Editing Divided differences (section)

==Definition==
Given ''n''&nbsp;+&nbsp;1 data points
<math display="block">(x_0, y_0),\ldots,(x_{n}, y_{n})</math>
where the <math>x_k</math> are assumed to be pairwise distinct, the '''forward divided differences''' are defined as:
<math display="block">\begin{align}
\mathopen[y_k] &:= y_k, && k \in \{ 0,\ldots,n\} \\
\mathopen[y_k,\ldots,y_{k+j}] &:= \frac{[y_{k+1},\ldots , y_{k+j}] - [y_{k},\ldots , y_{k+j-1}]}{x_{k+j}-x_k}, && k\in\{0,\ldots,n-j\},\ j\in\{1,\ldots,n\}.
\end{align}</math>

To make the recursive process of computation clearer, the divided differences can be put in tabular form, where the columns correspond to the value of ''j'' above, and each entry in the table is computed from the difference of the entries to its immediate lower left and to its immediate upper left, divided by a difference of corresponding ''x-''values:
<math display="block">
\begin{matrix}
x_0 & y_0 = [y_0] &           &               & \\
        &       & [y_0,y_1] &               & \\
x_1 & y_1 = [y_1] &           & [y_0,y_1,y_2] & \\
        &       & [y_1,y_2] &               & [y_0,y_1,y_2,y_3]\\
x_2 & y_2 = [y_2] &           & [y_1,y_2,y_3] & \\
        &       & [y_2,y_3] &               & \\
x_3 & y_3 = [y_3] &           &               & \\
\end{matrix}
</math>

=== Notation ===
Note that the divided difference <math>[y_k,\ldots,y_{k+j}]</math> depends on the values <math>x_k,\ldots,x_{k+j}</math> and <math>y_k,\ldots,y_{k+j}</math>, but the notation hides the dependency on the ''x''-values. If the data points are given by a function ''f'',
<math display="block">(x_0, y_0), \ldots, (x_{k}, y_n)
=(x_0, f(x_0)), \ldots, (x_n, f(x_n))</math>
one sometimes writes the divided difference in the notation
<math display="block">f[x_k,\ldots,x_{k+j}]
\ \stackrel{\text{def}}= \  [f(x_k),\ldots,f(x_{k+j})] 
= [y_k,\ldots,y_{k+j}].</math>Other notations for the divided difference of the function ''&fnof;'' on the nodes ''x''<sub>0</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> are:
<math display="block">f[x_k,\ldots,x_{k+j}]=\mathopen[x_0,\ldots,x_n]f= 
\mathopen[x_0,\ldots,x_n;f]=
D[x_0,\ldots,x_n]f.</math>