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

Higher order iterations for computing the matrix sign function of complex matrices are developed in this paper. The technique of generating higher order fixed point function produces the Newton and Halley methods as special cases for solving the equations S²=I, and such that SA=AS has all its eigenvalues in the right halfplane. The matrix sign function is used to compute the positive semidefinite solution of the algebraic Riccati and Lyapunov matrix equations. The performance of these methods is demonstrated by several examples