A Novel Approach to Find Pseudo–peripheral Vertices for Snay’s Heuristic

ABSTRACT The solution of linear systems represented by Ax=b is fundamental in many numerical simulations in science and engineering. Reducing the profile of A can reduce the storage requirements and time processing costs of solving such linear systems. In this work, we propose a generalized algorithm for finding pseudo-peripheral vertices for Snay’s heuristic. In experiment performed on 36 instances contained in the Harwell-Boeing and SuiteSparse matrix collections, it has been found that the number of pseudo-peripheral vertices selected in Snay’s heuristic may be suitable for small instances, but it is insufficient to obtain reasonable results in instances that are not small. This paper recommends to select up to 26% (0.3%) of pseudo-peripheral vertices in relation to the instance size when applied to instances smaller than 3,000 (larger than 20,000) vertices.

摘要 形如Ax=b的线性方程组的求解，是科学与工程领域诸多数值模拟的核心基础问题。缩减矩阵A的轮廓，可降低此类线性方程组求解的存储需求与处理时间开销。本研究提出一种针对Snay启发式（Snay’s heuristic）的伪外围顶点（pseudo-peripheral vertices）求解通用算法。本研究在Harwell-Boeing与SuiteSparse矩阵库收录的36个测试实例上开展实验，结果表明：Snay启发式中选取的伪外围顶点数量，对于小规模实例或可适用，但在非小规模实例中难以获得合理的求解效果。针对顶点数少于3000的实例，建议选取与实例规模相关的最高26%的伪外围顶点；而针对顶点数多于20000的实例，则建议选取最高0.3%的伪外围顶点。