Symplectic balancing of Hamiltonian matrices

We discuss the balancing of Hamiltonian matrices by structure preserving si milarity transformations. The method is closely related to balancing nonsym metric matrices for eigenvalue computations as proposed by Osborne [J. ACM, 7 (1960), pp. 338-345] and Parlett and Reinsch [Numer. Math., 13 (1969), p p. 296-304] and implemented in most linear algebra software packages. It is shown that isolated eigenvalues can be deflated using similarity transform ations with symplectic permutation matrices. Balancing is then based on equ ilibrating row and column norms of the Hamiltonian matrix using symplectic scaling matrices. Due to the given structure, it is sufficient to deal with the leading half rows and columns of the matrix. Numerical examples show t hat the method improves eigenvalue calculations of Hamiltonian matrices as well as numerical methods for solving continuous-time algebraic Riccati equ ations.