<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	B-Algorithm
Runge	I-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
(	O
)	O
are	O
a	O
family	O
of	O
implicit	B-Algorithm
and	I-Algorithm
explicit	I-Algorithm
iterative	O
methods	O
,	O
which	O
include	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
used	O
in	O
temporal	O
discretization	O
for	O
the	O
approximate	O
solutions	O
of	O
simultaneous	O
nonlinear	O
equations	O
.	O
</s>
<s>
The	O
most	O
widely	O
known	O
member	O
of	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
family	O
is	O
generally	O
referred	O
to	O
as	O
"	O
RK4	O
"	O
,	O
the	O
"	O
classic	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
"	O
or	O
simply	O
as	O
"	O
the	B-Algorithm
Runge	I-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
"	O
.	O
</s>
<s>
The	O
function	O
and	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
,	O
are	O
given	O
.	O
</s>
<s>
is	O
the	O
slope	O
at	O
the	O
beginning	O
of	O
the	O
interval	O
,	O
using	O
(	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
)	O
;	O
</s>
<s>
If	O
is	O
independent	O
of	O
,	O
so	O
that	O
the	O
differential	O
equation	O
is	O
equivalent	O
to	O
a	O
simple	O
integral	O
,	O
then	O
RK4	O
is	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
The	O
RK4	O
method	O
is	O
a	O
fourth-order	O
method	O
,	O
meaning	O
that	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
on	O
the	O
order	O
of	O
,	O
while	O
the	O
total	B-Algorithm
accumulated	I-Algorithm
error	I-Algorithm
is	O
on	O
the	O
order	O
of	O
.	O
</s>
<s>
The	O
family	O
of	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
RK4	O
method	O
mentioned	O
above	O
.	O
</s>
<s>
The	O
matrix	O
 [ aij ] 	O
is	O
called	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
matrix	O
,	O
while	O
the	O
bi	O
and	O
ci	O
are	O
known	O
as	O
the	O
weights	O
and	O
the	O
nodes	O
.	O
</s>
<s>
There	O
are	O
also	O
accompanying	O
requirements	O
if	O
one	O
requires	O
the	O
method	O
to	O
have	O
a	O
certain	O
order	O
p	O
,	O
meaning	O
that	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
O( hp+1	O
)	O
.	O
</s>
<s>
These	O
can	O
be	O
derived	O
from	O
the	O
definition	O
of	O
the	O
truncation	B-Algorithm
error	I-Algorithm
itself	O
.	O
</s>
<s>
In	O
general	O
,	O
if	O
an	O
explicit	B-Algorithm
-stage	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
has	O
order	O
,	O
then	O
it	O
can	O
be	O
proven	O
that	O
the	O
number	O
of	O
stages	O
must	O
satisfy	O
,	O
and	O
if	O
,	O
then	O
.	O
</s>
<s>
what	O
the	O
precise	O
minimum	O
number	O
of	O
stages	O
is	O
for	O
an	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
to	O
have	O
order	O
in	O
those	O
cases	O
where	O
no	O
methods	O
have	O
yet	O
been	O
discovered	O
that	O
satisfy	O
the	O
bounds	O
above	O
with	O
equality	O
.	O
</s>
<s>
A	O
slight	O
variation	O
of	O
"	O
the	O
"	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
also	O
due	O
to	O
Kutta	O
in	O
1901	O
and	O
is	O
called	O
the	O
3/8	O
-rule	O
.	O
</s>
<s>
However	O
,	O
the	O
simplest	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
the	O
(	O
forward	O
)	O
Euler	B-Algorithm
method	I-Algorithm
,	O
given	O
by	O
the	O
formula	O
.	O
</s>
<s>
This	O
is	O
the	O
only	O
consistent	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
with	O
one	O
stage	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
second-order	O
method	O
with	O
two	O
stages	O
is	O
provided	O
by	O
the	O
explicit	B-Algorithm
midpoint	B-Algorithm
method	I-Algorithm
:	O
</s>
<s>
In	O
this	O
family	O
,	O
gives	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
,	O
is	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
and	O
is	O
Ralston	O
's	O
method	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
the	O
two-stage	O
