The ultimate home computer sets pi record

[lazarus]

This computer just calculated pi to a world record 5 trillion digits.
Processor
2 x Intel Xeon X5680 @ 3.33 GHz - (12 physical cores, 24 hyperthreaded)
Memory
96 GB DDR3 @ 1066 MHz - (12 x 8 GB - 6 channels) - Samsung (M393B1K70BH1)
Motherboard
Asus Z8PE-D12
Hard Drives
1 TB SATA II (Boot drive) - Hitachi (HDS721010CLA332)
3 x 2 TB SATA II (Store Pi Output) - Seagate (ST32000542AS)
16 x 2 TB SATA II (Computation) - Seagate (ST32000641AS)
Raid Controller
2 x LSI MegaRaid SAS 9260-8i
Operating System
Windows Server 2008 R2 Enterprise x64
Built By
Shigeru Kondo

Reading further, it cost him $18,000 to build, and he used software from an American grad student.

The 5 trillionth digit of pi is, of course, 2.

[whistler162]

1. Oooops..... should someone tell him it is actually 7?

;)

[pokerfan]

2. What is pi?

Mathematician: Pi is the number expressing the relationship between the circumference of a circle and its diameter.
Physicist: Pi is 3.1415927plus or minus 0.00000005.
Engineer: Pi is about 3.

[struggle4progress]

3. Politician: Pi is 3

[hunter]

4. 355/113 is plenty close enough for most practical purposes.

Buffon's needle and all that.

Has any physicist ever tested some corner of the universe to see if the Hypothetical Programmer of This Universe (the "SF" as Erdös would say) uses this approximation anywhere?

[matt819]

5. Those Russian brothers in NY are going to be pissed

I can't remember their names - there was a New Yorker piece on them some time back. They built their machine in their apartment. Great story.

[pnorman]

6. Pi? Here's an algorith I came across a while back:

p=2: k=1
start:
s=sqr(s+2)
p=2*p/s
print p
k=k+1
goto start

I have a Palm Basic program in my cell-phone, and it converges to the correct value very rapidly. A while back, I ran it in a device with double-precision accuracy, and (except for the last digit) it also worked. I suspect that it'll be accurate to whatever precision the machine is capable of. Can anyone here explain the "math" behind that simple algorithm?

[struggle4progress]

7. I think it is probably based on an approximation of pi by regular 2^n-gons

Consider a regular 2^n-gon inscribed in the unit circle; it can be decomposed into 2^n isosceles triangles with sides 1, base b(n), and height h(n)

We have from Pythagoras
4*h(n)^2 + b(n)^2 = 4
and
b(n)^2 + (2 - 2*h(n))^2 = 4*b(n+1)^2

Adding the above appropriately and simplifying yields
b(n+1) = sqrt(2 - 2*h(n))
and plugging this back into the first Pythagoras relation above gives
2*h(n) = sqrt(2 + 2*h(n-1))

So if we start with the regular 2^2-gon, having b(2) = sqrt(2) and h(2) = sqrt(2)/2, the successive values of "s" in your algorithm will track 2*h(n)

If the nth approximation of pi is the half the circumference of the regular inscribed 2^n-gon
p(n) = 2^(n-1)*b(n)
then these approximations are obtained successively from p(2) = 2*sqrt(2) by the recursion
p(n+1) = p(n)*2*b(n+1)/b(n)

But
b(n+1)/b(n) = sqrt(2 - 2*h(n))/b(n) = sqrt(4 - 4*h(n)^2)/(b(n)*sqrt(2 + 2*h(n))) = b(n)/(b(n)*sqrt(2 + 2*h(n))) = 1/sqrt(2 + 2*h(n)) = 1/(2*h(n+1))
and so the successive multiplications by 2/s in your algorithm are simply these successive factors 2*b(n+1)/b(n)

[pnorman]

8. THANKS for that!

I'll save it for later and careful study. I had assumed that it had something to do with the Archimedes Solution: http://ualr.edu/lasmoller/pi.html, but I always thought of that as very slow to converge.

In any event, that simple algorithm should calculate to whatever accuracy the machine is capable of. I once had an HP-200 hand-held computer with Derive installed in it. That would have allowed me to specify the desired accuracy. If I can only find it, I'll try that algorithm!

[struggle4progress]

9. You can do it on your computer with a free program like this:

http://www.scilab.org/products/scilab/download

-->s = sqrt(2); p = 2*sqrt(2);
-->for k=1:5 do s = sqrt(2 + s), p = 2*p/s, end
s =
1.8477591
p =
3.0614675
s =
1.9615706
p =
3.1214452
s =
1.9903695
p =
3.1365485
s =
1.9975909
p =
3.1403312

[pnorman]

10. Thanks, I'll give it a try later.

Does it allow me to set it to ANY precision that I desire?

[struggle4progress]

11. It's double-precision arithmetic, whatever double-precision is on your box

I expect the language is rich enough to allow you to write your own (say) bitwise +,-,*,/, and sqrt algorithms for arrays if you really want arbitrary precision

On my machine, double-precision is something like sixteen decimal places

This BTW is not a terribly fast pi algorithm; much better algorithms are known from the 80s and 90s