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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares (Page 3)

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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares


=
i 1
N
t
i
2 2


=
i 1
N
t
i
y
i
+
2


=
i 1
N
t
i


=
i 1
N
t
i
2


=
i 1
N
t
i
3
N


=
i 1
N
t
i
3 2
-
+




=
i 1
N
t
i
2 3


=
i 1
N
t
i
4


=
i 1
N
t
i
2


=
i 1
N
t
i
4
N


=
i 1
N
t
i
2
+
+
-

C


=
i 1
N
t
i


=
i 1
N
t
i
2


=
i 1
N
t
i
y
i
N


=
i 1
N
t
i
3


=
i 1
N
t
i
y
i
-
+

=


=
i 1
N
t
i
3


=
i 1
N
y
i


=
i 1
N
t
i


=
i 1
N
t
i
2
y
i


=
i 1
N
t
i
2


=
i 1
N
t
i
2
y
i
N


=
i 1
N
t
i
2
-
+
-


=
i 1
N
t
i
2 2


=
i 1
N
y
i
+
2


=
i 1
N
t
i


=
i 1
N
t
i
2


=
i 1
N
t
i
3
N


=
i 1
N
t
i
3 2
-
+




=
i 1
N
t
i
2 3


=
i 1
N
t
i
4


=
i 1
N
t
i
2


=
i 1
N
t
i
4
N


=
i 1
N
t
i
2
+
+
-

This is a bit easier to look at if we make a few substitutions.


subs
,
,


seq
,
=
=
i 1
N
t
i
k
T
k
=
k
..
1 4


seq
,
=
=
i 1
N
y
i
t
i
k
S
k
=
k
..
0 2 sols
=
A
-
+
-
-
+
S
0
T
3
2
S
0
T
4
T
2
T
1
S
1
T
4
T
1
T
3
S
2
T
2
T
3
S
1
T
2
2
S
2
-
+
+
+
-
2 T
1
T
2
T
3
N T
3
2
T
2
3
T
4
T
1
2
T
4
N T
2
,

=
B
-
-
-
+
+
T
4
S
0
T
1
T
1
S
2
T
2
S
1
N T
4
S
0
T
2
T
3
N T
3
S
2
T
2
2
S
1
-
+
+
+
-
2 T
1
T
2
T
3
N T
3
2
T
2
3
T
4
T
1
2
T
4
N T
2
,
=
C
-
+
-
+
-
+
T
1
T
2
S
1
N T
3
S
1
T
3
S
0
T
1
S
2
T
1
2
S
2
N T
2
T
2
2
S
0
-
+
+
+
-
2 T
1
T
2
T
3
N T
3
2
T
2
3
T
4
T
1
2
T
4
N T
2

Collecting on the mixed terms
S
k
, we have the result
(
)
collect
,
,
% [
]
,
(
)
seq
,
S
k
=
k
..
0 2 N factor
for in
do
od
p
%
( )
print p
Page 3

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