<s>
In	O
numerical	O
linear	O
algebra	O
,	O
the	O
Bartels	B-Algorithm
–	I-Algorithm
Stewart	I-Algorithm
algorithm	I-Algorithm
is	O
used	O
to	O
numerically	O
solve	O
the	O
Sylvester	B-Algorithm
matrix	I-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Stewart	O
in	O
1971	O
,	O
it	O
was	O
the	O
first	O
numerically	B-Algorithm
stable	I-Algorithm
method	O
that	O
could	O
be	O
systematically	O
applied	O
to	O
solve	O
such	O
equations	O
.	O
</s>
<s>
It	O
remains	O
a	O
standard	O
approach	O
for	O
solving	O
Sylvester	B-Algorithm
equations	I-Algorithm
when	O
is	O
of	O
small	O
to	O
moderate	O
size	O
.	O
</s>
<s>
The	O
Bartels	B-Algorithm
–	I-Algorithm
Stewart	I-Algorithm
algorithm	I-Algorithm
computes	O
by	O
applying	O
the	O
following	O
steps	O
:	O
</s>
<s>
The	O
Hessenberg	O
–	O
Schur	O
algorithm	O
replaces	O
the	O
decomposition	O
in	O
step	O
1	O
with	O
the	O
decomposition	O
,	O
where	O
is	O
an	O
upper-Hessenberg	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
advantage	O
of	O
this	O
approach	O
is	O
that	O
can	O
be	O
found	O
using	O
Householder	B-Algorithm
reflections	I-Algorithm
at	O
a	O
cost	O
of	O
flops	O
,	O
compared	O
to	O
the	O
flops	O
required	O
to	O
compute	O
the	O
real	O
Schur	O
decomposition	O
of	O
.	O
</s>
<s>
For	O
large	O
systems	O
,	O
the	O
cost	O
of	O
the	O
Bartels	B-Algorithm
–	I-Algorithm
Stewart	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
prohibitive	O
.	O
</s>
<s>
When	O
and	O
are	O
sparse	O
or	O
structured	O
,	O
so	O
that	O
linear	O
solves	O
and	O
matrix	O
vector	O
multiplies	O
involving	O
them	O
are	O
efficient	O
,	O
iterative	B-Algorithm
algorithms	I-Algorithm
can	O
potentially	O
perform	O
better	O
.	O
</s>
<s>
These	O
include	O
projection-based	O
methods	O
,	O
which	O
use	O
Krylov	B-Algorithm
subspace	I-Algorithm
iterations	O
,	O
methods	O
based	O
on	O
the	O
alternating	B-Algorithm
direction	I-Algorithm
implicit	I-Algorithm
(	O
ADI	O
)	O
iteration	O
,	O
and	O
hybridizations	O
that	O
involve	O
both	O
projection	O
and	O
ADI	O
.	O
</s>
<s>
Iterative	B-Algorithm
methods	I-Algorithm
can	O
also	O
be	O
used	O
to	O
directly	O
construct	O
low	O
rank	O
approximations	O
to	O
when	O
solving	O
.	O
</s>
