. In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well. Key words. eigenvalues and eigenvectors, Davidson's method, Jacobi iterations, harmonic Ritz values AMS subject classifications. 65F15, 65N25 PII. S0036144599363084 1. Introduction. Suppose we want to compute one or more eigenvalues and their corresponding eigenvectors of the n n matrix A. Several iterative methods are available: Jacobi's diagonalization method [9], [23], the power method [9], the method of Lanczos [13], [23], Arnoldi's method [1], [26], and Davidson's method [4], ...

The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis, and makes it possible to get many linearly independent eigenvectors and principal vectors to pencils with multiple eigenvalues. Relations to iterative methods are established. Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane, the other a nonsymmetric matrix modeling an idealized aircraft stability problem.

Abstract. We provide a general frameworkfor the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This frameworkallows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Frayssé, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

We discuss approaches for an efficient handling of the correction equation in the Jacobi-Davidson method. The correction equation is effective in a subspace orthogonal to the current eigenvector approximation. The operator in the correction equation is a dense matrix, but it is composed from three factors that allow for a sparse representation. If the given matrix eigenproblem is sparse then one often aims for the construction of a preconditioner for that matrix. We discuss how to restrict this preconditioner effectively to the subspace orthogonal to the current eigenvector. The correction equation itself is formulated in terms of approximations for an eigenpair. In order to avoid misconvergence one has to make the right selection for the approximations, and this aspect will be discussed as well.

We provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual (MR) Krylov subspace methods due to restarting. We evaluate the popular practice of using nearly invariant subspaces to either augment Krylov subspaces or to construct preconditioners which invert on these subspaces. In the case where these spaces are exactly invariant, the augmentation approach is shown to be superior. We further show how a strategy recently introduced by de Sturler for truncating the approximation space of an MR method can be interpreted as a controlled loosening of the condition for global MR approximation based on the canonical angles between subspaces. For the special case of Krylov subspace methods, we give a concise derivation of the role of Ritz and harmonic Ritz values and vectors in the polynomial description of Krylov spaces as well as of the use of the implicitly updated Arnoldi method for manipulating Krylov spaces.

The Jacobi-Davidson subspace iteration method offers possibilities for solving a variety of eigenproblems. In practice one has to apply restarts because of memory limitations, in order to restrict computational overhead, and also if one wants to compute several eigenvalues. In general, restarting has negative effects on the convergence of subspace methods. We will show how effective restarts can be incorporated in the Jacobi-Davidson methods, very similar to the implicit restart procedure for the Arnoldi process. We will present two algorithms, JDQR for the standard eigenproblem, and JDQZ for the generalized eigenproblem, that are based on the iterative construction of the (generalized) partial Schur form with the Jacobi-Davidson approach. The algorithms are suitable for the efficient computation of several (even multiple) eigenvalues, and the corresponding eigenvectors, near a user-specified target value in the complex plane.

Abstract. We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi–Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. We give a few variants of this Jacobi–Davidson SVD (JDSVD) method with their theoretical properties. It is shown that the JDSVD can be seen as an accelerated (inexact) Newton scheme. We experimentally compare the method with some other iterative SVD methods. Key words. Jacobi–Davidson, singular value decomposition (SVD), singular values, singular vectors, norm, augmented matrix, correction equation, (inexact) accelerated Newton, improving singular values AMS subject classifications. 65F15 (65F35) PII. S1064827500372973

The irbleigs code is an implementation of an implicitly restarted block-Lanczos method for computing a few selected nearby eigenvalues and associated eigenvectors of a large, possibly sparse, Hermitian matrix A. The code only requires the evaluation of matrix-vector products with A; in particular, factorization of A is not demanded, nor is the solution of linear systems of equations with the matrix A. This, together with a fairly small storage requirement, makes the irbleigs code well suited for large-scale problems. Applications of the irbleigs code to certain generalized eigenvalue problems and to the computation of a few singular values and associated singular vectors are also discussed. Numerous computed examples illustrate the performance of the method and provide comparisons with other available codes.

. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the two-norm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...