Numerical stability and stabilization of Groebner basis computation
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
The approximate GCD of inexact polynomials
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computational Commutative Algebra 1
Computational Commutative Algebra 1
Stable border bases for ideals of points
Journal of Symbolic Computation
Structures of precision losses in computing approximate Gröbner bases
Journal of Symbolic Computation
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We present a heuristically certified form of floating-point arithmetic and its implementation in CoCoALib. This arithmetic is intended to act as a fast alternative to exact rational arithmetic, and is developed from the idea of paired floats expounded by Traverso and Zanoni (2002). As prerequisites we need a source of (pseudo-)random numbers, and an underlying floating-point arithmetic system where the user can set the precision. Twin-float arithmetic can be used only where the input data are exact, or can be obtained at high enough precision. Our arithmetic includes a total cancellation heuristic for sums and differences, and so can be used in classical algebraic algorithms such as Buchberger's algorithm. We also present a (new) algorithm for recovering an exact rational value from a twin-float, so in some cases an exact answer can be obtained from an approximate computation. The ideas presented here are implemented as a ring in CoCoALib, called RingTwinFloat, allowing them to be used easily in a wide variety of algebraic computations (including Grobner bases).