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From: crds@NCoast.ORG (Glenn A. Emelko)
Newsgroups: comp.misc
Subject: Here are FORMULAS for PI!
Message-ID: <1991Jan6.233340.808@NCoast.ORG>
Date: 6 Jan 91 23:33:40 GMT
Organization: North Coast Public Access *NIX, Cleveland, OH
Lines: 97

Okay, okay -- I waited to see if anyone was going to actually put forth
any formulas for Pi before I throw in my $1.85 worth of commentary.  Here
are some formulas for Pi which have historical significance, as best that
I can represent them in textual form:
-------------------------------------------------------------------------
John Wallis, 1665, VERY slow to converge:                 WALLIS' PRODUCT
 
Pi   2*2   4*4   6*6   8*8
-- = --- * --- * --- * --- ....
2    1*3   3*5   5*7   7*9
-------------------------------------------------------------------------
James Gregory, 1671, VERY slow to converge:              GREGORY'S SERIES
 
Pi       1   1   1   1
-- = 1 - - + - - - + - ....
4        3   5   7   9
-------------------------------------------------------------------------
John Machin, 1706, reasonable convergence:               MACHIN'S FORMULA
 
Pi
-- = 4 * arctan(1/5) - arctan(1/239)
4
 
		n^3   n^5   n^7
arctan(n) = n - --- + --- - --- + ....
		 3     5     7
-------------------------------------------------------------------------
John Gauss, about 1850, VERY slow to converge:
 
Pi^2        1     1     1     1
---- = 1 + --- + --- + --- + --- ....
  6        2*2   3*3   4*4   5*5
 
This is notable because Gauss succeeded in solving the elusive summation
of the reciprocals of the squares of the natural numbers -- other than
that it has little value.  Gauss went on to find similar equations for
Pi^3 thru Pi^26!
 
       infinity
Pi^2    ____    1
---- =  \      ---
  6     /___   n^2
         n=1
-------------------------------------------------------------------------
Srinivasa Ramanujan, 1914, fast convergance (about 8 digits per term):
 
             infinity
1    sqr(8)   ____    (4n)! * (1103+26390n)
-- = ------ * \       ---------------------
Pi    9801    /___      (n!)^4 * 396^(4n)
               n=0
 
Note that by definition, 0! = 1.
-------------------------------------------------------------------------
Johnathan M. Borwein and Peter B. Borwein, 1986, EXTREMELY fast:
 
(A computer iterative approach, the number of correct digits increases
 by a factor of four for each term!  This is not the fastest converging
 algorithm they developed, but it may be the most efficient.  This is
 *the* algorithm used on supercomputers to produce the 1986 record of
 29,360,000 decimal places on a Cray-2 by David H. Bailey, iterating
 it only 12 times, also to set the 1987 record of 134,217,000 places
 on a NEC SX-2 by Yasumasa Kanada and his colleagues.)
 
Y(0) = sqr(2) - 1         A(0) = 6 - 4sqr(2)
 
For each iteration (start with n=0):
 
         1 - (1-Y(n)^4)^(1/4)
Y(n+1) = --------------------
         1 + (1-Y(n)^4)^(1/4)
 
Compute the result Y(n+1) and then use it in the following:
 
A(n+1) = [(1+y(n+1)^4) * A(n)] - [2^(2n+3) * Y(n+1) * (1+Y(n+1)+Y(n+1)^2)]
 
If you wish to do another iteration, increment n and go for it.
 
When you are done, Pi = 1/A(n+1).
 
My sincere apologies to Borwein and Borwein for butchering this so badly
(it's accurate but hardly readible)
-------------------------------------------------------------------------
The article mentioned in the bibliography below about Ramanujan is very
interesting, and I highly recommend it for all to read.  It also contains
most of the the formulas I transcribed above, plus numerous others and
alot of the theory that went along with their development.
 
Enjoy!
 
Glenn A. Emelko
-------------------------------------------------------------------------
Bibliography:
 
Ramanujan and Pi, Jonathan M. Borwein and Peter B. Borwein, Scientific
     American, February 1988, Volume 258 Number 2
A History of Pi, Peter Beckmann, The Golam Press, 1977.