High schooler Develops new Integration Technique

[jedishrfu]

TL;DR Summary

Highschooler develops a new integration technique that works on 73% of the common integrals used in Calculus 2 named Maclaurin Integration.

https://www.wuft.org/news/2022/02/1...scovers-and-publishes-new-calculus-technique/

 

[mathman]

Details almost unreadable.

 

[berkeman]

Yeah, so far I'm not seeing a link to the paper. I could be missing it...

 

[pbuk]

https://arxiv.org/abs/2201.12717

 

[pbuk]

Haven't properly read it but I am afraid that if in the chart in section 3.3 the red line is supposed to be the antiderivative of the green line then there is something badly wrong.

Ref: arXiv:2201.12717

 

[pbuk]

Gotta give the copy editor credit for this though:

Where most integration techniques can only be applied to around 10% to 40% of integral problems, Bruda’s technique applies to approximately 73% of integrals. That means it is almost two times more effective than most mainstream techniques.

 

[jedishrfu]

The almost 73% is a curious number it means that 100 examples of different types were tested and almost 73 made it which I guess means 72 made it which implies they needed to only test 25 examples.

 

[Office_Shredder]

The paper itself is terribly written, which is not a surprise since it's by a high schooler. The idea is kind of neat, but they don't actually demonstrate any use for it. For example it feels like the paper is trying to claim the new series converges faster than a Taylor series, but they don't actually prove it, or even give an example where they show numerically that it's true.

You still need to compute all the derivatives just like you would for a Taylor series, so it's really not obvious what this is supposed to save you.

 

[martinbn]

I didn't see a single example, where anything was calculated explicitly, neither in closed form, nor as a series. It is just the formula that has the function and its derivatives. Worse actually it is the derivatives of $xf(x)$.

The 73% success rate reminds me of the joke. "The technique was used to build a rocket. The mission to the moon was great, the rocket traveled 73% of the distance before exploding."

 

[vela]

Haven't properly read it but I am afraid that if in the chart in section 3.3 the red line is supposed to be the antiderivative of the green line then there is something badly wrong.
View attachment 297398
Ref: arXiv:2201.12717

That behavior is caused by the fact Bruda truncated the inner summation, keeping only 11 terms. Mathematica was able to come up with a closed form for the infinite sum, and when I used that, I got the following plots:

Like Bruda, I summed only the first 7 terms of the series. The blue curve is a plot of the indefinite integral Mathematica calculated. He apparently added an integration constant to his results, which is why my orange curve is shifted vertically from his.

 

[pbuk]

So are you saying that you think that Bruda's method is useful, he just picked a bad example to demonstrate it? Catastrophically bad: his result is convex for x > 2 whereas the solution is concave?

 

[vela]

So are you saying that you think that Bruda's method is useful, he just picked a bad example to demonstrate it? Catastrophically bad: his result is convex for x > 2 whereas the solution is concave?

I was pointing out your suggestion that his method was flat out wrong is wrong. I'm not sure what you're saying about $x>2$, unless you're perhaps mistaking the graph of the function for the graph of its anti derivative. You seem to have something against the high school kid. Why so much hate?

My take is pretty much the same as @Office_Shredder's. I don't see what the supposed advantage of this method is.

 

[pbuk]

I'm not sure what you're saying about $x>2$, unless you're perhaps mistaking the graph of the function for the graph of its anti derivative.

I'm comparing the red line in this graph

with the orange line in this graph

which on the basis of the texts and this comment:

my orange curve is shifted vertically from his.

I expect to be similar (after a vertical shift) to some level of approximation.

You seem to have something against the high school kid. Why so much hate?

Not at all: if my comments could be construed as hateful then please suggest an edit or report them.

 

[ergospherical]

Summary:: Highschooler develops a new integration technique that works on 73% of the common integrals used in Calculus 2 named Maclaurin Integration.
https://www.wuft.org/news/2022/02/1...scovers-and-publishes-new-calculus-technique/

Nice idea and impressive work for a high-schooler! Kid has a bright future.

 

[PhDeezNutz]

I actually remember competing against this young man’s high school in math competitions (I graduated HS in 2007). Bucholz always walked away with tons of awards/trophies.

 

[valenumr]

So are you saying that you think that Bruda's method is useful, he just picked a bad example to demonstrate it? Catastrophically bad: his result is convex for x > 2 whereas the solution is concave?

