Definition:Finite Difference Operator

Definition

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

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

Standard Form

Forward Difference

The forward difference operator is defined as:

$\Delta f \left({x}\right) := f \left({x + 1}\right) - f \left({x}\right)$

Backward Difference

The backward difference operator is defined as:

$\nabla f \left({x}\right) := f \left({x}\right) - f \left({x - 1}\right)$

General Form

Generalized Forward Difference

The forward difference operator is defined as:

$\Delta_h f \left({x}\right) := f \left({x + h}\right) - f \left({x}\right)$

Generalized Backward Difference

The backward difference operator is defined as:

$\nabla_h f \left({x}\right) := f \left({x}\right) - f \left({x - h}\right)$

Central Difference

The central difference operator is defined as:

$\delta_h f \left({x}\right) := f \left({x + \dfrac h 2}\right) - f \left({x - \dfrac h 2}\right)$

Also see

Compare with derivative.