Conference rusure::math

Yes, I've heard the 16 million figure from an independent source
(non USENET).

Newsgroups: net.math
Path: decwrl!ucbvax!nike!riacs!ames!eugene
Subject: Blantant Public Relations
Posted: 22 Jan 86 08:04:22 GMT
Organization: NASA-Ames Research Center, Mtn. View, CA
 
I have been asked to post the following information.
 
We have calculated $pi$ to 29,360,128 decimal digits using one
processor of our Cray-2.  It took 28 hours and performed 12 trillion
operations over a three-day period. Memory requirements took
over 138 MWs (64-bit).  The result was determined by Borwein's algorithm.
[SIAM Rev. 26, 1984].  The check took 40 hours.  The technical report
will be issued shortly.  The work was performed by Dr. Dave Bailey,
a contractor with Informatics General Corp. of the Numerical
Aerodynamic Simulation Program Office, NASA Ames Research Center.
The previous large computation was done in Japan to about 10 million digits
on a Hitatchi 810.
 
If you want the algorithm, look it up at a library.
Don't ask me: I won't send it!
For a copy of the TR: Write Bailey at MS 233-1.  (Serious inquires, please.
The merely curious should wait for the press release.)
 
From the Rock of Ages Home for Retired Hackers:
--eugene miya
  NASA Ames Research Center
  {hplabs,ihnp4,dual,hao,decwrl,allegra}!ames!aurora!eugene
  emiya@ames-vmsb.ARPA

Article        20093
From: bnr@uc.msc.edu (Barry Rackner)
Newsgroups: comp.parallel,comp.arch,sci.math,sci.math.num-analysis
Subject: new pi record
Message-ID: <1991Aug20.115809.16602@hubcap.clemson.edu>
Date: 20 Aug 91 03:10:47 GMT
Organization: Minnesota Supercomputer Center
 
 
 
	David and Gregory Chudnovsky, of Columbia University, have succeeded 
        to compute 2,160,000,000 decimal digits of pi on their own parallel 
        machine.   Further information can be obtained from the Chudnovskys
        since I know nothing further at this point.
 
        The Chudnovskys held the previous record, at over a billion decimal
        digits, using a CRAY2 and an IBM 3090.  This computation was 
        documented in "Proc. Natl. Acad. Sci. USA 86 (1989)".
 
        As I am not an active reader of the net, please email inquiries
        to bnr@msc.edu.

It might be fun to get a copy of those digits to look for interesting
trivial facts about repetition.  For instance:

How long a string of the same digit repeating can we find ?

What's the longest repetition of TWO digits ?  For instance, can we find
12121212121212 ?

What's the shortest string of digits we *can't* find ?  Perhaps if there's
a particularly short string that's suspiciously missing, someone can try
to prove that such a string will never occur.

A very good book if you are interested in (among many other things!) pi...
Has anyone read it?

re .5
    
>                        -< read Carl Sagan's "CONTACT" >-
>
>A very good book if you are interested in (among many other things!) pi...
>Has anyone read it?

    Yes, but the SF conference may be a more appropriate place to ask.
    Press KP7 to add it to your notebook.

    
    i saw this in the net a while ago, to calculate PI. i retyped here
    because i only have a hard copy of it. 
    
    This program prints the first 800 digits of PI. 
    
    /*
    * from: dik@cwi.nl (Dik T. Winter) Date: 30 June 91
    * Organization: CWI, Amsterdam
    */
    
int a=10000, b,c=2800,d,e,f[2801],g;

main()
{

while(b-c) f[b++]= a/5;
for(;d=0,g=c*2;c -=14,printf("%.4d",e+d/a),e=d%a)
  for(b=c;d +=f[b]*a, f[b]= d% --g, d/=g--,--b;d*=b);
}
    

    
From: kbrannen@csfb1.fir.fbc.com (Kevin Brannen)
Subject: Repost of a `Digits of Pi'
Sender: news@csfb1.fir.fbc.com (Usenet News <Jim Freeman>)
Reply-To: uunet!csfb1!kbrannen
Organization: First Boston Corporation
Date: Fri, 15 Oct 1993 19:52:46 GMT
 
I've had 5 or 6 people ask for a copy and/or repost of a recent article on
calculating digits of Pi.  As it's not too long here it is in it's entirity.
 
Kevin
 
============================================================================
 
From: kmj@bilbo.baylor.edu (Kevin M. Johnson)
Subject: Re: How is PI calculated. <<<=================
Date: 14 Sep 1993 19:08:19 GMT
Organization: Baylor University
 
Andy P. Bajorinas (bajorap@pb.com) wrote:
: I am sure this is a FAQ but I could not find it in any FAQ list I have.
: How is pi calculated. (Besides measuring a REALLY big circle)
 
: Entries will be judged on brevity and clarity.  :^)
 
: bajorap@pb.com
 
I copied this out of a Popular Science magazine I think. The author 
said this was the latest (last year) best formula for calculating 
Pi. The article said that each term in the summation added on 14
correct digits to the decimal expansion of Pi, and the advantage of
this formula was that you could wait till you finished calculating, 
before you rounded in n digits. Enough chatter. Here it is.
 
 
 
 oo
 --]
 \   [C1 + n] * (6n)!             (-1)^n        (640320)^(3/2)         1
  >             ----------- * -------------  =  -------------------- * -- 
 /              (3n)!(n!)^3   (640320)^(3n)     163*8*27*7*11*19*127   Pi
 --]
 n=0
                     13591409
where     C1 = ------------------- 
               163*2*9*7*11*19*127
 
I think the article went on to say that they had calculated the first
billion digits of Pi in a relatively short amount of time, and because
of the nature of the formula, adding on digits would be very simple. I 
have to admit, I am impressed with 14 digits per term. But you might 
expect that with those factorials and powers) Just to convice you this 
is pretty cool. I will show you this for n=0. I get
 
             (640320)^(3/2) 
     Pi ~=  ---------------- ~= 3.14159265359....                 
	      163096908
 
on my calculator which only goes that far. I haven't tested it on 
a powerful math package, but I assume that they weren't lying to me.
Hope this helps you out.
 
Kevin Johnson 
Department of Mathematics
Baylor University
kmj@bilbo.baylor.edu
    
    

Newsgroups: sci.math,sci.answers,news.answers
From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: How to compute Pi?
Organization: University of Waterloo
Date: Thu, 30 Mar 1995 01:14:33 GMT
 
Archive-Name: sci-math-faq/specialnumbers/computePi
Last-modified: December 8, 1994
Version: 6.1
   
How to compute digits of pi ?
   
   Symbolic Computation software such as Maple or Mathematica can compute
   10,000 digits of pi in a blink, and another 20,000-1,000,000 digits
   overnight (range depends on hardware platform).
   
   It is possible to retrieve 1.25+ million digits of pi via anonymous
   ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and
   pi.dat.Z which reside in subdirectory doc/misc/pi. New York's
   Chudnovsky brothers have computed 2 billion digits of pi on a homebrew
   computer.
   
   There are essentially 3 different methods to calculate pi to many
   decimals.
   
    1. One of the oldest is to use the power series expansion of atan(x)
       = x - x^3/3 + x^5/5 - ... together with formulas like pi =
       16*atan(1/5) - 4*atan(1/239) . This gives about 1.4 decimals per
       term.
       
    2. A second is to use formulas coming from Arithmetic-Geometric mean
       computations. A beautiful compendium of such formulas is given in
       the book pi and the AGM, (see references). They have the advantage
       of converging quadratically, i.e. you double the number of
       decimals per iteration. For instance, to obtain 1 000 000
       decimals, around 20 iterations are sufficient. The disadvantage is
       that you need FFT type multiplication to get a reasonable speed,
       and this is not so easy to program.
       
    3. A third one comes from the theory of complex multiplication of
       elliptic curves, and was discovered by S. Ramanujan. This gives a
       number of beautiful formulas, but the most useful was missed by
       Ramanujan and discovered by the Chudnovsky's. It is the following
       (slightly modified for ease of programming):
       
       Set k_1 = 545140134; i k_2 = 13591409; k_3 = 640320; k_4 =
       100100025; k_5 = 327843840; k_6 = 53360;
       
       Then pi = (k_6 sqrt(k_3))/(S) , where
       
       S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 +
       nk_1))/(n!^3(3n)!(8k_4k_5)^n)
       
       The great advantages of this formula are that
       
       1) It converges linearly, but very fast (more than 14 decimal
       digits per term).
       
       2) The way it is written, all operations to compute S can be
       programmed very simply since it only involves
       multiplication/division by single precision numbers. This is why
       the constant 8k_4k_5 appearing in the denominator has been written
       this way instead of 262537412640768000. This is how the
       Chudnovsky's have computed several billion decimals.
       
   
   
   The following 160 character C program, written by Dik T. Winter at
   CWI, computes pi to 800 decimal digits.
   
 
     int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
     for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
     f[b]=d%--g,d/=g--,--b;d*=b);}
 
   
   
   
   
   References
   
   P. B. Borwein, and D. H. Bailey. Ramanujan, Modular Equations, and
   Approximations to pi American Mathematical Monthly, vol. 96, no. 3
   (March 1989), p. 201-220.
   
   
   
   J.M. Borwein and P.B. Borwein. The arithmetic-geometric mean and fast
   computation of elementary functions. SIAM Review, Vol. 26, 1984, pp.
   351-366.
   
   
   
   J.M. Borwein and P.B. Borwein. More quadratically converging
   algorithms for pi . Mathematics of Computation, Vol. 46, 1986, pp.
   247-253.
   
   
   
   Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the
   computation of pi Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2,
   1984, pp. 231-244.
   
   
   
   David Chudnovsky and Gregory Chudnovsky. The computation of classical
   constants. Columbia University, Proc. Natl. Acad. Sci. USA, Vol. 86,
   1989.
   
   
   
   Y. Kanada and Y. Tamura. Calculation of pi to 10,013,395 decimal
   places based on the Gauss-Legendre algorithm and Gauss arctangent
   relation. Computer Centre, University of Tokyo, 1983.
   
   
   
   Morris Newman and Daniel Shanks. On a sequence arising in series for
   pi . Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp.
   199-217.
   
   
   
   E. Salamin. Computation of pi using arithmetic-geometric mean.
   Mathematics of Computation, Vol. 30, 1976, pp. 565-570
   
   
   
   David Singmaster. The legal values of pi . The Mathematical
   Intelligencer, Vol. 7, No. 2, 1985.
   
   
   
   Stan Wagon. Is pi normal? The Mathematical Intelligencer, Vol. 7, No.
   3, 1985.
   
   
   
   
   
   A history of pi . P. Beckman. Golem Press, CO, 1971 (fourth edition
   1977)
   
   
   
   pi and the AGM - a study in analytic number theory and computational
   complexity. J.M. Borwein and P.B. Borwein. Wiley, New York, 1987.
   
   
   
   
     _________________________________________________________________
   
   
   
    alopez-o@barrow.uwaterloo.ca
    Sun Nov 20 20:45:48 EST 1994
 
 

   See also the recent article by Stan and Stan Wagon in the
   March 1995 Monthly:

	A Spigot Algorithm for the digits of PI
	American Mathematial Monthly, March 1995, pp 195-203

   - Jim

Article 18662 of sci.math.symbolic:
From: Simon Plouffe <plouffe@cecm.sfu.ca>
Newsgroups: sci.math.symbolic
Subject: PI CALCULATED TO 3.22 BILLION DIGITS
Organization: CECM
Lines: 108



Hello, we just received a mail from Yasumasa KANADA (japan) 
announcing the calculation of PI to 3.22 BILLION DIGITS.
The message was addressed to Jon and Peter BORWEIN at the
CECM, Centre for Experimental & Constructive Math, Simon Fraser
University. 

