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that approximates a given function y
D
y.t/, a
t
b, in the sense of least
squares. Let
F.˛;ˇ;/
D
Z
b
a
.˛
C
ˇt
C
t
2
y.t//
2
dt
(5.126)
and define ˛, ˇ,and to be the solution of the three equations:
@F
@˛
@F
@ˇ
@F
@
D
D
D
0:
(5.127)
Here,
D
2
Z
b
a
@F
@˛
.˛
C
ˇt
C
t
2
y.t//
dt
;
D
2
Z
b
a
@F
@ˇ
.˛
C
ˇt
C
t
2
y.t// t
dt
;
and
D
2
Z
b
a
@F
@
.˛
C
ˇt
C
t
2
y.t// t
2
dt
:
From (
5.127
), it follows that
3
.b
3
a
3
/
D
Z
b
.b
a/˛
C
1
1
2
.b
2
a
2
/ˇ
C
y.t/
dt
;
a
4
.b
4
a
4
/
D
Z
b
1
2
.b
2
a
2
/˛
C
1
1
3
.b
3
a
3
/ˇ
C
(5.128)
ty.t/
dt
;
a
5
.b
5
a
5
/
D
Z
b
1
3
.b
3
a
3
/˛
C
1
1
4
.b
4
a
4
/ˇ
C
t
2
y.t/
dt
:
a
The linear system (
5.128
) determines the coefficients ˛, ˇ,and in the quadratic
least squares approximation (
5.125
)ofy.t/.
Example 5.7.
For the function
y.t/
D
sin.t /;
0
t
=2;
(5.129)
the linear system (
5.128
) reads
0
@
1
A
0
@
1
A
D
0
@
1
A
=2
2
=8
3
=24
2
=8
3
=24
4
=64
3
=24
4
=64
5
=160
˛
ˇ
1
1
2
(5.130)
