Extrapolation to the limit

Let F(h) be some quantity, such as a numerical derivative or integral, depending on a finite step size h, where the limit of F(h) as File:Hepa img316.gif is wanted. If it is known that F(h) = F(0) + O(hⁿ), i.e., the order n of the error is known, then for any r (with 0<r<1) define

so as to obtain a smaller error as File:Hepa img316.gif ,

with m>n. If m is known, then the procedure can be repeated, with G instead of F and m instead of n.