## Quick Reference

Let *x*_{0}, *x*_{1}, *x*_{2},…, *x*_{n} be equally spaced values, so that *x*_{i}=*x*_{0} +*ih*, for *i*=1, 2,…, *n*. Suppose that the values *f*_{0}, *f*_{1}, …, *f*_{n} are known, where *f*_{i}=*f*(*x*_{i}), for some function *f*. The first differences are defined, for *i*=0, 1, 2,…, *n*−1, by Δ*f*_{i} =*f*_{i+1}−*f*_{i}. The second differences are defined by Δ^{2}*f*_{i}=Δ (Δ *f*_{i})=Δ *f*_{i+1}−Δ *f*_{i} and, in general, the *k*-th differences are defined by Δ^{k}*f*_{i}=Δ (Δ^{k−1}*f*_{i})=Δ^{k−1}*f*_{i+1}−Δ^{k−1}*f*_{i}. 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 *f*_{0}, *f*_{1}, *f*_{2},…, *f*_{n} 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.

*Subjects:*
Mathematics.