Determination of eigenstates via Lanczos-based forward substitution and filter-diagonalization

A filter-diagonalization scheme based on the Lanczos tridiagonalization algorithm is proposed for the determination of highly excited eigenstates in a prespecified energy range. Filtered states localized in the energy domain can be generated during the Lanczos iteration by forward substitution which formally corresponds to a Green function filter. Eigenstates in the energy range of interest are then obtained by diagonalizing a small Hamiltonian matrix in a subspace spanned by the filtered states. It is shown that the accuracy of this approach is affected neither by the loss of orthogonality among the Lanczos states nor by the instability in the forward substitution. Since the filtered states are assembled white the Lanczos states are generated, no storage of the Lanczos states is needed. The computational savings can be significant in calculating high energy eigenstates of large systems. The proposed scheme is shown numerically to have comparable accuracy with the basic Lanczos method. (C) 1997 Academic Press.