1. Visualizing the Error of Approximation of Interpolating Polynomials
We consider a few actual examples, showing the function, the interval, and the number of subdivision of the interval into equal parts. For some the approximations are very, very good, but others the approximations are poor.
All calculations and plots were generated with Maple. Mathematical text was generated with Scientific Notebook.
©2013 G. Donald Allen

2. The Set Up

3. TheQuestion

4. Error Dependency

5. The Basic Theorem
There is a fundamental theorem here, that tells all provided the function has enough continuous derivative. So, it will not apply to functions with corners, such as the absolute value function.

6. The Basic Theorem
We need to estimate the right side of this inequality.

7. Estimating

9. Note
It actually turns out that using equally spaced points is not necessarily the best strategy for interpolation.
It may also be that using a polynomial of high degree is not the best method for interpolation. A powerful competitor is the spline.
We save these topics for another day.

10. Examples We show the function f(x) The interval [a,b] The values of n.
Graphs of the function and interpolants.

11. Examples Original function in blue Interpolant in red
Note, the interpolant is so accurate, there is no apparent blue image.

12. The higher derivatives
For the previous example, look at the higher derivatives divided by n! As you can see they are bounded. This means we have good control of the error.

13. Example Original function in blue Interpolant in red
Note, the interpolant is so accurate, there is no apparent blue image.

14. Example
Note, the interpolant is not accurate at all, and apparently becoming more inaccurate near the endpoints.

15. The higher derivatives
For the previous example, look at the higher derivatives divided by n! – up to n= As you can see they are wild and going unbounded. This implies there may be little control of the error of approximation.
Below is the 12th derivative – messy!

16. Example
Very poor behavior near the endpoints. This function is not differentiable at zero. Thus we have little knowledge of the error on the basis of our theorem.