← Back to top
Finite difference coefficients
Here the is a list of the finite difference coefficients. All rows have been aligned by the point around which the derivative has been approximated. You can regard this page as an extension to the coefficients listed
on Wikipedia.
1st derivative
-1 1
-1 1
1/2 -2 3/2
-1/2 0 1/2
-3/2 2 -1/2
-1/3 3/2 -3 11/6
1/6 -1 1/2 1/3
-1/3 -1/2 1 -1/6
-11/6 3 -3/2 1/3
1/4 -4/3 3 -4 25/12
-1/12 1/2 -3/2 5/6 1/4
1/12 -2/3 0 2/3 -1/12
-1/4 -5/6 3/2 -1/2 1/12
-25/12 4 -3 4/3 -1/4
-1/5 5/4 -10/3 5 -5 137/60
1/20 -1/3 1 -2 13/12 1/5
-1/30 1/4 -1 1/3 1/2 -1/20
1/20 -1/2 -1/3 1 -1/4 1/30
-1/5 -13/12 2 -1 1/3 -1/20
-137/60 5 -5 10/3 -5/4 1/5
1/6 -6/5 15/4 -20/3 15/2 -6 49/20
-1/30 1/4 -5/6 5/3 -5/2 77/60 1/6
1/60 -2/15 1/2 -4/3 7/12 2/5 -1/30
-1/60 3/20 -3/4 0 3/4 -3/20 1/60
1/30 -2/5 -7/12 4/3 -1/2 2/15 -1/60
-1/6 -77/60 5/2 -5/3 5/6 -1/4 1/30
-49/20 6 -15/2 20/3 -15/4 6/5 -1/6
2nd derivative
1 -2 1
1 -2 1
1 -2 1
-1 4 -5 2
0 1 -2 1
1 -2 1 0
2 -5 4 -1
11/12 -14/3 19/2 -26/3 35/12
-1/12 1/3 1/2 -5/3 11/12
-1/12 4/3 -5/2 4/3 -1/12
11/12 -5/3 1/2 1/3 -1/12
35/12 -26/3 19/2 -14/3 11/12
-5/6 61/12 -13 107/6 -77/6 15/4
1/12 -1/2 7/6 -1/3 -5/4 5/6
0 -1/12 4/3 -5/2 4/3 -1/12
-1/12 4/3 -5/2 4/3 -1/12 0
5/6 -5/4 -1/3 7/6 -1/2 1/12
15/4 -77/6 107/6 -13 61/12 -5/6
137/180 -27/5 33/2 -254/9 117/4 -87/5 203/45
-13/180 31/60 -19/12 47/18 -17/12 -49/60 137/180
1/90 -1/15 1/12 10/9 -7/3 19/15 -13/180
1/90 -3/20 3/2 -49/18 3/2 -3/20 1/90
-13/180 19/15 -7/3 10/9 1/12 -1/15 1/90
137/180 -49/60 -17/12 47/18 -19/12 31/60 -13/180
203/45 -87/5 117/4 -254/9 33/2 -27/5 137/180
3th derivative
-1 3 -3 1
-1 3 -3 1
-1 3 -3 1
-1 3 -3 1
3/2 -7 12 -9 5/2
1/2 -3 6 -5 3/2
-1/2 1 0 -1 1/2
-3/2 5 -6 3 -1/2
-5/2 9 -12 7 -3/2
-7/4 41/4 -49/2 59/2 -71/4 17/4
-1/4 7/4 -11/2 17/2 -25/4 7/4
1/4 -7/4 7/2 -5/2 1/4 1/4
-1/4 -1/4 5/2 -7/2 7/4 -1/4
-7/4 25/4 -17/2 11/2 -7/4 1/4
-17/4 71/4 -59/2 49/2 -41/4 7/4
15/8 -13 307/8 -62 461/8 -29 49/8
1/8 -1 29/8 -8 83/8 -7 15/8
-1/8 1 -29/8 6 -35/8 1 1/8
1/8 -1 13/8 0 -13/8 1 -1/8
-1/8 -1 35/8 -6 29/8 -1 1/8
-15/8 7 -83/8 8 -29/8 1 -1/8
-49/8 29 -461/8 62 -307/8 13 -15/8
4th derivative
1 -4 6 -4 1
1 -4 6 -4 1
1 -4 6 -4 1
1 -4 6 -4 1
1 -4 6 -4 1
-2 11 -24 26 -14 3
-1 6 -14 16 -9 2
0 1 -4 6 -4 1
1 -4 6 -4 1 0
2 -9 16 -14 6 -1
3 -14 26 -24 11 -2
17/6 -19 107/2 -242/3 137/2 -31 35/6
5/6 -6 37/2 -92/3 57/2 -14 17/6
-1/6 1 -3/2 -2/3 7/2 -3 5/6
-1/6 2 -13/2 28/3 -13/2 2 -1/6
5/6 -3 7/2 -2/3 -3/2 1 -1/6
17/6 -14 57/2 -92/3 37/2 -6 5/6
35/6 -31 137/2 -242/3 107/2 -19 17/6