Richardson Extrapolation

We've illustrated above how higher order formulae become more complex and are affected by round-off error quicker then low order formulae. On the other hand, low order formulae are definitely less accurate for a given step size. Is there then a way to use results from a low order formula to obtain more accurate approximations?

Richardson Extrapolations attempts to do just that. It applies not only to finite difference formulae but to every formula of the general form

, with an unknown value to be approximated, and some constant.

Derivation

Example

Higher order corrections

Example