crds@NCoast.ORG (Glenn A. Emelko) (01/07/91)
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:
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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
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James Gregory, 1671, VERY slow to converge: GREGORY'S SERIES
Pi 1 1 1 1
-- = 1 - - + - - - + - ....
4 3 5 7 9
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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
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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
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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.
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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)
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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
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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.kdq@demott.com (Kevin D. Quitt) (01/08/91)
Why is it that all the good formulas for PI are really for 1/PI? Is inversion using an infinite/very-high precision package that easy? -- _ Kevin D. Quitt demott!kdq kdq@demott.com DeMott Electronics Co. 14707 Keswick St. Van Nuys, CA 91405-1266 VOICE (818) 988-4975 FAX (818) 997-1190 MODEM (818) 997-4496 PEP last
brnstnd@kramden.acf.nyu.edu (Dan Bernstein) (01/08/91)
In article <1991Jan7.213551.14050@demott.com> kdq@demott.COM (Kevin D. Quitt) writes: > Why is it that all the good formulas for PI are really for 1/PI? Not all of them are. > Is inversion using an infinite/very-high precision package that easy? Yeah, pretty much. It's a royal pain to get working the first time, but only a few times slower than multiplication after that. ---Dan
bnolan@ccvax.ucd.ie (Breanndan O Nuallain) (01/15/91)
In article <1991Jan6.233340.808@NCoast.ORG>, crds@NCoast.ORG (Glenn A. Emelko) writes: > 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 >... > 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! > Phew! He was some calculator, that Gauss fella! I reckon factorial twenty-six is about 4 * 10**26. That musta been his life's work! -- ____________________________________________________________ , , , Breanndan O Nuallain Department of Computer Science University College Dublin Phone: +353-1-693244 Belfield, Dublin 4 ext 2487 Ireland ------------------------------------------------------------
dbell@cup.portal.com (David J Bell) (01/16/91)
>> John Gauss, about 1850, VERY slow to converge: >> >> Gauss went on to find similar equations for >> Pi^3 thru Pi^26! >> >Phew! He was some calculator, that Gauss fella! I reckon >factorial twenty-six is about 4 * 10**26. That musta been >his life's work! , , , >Breanndan O Nuallain Department of Computer Science > University College Dublin And then, he went and raised PI to that power - with GAMMA, I made it about 6.8e28.... Dave dbell@cup.portal.com