# Finite difference coefficients

## How to calculate coefficients

In this example, I will calculate coefficients for DF4:

Use Taylor series:

So here:

Or in Matrix shape:

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:

So for our derivative f_n^1:

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

First derivative:

Second derivative:

## 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)