Newton polynomial

The principle of a difference engine is Newton polynomial of divided differences.

In this form, the Newton polynomials generate the Newton series.

The Newton Polynomial above can be expressed in a simplified form when are arranged consecutively with equal space.

So the Newton Polynomial above becomes:

If the nodes are reordered as , the Newton Polynomial becomes:

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

The choice of results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.

The resulting polynomial may have degree at most n(m + 1) 1, whereas the Newton polynomial has maximum degree n 1.

The polynomials here are called Newton polynomials (not, however, the Newton polynomials of interpolation theory).

The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using divided differences.

Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use.

A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials.

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form.

In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.