# 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.

## Central finite difference

This table contains the coefficients of the central differences, for several orders of accuracy:[1]

Derivative Accuracy −4 −3 −2 −1 0 1 2 3 4
1 2       −1/2 0 1/2
4     1/12 −2/3 0 2/3 −1/12
6   −1/60 3/20 −3/4 0 3/4 −3/20 1/60
8 1/280 −4/105 1/5 −4/5 0 4/5 −1/5 4/105 −1/280
2 2       1 −2 1
4     −1/12 4/3 −5/2 4/3 −1/12
6   1/90 −3/20 3/2 −49/18 3/2 −3/20 1/90
8 −1/560 8/315 −1/5 8/5 −205/72 8/5 −1/5 8/315 −1/560
3 2     −1/2 1 0 −1 1/2
4   1/8 −1 13/8 0 −13/8 1 −1/8
6 −7/240 3/10 −169/120 61/30 0 −61/30 169/120 −3/10 7/240
4 2     1 −4 6 −4 1
4   −1/6 2 −13/2 28/3 −13/2 2 −1/6
6 7/240 −2/5 169/60 −122/15 91/8 −122/15 169/60 −2/5 7/240

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

$\displaystyle f'''(x_{0}) \approx \displaystyle \frac{-\frac{1}{2}f(x_{-2}) + f(x_{-1}) -f(x_{+1}) +\frac{1}{2}f(x_{+2})}{h^3_x} + O\left(h_x^2 \right)$

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 order of accuracy:[1]

Derivative Accuracy 0 1 2 3 4 5 6 7 8
1 1 −1 1
2 −3/2 2 −1/2
3 −11/6 3 −3/2 1/3
4 −25/12 4 −3 4/3 −1/4
5 −137/60 5 −5 10/3 −5/4 1/5
6 −49/20 6 −15/2 20/3 −15/4 6/5 −1/6
2 1 1 −2 1
2 2 −5 4 −1
3 35/12 −26/3 19/2 −14/3 11/12
4 15/4 −77/6 107/6 −13 61/12 −5/6
5 203/45 −87/5 117/4 −254/9 33/2 −27/5 137/180
6 469/90 −223/10 879/20 −949/18 41 −201/10 1019/180 −7/10
3 1 −1 3 −3 1
2 −5/2 9 −12 7 −3/2
3 −17/4 71/4 −59/2 49/2 −41/4 7/4
4 −49/8 29 −461/8 62 −307/8 13 −15/8
5 −967/120 638/15 −3929/40 389/3 −2545/24 268/5 −1849/120 29/15
6 −801/80 349/6 −18353/120 2391/10 −1457/6 4891/30 −561/8 527/30 −469/240
4 1 1 −4 6 −4 1
2 3 −14 26 −24 11 −2
3 35/6 −31 137/2 −242/3 107/2 −19 17/6
4 28/3 −111/2 142 −1219/6 176 −185/2 82/3 −7/2
5 1069/80 −1316/15 15289/60 −2144/5 10993/24 −4772/15 2803/20 −536/15 967/240

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

$\displaystyle f'(x_{0}) \approx \displaystyle \frac{-\frac{11}{6}f(x_{0}) + 3f(x_{+1}) -\frac{3}{2}f(x_{+2}) +\frac{1}{3}f(x_{+3}) }{h_{x}} + O\left(h_{x}^3 \right),$
$\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{+1}) + 4f(x_{+2}) - f(x_{+3}) }{h_{x}^2} + O\left(h_{x}^2 \right),$

while the corresponding backward approximations are given by

$\displaystyle f'(x_{0}) \approx \displaystyle \frac{\frac{11}{6}f(x_{0}) - 3f(x_{-1}) +\frac{3}{2}f(x_{-2}) -\frac{1}{3}f(x_{-3}) }{h_{x}} + O\left(h_{x}^3 \right),$
$\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{-1}) + 4f(x_{-2}) - f(x_{-3}) }{h_{x}^2} + O\left(h_{x}^2 \right),$

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.