Is there an efficient computational algorithm for solving the generalized Sylvester's equation:
$\displaystyle \sum_{i=1}^{n}A_{i}XB_{i}=C$
The conventional Kronecker product approach to solve this equation is to vectorize both $X$ and $C$, and solve the linear equation
$\displaystyle (\sum_{i=1}^{n} B_{i}^{T} \bigotimes A_{i})\textrm{Vec}(X)=\textrm{Vec}(C)$. But, a direct solution of this linear equation is computationally infeasible due to the large size of the Kronecker product matrix appears on the left side. So, efficient methods are available for similar problems which use Kronecker products. Two common such problems are $AX+XB=C$ and $AXB=C$ (Solve for $X$ in both cases). I could see efficient methods proposed for both of these problems, but I could not find such an efficient algorithm for the above mentioned problem which involves the sum of matrices.
It would be greatly helpful for me if someone can point out an efficient algorithm for this type of matrix equations.
Best Answer
Here is a paper that addresses the problem you are looking at http://www.sciencedirect.com/science/article/pii/S0096300308006693 It uses Global FOM/GMRES, a generalization of Arnoldi process applied to blocks of vectors using a different inner-product.
It requires A and B to be sparse, or operators for which fast products are possible.