Warp wrote:
> Orchid XP v8 <voi### [at] devnull> wrote:
>> Well, the sine and cosine functions can't be computed by a finite number 
>> of additions, subtractions, multiplications, divisions and exponents. 
> 
>   They can if you allow the parameters to be complex numbers (which is why
> trigonometric functions are included in the set of elementary functions).

OK, but that just brings us back to the fact that the exponential 
operator has to be approximated. ;-)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Invisible wrote:
> http://office.microsoft.com/en-us/excel/HP052090051033.aspx

Nice, that link crashed my browser :) Clearly Microsoft and KDE don't like 
each other.

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Orchid XP v8 wrote:
> Warp wrote:
>> Orchid XP v8 <voi### [at] devnull> wrote:
>>> Well, the sine and cosine functions can't be computed by a finite 
>>> number of additions, subtractions, multiplications, divisions and 
>>> exponents. 
>>
>>   They can if you allow the parameters to be complex numbers (which is 
>> why
>> trigonometric functions are included in the set of elementary functions).
> 
> OK, but that just brings us back to the fact that the exponential 
> operator has to be approximated. ;-)
> 

Did you read the Wikipedia page on closed form expressions?  It's not 
very long and all of these points are addressed there.

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Nicolas Alvarez wrote:
> Invisible wrote:
>> http://office.microsoft.com/en-us/excel/HP052090051033.aspx
> 
> Nice, that link crashed my browser :) Clearly Microsoft and KDE don't like 
> each other.

Crashed it? Or just upset it?

Microsoft may use proprietry features and write terrible markup, but if 
a browser crashes when fed the wrong data, methinks the browser should 
be rewritten to fail gracefully. ;-)

The only unusual thing I can see about the page [other than too much 
JavaScript] is a flash advert.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Kevin Wampler wrote:

> Did you read the Wikipedia page on closed form expressions?  It's not 
> very long and all of these points are addressed there.

...OK, just read it. Doesn't seem to address very much.

Also, I've often wondered how the **** you compute something like the 
Gamma function or the Bessel-J function. I mean, have you *seen* the 
definition?!

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> Kevin Wampler wrote:
> 
>> Did you read the Wikipedia page on closed form expressions?  It's not 
>> very long and all of these points are addressed there.
> 
> ...OK, just read it. Doesn't seem to address very much.

I was thinking of:



trigonometric functions and inverse trigonometric functions)."

and

"For purposes of numeric computations, being in closed form is not in 
general necessary, as many limits and integrals can be efficiently 
computed."

Which seem to exactly cover both of the points you recently mentioned, 
but perhaps I misunderstood what points you were making.


> Also, I've often wondered how the **** you compute something like the 
> Gamma function or the Bessel-J function. I mean, have you *seen* the 
> definition?!

Yes.  If you take a look at the Wikipedia article for the gamma function 
you'll see that it includes a couple of nice representations in terms of 
infinite products.  The Bessel functions seem a bit more involved, but 
it has representations in terms of a sum over terms involving the gamma 
function, a hypergeometric series, and a recursive relation to a 
continued fraction -- so there seem to be many ways to go about 
computing it (I didn't look up what the standard approach in practice was).

I know that you say that you have trouble digesting the Wikipedia 
articles on these in full (entirely understandable, they are very dense 
and not always well written) but you still can answer these questions by 
spending a minute or two just skimming over them without trying to 
understand everything.

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>> ...OK, just read it. Doesn't seem to address very much.
> 
> I was thinking of:
> 


> trigonometric functions and inverse trigonometric functions)."
> 
> and
> 
> "For purposes of numeric computations, being in closed form is not in 
> general necessary, as many limits and integrals can be efficiently 
> computed."
> 
> Which seem to exactly cover both of the points you recently mentioned, 
> but perhaps I misunderstood what points you were making.

Oh, I see. I thought it was going to say something more insightful than 
that. (How the hell do you compute a limit or an integral anyway?)

>> Also, I've often wondered how the **** you compute something like the 
>> Gamma function or the Bessel-J function. I mean, have you *seen* the 
>> definition?!
> 
> Yes.  If you take a look at the Wikipedia article for the gamma function 
> you'll see that it includes a couple of nice representations in terms of 
> infinite products.

But you can't compute an infinite product.

