An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

. An implicitly restarted symplectic Lanczos method for the symplectic 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 some eigenvalues and eigenvectors of large and sparse symplectic matrices. Key words. eigenvalues, symplectic Lanczos method, implicit restarting, symplectic matrix. AMS subject classifications. 65F15, 65F50, 15A18 1. Introduction. We consider the numerical solution of the real symplectic eigenvalue problem Mx = x (1.1) where M 2 IR 2n\Θ2n is large and possibly sparse. A matrix M is called symplectic iff MJM T = J; (1.2) or equivalently, M T JM = J; where J = 0 I n \ n 0 (1.3) and I n is the n \Θ n identity matrix. The symplectic matrices form a group under multiplication. The eigenvalues of symplectic matrices occur in reciprocal pairs: If i...