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Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

  • Tateaki Sasaki
  • Fujio Kako
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

Abstract

Computation of Gröbner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for many years; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes loss of accuracy in various cases. On the basis of this analysis, we propose a practical method for computing Gröbner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a practical method of estimating the ill-conditionedness of the input system.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Tateaki Sasaki
    • 1
  • Fujio Kako
    • 2
  1. 1.Institute of MathematicsUniversity of TsukubaTsukuba-shiJapan
  2. 2.Department of Comp. Sci.Nara Women’s University, Nara-shiNaraJapan

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