Year: 2020
Author: Tao Tang, Jiang Yang
Communications in Mathematical Research , Vol. 36 (2020), Iss. 1 : pp. 93–112
Abstract
A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super- or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0005
Communications in Mathematical Research , Vol. 36 (2020), Iss. 1 : pp. 93–112
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Maximal eigenpair symmetric matrix dense matrix tridiagonal matrix Householder transformation complexity iteration.