A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem
Peter Benner, Heike Faßbender · Qucosa - Monarch (Chemnitz University of Technology) · 1998
A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. Breakdowns and near-breakdowns are overcome 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. Key words : symplectic Lanczos method, implicit restarting, Hamiltonian matrix, eigenvalues, low rank approximate solution, algebraic Riccati equation. AMS(MOS) subject classifications : 65F15, 65F50 1 Introduction Many applications require the numerical solution of the real Hamiltonian eigenvalue problem Hx = x (1) where H = " A G Q \\GammaA T # 2 IR 2n\\Theta2n is large and sparse and A; G = G T ; Q = Q T 2 IR n\\Thetan : The eigenvalues of Hamiltonian matrices are used in algorithms to c...