Higher-order finite difference: Difference between revisions

Latest revision as of 05:38, 9 May 2014

Definition

A higher-order finite difference for a function $f$ might refer to a function obtained by iterating one of the three standard forms of the finite difference: the forward difference, backward difference, or central difference. For a function $f$ at a point $x$ in the domain, and a positive integer $n$ , we have three cases:

Name	Symbol	Expression
$n^{t h}$ order forward difference	$Δ_{h}^{n} [f] (x)$	$\sum_{i = 0}^{n} (- 1)^{i} (\binom{n}{i}) f (x + (n - i) h)$
$n^{t h}$ order backward difference	$\nabla_{h}^{n} [f] (x)$	$\sum_{i = 0}^{n} (- 1)^{i} (\binom{n}{i}) f (x - i h)$
$n^{t h}$ order central difference	$δ_{h}^{n} [f] (x)$	$\sum_{i = 0}^{n} (- 1)^{i} (\binom{n}{i}) f (x + (\frac{n}{2} - i) h)$

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 ==Definition==
-A '''higher-order finite difference''' for a function <math>f</math> might refer to a function obtained by iterating the forward difference, backward difference, or central difference. For a function <math>f</math> at a point <math>x</math> in the domain, and a positive integer <math>n</math>, we have three cases:
+A '''higher-order finite difference''' for a function <math>f</math> might refer to a function obtained by iterating one of the three standard forms of the [[finite difference]]: the forward difference, backward difference, or central difference. For a function <math>f</math> at a point <math>x</math> in the domain, and a positive integer <math>n</math>, we have three cases:
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