Convergence of algorithms of decomposition type for the eigenvalue problem

Abstract

We develop the theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR, LR, SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm performs nested subspace iteration with a change of coordinate system at each step. Thus the convergence of the GR algorithm depends on the convergence of certain sequences of subspaces. It also depends on the quality of the coordinate transformation matrices, as measured by their condition numbers. We show that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate. For matrices possessing certain special types of structure, cubic convergence can be achieved.