  We propose efficient preconditioning algorithms for an eigenvalue problem
arising in quantum physics, namely the computation of a few interior
eigenvalues and their associated eigenvectors for the largest sparse real and
symmetric indefinite matrices of the Anderson model of localization. We compare
the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with
the shift-and-invert techniques in the implicitly restarted Lanczos method and
in the Jacobi-Davidson method. Our preconditioning approaches for the
shift-and-invert symmetric indefinite linear system are based on maximum
weighted matchings and algebraic multilevel incomplete $LDL^T$ factorizations.
These techniques can be seen as a complement to the alternative idea of using
more complete pivoting techniques for the highly ill-conditioned symmetric
indefinite Anderson matrices. We demonstrate the effectiveness and the
numerical accuracy of these algorithms. Our numerical examples reveal that
recent algebraic multilevel preconditioning solvers can accelerative the
computation of a large-scale eigenvalue problem corresponding to the Anderson
model of localization by several orders of magnitude.
