A method for solving the algebraic Riccati and Lyapunov equations using higher order matrix sign function algorithms

A set of higher order rational fixed point functions for computing the matrix sign function of complex matrices is developed. Our main focus is the representation of these rational functions in partial fraction form which in turn allows for a parallel implementation of the matrix sign function algorithms. The matrix sign function is then used to compute the positive semidefinite solution of the algebraic Riccati and Lyapunov matrix equations. It is also suggested that the proposed methods can be used to compute the invariant subspaces of a nonsingular matrix in any half plane. The performance of these methods is demonstrated by several examples.