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sci.math FAQ: How to compute Pi?


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Archive-Name: sci-math-faq/specialnumbers/computePi
Last-modified: December 8, 1994
Version: 6.2


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. The third, and perhaps the most elegant in its simplicity, arises
       from the construction of a large circle with known radius. The
       length of the circumference is then divided by twice the radius
       and pi is evaluated to the required accuracy. The most ambitious
       use of this method was successfully completed in 1993, when 
       H. G. Smythe produced 1.6 million decimals using high-precision
       measuring equipment and a circle with a radius of a staggering
       nine hundred and fifty miles.



   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
    Tue Apr 04 17:26:57 EDT 1995




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