Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations

Low-rank versions of the alternating direction implicit (ADI) iteration are popular and well estab- lished methods for the numerical solution of large-scale Sylvester and Lyapunov equations. Probably the biggest disadvantage of these methods is their dependence on a set of shift parameters that are crucial for fast convergence. Here we firstly review existing shift generation strategies that compute a number of shifts before the actual itera- tion. These approaches come with several disadvantages such as, e.g., expensive numerical computations and the difficulty to obtain necessary spectral information or data n eeded to initially setup their generation. Secondly, we propose two novel shift selection strategies with the motivation to resolve these issues at least partially. Both ap- proaches generate shifts automatically in the course of the ADI iterations. Extensive numerical tests show that one of these new approaches, based on a Galerkin projection onto the space spanned by the current ADI data, is superior to other approaches in the majority of cases both in terms of convergence speed and required execution time.