An algorithm to compute floating point Gro¨bner bases
Proceedings of the Maple summer workshop and symposium on Mathematical computation with Maple V : ideas and applications: ideas and applications
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Stabilization of polynomial systems solving with Groebner bases
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Remarks on automatic algorithm stabilization
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
Numerical stability and stabilization of Groebner basis computation
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Computing floating-point gröbner bases stably
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Hilbert stratification and parametric gröbner bases
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Intervals, syzygies, numerical gröbner bases: a mixed study
CASC'06 Proceedings of the 9th international conference on Computer Algebra in Scientific Computing
Term cancellations in computing floating-point Gröbner bases
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
Computing floating-point Gröbner bases accurately
ACM Communications in Computer Algebra
Computing a structured Gröbner basis approximately
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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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.