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Abstract.The simple Lanczos process is very effective for finding a few extreme eigenvalues of a large symmetric matrix along with the associated eigenvectors.Unfortunately, the process computes redundant copies of the outermost eigenvectors and has to be used with some skill. In this paper it is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step significantly. The degree of linear independence among the Lanczos vectors is controlled without the costly process of reorthogonalization.

This paper analyzes Davidson's method for computing a few eigenpairs of large sparse symmetric matrices. An explanation is given for why Davidson's method often performs well but occasionally performs very badly. Davidson's method is then generalized to a method which offers a powerful way of applying preconditioning techniques developed for solving systems of linear equations to solving eigenvalue problems. AMS(MOS) subject classifications. 65, 15 2. Davidson's method. Davidson [2] introduced a new method for computing a few eigenvalues of sparse symmetric matrices arising in quantum chemistry calculations.The standard solution technique for such problems is the Lanczos algorithm [7, Chap. 13] which is a clever implementation of the Rayleigh-Ritz procedure applied to a Krylov subspace (that is, a space of the form span (s, As,..., Aks)). Davidson's method also uses the Rayleigh-Ritz procedure (see [7, p. 213]) but on a non-Krylov subspace. Formally Davidson's method is as follows.

A new stable and efficient implementation of the Lanczos algorithm is presented.The Lanczos algorithm is a powerful method for finding a few eigenvalues and eigenvectors at one or both ends of the spectrum of a symmetric matrix A. The algorithm is particularly effective if A is large and sparse in that the only way in which A enters the calculation is through a subroutine which computes Av for any vector v. Thus the user is free to take advantage of any sparsity structure in A and A need not even be represented as a matrix at all.The simple Lanczos algorithm procedes as follows. an arbitrary unit vector, and define i3

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