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 vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones 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–Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.