Resources
AMSC/CMSC 460 - Spring 2018

1. Home
2. Resources
3. Node Polynomial

Node Polynomial

Below is an interactive application for the node polynomial

\[ \omega(x) = \prod_{i=0}^n (x-x_i), \]

for interpolation points $x_0,x_1,\ldots x_n$ contained in the interval $-1,1$. The optimal bounds on $\omega(x)$, $\pm 2^{1-n}$ are shown in red. You can select between equidistant nodes and Chebyshev nodes below. Note how for Chebyshev nodes, the node polynomial satisfies

\[ \max_{x\in[-1,1]} |\omega(x)| \leq 2^{1-n}, \]

while for the equidistant nodes $\omega(x)$ has large oscillations the further away from $0$ you get. Notice how these oscillations get worse and worse as n gets larger.

Try moving the nodes around yourself to see if you can get a better bound.

---