Advertisement

An Algorithm to Compute Floating Point Groebner Bases

  • Kiyoshi Shirayanagi

Abstract

Gröbner basis (GB) techniques are a valuable tool for solving many problems in polynomial ideal theory. As is well known, the process of computing a GB may involve large numbers of intermediate coefficients - say from a field k - even when the final GB does not involve many coefficients. In fact, the cost of performing exact arithmetic in k with the intermediate coefficients is a major factor determining the computational cost of computing the GB. This paper proposes a new approach using floating point computation that can be applied when k is a subfield of the real numbers.1

Keywords

Interval Arithmetic Floating Point Real Polynomial Float Point Arithmetic Exact Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Alefeld, G. and Herzberger, J., Introduction to Interval Computations, Computer Science and Applied Mathematics, Academic Press (1983).Google Scholar
  2. [2]
    Boege, W., Gebauer, R., and Kredel, H., Some Examples for Solving Systems of Algebraic Equations by Calculating Groebner Bases, J. Symb. Comp. 1 (1986), 83–98.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Buchberger, B., Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Chapter 6 in Multidimensional Systems Theory (N. K. Bose ed.), D. Reidel Publishing Company (1985), 184–232.Google Scholar
  4. [4]
    Buchberger, B., An algorithmical criterion for the solvability of algebraic systems of equations (German), Aequationes mathematicae 4 (3) (1970), 374–383.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., and Watt, S. M., First Leaves: A Tutorial Introduction to Maple V, Springer-Verlag (1992).Google Scholar
  6. [6]
    Dickson, L. E., Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, Am. J. of Math. 35 (1913), 413–426.MATHCrossRefGoogle Scholar
  7. [7]
    Knuth, D. E., The Art of Computer Programming, Vol. 2, Addison-Wesley (1969).Google Scholar
  8. [8]
    Moore, R. E., Interval Arithmetic and Automatic Error Analysis in Digital Computing, Ph.D. Thesis, Mathematics Dept., Stanford Univ., October (1962).Google Scholar
  9. [9]
    Petkovie, M., Iterative Methods for Simultaneous Inclusion of Polynomial Zeros, L. N. Math., Springer-Verlag 1387 (1989).Google Scholar
  10. [10]
    Sasaki, T. and Takeshima, T., A Modular Method for Gröbner-basis Construction over Q and Solving System of Algebraic Equations, J. Information Processing 12 (4) (1989), 371–379.MathSciNetMATHGoogle Scholar
  11. [11]
    Trinks, W., On Improving Approximate Results of Buchberger’s Algorithm by Newton’s Method, SIGSAM Bull. 18 (3) (1984), 7–11.MATHCrossRefGoogle Scholar
  12. [12]
    Winkler, F., A p-adic Approach to the Computation of Gröbner Bases, J. Symb. Comp. 6/2, 3 (1988), 287–304.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Kiyoshi Shirayanagi
    • 1
  1. 1.NTT Communication Science LaboratoriesKyotoJapan

Personalised recommendations