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How Many Decimals of Pi Do We Really Need?
101–110 of 134 posts
Re: How Many Decimals of Pi Do We Really Need?
#102Tl;dr: 15
Re: How Many Decimals of Pi Do We Really Need?
#103Earlier quoted context omitted.
A branch of physics used to be taught a long time ago called "numerical analysis" to deal with this issue. We even used to be careful about the difference between 'precise and exact'. Pi = acos(0) is absolutely exact. But computer don't know about symbolic calculus. So to put the value in a register we used tricks. Pi as a the converging value at the infinite of the Taylor development is awesome. But computer don't k…
I can't speak for other nations, but they still teach numeric analysis in Chinese universities as an undergraduate course. In my university it is a required subject. Many of us have countless dreadful memories of Runge-Kutta method, Euler's method, Newton's method, rate of convergence, numerical stability and error margins, just to name a few of the dreads...
Taught to me at university. Far more important in the age of computers than before.
Re: How Many Decimals of Pi Do We Really Need?
#104Earlier quoted context omitted.
Never heard it stated that way. Can you elaborate on "the smallest possible distance is Plank's length"? Is that the smallest observable distance?
If you have some time to spare, watch this talk "The astonishing simplicity of everything" by Neil Turok. He explains all this so beautifully. https://www.youtube.com/watch?v=f1x9lgX8GaE
Well presented and really gets you thirsty for more, excellent talk – thank you very much for sharing!
Re: How Many Decimals of Pi Do We Really Need?
#105Re: How Many Decimals of Pi Do We Really Need?
#106Earlier quoted context omitted.
Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.
It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the pr…
If the universe were a computer, it would not have to compute π to "simulate" physical processes. Mechanical processes don't depend on the value of π directly.
If universe had to know every constant somehow involved in the process, it would not be able to simulate computers, because it involves uncomputable numbers: https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputa...
Re: How Many Decimals of Pi Do We Really Need?
#107Earlier quoted context omitted.
I agree, that's a poor answer by NASA director and chief engineer. Here is a better answer: The precision used for calculations is dependent on the number of "steps" required to get to the final result. Roughly, for N repeated calculations you lose somewhere between sqrt(N) * eps to N * eps of precision (eps=2e-16 for IEEE64). Here are some actual examples: IEEE64 (~16 decimal digits) is OK for interplanetary navigat…
Here are some actual examples: Something to add to your list of examples: During the first Gulf war, 28 US soldiers died due to accumulated rounding errors in the Patriot Missile battery computers: https://www.ima.umn.edu/~arnold/disasters/patriot.html (This was in fact a known issue, and operators had been instructed to reboot the computers every 8 hours. Unfortunately this instruction ignored the fact that, in the…
Yeah, that doesn't surprise me in the least, many high-tech military systems have MTBF/MTTF of a few hours at best. Also, that's what you get when you try to do radar time computations using 24-bit fixed point in Ada.
Back to astronomy, in many astronomy libraries (such as astropy library) computations regarding time are done using 2 doubles (about 106 bit precision). 1 double is not enough.
_brandmeyer_ also mentioned something important that I totally forgot - any trigonometric computation requires computing modulo-pi to an accuracy of 1 ulp, which requires storing PI to ~1144 bits for double precision (for numbers near pi) (see Kahan argument reduction paper).
Since Intel processsors don't reach the required precision for IEEE standard above pi/2, this modulo reduction is done in software to this day. gcc maintains a 1144 bit PI constant and does a 1144 bit modulo every time you compute a sine/cosine above pi.
TLDR - 344 decimal digits of PI are used. High-precision PI computation is surprisingly more common than we expect...
[1] http://docs.astropy.org/en/stable/time/index.html
[2] https://software.intel.com/en-us/blogs/2014/10/09/fsin-docum...
[3] https://gcc.gnu.org/gcc-4.3/changes.html
[4] http://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pd...
Re: How Many Decimals of Pi Do We Really Need?
#108I think this is a bit of an oversimplification. You must consider compounding when talking about rounding errors. A single matrix operation with hundreds of rows and columns can easily have millions of multiplications. At every multiplication the previous error gets multiplied. That's why I don't feel the answer was exhaustive.
Re: How Many Decimals of Pi Do We Really Need?
#109I think this is a bit of an oversimplification. You must consider compounding when talking about rounding errors. A single matrix operation with hundreds of rows and columns can easily have millions of multiplications. At every multiplication the previous error gets multiplied. That's why I don't feel the answer was exhaustive.
> At every multiplication the previous error gets multiplied
This is a bit of an oversimplification as well, it's not like you keep multiplying pi with itself over and over again and it's not like the error you introduce is random, if you've rounded pi once, you're gonna keep make a slight error in the same direction.
If you were right there'd be no hope of ever getting sane results when multiplying largish matrices of doubles regardless of the presence of pi.
I'm not saying that accumulation of error doesn't exist, I'm just saying that it's not to the extremes you're describing.
Re: How Many Decimals of Pi Do We Really Need?
#110Earlier quoted context omitted.
I agree, that's a poor answer by NASA director and chief engineer. Here is a better answer: The precision used for calculations is dependent on the number of "steps" required to get to the final result. Roughly, for N repeated calculations you lose somewhere between sqrt(N) * eps to N * eps of precision (eps=2e-16 for IEEE64). Here are some actual examples: IEEE64 (~16 decimal digits) is OK for interplanetary navigat…
So don't use a lossy floating point representation and use arbitrary precision instead. Just because π is estimated to 15 significant digits doesn't mean the entire calculation needs to be.
Arbitrary precision is not a valid answer: every multiplication doubles the number of mantissa bits. Starting with a 64-bit precision, after just 40 multiplications, you will have consumed 800 GB of RAM just for storing a single number, at which point you'll ask yourself: how many decimal digits do I really need? Which was the initial question...