Page 128 - Linear Algebra Done Right

Orthogonal Projections and Minimization Problems
3
2
4
5
to the basis (1,x,x ,x ,x ,x ) of U, producing an orthonormal basis
(e 1 ,e 2 ,e 3 ,e 4 ,e 5 ,e 6 ) of U. Then, again using the inner product given
perform integrations is
by 6.39, compute P U v using 6.35 (with m = 6). Doing this computation A machine that can 115
shows that P U v is the function useful here.
3
5
6.40 0.987862x − 0.155271x + 0.00564312x ,
where the π’s that appear in the exact answer have been replaced with
a good decimal approximation.
By 6.36, the polynomial above should be about as good an approxi-
mation to sin x on [−π, π] as is possible using polynomials of degree
at most 5. To see how good this approximation is, the picture below
shows the graphs of both sin x and our approximation 6.40 over the
interval [−π, π].
1

0.5

-3 -2 -1 1 2 3

-0.5

-1
Graphs of sin x and its approximation 6.40

Our approximation 6.40 is so accurate that the two graphs are almost
identical—our eyes may see only one graph!
Another well-known approximation to sin x by a polynomial of de-
gree 5 is given by the Taylor polynomial

x 3 x 5
6.41 x − + .
3! 5!

To see how good this approximation is, the next picture shows the
graphs of both sin x and the Taylor polynomial 6.41 over the interval
[−π, π].