A stable simultaneous vector inverse iteration method with shift

A numerically stable technique to remove the limitation in choosing a shift in the simultaneous vector inverse iteration step of the subspace iteration method with shift is presented. A major difficulty of the subspace iteration method with shift is that, because of the singularity problem, a shift close to an eigenvalue cannot be used, resulting in slower convergence. In this paper, the above singularity problem has been solved by using side conditions without sacrifice of convergence. The proposed method is always non-singular even if a shift is on an exact eigenvalue. This is one of the significant characteristics of the proposed method. The non-singularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of the above two methods are almost the same for large structures. To show the effectiveness of the method proposed, two numerical examples are considered.