Editing Divided differences (section)

===Forward and backward differences===
{{details|Finite difference}}

When the data points are equidistantly distributed we get the special case called '''forward differences'''. They are easier to calculate than the more general divided differences.

Given ''n''+1 data points
<math display="block">(x_0, y_0), \ldots, (x_n, y_n)</math>
with
<math display="block">x_{k} = x_0 + k h,\ \text{ for } \ k=0,\ldots,n \text{ and fixed } h>0</math>
the forward differences are defined as
<math display="block">\begin{align}
\Delta^{(0)} y_k &:= y_k,\qquad k=0,\ldots,n \\
\Delta^{(j)}y_k &:= \Delta^{(j-1)}y_{k+1} - \Delta^{(j-1)}y_k,\qquad k=0,\ldots,n-j,\ j=1,\dots,n.
\end{align}</math>whereas the backward differences are defined as:
<math display="block">\begin{align}
\nabla^{(0)} y_k &:= y_k,\qquad k=0,\ldots,n \\
\nabla^{(j)}y_k &:= \nabla^{(j-1)}y_{k} - \nabla^{(j-1)}y_{k-1},\qquad k=0,\ldots,n-j,\ j=1,\dots,n.
\end{align}</math>
Thus the forward difference table is written as:<math display="block">
\begin{matrix}
y_0 &               &                   &                  \\
    & \Delta y_0 &                   &                  \\
y_1 &               & \Delta^2 y_0 &                  \\
    & \Delta y_1 &                   & \Delta^3 y_0\\
y_2 &               & \Delta^2 y_1 &                  \\
    & \Delta y_2 &                   &                  \\
y_3 &               &                   &                  \\
\end{matrix}
</math>whereas the backwards difference table is written as:<math display="block">
\begin{matrix}
y_0 &               &                   &                  \\
    & \nabla y_1 &                   &                  \\
y_1 &               & \nabla^2 y_2 &                  \\
    & \nabla y_2 &                   & \nabla^3 y_3\\
y_2 &               & \nabla^2 y_3 &                  \\
    & \nabla y_3 &                   &                  \\
y_3 &               &                   &                  \\
\end{matrix} 
</math>

The relationship between divided differences and forward differences is<ref>{{cite book|last1=Burden|first1=Richard L.| last2=Faires|first2=J. Douglas | title=Numerical Analysis |url=https://archive.org/details/numericalanalysi00rlbu |url-access=limited | date=2011|page=[https://archive.org/details/numericalanalysi00rlbu/page/n146 129]|publisher=Cengage Learning | isbn=9780538733519| edition=9th}}</ref>
<math display="block">[y_j, y_{j+1}, \ldots , y_{j+k}] = \frac{1}{k!h^k}\Delta^{(k)}y_j,    </math>whereas for backward differences:{{Citation needed|date=December 2023}}<math display="block">[{y}_{j}, y_{j-1},\ldots,{y}_{j-k}] = \frac{1}{k!h^k}\nabla^{(k)}y_j.       </math>