<s>
Unlike	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
,	O
this	O
algorithm	O
does	O
not	O
require	O
the	O
matrix	B-Architecture
to	O
be	O
self-adjoint	B-Algorithm
,	O
but	O
instead	O
one	O
needs	O
to	O
perform	O
multiplications	O
by	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
The	O
biconjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
is	O
numerically	B-Algorithm
unstable	I-Algorithm
(	O
compare	O
to	O
the	O
biconjugate	B-Algorithm
gradient	I-Algorithm
stabilized	I-Algorithm
method	I-Algorithm
)	O
,	O
but	O
very	O
important	O
from	O
a	O
theoretical	O
point	O
of	O
view	O
.	O
</s>
<s>
are	O
then	O
orthogonal	O
to	O
the	O
residuals	B-Algorithm
:	O
</s>
<s>
If	O
is	O
self-adjoint	B-Algorithm
,	O
and	O
,	O
then	O
,	O
,	O
and	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
produces	O
the	O
same	O
sequence	O
at	O
half	O
the	O
computational	O
cost	O
.	O
</s>
<s>
The	O
sequences	O
produced	O
by	O
the	O
algorithm	O
are	O
biorthogonal	B-Algorithm
,	O
i.e.	O
,	O
for	O
.	O
</s>
<s>
The	O
algorithm	O
thus	O
produces	O
projections	B-Algorithm
onto	O
the	O
Krylov	O
subspace	O
.	O
</s>
