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From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: How to compute Pi?
Summary: Part 12 of many, New version,
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Archive-Name: sci-math-faq/specialnumbers/computePi
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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