The Newton polynomial is supposed to be numerically stable. But I can't really see why.
Having
$\sum_{j=0}^{k} a_{j} n_{j}(x)$
with
$a_j = y[t_0, \dots, t_j ] = [y_\nu,\ldots,y_{\nu+j}] = \frac{[y_{\nu+1},\ldots , y_{\nu+j}] - [y_{\nu},\ldots , y_{\nu+j-1}]}{x_{\nu+j}-x_\nu}$
Choosing the divided differences in a bad way i.e. $y[t_0,t_1]=\frac{y[t_1]-y[t_0]}{t_1-t_0}$ with $t_0 \approx t_1$ would create numerical instability wouldn't it?