Some insights on restarting symmetric eigenvalue methods with Ritz and harmonic Ritz vectors

Eigenvalue iterative methods, such as Arnoldi and Jacobi-Davidson, are typically used with restarting. This has signiicant performance shortcomings, since important components of the invariant subspace may be discarded. One way of saving more information at restart is the idea of \thick" restarting which keeps more Ritz vectors than needed. Our previously proposed dynamic thick restarting chooses these vectors in a way that has proved eecient on a wide variety of matrices. It is also possible to keep this information more compactly by combining thick restarting with a technique based on a three term recurrence. In this paper, we give strong experimental evidence that saving more information with thick restarting is not necessarily beneecial, and provide an explanation to the eeciency of the dynamic scheme. In addition, we show through a variety of experiments that restarting with harmonic Ritz instead of Ritz vectors does not improve the convergence of symmetric eigenvalue methods when an extreme part of the spectrum or some eigenpair within this part is needed. However, for computing highly interior eigenpairs, harmonic Ritz vectors may be the only viable alternative.