How would I solve the following equation, which is similar to an algebraic Riccati equation or a nonlinear sylvester equation?

Question

I have the following matrix equation that I would like to solve for $X$:

$0 = AX + XB + XCX + D$

In general, $X$ will be rectangular, with $(m\times n)$ dimensions. So if I write the equation out with indices, it is:

$0 = A_{mm}X_{mn} + X_{mn}B_{nn} + X_{mn}C_{nm}X_{mn} + D_{mn}$

I assume everything to be real, and $m,n$ are dimensions small enough that a diagonalization of an $m\times n$ matrix is computationally feasible.

I see that if $C=0$, then it is just a linear Sylvester equation, and if $A=B$ then it seems to be an Algebraic Riccati equation, but neither of these assertions can be made.

I appreciate any guidance towards a solution. Perhaps this is an equation with well known properties (I'm not a mathematician)?

Thanks in advance!

Answer 1

This is known as a nonsymmetric algebraic ricatti equation (NARE). You may google this term to find out the latest development.