---
tag: 752Z7DRT7P
kind: concept
name: Finite difference coefficients
msc: [65D25, 65N06]
specializes: [NVH3F8FTYS]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_difference_coefficient
same_as:
  wikidata: Q1747602
  wikipedia: Finite difference coefficient
---

## Summary

Finite difference coefficients are the weights that turn values of a function on a stencil of points into an approximation of a derivative.
For a given derivative and stencil, they are found by matching Taylor expansions.
This leads to a Vandermonde system with a unique solution.
They are the building blocks of [[numerical differentiation|NVH3F8FTYS]] and of the [[finite difference method|X7HYEBA5SA]] for differential equations.
Tables of central, forward and backward coefficients of various orders are standard.

## Explanations

### L1 human

To estimate the second derivative of $f$ at $x$ from the values at $x - h$, $x$ and $x + h$, one uses the weights $1, -2, 1$ and divides by $h^2$.
For $f(x) = x^2$ at $x = 0$ with $h = 1$, this gives $1 - 0 + 1 = 2$, the true second derivative.
Other derivatives and other sets of points need other weights.
Finite difference coefficients are these weights, and there is a systematic way to compute them for any stencil.