Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In the present paper we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a lowrank perturbation of a standard Lyapunov operator; it is related to the Sherman-Morrison-Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an ADI-iteration as preconditioner, and Krylov subspace methods. Copyright c © 2000 John Wiley & Sons, Ltd.