Example 4.  Error Analysis.  Investigate the error for the Chebyshev polynomial approximations in Example 3.

Solution 4.

4 (a).  Investigate the error for the Chebyshev interpolation polynomial ,  of degree n = 2.

Compare the maximum error with the theoretical error bound:

          

The error bound is about 1.15 times as large as the maximum error.  This is to be expected, after all it is an "error bound."

4 (b).  Investigate the error for the Chebyshev interpolation polynomial ,  of degree n = 2.

Compare the maximum error with the theoretical error bound:

          

The error bound is about 1.6 times as large as the maximum error.  This is to be expected, after all it is an "error bound."

4 (c).  Investigate the error for the Chebyshev interpolation polynomial ,  of degree n = 3.

Compare the maximum error with the theoretical error bound:

          

The error bound is about 1.11 times as large as the maximum error.  This is to be expected, after all it is an "error bound."

4 (d).  Investigate the error for the Chebyshev interpolation polynomial ,  of degree n = 4.

Compare the maximum error with the theoretical error bound:

          

The error bound is about 1.6 times as large as the maximum error.  This is to be expected, after all it is an "error bound."

4 (e).  Investigate the error for the Chebyshev interpolation polynomial ,  of degree n = 5.

Compare the maximum error with the theoretical error bound:

          

The error bound is about 1.11 times as large as the maximum error.  This is to be expected, after all it is an "error bound."

(c) John H. Mathews 2004