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Perturbation analysis of the matrix equation X=Q+A H (X ^-C) -1 A. (English) Zbl 1032.15009

The maximal solution (unique Hermitian solution) X to the nonlinear matrix equation

X=Q+A H (X ^-C) -1 A

is discussed. Here in this equation, Q stands for an n×n Hermitian matrix, A H =A -T , A>0 (A0) with A being an mn×n Hermitian positive (positive semidefinite) matrix, C is an mn×mn Hermitian positive semidefinite matrix and X ^ is the m×m block diagonal matrix defined by X ^=diag(X,X,,X), where X is an n×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.

15A24Matrix equations and identities
65F30Other matrix algorithms