## Appendix C

Richardson Extrapolation

Digital computers cannot, in general, evaluate values with inﬁnite precision and thus need to rely on
approximations. It is, of course, among the goals of any scheme resulting in such an approximate value
to have as little an error as possible. Numerical schemes to reduce this error, such as the Richardson
extrapolation presented here, are therefore of great interest.

The Richardson extrapolation scheme assumes that a function can be expanded around a point of
evaluation depending on a parameter in a form

such
that the point of evaluation corresponds to the parameter . Expanding the function at two
diﬀerent values of the parameter and gives
Multiplying Equation by and subtracting Equation allows to eliminate the terms of order

Reordering this expression shows the approximation for explicitly as
An
approximation which now is of order instead of the original order . This procedure can be applied
repeatedly to further increase the quality of the approximation.
Applications using digital computers usually utilize factors corresponding to powers of . This
allows the following settings

which further illustrate the recursive nature of the procedure.