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Estimating PI
- Thread starter pchoate
- Start date
The easy formula is
4/1 - 4/3 + 4/5 - 4/7 + 4/9....
But it converges very slowly. 16 decimal places takes adding a billion fractions or something. So I took a weighted average using excel and it drills down towards the answer very quickly, like a few dozen steps to get 16 places...
Can I post a worksheet in this forum?
This message was edited by pchoate on 2003-02-17 00:42
4/1 - 4/3 + 4/5 - 4/7 + 4/9....
But it converges very slowly. 16 decimal places takes adding a billion fractions or something. So I took a weighted average using excel and it drills down towards the answer very quickly, like a few dozen steps to get 16 places...
Can I post a worksheet in this forum?
This message was edited by pchoate on 2003-02-17 00:42
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No - I just don't have much time on the PC right now. I just tried Colo's utility but I don't have forms loaded (?) so it won't work.
Okay I can try to describe it. The formula
Pi=4/1 - 4/3 + 4/5 - 4/7...can be thought of as a sequence of points on an x-y chart oscilating around and converging on the x-axis as x (the denominator 1,3,5,7,...) goes to infinity.
My idea is similar to connecting the dots between these points, and then using their midpoints to find new dots. Then you connect the midpoints to get new a third set, then do the same to get a fourth set.
It turns out that using this method there is enough information in the first 30 terms or so of the sequence to approximate PI to 16 places.
If you did just with the sequence it would take 5,000,000,000,000 terms or something to get that close, because the denominator is directly proportional to the accuracy.
I'm not a number theorist, so I can't do a proof on why this works, or at least I'm not going to spend my time trying. :wink: I tried to put it into a "formula" but it's iterative, so it's pretty hard to write down. The spreadsheet itself is fairly simple. I came up with it when my son asked me for a good way to calculate Pi. I noticed how badly the sequence performed and so I tried a weighted average approach and to my surprise it works incredibly well in comparison.
I showed it to a friend who got a BA in math with me, he thought it was interesting and passed it on, but I haven't shown it to anyone who would really know if it's worth anything.
Where can I post this file? Is there a good anonymous FTP? I can email it.
This message was edited by pchoate on 2003-02-17 10:44
Okay I can try to describe it. The formula
Pi=4/1 - 4/3 + 4/5 - 4/7...can be thought of as a sequence of points on an x-y chart oscilating around and converging on the x-axis as x (the denominator 1,3,5,7,...) goes to infinity.
My idea is similar to connecting the dots between these points, and then using their midpoints to find new dots. Then you connect the midpoints to get new a third set, then do the same to get a fourth set.
It turns out that using this method there is enough information in the first 30 terms or so of the sequence to approximate PI to 16 places.
If you did just with the sequence it would take 5,000,000,000,000 terms or something to get that close, because the denominator is directly proportional to the accuracy.
I'm not a number theorist, so I can't do a proof on why this works, or at least I'm not going to spend my time trying. :wink: I tried to put it into a "formula" but it's iterative, so it's pretty hard to write down. The spreadsheet itself is fairly simple. I came up with it when my son asked me for a good way to calculate Pi. I noticed how badly the sequence performed and so I tried a weighted average approach and to my surprise it works incredibly well in comparison.
I showed it to a friend who got a BA in math with me, he thought it was interesting and passed it on, but I haven't shown it to anyone who would really know if it's worth anything.
Where can I post this file? Is there a good anonymous FTP? I can email it.
This message was edited by pchoate on 2003-02-17 10:44
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