Forward Difference Notation

Using the forward difference notation and or , Aitken's Method can be written as .

The advantage of this notation is most noticeable for calculations by hand, where the forward differences can easily be calculated in table form:

> f:=x->exp(x)-x-1;

> p[0]:=1.0;

> for i from 1 to 10 do p[i]:=p[i-1]-f(p[i-1])/(D(f)(p[i-1])): Deltap[i]:=p[i+1]-p[i]: Delta2p[i]:=Deltap[i+1]-Deltap[i]: if i<= 8 then print(i,p[i],Deltap[i],Delta2p[i]); fi; od:

> for i from 1 to 8 do pe1[i]:=p[i]-(p[i+1]-p[i])^2/(p[i+2]-2*p[i+1]+p[i]):pe2[i]:=p[i]-(Deltap[i])^2/Delta2p[i]: print(i,pe1[i],pe2[i]); od:

The calculation of the forward difference table only involves taking differences. The approximation for each value of i is then found by a much simpler calculation.