> The Bessel functions seem a bit more involved, but 
> it has representations in terms of a sum over terms involving the gamma 
> function, a hypergeometric series, and a recursive relation to a 
> continued fraction -- so there seem to be many ways to go about 
> computing it (I didn't look up what the standard approach in practice was).

Uh...OK.

> I know that you say that you have trouble digesting the Wikipedia 
> articles on these in full (entirely understandable, they are very dense 
> and not always well written) but you still can answer these questions by 
> spending a minute or two just skimming over them without trying to 
> understand everything.

Heh. Some of them explain things quite clearly, but others are 
incomprehensible. That's the trouble with a reference source written by 
bored Internet surfers; first it's a reference, not an introduction, and 
second the quality is *highly* variable. ;-)

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>> Not really more complex, just a game that is played repeatedly and you 
>> have a certain % chance of winning (say 57%).  I just want to know if you 
>> play 10,100, 1000 times what is the *likely* number of wins in some kind 
>> of worst case and best case scenario.  ie it's obviously possible to win 
>> every single game, but that is *really* unlikely, I want to know how many 
>> you'd win with 90% or 99% confidence.
>
> Right. So it's ye olde "2SD = 95% confidence" rule then.

Indeed.  It turns out in my game that you break even if you win 54% of the 
time, and in practise you can only realistically expect to win 57% of the 
time if you are good at the game.  These two facts mean that after even 1000 
games there is still a huge variation in how much profit you can expect to 
make - 95% confidence you will make between -7 and 607 arbitrary units of 
profit.

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Invisible wrote:
> Oh, I see. I thought it was going to say something more insightful than 
> that. (How the hell do you compute a limit or an integral anyway?)

Ahh, no nothing more insightful, just that it did address the major 
points you were worrying about.

Also, I'll assume you mean "how do you numerically compute a limit or 
integral?":

If it's a limit, just compute it for numbers as close as possible to the 
limiting value (or really really big numbers if the limiting value is 
infinite).  The closer (or the bigger) the number you compute it for the 
closer your answer will be.

If it's an integral do something like this: 
http://en.wikipedia.org/wiki/Riemann_sum

> But you can't compute an infinite product.

If the product converges (which it does, otherwise you couldn't define a 
number/function with it) then you can by definition get an arbitrarily 
good approximation by computing the product of the first n terms for a 
large enough n.

You might want to read up on these, they're pretty much fundamental to 
all of calculus and many many other ways of defining things over the 
real (or complex) numbers (Taylor expansions, Fourier series, numeric 
optimization and root finding, etc.):

http://en.wikipedia.org/wiki/Limit_(mathematics)
http://en.wikipedia.org/wiki/Limit_of_a_sequence

Out of curiosity have you ever had a calculus class?  I'd think that 
these things should have been covered there, and it makes me wonder if 
maybe the UK curriculum is somewhat different that what I'm used to.

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>> Oh, I see. I thought it was going to say something more insightful 
>> than that. (How the hell do you compute a limit or an integral anyway?)
> 
> Ahh, no nothing more insightful, just that it did address the major 
> points you were worrying about.
> 
> Also, I'll assume you mean "how do you numerically compute a limit or 
> integral?":
> 
> If it's a limit, just compute it for numbers as close as possible to the 
> limiting value (or really really big numbers if the limiting value is 
> infinite).  The closer (or the bigger) the number you compute it for the 
> closer your answer will be.

Wouldn't that be increadibly unstable, numerically?

> If it's an integral do something like this: 
> http://en.wikipedia.org/wiki/Riemann_sum

I see. (Although I'm still not sure how you compute an infinite integral 
this way...)

>> But you can't compute an infinite product.
> 
> If the product converges (which it does, otherwise you couldn't define a 
> number/function with it) then you can by definition get an arbitrarily 
> good approximation by computing the product of the first n terms for a 
> large enough n.

I don't see how that is the case.

If you have an infinite *sum*, then as long as the terms get 
progressively more tiny and never get larger again, you can disregard 
all the terms after a certain point. But if you're taking a *product* 
then any term, anywhere in the series could radically alter the final 
result.

> Out of curiosity have you ever had a calculus class?

I've never had *any* maths class!

(Unless you count what we did at school. This simply involved filling 
out hundreds of thousands of pages of long-division problems over a 
7-year period...)

Hypothetically I shouldn't be able to do algebra at all...

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