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 established methods for the numerical solution of large-scale Sylvester and Lyapunov equations. Probably the largest disadvantage of these methods is their dependence on a set of shift parameters that are crucial for a fast convergence. Here we compare existing shifts generation strategies that compute a number of shifts before the actual iteration. These approaches come with several disadvantages such as, e.g., expensive numerical computations and difficult to obtain necessary spectral or setup data. We propose two novel shift strategies whose motivation is to solve these issues at least partly. They 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 current ADI data, seems to be superior to other approaches in the majority of cases, both in terms of convergence speed and required execution time.