<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
scientific	O
computing	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
a	O
numerical	B-Algorithm
method	I-Algorithm
to	O
solve	O
ordinary	O
differential	O
equations	O
derived	O
from	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
for	O
computing	O
integrals	O
.	O
</s>
<s>
The	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
an	O
implicit	B-Algorithm
second-order	O
method	O
,	O
which	O
can	O
be	O
considered	O
as	O
both	O
a	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
and	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
:	O
the	O
value	O
appears	O
on	O
both	O
sides	O
of	O
the	O
equation	O
,	O
and	O
to	O
actually	O
calculate	O
it	O
,	O
we	O
have	O
to	O
solve	O
an	O
equation	O
which	O
will	O
usually	O
be	O
nonlinear	O
.	O
</s>
<s>
We	O
can	O
use	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
to	O
get	O
a	O
fairly	O
good	O
estimate	O
for	O
the	O
solution	O
,	O
which	O
can	O
be	O
used	O
as	O
the	O
initial	O
guess	O
of	O
Newton	O
's	O
method	O
.	O
</s>
<s>
Cutting	O
short	O
,	O
using	O
only	O
the	O
guess	O
from	O
Eulers	B-Algorithm
method	I-Algorithm
is	O
equivalent	O
to	O
performing	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Now	O
combine	O
both	O
formulas	O
and	O
use	O
that	O
and	O
to	O
get	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
for	O
solving	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
It	O
follows	O
from	O
the	O
error	O
analysis	O
of	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
for	O
quadrature	O
that	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
of	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
for	O
solving	O
differential	O
equations	O
can	O
be	O
bounded	O
as	O
:	O
</s>
<s>
Thus	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
a	O
second-order	O
method	O
.	O
</s>
<s>
This	O
includes	O
the	O
left-half	O
plane	O
,	O
so	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
A-stable	B-Algorithm
.	O
</s>
<s>
The	O
second	O
Dahlquist	O
barrier	O
states	O
that	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
the	O
most	O
accurate	O
amongst	O
the	O
A-stable	B-Algorithm
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
that	O
is	O
A-stable	B-Algorithm
has	O
at	O
most	O
order	O
two	O
,	O
and	O
the	O
error	O
constant	O
of	O
a	O
second-order	O
A-stable	B-Algorithm
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
cannot	O
be	O
better	O
than	O
the	O
error	O
constant	O
of	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
region	B-Algorithm
of	I-Algorithm
absolute	I-Algorithm
stability	I-Algorithm
for	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
precisely	O
the	O
left-half	O
plane	O
.	O
</s>
<s>
This	O
means	O
that	O
if	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
applied	O
to	O
the	O
linear	O
test	O
equation	O
y'''	O
=	O
λy''	O
,	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
decays	O
to	O
zero	O
if	O
and	O
only	O
if	O
the	O
exact	O
solution	O
does	O
.	O
</s>
