Nonlinear matrix equations $X^{p} =A+\sum \limits _{i = 1}^{m} {M_{i}^{*}(B+X^{-1})^{-1}{M_{i}}} $ have extensive applications in control theory, ladder networks, dynamic programming, and stochastic filtering. In this paper a fixed-point acceleration iteration algorithm is proposed and, based on the basic characteristics of the Thompson metric space, the convergence and error estimator of the proposed algorithm are proved. Numerical comparison experiments show that the proposed algorithm is feasible and effective.
