There has been a flurry of activity in recent years in the area of
solution of matrix equations. In particular,
a good understanding
has been reached
on how to approach the solution of large scale
Lyapunov equations;
see, e.g.,
[1]-[3].
An effective way to solve Lyapunov equations of the form
, where
and
are
,
is to use Galerkin projection with appropriate extended or rational
Krylov subspaces. These methods work in part because the solution is known
to be symmetric positive definite with rapidly decreasing singular values, and therefore
it can be approximated by
a low rank matrix
.
Thus the computations are performed usually with storage which is lower rank,
i.e., much lower than order of
.
Generalized Lyapunov equations have additional terms. In this paper, we concentrate on equations of the following form
In the present work, we propose a return to classical iterative methods, and consider
instead stationary iterations, where the iterate
is the solution of
One of the advantages of this classical approach is that only the
data and the low-rank factors of the old and new
iterates
and
need to be kept in storage. Furthermore, the
solutions of the
Lyapunov equations (
)
can be performed
with the Galerkin projection methods mentioned above, where the growth
of rank can usually be well contained.
In our work, we developed a general srategy for augmented Krylov projection methods for
sequences of Lyapunov equations with converging right-hand sides, and we
apply it to the sequence
(), whose solutions converge to the solution of
(
).
Numerical experiments show the competitiveness of the proposed approach.
This is joint work with Stephen D. Shank and Valeria Simoncini.