NAOct 13, 2011
Bounds on the convergence of Ritz values from Krylov subspaces to interior eigenvalues of Hermitean matricesChris Johnson, A. D. Kennedy
We consider bounds on the convergence of Ritz values from a sequence of Krylov subspaces to interior eigenvalues of Hermitean matrices. These bounds are useful in regions of low spectral density, for example near voids in the spectrum, as is required in many applications. Our bounds are obtained by considering the usual Kaniel-Paige-Saad formalism applied to the shifted and squared matrix.
HEP-LATNov 6, 2012
Numerical determination of partial spectrum of Hermitian matrices using a Lanczos method with selective reorthogonalizationChris Johnson, A. D. Kennedy
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 (γ_5D) in lattice quantum chromodynamics, and compare it with previous methods.