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Finite difference coefficient

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In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward.

Contents

Central finite difference

This table contains the coefficients of the central differences, for several orders of accuracy and with uniform grid spacing:

For example, the third derivative with a second-order accuracy is

f ( x 0 ) 1 2 f ( x 2 ) + f ( x 1 ) f ( x + 1 ) + 1 2 f ( x + 2 ) h x 3 + O ( h x 2 )

where h x represents a uniform grid spacing between each finite difference interval.

Forward and backward finite difference

This table contains the coefficients of the forward differences, for several orders of accuracy and with uniform grid spacing:

For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are

f ( x 0 ) 11 6 f ( x 0 ) + 3 f ( x + 1 ) 3 2 f ( x + 2 ) + 1 3 f ( x + 3 ) h x + O ( h x 3 ) , f ( x 0 ) 2 f ( x 0 ) 5 f ( x + 1 ) + 4 f ( x + 2 ) f ( x + 3 ) h x 2 + O ( h x 2 ) ,

while the corresponding backward approximations are given by

f ( x 0 ) 11 6 f ( x 0 ) 3 f ( x 1 ) + 3 2 f ( x 2 ) 1 3 f ( x 3 ) h x + O ( h x 3 ) , f ( x 0 ) 2 f ( x 0 ) 5 f ( x 1 ) + 4 f ( x 2 ) f ( x 3 ) h x 2 + O ( h x 2 ) ,


In general, to get the coefficients of the backward approximations, give all odd derivatives listed in the table the opposite sign, whereas for even derivatives the signs stay the same. The following table illustrates this:

References

Finite difference coefficient Wikipedia


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