Divided differences

Definition

For a set of pairs of data points

Given $k+1$ data points:

$(x_0,y_0),(x_1,y_1),\dots ,(x_k,y_k)$

The forward divided differences are defined as:

$[y_{\nu }]:=y_{\nu },\nu \in \{0,\dots ,k\}$

$[y_{\nu },\dots ,y_{\nu +j}]:=\frac{[y_{\nu +1},\dots ,y_{\nu +j}]-[y_{\nu },\dots ,y_{\nu +j-1}]}{x_{\nu +j}-x_{\nu }},\nu \in \{0,\dots ,k-j\},j\in \{1,\dots ,k\}.$

The backward divided differences are defined as:

$[y_{\nu }]:=y_{\nu },\nu \in \{0,\dots ,k\}$

$[y_{\nu },\dots ,y_{\nu -j}]:=\frac{[y_{\nu },\dots ,y_{\nu -j+1}]-[y_{\nu -1},\dots ,y_{\nu -j}]}{x_{\nu }-x_{\nu -j}},\nu \in \{j,\dots ,k\},j\in \{1,\dots ,k\}.$

For a function

Suppose $f$ is a function and $x_0,x_1,\dots ,x_k$ are points in the domain of $f$. The (forward or backward) divided difference of $f$ for these points, denoted in any of these ways: $[x_0,x_1,\dots ,x_k]f$, $[x_0,x_1,\dots ,x_k;f]$, $D[x_0,x_1,\dots ,x_k]f$, is defined as the (forward or backward respectively) divided difference for the set of pairs:

$(x_0,f(x_0)),(x_1,f(x_1)),\dots ,(x_k,f(x_k))$