Analysis of the solution of the Sylvester equation using low-rank ADI with exact shifts

The solution to a general Sylvester equation A X − X B = G F ∗ with a low-rank right-hand side is analyzed quantitatively through the Low-rank Alternating-Directional-Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right-hand side factors G and F . Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones.