Category:Backward Difference Operator

This category contains results about Backward Difference Operator.
Definitions specific to this category can be found in Definitions/Backward Difference Operator.

$\nabla \map f {x_r} := \map f {x_r} - \map f {x_{r - 1} }$

for $r = 1, 2, \ldots, n$

$\ds \map {\nabla^2 f} {x_r}$

$=$

$\ds \map \nabla {\map {\nabla f} {x_r} }$

$\ds $

$=$

$\ds \nabla \map f {x_r} - \Delta \map f {x_{r - 1} }$

$\ds $

$=$

$\ds \map f {x_r} - 2 \Delta \map f {x_{r - 1} } + \Delta \map f {x_{r - 2} }$

for $r = 2, 3, \ldots, n$

$\ds \map {\nabla^k f} {x_i}$

$=$

$\ds \map \nabla {\map {\nabla^{k - 1} f} {x_i} }$

$\ds $

$=$

$\ds \nabla^{k - 1} \map f {x_i} - \nabla^{k - 1} \map f {x_{i - 1} }$

$\ds $

$=$

$\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i - s}$

for $i = k, k + 1, k + 2, \ldots, n$

Pages in category "Backward Difference Operator"

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