Lab for Chebyshev Polynomial Approximation

Module for Chebyshev Polynomials

Background. Chebyshev Approximation. To construct and evaluate the Chebyshev interpolating polynomial for f(x),
of degree n over the interval , where is based on the nodes
.

In practice, all we need to know is that the Chebyshev polynomial uses the interpolation nodes
. It will suffice to construct these data points and
use Mathematica's built in "Fit" procedure.

Load in Mathematica's graphics package "Colors".

Report to be handed in.

Computer Exercises
 




 

Exercise 1. Investigate the Chebyshev polynomial of degree n = 4 for .

Be assured that our method of constructing the Chebyshev polynomial is consistent with the traditional method.
Indeed, Mathematica has a built in procedure for this construction. It is invoked with the following command:

Find the error for the Chebyshev approximation.

Find the error bound for the Chebyshev approximation.
The error at the right endpoint is:

The error at the local minimum is:

The error bound for the Chebyshev polynomial is:

Construct the Taylor polynomial approximation for f[x] of degree n = 4,
and compare the error bounds.

Find the error for the Taylor approximation.

Find the error bound for the Taylor approximation.
The error at the left endpoint is:

The error at the right endpoint is:

The error bound for the Taylor polynomial is:

The error for the Chebyshev polynomial is about 6% of the error for the Taylor polynomial,
as shown by the following computation.

What do you conclude ?

(c) John H. Mathews, 1998