Why can't some problems be solved analytically? What makes certain problems difficult or impossible to solve beyond numeric approximation?

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# Why can't some problems be solved analytically?

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Why can't some problems be solved analytically? What makes certain problems difficult or impossible to solve beyond numeric approximation?

Can you name a single one though?

Well, I was principally thinking of those types differential equations that don't have exact solutions.

Name a single one

Burger's equation or Navier stokes? I don't know why you're insisting on specific problems because I was asking why this happens in general.

Because their differential equations describe properties that cannot be replicated using normal functions. You can define any arbitrary way to relate a function to its derivative(s) but the only way you will get an analytical solution is if those properties match that of some combination of elementary functions.

If I tell you y' + y = 0 it means the solution is one where the function's derivative is equal to the negative of itself, which is the defining trait of an exponential decay. However I can just make up some bullshit relation that y'''3 - log(arcsec(x*y'))*y" = x^y that shares no properties in common with any elementary functions, and you won't be able to get a solution to it beyond defining a new function as the solution.

>beyond defining a new function as the solution

But wouldn't that still be an exact solution?

In a way, sure, but in practice, no.

If you can only describe it as 'the function that is the solution to X' that's not really feasible for actual use. Fundamentally it implies you have to use a computational method every time you want to evaluate the function of use it any meaningful way.

Looks like nope.

You have two unit circles. What is the distance between their center points if the circles overlap exactly half of each other's area?

Is this a classic problem? I've never seen it before and it's unintuitive to me that there wouldn't be some polynomial of pi solution.

Dunno how classic it is or isn't. But I'm not the first one to ever ask this problem. You can google the first few decimals of this problem's numerical solution and see that the problem shows up on google and on reddit and all kinds of places.

looks like the distance would be 2sinx, where x is the solution to 2x + sin(2x) = 1/2 . If x were some multiple of pi, then sin(2x) is some fraction, but that would mean both 2x isn't and is rational - a contradiction. If x is a is a nonzero fraction, then obviously we get rational+irrational = rational , also impossible. x=0 doesn't fit the eq. Which means x is some non-pi irrational number.

At that point the problem is the same as

. You'd need a numerical method to approximate it.

I mean, honestly. Since pi is irrational, per OP's question you can't even write this out "analytically". The basic equation sin(x) = 1 has the solution pi/4. You need a numerical method to approximate it. We just give the solution the label pi because pi is so ubiquitous and important.

pi/2 im sorry

This post just makes me wish I had the time to do numerical analysis in grad school.

From my understanding, most numerical analysis people don't consider e and π to be "not analytic" because we have closed form solutions to series which produce them exactly.

When I think of a non-analytic solution I think of something like a local minima to a convex non-quadratic function where there may in general be no closed form analytical solution, despite the convexity guaranteeing the existence of at least one proper local minima.

>don't consider e and π to be "not analytic"

Why would anyone even be consider then to be non-analytic at all?

Who should I marry? Should I get married? Is Becky or Sarah the girl for me? What should I do for a living? How much tax should which people pay? What should it be spent on?

Compute an answer to these problems, I implore you.

Why can't some numbers be written as a fraction? What makes certain numbers difficult or impossible to write as a fraction beyond an approximation?

inb4 the Wildebergers show up sperg out about only the rationals existing.

Out of all the homosexuals on this board, those memelords are least of IQfy's problems.

This is true. I'd rather deal with them than a strict determinist or strict materialist (which are truthfully fairly similar as you pretty much have to be a strict materialist to believe in strict determinism without invoking some sort of God-like entity).

I'm a finitist materialist. Now what?

Idk man, continue to be moronic I guess?

Imagine being an infinite believing non materialist. That's the real moron.

Let me know when you figure out what the "biggest number" is. Similarly, let me know when you figure out what materially constrains the truth value of "2+2=4."

You shut your mouth curr! Wildberger did nothing wrong!

Horsefrickers? Is there a meme I missed?

https://implyingrigged.info/wiki/2022_IQfy_Autumn_Babby_Cup#Group_C

Ah, so it's mainly a /vg/ thing? Okay. I've genuinely never noticed those threads anywhere and don't go on /vg/.

It's a IQfy wide event, /vg/ is just where the bulk discussion happens, though most boards try to make a gameday thread during an active tournament.

This was one of IQfy's from last autumn https://warosu.org/sci/thread/15851815#p15851815

Last autumn I was probably too busy schizoposting in a determinism thread to notice.

Usually it has to do with being non-linear. Nonlinear pdes don't have general solutions much like polynomials of degree 5 or more don't have general solutions. Although there's a proof for polynomials but there's no proof for pdes as far as I know

So what generalizations can be made about problems with computational solutions in other fields?

Aren't non-linear analytic equations representable as a Taylor series of polynomials? Meaning, each nonlinear function is implicitly a 5+ degree polynomial

A Taylor series is an approximation using finite terms, its the power series that are exact/

Skill issue

It all comes down to inverses. Some functions simply do not have inverses and thus cannot be linearized.

It's fairly common in fact. All computational methods, btw, simply utilize small, invertible or linear functions and then stitches shit back together.

The difficulty of solving a problem rises exponentially with its dimensionality, so highly dimensional problems can only be solved through dimensionality reduction.

Dimensionality reduction leads to losing information

That wouldn't contradict my answer even if you could prove it.

How so?

Because it would still be a viable way to approach problems that are too complex for an analytic solution, even if the solution can never be exact.

Non-linear bros, what's our response?

Monte Carlo numerical approx for higher dim PDEs

But that's still a computational method

It's an approximation method. It's not an exact solution.

Brainlet take. Solve it exactly

What is the brainlet take? I'm not saying that monte carlo methods are the only way or something. The point of

was to point out that Monte Carlo methods are an approximate optimization approach.

They aren't like gradient descent where (if your function is locally convex around a minima in a region of measure greater than zero) you might actually get the exact analytical limit as your number of iterations goes to infinity. At best with an MC method you can get a computationally stable asymptotic average (which isn't the same as an analytical solution).

The point of

oops.

Just use AI bro

Fart on me

Hmm, nyo.

Wait until you find out there are noncomputable PDEs.

Enlighten me.