Editing Polynomial interpolation (section)

==== Newton forward formula ====
The Newton polynomial can be expressed in a simplified form when <math>x_0, x_1, \dots, x_k</math> are arranged consecutively with equal spacing.

If <math>x_0, x_1, \dots, x_k</math> are consecutively arranged and equally spaced with <math>{x}_{i}={x}_{0}+ih  </math> for ''i'' = 0, 1, ..., ''k'' and some variable x is expressed as <math>{x}={x}_{0}+sh</math>, then the difference <math>x-x_i</math> can be written as <math>(s-i)h</math>. So the Newton polynomial becomes

: <math>\begin{align}
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{align}</math>

Since the relationship between divided differences and [[Divided differences#Forward and backward differences|forward differences]] is given as:<ref>{{cite book |last1=Burden |first1=Richard L. |url=https://archive.org/details/numericalanalysi00rlbu |title=Numerical Analysis |last2=Faires |first2=J. Douglas |date=2011 |isbn=9780538733519 |edition=9th |page=[https://archive.org/details/numericalanalysi00rlbu/page/n146 129] |publisher=Cengage Learning |url-access=limited}}</ref><math display="block">[y_j, y_{j+1}, \ldots , y_{j+n}] = \frac{1}{n!h^n}\Delta^{(n)}y_j,</math>Taking <math>y_i=f(x_i)</math>, if the representation of x in the previous sections was instead taken to be <math>x=x_j+sh</math>, the '''Newton forward interpolation formula''' is expressed as:<math display="block">f(x) \approx N(x)=N(x_j+sh) = \sum_{i=0}^{k}{s \choose i}\Delta^{(i)} f(x_j)
</math>which is the interpolation of all points after <math>x_j</math>. It is expanded as:<math display="block">f(x_j+sh)=f(x_j)+\frac{s}{1!}\Delta f(x_j)+ \frac{s(s-1)}{2!}\Delta^2 f(x_j)+\frac{s(s-1)(s-2)}{3!}\Delta^3 f(x_j)+\frac{s(s-1)(s-2)(s-3)}{4!}\Delta^4 f(x_j)+\cdots                      
</math>