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Proof of expanded divided difference?

  1. Feb 28, 2016 #1
    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. jcsd
  3. Feb 28, 2016 #2
    Just used Lagrange on the same points, it also gave -6101 for x=6, must be standard.
  4. Feb 28, 2016 #3
    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.
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