In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions.
[citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.
[1] 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.
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:
Note that the divided difference
, 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:
Thus, the table corresponding to these terms upto two columns has the following form:
The divided difference scheme can be put into an upper triangular matrix:
contains the divided difference scheme for the identity function with respect to the nodes
contains the divided differences for the power function with exponent
Consequently, you can obtain the divided differences for a polynomial function
be a function which corresponds to a power series.
You can compute the divided difference scheme for
by applying the corresponding matrix series to
, the divided differences can be expressed as[4]
This is a consequence of the Peano kernel theorem; it is called the Peano form of the divided differences and
is the Peano kernel for the divided differences, all named after Giuseppe Peano.
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.
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:
whereas for backward differences:[citation needed]