Spectral factorization via Lyapunov equation based Newton-Raphson iteration

A fast, effective method is proposed for computing canonical factorizations of real polynomial matrices that are para-Hermitian and positive definite or nonnegative definite on the imaginary axis. Key to this technique is the extraction of the leading coefficients right at the initialization, while the remaining coefficients are subject to iteration. In each step, an approximate factor is decoupled from its conjugate when a minimal state space description is reduced to block diagonal form using a Lyapunov equation the size of the spectral factor determinantal degree. The tools are Cholesky factorization, real Schur decomposition, and backward substitution of triangular systems, together with connections between realizations in state space and polynomial matrix fractions. The convergence is quadratic and deteriorates if the given matrix has zeros on or near the imaginary axis.