Subspace Methods for Large Sparse Interior Eigenvalue Problems

The calculation of a few interior eigenvalues of a matrix has not received much attention in the past, most methods being some spin-off of either the complete eigenvalue calculation or a subspace method designed for the extremal part of the spectrum. The reason for this could be the rather chaotic behaviour of most methods tried. Only 'shift and invert' and polynomial iteration seemed to have a predictable behavior. However, polynomial iteration is reasonably fast only for extremal eigenvalues of a matrix where all eigenvalues are close to a known line, and inverting a large sparse indefinite system is tricky, while any inaccuracy in the inverse carries through to the eigenvector. By now, subspace methods have been developed to a state where they can be applied with benefit to the calculation of inner eigenpairs (eigenvalues and-vectors). This is achieved by using a combination of improved approximate residual correction (Jacobi-Davidson method) with new methods to extract approximations to inner eigenvectors of a large (dimension n) matrix from a low dimensional (dimension m √ n) subspace. Suited to the needs of practical applications, the selection of eigenpairs requested may be specified by very different means-an eigenvalue range, closeness of the eigenvectors to a given selection of approximate eigenvec-tors, or special patterns of the eigenvectors like the number of local extrema of the components. Depending on information generated with little overhead relative to standard subspace computations, the extraction method may be switched between standard Ritz projection, inverse Ritz projection and residual minimization. Tests indicate reliable and predictable convergence, while performance depends heavily on the quality of the approximate inverse applied. Some applications are in theoretical chemistry and from accelerator design. In these cases, the eigenpairs requested are typically from the lower part of the spectrum , but too far from the end to be solved by conventional subspace methods.