On the numerical solution of matrix polynomial equations (Q2729357)

The authors present certain direct and iterative methods for solving matrix polynomial equations (appearing in the problem of parameter identification for a stationary linear stochastic system) of the form NEWLINE\[NEWLINE AX + AX^2 + \dots + AX^n = C. NEWLINE\]NEWLINE Here \(X\in\mathbb R^{n\times n}\) is the matrix which defines the dynamics of a stochastic system, \(A\in\mathbb R^{m\times n}\) is the matrix of observations of this system, and \(C = \sum_{i=1}^n C_i\), where \(C_i\in\mathbb R^{m\times n}\).NEWLINENEWLINENEWLINEThe authors prove that, for the problem under consideration, there exists an equivalent representation which admits an effective and numerically stable solution under different ways of describing the original problem. A local convergence theorem of iterative algorithms is given, and the restrictions involved by this theorem are discussed. In the case of the problem of the so-called total rank, the authors expose an algorithm for constructing the set of all solutions to the equation. Estimations of the convergence rate of the iterative methods are established and, moreover, certain recommendations for their numerical realizations are presented.