Skip to main content
SpringerLink
Log in

Some performance studies in exact linear algebra

  • Conference paper
  • First Online:
Workshop on wide area networks and high performance computing

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 249))

  • 58 Accesses

Abstract

We consider parallel algorithms for computing the Hermite normal form of matrices over Euclidean rings. We use standard types of reduction methods which are the basis of many algorithms for determining canonical forms of matrices over various computational domains. Our implementations take advantage of well-performing sequential code and give very good performance.

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. W. A. Blankinship. A new version of the Euclidian algorithm. Amer. Math. Monthly 70 (1963) 742–745.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. H. Bradley. Algorithm and bound for the greatest common divisor of n integers. Comm. ACM 13 (1970) 433–436.

    Article  MATH  MathSciNet  Google Scholar 

  3. T-W.J. Chou and G.E. Collins. Algorithms for the solution of systems of linear Diophantine equations. SIAM J. Comput. 11 (1982) 687–708.

    Article  MATH  MathSciNet  Google Scholar 

  4. X.G. Fang and G. Havas. On the worst-case complexity of integer gaussian elimination. ISSAC'97 (Proc. 1997 Internat. Sympos. Symbolic Algebraic Comput.), ACM Press (1997) 28–31.

    Google Scholar 

  5. K.O. Geddes, S.R. Czapor and G. Labahn. Algorithms for Computer Algebra. Kluwer Academic Publishers, 1992.

    Google Scholar 

  6. M. Giesbrecht. Fast computation of the Smith normal form of an integer matrix. ISSAC'95 (Proc. 1995 Internat. Sympos. Sympolic Algebraic Comput.), ACM Press (1995) 110–118.

    Google Scholar 

  7. B. Hartley and T.O. Hawkes. Rings, Modules and Linear Algebra. Chapman and Hall, 1976.

    Google Scholar 

  8. G. Havas, D.F. Holt and S. Rees. Recognizing badly presented Z-modules. Linear Algebra Appl 192 (1993) 137–163.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Havas and B.S. Majewski. Hermite normal form computation for integer matrices. Congressus Numerantium 105 (1994) 87–96.

    MATH  MathSciNet  Google Scholar 

  10. G. Havas and B.S. Majewski. Extended gcd calculation. Congressus Numerantium 111 (1995) 104–114.

    MATH  MathSciNet  Google Scholar 

  11. G. Havas and B.S. Majewski. Integer matrix diagonalization. J. Symbolic Computation 24 (1997) 399–408.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Havas, B.S. Majewski and K.R. Matthews. Extended gcd and Hermite normal form algorithms via lattice basis reduction. Experimental Mathematics 7 (1998) 125–135.

    MATH  MathSciNet  Google Scholar 

  13. G. Havas and C. Wagner. Matrix reduction algorithms for Euclidean rings. Proc. 1998 Asian Symposium on Computer Mathematics, Lanzhou University Press (1998) 65–70.

    Google Scholar 

  14. R. Kannan and A. Bachem. Polynomial algorithms for computing Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8 (1979) 499–507.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Kannan. Solving systems of linear equations over polynomials. Theoretical Computer Science 39 (1985) 69–88.

    Article  MATH  MathSciNet  Google Scholar 

  16. E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Parallel algorithms for matrix normal forms. Linear Algebra Appl. 136 (1990) 189–208.

    Article  MATH  MathSciNet  Google Scholar 

  17. B. S. Majewski and G. Havas. A solution to the extended gcd problem. ISSAC'95 (Proc. 1995 Internat. Sympos. Symbolic Algebraic Comput.), ACM Press (1995) 248–253.

    Google Scholar 

  18. M. Newman. Integral Matrices. Academic Press, 1972.

    Google Scholar 

  19. C.C. Sims. Computation with finitely presented groups. Cambridge University Press (1994).

    Google Scholar 

  20. A. Storjohann. Near optimal algorithms for computing Smith normal forms of integer matrices. ISSAC'96 (Proc. 1996 Internat. Sympos. Symbolic Algebraic Comput.), ACM Press (1996) 267–274.

    Google Scholar 

  21. A. Storjohann. Computing Hermite and Smith Normal Forms of Triangular Integer Matrices. Linear Algebra Appl. 282 (1998) 25–45.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Storjohann and G. Labahn. Asymptotically fast computation of Hermite normal forms of integer matrices. ISSAC'96 (Proc. 1996 Internat. Sympos. Symbolic Algebraic Comput.), ACM Press (1996) 259–266.

    Google Scholar 

  23. C. Wagner. Normalformberechnung von Matrizen über euklidischen Ringen. PhD thesis, Institut für Experimentelle Mathematik, Universität-GH Essen, 1997. Published by Shaker-Verlag, 52013 Aachen/Germany, 1998.

    Google Scholar 

  24. C. Wagner. Fast parallel Hermite normal form computation of matrices over F[x]. Euro-Par'98 Parallel Processing, Lecture Notes Comput. Sci. 1470 (1998) 821–830.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Discrete Mathematics and Computing Department of Computer Science and Electrical Engineering, The University of Queensland, 4072, Queensland, Australia

    George Havas

  2. Fachgruppe Praktische Informatik Fachbereich Elektrotechnik und Informatik, Universität-GHS Siegen, D-57078, Siegen, Germany

    Clemens Wagner

Authors
  1. George Havas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clemens Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

G. Cooperman (Professor)E. Jessen (Professor)G. Michler (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag London Limited

About this paper

Cite this paper

Havas, G., Wagner, C. (1999). Some performance studies in exact linear algebra. In: Cooperman, G., Jessen, E., Michler, G. (eds) Workshop on wide area networks and high performance computing. Lecture Notes in Control and Information Sciences, vol 249. Springer, London. https://doi.org/10.1007/BFb0110086

Download citation

  • DOI: https://doi.org/10.1007/BFb0110086

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-642-4

  • Online ISBN: 978-1-84628-578-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation