Suppose that is an approximation to for every and that
for some constants , , ,..
Now for given values , and we will produce an approximation to . Indeed, from
we firstly get
Similarly, we obtain
Now we easily get
which implies that
For example, for given , , one can compute that
In general, Let be an approximation of that depends on a positive step size with an error formula of the form
where the are unknown constants and the are known constants such that . The exact value sought can be given by
which can be simplified with Big O notation to be
Using the step sizes and the two formulas for are:
Multiplying the second equation by and subtracting the first equation gives
Which can be solved for to give
By this process, we have achieved a better approximation of by subtracting the largest term in the error which was . This process can be repeated to remove more error terms to get even better approximations. A general recurrence relation can be defined for the approximations by
such that with .