Journal of Applied Mathematics
Volume 2003 (2003), Issue 6, Pages 277-303
Direct methods for matrix Sylvester and Lyapunov equations
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
and Hammarling's method for
with 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 with more columns than rows as well as the
usual case of a 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.