From msuinfo!agate!usenet.ins.cwru.edu!news.ecn.bgu.edu!siemens!princeton!phoenix.Princeton.EDU!rwsayer Sun Oct 17 00:15:06 1993
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From: rwsayer@phoenix.Princeton.EDU (Ronald Winston Sayer)
Subject: Re: Digits of pi
Message-ID: <1993Oct13.010448.20634@Princeton.EDU>
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Date: Wed, 13 Oct 1993 01:04:48 GMT
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Xref: msuinfo sci.crypt:20364 sci.math:54578
>Does someone have a simple formula for computing the digits of
>pi that can be implemented using a basic language like C??
>
>I know its a loaded question, but I'll take any help I can get.
>
>Tod
Well, there's the venerable
PI/4 = 1 - 1/3 +1/5 -1/7 +1/9 - ......
but that's rediculously wimpy.
Try this
PI/4 = 4*ATAN (1/5) - ATAN(1/239)
where ,
ATAN(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + .....
This converges nicely, and will easily give you 10,000
didgits of PI on any reasonable workstation, or maybe even a PC.
There are more complicated ways to calculate PI that converge
much more rapidly, but this is one of my favorite due to its
simplicity.
Ron Sayer
rwsayer@phoenix.princeton.edu
From msuinfo!agate!usenet.ins.cwru.edu!magnus.acs.ohio-state.edu!math.ohio-state.edu!cs.utexas.edu!uunet!olivea!decwrl!pa.dec.com!e2big.mko.dec.com!peavax.mlo.dec.com!3d.enet.dec.com!roth Sun Oct 17 00:15:06 1993
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From: roth@3d.enet.dec.com (Jim Roth)
Subject: Re: Digits of pi
Message-ID: <1993Oct13.162340.4373@peavax.mlo.dec.com>
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References: <CErF0o.4nA@chinet.chinet.com> <29fg2e$55o@news.delphi.com> <HENRIK.JOHANSSON.93Oct13140722@bond.nexus.comm.se>
Date: Wed, 13 Oct 1993 17:26:35 GMT
Lines: 51
Xref: msuinfo sci.crypt:20374 sci.math:54604
In article <HENRIK.JOHANSSON.93Oct13140722@bond.nexus.comm.se>, Henrik.Johansson@nexus.comm.se (Henrik Johansson) writes...
>
>>Try this
>>
>>PI/4 = 4*ATAN (1/5) - ATAN(1/239)
>>There are more complicated ways to calculate PI that converge
>>much more rapidly, but this is one of my favorite due to its
>>simplicity.
>Do you happen to have online descriptions of those more rapidly
>converging calculations? If so - please make them available in some
>form! Either by email to me directly, or some math newsgroup.
In the latest revision of Numerical Recipes, there is a Brent-Salamin
AGM algorithm to compute PI, but I don't have it online.
For just 10K digits of PI, a simple program like this will suffice.
- Jim
/* 1000 digits of PI */
/* 'Spigot' algorithm origionally due to Stanly Rabinowitz */
#include <stdio.h>
main()
{
int d = 4, r = 10000, n = 251, m = 3.322*n*d;
int i, j, k, q;
static int a[3340];
for (i = 0; i <= m; i++) a[i] = 2;
a[m] = 4;
for (i = 1; i <= n; i++) {
q = 0;
for (k = m; k > 0; k--) {
a[k] = a[k]*r+q;
q = a[k]/(2*k+1);
a[k] -= (2*k+1)*q;
q *= k;
}
a[0] = a[0]*r+q;
q = a[0]/r;
a[0] -= q*r;
printf("%04d%s",q, i & 7 ? " " : "\n");
}
}
From msuinfo!uwm.edu!cs.utexas.edu!math.ohio-state.edu!darwin.sura.net!europa.eng.gtefsd.com!news.ans.net!malgudi.oar.net!witch!cyberg!fb Sun Oct 17 00:15:06 1993
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Reply-To: fb@cyberg.win.net (Francis Barrett)
From: fb@cyberg.win.net (Francis Barrett)
Date: Wed, 13 Oct 1993 15:33:06 GMT
Subject: Re: Digits of pi
Lines: 56
Xref: msuinfo sci.crypt:20379 sci.math:54608
In article <29fg2e$55o@news.delphi.com>,
TODANIELS@DELPHI.COM (todaniel@news.delphi.com) writes:
>Does someone have a simple formula for computing the digits of
>pi that can be implemented using a basic language like C??
>I know its a loaded question, but I'll take any help I can get.
Providing you have an efficient subroutine package for multiprecision
arithmetic, there are a number of quickly convergent iterations which
can be employed to calculate many digits of Pi.
My favorite is a quartic iteration based on an elliptic function called
the Singular Value Function of the Second Kind. It converges exponentially
to 1/Pi as its argument increases. There is a rich set of "modular
equations" which algebraically relate values of this function for
different values of its argument. Repeating these formulas as iterations
allows us to compute the value of this function for increasingly large
arguments and consequently the value of Pi.
