Astronomical Applications Department, U.S. Naval Observatory Linear Least Squares Page 3
=
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