step 1: I am implementing a fast solver and the idea is to solve a Sylvester equation of the form $$A_1 X+X B_1=S.$$
step 2: If the computed matrix X does not meet a tolerance criteria, I augment the matrices A_1 and B_1 and resolve the following Sylvester equation $$\begin{pmatrix}A_1 & A_2\end{pmatrix}X'+X' \begin{pmatrix}B_1 \\ B_2\end{pmatrix}= S. $$
Is there a fast way for using the previously computed matrix $X$ from step 1 without having to solve the full augmented Sylvester equation from step 2?
Thank you in advance for your help!
Update:
Lutz, thank you for your response. You are right. The first equation is as follows:
$$ (I_{r_1} - A)X + X(I_{r_2} - B) = S,$$
where $A$ is an $r_1 \times r_1$ matrix and $B$ is an $r_2 \times r_2$ matrix.
The matrix $A$ is given by $A_1 \cdot D_1 \cdot A_1^T$, and $B$ is given by $B_1 \cdot D_1 \cdot B_1^T$, and $S$ is given by $A_1^T \cdot D_2 \cdot B_1$.
Then I augment the matrix $A_1$ as $A_{11} = [A_1, A_2]$, and matrix $B_1$ as $B_{11} = [B_1, B_2]$.
Then the new Sylvester equation to solve is given by:
$$ (I_{r_1^*} - A_{\text{new}})X + X(I_{r_2^*} - B_{\text{new}}) = S_{\text{new}}, $$
where $A_{\text{new}} = A_{11} \cdot D_1 \cdot A_{11}^T$, $B_{\text{new}} = B_{11} \cdot D_1 \cdot B_{11}^T$, and $S_{\text{new}} = A_{11}^T \cdot D_2 \cdot B_{11}$.
