Topic 12.1: Centred Divided-Difference Formulae (HOWTO)

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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:

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.