In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. Divided differences is a recursive division process. Given a sequence of data points, the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.

Definition

Given n + 1 data points where the x_k are assumed to be pairwise distinct, the forward divided differences are defined as: To make the recursive process of computation clearer, the divided differences can be put in tabular form, where the columns correspond to the value of j above, and each entry in the table is computed from the difference of the entries to its immediate lower left and to its immediate upper left, divided by a difference of corresponding x-values:

Notation

Note that the divided difference depends on the values and, but the notation hides the dependency on the x-values. If the data points are given by a function f, one sometimes writes the divided difference in the notation Other notations for the divided difference of the function ƒ on the nodes x0, ..., xn are:

Example

Divided differences for k=0 and the first few values of j: Thus, the table corresponding to these terms upto two columns has the following form:

Properties

Matrix form

The divided difference scheme can be put into an upper triangular matrix: Then it holds

Polynomials and power series

The matrix contains the divided difference scheme for the identity function with respect to the nodes, thus J^m contains the divided differences for the power function with exponent m. Consequently, you can obtain the divided differences for a polynomial function p by applying p to the matrix J: If and then This is known as Opitz' formula. Now consider increasing the degree of p to infinity, i.e. turn the Taylor polynomial into a Taylor series. Let f be a function which corresponds to a power series. You can compute the divided difference scheme for f by applying the corresponding matrix series to J: If and then

Alternative characterizations

Expanded form

With the help of the polynomial function this can be written as

Peano form

If and n\geq 1, the divided differences can be expressed as where f^{(n)} is the n-th derivative of the function f and B_{n-1} is a certain B-spline of degree n-1 for the data points, given by the formula This is a consequence of the Peano kernel theorem; it is called the Peano form of the divided differences and B_{n-1} is the Peano kernel for the divided differences, all named after Giuseppe Peano.

Forward and backward differences

When the data points are equidistantly distributed we get the special case called forward differences. They are easier to calculate than the more general divided differences. Given n+1 data points with the forward differences are defined as whereas the backward differences are defined as: Thus the forward difference table is written as:whereas the backwards difference table is written as: The relationship between divided differences and forward differences is whereas for backward differences: