The Method of Least Squares - Elements of Scientific Computing - page 157

so we arrive at the equation

55˛ C 385ˇ D 20: (5.26)

We n ow h ave a 2 2 system of linear equations that determines ˛ and ˇ:
10 55
55 385

! ˛
ˇ

! D 3:12
20

! :

(5.27)

By the formula (3.63)onpage89,wehave

10 55
55 385

! 1 D 385 55

55 10

! ;

1
825

(5.28)

so

˛
ˇ

! D 385 55

55 10

! 3:12
20

! 0:123
0:034

! :

1
825

(5.29)

Hence we have the linear model

p.t/ D 0:123 C 0:034t: (5.30)

In Fig. 5.9 we have plotted all the data from 1991 to 2000 together with the
constant approximation

p 0 .t / D 0:312 (5.31)

and the linear approximation

p 1 .t / D 0:123 C 0:034t: (5.32)

5.1.3 Approximation by a Quadratic Function

We have now seen approximations by constants and linear functions, and we pro-
ceed by using a quadratic function. It is tempting to assume that we can simply go on
forever and increase the degree of the approximating polynomial as high as we want.
In practice, that is not a good idea. When more accuracy is needed, it is common to
glue together pieces of polynomials of rather low degree. It turns out that this gives
computations that are better suited for computers. However, quadratic polynomials
are just fine and we now consider approximations of the form