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Cylindrical algebraic decomposition using validated numerics. (English) Zbl 1124.68123
Summary: We present a version of the Cylindrical Algebraic Decomposition (CAD) algorithm which uses interval sample points in the lifting phase, whenever the results can be validated. This gives substantial time savings by avoiding computations with exact algebraic numbers. We use bounds based on Rouche’s theorem combined with information collected during the projection phase and during construction of the current cell to validate the singularity structure of roots. We compare empirically our implementation of this variant of CAD with implementations of CAD using exact algebraic sample points (our and QEPCAD) and with our implementation of CAD using interval sample points with validation based solely on interval data.

MSC:
68W30 Symbolic computation and algebraic computation
14P99 Real algebraic and real-analytic geometry
Software:
Mathematica; QEPCAD
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