Top Eigenpairs of Large Scale Matrices

Abstract

This paper is devoted to the study of an extended global algorithm on computing the top eigenpairs of a large class of matrices. Three versions of the algorithm are presented that includes a preliminary version for real matrices, one for complex matrices, and one for large scale sparse real matrix. Some examples are illustrated as powerful applications of the algorithms. The main contributions of the paper are two localized estimation techniques, plus the use of a machine learning inspired approach in terms of a modified power iteration. Based on these new tools, the proposed algorithm successfully employs the inverse iteration with varying shifts (a very fast "cubic algorithm") to achieve a superior estimation accuracy and computation efficiency to existing approaches under the general setup considered in this work.