Computing eigenvalue bounds for iterative subspace matrix methods

A procedure is presented for the computation of bounds to eigenvalues of the generalized hermitian eigenvalue problem and to the standard hermitian eigenvalue problem. This procedure is applicable to iterative subspace eigenvalue methods and to both outer and inner eigenvalues. The Ritz values and their corresponding residual norms, all of which are computable quantities, are needed by the procedure. Knowledge of the exact eigenvalues is not needed by the procedure, but it must be known that the computed Ritz values are isolated from exact eigenvalues outside of the Ritz spectrum and that there are no skipped eigenvalues within the Ritz spectrum range. A multipass refinement procedure is described to compute the bounds for each Ritz value. This procedure requires O(m) effort where m is the subspace dimension for each pass.