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.

Thanks in advance!

  • 2
    $\begingroup$ please don't cross post. at the very least, you should include a link to the identical question you've posted on math.stackexchange. $\endgroup$ – GoHokies Feb 13 '18 at 17:13
  • $\begingroup$ Could perchance your equation be rewritten as a nonsymmetric algebraic Riccati equation $AX-XB = XCX+D$? There is a lot of theory on those. $\endgroup$ – Federico Poloni Feb 14 '18 at 13:36
  • $\begingroup$ Sorry, GoHokies, my mistake! $\endgroup$ – Nico F. Feb 14 '18 at 17:43
  • $\begingroup$ Federico Poloni, thanks for the comment. I've one particular case which can be written in that way. Any relevant link to recommend ? $\endgroup$ – Nico F. Feb 14 '18 at 17:44
  • $\begingroup$ @NicoF. You need to use the 'at' symbol if you want me to get notified. Basically, solutions can be computed through certain invariant subspaces, and there are several methods to get them. If you don't have special sign/symmetry properties, I suggest the Schur method in sciencedirect.com/science/article/pii/S0377042705003560. You may check also riccati.dm.unipi.it/nsare/index.html, and if you wish to see other methods also epubs.siam.org/doi/abs/10.1137/050647669 and link.springer.com/article/10.1007/s00211-005-0673-7. $\endgroup$ – Federico Poloni Feb 14 '18 at 23:08

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