Definition:Finite Difference Operator

Definition

Let $f: \R \to \R$ be a real function.

Let $y = \map f x$ have known values:

$y_k = \map f {x_k}$

for $x_k \in \set {x_0, x_1, \ldots, x_n}$ defined as:

$x_k = x_0 + k h$

for some $h \in \R_{>0}$.

The (finite) difference operator on $f$ comes in a number of forms, as follows.

First Forward Difference Operator

The first forward difference operator on $f$ is defined as:

$\Delta \map f {x_i} := \map f {x_{i + 1} } - \map f {x_i}$

for $i = 0, 1, 2, \ldots, n - 1$

Second Forward Difference Operator

The second forward difference operator on $f$ is defined as:

\(\ds \map {\Delta^2 f} {x_i}\)

\(=\)

\(\ds \map \Delta {\map {\Delta f} {x_i} }\)

\(\ds \)

\(=\)

\(\ds \Delta \map f {x_{i + 1} } - \Delta \map f {x_i}\)

\(\ds \)

\(=\)

\(\ds \map f {x_{i + 2} } - 2 \Delta \map f {x_{i + 1} } + \Delta \map f {x_i}\)

for $i = 0, 1, 2, \ldots, n - 2$

$k$th Forward Difference Operator

The $k$th forward difference operator on $f$ is defined as:

\(\ds \map {\Delta^k f} {x_i}\)

\(=\)

\(\ds \map \Delta {\map {\Delta^{k - 1} f} {x_i} }\)

\(\ds \)

\(=\)

\(\ds \Delta^{k - 1} \map f {x_{i + 1} } - \Delta^{k - 1} \map f {x_i}\)

\(\ds \)

\(=\)

\(\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i + s}\)

for $i = 0, 1, 2, \ldots, n - k$

First Backward Difference Operator

The first backward difference operator on $f$ is defined as:

$\nabla \map f {x_r} := \map f {x_r} - \map f {x_{r - 1} }$

for $r = 1, 2, \ldots, n$

Second Backward Difference Operator

The second backward difference operator on $f$ is defined as:

\(\ds \map {\nabla^2 f} {x_r}\)

\(=\)

\(\ds \map \nabla {\map {\nabla f} {x_r} }\)

\(\ds \)

\(=\)

\(\ds \nabla \map f {x_r} - \Delta \map f {x_{r - 1} }\)

\(\ds \)

\(=\)

\(\ds \map f {x_r} - 2 \Delta \map f {x_{r - 1} } + \Delta \map f {x_{r - 2} }\)

for $r = 2, 3, \ldots, n$

$k$th Backward Difference Operator

The $k$th backward difference operator on $f$ is defined as:

\(\ds \map {\nabla^k f} {x_i}\)

\(=\)

\(\ds \map \nabla {\map {\nabla^{k - 1} f} {x_i} }\)

\(\ds \)

\(=\)

\(\ds \nabla^{k - 1} \map f {x_i} - \nabla^{k - 1} \map f {x_{i - 1} }\)

\(\ds \)

\(=\)

\(\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i - s}\)

for $i = k, k + 1, k + 2, \ldots, n$

First Central Difference Operator

The first central difference operator on $f$ is defined as:

$\delta_{i + 1/2} := \map f {x_i + \dfrac h 2} - \map f {x_i - \dfrac h 2}$

for $i = 1, 2, \ldots, n - 1$

Also see

Compare with derivative.

• Results about finite difference operators can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): finite differences
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): finite differences