From: Simon Plouffe
Newsgroups: sci.math
Subject: Computation of the n'th digit of Pi in base 10.
Date: Mon, 20 Apr 1998 07:07:18 GMT
Hello, (in answer to many postings about Pi).
I see that there are some debates about the possibility of
computing Pi in other bases than 2.
First of all, we (me and D. Bailey and P. Borwein) have found a way
in September 1995 to compute Pi in base 2 (or 16), in our
original article, we mention that we did not found similar
formulas for base 10. BUT we searched a certain class of formulas
only.
See http://www.lacim.uqam.ca/plouffe/Simon/MOCpi.ps
for the original article or
http://www.lacim.uqam.ca/plouffe/Simon/BaileyBorweinPlouffe.pdf
in pdf format.
Later in November 1996, I have found a way to compute Pi
in ANY base with a different algorithm that uses a formula
for Pi known since Newton as explained in
http://www.lacim.uqam.ca/plouffe/Simon/articlepi.html
The algorithm is 'more' general but the price to pay
is that it is fairly slow to actually compute it,
the time taken is roughly O(n^3) compared to O(n*log(n))
for the algorithm in base 2.
Still : the algorithm can compute (let's say) the 10000'th
digit of Pi without having to compute the previous digits
and with very little memory.
Now, what I have found is only a way that proves that
it is possible to do it, it is more theoretical than
practical, it surely cannot go beyond the 51.5 billion
digits that were recently computed by Kanada in Japan.
That late algorithm uses those n'th order Borwein-like
algorithms and uses a lot of memory and a lot of CPU
power but it is the fastest way to do it so far in
base 10.
Again, it might be that there is actually a better
way to compute it very fast (in a probably different
way). I am working on this but it might take several
weeks, months, centuries, I don't know! :)
(see http://www.lacim.uqam.ca/pi/records.html for the
latest records of computations including the ones
about the n'th binary digits).
Simon Plouffe
http://www.lacim.uqam.ca/plouffe
Plouffe's Inverter at http://www.lacim.uqam.ca/pi/