# Proof of expanded divided difference?

1. Feb 28, 2016

### Superposed_Cat

Hey all, since I was programming a polynomial interpolater i found it easier to use the expanded divided difference $$f[x_0 ,...,x_n] = \sum_{j=0}^{n} \frac{f(x_j)}{\Pi_{k}^{n,j \neq k} (x_j - x_k)}$$ , it works, but I can find no proof, any help/ references appreciated.

Second question: how accurate is Newton interpolating polynomial supposed to be? I gave it points from the function $$-x^5 +x^4 +x^3 +x^2 +x+1$$,

(1, 4), (2, -1),(3, -122),(4, -683),(5, -2344)

and it re-interpolated them correctly, but when I gave it the unknown point (6, -6221) it gave (6,-6101), is this error unnaturally large?

2. Feb 28, 2016

### Superposed_Cat

Just used Lagrange on the same points, it also gave -6101 for x=6, must be standard.

3. Feb 28, 2016

### MrAnchovy

5 points are interpolated by a quartic: there is no information to derive the coefficient of x5 so extrapolation outside the rnage of the data points is inaccurate to an arbitrary extent.