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?

reallyneed the generic case, or can you get away with a simpler structure? As noted by Suvrit, the cases $AX+XD=B+XCX$ and $AX^2+BX+C=0$ are well-studied; if some of your matrices are invertible you can reduce some more cases to this form. The more general problem "here's a bunch of quadratic equations, give me a solution" is known to be NP-hard on a finite field, so you may have little luck in the generic case. $\endgroup$ – Federico Poloni Nov 18 '11 at 12:42