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Then
48
48
48
3
/
4
EI
L
4
and
A
=
B
=
p
0
1
/
2
(3)
1
/
4
And from Eq. (7.19), the linear equation for
w
1
is
48
EI
L
4
2
p
0
L
4
96
EI
48
EI
L
4
3
2
p
0
·
3
w
=
·
or
w
=
(4)
1
1
This is the same result as that obtained in Example 7.1.
7.3.3
Minimax Method
One possibility of minimizing the residual is to find the free parameters
u
such that the
maximum residual at selected points is minimized (
L
∞
norm). This approach is often
referred to as the
min-max, minimax, minimum absolute error,
or
Tchebychev
4
fit method
.
,n
locations (in
V
). Often, it
makes sense to determine these points in the same fashion as with orthogonal collocation.
Then with the minimax method the coefficients
Suppose the residual
R
is sampled at
x
=
x
j
,j
=
1
,
2
,
...
u
i
are selected such that the maximum of
the absolute value of these residuals is a minimum, i.e.,
max
|
R
(
x
j
)
|
is a minimum
j
=
1
,
2
,
...
,n
or
min
max
|
R
(
x
j
)
|=
min
max
|
R
j
|
(7.20)
j
=
1
,
2
,
...
,n
j
=
1
,
2
,
...
,n
As with the least squares method,
n
is not necessarily equal to
m,
the number of unknown
coefficients.
This is a useful method since it can be reduced to a problem in linear programming. To
convert the problem defined by Eq. (7.20) to a linear programming form, set an unknown
number
φ
=
...
equal to the (unknown) maximum value of
R
j
,j
1
,
2
,
,n,
that is, let
φ
=
max
|
R
j
|
,
j
=
1
,
2
,
...
,n
(7.21)
This is equivalent to requiring that
R
j
satisfy
|
|≤
φ
−
φ
≤
≤
φ
R
j
or
R
j
(7.22a)
or
R
j
−
φ
≤
0
and
φ
+
R
j
≥
0
(7.22b)
Now the min-max approximation problem is one of finding the unknown coefficients
u
i
such that
φ
is minimized
(7.23)
4
Patnutil Lvovich Tchebychev (1821-1894) was a Russian mathematician, who left an imprint in many areas
of mathematics. These included the theories of integrals and numbers, quadratic forms, polynomials, motion
theorems, and rectilinear motion. His collected works were published in two volumes which appeared in French
in 1900 and 1907.
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