The error in polynomial interpolation

Theorem 8.4.1 in Greenbaum and Chartier states the following.


Assume $f$ is $n+1$ times continuously differentiable in a region $[a,b]$, and that $x_0, \ldots, x_n$ are distinct points in $[a,b]$.

Let $p(x)$ be the unique polynomial of degree $n$ that interpolates $f$ and $x_0, \ldots, x_n$.

Then $$f(x) - p(x) = \frac{1}{(n+1)!} f^{(n+1)}(\xi_x) \prod_{j=0}^n (x-x_i)$$ for some point $\xi_x$ in $[a,b]$ that depends on $x$.

This theorem gives us three terms that are fighting with each other: $$|f(x) - p(x)| \le \frac{1}{(n+1)!} \cdot (\text{worst derivative}) \cdot (\text{polynomial on points}) The first term $1/(n+1)!$ is going down with $n$, so we want $n$ to be large. The second term $f^{(n+1)}$ says we want the derivatives to be controlled, so this will be bad if we have large derivatives. The final term depends on the points $xi$ that are used to interpolate. We want the largest value of $\prod{j=0}^n (x-x_i)$ to be small.

Which part of the Runge function makes the approximation bad?

Recall that the Runge function is $$ f(x) = \frac{1}{1+x^2} $$ and this results in an extremely bad functional approximation when we use uniformly spaced points. So we want to understand what is causing that.

In [1]:
using Plots
using ApproxFun
using Interact
In [14]:
# Is it the polynomial that results from uniformly spaced points?
@manipulate for N=3:25
    xx1 = linspace(-5,5,N)
    x = Fun(x -> x, [-5.,5.])
    f = 0.0*x + 1.0
    for i=1:N
        f = f.*(x-xx1[i])
    title!(@sprintf("max = %g, N! = %g",
        maximum(abs(f)), factorial(N)))
In [15]:
# Or the derivative?
@manipulate for N=3:25
    xx1 = linspace(-5,5,N)
    x = Fun(x -> x, [-5.,5.])
    f = 1./(1+x.^2);
    d = differentiate(f,N)
    title!(@sprintf("max = %g, N! = %g",
        maximum(d), factorial(N)))