Algorithm 800: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices I: The square-reduced method

This article describes LAPACK-based Fortran 77 subroutines for the reductio n of a Hamiltonian matrix to square-reduced form and the approximation of a ll its eigenvalues using the implicit version of Van Loan's method. The tra nsformation of the Hamiltonian matrix to a square-reduced form transforms a Hamiltonian eigenvalue problem of order 2n to a Hessenberg eigenvalue prob lem of order Iz,The eigenvalues of the Hamiltonian matrix are the square ro ots of those of the Hessenberg matrix. Symplectic scaling and norm scaling are provided, which, in some cases, improve the accuracy of the computed ei genvalues. We demonstrate the performance of the subroutines for several ex amples and show how they can be used to solve some control-theoretic proble ms.