Pi approximation games

But that isn't how the trick was done. There is a cute expression for the Nth hexadecimal digit representation of pi detailed at:

No-one in their right mind would start from that crude series today - its convergence for x=1 is absolutely hopeless. See for example:

The algorithm that can supply the Nth digits of pi lazily *only* works for hexadecimal digits (or base 2^N for certain N).

It doesn't work at all for the *decimal* representation of pi.

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Regards,
Martin Brown

I have in my hands an "Nelson's Encyclopedia" printed in 1951 (12 years before the international inch) it says the metre (then defined by scratches on a bar in Paris) is is about 1.0936143 yards (then defined by scratches on a bar in London)

1 / 36 / 1.0936143 comes to .025399977m

so, yes the American inch was the bigger of the two.

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I think you missed the earlier post in which I said that I /did/ do this, as a project at university. There may be other ways to calculate pi that are more efficient in base 16 - but that's a different matter entirely.

As I said, it was a university project. The aim was not to calculate pi, but to understand the usage of infinite lists, infinite precision numbers, ways to manipulate them, and ways to prove that the code to do so is correct. The rate of convergence is utterly irrelevant.

Just for fun, I figured out an alternative series that was faster to converge (possibly (atan(1/2) + atan(1/3)), though I can't remember for sure).

Nonsense. I did it, and I didn't find it particularly hard.

Perhaps you misunderstand what "lazy evaluation" actually is. The algorithm you reference is a way to calculate a hexadecimal digit of pi without having to calculate the previous digits - no such method is known for calculating the decimal digits (though that does not mean one does not exist). But that has nothing at all to do with lazy evaluation, which simply means that the relevant numbers are calculated when needed rather than in advance. It means you can work with an unlimited list, and treat it as though it were a fully calculated infinite list, without the computer actually having to calculate anything until the last minute.

We just have to slightly increase the value of 1, so pi will equal 3. :-)

-jm

An YES answer to a question that I have had rattling around for a while, to wit -

Are there properties (aside from trivial ones) of base-n numbers not shared by base-a-different-n numbers.

*Why* can you skip ahead to a hex digit of pi but not a dec digit?

Is this unique to pi and base 2^x, or is it true for certain irrational numbers, or all? If only some, are there similar (lack of) algorithms for other combinations of base and irrational number(s)?

I just copied and pasted from a web page, and didn't notice "Gott" vs. "Gatt".

I'm very much afraid to look up what "Gatt" might be in German.

DTA

Hey Sinner,

I was just messing with you to throw in a quote from a number theorist indicating that God approves of the integers only ...

Floating point is trash for the masses (and for prototypes and student projects). It has no place in serious embedded systems.

Issues:

a)Far slower on low-end micros.

b)Hard to analytically bound the error that arises from calculations (integers are more friendly in that regard--most errors can be reduced to the floor() function).

c)God does not approve (see the quote in German above).

DTA

Proving properties of numbers, such as whether they are rational, irrational, transcendental, normal, etc., is generally very difficult. And many of their properties probably have no better explanation than coincidence.

Pi is known to be "normal", meaning that if you write out its "decimal expansion" in any base, the digits will be evenly distributed amongst all possible digits.

Because an infinite series expansion for PI is known where all the components being summed together are of the form sum ( 1/16^k.f(k) )

It is therefore possible to start from a chosen digit position and compute just the digits of interest from there on. The URL I posted describes a bit more of the details of the algorithm.

It is entirely possible that some other irrationals may have an elegant series expression in some base or other, but off hand I don't know of any other commonly known examples of this. A quick back of the envelope playing around suggests that the golden ratio phi may well have an expansion base 8 that is amenable to the same sort of trick.

phi = (1 + sqrt(5))/2 = 1/2 + sqrt(1 + 1/4)

series expansion for sqrt(1+x) with x = 1/4

1/2 + 1 + x/2 - x^2/2!/4 + 3x^3/3!/8 - ... 1 + 1/2 + 1/8 - sum[k=2,inf]{ 1/(-4^k) ((2k-3)!/((k-2)!k!) }

as ever subject to mistakes, typos and algebra errors.

--
Regards,
Martin Brown

You surely mean inhabitants of the United States of America. Mexicans, Argentinians and Cubans, though Americans themselves, don't confuse English and British. Another common mistake made by yankees :-)

Interesting!

Taylor Maclaurin expansions are great for getting insight into nonlinear stuff.

Of course MATLAB and such like will do this, but they don't work for free.

I only seem to be using ARMs now, Cortex M3 or M4 in new projects.

It is also hard to work with integer representations of real-world quantities. You have to scale everything to fit, keep track of units and scaling factors. Keep track of overflows and underflows, loss of precision. Decide what register width to use for each step of the calculation, and use the correct types to achieve this. Rewrite the calculation so as to minimise the dynamic range of intermediate values. With floats you can usually just convert to standard units at the start of the calculation, and not worry too much about it. If you really need high precision then there are doubles.

Also there is worrying about floating point registers in context switches, non-atomic operations and inconsistent or ambiguous data formats.

I do still keep most things integer until they are "used" - an array of ADC values say. But it is kind of a load off to be able to just work directly in the applications natural units when I want to.

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John Devereux

Scilab is free...

I've some Canadian relatives who were very vocal on their right to be called American.

Yeah, but you'd have an irrational number of fingers.

Jeroen Belleman

And I would not be surprised that floating-point numbers were also an ABOMINATION UNTO NUGGAN.

--------------------------------------- Posted through

Try sqrt(2). 7/5 is fair. 17/12 is good enough for carpentry. 41/29 has less than one fifth of that error, but 99/70 is much better yet. In fact, the best (in terms of accuracy per bit expended) have an odd numerator and an even denominator.

Jerry

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Engineering is the art of making what you want from things you can get.
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Mac, you clearly haven't learned about the continual march of technological progress.

It won't get better if you Pickett.

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Rob Gaddi, Highland Technology -- www.highlandtechnology.com
Email address domain is currently out of order.  See above to fix.

BBP's original paper (link below) lists a number of other examples, including log(2), log (9/10), pi**2, many base 2 logs, ... Others have computed other constants (see Wiki article for links)