Richardson Extrapolation to h = 0

We observe that the Euler method gives an error of the form , where depends on the differential equation only if round–off errors are ignored. To halve the error produced by the Euler method, it is necessary to halve the step size if the term can be neglected. Let us consider two integrations of a differential equation, one using step size h and one using step size . If the answers are and respectively, we can write

(3.3)

Eliminating from (3.3), we get

(3.4)

This can be used as a better numerical approximation for it is more accurate than the Euler formula, and that it gains accuracy more rapidly as is decreased because the error is rather than . This process of deriving a higher order method from a lower order method is also called the deferred approach to the limit.