## Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenproblems

Gerard L. G. Sleijpen, Henk A. van der Vorst, and Ellen Meijerink

### Abstract

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.

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### Key words

linear eigenproblems, generalized eigenproblems, Jacobi-Davidson, harmonic Ritz values, preconditioning.

65F15.

### Links to the cited ETNA articles

 [13] Vol. 1 (1993), pp. 11-32 Gerard L. G. Sleijpen and Diederik R. Fokkema: BiCGstab($l$) for linear equations involving unsymmetric matrices with complex spectrum [20] Vol. 7 (1998), pp. 202-214 Kesheng Wu, Yousef Saad, and Andreas Stathopoulos: Inexact Newton preconditioning techniques for large symmetric eigenvalue problems

### ETNA articles which cite this article

 Vol. 20 (2005), pp. 235-252 Michiel E. Hochstenbach: Generalizations of harmonic and refined Rayleigh-Ritz

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