Alternating direction implicit iteration solution of Lyapunov equations

In this thesis, a new procedure is presented for solving the Lyapunov equations. First, the system is reduced to tridiagonal form with Gaussian similarity transformations, then the resulting system is solved with Alternating-Direction-Implicit (ADI) iteration. A matrix commutation property essential for "model problem" convergence of ADI iteration applied to elliptic difference equations is not needed for this application. All stable Lyapunov matrix equations are model ADI problems.