耦合非线性矩阵方程组$X_i+{\displaystyle \sum _{j=1}^3{A}_{ij}^{\ast }}{X}_j^{-1}{A}_{ij}=I\left(i=1,2,3\right)$的正定解

On the Positive Definite Solutions of a System of Coupled Nonlinear Matrix Equations $X_i+{\displaystyle \sum _{j=1}^3{A}_{ij}^{\ast }}{X}_j^{-1}{A}_{ij}=I\left(i=1,2,3\right)$

摘要: 本文研究非线性矩阵方程组$X_i+{\displaystyle \sum _{j=1}^3{A}_{ij}^{\ast }}{X}_j^{-1}{A}_{ij}=I$ $\left(i=1,2,3\right)$ 的数值求解问题，其中$X_i$ 是待求解的矩阵，$A_{ij}$ $\left(i=1,2,3,j=1,2,3\right)$ 为任意$n\times n$ 阶矩阵，$I$ 表示$n$ 阶单位矩阵。本文提出一种免逆迭代算法，进行了收敛性和稳定性分析，数值实验表明该方法在迭代次数和计算时间上优于不动点算法。

Abstract: This paper investigates the numerical solution of the nonlinear matrix system$X_i+{\displaystyle \sum _{j=1}^3{A}_{ij}^{\ast }}{X}_j^{-1}{A}_{ij}$ $=I\left(i=1,2,3\right)$ . where $X_i$ denotes the matrix to be solved，$A_{ij}$ $\left(i=1,2,3,j=1,2,3\right)$ are arbitrary $n\times n$ matrices, and $I$ represents the $n$ -order identity matrix. An inversion-free iterative algorithm is proposed in this study, followed by the analysis of its convergence and stability. Numerical experiments demonstrate that the proposed method outperforms the fixed-point algorithm in terms of the number of iterations and computational time.