Implementing the Iakoubovski-Merino spectral factorization algorithm using state-space methods

Abstract

The necessity of factoring spectral matrices arises in stationary control settings. The optimality conditions for quadratic optimization of stochastic objectives are Wiener-Hopf equations that cannot be solved algebraically because of the asymmetry induced by the unforecastability of future stochastic realizations. Iakoubovski and Merino [M. Iakoubovski, O. Merino, Calculating spectral factors of low and full rank matrix-valued functions on the unit circle, in: Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 505–506] develop an algorithm to factor a Hermitian matrix f. They use a Newton iteration method that is similar in spirit to the method of Wilson [G.N. Wilson, The factorization of matricial spectral densities, SIAM Journal of Applied Mathematics 23 (4) (1972) 420–426]. Iakoubovski and Merino prove convergence however, and also provide an extended algorithm that allows for factorization when the matrix to be factored has less than full rank. This note explores their algorithm using state space methods. An algebraic version of the algorithm is stated and numerical examples are provided. Conventional factorizations that solve a Riccati equation are sensitive to slight deviations from Hermitianness of the matrix to be factored. This is because a Cholesky decomposition is required in the final stage of the factorization, and Cholesky decomposition requires the subject matrix to be Hermitian. The Iacoubovski-Merino state space method finds a factor even when there are slight deviations from Hermitianness.