Maybe you can point me to some results already developed for this.
I have to solve for $X$ the following "Sylvester-like" equation:
$$ AX - XB = F(X)$$
where $A\in\mathbb{R}^{a\times n}$, $B\in\mathbb{R}^{p\times b}$, $X\in\mathbb{R}^{n\times p}$ and $F(X)$ is a nonlinear function of the elements of X.
I know I can solve this using using generic numerical algorithms to solve for the roots of $ AX - XB - F(X) = 0$. Nevertheless, I would like to know if there is any algorithm or procedure that actually exploits the "Sylvester-like" structure of the equation.
Thanks in advance!
I have studied the equation $AX-XA=F(X)$, for $F(X)=X^p$, see here. In this case methods from the algebraic Riccati equation, Jordan chains and other methods could be applied. It is possible, that one can do this also for your more general equation.