We describe several methods based on combinations of Krylov subspace techniques, deflation procedure and preconditionings, for computing a small number of eigenvaluea and eigenvectors or Schur vectors of large sparse matrices. The most effective techniques for solving realistic problems from applications are those methods based on so,e form of preconditioning and one of several Krylov subspace techniques, such as Arnoldi's method or the Lanczos procedure. We consider two forms of preconditionings: shift-and-invert and polynomial acceleration. The latter presents some advantages for parallel/vector processing but may be ineffective if eigenvalues inside the spectrum are sought. We provide some algorithmic details that improve the reliability and effectiveness of these techniques.
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* This work was supported in part by USRA under NASA Grant Number NCC 2-387 and in part by ARO under contract DAALO3-88-K-0085.