How to Implement the Spectral Transformation

The general, linear eigenvalue equations $({\mathbf {H}} - \lambda {\mathbf {M}}){\mathbf {z}} = 0$, where H and M are real symmetric matrices with M positive semidefinite, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs $(\lambda ,{\mathbf {z}})$. 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 $\lambda$ 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.