Iterative solution of two matrix equations. (English) Zbl 0940.65036

Iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations \(X+ A^*X^{-1}A= Q\) and \(X- A^*A^{-1}A= Q\) are studied. Here \(Q\) is Hermitian positive definite and the ordering \(X\leq Y\) if \(X- Y\) is positive semidefinite. The convergence rates of various algorithms depend on the eigenvalues of \((X_+)^{-1}A\), where \(X_+\) is a solution. These eigenvalues are related to the eigenvalues of a matrix pencil which is independent of \(X_+\).


65F10 Iterative numerical methods for linear systems
65F30 Other matrix algorithms (MSC2010)
15A22 Matrix pencils
15A24 Matrix equations and identities
Full Text: DOI


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