Preconditioned Iterative Methods for Eigenvalue Counts

We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications. It can also be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval. We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations.