The Method of Least Squares - Elements of Scientific Computing - page 180

defined on 0 t 1. The linear system ( 5.116 ) now reads
1

! ˛
ˇ

D e C e 1 2
2e 1

1
2

(5.121)

1
2

1
3

and the solution is

˛ D 4e 8e 1 8 0:070;
ˇ D 18e 1 6e C 12 2:312:

Hence, the linear least squares approximation of ( 5.120 )isgivenby

p.t/ D 0:070 C 2:312 t: (5.122)

Example 5.6. Consider

1
10 cos.t /

y.t/ D t 2 C

(5.123)

defined on 0 t 1. The linear system ( 5.116 ) then reads
1 2
1
2

! ˛
ˇ

D

!

1
3

C 10 sin.1/

1
3

3
20

C 10 cos.1/ C 10

sin.1/

and thus

˛ 0:059;
ˇ 0:953;

and we conclude that the linear least squares approximation of ( 5.123 )isgivenby

p.t/ 0:059 C 0:953 t: (5.124)

5.3.3 Approximation Using Quadratic Functions

We seek a quadratic function

p.t/ D ˛ C ˇ C t 2

(5.125)