On the ADI method for Sylvester equations

This paper is concerned with the numerical solution of large scale Sylvester equations AX −XB = 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. We also demonstrate that often much more accurate solutions than ADI solutions can be obtained by performing Galerkin projection via the column space and row space of the computed approximate solutions.