In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is made up of the point itself together with its four "neighbors". It is used to write finite difference approximations to derivatives at grid points.
In one dimension, if the spacing between points in the grid is h, then the five-point stencil of a point x in the grid is
Obtaining the formula
This formula can be obtained by writing out the four Taylor series of ƒ(x ± h) and ƒ(x ± 2h) up to terms of h 3 (or up to terms of h 5 to get an error estimation as well) and solving this system of four equations to get ƒ ′(x). Actually, we have at points x + h and x − h:
Evaluating (E 1+) − (E 1−) gives us
Note that the residual term O1(h 4) should be of the order of h 5 instead of h 4 because if the terms of h 4 had been written out in (E 1+) and (E 1−), it can be seen that they would have canceled each other out by ƒ(x + h) − ƒ(x − h). But for this calculation, it is left like that since the order of error estimation is not treated here (cf below).
Similarly, we have
and gives us
In order to eliminate the terms of ƒ (3)(x), calculate 8 × (E1) − (E2)
thus giving the formula as above. Note: the coefficients of f in this formula, (8, -8,-1,1), represent a specific example of the more general Savitzky-Golay filter
The error in this approximation is of order h 4. That can be seen from the expansion
The centered difference formulas for five-point stencils approximating second, third, and fourth derivatives are
The errors in these approximations are O(h 4), O(h 2) and O(h 2) respectively.
Relationship to Lagrange interpolating polynomials
As an alternative to deriving the finite difference weights from the Taylor series, they may be obtained by differentiating the Lagrange polynomials
where the interpolation points are
Then, the quartic polynomial interpolating ƒ(x) at these five points is
and its derivative is
So, the finite difference approximation of ƒ ′(x) at the middle point x = x2 is
Evaluating the derivatives of the five Lagrange polynomials at x=x2 gives the same weights as above. This method can be more flexible as the extension to a non-uniform grid is quite straightforward.
In two dimensions, if for example the size of the squares in the grid is h by h, the five point stencil of a point (x, y) in the grid is
The error in this approximation is O(h 2), which may be explained as follows:
From the 3 point stencils for the second derivative of a function with respect to x and y:
If we assume :
- Abramowitz & Stegun, Table 25.2
- Abramowitz & Stegun, 25.3.30
- Abramowitz, Milton; Stegun, Irene A. (1970), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover. Ninth printing. Table 25.2.