Journal of Applied Mathematics
Volume 2003 (2003), Issue 6, Pages 277-303

Direct methods for matrix Sylvester and Lyapunov equations

Danny C. Sorensen1 and Yunkai Zhou1,2

1Department of Computational and Applied Mathematics, Rice University, Houston 77005-1892, TX, USA
2Argonne National Laboratory, Math and Computer Science Division, Argonne 60439, IL, USA

Received 12 December 2002; Revised 31 January 2003

Copyright © 2003 Danny C. Sorensen and Yunkai Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for A1X+XA2+D=0 and Hammarling's method for AX+XAT+BBT=0 with A stable. We construct three schemes for solving the unitarily reduced quasitriangular systems. We also construct a new rank-1 updating scheme in Hammarling's method. This new scheme is able to accommodate a B with more columns than rows as well as the usual case of a B with more rows than columns, while Hammarling's original scheme needs to separate these two cases. We compared all of our schemes with the Matlab Sylvester and Lyapunov solver lyap.m; the results show that our schemes are much more efficient. We also compare our schemes with the Lyapunov solver sllyap in the currently possibly the most efficient control library package SLICOT; numerical results show our scheme to be competitive.