This technique is based on the work of the eccentric Indian
mathematician Srinivasa Ramanujan.
The basic iteration goes like this...
Y[N+1] = (1-(1-Y[N]^4)^0.25)/(1+(1-Y[N]^4)^0.25)
A[N+1] = (1+Y[N+1])^4*A[N]-4^(N+1)*SQRT(R)*Y[N+1]*(1+Y[N+1]+Y[N+1]^2)
This will compute approximately 45 million decimal digits of Pi in
only 12 iterations with R = 4.
This formula works for any positive R, but the initial values of the
iteration, Y[0] and A[0] are dependent on the R chosen. A[0] should
be the Singular Value Function of the Second Kind evaluated at R and
Y[0] should be the square root of the another function, called the
Singular Value Function of the First Kind, again evaluated at R.
For instance, if we choose R = 2, the correct starting values turn out
to be A[0] = SQRT(2)-1 and Y[0] = SQRT(SQRT(2)-1).
Then the iteration looks like this...
N Y[N] A[N] 1/A[N]
0 0.6435942530 0.414213562 2.41421356
1 0.0235239602 0.318310704 3.14158459
2 0.0000000383 0.318309886 3.14159265
As you can see, the convergence to Pi is quite rapid, and each iteration
gives approximately four times the number of decimal places as the
previous one.
Using techniques like these, and an efficient multiprecision package,
computation of Pi to millions of places is well within the computational
power of the typical personal computer.
---------------------------------------------------------------
Francis Barrett, F.R.C. | Thou canst not travel on the path |
The Cybernetics Guild | before thou hast become the Path |
fb@cyberg.win.net | itself. |
---------------------------------------------------------------
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Message-ID: <16@cyberg.win.net>
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Reply-To: fb@cyberg.win.net (Francis Barrett)
From: fb@cyberg.win.net (Francis Barrett)
Date: Thu, 14 Oct 1993 19:55:32 GMT
Subject: Re: Digits of pi
Lines: 24
Xref: msuinfo sci.crypt:20427 sci.math:54715
In article <140722@bond.nexus.comm.se>,
Henrik Johansson (Henrik.Johansson@nexus.comm.se) writes:
>>Try this
>>PI/4 = 4*ATAN (1/5) - ATAN(1/239)
Before the latest approximations to Pi using modular equations were
developed, Taylor series expansions of arctans were the state of the
art. There are a huge number of these, all designed to make the
denominators of the series terms grow as rapidly as possible.
Besides the one above, the following are also popular...
Pi = 20*arctan(1/7) + 8*arctan (3/79)
Pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
Pi/4 = 3*arctan(1/4) + arctan(1/20) + arctan(1/1985)
Pi = 24*arctan(1/8) + 8*arctan(1/57) + 4*arctan(1/239)
pi = 48*arctan(1/18) + 32*arctan(1/57) - 20*arctan(1/239)
Pi has so far been computed to over a billion places and no pattern in
the digits has been discovered, other than the obvious one.
---------------------------------------------------------------
Francis Barrett, F.R.C. | Thou canst not travel on the path |
The Cybernetics Guild | before thou hast become the Path |
fb@cyberg.win.net | itself. |
---------------------------------------------------------------
From msuinfo!agate!howland.reston.ans.net!pipex!sunic!kth.se!vana!TORDM Sun Oct 17 00:15:06 1993
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From: tordm@vana (Tord G.M. Malmgren)
Subject: Re: Digits of pi
Message-ID: <1993Oct16.104528.6972@kth.se>
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Date: Sat, 16 Oct 1993 10:45:28 GMT
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Xref: msuinfo sci.crypt:20457 sci.math:54801
In article <16@cyberg.win.net>, fb@cyberg.win.net (Francis Barrett) writes:
>Besides the one above, the following are also popular...
>
> Pi = 20*arctan(1/7) + 8*arctan (3/79)
> Pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
> Pi/4 = 3*arctan(1/4) + arctan(1/20) + arctan(1/1985)
> Pi = 24*arctan(1/8) + 8*arctan(1/57) + 4*arctan(1/239)
> pi = 48*arctan(1/18) + 32*arctan(1/57) - 20*arctan(1/239)
and also two "faster" ones;
pi/4 = 22*arctan(1/28)+2*arctan(1/443)-5*arctan(1/1393)-10*arctan(1/11018)
pi/4 = 44*arctan(1/57)+7*arctan(1/239)-12*arctan(1/682)+24*arctan(1/12943)
which someone posted before (don't remember who.) To get even better
ones read "An Algorithm for The Calculation of \pi" by George Miel...
(I think "Classroom Notes" October 1979.)
---------------+--------------------------------+----------------------------
Tord Malmgren | InterNet: TordM@VanA.PhySto.SE | These opinions are my OWN,
| BITNet : TordM@SESUF51 | and NOT of this department!
---------------+--------------------------------+----------------------------
Department of Physics, University of Stockholm -- Sweden (Scandinavia)