<s>
In	O
mathematics	O
,	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	I-Algorithm
Fehlberg	I-Algorithm
method	I-Algorithm
(	O
or	O
Fehlberg	B-Algorithm
method	I-Algorithm
)	O
is	O
an	O
algorithm	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
for	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
It	O
was	O
developed	O
by	O
the	O
German	O
mathematician	O
Erwin	O
Fehlberg	B-Algorithm
and	O
is	O
based	O
on	O
the	O
large	O
class	O
of	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
novelty	O
of	O
Fehlberg	B-Algorithm
's	O
method	O
is	O
that	O
it	O
is	O
an	O
embedded	O
method	O
from	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
family	I-Algorithm
,	O
meaning	O
that	O
identical	O
function	O
evaluations	O
are	O
used	O
in	O
conjunction	O
with	O
each	O
other	O
to	O
create	O
methods	O
of	O
varying	O
order	O
and	O
similar	O
error	O
constants	O
.	O
</s>
<s>
The	O
method	O
presented	O
in	O
Fehlberg	B-Algorithm
's	O
1969	O
paper	O
has	O
been	O
dubbed	O
the	O
RKF45	B-Algorithm
method	O
,	O
and	O
is	O
a	O
method	O
of	O
order	O
O(h4 )	O
with	O
an	O
error	O
estimator	O
of	O
order	O
O(h5 )	O
.	O
</s>
<s>
By	O
performing	O
one	O
extra	O
calculation	O
,	O
the	O
error	O
in	O
the	O
solution	O
can	O
be	O
estimated	O
and	O
controlled	O
by	O
using	O
the	O
higher-order	O
embedded	O
method	O
that	O
allows	O
for	O
an	O
adaptive	B-Algorithm
stepsize	I-Algorithm
to	O
be	O
determined	O
automatically	O
.	O
</s>
<s>
Any	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
uniquely	O
identified	O
by	O
its	O
Butcher	O
tableau	O
.	O
</s>
<s>
The	O
coefficients	O
found	O
by	O
Fehlberg	B-Algorithm
for	O
Formula	O
1	O
(	O
derivation	O
with	O
his	O
parameter	O
α2	O
=	O
1/3	O
)	O
are	O
given	O
in	O
the	O
table	O
below	O
,	O
using	O
array	O
indexing	O
of	O
base	O
1	O
instead	O
of	O
base	O
0	O
to	O
be	O
compatible	O
with	O
most	O
computer	O
languages	O
:	O
</s>
<s>
Fehlberg	B-Algorithm
outlines	O
a	O
solution	O
to	O
solving	O
a	O
system	O
of	O
n	O
differential	O
equations	O
of	O
the	O
form	O
:	O
</s>
<s>
where	O
h	O
is	O
an	O
adaptive	B-Algorithm
stepsize	I-Algorithm
to	O
be	O
determined	O
algorithmically	O
:	O
</s>
<s>
The	O
coefficients	O
found	O
by	O
Fehlberg	B-Algorithm
for	O
Formula	O
2	O
(	O
derivation	O
with	O
his	O
parameter	O
α2	O
=3	O
/8	O
)	O
are	O
given	O
in	O
the	O
table	O
below	O
,	O
using	O
array	O
indexing	O
of	O
base	O
1	O
instead	O
of	O
base	O
0	O
to	O
be	O
compatible	O
with	O
most	O
computer	O
languages	O
:	O
</s>
<s>
In	O
another	O
table	O
in	O
Fehlberg	B-Algorithm
,	O
coefficients	O
for	O
an	O
RKF4(5 )	O
derived	O
by	O
D	O
.	O
Sarafyan	O
are	O
given	O
:	O
</s>
