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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares (Page 12)

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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares
end
Polynomial of degree 3
In this example, we will not show the solution, since it is a bit unwieldy.




leastsqrs
, , ,
=
Y
+
+
+
A B t
i
C t
i
2
D t
i
3
t i [
]
, , ,
A B C D
leastsqrs[0]: normal equations
=
-
+
+
+
+
2 S
0
2 A N 2 B T
1
2 C T
2
2 D T
3
0
=
-
+
+
+
+
2 S
1
2 A T
1
2 B T
2
2 C T
3
2 D T
4
0
=
-
+
+
+
+
2 S
2
2 A T
2
2 B T
3
2 C T
4
2 D T
5
0
=
-
+
+
+
+
2 S
3
2 A T
3
2 B T
4
2 C T
5
2 D T
6
0
leastsqrs[0]: Solving the normal equations...
leastsqrs[1]: substitution list


=
i 1
N
t
i
6


=
i 1
N
t
i
2 3
2


=
i 1
N
t
i
2 2


=
i 1
N
t
i
3


=
i 1
N
t
i
5


=
i 1
N
t
i
2 2


=
i 1
N
t
i
4 2
-
-
=



=
i 1
N
t
i
2


=
i 1
N
t
i
5 2
N


=
i 1
N
t
i
2


=
i 1
N
t
i
6


=
i 1
N
t
i
4
N
+
-
2


=
i 1
N
t
i
2


=
i 1
N
t
i


=
i 1
N
t
i
4


=
i 1
N
t
i
5
3


=
i 1
N
t
i
2


=
i 1
N
t
i
3 2


=
i 1
N
t
i
4
+
+
2


=
i 1
N
t
i
2


=
i 1
N
t
i


=
i 1
N
t
i
6


=
i 1
N
t
i
3


=
i 1
N
t
i
4 3
N


=
i 1
N
t
i
6


=
i 1
N
t
i
3 2
N
-
+
+
2


=
i 1
N
t
i
4
N


=
i 1
N
t
i
3


=
i 1
N
t
i
5
2


=
i 1
N
t
i


=
i 1
N
t
i
3


=
i 1
N
t
i
4 2
-
-
2


=
i 1
N
t
i


=
i 1
N
t
i
3 2


=
i 1
N
t
i
5


=
i 1
N
t
i
2


=
i 1
N
t
i
6


=
i 1
N
t
i
4


=
i 1
N
t
i
2


=
i 1
N
t
i
5 2
+
+
-


=
i 1
N
t
i
3 4
-
=
T
6
=
i 1
N
t
i
6
=
T
5
=
i 1
N
t
i
5
=
S
0
=
i 1
N
Y
i
=
T
2
=
i 1
N
t
i
2
=
T
1
=
i 1
N
t
i
=
T
3
=
i 1
N
t
i
3
,
,
,
,
,
,
,
=
S
1
=
i 1
N
t
i
Y
i
=
S
3
=
i 1
N
t
i
3
Y
i
=
T
4
=
i 1
N
t
i
4
=
S
2
=
i 1
N
t
i
2
Y
i
,
,
,

leastsqrs[2]: Verifying the solution...
(
)
map
,
cost %
Page 12

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