Finite difference: Difference between revisions

Revision as of 05:25, 9 May 2014

Definition

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

@@ Line 10: / Line 10: @@
 {| class="sortable" border="1"
-! Name !! Expression !! Value of <math>a</math> !! Value of <math>b</math> !! Limit as <math>h \to 0^+</math>
+! Name !! Symbol !! Expression !! Value of <math>a</math> !! Value of <math>b</math> !! Limit as <math>h \to 0^+</math>
 |-
-| 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>.
+| forward difference || <math>\Delta_h[f](x)</math> || <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> || <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>\nabla_h[f](x)</math> || <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\left(x + \frac{h}{2}\right) - f\left(x - \frac{h}{2}\right)</math> || <math>\frac{-h}{2}</math> || <math>\frac{h}{2}</math> || If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
+| central difference || <math>\delta_h[f](x)</math> || <math>f\left(x + \frac{h}{2}\right) - f\left(x - \frac{h}{2}\right)</math> || <math>\frac{-h}{2}</math> || <math>\frac{h}{2}</math> || If <math>f</math> is differentiable at <math>x</math>, then this equals <math>f'(x)</math>.
 |}
 ==See also==
 * [[Higher-order finite difference]]