Newton's Forward Difference Formula

Newton’s forward difference formula is “finite difference” identity capable of giving an interpolated value between the tabulated points {f_k} in terms of the first value f₀ and powers of the “forward difference Δ”.  

For

x_k ε [0. 1]

Let h be the step length in the given data,

In terms of the divided differences, we have the interpolation formula as

Using the relations for the divided differences

we get,

..............................1.1

This relation is called the Newton’s forward difference interpolation formula.

Suppose that we want to interpolate near the point x₀ . Set x = x₀ sh. Then

Therefore,

Substituting in (1.1), we obtain

..........................1.2

We note that the coefficients are the binomial coefficients sC₀ , sC₁,..., sC_n .

Hence, we can write the formula (1.2) as

Note that

This is an alternate form of the Newton’s forward difference interpolation formula.