Pi to x Million Decimal PlacesDate: 1 Aug 1995 18:14:05 -0400 From: John Wallbank Subject: Pi Calculation Please explain _how_ Pi is calculated to X million decimal places? Date: 3 Aug 1995 14:05:53 -0400 From: Dr. Ken Subject: Re: Pi Calculation Hello there! Here's something I pulled off the FAQ file for the newsgroup sci.math. If you want to see what else they've got, you can either use the WWW URL ftp://ftp.belnet.be/pub/usenet-faqs/usenet-by-hierarchy/ news/answers/sci-math-faq/ or you can ftp directly to ftp.belnet.be and find it in the same directory. What I got is ftp://ftp.belnet.be/pub/usenet-faqs/usenet-by-hierarchy/ news/answers/sci-math-faq/specialnumbers/computePi Also, there's a really nice article in a _New Yorker_ from a few years back about the Chudnovsky brothers. I'm not sure when it was, though. Maybe '91? Here it is! Date: Tue, 25 Apr 1995 17:42:10 GMT From: Alex Lopez-Ortiz Subject: sci.math FAQ: How to compute Pi? Newsgroups: sci.math,sci.answers,news.answers 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; 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. _________________________________________________________________ -K |
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