A Practical Method for Floating-Point Gröbner Basis Computation

Tateaki Sasaki ()
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Tateaki Sasaki: University of Tsukuba, Institute of Mathematics

A chapter in Computer Mathematics, 2014, pp 109-124 from Springer

Abstract: Abstract Computing Gröbner bases with inexact coefficients is eagerly desired in industrial applications, but the computation with floating-point numbers is quite unstable if performed naively. In previous papers, the present author clarified that large term-cancellations occur frequently making the computation unstable, and he proposed a method of removing the harm of exact term-cancellations and a method of estimating the amounts of inexact term-cancellations. However, the estimation is very rough and never satisfactory. The inexact cancellations cause the accuracy loss of the output system, hence it is important to estimate their amounts precisely. In this chapter, we propose a practical method of estimating the amounts of inexact cancellations fairly well and very simply. We have tested our method by several examples, and obtained reasonable results. We also propose a simple device to reduce the amounts of exact term-cancellations.

Keywords: Inexact Cancellation; High-precision Method; Initial Accuracy; Input Polynomials; Polynomial Remainder Sequences (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-43799-5_10

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DOI: 10.1007/978-3-662-43799-5_10

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