A Multilevel Spectral Indicator Method for Eigenvalues of Large Non-Hermitian Matrices

TL;DR

This paper introduces a memory-efficient multilevel spectral indicator method for computing many eigenvalues of large non-Hermitian matrices, improving upon existing SIMs by leveraging Cayley transformation and Krylov subspaces.

Contribution

It proposes a novel multilevel eigensolver that reduces memory usage and enhances efficiency for large sparse non-Hermitian matrices, building on spectral indicator methods.

Findings

Uses less memory than earlier SIMs

Effective for large sparse matrices

Demonstrated with multiple examples

Abstract

Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region contains eigenvalue(s). Then the region containing eigenvalues(s) is subdivided and tested. The procedure is repeated until the eigenvalues are identified within a specified precision. In this paper, using Cayley transformation and Krylov subspaces, a memory efficient multilevel eigensolver is proposed. The method uses less memory compared with the early versions of SIMs and is particularly suitable to compute many eigenvalues of large sparse (non-Hermitian) matrices. Several examples are presented for demonstration.