divided difference

Let f be a real (or complex) function. Given distinct real (or complex) numbers x0,x1,x2,, the divided differencesMathworldPlanetmath of f are defined recursively as follows:

Δ1f[x0,x1] =f(x1)-f(x0)x1-x0
Δn+1f[x0,x1,,xn+1] =Δnf[x1,x2,,xn+1]-Δnf[x0,x2,,xn+1]x1-x0

It is also convenient to define the zeroth divided difference of f to be f itself:


Some important properties of divided differences are:

  1. 1.

    Divided differences are invariant under permutationsMathworldPlanetmath of x0,x1,x2,

  2. 2.

    If f is a polynomial of order m and m<n, then the n-th divided differences of f vanish identically

  3. 3.

    If f is a polynomial of order m+n, then Δn(x,x1,,xn) is a polynomial in x of order m.

Divided differences are useful for interpolating functions when the values are given for unequally spaced values of the argument.

Becuse of the first property listed above, it does not matter with respect to which two arguments we compute the divided difference when we compute the n+1-st divided difference from the n-th divided difference. For instance, when computing the divided difference table for tabulated values of a function, a convenient choice is the following:

Title divided difference
Canonical name DividedDifference
Date of creation 2013-03-22 14:40:59
Last modified on 2013-03-22 14:40:59
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Definition
Classification msc 39A70