= Symmetry of divided differences
{title2=$f[x_{\sigma(0)},\ldots,x_{\sigma(k)}]=f[x_0,\ldots,x_k]$}
A <divided difference> depends on the chosen <interpolation nodes> and their data, but is unchanged by any <permutation> $\sigma$ of those same <interpolation nodes>. The explicit <sum> in <divided difference> makes this visible: a <permutation> merely rearranges its summands.<NIST Digital Library of Mathematical Functions/3.3 Interpolation>[NIST Digital Library of Mathematical Functions; §3.3(iii), equation 3.3.35: symmetric explicit expression]{ref}
There is also a short explanation using <Lagrange cardinal polynomials>. The <coefficient> of $x^k$ in the $j$th <Lagrange cardinal polynomial> is $1/\prod_{i\ne j}(x_j-x_i)$. Taking that <coefficient> in their weighted <sum> gives exactly the explicit <divided difference> formula. Since <polynomial interpolation> has a unique answer for a fixed set of distinct <interpolation nodes>, this <coefficient> cannot depend on their ordering. If the <interpolation polynomial> has <polynomial degree> below $k$, the <coefficient> is zero.
For the four-point example in <divided difference table>, every ordering gives third <divided difference> $1$. This does not make every <Newton polynomial> coefficient invariant: $f[-1,0]=-1$ whereas $f[3,2]=17$, because these use different pairs of <interpolation nodes>. Even at second order, $f[-1,0,2]=1$ and $f[3,2,0]=5$ use different triples. The <node order in Newton interpolation> determines which subsets occur along the top row.
Back to article page
