Controllability Properties of Numerical Eigenvalue Algorithms

We analyze controllability properties of the inverse iteration and the QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and suucient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditions for the solvability of this matrix equation are given. We discuss an interpretation of the system in terms of control systems on rational functions. Finally, rst results on the extension to inverse Rayleigh iteration on Grassmann manifolds using complex shifts is discussed. For many numerical matrix eigenvalue methods such as the QR algorithm or inverse iterations shift strategies have been introduced in order to design algorithms that have faster (local) convergence. The shifted inverse iteration is studied in 3,4,15] and in 17,18], where the latter references concentrate on complex shifts. For an algorithm using multidimensional shifts for the QR-algorithm see the paper of Absil, Mahony, Sepulchre and van Dooren in this book. In this paper we interpret the shifts as control inputs to the algorithm. With this point of view standard shift strategies as the well known Rayleigh iteration can be interpreted as feedbacks for the control system. It is known (for instance in the case of the inverse iteration or its multidimensional analogue , the QR-algorithm) that the behavior of the Rayleigh shifted algorithm can be very complicated, in particular if it is applied to non-Hermitian matrices A 4]. It is therefore of interest to obtain a better understanding of the underlying control system, which up to now has been hardly studied. Here we focus on controllability properties of the corresponding systems on projective space for the case of inverse iteration, respectively the Grass-mannian manifold for the QR-algorithm. As it turns out the results depend heavily on the question whether one uses real or complex shifts. The control-lability of the inverse iteration with complex shifts has been studied in 13], while the real case is treated in 14]. ? This paper was written while Fabian Wirth was a guest at the Centre Automa-tique et Syst emes, Ecole des Mines de Paris, Fontainebleau, France. The hospitality of all the members of the centre is gratefully acknowledged.