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7.3.2
Least Squares Collocation
The estimation of parameters using least squares is usually attributed to Gauss's work of
1795. However, since this work was not published until 1809, there was some controversy
because Legendre published similar concepts in 1806.
A useful application of the least squares technique is to couple it with collocation. This
method minimizes the sum of the squares of the residuals at the collocation points. In
this case, the number of collocation points is not necessarily equal to the number of free
parameters. Suppose
n
collocation points are selected at
x
j
,j
=
1
,
2
,
...
,n,
then the least
squares method requires that
R
n
2
(
x
j
)
be a minimum
(7.18a)
or
C
T
C
be a minimum
(7.18b)
where
C
T
=
[
R
(
x
1
)
R
(
x
2
)...
R
(
x
n
)
]. With the trial solution of Eq. (7.9) as the approximation
of
u, R
(
x
j
)
=
L
u
(
x
j
)
−
f
(
x
j
)
.If
L
is linear,
R
(
x
j
)
is a linear equation in terms of
u
i
,i
=
a
j
1
,
2
,
...
,m
.
Let this equation be
R
(
x
j
)
=
u
−
b
j
,
then
C
=
A
u
−
B
where
a
1
a
2
a
n
b
1
b
2
b
n
A
=
and
B
=
Equation (7.18b) becomes
C
T
C
u
T
A
T
A
u
T
A
T
B
B
T
B
=
−
+
u
2
∂
∂
The necessary condition to make
C
T
C
a minimum is
u
(
C
T
C
)
=
0
,
i.e.,
A
T
A
A
T
B
=
u
(7.19)
This constitutes a set of simultaneous linear equations from which
u
can be determined.
For a vector residual
R
the minimization process is repeated
p
times.
EXAMPLE 7.2 Beam with Linearly Varying Load
Return to the beam of Fig. 7.1 and use the same trial function as in Example 7.1.
2
3
4
w
=
w
1
(
3
ξ
−
5
ξ
+
2
ξ
)ξ
=
x
/
L
(1)
Substitute this into the beam governing equation, and use
x
1
=
L
/
4
,x
2
=
L
/
2
,
and
x
3
=
3
L
/
4 as the collocation points. The residuals at these points are
EI
48
3
p
0
4
R
(
x
1
)
=
L
4
w
−
1
(2)
EI
48
p
0
2
R
(
x
2
)
=
L
4
w
1
−
and
EI
48
p
0
4
(
)
=
L
4
w
−
R
x
3
1
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