On the ADI method for Sylvester equations

Authors:
Peter Benner;Ren-Cang Li;Ninoslav Truhar
Affiliations:
Mathematics in Industry and Technology, Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany;Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX 76019-0408, USA;Department of Mathematics, J.J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

This paper is concerned with the numerical solution of large scale Sylvester equations AX-XB=C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) [22] and Li and White (2002) [20] demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz's shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.