====== Finite difference coefficients ======
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===== How to calculate coefficients =====

In this example, I will calculate coefficients for DF4:

Use Taylor series:
{{ :software:algo:df41.png?250 |}}

So here:
{{ :software:algo:df42.png?450 |}}

Or in Matrix shape:
{{ :software:algo:df43.png?450 |}}

Here, we are looking for first derivative, so f_n^1. We only need to invert system to get coefficients. Trick is to move \Delta_x^k on right vector. Resulting matrix is then easy to solve. At the end, we have:

{{ :software:algo:df44.png?450 |}}

So for our derivative f_n^1:
{{ :software:algo:df45.png?300 |}}

With the same method, it is possible to get coefficients for all type of derivative, centered and uncentered.

First derivative:
{{ :software:algo:df46.png?500 |}}

Second derivative:
{{ :software:algo:df47.png?500 |}}


===== Coefficients =====

==== Central finite difference ====

This table contains the coefficients of the central differences, for several orders of accuracy.

^ 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 |
| 5 | 2 |   | -1/2 | 2 | -5/2 | 0| 5/2| -2 | 1/2 |   |

==== Forward and backward finite difference ====

This table contains the coefficients of the forward differences, for several order of accuracy.

^ 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 |

Backward can be obtained by inverting signs.

(source : http://en.wikipedia.org/wiki/Finite_difference_coefficient)