Iterative algorithm for solving a system of nonlinear matrix equations. (English) Zbl 1268.65058

Summary: We discuss the positive definite solutions for the system of nonlinear matrix equations \(X - A^\ast Y^{-n} A = I\) and \(Y - B^\ast X^{\-m} B = I\), where \(n, m\) are two positive integers. Some properties of solutions are studied, and the necessary and sufficient conditions for the existence of positive definite solutions are given. An iterative algorithm for obtaining positive definite solutions of the system is proposed. Moreover, the error estimations are found. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.


65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
65H10 Numerical computation of solutions to systems of equations
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