When using Richardson extrapolation for numerical integration, are there any criteria whether ratio between the steps should be or does it not matter what step size I use?
For an integral I can write
$$I = I(h_j) + \frac{I(h_j) - I(h_i)}{\frac{h_i}{h_j}-1}$$
in which $h_j$ and $h_i$ are different step sizes, but is there any specific size difference between them.. I could it just be $1$ and $0.00001$ or?
Theoretically, $h_j/h_i$ could be any positive number except $1$. (You want the steps to be different after all).
Practically, it should be an integer (or reciprocal of an integer, depending on the order of division). The expensive part of numerical integration is evaluation of the integrand, so one wants to reduce the number of such evaluations if possible. With an integer ratio, such as $2$, the sample points for one step are a subset of the sample points for the other step. Thus, one can reuse already available function values at those points.
And since one may want to use Richardson extrapolation iteratively, by combining more than two step sizes, it is convenient to have the ratios be powers of some fixed integer, usually the powers of $2$.
Finally, extremely large ratios are not practical, since they combine a very coarse grid (imprecise value) with a very fine one (expensive computation).