An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem

An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors, and invariant subspaces of large and sparse Hamiltonian matrices and low-rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.