This is a bi-lingual blog of the members of the ADAMIS team at Laboratoire APC and invited guests. We comment on selected papers and events exploring, or relevant to, the interface between physics, cosmology, applied math, statistics, and numerical algorithms and which we have found interesting.

The opinions expressed in this blog reflect those of their authors and neither that of the ADAMIS group as a whole nor of Laboratoire APC.

Monday, September 20, 2010

Two quadrillionth digit of π ...

has been just computed. That is we know as of now the 2,000,000,000,000,000th digit of the ever mysterious π and it turns out to be ... 0 ... Quite a let-down one would think. However this news, which hit the media last week (check here BBC commentary) is certainly of some interest, even if none of us probably bit his/her fingers waiting for this moment in suspense.


First of all it may have come as a surprise to some of us, in particular those brought up on the standard series expansions available for π in any mathematical handbook, that it was not an entire sequence of two quadrillion digits, which have been computed but just a specific, single digit, which occupies two quadrillionth place after the decimal place. In fact the 'record' number of consecutive π digits is "just" above 2 trillions ... a pittance in comparison. Though it may boggle one's mind how one can get a subsequent digit without first, or rather simultaneously, tediously calculating the previous ones, the relevant formalism has been known since the late 90ies of the last (XX) century. Further away from the decimal point the digit is - longer the computation is going to take as more operations have to be performed. However, there is no need to get all the preceding numbers at least explicitly. Boggling it may be, but no magic is involved and those curious of technical details can find them here.

As heavy calculations are however involved use of computers is unavoidable. And the big ones for that matter - yet another sign of our times. The computations in this case took just over 3 weeks and used 1,000 processors of a state-of-the-art computer. (Equivalent roughly to just over 50 years of your single processor desktop.) Even that would not suffice if special software has not been used. First of all, suitable software had to be parallel to allow to harness the power of all the available processors simultaneously. Second, the software needed to be efficient to make the computation possible at all. Third, the software had to be fault tolerant, so a failure of one or another of the many processors running continuously over many days did not interrupt the computation. Instead of developing their own tools from the scratch the programmer(s) opted for off-shelf software referred to as MapReduce. This is a general, parallel framework in which many real life and scientific problems can be phrased and implemented and which by its virtue assures high performance, while saving the programmer loads of work. The software is commercial as I understand and Maude certainly abhors. The details of it can be found here. In essence one processor plays a role of a 'work supervisor', who distributes the work load between all the other processors, called 'workers'. It also checks periodically if a worker is down (though 'gone for lunch' better fits the French reality) and if so it will bypass it in the work distribution. Each of the workers performs assigned tasks, which are either of a type 'map' or 'reduce'. The map tasks are a set of basic calculations resulting usually in some intermediate and smaller by data volume result. The reduce tasks take the intermediate results as an input and combine those together to produce the final output. Being conceptually rather simple this is potentially an interesting step in making parallel computers power more readily accessible for a general user, without that all inevitable nitty-gritty. Now at least the "π test" seems to have been passed.

All in all for a pastime activity, as calculating the digits of π is often construed, ..., it is quite a 'boulot', isn't it ?!

However, a pastime it may be but it is also a sign of a progressively bigger importance of computers in theoretical math (theoretical as opposed to the applied one, where numerical calculations are just bread-and-butter.). This has grown enough recently to the extent that a new subfield referred often to as experimental mathematics seems to be emerging on our watch ... And think that just a few years ago that would have been nothing more than an oxymoron.

No comments:

Post a Comment