Given a set of k + 1 data points
where no two xj 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
The coefficients are defined as
![a_{j}:=[y_{0},\ldots ,y_{j}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40744c93b8380df3b93026cedb3e03e51e3dd2e)
where
![[y_{0},\ldots ,y_{j}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e845258d4c5470e5c7fdfdaab88ea5e9734aaf16)
is the notation for divided differences.
Thus the Newton polynomial can be written as
![N(x)=[y_{0}]+[y_{0},y_{1}](x-x_{0})+\cdots +[y_{0},\ldots ,y_{k}](x-x_{0})(x-x_{1})\cdots (x-x_{{k-1}}).](https://wikimedia.org/api/rest_v1/media/math/render/svg/b68037ee2fd3c52e2f564605fcd6308048dee2f1)
The Newton Polynomial above can be expressed in a simplified form when 
![{\begin{aligned}N(x)&=[y_{0}]+[y_{0},y_{1}]sh+\cdots +[y_{0},\ldots ,y_{k}]s(s-1)\cdots (s-k+1){h}^{{k}}\\&=\sum _{{i=0}}^{{k}}s(s-1)\cdots (s-i+1){h}^{{i}}[y_{0},\ldots ,y_{i}]\\&=\sum _{{i=0}}^{{k}}{s \choose i}i!{h}^{{i}}[y_{0},\ldots ,y_{i}]\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38f6ea4946a36e94e6aaff2deb6dafa98db65ba0)
is called the Newton Forward Divided Difference Formula.
If the nodes are reordered as {\displaystyle {x}_{k},{x}_{k-1},\dots ,{x}_{0}}
![N(x)=[y_{k}]+[{y}_{{k}},{y}_{{k-1}}](x-{x}_{{k}})+\cdots +[{y}_{{k}},\ldots ,{y}_{{0}}](x-{x}_{{k}})(x-{x}_{{k-1}})\cdots (x-{x}_{{1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0920f11e9b22550334ffcfa07157b8861597792)
If 
![{\begin{aligned}N(x)&=[{y}_{{k}}]+[{y}_{{k}},{y}_{{k-1}}]sh+\cdots +[{y}_{{k}},\ldots ,{y}_{{0}}]s(s+1)\cdots (s+k-1){h}^{{k}}\\&=\sum _{{i=0}}^{{k}}{(-1)}^{{i}}{-s \choose i}i!{h}^{{i}}[{y}_{{k}},\ldots ,{y}_{{k-i}}]\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d396dd13198ef0342ed29a4b0c2d4061b06f412)
is called the Newton Backward Divided Difference Formula.
IPUNOTES 