On a class of nonlinear matrix equations $X\pm A^{\small H}f(X)^{-1}A=Q$

TL;DR

This paper thoroughly studies a class of nonlinear matrix equations relevant to control and engineering, establishing conditions for maximal positive definite solutions and proposing an accelerated iterative method with superlinear convergence.

Contribution

It introduces a complete analysis of the equations, proves existence of maximal solutions, and develops an efficient accelerated iterative algorithm.

Findings

Existence of maximal positive definite solutions under certain conditions

Development of an R-superlinear convergent iterative method

Enhanced computational efficiency for solving the equations

Abstract

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.