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 $S_{\text{new}} = A_{11}^T \cdot D_2 \cdot B_{11}$.
Best Answer
I am afraid that your update is at odds/incompatible with many (all?) known techniques for solving Sylvester matrix equations.
Your update procedure is a fairly general update of the coefficient matrices that define the Sylvester equation. Without any restrictions on the extensions of $A_1$, $B_1$ there is no guarantee that the new equation is even uniquely solvable. The update is incompatible with Schur form of $A$ and $B$, which eliminates the possibility of recycling work done for the Bartel-Stewart method. The update can change the action of the coefficient matrices dramatically and this is likely to precludes the use of the Arnoldi methods by Jamoukha and Kasenally or the extended Krylov subspace method by Simoncini. There is chance that low-rank ADI method by Penzl can use the solution of the old equation as an initial guess for the solution to the new equation, but it is not at all certain that the old shift parameters will remain good. This will hinge on the change in the spectrum of $A$ and $B$. Nevertheless, this is probably the best place to start looking.
References:
Obviously, this does not mean that what you seek is impossible, but if you find a solution, then it will be worthy of publication.