<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
scientific	O
computing	O
,	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
(	O
or	O
implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
)	O
is	O
one	O
of	O
the	O
most	O
basic	O
numerical	B-Algorithm
methods	I-Algorithm
for	I-Algorithm
the	I-Algorithm
solution	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
It	O
is	O
similar	O
to	O
the	O
(	O
standard	O
)	O
Euler	B-Algorithm
method	I-Algorithm
,	O
but	O
differs	O
in	O
that	O
it	O
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
has	O
error	O
of	O
order	O
one	O
in	O
time	O
.	O
</s>
<s>
This	O
differs	O
from	O
the	O
(	O
forward	O
)	O
Euler	B-Algorithm
method	I-Algorithm
in	O
that	O
the	O
forward	O
method	O
uses	O
in	O
place	O
of	O
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
:	O
the	O
new	O
approximation	O
appears	O
on	O
both	O
sides	O
of	O
the	O
equation	O
,	O
and	O
thus	O
the	O
method	O
needs	O
to	O
solve	O
an	O
algebraic	O
equation	O
for	O
the	O
unknown	O
.	O
</s>
<s>
Finally	O
,	O
use	O
that	O
is	O
supposed	O
to	O
approximate	O
and	O
the	O
formula	O
for	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
follows	O
.	O
</s>
<s>
The	O
same	O
reasoning	O
leads	O
to	O
the	O
(	O
standard	O
)	O
Euler	B-Algorithm
method	I-Algorithm
if	O
the	O
left-hand	O
rectangle	O
rule	O
is	O
used	O
instead	O
of	O
the	O
right-hand	O
one	O
.	O
</s>
<s>
The	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
(	O
defined	O
as	O
the	O
error	O
made	O
in	O
one	O
step	O
)	O
of	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
Method	I-Algorithm
is	O
,	O
using	O
the	O
big	O
O	O
notation	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
method	O
with	O
LTE	O
(	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
)	O
is	O
said	O
to	O
be	O
of	O
kth	O
order	O
.	O
</s>
<s>
The	O
region	O
of	O
absolute	O
stability	O
for	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
the	O
complement	O
in	O
the	O
complex	O
plane	O
of	O
the	O
disk	O
with	O
radius	O
1	O
centered	O
at	O
1	O
,	O
depicted	O
in	O
the	O
figure	O
.	O
</s>
<s>
This	O
includes	O
the	O
whole	O
left	O
half	O
of	O
the	O
complex	O
plane	O
,	O
making	O
it	O
suitable	O
for	O
the	O
solution	O
of	O
stiff	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
even	O
L-stable	B-Algorithm
.	O
</s>
<s>
The	O
region	O
for	O
a	O
discrete	O
stable	O
system	O
by	O
Backward	B-Algorithm
Euler	I-Algorithm
Method	I-Algorithm
is	O
a	O
circle	O
with	O
radius	O
0.5	O
which	O
is	O
located	O
at	O
(	O
0.5	O
,	O
0	O
)	O
in	O
the	O
z-plane	O
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
a	O
variant	O
of	O
the	O
(	O
forward	O
)	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Other	O
variants	O
are	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
and	O
the	O
exponential	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
with	O
one	O
stage	O
,	O
described	O
by	O
the	O
Butcher	O
tableau	O
:	O
</s>
<s>
The	O
method	O
can	O
also	O
be	O
seen	O
as	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
with	O
one	O
step	O
.	O
</s>
<s>
It	O
is	O
the	O
first	O
method	O
of	O
the	O
family	O
of	O
Adams	O
–	O
Moulton	O
methods	O
,	O
and	O
also	O
of	O
the	O
family	O
of	O
backward	B-Algorithm
differentiation	I-Algorithm
formulas	I-Algorithm
.	O
</s>
