Finite difference coefficient

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

${\displaystyle f^{‴}(x_0)\approx \frac{-\frac{1}{2}f(x_{-2})+f(x_{-1})-f(x_{+1})+\frac{1}{2}f(x_{+2})}{h_x^3}+O\left(h_x^2\right),}$

where ${\displaystyle h_x}$ represents a uniform grid spacing between each finite difference interval, and ${\displaystyle x_n=x_0+n{h}_x}$.

For the ${\displaystyle m}$-th derivative with accuracy ${\displaystyle n}$, there are ${\displaystyle 2p+1=2\lfloor \frac{m+1}{2}\rfloor -1+n}$ central coefficients ${\displaystyle a_{-p},a_{-p+1},...,a_{p-1},a_p}$. These are given by the solution of the linear equation system

${\displaystyle \left(\begin{array}{ccccc}1& 1& ...& 1& 1\\ -p& -p+1& ...& p-1& p\\ (-p{)}^2& (-p+1{)}^2& ...& (p-1{)}^2& p^2\\ ...& ...& ...& ...& ...\\ ...& ...& ...& ...& ...\\ ...& ...& ...& ...& ...\\ (-p{)}^{2p}& (-p+1{)}^{2p}& ...& (p-1{)}^{2p}& p^{2p}\end{array}\right)\left(\begin{array}{c}{a}_{-p}\\ a_{-p+1}\\ a_{-p+2}\\ ...\\ ...\\ ...\\ a_p\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\\ ...\\ m!\\ ...\\ 0\end{array}\right),}$

where the only non-zero value on the right hand side is in the ${\displaystyle (m+1)}$-th row.

An open source implementation for calculating finite difference coefficients of arbitrary derivates and accuracy order in one dimension is available.
Given that the left-hand side matrix ${\displaystyle {\mathbf{J}}^T}$ is a transposed Vandermonde matrix, a rearrangement reveals that the coefficients are basically computed by fitting and deriving a ${\displaystyle 2p}$-th order polynomial to a window of ${\displaystyle 2p+1}$ points. Consequently, the coefficients can also be computed as the ${\displaystyle m}$-th order derivative of a fully determined Savitzky–Golay filter with polynomial degree ${\displaystyle 2p}$ and a window size of ${\displaystyle 2p+1}$. For this, open source implementations are also available. There are two possible definitions which differ in the ordering of the coefficients: a filter for filtering via discrete convolution or via a matrix-vector-product. The coefficients given in the table above correspond to the latter definition.

The theory of Lagrange polynomials provides explicit formulas for the finite difference coefficients. For the first six derivatives we have the following:

where ${\displaystyle H_{n,m}}$ are generalized harmonic numbers.