Topic 12.3: Richardson Extrapolation (HOWTO)

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Problem

Approximate the derivative of a univariate function f(x) at a point x0. We will assume that we can calculate f(x) for arbitrary values of x.

Assumptions

We will assume that the function is sufficiently differentiable.

Tools

We will use interpolation and extrapolation.

Initial Values

We will start with an initial value h.

Process

For i = 0, 1, 2, let Ri, 0 be the centred divided- difference formula with a step of h/2i. Then, for j = 1, 2, ..., i, calculate

Halting Conditions

There are two conditions which may cause the iteration process to halt:

  1. We halt if |Ri, i − Ri − 1, i − 1| < εstep, or
  2. If we have iterated some maximum number of times, say N, and have not met Condition 1, we halt and indicate that a solution was not found.

If we halt due to Condition 1, we state that Ri, i is our approximation to the derivative.

If we halt due to Condition 2, we may state that a solution may not exist.

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