Parallel Reduction of Matrices in Gröbner Bases Computations

  • Severin Neumann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

In this paper we provide an parallelization for the reduction of matrices for Gröbner basis computations advancing the ideas of using the special structure of the reduction matrix [4]. First we decompose the matrix reduction in three steps allowing us to get a high parallelization for the reduction of the bigger part of the polynomials. In detail we do not need an analysis of the matrix to identify pivot columns, since they are obvious by construction and we give a rule set for the order of the reduction steps which optimizes the matrix transformation with respect to the parallelization. Finally we provide benchmarks for an implementation of our algorithm. This implementation is available as open source.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Björck, G., Haagerup, U.: All cyclic p-roots of index 3, found by symmetry-preserving calculations (2008)Google Scholar
  2. 2.
    Faugére, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139(1-3), 61–88 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Faugére, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC 2002, New York, NY, USA, pp. 75–83 (2002)Google Scholar
  4. 4.
    Faugére, J.-C., Lachartre, S.: Parallel Gaussian Elimination for Gröbner bases computations in finite fields. In: Proceedings of the 4th International Workshop on Parallel and Symbolic Computation, PASCO 2010, New York, USA, pp. 89–97 (July 2010)Google Scholar
  5. 5.
    Gao, S., Volny IV, F., Wang, M.: A New Algorithm for Computing Gröbner Bases (2010)Google Scholar
  6. 6.
    Gebauer, R., Michael Möller, H.: On an installation of Buchberger’s algorithm. Journal of Symbolic Computation 6, 275–286 (1988)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Giovini, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C.: One sugar cube, please or selection strategies in the Buchberger algorithm. In: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISAAC 1991, New York, USA, pp. 49–54 (1991)Google Scholar
  8. 8.
    Huynh, D.T.: A superexponential lower bound for Gröbner bases and Church-Rosser Commutative Thue systems. Inf. Control 68, 196–206 (1986)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Katsura, S., Fukuda, W., Inawashiro, S., Fujiki, N., Gebauer, R.: Distribution of effective field in the ising spin glass of the ± J model at T = 0. Cell Biochemistry and Biophysics 11, 309–319 (1987)Google Scholar
  10. 10.
    McKay, C.E.: An analysis of improvements to Buchberger’s algorithm for Gröbner basis computation. Master thesis, University of Maryland, USA (2004)Google Scholar
  11. 11.
    Ponder, C.G.: Evaluation of “performance enhancements” in algebraic manipulation systems. PhD thesis. University of California, USA (1988)Google Scholar
  12. 12.
    Quinn, M.J.: Parallel Programming in C with MPI and OpenMP. McGraw-Hill Education Group (September 2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Severin Neumann
    • 1
  1. 1.Fakultät für Mathematik und InformatikUniversität PassauPassauGermany

Personalised recommendations