This paper is concerned with the numerical solution of large scale Sylvester equations A X - X B = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000)  and Li and White (2002)  demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz's shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions. © 2009 Elsevier B.V.
Benner, P., Li, R. C., & Truhar, N. (2009). On the ADI method for Sylvester equations. Journal of Computational and Applied Mathematics, 233(4), 1035–1045. https://doi.org/10.1016/j.cam.2009.08.108