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.
Work supported in part by Japan Society for the Promotion of Science under Grants 19300001.
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Sasaki, T., Kako, F. (2008). Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_23
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DOI: https://doi.org/10.1007/978-3-540-87827-8_23
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