Numerical determination of partial spectrum of Hermitian matrices using a Lanczos method with selective reorthogonalization

TL;DR

This paper presents a new selective reorthogonalization algorithm based on LANSO for efficiently computing eigenvalues and eigenvectors of Hermitian matrices within a specific region, with applications in lattice QCD.

Contribution

It introduces a novel bound for selective reorthogonalization that improves eigenpair computation accuracy and efficiency in finite-precision arithmetic.

Findings

The new algorithm effectively computes partial spectra of Hermitian matrices.

Performance comparison shows improvements over previous methods in lattice QCD applications.

Selective reorthogonalization reduces unnecessary computations while maintaining accuracy.

Abstract

We introduce a new algorithm for finding the eigenvalues and eigenvectors of Hermitian matrices within a specified region, based upon the LANSO algorithm of Parlett and Scott. It uses selective reorthogonalization to avoid the duplication of eigenpairs in finite-precision arithmetic, but uses a new bound to decide when such reorthogonalization is required, and only reorthogonalizes with respect to eigenpairs within the region of interest. We investigate its performance for the Hermitian Wilson--Dirac operator (\gamma_5D) in lattice quantum chromodynamics, and compare it with previous methods.