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Concept
Finite difference coefficient
Summary
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 and with uniform grid spacing:
For example, the third derivative with a second-order accuracy is
: f'''(x_{0}) \approx \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, and x_n = x_0 + n h_x.
For the m-th derivative with accuracy n, there are 2p + 1 = 2 \left\lfloor \frac{m+1}{2} \right\rfloor - 1 + n central coefficients a_{-p}, a_{-p+1}, ..., a_{p-1}, a_p. These are given by the solution of the linear equation system
:
\begin{
Official source
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