Let x0, x1, x2,…, xn be equally spaced values, so that xi=x0 +ih, for i=1, 2,…, n. Suppose that the values f0, f1, …, fn are known, where fi=f(xi), for some function f. The first differences are defined, for i=0, 1, 2,…, n−1, by Δfi =fi+1−fi. The second differences are defined by Δ2fi=Δ (Δ fi)=Δ fi+1−Δ fi and, in general, the k-th differences are defined by Δkfi=Δ (Δk−1fi)=Δk−1fi+1−Δk−1fi. For a polynomial of degree n, the (n+1)-th differences are zero.
These finite differences may be displayed in a table, as in the following example. Alongside it is a numerical example.With such tables it should be appreciated that if the values f0, f1, f2,…, fn are rounded values then increasingly serious errors result in the succeeding columns.
Numerical methods using finite differences have been extensively developed. They may be used for interpolation, as in the Gregory–Newton forward difference formula, for finding a polynomial that approximates to a given function, or for estimating derivatives from a table of values.