On the residual of large-scale Lyapunov equations for Krylov-based approximate solutions

In this paper a new formulation of the residual of large-scale Lyapunov equations is presented, which results from the approximate solution using projections by Krylov subspaces. The formulation is based on low-rank factors, which allows an efficient numerical treatment of the residual even for large-scale Lyapunov equations. It is shown, how the matrix 2-norm can be computed with low numerical effort. The results are presented for the most general case, which means that generalized Lyapunov equations are considered and that oblique projections are utilized for approximately solving the Lyapunov equation. With this regard, this paper also presents generalizations to Krylov-based methods that are available in the literature. Furthermore, based on the new results, the suitability of the residual as a stopping criterion in iterative methods is discussed and an upper bound on the approximation is reviewed. Numerical examples illustrate the contributions.