Perturbation analysis for solutions of \(X \pm A^{\ast}X^{-n} A = Q\)
The paper is devoted to the properties of positive definite solutions of the nonlinear matrix equations: \[ X + A^*X^{(-n)}A = Q \] and \[ X - A^*X^{(-n)}A = Q, \] where \(A, Q\) are complex \(m \times m\) matrices, \(Q\) is a positive definite matrix, and \(n\) is a natural number. New, more effective perturbation estimates for a positive definite solution are proposed. The topic has been previously introduced by the author and \textit{V. I. Hasanov} in [Appl. Math. Comput. 156, 513--525 (2004; Zbl 1063.15012)], and in [Mathematics and Education in Mathematics, Proceedings 32nd Spring Conference of the Union of Bulgarian Mathematicians, 258--262 (2003)]. The new bounds are compared with earlier results, and are summarized in a table. Some advantages are pointed out. For the numerical results the computer system MATLAB has been used.
scientific article; zbMATH DE number 2137804