second-order	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
with	O
α	O
=	O
2/3	O
,	O
also	O
known	O
as	O
Ralston	O
method	O
.	O
</s>
<s>
The	O
numerical	B-General_Concept
solutions	I-General_Concept
correspond	O
to	O
the	O
underlined	O
values	O
.	O
</s>
<s>
Adaptive	O
methods	O
are	O
designed	O
to	O
produce	O
an	O
estimate	O
of	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
of	O
a	O
single	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
step	O
.	O
</s>
<s>
The	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	I-Algorithm
Fehlberg	I-Algorithm
method	I-Algorithm
has	O
two	O
methods	O
of	O
orders	O
5	O
and	O
4	O
.	O
</s>
<s>
However	O
,	O
the	O
simplest	O
adaptive	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
involves	O
combining	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
which	O
is	O
order	O
2	O
,	O
with	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
which	O
is	O
order	O
1	O
.	O
</s>
<s>
Other	O
adaptive	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
are	O
the	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
(	O
orders	O
3	O
and	O
2	O
)	O
,	O
the	O
Cash	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
method	I-Algorithm
and	O
the	O
Dormand	B-Algorithm
–	I-Algorithm
Prince	I-Algorithm
method	I-Algorithm
(	O
both	O
with	O
orders	O
5	O
and	O
4	O
)	O
.	O
</s>
<s>
A	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
said	O
to	O
be	O
nonconfluent	O
if	O
all	O
the	O
are	O
distinct	O
.	O
</s>
<s>
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	O
Nyström	O
methods	O
are	O
specialized	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
that	O
are	O
optimized	O
for	O
second-order	O
differential	O
equations	O
of	O
the	O
following	O
form	O
:	O
</s>
<s>
All	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
mentioned	O
up	O
to	O
now	O
are	O
explicit	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
are	O
generally	O
unsuitable	O
for	O
the	O
solution	O
of	O
stiff	B-Algorithm
equations	I-Algorithm
because	O
their	O
region	O
of	O
absolute	O
stability	O
is	O
small	O
;	O
in	O
particular	O
,	O
it	O
is	O
bounded	O
.	O
</s>
<s>
This	O
issue	O
is	O
especially	O
important	O
in	O
the	O
solution	O
of	O
partial	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
instability	O
of	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
motivates	O
the	O
development	O
of	O
implicit	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
difference	O
with	O
an	O
explicit	B-Algorithm
method	I-Algorithm
is	O
that	O
in	O
an	O
explicit	B-Algorithm
method	I-Algorithm
,	O
the	O
sum	O
over	O
j	O
only	O
goes	O
up	O
to	O
i	O
−	O
1	O
.	O
</s>
<s>
This	O
also	O
shows	O
up	O
in	O
the	O
Butcher	O
tableau	O
:	O
the	O
coefficient	O
matrix	O
of	O
an	O
explicit	B-Algorithm
method	I-Algorithm
is	O
lower	O
triangular	O
.	O
</s>
<s>
See	O
Adaptive	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
above	O
for	O
the	O
explanation	O
of	O
the	O
row	O
.	O
</s>
<s>
This	O
can	O
be	O
contrasted	O
with	O
implicit	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
(	O
the	O
other	O
big	O
family	O
of	O
methods	O
for	O
ODEs	O
)	O
:	O
an	O
implicit	O
s-step	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
needs	O
to	O
solve	O
a	O
system	O
of	O
algebraic	O
equations	O
with	O
only	O
m	O
components	O
,	O
so	O
the	O
size	O
of	O
the	O
system	O
does	O
not	O
increase	O
as	O
the	O
number	O
of	O
steps	O
increases	O
.	O
</s>
<s>
The	O
simplest	O
example	O
of	O
an	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
which	O
can	O
be	O
re-arranged	O
to	O
get	O
the	O
formula	O
for	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
listed	O
above	O
.	O
</s>
<s>
Another	O
example	O
for	O
an	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
