On the matrix sign function method for the computation of invariant subspaces

There is some concern about the numerical stability of algorithms that use the matrix sign function to solve Riccati and Lyapunov equations and to find bases of invariant subspaces. In this paper, we demonstrate that evaluating the matrix sign function is a more ill-conditioned computational problem than the problem of finding bases of the two invariant subspaces. Nevertheless, we also give perturbation and error analyses, which show that the accuracy of the Newton iteration with the scaling for the computation of the invariant subspaces in most circumstances is competitive with conventional methods.