Approximation of pi

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evinda
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Approximation of pi

Postby evinda » Tue Oct 29, 2013 10:58 pm UTC

Hey!
I am writing a code in C that finds an approximation of pi,with single and double precision, using the relation the sum Σ{1/k^2, k=1....n}, knowing that pi2/6=Σ{1/k^2, k=1....oo}.

I used a for loop to calculate the sum s=Σ{1/k^2, k=1....n}.

For n=100, when I declared the variables as float I found:3.1320766431412150. And when the variables are double, my output is: 3.1320765318091053
Could you tell me if these results are right?

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z4lis
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Re: Approximation of pi

Postby z4lis » Wed Oct 30, 2013 1:34 am UTC

Since Google says that the first few digits of pi are 3.14159265359 and you didn't post any of your code for us to look at, the only reasonable answer I can think to give you is... no. But I do wonder why it's so far off. What happens for higher value of n?
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Derek
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Re: Approximation of pi

Postby Derek » Wed Oct 30, 2013 1:45 am UTC

Wolfram Alpha says that the approximation you got is about right for your method. The sum of inverse squares approximation converges very slowly, 100 terms just isn't enough to get the first three digits. I suggest using a different method to calculate the digits.

fishfry
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Re: Approximation of pi

Postby fishfry » Wed Oct 30, 2013 3:21 am UTC

I did this once with the series pi/4 = 1/3 - 1/5 + 1/7 - 1/9 ...

The convergence is astonishingly slow. I ran my program for days just to get 10 or 12 decimal places.

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Jplus
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Re: Approximation of pi

Postby Jplus » Wed Oct 30, 2013 9:34 am UTC

Because of the way computers round floats, you might get more accurate results if you start with the greatest value of k (so the least value of 1/k2).
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PM 2Ring
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Re: Approximation of pi

Postby PM 2Ring » Wed Oct 30, 2013 9:44 am UTC

Python agrees.
>>> n=100;s=sum(1.0/(i*i) for i in xrange(1,n+1));(6*s)**0.5
3.1320765318091053

>>> n=1000000;s=sum(1.0/(i*i) for i in xrange(1,n+1));(6*s)**0.5
3.1415916986605099


fishfry wrote:I did this once with the series pi/4 = 1/3 - 1/5 + 1/7 - 1/9 ...

The convergence is astonishingly slow. I ran my program for days just to get 10 or 12 decimal places.

Well that's the slowest converging arctan Taylor series, so one shouldn't expect it to run quickly. :) And with so many terms, you tend to get a lot of rounding error, although the alternating terms do help a bit.

If you want to calculate pi using arctan series, it's a Good Idea to use one of the Machin-Like formulae.

FWIW, I once used arctan Taylor series and
arctan(1) = 4 * (2*arctan(1/10) - arctan(1/515)) - arctan(1/239)
to compute pi to 10 decimals, by hand.

Also see http://turner.faculty.swau.edu/mathemat ... forms.html

jedelmania
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Re: Approximation of pi

Postby jedelmania » Wed Oct 30, 2013 10:13 am UTC

fishfry wrote:I did this once with the series pi/4 = 1/3 - 1/5 + 1/7 - 1/9 ...

The convergence is astonishingly slow. I ran my program for days just to get 10 or 12 decimal places.


You can convert the series pi/4 = 1/3 - 1/5 + 1/7 - 1/9 ...

into

pi/2 = sum(n=0 to infinity) { [(2n)! (2n-2)!] / [ (2n+1)! (2n-1)!] x [1/2]^n } [for notational convenience, write (-1)!=(-2)!=1].

This converges pretty fast, although the large factorials may blow up your computer.

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Qaanol
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Re: Approximation of pi

Postby Qaanol » Wed Oct 30, 2013 1:16 pm UTC

On a tangentially-related note, I recently learned a neat proof that 22/7 exceeds π:

[; \int_0^1 \frac{x^4(1-x)^4}{1+x^2} dx ;]

The integrand is positive on (0, 1), and by polynomial division reduces to a polynomial plus a remainder term that integrates to a nice arctan value, and the result evaluates to 22/7 - π.

Also, less related, I have memorized some best-rational-approximations that are within half a millionth of the correct values (digits in black agree, the rest are grey):

355/113 = 3.14159292
π = 3.14159265

2721/1001 = 2.71828171
e = 2.71828182

1597/987 = 1.61803444
φ = 1.61803398

1393/985 = 1.41421319
√2 = 1.41421356

228/395 = 0.57721518
γ = 0.57721566

Of course, for most purposes I am still a big fan of sevenths:

22/7 = 3.14285714
19/7 = 2.71428571
11⅓/7 = 1.61904761
9.9/7 = 1.41428571
4/7 = 0.57142857

Edit: fixed integrand (thanks Nicias)
Last edited by Qaanol on Thu Oct 31, 2013 7:07 pm UTC, edited 2 times in total.
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Nicias
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Re: Approximation of pi

Postby Nicias » Wed Oct 30, 2013 1:54 pm UTC

Qaanol wrote:On a tangentially-related note, I recently learned a neat proof that 22/7 exceeds π:

[; \int_0^1 \frac{x^4(1-x)^4}{1+x} dx ;]

You forgot a ^2.

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PM 2Ring
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Re: Approximation of pi

Postby PM 2Ring » Thu Oct 31, 2013 6:57 am UTC

jedelmania wrote:You can convert the series pi/4 = 1/3 - 1/5 + 1/7 - 1/9 ...

into

pi/2 = sum(n=0 to infinity) { [(2n)! (2n-2)!] / [ (2n+1)! (2n-1)!] x [1/2]^n } [for notational convenience, write (-1)!=(-2)!=1].

This converges pretty fast, although the large factorials may blow up your computer.

I don't think that's right. And the factorials in the terms of that series can be canceled.


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