Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$?

Especially, 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 a algorithm can solve this kind equations by provide one solution $X$.

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. – Federico Poloni Nov 18 '11 at 12:42