How to Implement the Spectral Transformation

The general, linear eigenvalue equations (H XM)z = 0, where H and M are real symmetric matrices with M positive semidefimte, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs (X,z). When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues a near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction. The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration. A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when M is ill-conditioned or singular. We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well. 1. Overview. This contribution is an addendum to the paper by Ericsson and Ruhe [1] and also [7]. The value of the spectral transformation is reiterated in a later section. Here we outline our implementation of this transformation. The equation to be solved, for an eigenvalue À and eigenvector z, is (1) (H AM)z = 0, H and M are real symmetric n X n matrices, and M is positive semidefinite. A practical instance of (1) occurs in dynamic analysis of structures, where H and M are the stiffness and mass matrices, respectively. We assume that a linear combination of H and M is positive definite. It then follows that all eigenvalues X are real. In addition, one has a real scalar a, distinct from any eigenvalue, and we seek a few eigenvalues X close to a, together with their eigenvectors z. Ericsson and Ruhe replace (1) by a standard eigenvalue equation (2) [c(H-aM)"1CT-rl]y = 0, where C is the Choleski factor of M; M = CTC and y = Cz. If M is singular then so is C, but fortunately the eigenvector z can be recovered from y via z = (H oM)"'CTy. Of course, there is no intention to invert (H oM) explicitly. The Received May 14, 1984; revised December 20, 1985. 1980 Mathematics Subject Classification. Primary 65F15. *This research was supported in part by the AFOSR contract F49620-84-C-0090. The third author was also supported in part by the Swedish Natural Science Research Council. **The paper was written while this author was visiting the Center for Pure and Applied Mathematics, University of California, Berkeley, California 94720. ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page