A Thick-Restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems

TL;DR

This paper introduces a novel Thick-Restart Lanczos algorithm enhanced with polynomial filtering and deflation techniques, enabling efficient computation of eigenvalues in specific spectral intervals for large Hermitian matrices.

Contribution

It combines polynomial filtering with a Thick-Restart Lanczos method and a new least-squares polynomial filter, improving spectrum-slicing for large eigenvalue problems.

Findings

Effective eigenvalue computation in specified spectral intervals

Enhanced algorithm performance with polynomial filters and deflation

Applicable to large Hermitian matrices in spectrum-slicing

Abstract

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation (`locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a `spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.