On the numerical solution of matrix polynomial equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 0973.65039
author: E. V. Dulov
author: N. A. Andrianova
publication date: 2001-07-22T00:00:00Z
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 1622452
zbMATH Keywords: matrix polynomial equation
zbMATH Keywords: parameter identification
zbMATH Keywords: direct and iterative algorithms
zbMATH Keywords: convergence
zbMATH Keywords: stochastic system
MaRDI profile type: Publication
title: On the numerical solution of matrix polynomial equations
published in: Sibirskiĭ Zhurnal Vychislitel'noĭ Matematiki
review text: 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.
reviewed by: Vladimir N. Grebenev
description: scientific article; zbMATH DE number 1622452