See reader questions & answers on this topic!  Help others by sharing your knowledge ArchiveName: scimathfaq/specialnumbers/computePi Lastmodified: 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 20,0001,000,000 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 ArithmeticGeometric 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 highprecision 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. 201220. J.M. Borwein and P.B. Borwein. The arithmeticgeometric mean and fast computation of elementary functions. SIAM Review, Vol. 26, 1984, pp. 351366. J.M. Borwein and P.B. Borwein. More quadratically converging algorithms for pi . Mathematics of Computation, Vol. 46, 1986, pp. 247253. Shlomo Breuer and Gideon Zwas Mathematicaleducational aspects of the computation of pi Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231244. 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 GaussLegendre 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. 199217. E. Salamin. Computation of pi using arithmeticgeometric mean. Mathematics of Computation, Vol. 30, 1976, pp. 565570 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. _________________________________________________________________ alopezo@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 User Contributions:
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