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

The solution to a general Sylvester equation AX−XB=GF∗AX−XB=GF∗ 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 XX are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of AA and BB and the right-hand side factors GG and FF. Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones.