A Convergence proof of an Iterative SubspaceMethod for Eigenvalues Problems ?

The generalized Davidson algorithm can be seen as a method which uses preconditioned residuals to create a subspace where it is easier to nd the smallest eigenvalue and its eigenvector. In this paper theoretical results proving convergence rates are shown. In addition, we investigate the use of multigrid as a preconditioner for this method and describe a new algorithm for calculating some other eigenvalue{eigenvector pairs as well, while avoiding problems of misconvergence. The advantages of implicit restarts are also investigated. The Davidson algorithm 6] and its variants provide a means to calculate ei-genvalues of large, sparse, or structured symmetric matrices. While it has been known to, and used by, quantum chemists, this algorithm has only recently come to the attention of mathematicians through the work of Saad 14] and others. In this paper the term \Davidson algorithm" is used to mean the generalized Dav-idson algorithm which allows any matrix M as a preconditioner, rather than restricting the choice of preconditioner to the diagonal preconditioner in 6]. The Generalized Davidson algorithm is shown in Figure 1. Quantum chemists have frequently studied matrices that are strongly diagonally dominant, and so the \diagonal preconditioner" M = (D ? I) ?1 with D = diag(A) gives extremely rapid convergence. For more general cases we use other preconditioners. Numerical results with a multigrid preconditioner applied to a PDE eigenproblem are shown in the last Section. In the Davidson method, if the preconditioner used is M = I, 6 = 0 the method is equivalent to the Lanczos algorithm in exact arithmatic. In fact, with this preconditioner, the Davidson algorithm is essentially an expensive implementation of the Lanczos algorithm with complete reorthogonalization 7, x9.2.3]. Both the Lanczos and Davidson algorithms involve the computation of an or-thonormal basis for a subspace from a given starting vector x 0. For the Davidson algorithm the subspace also depends on the preconditioner used. The Lanczos method generates successive bases for Krylov subspaces