Which sequence acceleration method should I use?

[japplepie]

The sequence that I'm working with is sort of monotonic.

It's monotonic most of the time, sometimes there's one number that ruins the trend like 0.0001001 in
0.1, 0.01, 0.001, 0.0001001, 0.0001, ...
but those are very rare. Btw, the sequence above is just an example.

Is Richardson extrapolation the best method to use in this case?

I also have a question about the Richardson extrapolation .
I'm going to call the nth Richardson extrapolation, R(n)

Why is it that when I test R(10) on a 30 term sequence, it gives me ridiculous approximations like -5.24 when the actual answer is 1.68?

 

[wabbit]

I think you'll be better off just dropping those parasite terms and working with the other terms alone, assuming they are well behaved, e.g. as in your example $|x_{n+1}-x_n|\sim a^n$ - typical acceleration schemes work only if the convergence is regular, what they do is leverage that regularity to predict the limit. They will produce worse results than no acceleration if applied to a generic random sequence.

If you applied a high order scheme to your original sequence, the result is not surprising - that sequence has wild variations for instance in the ratio $\frac{x_{n+1}-x_n}{x_n-x_{n-1}}$ and extrapolating based on an assumption of regular behaviour is bound to give poor results.