Nonlinear Sylvester-Like Equation

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 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.

• Could perchance your equation be rewritten as a nonsymmetric algebraic Riccati equation $AX-XB = XCX+D$? There is a lot of theory on those. – Federico Poloni Feb 14 '18 at 13:36