Newton form

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

(The other difference formulas, such as those of Gauss, Bessel and Stirling, can be derived from Newton's, by renaming of the x-values of the data points.)

For any given finite set of data points, there is only one polynomial, of least possible degree, that passes through all of them. Thus, it is more appropriate to speak of "the Newton form of the interpolation polynomial" rather than of "the Newton interpolation polynomial". Like the Lagrange form, it is merely another way to write the same polynomial.

Given a set of k + 1 data points

where no two x_j are the same, the interpolation polynomial in the Newton form is a linear combination of Newton basis polynomials

with the Newton basis polynomials defined as

for j > 0 and ${\displaystyle n_{0}(x)\equiv 1}$.

The coefficients are defined as

where

is the notation for divided differences.

Thus the Newton polynomial can be written as

The Newton Polynomial above can be expressed in a simplified form when ${\displaystyle x_{0},x_{1},\dots ,x_{k}}$ are arranged consecutively with equal space. Introducing the notation ${\displaystyle h=x_{i+1}-x_{i}}$ for each ${\displaystyle i=0,1,\dots ,k-1}$ and ${\displaystyle x=x_{0}+sh}$, the difference ${\displaystyle x-x_{i}}$ can be written as ${\displaystyle (s-i)h}$. So the Newton Polynomial above becomes:

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