Perturbation analysis of the matrix equation \(X=Q+A^{\text H}(\widehat X-C)^{-1}A\)
type: Item
instance of: scholarly article
title: Perturbation analysis of the matrix equation \(X=Q+A^{\text H}(\widehat X-C)^{-1}A\)
zbMATH Open document ID: 1032.15009
DOI: 10.1016/S0024-3795(03)00491-9
author: Ji-guang Sun
published in: Linear Algebra and its Applications
publication date: 2003-10-15T00:00:00Z
review text: The maximal solution (unique Hermitian solution) \(X\) to the nonlinear matrix equation \[ X=Q+A^H(\widehat X-C)^{-1}A \] is discussed. Here in this equation, \(Q\) stands for an \(n\times n\) Hermitian matrix, \(A^H=A^{-T}\), \(A>0\) (\(A\geq 0\)) with \(A\) being an \(mn\times n\) Hermitian positive (positive semidefinite) matrix, \(C\) is an \(mn\times mn\) Hermitian positive semidefinite matrix and \(\widehat X\) is the \(m\times m\) block diagonal matrix defined by \(\widehat X =\text{diag}(X,X,\dots,X)\), where \(X\) is an \(n\times n\) matrix. This matrix equation has significance in interpolation theory. Some perturbation results (perturbation bounds and condition numbers) for the maximal solution \(X\) to this matrix equation are presented and residual bounds for an approximate solution to the maximal solution are given. Two numerical examples with \(n=2\), \(m=3\) are given as well-conditioned and moderately ill-conditioned numerical cases to support the main presented results.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 1993117
zbMATH Keywords: nonlinear matrix equation
zbMATH Keywords: perturbation bound
zbMATH Keywords: condition number
zbMATH Keywords: residual bound
zbMATH Keywords: maximal solution
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
cites work: Q3702392
cites work: On the Singular “Vectors” of the Lyapunov Operator
cites work: Q3961697
cites work: Q5652137
cites work: The symmetric linear matrix equation
cites work: A Theory of Condition
cites work: Q4369403
cites work: Perturbation Theory for Algebraic Riccati Equations
cites work: Condition numbers of algebraic Riccati equations in the Frobenius norm
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description: scientific article; zbMATH DE number 1993117