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!

  • 2
    $\begingroup$ 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
  • 3
    $\begingroup$ 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

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

  • 2
    $\begingroup$ [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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.