Perturbation analysis of the matrix equation \(X=Q+A^{\text H}(\widehat X-C)^{-1}A\) (Q1410709)

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.