The DividedDifferenceTable command takes an interpolation structure and computes the associated divided difference table.

This command can only be used on interpolation structures that were computed with Hermite or Newton methods.

The POLYINTERP structure is created using the PolynomialInterpolation command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\:$

$\mathrm{xy}≔\left[\left[1.0\,0.7651977\right]\,\left[1.3\,0.6200860\right]\,\left[1.6\,0.4554022\right]\,\left[1.9\,0.2818186\right]\right]$

$\mathrm{xy}≔\left[\left[1.0\,0.7651977\right]\,\left[1.3\,0.6200860\right]\,\left[1.6\,0.4554022\right]\,\left[1.9\,0.2818186\right]\right]$

$\mathrm{p1}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xy}\,\mathrm{independentvar}='x'\,\mathrm{method}=\mathrm{newton}\right)\:$

$\mathrm{DividedDifferenceTable}\left(\mathrm{p1}\right)$

$\left[\begin{array}{cccc}0.7651977& 0& 0& 0\\ 0.6200860& \mathrm{-0.4837056667}& 0& 0\\ 0.4554022& \mathrm{-0.5489460000}& \mathrm{-0.1087338888}& 0\\ 0.2818186& \mathrm{-0.5786120000}& \mathrm{-0.04944333333}& 0.06587839497\end{array}\right]$

$\mathrm{p1a}≔\mathrm{AddPoint}\left(\mathrm{p1}\,\left[1.8\,0.3920223\right]\right)\:$

$\mathrm{DividedDifferenceTable}\left(\mathrm{p1a}\right)$

$\left[\begin{array}{ccccc}0.7651977& 0& 0& 0& 0\\ 0.6200860& \mathrm{-0.4837056667}& 0& 0& 0\\ 0.4554022& \mathrm{-0.5489460000}& \mathrm{-0.1087338888}& 0& 0\\ 0.2818186& \mathrm{-0.5786120000}& \mathrm{-0.04944333333}& 0.06587839497& 0\\ 0.3920223& \mathrm{-1.102037000}& \mathrm{-2.617125000}& \mathrm{-5.135363334}& \mathrm{-6.501552161}\end{array}\right]$

$\mathrm{xyyp}≔\left[\left[1\,1.105170918\,0.2210341836\right]\,\left[1.5\,1.252322716\,0.3756968148\right]\,\left[2\,1.491824698\,0.5967298792\right]\right]$

$\mathrm{xyyp}≔\left[\left[1\,1.105170918\,0.2210341836\right]\,\left[1.5\,1.252322716\,0.3756968148\right]\,\left[2\,1.491824698\,0.5967298792\right]\right]$

$\mathrm{p2}≔\mathrm{PolynomialInterpolation}\left(\mathrm{xyyp}\,\mathrm{method}=\mathrm{hermite}\,\mathrm{function}=\exp \left(0.1x^2\right)\,\mathrm{independentvar}='x'\,\mathrm{errorboundvar}='\mathrm{\xi }'\,\mathrm{digits}=5\right)\:$

$\mathrm{DividedDifferenceTable}\left(\mathrm{p2}\right)$

$\left[\begin{array}{cccccc}1.1052& 0& 0& 0& 0& 0\\ 1.1052& 0.22103& 0& 0& 0& 0\\ 1.2523& 0.29420& 0.14634& 0& 0& 0\\ 1.2523& 0.37570& 0.16300& 0.033320& 0& 0\\ 1.4918& 0.47900& 0.20660& 0.043600& 0.010280& 0\\ 1.4918& 0.59673& 0.23546& 0.057720& 0.014120& 0.0038400\end{array}\right]$