  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.In particular, it can 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.
