<s>
In	O
mathematics	O
and	O
computational	O
science	O
,	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
may	O
refer	O
to	O
the	O
improved	O
or	O
modified	B-Algorithm
Euler	I-Algorithm
's	I-Algorithm
method	I-Algorithm
(	O
that	O
is	O
,	O
the	O
explicit	B-Algorithm
trapezoidal	I-Algorithm
rule	I-Algorithm
)	O
,	O
or	O
a	O
similar	O
two-stage	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
It	O
is	O
named	O
after	O
Karl	O
Heun	O
and	O
is	O
a	O
numerical	B-General_Concept
procedure	O
for	O
solving	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
with	O
a	O
given	O
initial	O
value	O
.	O
</s>
<s>
Both	O
variants	O
can	O
be	O
seen	O
as	O
extensions	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
into	O
two-stage	O
second-order	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
procedure	O
for	O
calculating	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
to	O
the	O
initial	O
value	O
problem	O
:	O
</s>
<s>
by	O
way	O
of	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
is	O
to	O
first	O
calculate	O
the	O
intermediate	O
value	O
and	O
then	O
the	O
final	O
approximation	O
at	O
the	O
next	O
integration	O
point	O
.	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
used	O
as	O
the	O
foundation	O
for	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
uses	O
the	O
line	O
tangent	O
to	O
the	O
function	O
at	O
the	O
beginning	O
of	O
the	O
interval	O
as	O
an	O
estimate	O
of	O
the	O
slope	O
of	O
the	O
function	O
over	O
the	O
interval	O
,	O
assuming	O
that	O
if	O
the	O
step	O
size	O
is	O
small	O
,	O
the	O
error	O
will	O
be	O
small	O
.	O
</s>
<s>
Heun	B-Algorithm
's	I-Algorithm
Method	I-Algorithm
addresses	O
this	O
problem	O
by	O
considering	O
the	O
interval	O
spanned	O
by	O
the	O
tangent	O
line	O
segment	O
as	O
a	O
whole	O
.	O
</s>
<s>
If	O
the	O
tangent	O
line	O
at	O
the	O
right	O
end	O
point	O
is	O
considered	O
(	O
which	O
can	O
be	O
estimated	O
using	O
Euler	B-Algorithm
's	I-Algorithm
Method	I-Algorithm
)	O
,	O
it	O
has	O
the	O
opposite	O
problem	O
.	O
</s>
<s>
Heun	B-Algorithm
's	I-Algorithm
Method	I-Algorithm
considers	O
the	O
tangent	O
lines	O
to	O
the	O
solution	O
curve	O
at	O
both	O
ends	O
of	O
the	O
interval	O
,	O
one	O
which	O
overestimates	O
,	O
and	O
one	O
which	O
underestimates	O
the	O
ideal	O
vertical	O
coordinates	O
.	O
</s>
<s>
A	O
prediction	O
line	O
must	O
be	O
constructed	O
based	O
on	O
the	O
right	O
end	O
point	O
tangent	O
's	O
slope	O
alone	O
,	O
approximated	O
using	O
Euler	B-Algorithm
's	I-Algorithm
Method	I-Algorithm
.	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
Method	I-Algorithm
is	O
used	O
to	O
roughly	O
estimate	O
the	O
coordinates	O
of	O
the	O
next	O
point	O
in	O
the	O
solution	O
,	O
and	O
with	O
this	O
knowledge	O
,	O
the	O
original	O
estimate	O
is	O
re-predicted	O
or	O
corrected	O
.	O
</s>
<s>
The	O
scheme	O
can	O
be	O
compared	O
with	O
the	O
implicit	B-Algorithm
trapezoidal	B-Algorithm
method	I-Algorithm
,	O
but	O
with	O
replaced	O
by	O
in	O
order	O
to	O
make	O
it	O
explicit	O
.	O
</s>
<s>
is	O
the	O
result	O
of	O
one	O
step	O
of	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
on	O
the	O
same	O
initial	O
value	O
problem	O
.	O
</s>
<s>
So	O
,	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
predictor-corrector	B-Algorithm
method	I-Algorithm
with	O
forward	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
as	O
predictor	O
and	O
trapezoidal	B-Algorithm
method	I-Algorithm
as	O
corrector	O
.	O
</s>
<s>
The	O
improved	B-Algorithm
Euler	I-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
two-stage	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
,	O
and	O
can	O
be	O
written	O
using	O
the	O
Butcher	O
tableau	O
(	O
after	O
John	O
C	O
.	O
Butcher	O
)	O
:	O
</s>
<s>
The	O
other	O
method	O
referred	O
to	O
as	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
(	O
also	O
known	O
as	O
Ralston	O
's	O
method	O
)	O
has	O
the	O
Butcher	O
table	O
:	O
</s>
