ADI preconditioned Krylov methods for large Lyapunov matrix equations

In the present paper, we propose preconditioned Krylov methods for solving large Lyapunov matrix equations AX+XAT+BBT=0. Such problems appear in control theory, model reduction, circuit simulation and others. Using the Alternating Direction Implicit (ADI) iteration method, we transform the original Lyapunov equation to an equivalent symmetric Stein equation depending on some ADI parameters. We then define the Smith and the low rank ADI preconditioners. To solve the obtained Stein matrix equation, we apply the global Arnoldi method and get low rank approximate solutions. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches.