Volume 43, pp. 142-162, 2014-2015.

Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations

Peter Benner, Patrick Kürschner, and Jens Saak

Abstract

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 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 iteration. These approaches come with several disadvantages such as, e.g., expensive numerical computations and the difficulty to obtain necessary spectral information or data needed to initially setup their generation. Secondly, we propose two novel shift selection strategies with the motivation to resolve these issues at least partially. Both approaches 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.

Full Text (PDF) [337 KB]

Key words

Lyapunov equation, Sylvester equation, alternating directions implicit, shift parameters

AMS subject classifications

65F10, 65F30, 15A06

Links to the cited ETNA articles

[11]Vol. 29 (2007-2008), pp. 136-149 Peter Benner, Hermann Mena, and Jens Saak: On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations

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