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.

The current record is held by Yasumasa Kanada and Daisuke Takahashi
from the University of Tokyo
with 51 billion digits of *pi * (51,539,600,000 decimal
digits to be precise).

Nick Johnson-Hill has an interesting page of *pi *
trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm

This computations were made by Yasumasa Kanada, at the University of Tokyo.

There are essentially 3 different methods to calculate *pi *
to many decimals.

- 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. - 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. -
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;**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. 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.

An interesting new method was recently proposed by David Bailey,
Peter Borwein and
Simon Plouffe. It can compute the *N*th ** hexadecimal** digit of Pi
efficiently without the previous *N-1* digits. The method is based
on the formula:

*pi = sum_(i = 0)^oo (1 16^i) ((4 8i + 1) - (2 8i + 4) - (1 8i + 5) - (1 8i + 6))*

in *O(N)* time and *O(log N)* space. (See references.)

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);}

* P. B. Borwein, and D. H. Bailey.* ** Ramanujan, Modular Equations,
and Approximations to pi **

* D. H. Bailey, P. B. Borwein, and S. Plouffe.* ** A New Formula for
Picking off Pieces of Pi,** * Science News,* v 148, p 279 (Oct 28, 1995).
also at ` http://www.cecm.sfu.ca/~pborwein `

* 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 .**

* Shlomo Breuer and Gideon Zwas* ** Mathematical-educational aspects of the computation of pi **

* David Chudnovsky and Gregory Chudnovsky.* ** The computation of classical constants.** * Columbia University,
Proc. Natl. Acad. Sci. USA,* Vol. 86, 1989.

* Classical Constants and Functions: Computations and Continued
Fraction Expansions* * D.V.Chudnovsky, G.V.Chudnovsky, H.Cohn, M.B.Nathanson, eds.* Number Theory, New York Seminar 1989-1990.

* Y. Kanada and Y. Tamura.* ** Calculation of pi to 10,013,395 decimal places based on the
Gauss-Legendre algorithm and Gauss arctangent relation.**

* Morris Newman and Daniel Shanks.* ** On a sequence arising in series for pi .**

* E. Salamin.* ** Computation of pi using arithmetic-geometric mean.**

* David Singmaster.* ** The legal values of pi .**

* Stan Wagon.* ** Is pi normal?**

* A history of pi .*

* pi and the AGM - a study in analytic number theory and
computational complexity.*

Fri Feb 20 21:45:30 EST 1998