A symplectic Lanczos-type algorithm to compute the eigenvalues of positive definite Hamiltonian matrices

We consider the application of symmetric Boundary Value Methods to linear autonomous Hamiltonian systems. The numerical approximation of the Hamiltonian function exhibits a superconvergence property, namely its order of convergence is p + 2 for a p order symmetric method. We exploit this result to define a natural projection procedure that slightly modifies the numerical solution so that, without changing the convergence properties of the numerical method, it provides orbits lying on the same quadratic manifold as the continuous ones. A numerical test is also reported.