Iterative solution to a system of matrix equations
type: Item
instance of: scholarly article
title: Iterative solution to a system of matrix equations
zbMATH Open document ID: 1297.65046
DOI: 10.1155/2013/124979
author: Yong Lin
author: Qing-Wen Wang
published in: Abstract and Applied Analysis
publication date: 2014-06-20T00:00:00Z
review text: Summary: An efficient iterative algorithm is presented to solve a system of linear matrix equations \(A_1X_1B_1+A_2X_2B_2=E\), \(C_1X_1D_1+C_2X_2D_2=F\) with real matrices \(X_1\) and \(X_2\). By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices \(X^0_1\) and \(X^0_2\), a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions \(\hat{X}_1\) and \(\hat{X}_2\) to the given matrices \(\tilde{X}_1\) and \(\tilde{X}_2\) in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations \(A_1\bar{X}_1B_1+A_2\bar{X}_2B_2=\bar{E}\), \(C_1\bar{X}_1D_1+C_2\bar{X}_2D_2=\bar{F}\), where \(\bar{E}=E-A_1\tilde{X}_1B_1-A_2\tilde{X}_2B_2\), \(\bar{F}=F-C_1\tilde{X}_1D_1-C_2\tilde{X}_2D_2\). The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices \(A_1,A_2,B_1,B_2,C_1,C_2,D_1,D_2\) are large, our algorithm is efficient as well.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 6306035
zbMATH Keywords: iterative algorithm
zbMATH Keywords: system of linear matrix equations
zbMATH Keywords: least norm solution
zbMATH Keywords: numerical example
Wikidata QID: Q58915305
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/124979
OpenAlex ID: W1992355342
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description: scientific article; zbMATH DE number 6306035