Deflation Techniques for an Implicitly Restarted Arnoldi Iteration

A deeation procedure is introduced that is designed to improve the convergence of an implicitly restarted Arnoldi iteration for computing a few eigenvalues of a large matrix. As the iteration progresses the Ritz value approximations of the eigenvalues of A converge at diierent rates. A numerically stable scheme is introduced that implicitly deeates the converged approximations from the iteration. We present two forms of implicit deeation. The rst, a locking operation, decouples converged Ritz values and associated vectors from the active part of the iteration. The second, a purging operation, removes unwanted but converged Ritz pairs. Convergence of the iteration is improved and a reduction in computational eeort is also achieved. The deeation strategies make it possible to compute multiple or clustered eigenvalues with a single vector restart method. A Block method is not required. These schemes are analyzed with respect to numerical stability and computational results are presented.