关于非线性矩阵方程类 $X\pm A^{\small H}f(X)^{-1}A=Q$ 的研究
On a class of nonlinear matrix equations $X\pm A^{\small H}f(X)^{-1}A=Q$
AI总结：
本文研究了一类非线性矩阵方程的性质与求解方法，利用Sherman Morrison Woodbury公式证明了方程的最大正定解的存在性，并设计了具有R超线性收敛的迭代方法以高效求解。
摘要：
非线性矩阵方程在控制和工程问题中广泛出现。本文对一类非线性矩阵方程进行了全面研究。借助Sherman Morrison Woodbury公式，我们证明了在一定条件下，此类方程具有最大正定解。此外，本文还详细研究了该类矩阵方程的性质。然后，设计了一种具有R超线性收敛性（收敛阶$r>1$）的迭代方法，以高效求解最大正定解。
英文摘要：
Nonlinear matrix equations are encountered in many applications of control and engineering problems. In this work, we establish a complete study for a class of nonlinear matrix equations. With the aid of Sherman Morrison Woodbury formula, we have shown that any equation in this class has the maximal positive definite solution under a certain condition. Furthermore, A thorough study of properties about this class of matrix equations is provided. An acceleration of iterative method with R-superlinear convergence with order $r>1$ is then designed to solve the maximal positive definite solution efficiently.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
