Editing Finite difference (section)

==Arbitrarily sized kernels==
{{main|Finite difference coefficient}}
{{further|Five-point stencil}}

Using [[linear algebra]] one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the [[Taylor expansion]] of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid.<ref>{{cite journal|last1=Fraser|first1=Duncan C.|title=On the Graphic Delineation of Interpolation Formulæ| journal=Journal of the Institute of Actuaries|date=1 January 1909|volume=43|issue=2|pages=235–241|doi=10.1017/S002026810002494X| url=https://archive.org/stream/journal43instuoft#page/236/mode/2up|access-date=17 April 2017}}</ref>
This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side.<ref>[http://commons.wikimedia.org/wiki/File:FDnotes.djvu notes]</ref>
Finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order may be constructed.<ref>[http://web.media.mit.edu/~crtaylor/calculator.html Finite Difference Coefficients Calculator]</ref>

===Properties===
* For all positive {{mvar|k}} and {{mvar|n}} <math display="block">\Delta^n_{kh} (f, x) = \sum\limits_{i_1=0}^{k-1} \sum\limits_{i_2=0}^{k-1} \cdots \sum\limits_{i_n=0}^{k-1} \Delta^n_h \left(f, x+i_1h+i_2h+\cdots+i_nh\right).</math>
* [[Leibniz rule (generalized product rule)|Leibniz rule]]: <math display="block">\Delta^n_h (fg, x) = \sum\limits_{k=0}^n \binom{n}{k} \Delta^k_h (f, x) \Delta^{n-k}_h(g, x+kh).</math>