The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectorsmay fail to converge. To overcome this potential problem,weminimize residuals formedwith periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the correspondingwell-knownones for Rayleigh–Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh–Ritzmethodwith refinement. Thenumerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors. © 2010 Elsevier B.V. All rights reserved.