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The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation of a look-ahead version of the Lanczos algorithm is presented that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead.

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for nonHermitian matrices.

Three leading iterative methods for the solution of nonsymmetric systems of linear equations are CGN (the conjugate gradient iteration applied to the normal equations), GMRES (residual minimization in a Krylov space), and CGS (a biorthogonalization algorithm adapted from the biconjugate gradient iteration). Do these methods differ fundamentally in capabilities? If so, which is best under which circumstances? The existing literature, in relying mainly on empirical studies, has failed to confront these questions systematically. In this paper it is shown that the convergence of CGN is governed by singular values and that of GMRES and CGS by eigenvalues or pseudo-eigenvalues. The three methods are found to be fundamentally different, and to substantiate this conclusion, examples of matrices are presented for which each iteration outperforms the others by a factor of size O(V) or O(N) where N is the matrix dimension. Finally, it is shown that the performance of iterative methods for a particular matrix cannot be predicted from the properties of its symmetric part.

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