  Our contribution is two-folded. First, starting from the known fact that
every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give
a complete characterization of the square roots of a real skew-Hamiltonian
matrix W. Second, we propose a structure exploiting method for computing square
roots of W. Compared to the standard real Schur method, which ignores the
structure, our method requires significantly less arithmetic.
