The implicit application of a rational filter in the RKS method (Q1378473)

The paper presents a study of the rational Krylov subspace (RKS) method. This method is used for the calculation of a few eigenvalues of \(Ax= \lambda Bx\) with \(A\) and \(B\) real \(n\times n\) matrices, and where \(B\) may be singular. The aim is to throw away unwanted directions from the rational Krylov subspace by applying a rational filter to its basis. The implicitly restarted Arnoldi method applies a polynomial filter to the Arnoldi vectors by use of orthogonal transformations. Two applications are considered. The first one is the filtering of unwanted eigenvalues using exact shifts. Second, eigenvalue problems can have an infinite eigenvalue without physical relevance. This infinite eigenvalue can corrupt the eigensolution. An implicit filtering is proposed for avoiding such corruptions.