High degree interpolation polynomial in Newton form

Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.