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Old 12-09-2005, 05:03 PM   #1
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Default calculating pi

is there a way to have ur comp just like calculate pi to like the end?

just like keep calculating it and saving it to a wordpad file or something?
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Old 12-09-2005, 05:37 PM   #2
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Default Re: calculating pi

There is no end
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Old 12-09-2005, 05:49 PM   #3
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Default Re: calculating pi

Quote:
Originally Posted by MarxSoccer
There is no end
No one has yet proven pi to be an irrational number, hence there may or may not be an end: that is the question
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Old 12-09-2005, 06:10 PM   #4
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Default Re: calculating pi

well if you got alot of spare times i suppose you could attempt to find an end if there is one
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Old 12-09-2005, 06:22 PM   #5
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Default Re: calculating pi

There are super computers that have been trying to calculate pi for years now. I have no idea what the current number of digits is, since it is growing every second. If you really want to try to start working on it, here's your match to the fire ..





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.

The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of pi (51,539,600,000 decimal digits to be precise).

Nick Johnson-Hill has an interesting page of pi trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm

This computations were made by Yasumasa Kanada, at the University of Tokyo.

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. 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.

An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. The method is based on the formula:

pi = sum_(i = 0)^oo (1 16^i) ((4 8i + 1) - (2 8i + 4) - (1 8i + 5) - (1 8i + 6))

in O(N) time and O(log N) space. (See references.)

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-cf[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.

D. H. Bailey, P. B. Borwein, and S. Plouffe. A New Formula for Picking off Pieces of Pi, Science News, v 148, p 279 (Oct 28, 1995). also at http://www.cecm.sfu.ca/~pborwein

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.

Classical Constants and Functions: Computations and Continued Fraction Expansions D.V.Chudnovsky, G.V.Chudnovsky, H.Cohn, M.B.Nathanson, eds. Number Theory, New York Seminar 1989-1990.

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.
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Old 12-09-2005, 06:23 PM   #6
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Default Re: calculating pi

And here:



How to compute digits of ?

Symbolic Computation software such as Maple or Mathematica can compute 10,000 digits of 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 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 on a homebrew computer.

The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of (51,539,600,000 decimal digits to be precise).

Nick Johnson-Hill has an interesting page of  trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm

The new record for the number of digits of is 4.29496 billion decimal digits of pi were calculated and verified by 28th August '95.

Related documents are available with anonymous ftp to www.cc.u-tokyo.ac.jp.

ftp://www.cc.u-tokyo.ac.jp/

This computations were made by Yasumasa Kanada, at the University of Tokyo.

There are essentially 3 different methods to calculate to many decimals.

1. One of the oldest is to use the power series expansion of together with formulas like . 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 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

Then , where

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. This is why the constant appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals.

An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. The method is based on the formula:

in O(N) time and O(log N) space. (See references.)

The following 160 character C program, written by Dik T. Winter at CWI, computes to 800 decimal digits.

int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-cf[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 American Mathematical Monthly, vol. 96, no. 3 (March 1989), p. 201-220.

D. H. Bailey, P. B. Borwein, and S. Plouffe. A New Formula for Picking off Pieces of Pi, Science News, v 148, p 279 (Oct 28, 1995). also at http://www.cecm.sfu.ca/~pborwein

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 . Mathematics of Computation, Vol. 46, 1986, pp. 247-253.

Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computation of 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.

Classical Constants and Functions: Computations and Continued Fraction Expansions D.V.Chudnovsky, G.V.Chudnovsky, H.Cohn, M.B.Nathanson, eds. Number Theory, New York Seminar 1989-1990.

Y. Kanada and Y. Tamura. Calculation of 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 . Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp. 199-217.

E. Salamin. Computation of using arithmetic-geometric mean. Mathematics of Computation, Vol. 30, 1976, pp. 565-570

David Singmaster. The legal values of . The Mathematical Intelligencer, Vol. 7, No. 2, 1985.

Stan Wagon. Is normal? The Mathematical Intelligencer, Vol. 7, No. 3, 1985.

A history of . P. Beckman. Golem Press, CO, 1971 (fourth edition 1977)

and the AGM - a study in analytic number theory and computational complexity. J.M. Borwein and P.B. Borwein. Wiley, New York, 1987.
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Old 12-09-2005, 06:33 PM   #7
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Default Re: calculating pi

Look at my Super Pi thread That does calculate Pi to up to 32million places. You can also apply for a program which will do more
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Old 12-09-2005, 06:36 PM   #8
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Default Re: calculating pi

That program is stupid. Pi is alot longer than that program computes. Sure if you just want so many digits, but..
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Old 12-09-2005, 06:47 PM   #9
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Default Re: calculating pi

Quote:
Originally Posted by MarxSoccer
That program is stupid. Pi is alot longer than that program computes. Sure if you just want so many digits, but..
its may be lot longer but really, whats the point??

if you have pi to 30 billion places? you can not possibly comprehend the tiny difference between that and *only* 25 billion deciaml places! unless your trying to calculate the size of the universe whats the point exactly?! and what in the end...? you might be a few miles out in millions of light years?!

sorry that sounds like a rant its not really meant to be! lol and yes it is a bit sarcastic...
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Old 12-09-2005, 06:51 PM   #10
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Default Re: calculating pi

Quote:
Originally Posted by MarxSoccer
That program is stupid. Pi is alot longer than that program computes. Sure if you just want so many digits, but..
If you contact the makers, you can apply to have the program which will do Pi to the world record, which is 4.2 billion places Long enough?
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