<s>
The	O
backward	B-Algorithm
differentiation	I-Algorithm
formula	I-Algorithm
(	O
BDF	O
)	O
is	O
a	O
family	O
of	O
implicit	B-Algorithm
methods	I-Algorithm
for	O
the	O
numerical	B-Algorithm
integration	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
They	O
are	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
that	O
,	O
for	O
a	O
given	O
function	O
and	O
time	O
,	O
approximate	O
the	O
derivative	O
of	O
that	O
function	O
using	O
information	O
from	O
already	O
computed	O
time	O
points	O
,	O
thereby	O
increasing	O
the	O
accuracy	O
of	O
the	O
approximation	O
.	O
</s>
<s>
These	O
methods	O
are	O
especially	O
used	O
for	O
the	O
solution	O
of	O
stiff	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Since	O
is	O
evaluated	O
for	O
the	O
unknown	O
,	O
BDF	O
methods	O
are	O
implicit	B-Algorithm
and	O
possibly	O
require	O
the	O
solution	O
of	O
nonlinear	O
equations	O
at	O
each	O
step	O
.	O
</s>
<s>
The	O
stability	O
of	O
numerical	O
methods	O
for	O
solving	O
stiff	B-Algorithm
equations	I-Algorithm
is	O
indicated	O
by	O
their	O
region	O
of	O
absolute	O
stability	O
.	O
</s>
<s>
Ideally	O
,	O
the	O
region	O
contains	O
the	O
left	O
half	O
of	O
the	O
complex	O
plane	O
,	O
in	O
which	O
case	O
the	O
method	O
is	O
said	O
to	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
However	O
,	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
with	O
an	O
order	O
greater	O
than	O
2	O
cannot	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
The	O
BDF	O
methods	O
are	O
the	O
most	O
efficient	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
of	O
this	O
kind	O
.	O
</s>
