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:

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

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

Alternative closed form expression

Rather than the iterative definition, the following closed form definition is sometimes preferred for the forward divided difference:

$[y_0,y_1,\dots ,y_k]=\sum _{i=0}^k(\frac{y_i}{{\prod }_{j\ne i}(x_i-x_j)})$

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:

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

By definition, divided difference refers to the forward divided difference.

Alternative closed form expression

$[x_0,x_1,\dots ,x_k;f]=\sum _{i=0}^k(\frac{f(x_i)}{{\prod }_{j\ne i}(x_i-x_j)})$

Definition as a function

Consider a function $f$ with domain a subset $S$ of $\mathbb{R}$. Suppose $k$ is a positive integer. Denote by $[S{]}_k$ the set of $k$-tuples of pairwise distinct elements of $S$. The $k$-fold forward divided difference function is a function:

$[S{]}_k\to \mathbb{R}$

defined as:

$(x_0,x_1,\dots ,x_k)\mapsto [x_0,x_1,\dots ,x_k;f]$

Relation with operations on functions

Method for constructing new functions from old	In symbols	Difference quotient in terms of the old functions and their difference quotients	Proof
pointwise sum	$f+g$ is the function $x\mapsto f(x)+g(x)$ $f_1+f_2+\dots +f_n$ is the function $x\mapsto f_1(x)+f_2(x)+\dots +f_n(x)$	Divided difference of sum is sum of divided differences	divided differences are linear
pointwise difference	$f-g$ is the function $x\mapsto f(x)-g(x)$	Divided difference of difference is difference of divided differences	divided differences are linear
scalar multiple by a constant	$af$ is the function $x\mapsto af(x)$ where $a$ is a real number	$x\mapsto a\mathrm{\Delta }f(x)$	divided differences are linear
pointwise product	$f\cdot g$ (sometimes denoted $fg$) is the function $x\mapsto f(x)g(x)$ $f_1\cdot f_2\cdot \dots f_n$ (sometimes denoted $f_1{f}_2\dots f_n$ is the function $x\mapsto f_1(x)f_2(x)\dots f_n(x)$	See product rule for divided differences	product rule for divided differences
pointwise quotient	$f/g$ is the function $x\mapsto f(x)/g(x)$	?	?
composite of two functions	$f\circ g$ is the function $x\mapsto f(g(x))$	?	chain rule for divided differences