Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Algebra and Parallelism

CAP 1990: Computer Algebra and Parallelism pp 101–114Cite as

Boolean Gröbner bases and their MIMD implementation

  • Conference paper
  • First Online:

  • 147 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 584))

Abstract

We present two methods to compute Gröbner bases in parallel, both based on Buchberger's sequential algorithm. A distributed memory MIMD computer (the FPS 140) gives experimental results obtained with boolean polynomials.

The algorithms were implemented on the FPS T40 connected as a ring and as a hypercube of processors. The first implementation shows the interest of the parallelization. The second one, based on a divide and conquer strategy, has a behavior very close to the sequential algorithm.

We evaluate the contribution of the parallelism by a direct comparison of sequential and parallel times without references to complexity.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. M. Aroca, H. Hironaka, J. L. Vincente, “The theory of Maximal Contact,” Memorias de Mathematica del Instituto “Jorge Juana,” 29, (1975).

    Google Scholar 

  2. B. Buchberger, “Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungsystems,” Aequationes Math, 4, pp. 374–383, (1970).

    Article  Google Scholar 

  3. B. Buchberger, “A criterion for detecting unnecessary reductions in the construction of Gröbner bases,” Proc. EUROSAM 79 (Ed. W. Ng), Springer LNCS 72, pp. 3–21, (1979).

    Google Scholar 

  4. B. Buchberger, “Gröbner Bases: an algorithmic method in polynomial ideal theory,” Recent Trends in Multidimensional Systems Theory (Ed. N. K. Bose), Reidel, (1985).

    Google Scholar 

  5. H. Melenk, H. M. Möller, W. Neun, “On Gröbner Basis Computation on a Supercomputer Using Reduce,” FB Mathematik und Informatik der Fernuniversität Hagen, Preprint SC 88-2, (1988).

    Google Scholar 

  6. C. Ponder, “Parallelism Algorithms for Gröbner Basis Reduction,” Electrical Engineering and Computer Sciences Dept, University of California, Berkeley, CA, (1988).

    Google Scholar 

  7. P. Sénéchaud, “Bases de Gröbner Booléennes Méthodes de Calcul; Applications; Parallelisation,” Thèse INPG, (1990).

    Google Scholar 

  8. W. Trinks, “On B. Buchberger's Method for Solving Algebraic Equations,” J. Number Theory, 10/4, pp. 475–488, (1978).

    Article  Google Scholar 

  9. J. P. Vidal, “The Computation of Gröbner Bases on a shared memory multiprocessor,” Technical Report, School of Computer Science, Carnegie Mellon University, to appear.

    Google Scholar 

  10. S. Watt Bounded Parallelism in Computer Algebra, Ph. D. Thesis, University of Waterloo, Waterloo, Ontario (1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences, Dept. de mathematique, 123 Av. A. Thomas, 87060, Limoges Cedex, France

    Pascale Senechaud

Authors
  1. Pascale Senechaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Richard E. Zippel

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Senechaud, P. (1992). Boolean Gröbner bases and their MIMD implementation. In: Zippel, R.E. (eds) Computer Algebra and Parallelism. CAP 1990. Lecture Notes in Computer Science, vol 584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55328-2_8

Download citation

  • DOI: https://doi.org/10.1007/3-540-55328-2_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55328-1

  • Online ISBN: 978-3-540-47026-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation