Re: Re: Pi upto a Billion Digits

*To*: mathgroup at smc.vnet.net*Subject*: [mg75725] Re: [mg75672] Re: Pi upto a Billion Digits*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 9 May 2007 04:33:37 -0400 (EDT)*References*: <f1msfm$rnf$1@smc.vnet.net> <200705080954.FAA18685@smc.vnet.net>

Szabolcs wrote: > 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 I'll comment on that last part. First we rerun the computation. I'm using version 6 on a Linux machine, around 3 GHz. In[8]:= d = Table[ClearSystemCache[]; First[Timing[v[k]=N[Pi,2^k*10^5]]], {k,1,7}] Out[8]= {0.47603, 1.08807, 2.56416, 6.09238, 14.2249, 31.874, 73.5246} What we see is "soft" linear time, roughly that associated with the (bit) complexity of multiplication. Now we convert to strings of digits. In[12]:= f = Table[First[Timing[s10[k]=RealDigits[v[k]]]], {k,7}] Out[12]= {0.176011, 0.47603, 1.12407, 2.77217, 6.65242, 15.861, 37.3623} We see similar complexity. Not surprising, because the conversion from binary to decimal involves division, which has the same complexity as multiplication. If we avoid the b2d conversion it becomes a byte faster. In[13]:= g = Table[First[Timing[s16[k]=RealDigits[v[k],16]]], {k,7}] Out[13]= {0.004, 0.008002, 0.012, 0.020002, 0.036002, 0.076004, 0.15201} Outputting the results e.g. to a file will show similar behavior to RealDigits; slow if in decimal and relatively fast if in hex. Upshot: This has little to do with I/O, and more to do with conversion of large numbers from binary to decimal form. Daniel Lichtblau Wolfram Research

**References**:**Re: Pi upto a Billion Digits***From:*Szabolcs <szhorvat@gmail.com>