Finite difference: Difference between revisions

Revision as of 05:23, 9 May 2014

Given a function $f$ , a finite difference for $f$ with parameters real numbers $a$ and $b$ is the function:

$x \mapsto f (x + b) - f (x + a)$

The quotient of this by the value $b - a$ is a difference quotient expression.

There are three main types of finite differences parametrized by a positive real number $h$

Name	Expression	Value of $a$	Value of $b$	Limit as $h \to 0^{+}$
forward difference	$f (x + h) - f (x)$	0	$h$	right-hand derivative $f'_{+} (x)$ . If $f$ is differentiable at $x$ , then this equals $f^{'} (x)$ .
backward difference	$f (x) - f (x - h)$	$- h$	0	left-hand derivative $f'_{-} (x)$ . If $f$ is differentiable at $x$ , then this equals $f^{'} (x)$ .
central difference	$f (x + \frac{h}{2}) - f (x - \frac{h}{2})$	$- h$	$h$	If $f$ is differentiable at $x$ , then this equals $f^{'} (x)$ .

@@ Line 14: / Line 14: @@
 | forward difference || <math>f(x + h) - f(x)</math> || 0 || <math>h</math> || right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
 |-
-| backward difference || <math>f(x) - f(x - h)</math> || -h</math> || 0 || left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
+| backward difference || <math>f(x) - f(x - h)</math> || <math>-h</math> || 0 || left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
 |-
-| central difference || <math>f(x + h) - f(x - h)</math> || <math>-h</math> || <math>h</math> || If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
+| central difference || <math>f(x + \frac{h}{2}) - f(x - \frac{h}{2})</math> || <math>-h</math> || <math>h</math> || If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
 |}
 ==See also==
 * [[Higher-order finite difference]]