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0.15
0.1
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
*=0.345
α
Fig. 5.7
A graph of G
D
G.˛/ given by ( 5.8 )
Note that there arise two different constant approximations from minimizing F
given by ( 5.2 )andG given by ( 5.8 ). It is not correct to say one approximation is
better than the other. They are both the best approximation we can get according
to two different criteria. There exists many other criteria as well, each leading to
slightly different constant approximations.
In Fig. 5.8 we have plotted all the data in Table 5.1 together with the constant
functions
p.t/ D 0:312
and
q.t/ D 0:345
obtained by minimizing F.˛/ (see ( 5.2 )) and G.˛/ (see ( 5.8 )), respectively.
We notice that minimizing the quadratic function F leads to a problem that can
easily be solved. The alternative, G, results in a nonlinear problem that needs to be
solved and we had to appeal to Newton's method. This observation is fairly general
and we will therefore rely on minimizing quadratic deviations in the rest of this
chapter. That is, we will focus on the method of least squares.
5.1.2
Approximation by a Linear Function
Next, we will model the data in Table 5.1 using a linear function. More precisely,
we will try to determine two constants ˛ and ˇ such that
 
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