A refined CJ--SS--RR method with a reliable removal approach of spurious Ritz values for the Hermitian eigenvalue problem

Abstract

Under the hypothesis that the deviations of the desired eigenvectors of the matrix $A$ from the underlying subspace tend to zero, the Ritz vectors may not converge and have poor or little accuracy. This phenomenon is not unusual and particularly occurs when the associated Ritz values are close, which is independent of the eigenvalue distribution of $A$. For the (block) SS--RR methods, there are possibly {\em more} Ritz values that converge to the same desired eigenvalue(s) counting multiplicity in the region of interest, meaning that some of the Ritz values must be spurious and the corresponding residual norms of the Ritz pairs may not be small. Consequently, the (block) SS--RR methods including the CJ--SS--RR method cannot base on the corresponding residual norms to effectively identify if the Ritz values in the region are genuine or spurious. This paper proposes refined SS--RR, abbreviated as SS--RRR, methods based on the refined Rayleigh--Ritz projection that compute the eigenpairs of large matrices with the eigenvalues located in the given region. We present a new approach to accurately implement the RRR methods more efficiently than ever before for a general subspace.Exploiting the unconditional convergence of the refined Ritz vectors when the subspace is sufficiently accurate, we propose a tune-free removal approach to effectively remove spurious Ritz values with a rigorous theory supported, and develop a restarted CJ--SS--RRR algorithm. Numerical experiments show that the restarted CJ--SS--RRR algorithm is more efficient and effective than the restarted CJ--SS--RR algorithm.