Hypergraph edge elimination - A symbolic phase for Hermitian eigensolvers based on rank-1 modifications

Electronic Transactions on Numerical Analysis

article

Karsten Kahl¹  Bruno Lang¹ 

[1] University of Wuppertal, IMACM, Mathematics and Natural Sciences

关键词: symmetric eigenvalue problem;    hypergraphs;    Gaussian elimination;   

DOI  :  10.1553/etna_vol54s51

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

It is customary to identify sparse matrices with the corresponding adjacency or incidence graphs. For the solution of a linear system of equations using Gaussian elimination, the representation by its adjacency graph allows a symbolic factorization that can be used to predict memory footprints and enables the determination of near-optimal elimination orderings based on heuristics. The Hermitian eigenvalue problem on the other hand seems to evade such treatment at first glance due to its inherent iterative nature. In this paper we prove this assertion wrong by revealing a tight connection of Hermitian eigensolvers based on rank-$1$ modifications with a symbolic edge elimination procedure. A symbolic calculation based on the incidence graph of the matrix can be used in analogy to the symbolic phase of Gaussian elimination to develop heuristics which reduce memory footprint and computations. Yet, we also show that the question of an optimal elimination strategy remains NP-complete, in analogy to the linear systems case.

【 授权许可】

Unknown