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.

Forward Difference

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

$\map {\Delta_h f} x := \map f {x + h} - \map f x$

Backward Difference

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

$\map {\nabla_h f} x := \map f x - \map f {x - h}$

Central Difference

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

$\map {\delta_h f} x := \map f {x + \dfrac h 2} - \map f {x - \dfrac h 2}$

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