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.