Error Analysis of Symplectic Lanczos Method for Hamiltonian Eigenvalue Problem

Abstract:

      A rounding error analysis of the symplectic Lanczos method is given for the Hamil-tonian eigenvalue problem. It is applicable when no break down occurs and shows that the restriction of preserving the Hamiltonian structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. All analysis follows the lines of Bai's analysis of the nonsymmetric Lanczos algorithm and the lines of H.FaBbender's analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem. As is expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.