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TitleFloating-point Gr\"obner basis computation with ill-conditionedness estimation.
Author(s) Kako Fujio, Tateaki Sasaki
TypeBook, Chapter in Book, Conference Proceeding
AbstractComputation 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.
URL http://link.springer.com/chapter/10.1007%2F978-3-540-87827-8_23
PublisherBerlin: Springer
Translation No
Refereed No