<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
scientific	O
computing	O
,	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
are	O
a	O
family	O
of	O
numerical	B-Algorithm
methods	I-Algorithm
for	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
are	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
More	O
specifically	O
,	O
they	O
are	O
collocation	B-Algorithm
methods	I-Algorithm
based	O
on	O
the	O
points	O
of	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
based	O
on	O
s	O
points	O
has	O
order	O
2s	O
.	O
</s>
<s>
All	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
are	O
A-stable	O
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
of	O
order	O
two	O
is	O
the	O
implicit	B-Algorithm
midpoint	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
of	O
order	O
four	O
has	O
Butcher	O
tableau	O
:	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
of	O
order	O
six	O
has	O
Butcher	O
tableau	O
:	O
</s>
<s>
The	O
computational	O
cost	O
of	O
higher-order	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
is	O
usually	O
excessive	O
,	O
and	O
thus	O
,	O
they	O
are	O
rarely	O
used	O
.	O
</s>
<s>
Gauss-Legendre	O
Runge-Kutta	B-Algorithm
(	O
GLRK	O
)	O
methods	O
solve	O
an	O
ordinary	O
differential	O
equation	O
with	O
.	O
</s>
<s>
The	O
distinguishing	O
feature	O
of	O
GLRK	O
is	O
the	O
estimation	O
of	O
with	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
The	O
Runge-Kutta	B-Algorithm
implicit	O
definition	O
is	O
invoked	O
to	O
accomplish	O
this	O
.	O
</s>
<s>
The	O
values	O
of	O
the	O
Runge-Kutta	B-Algorithm
parameters	O
can	O
be	O
determined	O
from	O
a	O
Taylor	O
series	O
expansion	O
in	O
.	O
</s>
<s>
The	O
Gauss-Legendre	B-Algorithm
methods	I-Algorithm
are	O
implicit	O
,	O
so	O
in	O
general	O
they	O
cannot	O
be	O
applied	O
exactly	O
.	O
</s>
<s>
Below	O
is	O
a	O
Matlab	O
function	O
which	O
implements	O
the	O
Gauss-Legendre	B-Algorithm
method	I-Algorithm
of	O
order	O
four	O
.	O
</s>
<s>
The	O
only	O
extra	O
work	O
compared	O
to	O
explicit	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
is	O
the	O
computation	O
of	O
the	O
Jacobian	O
.	O
</s>
<s>
In	O
these	O
methods	O
,	O
the	O
averaged	O
position	O
is	O
used	O
in	O
computing	O
instead	O
of	O
just	O
the	O
initial	O
position	O
in	O
standard	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
method	O
of	O
order	O
2	O
is	O
just	O
an	O
implicit	O
midpoint	B-Algorithm
method	I-Algorithm
.	O
</s>
