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
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The first derivative of a function f of a real variable at a point x can be approximated using a five-point stencil as:[1]
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The center point f(x) itself is not involved, only the four neighboring points.
This formula can be obtained by writing out the four Taylor series of f(x ± h) and f(x ± 2h) up to terms of h3 (or up to terms of h5 to get an error estimation as well) and solving this system of four equations to get f ′(x). Actually, we have at points x + h and x − h:
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Evaluating gives us
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The residual term O1(h4) should be of the order of h5 instead of h4 because if the terms of h4 had been written out in (E1+) and (E1−), it can be seen that they would have canceled each other out by f(x + h) − f(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
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and gives us
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In order to eliminate the terms of ƒ(3)(x), calculate 8 × (E1) − (E2)
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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[2]
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which can be obtained by expanding the left-hand side in a Taylor series. Alternatively, apply Richardson extrapolation to the central difference approximation to on grids with spacing 2h and h.
1D higher-order derivatives
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The centered difference formulas for five-point stencils approximating second, third, and fourth derivatives are
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The errors in these approximations are O(h4), O(h2) and O(h2) respectively.[2]
Relationship to Lagrange interpolating polynomials
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As an alternative to deriving the finite difference weights from the Taylor series, they may be obtained by differentiating the Lagrange polynomials
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where the interpolation points are
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Then, the quartic polynomial interpolating f(x) at these five points is
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and its derivative is
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So, the finite difference approximation of f ′(x) at the middle point x = x2 is
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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
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forming a pattern that is also called a quincunx. This stencil is often used to approximate the Laplacian of a function of two variables:
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The error in this approximation is O(h 2),[3] which may be explained as follows:
From the 3 point stencils for the second derivative of a function with respect to x and y:
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If we assume :
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