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list1
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Re: Close form formula vs Simuations

September 9th, 2017, 7:02 pm

Yes, eg if we have the equation A = pi r^2 then we can't compute it exactly for most r.
If the answer of a problem is represented in the form [$]\pi r^2[$] then it is the closed form solution.
 
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outrun
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Re: Close form formula vs Simuations

September 9th, 2017, 7:46 pm

Yes, eg if we have the equation A = pi r^2 then we can't compute it exactly for most r.
If the answer of a problem is represented in the form [$]\pi r^2[$] then it is the closed form solution.
You need an algorithm or series expansion to compute pi. Iifinite time to fill it. And that's just the start, you need an infinite long memory for both pi *and* (most values of) r. To compute the square and product you need to apply algorithms that manipulate the representation symbols (0s and 1s for binary representation).
Last edited by outrun on September 9th, 2017, 7:51 pm, edited 1 time in total.
 
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Cuchulainn
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Re: Close form formula vs Simuations

September 9th, 2017, 7:47 pm

Quiz: Let's suppose that you don't know the formula [$]\pi r^2[$] (go on, pretend) but we do know the circumference [$]2\pi r[$].

Question: how can I find accurate approximation to the area (and even get \pi)?).

Hint: what would Euler have done?
 
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outrun
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Re: Close form formula vs Simuations

September 9th, 2017, 7:55 pm

If you had a smartphone you would have liked the Euclidia app, it has all kinds of puzzles like that, I use it to kill time on a plane.
 
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Cuchulainn
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Re: Close form formula vs Simuations

September 9th, 2017, 8:23 pm

If you had a smartphone you would have liked the Euclidia app, it has all kinds of puzzles like that, I use it to kill time on a plane.
I got a new Nokia 3310 (orange coloured) for my birthday. Does it support Euclid?
 
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list1
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Re: Close form formula vs Simuations

September 9th, 2017, 8:33 pm

Yes, eg if we have the equation A = pi r^2 then we can't compute it exactly for most r.
If the answer of a problem is represented in the form [$]\pi r^2[$] then it is the closed form solution.
You need an algorithm or series expansion to compute pi. Iifinite time to fill it. And that's just the start, you need an infinite long memory for both pi *and* (most values of) r. To compute the square and product you need to apply algorithms that manipulate the representation symbols (0s and 1s for binary representation).
Close form solution and how to find approximation of an irrational numbers are two different problems. In such a way to find approximation of [$] \pi [$] or [$]\sqrt 2[$] are similar problem. Solution in the form [$]y = \pi r^2[$] or y = sin x are examples of the solutions in closed form.
 
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outrun
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Re: Close form formula vs Simuations

September 9th, 2017, 9:46 pm

If the answer of a problem is represented in the form [$]\pi r^2[$] then it is the closed form solution.
You need an algorithm or series expansion to compute pi. Iifinite time to fill it. And that's just the start, you need an infinite long memory for both pi *and* (most values of) r. To compute the square and product you need to apply algorithms that manipulate the representation symbols (0s and 1s for binary representation).
Close form solution and how to find approximation of an irrational numbers are two different problems. In such a way to find approximation of [$] \pi [$] or [$]\sqrt 2[$] are similar problem. Solution in the form [$]y = \pi r^2[$] or y = sin x are examples of the solutions in closed form.
What if A is the payoff function of an exotic option, r is a strike factor, pi is the price of an American out with some special parameter values. Now A is no longer closed form by inductions?
 
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outrun
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Re: Close form formula vs Simuations

September 9th, 2017, 9:47 pm

If you had a smartphone you would have liked the Euclidia app, it has all kinds of puzzles like that, I use it to kill time on a plane.
I got a new Nokia 3310 (orange coloured) for my birthday. Does it support Euclid?
Wow, that very fasioable! Research show it going to be the best selling phone in 2018, Hugh expectations!
 
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outrun
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Re: Close form formula vs Simuations

September 9th, 2017, 11:29 pm

..why is a formula with pi in it closed form? Pi itself isn't, you need to iterate a converging numerical method. I also have a fast converging numerical for the American put. I don't see a difference between the two when I look at the electrons moving through my CPU.

What exactly in the criterium list1? Does it involve computational resource bounds like computing time, memory usage, convergence speed, error tolerance levels? I don't thinks so.. So why is pi closed form and the American put not? It's arbitrary and non exact: it's a non-mathematical, non-numerical criteria,.. maybe practical? It's like Alan said. 

In computer science and numerical methods there are much clearer measures: P vs NP, co-NP, NP-hard etc etc an onion of complexity classes. Then you have O() complexity for error, time and memory resources for algorithms.
 
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Cuchulainn
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Re: Close form formula vs Simuations

September 10th, 2017, 11:49 am

Yes, eg if we have the equation A = pi r^2 then we can't compute it exactly for most r.
If the answer of a problem is represented in the form [$]\pi r^2[$] then it is the closed form solution.
I don't agree. Just because you have a cute symbol [$]\pi[$] does not make it closed form. It is a transcendental number and even Archimedes uses approximation.
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e.

And see the [$]e^5[$] thread, a series for it is not closed form.

You have maybe persuaded yourself that it looked closed. Happens a lot.

From Wolfram
An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.
Last edited by Cuchulainn on September 10th, 2017, 12:09 pm, edited 1 time in total.
 
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Cuchulainn
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Re: Close form formula vs Simuations

September 10th, 2017, 11:57 am

If you had a smartphone you would have liked the Euclidia app, it has all kinds of puzzles like that, I use it to kill time on a plane.
I got a new Nokia 3310 (orange coloured) for my birthday. Does it support Euclid?
Wow, that very fasioable! Research show it going to be the best selling phone in 2018, Hugh expectations!
I bought it because it does what I want, + cheap + none of that fancy crap. Too much consumer brainwashing going on.
 
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list1
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Re: Close form formula vs Simuations

September 10th, 2017, 12:12 pm

There is no low which specifies application of the notion closed form. It is not common to say closed form of a number [$]\pi , \sqrt 3[$] or others. In finance we use closed form formula though ib math in a similar situation we say analytic, exact, explicit formula or solution. When I first read explicit form I was emberaced why people in finance invented new notion while there are old well known notions in math which used for a long time.Nevertheless it is not a big deal. If we arrive at well or not well known ( it can be a quite subjective issue too ) answer with known properties we can call it closed form. But it is only terminology whether a formula can be called closed form or not. 
I think the good illustration are Bessel functions. Before they were well known these are solution of equations. We can denote them as we wish. It difficult to call them closed form though say cos x was well known from geometrical, or calculus points of views. Cos x can be called in closed form  while a Bessel function was difficult. Now one can find similar properties of the Bessel function. Of course it looks much more complex than cos x. Nevertheless answer presented in Bessel function can be called in closed form. 
But again it is somewhat conditional on subjective point and its only terminology problem and sometimes it is easier to agree in opposite point than to insist and proves yours.
 
cfrm17
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Re: Close form formula vs Simuations

March 14th, 2018, 5:57 pm

Close form solution is always good. But unfortunately it doesn't exist for some exotic products.