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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 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!

controllystuff
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I have studied the equation $AX-XA=F(X)$, for $F(X)=X^p$, see here. In this case methods from the algebraic Riccati equation, Jordan chains and other methods could be applied. It is possible, that one can do this also for your more general equation.

Dietrich Burde
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  • Thanks for this. At least it provides a starting point!!! – controllystuff Feb 13 '18 at 15:57
  • You are welcome. Already the case $F(X)=X^p$ is non-trivial, so it might be difficult to solve this equation for general non-linear $F(X)$. – Dietrich Burde Feb 13 '18 at 17:20
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    I'm afraid so... Actually, I'm not expecting to find an analytical solution in this case, but to find an efficient algorithm to solve it numerically. That equation is involved in a real-time calculation, and using regular solvers based on, for example, Newton's method, does not bring satisfactory results in terms of computational cost. That's why I was wondering if there is something out there that actually exploits this "Sylvester-like" form. Thanks anyway! I'm on your article right now and it is extremely interesting! – controllystuff Feb 13 '18 at 17:24