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In-Depth Information
we can derive from (
5.55
) the following 2
2 system of linear equations determining
˛ and ˇ:
0
@
1
A
0
@
1
A
n
X
n
X
0
@
1
A
n
t
i
y
i
˛
i D1
i D1
D
:
(5.58)
X
n
X
n
X
n
t
i
t
i
t
i
y
i
ˇ
i D1
i D1
i D1
Approximation by a Quadratic Function
Finally, we will seek a quadratic approximation on the form
p
2
.t /
D
˛
C
ˇt
C
t
2
(5.59)
ofthedatagivenin(
5.47
). Using
n
X
˛
C
ˇt
i
y
i
2
;
C
t
i
F.˛;ˇ;/
D
(5.60)
i D1
we appeal to the principle of least squares and require that ˛, ˇ,and are such that
@F
@˛
@F
@ˇ
@F
@
D
D
D
0:
(5.61)
Since
n
X
@F
@˛
C
t
i
D
2
.˛
C
ˇt
i
y
i
/
(5.62)
i D1
n
X
@F
@ˇ
C
t
i
D
2
.˛
C
ˇt
i
y
i
/t
i
;
(5.63)
i D1
n
X
@F
@
C
t
i
y
i
/t
i
;
D
2
.˛
C
ˇt
i
(5.64)
i D1
(
5.61
) leads to the following 3
3 linear system of equations that determines ˛, ˇ,
and :
0
1
0
1
n
X
n
X
n
X
t
i
n
t
i
y
i
@
A
@
A
0
1
i D1
i D1
i D1
˛
ˇ
n
X
n
X
n
X
n
X
t
i
t
i
@
A
D
t
i
y
i
t
i
:
(5.65)
i D1
i D1
i D1
i D1
X
n
X
n
X
n
X
n
t
i
t
i
t
i
y
i
t
i
i D1
i D1
i D1
i D1
