<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
a	O
branch	O
of	O
applied	O
mathematics	O
,	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
a	O
one-step	O
method	O
for	O
numerically	B-Algorithm
solving	O
the	O
differential	O
equation	O
,	O
</s>
<s>
for	O
Here	O
,	O
is	O
the	O
step	O
size	O
a	O
small	O
positive	O
number	O
,	O
and	O
is	O
the	O
computed	O
approximate	O
value	O
of	O
The	O
explicit	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
sometimes	O
also	O
known	O
as	O
the	O
modified	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
the	O
implicit	O
method	O
is	O
the	O
most	O
simple	O
collocation	B-Algorithm
method	I-Algorithm
,	O
and	O
,	O
applied	O
to	O
Hamiltonian	O
dynamics	O
,	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
modified	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
can	O
refer	O
to	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
for	O
further	O
clarity	O
see	O
List	B-Algorithm
of	I-Algorithm
Runge	I-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
the	O
basic	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
the	O
tangent	O
of	O
the	O
curve	O
at	O
is	O
computed	O
using	O
.	O
</s>
<s>
Instead	O
,	O
this	O
tangent	O
is	O
estimated	O
by	O
using	O
the	O
original	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
to	O
estimate	O
the	O
value	O
of	O
at	O
the	O
midpoint	O
,	O
then	O
computing	O
the	O
slope	O
of	O
the	O
tangent	O
with	O
.	O
</s>
<s>
The	O
local	O
error	O
at	O
each	O
step	O
of	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
of	O
order	O
,	O
giving	O
a	O
global	O
error	O
of	O
order	O
.	O
</s>
<s>
Thus	O
,	O
while	O
more	O
computationally	O
intensive	O
than	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
's	O
error	O
generally	O
decreases	O
faster	O
as	O
.	O
</s>
<s>
The	O
methods	O
are	O
examples	O
of	O
a	O
class	O
of	O
higher-order	O
methods	O
known	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
solution	O
is	O
then	O
to	O
use	O
a	O
Taylor	O
series	O
expansion	O
exactly	O
as	O
if	O
using	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
to	O
solve	O
for	O
:	O
</s>
<s>
and	O
the	O
explicit	O
midpoint	B-Algorithm
method	I-Algorithm
(	O
1e	O
)	O
.	O
</s>
<s>
which	O
contains	O
the	O
implicit	O
Euler	B-Algorithm
method	I-Algorithm
with	O
step	O
size	O
as	O
its	O
first	O
part	O
.	O
</s>
<s>
Replacing	O
the	O
implicit	O
with	O
the	O
explicit	B-Algorithm
Euler	I-Algorithm
method	O
in	O
the	O
determination	O
of	O
results	O
again	O
in	O
the	O
explicit	O
midpoint	B-Algorithm
method	I-Algorithm
.	O
</s>
