TITLE: Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
QUESTION [5 upvotes]: Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix satisfying
$$X=A+BXC+DXEXF$$
where $A,B,C,D,E,F$ are given matrices.
Of course, the equation could be in higher degree, such as
$$X=X^n+X^{n-1}+A$$
Is there an algorithm that can solve this kind of matrix equations?

REPLY [4 votes]: You could solve your equations by suitably extending the methods for solving Nonsymmetric Riccati equations, see e.g.,


Nonsymmetric algebraic Riccati equations and Wiener-Hopf Factorization for M-Matrices by C.-H. Guo