Constructing fewer open cells by GCD computation in CAD projection

Han, Jingjun; Dai, Liyun; Xia, Bican

doi:10.1145/2608628.2608676

Cited by 20 publications

(14 citation statements)

References 17 publications

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“…Although we have not been able to provide a full justification of why this modified approach works, in Section 4 we give some ideas of why this argument rules and no information is lost. We recently knew that the consideration of only the square free part of the first resultant has been also used in the context of cylindrical algebraic decomposition, see [18].…”

Section: Introduction the Double Resultantmentioning

confidence: 99%

Detection of Special Curves Via the Double Resultant

Ferragut

García-Saldaña

Gasull

2015

Qual. Theory Dyn. Syst.

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Abstract. We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.

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Section: Introduction the Double Resultantmentioning

confidence: 99%

Detection of Special Curves Via the Double Resultant

Ferragut

García-Saldaña

Gasull

2015

Qual. Theory Dyn. Syst.

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“…In [8], Han et al propose an algorithm, HpTwo, for constructing Open CADs that reduces the size of the projection factor set, and thus the number of cells, by computing the intersection of projections produced by different variable orders. Their algorithm produces a (weak) decomposition into regions in which the input polynomials have constant sign.…”

Section: Discussionmentioning

confidence: 99%

“…with input of the form f σ0, is where Open NuCAD provides the least benefit vs. the usual Open CAD. [8] provides two sets of experiments demonstrating HpTwo: first constructing sign-invariant Open CADs for polynomials generated by the Maple command randpoly([x,y,z,w],degree=4)-1, the second for polynomials generated by the Maple command randpoly([seq(x[i],i=1..5)],degree=3) + add( x[i]2,i=1..5)-1. Compared to the usual Open CAD, it produced an average of 46% fewer cells in the first set, and 63% fewer in the second set.…”

Section: Discussionmentioning

confidence: 99%

Open Non-uniform Cylindrical Algebraic Decompositions

Brown

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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This paper introduces the notion of an Open Non-uniform Cylindrical Algebraic Decomposition (NuCAD), and presents an efficient model-based algorithm for constructing an Open NuCAD from an input formula. Using a limited experimental implementation of the algorithm, we demonstrate the effectiveness of the approach. NuCAD generalizes Cylindrical Algebraic Decomposition (CAD) as defined by Collins in his seminal work from the early 1970s, and extended in concepts like Hong's partial CAD. A NuCAD, like a CAD, is a decomposition of R n into cylindrical cells. But unlike a CAD, the cells in a NuCAD need not be arranged cylindrically. It is in this sense that NuCADs are not uniformly cylindrical. However, NuCADs -like CADs -carry a treelike structure that relates different cells. It is a very different tree but, as with the CAD tree structure, it allows some operations to be performed efficiently, for example locating the containing cell for an arbitrary input point.

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“…A key improvement to CAD has been in the projection operator to reduce the number of projection polynomials computed (Hong, 1990;McCallum, 1998McCallum, , 1999bMcCallum, , 2001Brown, 2001;Bradford et al, 2013a;Han et al, 2014;Bradford et al, 2016).…”

Section: Projection Operatorsmentioning

confidence: 99%

Cylindrical algebraic decomposition with equational constraints

England

Bradford

Davenport

2020

Journal of Symbolic Computation

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Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear real arithmetic.The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree.

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Constructing fewer open cells by GCD computation in CAD projection

Cited by 20 publications

References 17 publications

Detection of Special Curves Via the Double Resultant

Detection of Special Curves Via the Double Resultant

Open Non-uniform Cylindrical Algebraic Decompositions

Cylindrical algebraic decomposition with equational constraints

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