Equidistant data points

If the data points are equidistant, i.e. [Maple Math] , then Newton's divided difference formula can be rewritten. Indeed, if you write [Maple Math] , with [Maple Math] a real number, then [Maple Math] and so, e.g., [Maple Math] .

Newton's divided difference formula can then be written as [Maple Math] .

Or, using binomial coefficients:

> matrix(2,1,[s,k])=(s*(s-1)*(s-2)*`...`*(s-k+1))/k!;

[Maple Math]

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);

[Maple Math]

In addition, one can write the divided differences using forward difference notation: [Maple Math] , [Maple Math] or [Maple Math] .

In general, [Maple Math] .

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);

[Maple Math]

But do remember that this version only applies to equidistant points!