Numerical differentiation: Difference between revisions

Revision as of 05:42, 9 May 2014

Definition

Numerical differentiation refers to a method for computing the approximate numerical value of the derivative of a function at a point in the domain as a difference quotient. Explicitly, the numerical derivative $f^{'} (x)$ of a function $f$ at a point $x$ may be computed using either of these three formulas, for $h$ a sufficiently small positive real number:

Expression	Interpretation of limit as $h \to 0$
Forward difference quotient $\frac{f (x + h) - f (x)}{h}$ , comes from the forward difference form of the finite difference	The right-hand derivative $f'_{+} (x)$ . If $f$ is differentiable at $x$ , this equals the two-sided derivative $f^{'} (x)$ .
Backward difference quotient $\frac{f (x) - f (x - h)}{h}$ , comes from the backward difference form of the finite difference	The left-hand derivative $f'_{-} (x)$ . If $f$ is differentiable at $x$ , this equals the two-sided derivative $f^{'} (x)$ .
Central difference quotient $\frac{f (x + h) - f (x - h)}{2 h}$ , comes from the central difference form of the finite difference	If $f$ is differentiable at $x$ , this equals the two-sided derivative $f^{'} (x)$ . Otherwise, however, it does not have any direct interpretation as a one-sided derivative of $f$ .

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 ! Expression !! Interpretation of limit as <math>h \to 0</math>
 |-
-| Forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> || The right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
+| Forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math>, comes from the forward difference form of the [[finite difference]] || The right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
 |-
-| Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> || The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
+| Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math>, comes from the backward difference form of the [[finite difference]] || The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
 |-
-| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math> || If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
+| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math>, comes from the central difference form of the [[finite difference]] || If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
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