<s>
In	O
numerical	O
linear	O
algebra	O
,	O
the	O
biconjugate	B-Algorithm
gradient	I-Algorithm
stabilized	I-Algorithm
method	I-Algorithm
,	O
often	O
abbreviated	O
as	O
BiCGSTAB	B-Algorithm
,	O
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
developed	O
by	O
H	O
.	O
A	O
.	O
van	O
der	O
Vorst	O
for	O
the	O
numerical	O
solution	O
of	O
nonsymmetric	O
linear	O
systems	O
.	O
</s>
<s>
It	O
is	O
a	O
variant	O
of	O
the	O
biconjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
(	O
BiCG	B-Algorithm
)	O
and	O
has	O
faster	O
and	O
smoother	O
convergence	O
than	O
the	O
original	O
BiCG	B-Algorithm
as	O
well	O
as	O
other	O
variants	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
squared	I-Algorithm
method	O
(	O
CGS	O
)	O
.	O
</s>
<s>
It	O
is	O
a	O
Krylov	B-Algorithm
subspace	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Unlike	O
the	O
original	O
BiCG	B-Algorithm
method	O
,	O
it	O
does	O
n't	O
require	O
multiplication	O
by	O
the	O
transpose	O
of	O
the	O
system	O
matrix	O
.	O
</s>
<s>
To	O
solve	O
a	O
linear	O
system	O
,	O
BiCGSTAB	B-Algorithm
starts	O
with	O
an	O
initial	O
guess	O
and	O
proceeds	O
as	O
follows	O
:	O
</s>
<s>
Preconditioners	O
are	O
usually	O
used	O
to	O
accelerate	O
convergence	O
of	O
iterative	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
To	O
solve	O
a	O
linear	O
system	O
with	O
a	O
preconditioner	O
,	O
preconditioned	O
BiCGSTAB	B-Algorithm
starts	O
with	O
an	O
initial	O
guess	O
and	O
proceeds	O
as	O
follows	O
:	O
</s>
<s>
In	O
BiCG	B-Algorithm
,	O
the	O
search	O
directions	O
and	O
and	O
the	O
residuals	O
and	O
are	O
updated	O
using	O
the	O
following	O
recurrence	O
relations	O
:	O
</s>
<s>
It	O
is	O
unnecessary	O
to	O
explicitly	O
keep	O
track	O
of	O
the	O
residuals	O
and	O
search	O
directions	O
of	O
BiCG	B-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
BiCG	B-Algorithm
iterations	O
can	O
be	O
performed	O
implicitly	O
.	O
</s>
<s>
This	O
can	O
also	O
be	O
derived	O
from	O
the	O
BiCG	B-Algorithm
relations	O
:	O
</s>
<s>
Now	O
it	O
remains	O
to	O
determine	O
the	O
BiCG	B-Algorithm
constants	O
and	O
and	O
choose	O
a	O
suitable	O
.	O
</s>
<s>
Since	O
BiCGSTAB	B-Algorithm
does	O
not	O
explicitly	O
keep	O
track	O
of	O
or	O
,	O
is	O
not	O
immediately	O
computable	O
from	O
this	O
formula	O
.	O
</s>
<s>
In	O
BiCG	B-Algorithm
,	O
</s>
<s>
Finally	O
,	O
BiCGSTAB	B-Algorithm
selects	O
to	O
minimize	O
in	O
-norm	O
as	O
a	O
function	O
of	O
.	O
</s>
<s>
BiCGSTAB	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
combination	O
of	O
BiCG	B-Algorithm
and	O
GMRES	O
where	O
each	O
BiCG	B-Algorithm
step	O
is	O
followed	O
by	O
a	O
GMRES( )	O
(	O
i.e.	O
,	O
GMRES	O
restarted	O
at	O
each	O
step	O
)	O
step	O
to	O
repair	O
the	O
irregular	O
convergence	O
behavior	O
of	O
CGS	O
,	O
as	O
an	O
improvement	O
of	O
which	O
BiCGSTAB	B-Algorithm
was	O
developed	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
BiCGSTAB	B-Algorithm
is	O
likely	O
to	O
stagnate	O
,	O
as	O
confirmed	O
by	O
numerical	O
experiments	O
.	O
</s>
<s>
This	O
gives	O
rise	O
to	O
algorithms	O
including	O
BiCGSTAB2	O
and	O
the	O
more	O
general	O
BiCGSTAB( )	O
.	O
</s>
<s>
In	O
BiCGSTAB( )	O
,	O
a	O
GMRES( )	O
step	O
follows	O
every	O
BiCG	B-Algorithm
steps	O
.	O
</s>
<s>
BiCGSTAB2	O
is	O
equivalent	O
to	O
BiCGSTAB( )	O
with	O
.	O
</s>
