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

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.