Re: Convergence Acceleration for Series

[Dirk Laurie <dirk.laurie@gmail.com>]

2018-01-15 16:22 GMT+02:00 kevin lucas <lucaskevin296@gmail.com>:

> Mr. Laurie, your chapter in the SIAM 100-digit challenge was one of the
> first places I initially checked for a solution. I'm new to PARI and could
> not find a working implementation of Levin transforms in any language, so
> thank you very much for the script. It gets 16 places, which is enough for
> the OEIS to identify it (https://oeis.org/search?q=0.0840695087276559). Is
> there a systematic way to pick out where to stop the summation for similar
> problems? I assume we picked 40 here because the values obtained didn't
> agree with the expected value past there, but what if one did not know that
> beforehand?

That chapter explains how to estimate accumulated roundoff error, and
suggests that one should stop as soon as the difference between two
extrapolated values is less than the roundoff estimate. Practically that
means: treat the accelerated sequence as asymptotic, and stop
immediately after the smallest difference. But the chapter also cautions
against any attempt to automate such decisions.

I note that my previous reply did not go to the list toom so I attach the
file levin.gp again.

Attachment: levin.gp
Description: application/gnuplot