  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.
