Gauss's Forward Formula

Author

Eric W. Weisstein
September 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumericalMath/GausssForwardFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GausssForwardFormula.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

g[p_,n_?EvenQ]:=Binomial[p+n/2-1,n];
g[p_,n_?OddQ]:=Binomial[p+(n-1)/2,n]

δ[f_,p_,p0_,ord_Integer]:=DifferenceDelta[f/.p->p-ord/2,{p,ord}]/.p->p0

GaussForward[f_,p_,ord_]:=f[0]+p δ[f[x],x,1/2,1]+Total[Table[g[p,n] δ[f[x],x,Mod[n,2]/2,n],{n,2,ord}]]

Plot

Plot[{GaussForward[Exp[#]&,x,20],Exp[x]}//Evaluate,{x,0,1}]