Carrying Out and Monitoring Searches

A practical, relevant question for carrying out searches concerns the dependence on the pool size. This simple example shows a trend that is also visible in much larger pool sizes--a high sensitivity to the actual pool size. This sensitivity occurs because a sequence that at iteration shows a relatively poor performance might show a much better performance at iteration . When the pool size is not large enough, this sequence might be discarded too early. In general, the larger the pool size the better the results. But the larger the pool size, the longer the search will run and with pool sizes too large, we might run out of memory.

For nonsimple continued fractions we see a similar, non-monotonic dependence on the pool size.

Here a larger search is carried out. We do not restrict the possible initial sequences and use a pool size of 1000.

We achieved about 60 coinciding digits in about 15 minutes. The next three graphics show the number of coinciding digits of the optimal sequence, the number of different groups of sequences with different initial six digits, and the position of the first different digit across the pool of sequences as a function of the iteration number. We see that on average we gain a constant number of digits per iteration as long as we gain digits. We also see how the numbers in the pool converge towards a single number.

And here is the equivalent of the last calculation for nonsimple continued fractions. Again we use a pool size of 1000. This time, we obtain more than 100 identical digits. And again we see on average a linear increase in the digits gained as well as convergence on one final number (after about 40 iterations).