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!