Sylvester's equation: Accuracy and computational stability
type: Item
instance of: scholarly article
title: Sylvester's equation: Accuracy and computational stability
zbMATH Open document ID: 0840.65026
DOI: 10.1016/0377-0427(94)00053-4
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 1996-02-01T00:00:00Z
review text: Two problems related to the equation \(AX + XB = C\), known as the Sylvester matrix equation, are investigated. The Sylvester equation in which \(A\), \(B\) and \(C\) are \(m \times m\), \(n \times n\) and \(m \times n\) matrices, respectively, appears frequently in the theory of linear dynamic systems in relation to problems of control and stability. The first problem deals with the accuracy of the computed solution \(\widehat{X}\). The second handles the stability aspects of the solution and discusses ways for computing the backward error. A scheme is proposed using the residual \(R\) formed as \(A\widehat{X} + \widehat{X} B - C\) to ameliorate the solution of Sylvester's equation based on an iterative improvement technique similar to the famous one used in linear equations' solvers. The convergence of the proposed scheme is proved. The results apply automatically to the special case when \(A = B^T\) (Lyapunov equation).
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 65K10
zbMATH DE Number: 825278
zbMATH Keywords: matrix norms
zbMATH Keywords: conditioning
zbMATH Keywords: scaling
zbMATH Keywords: error analysis
zbMATH Keywords: Sylvester matrix equation
zbMATH Keywords: linear dynamic systems
zbMATH Keywords: control
zbMATH Keywords: stability
zbMATH Keywords: iterative improvement
zbMATH Keywords: convergence
zbMATH Keywords: Lyapunov equation
reviewed by: George S. Stavrakakis
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)00053-4
OpenAlex ID: W2025570567
cites work: Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]
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description: scientific article; zbMATH DE number 825278