On Hamiltonian and symplectic Lanczos processes

Abstract

Large sparse Hamiltonian eigenvalue problems arise in a variety of contexts. These problems can be attacked directly, or they can first be transformed to problems having some related structure, such as symplectic or skew-Hamiltonian. In the interest of efficiency, stability, and accuracy, such problems should be solved by methods that preserve the structure, whether it be Hamiltonian, skew-Hamiltonian, or symplectic. The present work outlines Krylov subspace methods for computing partial eigensystems of skew-Hamiltonian, Hamiltonian, and symplectic matrices and records some of the relationships between them. The ordinary unsymmetric Lanczos process is a structure-preserving method for skew-Hamiltonian matrices. The Hamiltonian and symplectic Lanczos processes developed here are condensed versions of the processes that have been published previously. The condensed Hamiltonian Lanczos process applied to H is equivalent to the unsymmetric Lanczos process applied to the skew-Hamiltonian H 2 but costs half as much to execute. The condensed symplectic Lanczos process applied to S is equivalent to the unsymmetric Lanczos process applied to the skew-Hamiltonian matrix S+ S −1 but also costs half as much to execute. Implicit restarts of the Hamiltonian and symplectic Lanczos processes can be effected by the SR algorithm. Because of the known relationship between the SR and HR algorithms, the much simpler HR algorithm can be used to restart the condensed symplectic and Hamiltonian Lanczos processes. Each of these restart procedures is equivalent to restarting the unsymmetric Lanczos process using the HR algorithm. The HR algorithm is most effectively implemented as an implicit (bulge-chasing) HZ algorithm on a symmetric, tridiagonal–diagonal pencil T− λD.