Kanada employed 2 different methods , the Borwein's quartic convergent
algorithm and the Gauss-Legendre algorithm. the value of 1/Pi was also
obtained. It CONFIRMS the value of 2.16 billion digits obtained
 earlier by the Chudnovsky brothers. 


here is the DETAILS, not the digits (sorry folk's), a value of
10 MILLION digits will be available SOON at our site from KANADA
special cd-rom. SEE http://www.cecm.sfu.ca

---------------------------------------------------------------
ANNOUNCEMENT-ANNOUNCEMENT-
  And now the details...Jon

   ====================================
   Dear folks,

   Our latest record was established as the followings;

   Declared record:
    3,221,220,000 decimal digits
    Two independent calculation based on two different algorithms 
generated
   3,221,225,472 (=3*2^30) decimal digits of pi and comparison of two 
generated
   sequences matched 3,221,225,466 decimal digits, e.g. 6 decimal digits
   difference.  Then we are declaringt 3,221,220,000 decimal digits as 
the
   new world record.

   Main program run:
    Job start    : 16th June 1995 22:41
    Job end      : 18th June 1995 11:33
    Elapsed time : 36:52:28
    Vector CPU   : 44:30:50
    Main memory  : 1888.5 MB
    ES memory    : 26608 MB
    Algorithm    : Borwein's 4-th order convergent algorithm

   Verification program run:
    Job start    : 24th June 1995 01:13
    Job end      : 26th June 1995 06:56
    Elapsed time : 53:43:46
    Vector CPU   : 48:28:43
    Main memory  : 1891.25 MB
    ES memory    : 26612 MB
    Disk storage : 164 MB * 5 = 820 MB
    Algorithm    : Gauss-Legendre algorithm

   3,000,000,000-th digits of pi and 1/pi:
    pi  : 37608 19468 51598 04548
    1/pi: 43270 91443 13660 42701
                                ^
                   3,000,000,000-th
    (First digit '3' for pi or '0' for 1/pi is not included in the above 
count.)

   Frequency distribution for pi-3 up to 3,000,000,000 decimal places:
    '0' : 299999143; '1' : 299995932; '2' : 299989126; '3' : 299992290
    '4' : 300002257; '5' : 299979016; '6' : 300025447; '7' : 299975510
    '8' : 300016550; '9' : 300024729; Chi square = 9.24

   Frequency distribution for 1/pi up to 3,000,000,000 decimal places:
    '0' : 300009029; '1' : 300002431; '2' : 299992729; '3' : 299998519
    '4' : 299970883; '5' : 299980175; '6' : 300010123; '7' : 300025696
    '8' : 300008447; '9' : 300001968; Chi square = 7.40

   3,221,220,000-th digits of pi and 1/pi;
    pi  : 34929 13958 77673 90319
    1/pi: 21355 75330 20722 29175
                                ^
                        3,221,220,000-th

   (First digit '3' for pi or '0' for 1/pi is not included in the above 
count.)

  Programs were written by Mr. Daisuke TAKAHASHI, a member of Kanada 
Lab.
  CPU used was HITAC S-3800/480 at the Computer Centre, University of 
Tokyo.
  Two CPU were definitely used through single job parallel processing 
for
  both program run.

   Yasumasa KANADA
   Computer Centre, University of Tokyo
    Bunkyo-ku Yayoi 2-11-16
    Tokyo 113 Japan
    Fax   : +81-3-3814-7231 (office)
    E-mail: kanada@pi.cc.u-tokyo.ac.jp
    ========================

     As you can see from the above text, your algorithm is faster than
    Gauss-Legendre algorithm because Mr. Takahashi developed new faster
    algorithm for high-end pi calculation.

    Yasumasa

    Yasumasa Kanada (kanada@pi.cc.u-tokyo.ac.jp) just announced computation
    and verification of 4.29496 billion decimal digits. Documents at
    ftp://www.cc.u-tokyo.ac.jp.
    
    Interesting that they computed pi to billions of decimal places but
    reported the number of decimal places to only five decimal places.
    
    
    				-- edp
    
    
Public key fingerprint:  8e ad 63 61 ba 0c 26 86  32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.

    The ten-billionth hexadecimal digit of pi is 9, according to Simon
    Plouffe, Peter Borwein, and David Bailey.  They have algorithms for
    computing the d-th digit with very little memory and run times on the
    order of d.  More inform is at http://www.cecm.sfu.ca/~pborwein and
    http://www.mathsoft.com/asolve/plouffe/plouffe.html.
    
    
    				-- edp
    
    
Public key fingerprint:  8e ad 63 61 ba 0c 26 86  32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.