Quasi-kernel polynomials and their use in non-Hermitian matrix iterations

Abstract

Some of the most efficient iterative algorithms for large sparse Hermitian matrix computations are based on orthogonal or kernel polynomials. For the case of non-Hermitian matrices, methods based on orthogonal or kernel polynomials are less satisfactory, in that the resulting algorithms involve long recurrences. Consequently, it is usually too expensive to run the full algorithms and restarts are necessary. A typical example is the generalized minimal residual method (GMRES) for solving non-Hermitian linear systems, where work and storage per iteration grow linearly with the iteration number. Recently, two quasi-minimal residual methods (QMR) for solving non-Hermitian linear systems have been proposed, which — unlike GMRES — are based on short recurrences and hence can be used as true iterative schemes, without restarts. In this paper, the concept of quasi-kernel polynomials is introduced. Some general theory for quasi-kernel polynomials is developed, such as recurrence relations and a characterization of roots of quasi-kernel polynomials as generalized eigenvalues. It is pointed out that the QMR approaches are based on two particular instances of quasi-kernel polynomials. Also, the use of quasi-kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non-Hermitian matrices is briefly discussed.