<s>
In	O
mathematics	O
,	O
a	O
collocation	B-Algorithm
method	I-Algorithm
is	O
a	O
method	O
for	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
ordinary	O
differential	O
equations	O
,	O
partial	O
differential	O
equations	O
and	O
integral	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
idea	O
is	O
to	O
choose	O
a	O
finite-dimensional	O
space	O
of	O
candidate	O
solutions	O
(	O
usually	O
polynomials	O
up	O
to	O
a	O
certain	O
degree	O
)	O
and	O
a	O
number	O
of	O
points	O
in	O
the	O
domain	O
(	O
called	O
collocation	B-Algorithm
points	I-Algorithm
)	O
,	O
and	O
to	O
select	O
that	O
solution	O
which	O
satisfies	O
the	O
given	O
equation	O
at	O
the	O
collocation	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
at	O
all	O
collocation	B-Algorithm
points	I-Algorithm
for	O
.	O
</s>
<s>
All	O
these	O
collocation	B-Algorithm
methods	I-Algorithm
are	O
in	O
fact	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
coefficients	O
ck	O
in	O
the	O
Butcher	O
tableau	O
of	O
a	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
are	O
the	O
collocation	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
However	O
,	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>
Pick	O
,	O
as	O
an	O
example	O
,	O
the	O
two	O
collocation	B-Algorithm
points	I-Algorithm
c1	O
=	O
0	O
and	O
c2	O
=	O
1	O
(	O
so	O
n	O
=	O
2	O
)	O
.	O
</s>
<s>
This	O
method	O
is	O
known	O
as	O
the	O
"	O
trapezoidal	B-Algorithm
rule	I-Algorithm
"	O
for	O
differential	O
equations	O
.	O
</s>
<s>
and	O
approximating	O
the	O
integral	O
on	O
the	O
right-hand	O
side	O
by	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
for	O
integrals	O
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
methods	I-Algorithm
use	O
the	O
points	O
of	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
as	O
collocation	B-Algorithm
points	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>
In	O
fact	O
,	O
one	O
can	O
show	O
that	O
the	O
order	O
of	O
a	O
collocation	B-Algorithm
method	I-Algorithm
corresponds	O
to	O
the	O
order	O
of	O
the	O
quadrature	O
rule	O
that	O
one	O
would	O
get	O
using	O
the	O
collocation	B-Algorithm
points	I-Algorithm
as	O
weights	O
.	O
</s>
<s>
In	O
direct	O
collocation	B-Algorithm
method	I-Algorithm
,	O
we	O
are	O
essentially	O
performing	O
variational	O
calculus	O
with	O
the	O
finite-dimensional	O
subspace	O
of	O
piecewise	O
linear	O
functions	O
(	O
as	O
in	O
trapezoidal	B-Algorithm
rule	I-Algorithm
)	O
,	O
or	O
cubic	O
functions	O
,	O
or	O
other	O
piecewise	O
polynomial	O
functions	O
.	O
</s>
<s>
In	O
orthogonal	O
collocation	B-Algorithm
method	I-Algorithm
,	O
we	O
instead	O
use	O
the	O
finite-dimensional	O
subspace	O
spanned	O
by	O
the	O
first	O
N	O
vectors	O
in	O
some	O
orthogonal	O
polynomial	O
basis	O
,	O
such	O
as	O
the	O
Legendre	O
polynomials	O
.	O
</s>
