Equidistant data points
If the data points are equidistant, i.e.
, then Newton's divided difference formula can be rewritten. Indeed, if you write
, with
a real number, then
and so, e.g.,
.
Newton's divided difference formula can then be written as
.
Or, using binomial coefficients:
> matrix(2,1,[s,k])=(s*(s-1)*(s-2)*`...`*(s-k+1))/k!;
so that
> P[n](x[0]+s*h)=Sum(matrix(2,1,[s,k])*k!*h^k*f*[x[0],x[1],`...`,x[k]],k=0..n);
In addition, one can write the divided differences using forward difference notation:
,
or
.
In general,
.
Newton's divided difference formula can then be written as
> P[n](x)=Sum(matrix(2,1,[s,k])*Delta^k*f(x[0]),k=0..n);
But do remember that this version only applies to equidistant points!