Zahlen. Computing, 7:281–292, 1971

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, 56126, Pisa, Italy

    Dario Bini

  2. Istituto di Matematica Computazionale, CNR, 56126, Pisa, Italy

    Gianna M. Del Corso & Giovanni Manzini

  3. Istituto di Elaborazione dell'Informazione, CNR, 56126, Pisa, Italy

    Gianna M. Del Corso

  4. Dipartimento di Scienze e Tecnologie Avanzate, Università di Torino, 15100, Alessandria, Italy

    Giovanni Manzini

  5. Dipartimento di Scienze dell'Informazione, Università di Bologna, 40127, Bologna, Italy

    Luciano Margara

Authors
  1. Dario Bini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianna M. Del Corso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giovanni Manzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luciano Margara
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Kim G. Larsen Sven Skyum Glynn Winskel

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bini, D., Del Corso, G.M., Manzini, G., Margara, L. (1998). Inversion of circulant matrices over Zm . In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055096

Download citation

  • DOI: https://doi.org/10.1007/BFb0055096

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64781-2

  • Online ISBN: 978-3-540-68681-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation