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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!

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    $\begingroup$ Two remarks: 1. The equation is known to have many solutions. Are you interested in a particular one? (Tagging control-theory, maybe in the stabilizing solution?) 2. Are you familiar with the technique used in the symmetric case (Hamiltonian matrix, stable-unstable invariant subspace decomposition etc)? The nonsymmetric case is quite similar (see, for example, here) $\endgroup$ – A.Γ. Oct 5 '15 at 19:49
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    $\begingroup$ Relevant $\endgroup$ – Lonidard Oct 5 '15 at 22:09
  • $\begingroup$ A.G: Thanks! While I expected this to have multiple solutions, I don't have a good way to determine which solution is best. I have arrived at this equation as a intermediate step in the solution of a much larger and "more nonlinear" equation (D is a complicated function of X) - such that I am going to solve the above equation to obtain a better guess for the next iteration of my larger equation. I appreciate your help, and it has given me a starting point. $\endgroup$ – Nick Oct 6 '15 at 15:05
  • $\begingroup$ bharb: Thank you for the link! Looks very relevant and helpful! $\endgroup$ – Nick Oct 6 '15 at 15:06
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This is known as a nonsymmetric algebraic ricatti equation (NARE). You may google this term to find out the latest development.

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    $\begingroup$ [Here][1] you will find a paper with numerical methods to solve the NARE. [1]: drops.dagstuhl.de/volltexte/2008/1395/pdf/… $\endgroup$ – WG- Oct 6 '15 at 6:54
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    $\begingroup$ I didn't notice that A.G. had provided a (very subtle) link in his/her comment too. The book chapter he/she linked to seems very useful. $\endgroup$ – user1551 Oct 6 '15 at 7:33

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