The nonlinear matrix equation \(X^p=R+M^T(X^{-1}+B)^{-1}M\), where p is a positive integer, M is an arbitrary \(n\times n\) real matrix, R and B are symmetric positive semidefinite matrices, is considered. When \(p=1\), this matrix equation is the well-known discrete-time algebraic Riccati equation (DARE), we study the convergence rate of an iterative method which was proposed in Meng and Kim (J Comput Appl Math 322:139–147, 2017). For the generalized case \(p\ge 1\), a structured condition number based on the classic definition of condition number is defined and its explicit expression is obtained. Finally, we give some numerical examples to show the sharpness of the structured condition number.

中文翻译：

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非线性矩阵方程的进一步研究

非线性矩阵方程\（X ^ p = R + M ^ T（X ^ {-1} + B）^ {-1} M \），其中p是正整数，M是任意\（n \ times考虑实数矩阵R和B是对称正半定矩阵。当\（p = 1 \）时，此矩阵方程是众所周知的离散时间代数Riccati方程（DARE），我们研究了Meng和Kim（J Comput Appl Math 322： 139–147，2017年）。对于一般情况\（p \ ge 1 \），根据条件数的经典定义，定义了结构化条件数，并获得了其显式表达式。最后，我们给出一些数值例子来说明结构化条件数的清晰度。