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MathGroup Archive 2007

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Re: Pi upto a Billion Digits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75789] Re: Pi upto a Billion Digits
  • From: Raj <rajanikanth at gmail.com>
  • Date: Thu, 10 May 2007 05:23:40 -0400 (EDT)
  • References: <f1msfm$rnf$1@smc.vnet.net>

On May 9, 2:06 am, DrMajorBob <drmajor... at bigfoot.com> wrote:
> Calculating N[Pi,2*10^8], I got an abnormal termination from the
> C++ library after ten or fifteen minutes.
>
> So... 10^9 is out of the question on my machine.
>
> Bobby
>
> On Tue, 08 May 2007 11:29:51 -0500, Szabolcs Horv=E1t <szhor... at gmail.com>
> wrote:
>
>
>
> > On 08/05/07, DrMajorBob <drmajor... at bigfoot.com> wrote:
> >> On my machine (WinXP, Athlon 3200+, 2GB RAM, Mathematica 6), I estimate
> >> 3.4 hours:
>
> > Did you run the calculation?
>
> >> Timing[data =
> >>    Table[Update[];
> >>     Log@{#, First@Timing[N[Pi, #]]} &[2^k 10^5], {k, 1, 8}]]
>
> >> {231.921, {{Log[200000], -1.06711}, {Log[400000], -0.226901}, {Log[
> >>      800000], 0.669879}, {Log[1600000], 1.50741}, {Log[3200000],
> >>     2.37052}, {Log[6400000], 3.21451}, {Log[12800000],
> >>     4.05123}, {Log[25600000], 4.87675}}}
>
> >> (I doubled two steps more than Szabolcs did.)
>
> >> The log-log data is VERY linear:
>
> >> Needs["LinearRegression`"]
> >> AdjustedRSquared /. Regress[data, x, x]
>
> >> 0.999901
>
> >> Clear[a, b, z, logSecs, hours]
> >> model = a + b z;
> >> logSecs[z_] = model /. FindFit[data, model, {a, b}, z]
> >> hours[z_] := Exp[logSecs[Log[z]]]/60^2
> >> hours[10^9]
>
> >> -16.0424 + 1.22792 z
>
> >> 3.37132
>
> >> Also note, omitting Update[] has NO effect on the timings:
>
> >> Timing[data =
> >>    Table[Log@{#, First@Timing[N[Pi, #]]} &[2^k 10^5], {k, 1, 8}]]
>
> >> {231.703, {{Log[200000], -1.07002}, {Log[400000], -0.207024}, {Log[
> >>      800000], 0.628609}, {Log[1600000], 1.52126}, {Log[3200000],
> >>     2.36471}, {Log[6400000], 3.21133}, {Log[12800000],
> >>     4.04989}, {Log[25600000], 4.8771}}}
>
> >> Bobby
>
> > Update[] forces Mathematica to recalculate everything instead of using
> > cached values. If it has no effect in Mathematica 6 that is very
> > strange (it would mean that Mathematica 6 doesn't cache results in
> > this case, and that the performance may be worse than in 5.2). Does it
> > have an effect when you don't use it in Table[], but evaluate the
> > expressions separately?
>
> > In 5.2 it has a very noticeable effect:
>
> > In[178]:=
> > Update[]
>
> > In[179]:=
> > Timing[N[Pi,4*10^5];]
>
> > Out[179]=
> > {3.25 Second,Null}
>
> > In[180]:=
> > Timing[N[Pi,8*10^5];]
>
> > Out[180]=
> > {5.579 Second,Null}
>
> > In[181]:=
> > Update[]
>
> > In[182]:=
> > Timing[N[Pi,8*10^5];]
>
> > Out[182]=
> > {8.187 Second,Null}
>
> > In[183]:=
> > Update[]
>
> > In[184]:=
> > Table[Timing[N[Pi,k 4*10^5];],{k,2}]
>
> > Out[184]=
> > {{3.156 Second,Null},{5.625 Second,Null}}
>
> > In[185]:=
> > Update[]
>
> > In[186]:=
> > Table[Update[];Timing[N[Pi,k 4*10^5];],{k,2}]
>
> > Out[186]=
> > {{3.11 Second,Null},{8.062 Second,Null}}
>
> > Szabolcs
>
> --
> DrMajor... at bigfoot.com

Thanks for the replies. It seemed to take a long time(> 1 hour) on my
computer...so  I was curious if someone already evaluated it.

Thanks,

Raj



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