The Solution of Large Symmetric Eigenproblems by Sectioning

Abstract

When a relatively few eigenvalues are desired for a very large symmetric matrix eigenvalue problem, direct methods such as Householder reduction tend to be inefficient. Inverse iteration works reasonably well but runs into difficulties when eigenvalues are clustered. This paper presents a method for determining the eigenvalues lying in a “section” $\alpha < \lambda < \beta $ of the eigenvalue spectrum together with the corresponding eigenvectors. In contrast with inverse iteration, the sectioning method works particularly well for clustered eigenvalues. The sectioning method proceeds in three phases : first, a basis for the invariant subspace corresponding to the spectral section $\alpha < \lambda < \beta $ is computed, next this basis is used to reduce the eigenproblem by the Ritz process, and finally, the reduced problem is solved in high precision by a fairly standard Householder technique.