Newton Interpolation

Exercise: Inductive argument?

Example: Given points $(1,2),(3,4),(5,8)$, find a 2-degree Newton polynomial.

$[y_0,y_1]=\frac{4-2}{3-1}=2$.
$[y_1,y_2]=\frac{8-5}{4-3}=3$

$[y_0,y_1,y_2]=\frac{[y_2,y_1]-[y_1,y_0]}{x_2-x_0}=\frac{3-2}{4-1}=\frac{1}{3}$.

Runge Phenomenon

Fitting a single high-degree polynomial to many data points can lead to erroneous oscillations; the solution is to use several piece-wise low-degree polynomials!