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Chordal graphs in triangular decomposition in top-down style
Journal of Symbolic Computation ( IF 0.7 ) Pub Date : 2019-10-18 , DOI: 10.1016/j.jsc.2019.10.011
Chenqi Mou , Yang Bai , Jiahua Lai

In this paper, we first prove that when the associated graph of a polynomial set is chordal, a particular triangular set computed by a general algorithm in top-down style for computing the triangular decomposition of this polynomial set has an associated graph as a subgraph of this chordal graph. Then for Wang's method and a subresultant-based algorithm for triangular decomposition in top-down style and for a subresultant-based algorithm for regular decomposition in top-down style, we prove that all the polynomial sets appearing in the process of triangular decomposition with any of these algorithms have associated graphs as subgraphs of this chordal graph. These theoretical results can be viewed as non-trivial polynomial generalization of existing ones for sparse Gaussian elimination, inspired by which we further propose an algorithm for sparse triangular decomposition in top-down style by making use of the chordal structure of the polynomial set. The effectiveness of the proposed algorithm for triangular decomposition, when the polynomial set is chordal and sparse with respect to the variables, is demonstrated by preliminary experimental results.



中文翻译:

自上而下的三角分解中的和弦图

在本文中,我们首先证明,当多项式集的关联图是弦的时,由通用算法以自顶向下样式计算的用于计算该多项式集的三角分解的特定三角形集具有关联图作为的子图。这个和弦图。然后,对于Wang的方法和以子结果为基础的自顶向下样式的三角分解算法,以及以子结果为基础的自顶向下样式的常规分解算法,我们证明了在多项式分解过程中出现的所有多项式集这些算法中有一个将相关图作为此弦图的子图。这些理论结果可被视为现有稀疏高斯消去的非平凡多项式一般化,受此启发,我们进一步提出了一种利用多项式集合的弦结构以自顶向下样式进行稀疏三角分解的算法。初步的实验结果证明了所提出的三角分解算法在多项式集相对于变量为弦和稀疏的情况下的有效性。

更新日期:2019-10-18
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