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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares (Page 2)

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Astronomical Applications Department, U.S. Naval Observatory - Linear Least Squares
Next, we calculate the partial derivatives of
2
with respect to the model parameters, setting these
derivatives to zero and thus forming the normal equations.
=
A
( )
rhs %
0
=
B
(
)
rhs %%
0
=
C
(
)
rhs %%%
0
=
=
i 1
N




-
+
+
+
2 y
i
2 A 2 B t
i
2 C t
i
2
0
=
=
i 1
N




-
2




- -
-
y
i
A B t
i
C t
i
2
t
i
0
=
=
i 1
N




-
2




- -
-
y
i
A B t
i
C t
i
2
t
i
2
0
:=
NormalEqs
{
}
,
,
% %% %%%
Finally, we solve the normal equations for the parameter values. These values are the linear least
squares solution.
:=
sols
(
)
sortsols
,
(
)
solve
,
(
)
value NormalEqs {
}
, ,
A B C
[
]
, ,
A B C
for in
do
od
p
sols
( )
print p
A


=
i 1
N
y
i


=
i 1
N
t
i
3 2


=
i 1
N
y
i


=
i 1
N
t
i
4


=
i 1
N
t
i
2


=
i 1
N
t
i


=
i 1
N
t
i
y
i


=
i 1
N
t
i
4
-
+

=


=
i 1
N
t
i


=
i 1
N
t
i
3


=
i 1
N
t
i
2
y
i


=
i 1
N
t
i
2


=
i 1
N
t
i
3


=
i 1
N
t
i
y
i
-
-


=
i 1
N
t
i
2 2


=
i 1
N
t
i
2
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
+
+
-

B


=
i 1
N
t
i
4


=
i 1
N
y
i


=
i 1
N
t
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
y
i
N


=
i 1
N
t
i
4


=
i 1
N
y
i


=
i 1
N
t
i
2


=
i 1
N
t
i
3
N


=
i 1
N
t
i
3


=
i 1
N
t
i
2
y
i
-
-
+
Page 2

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