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Re: Question on version 4 and long Pi calculations.....
*To*: mathgroup at smc.vnet.net
*Subject*: [mg23540] Re: Question on version 4 and long Pi calculations.....
*From*: Mark Sofroniou <marks at wolfram.com>
*Date*: Sat, 20 May 2000 03:10:19 -0400 (EDT)
*Organization*: Wolfram Research Inc
*References*: <8fqrvb$h66@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
zeno at magicnet.net wrote:
> I know that version 4 can calculate Pi much faster than previous
> versions..the web site though mentions "up to 10 million" places, etc. Does
> that mean the extreme speed increase is for only up to the first 10 million
> places? Or is the speed increase the same for any amount of places, and the
> only limit is ones memeory? If one had enough memory, could one calculte Pi
> to 50 milion places with a great speed increase in addition to 1 milion
> places or 10 million?
There are two classes of asymptotically fast algorithms currently
used to approximate Pi numerically.
The first class are based on arithmetic-geometric mean methods.
These are asymptotically the fastest known methods and were used
to establish the current computational record of about 206 * 10^9
digits.
http://www.lacim.uqam.ca/pi/records.html
The second class of methods are based on binary splitting of
Chudnovsky or Ramanujan formulae:
http://xavier.gourdon.free.fr/Constants/Algorithms/splitting.html
The binary splitting methods have a slightly higher asymptotic
complexity but are faster at lower precisions because the constant
of proportionality in the asymptotic estimate is lower than the
arithmetic geometric mean methods.
Mathematica notebooks that implement the two classes of methods
can be found at:
http://library.wolfram.com/conferences/conference98/abstracts/high_precision_computations.html
Version 4.0 of Mathematica uses the binary splitting approach.
At the time when the documentation was written it was not known
in practice exactly when the arithmetic-geometric mean methods
would be faster, so the documentation conservatively stated
that the binary splitting method would be used below 10 million
digits. Above 10 million digits we anticipated a switch to
an arithmetic-geometric mean type method. However it has since
become clear that the binary splitting technique is more efficient
even for 10^9 digits or more.
So in summary the "10 million digits or more" refers to the particular
method used to compute Pi. Assuming you have sufficient memory in your
computer the only restriction on the numerical approximation Pi is
the maximum number of digits that can be represented in Mathematica.
An indication of the latter is the magnitude of $MaxNumber.
In version 4.0 the maximum number of digits that you can work with is
around 323 million.
In[1]:= $MaxNumber
323228010
Out[1]= 1.440397193981785 10
In the development version this has been increased to around 646
million digits.
Mark Sofroniou,
Wolfram Research.
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