On equations that are equivalent to the nonlinear matrix equation \(X+A^{*}X ^{-\alpha}A=Q\)
The authors consider the following nonlinear matrix equations \[ X^{-1}+A^*X^{\alpha}A=Q\quad (0<\alpha\leq 1)\tag{1} \] and \[ X^{-1}+(AXA^*)^{1/\alpha}=Q\quad (1<\alpha),\tag{2} \] where \(A\) is a nonsingular matrix, \(A^*\) is the conjugate transpose of \(A\), and \(Q\) is an Hermitian positive definite matrix. Necessary and sufficient conditions for the existence of a positive definite solution of Equations (1) and (2) are presented, respectively. In the process of proving the main results, the authors obtain two iterative algorithms and derive error estimates. Finally, three numerical examples are given to demonstrate that these algorithms are applicable.
scientific article; zbMATH DE number 5724563
scientific article