divided differences

The divided differences can be written in the form of a table.

Note that there is also a connection to divided differences.

They are easier to calculate than the more general divided differences.

Now, we create a divided differences table for the points .

It can be generalized to more variables according by the mean value theorem for divided differences.

You can represent partial fractions using the expanded form of divided differences.

In mathematics, divided differences is a recursive division process.

Divided differences of polynomials are particularly interesting, because they can benefit from the Leibniz rule.

The principle of a difference engine is Newton's method of divided differences.

Furthermore if the x are distributed equidistantly the calculation of the divided differences becomes significantly easier.

The Taylor series or any other representation with function series can in principle be used to approximate divided differences.

The forward divided differences are defined as:

When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times.

If limits of the divided differences are accepted, then this connection does also hold, if some of the coincide.

Taylor series for divided differences:

If you need to compute a whole divided difference scheme with respect to a Taylor series, see the section about divided differences of power series.

An example is the set of matrices of divided differences with respect to a fixed set of nodes.

Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences.

The name derives from the method of divided differences, a way to interpolate or tabulate functions by using a small set of polynomial coefficients.

You can generalize the mean to variables by considering the mean value theorem for divided differences for the th derivative of the logarithm.

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.

The generated Hermite polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences.

The limit of the Newton polynomial if all nodes coincide is a Taylor polynomial, because the divided differences become derivatives.

One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients, e.g. Neville's algorithm.

This prefix based approach can also be used to obtain the generalized divided differences for (confluent) Hermite interpolation as well as for parallel algorithms for Vandermonde systems.