Plane Geometry - Bruce E. Shapiro

SECTION 44. ESTIMATING π 249algorithm,whereS =π = k √6 k3S∞∑(−1) n (6n)!(k 2 + nk 1 )n! 3 (3n)!(8k 4 k 5 ) nn=0and the constants are k 1 = 545140134, k 2 = 13591409, k 3 = 640320, k 4 =100100025, k 5 = 327843840 and k 6 = 53360. The Chudnuvskys used thisformula to calculate over a billion digits in the 1980’s. The original formulais actually due to Srinvasa Ramaujan.The first 100 billion digits of ı are posted at http://ja0hxv.calico.jp/pai/epivalue.html.An iterative based on results of Gauss and Legendre was developed byRichard Brent and Eugene Salamin in 1975 and forms the basis of thepopular superpi program that runs under Windows. The Salamin-BrentAlgorithm sets initial values a 0 = 1, b 0 = 1/ √ 2, t 0 = 1/4, and p 0 = 1, thenrepeats the following calculation until the desired accuracy is reached:a n+1 = a n + b n2b n+1 = √ a n b nt n+1 = t n − p n (a n − a n+1 ) 2p n+1 = 2p nπ n+1 = (a n + b n ) 24t nThis algorithm converges very quickly, with the number of correct digitsapproximately doubling with every iteration. Several versions of thisprogram have been posted on Yasumasa Kanada’s website (http://www.super-computing.org). has used supercomputers to calculate over a trilliondigits.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.