Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem

Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the rational Krylov method for this application. This method does not require a complete restart after a change of pole. The contribution of this paper is threefold. First, it is shown that the change of pole can be considered as a change of Lanczos basis. Second, moving the pole near a locked eigenvalue may prevent other eigenvalues to be computed accurately. Third, we show that a pole chosen close to an eigenvalue may lead to a loss of numerical stability. Usually, this will not lead to disaster but reduce the accuracy of the computed eigenvalues away from the pole. Numerical examples illustrate the theory.