A rational QR-iteration
Raf Vandebril, Marc Van Barel, Nicola Mastronardi · 2007
In this manuscript a new type of QR-iteration will be presented. Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A−κI = RQ will be computed, followed by a QR-factorization of the matrix (A−σI)Q H. Applying the unitary similarity transformation defined by the QR-factorization of the transformed matrix (A−σI)Q H, will yield interesting convergence properties. It will be shown that the convergence behavior is related to a subspace iteration based on a rational function in A namely (A − σI)(A − κI) −1. Convergence properties of this new iteration will be investigated and examples will be presented, illustrating the effectiveness of this approach with respect to some specific classes of matrices