On a numerical solution of matrix polynomial equations

In this paper, we propose some direct and iterative algorithms for the solution of matrix polynomial equations of the form $AX+AX^2+\dots+AX^n=C$. A local convergence theorem of iterative algorithms is given, and the restrictions involved by this theorem are discussed. We give an estimation of convergence speed for these methods and make a number of useful notes for it's effective numerical implementation. Explicitly we discuss a special case, arising in problems of parameter estimation of linear dynamic stochastic systems.