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:

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

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

The backward divided differences are defined as:

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

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

Alternative closed form expression

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

$[y_{0},y_{1},\dots ,y_{k}]=\sum _{i=0}^{k}\left({\frac {y_{i}}{\prod _{j\neq i}(x_{i}-x_{j})}}\right)$

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

$[x_{0},x_{1},\dots ,x_{k};f]=\sum _{i=0}^{k}\left({\frac {f(x_{i})}{\prod _{j\neq i}(x_{i}-x_{j})}}\right)$

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	$x\mapsto a\Delta f(x)$	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