On a class of nonlinear matrix equations $X\pm A^{\small H}f(X)^{-1}A=Q$
For researchers in control and engineering dealing with nonlinear matrix equations, this provides theoretical guarantees and an efficient solver, though the contribution is incremental.
The paper studies a class of nonlinear matrix equations, proving existence of a maximal positive definite solution under certain conditions and designing an accelerated iterative method with R-superlinear convergence of order r>1.
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.