Open Weak CAD and Its Applications

@article{Dai2017OpenWC,
  title={Open Weak CAD and Its Applications},
  author={Liyun Dai and Jingjun Han and Hoon Hong and Bican Xia},
  journal={ArXiv},
  year={2017},
  volume={abs/1507.03834}
}
Abstract The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea is to compute the intersection of projection factor sets produced by different projection orders. The resulting open weak CAD often has smaller number of sample points than open CADs. The algorithm can be used for computing sample points for all open… Expand
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References

SHOWING 1-10 OF 36 REFERENCES
Proving inequalities and solving global optimization problems via simplified CAD projection
TLDR
A simplified Brown-McCallum's CAD projection operator, Nproj, is proposed, of which the projection scale is always no larger than that of Brown- McCallum's, and the lifting phase is also simplified. Expand
Constructing fewer open cells by GCD computation in CAD projection
TLDR
It is proved that the new projection operator based on cylindrical algebraic decomposition can be used for testing semi-definiteness of polynomials and still guarantees obtaining at least one sample point from every connected component of the highest dimension. Expand
Testing Sign Conditions on a Multivariate Polynomial and Applications
  • M. S. E. Din
  • Mathematics, Computer Science
  • Math. Comput. Sci.
  • 2007
TLDR
The paper shows how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semi-algebraic set defined by a single inequality or a single inequation. Expand
An Improved Projection Operation for Cylindrical Algebraic Decomposition of Three-Dimensional Space
  • S. McCallum
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1988
TLDR
It is shown, using a theorem from complex analytic geometry, that the original projection set for trivariate polynomials that Collins used can be substantially reduced in size, without affecting its essential properties. Expand
Polar varieties and computation of one point in each connected component of a smooth real algebraic set
TLDR
An algorithm is deduced that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations and its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity. Expand
Partial Cylindrical Algebraic Decomposition for Quantifier Elimination
TLDR
This paper presents a method which intermingles CAD construction with truth evaluation so that parts of the CAD are constructed only as needed to further truth evaluation and aborts CAD construction as soon as no more truth evaluation is needed. Expand
On the combinatorial and algebraic complexity of quantifier elimination
TLDR
This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data, and new and improved algorithms for deciding a sentence in the first order theory over real closed fields, are obtained. Expand
Solving Systems of Strict Polynomial Inequalities
TLDR
An algorithm for finding an explicit description of solution sets of systems of strict polynomial inequalities, correct up to lower dimensional algebraic sets is presented, based on the cylindrical algebraic decomposition algorithm. Expand
Improved Projection for Cylindrical Algebraic Decomposition
TLDR
A simple theorem is presented showing that the mathematics in McCallum?s paper actually point to a better projection operator than he proposes, which has the potential to not simply speed up CAD computation for problems that are currently solvable in practice, but actually increase the scope of problems that can realistically be attacked via CADs. Expand
Variant quantifier elimination
TLDR
The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals, and it is found that the algorithm can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. Expand
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4
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