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Problem
Approximate the derivative of a univariate function f(x) at a point x0. We will assume that we are given a sequence of points (xi, f(xi)). We will not look at iteration because the process of Richardson extrapolation is significantly better.
Assumptions
We need to assume the function has a second derivative if we are to bound the error on our approximation.
Tools
We will use interpolation.
Process
If we are to evaluate the derivative at the point (xi, f(xi)) and have access to the two surrounding points, (xi − 1, f(xi − 1)) and (xi + 1, f(xi + 1)), then we may calculate:
This is simply another form of the formula
where h is the distance between the points, that is, h = xi - xi − 1.
If we have access to two points on either side of xi, we can calculate
where h = xi - xi − 1.
This is another form of the formula:
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