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

which gives

i.e.,

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 .

## Leave a Reply