The	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
a	O
collocation	B-Algorithm
method	I-Algorithm
(	O
as	O
discussed	O
in	O
that	O
article	O
)	O
.	O
</s>
<s>
All	O
collocation	B-Algorithm
methods	I-Algorithm
are	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
,	O
but	O
not	O
all	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
are	O
collocation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
form	O
a	O
family	O
of	O
collocation	B-Algorithm
methods	I-Algorithm
based	O
on	O
Gauss	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
A	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
with	O
s	O
stages	O
has	O
order	O
2s	O
(	O
thus	O
,	O
methods	O
with	O
arbitrarily	O
high	O
order	O
can	O
be	O
constructed	O
)	O
.	O
</s>
<s>
The	O
advantage	O
of	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
over	O
explicit	B-Algorithm
ones	O
is	O
their	O
greater	O
stability	O
,	O
especially	O
when	O
applied	O
to	O
stiff	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Explicit	B-Algorithm
methods	I-Algorithm
have	O
a	O
strictly	O
lower	O
triangular	O
matrix	O
A	O
,	O
which	O
implies	O
that	O
det( I	O
−	O
zA	O
)	O
=	O
1	O
and	O
that	O
the	O
stability	O
function	O
is	O
a	O
polynomial	O
.	O
</s>
<s>
The	O
numerical	B-General_Concept
solution	I-General_Concept
to	O
the	O
linear	O
test	O
equation	O
decays	O
to	O
zero	O
if	O
|	O
r(z )	O
|	O
<	O
1	O
with	O
z	O
=	O
hλ	O
.	O
</s>
<s>
The	O
stability	O
function	O
of	O
an	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
a	O
polynomial	O
,	O
so	O
explicit	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
can	O
never	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
A	O
Padé	O
approximant	O
with	O
numerator	O
of	O
degree	O
m	O
and	O
denominator	O
of	O
degree	O
n	O
is	O
A-stable	B-Algorithm
if	O
and	O
only	O
if	O
m	O
≤	O
n	O
≤	O
m	O
+	O
2	O
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
with	O
s	O
stages	O
has	O
order	O
2s	O
,	O
so	O
its	O
stability	O
function	O
is	O
the	O
Padé	O
approximant	O
with	O
m	O
=	O
n	O
=	O
s	O
.	O
It	O
follows	O
that	O
the	O
method	O
is	O
A-stable	B-Algorithm
.	O
</s>
<s>
This	O
shows	O
that	O
A-stable	B-Algorithm
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
can	O
have	O
arbitrarily	O
high	O
order	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
order	O
of	O
A-stable	B-Algorithm
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
cannot	O
exceed	O
two	O
.	O
</s>
<s>
The	O
corresponding	O
concepts	O
were	O
defined	O
as	O
G-stability	O
for	O
multistep	B-Algorithm
methods	I-Algorithm
(	O
and	O
the	O
related	O
one-leg	O
methods	O
)	O
and	O
B-stability	O
(	O
Butcher	O
,	O
1975	O
)	O
for	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
A	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
applied	O
to	O
the	O
non-linear	O
system	O
,	O
which	O
verifies	O
,	O
is	O
called	O
B-stable	O
,	O
if	O
this	O
condition	O
implies	O
for	O
two	O
numerical	B-General_Concept
solutions	I-General_Concept
.	O
</s>
<s>
Let	O
,	O
and	O
be	O
three	O
matrices	O
defined	O
by	O
A	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
said	O
to	O
be	O
algebraically	O
stable	O
if	O
the	O
matrices	O
and	O
are	O
both	O
non-negative	O
definite	O
.	O
</s>
<s>
In	O
general	O
a	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
of	O
order	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
We	O
develop	O
the	O
derivation	O
for	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
fourth-order	O
method	O
using	O
the	O
general	O
formula	O
with	O
evaluated	O
,	O
as	O
explained	O
above	O
,	O
at	O
the	O
starting	O
point	O
,	O
the	O
midpoint	O
and	O
the	O
end	O
point	O
of	O
any	O
interval	O
;	O
thus	O
,	O
we	O
choose	O
:	O
</s>
