A new backward stable, structure preserving method of complexity O(n3) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix ℋ and the extended matrix (figure presented) and makes use of the symplectic URV-like decomposition that was recently introduced by the authors.
Benner, P., Mehrmann, V., & Xu, H. (1997). A new method for computing the stable invariant subspace of a real Hamiltonian matrix. Journal of Computational and Applied Mathematics, 86(1), 17–43. https://doi.org/10.1016/S0377-0427(97)00146-5