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MathGroup Archive 2007

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Re: Pi upto a Billion Digits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75672] Re: Pi upto a Billion Digits
  • From: Szabolcs <szhorvat at gmail.com>
  • Date: Tue, 8 May 2007 05:54:55 -0400 (EDT)
  • Organization: University of Bergen
  • References: <f1msfm$rnf$1@smc.vnet.net>

Raj wrote:
> hi!
> 
> Could somebody tell me if they ever tried finding Pi upto a billion
> digits using the N function:
> N[Pi,10^9] and how long did it take?
> 
> Thanks,
> 
> Raj

I haven't tried it, but you can estimate how long it takes, like this:

d = Table[Update[]; First@Timing[N[Pi, 2^k 10^5]], {k, 1, 6}]/Second

{1.437, 3.579, 9.14, 23.375, 58.453, 143.391}

(Results are from Mathematica 5.2, 1.7 GHz processor.)

ListPlot[Log@d]

Now we can extrapolate. Find the slope of Log@d

slope = Mean@ListConvolve[{1, -1}, Log@d]

0.920604

10^9 is approximately 2^12 * 10^5, so you can expect

Exp[11*slope + d[[1]] ]

105202.

This is approximately 30 hours. Of course this result may be an order of 
magnitude off, but I suspect that the precise result is less than 30 
hours, not more. With a little patience you can get a much better estimate.

Or you can leave the calculation running for a day and hope that you 
don't run out of memory (memory usage may be a bigger problem than CPU 
time). If you do run the calculation, please let us know your findings.

But note that the time to export the result to a file may be comparable 
to the time of the calculation (or even much longer -- Mathematica I/O 
is pretty slow).

Szabolcs


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