Iterative diagonalization of a large sparse matrix using spectral transformation and filter diagonalization

Abstract

In this review we consider iterative diagonalization of a large sparse matrix using spectral transformation and filter diagonalization. The emphasis is on real symmetric and complex hermitian matrices. Classical orthogonal polynomial expansions, Davidson iteration and Krylov subspace methods for diagonalization are briefly described. The standard Lanczos method is discussed in detail. It has two main drawbacks, ghosting, i.e. generation of spurious and duplicate eigenvalues due to loss of orthogonality among the Lanczos vectors, and slow convergence for dense interior regions of the eigenvalue spectrum. Filter diagonalization, in particular combined with the minimum residual algorithm, and reorthogonalization are discussed as ways to rectify the ghosting problem. It is described in detail how spectral transformation can be used to turn the convergence problem into an advantage for the user. Applications to quantum scattering treatments of chemical reactions and a bound states calculation of a molecule are given.