The Method of Least Squares - Elements of Scientific Computing - page 192

(f) Set D ˛=ˇ and show that ( 5.149 ) can be written in the form

y 0 .t /
y.t/

D ˛ C y.t /: (5.159)

(g) Use ( 5.153 )and( 5.159 ) to show that
ln.y.t nC1 // ln.y.t n // D ˛.t nC1 t n / C Z t nC1
t n

y.t/ dt : (5.160)

(h) Use ( 5.160 ) to show that

ln .y.t nC1 /=y.t n //
t nC1 t n 2 .y.t nC1 / C y.t n // :

˛ C (5.161)

(i) We want to reconsider the problem of computing a model of the future growth of
the world population. We therefore want to use the data in Table 5.4 on page 170
in order to determine the coefficients ˛ and in ( 5.161 ). In ( 5.161 ) we note that

t nC1 t n D 1:

In addition, we define

c n D ln .y.t nC1 /=y.t n // (5.162)

and

1
2 .y.t nC1 / C y.t n // :

d n D (5.163)

From the data in Table 5.3 , c n and d n are readily computed. Make a table con-
sisting of columns for the year, n, y.t n /, c n ,andd n , in the same style as in
Tab le 5.3 .
(j) Using the definitions ( 5.162 )and( 5.163 ) together with ( 5.161 ), we have

c n ˛ C d n ; D 0;1;:::;9; (5.164)

where c n , d n are given and where ˛ and are to be determined. We want to
compute ˛ and ˇ using the method of least squares. In order to do this, we
define

9 X

.˛ C d n c n / 2 : F.˛;/ D (5.165)

nD0

Explain why we want to compute the values of ˛ and that minimize F .