I didn't really get through the whole thing, but I seem to remember something about the method being valid over the domain (0, 2) for the opening example.

 

[pbuk]

I didn't really get through the whole thing, but I seem to remember something about the method being valid over the domain (0, 2) for the opening example.

Ah yes, this is indeed the case and is mentioned in a couple of places, but critically not in relation to the chart in section 3.3 which confused me.

So are you saying that you think that Bruda's method is useful, he just picked a bad example to demonstrate it? Catastrophically bad: his result is convex for x > 2 whereas the solution is concave?

So yes, it was a bad example - or rather a poorly illustrated example. I have sent the author a constructive note.

 

[nuuskur]

The short note on arxiv contains some odd comment.

Compared to other integration techniques such as Trigonometric substitution, Integration by Partial Fractions, or Integration by Parts, Maclaurin Integration requires by far the least amount of labor to utilize

We've used this technique in our analysis tutorials. Secondly, there is a lot of labor involved in computing the derivatives. Thirdly, nothing is said about its convergence rate. Why would this technique be preferable to Taylor series, for instance?

Since this set of conditions are fairly liberal

What does this mean? That there are are lot of $f$ for which $xf(x)$ is smooth? I don't think so. Even the class of continuous maps is tiny.


But most disturbing to me is the final section regarding "accuracy" based on a single example. Doing precisely what I teach my students Not to do: cherry pick examples to test hypothesis and regard it as proof. Numerical integration is very well understood by now. Test your formula against the known artillery!

 

[WWGD]

Interesting take on the space of all integrble functions, my Bruder.

 

[jasonRF]

This young man has way more going on than I did at his age, and I wish him the best. His approach is interesting, and I don't fault him for his youthful enthusiasm in writing about the utility of the method.


EDIT: PLEASE IGNORE MY STATEMENTS BELOW! I read his formula wrong (my old eyes confused the $u$ and $v$) so was looking at the wrong series!
EDIT take 2: I think my comments below are correct. I get that the inner sum is $$
\sum_{n=0}^{\infty} \frac{n! \, (1-x)^{n+u+1}}{(n + u + 1)!}$$
and if that is correct, then I'm pretty sure the formula diverges outside of the interval $x\in (0,2)$.

I didn't really get through the whole thing, but I seem to remember something about the method being valid over the domain (0, 2) for the opening example.

This. While he did mention these assumptions during the derivation, I'm not sure whether he clearly stated that the formula he derives is also only valid for $x \in (0, 2)$, and I'm pretty sure he didn't mention that the formula diverges outside of that interval. The fact that he plots his example in regions of divergence may imply that he hadn't thought things through to that level. If he had included more terms in the inner sumation he may have noticed the trouble empirically. Again, his work is far better than anything I did at his age and I'm not trying to belittle his efforts here.

So my friendly suggestion for improvement is for him to prove to himself where his formula is valid, make this more explicit in the paper, and only plot examples in regions where the formula is valid.

jason

 

[vela]

I expect to be similar (after a vertical shift) to some level of approximation.

I'm not sure why you'd expect this when he truncated one of the series whereas I didn't. It's like expecting $x-x^3/6$ to behave similarly to the full series for $\sin x$ for $x \gg 1$.

Not at all: if my comments could be construed as hateful then please suggest an edit or report them.

It's just that your comments came across as unnecessarily negative and dismissive. (It could just be me.)

 

[pbuk]

So my friendly suggestion for improvement is for him to prove to himself where his formula is valid, make this more explicit in the paper, and only plot examples in regions where the formula is valid.

This is pretty much what I wrote to him, and he has acknowledged.

 

[Ssnow]

It is a good idea!, especially if you consider it came from a young student of the high school ... it seems a laborious method with higher derivatives ... one time I wrote a simple article of the same type involving the Tetration function with application in calculating integrals, it has been rejected from ArXiv and it was less empirical than this. But this is a different case, the youngness will say a lot ...😄

Ssnow

 

[MevsEinstein]

Maclaurin Integration's formula is too big and lengthy, but computer scientists can make an algorithm based on it to find the integrals of functions that can't be integrated by other techniques.

Nice job Glenn Bruda!

 

[ohwilleke]

Honestly, I'm a little surprised he's going to U of Florida. Nothing wrong with it, but he would probably have no problem getting into CalTech, MIT or Princeton, and would benefit from having more brilliant peers at schools like those.

 

[jedishrfu]

Many factors likely played into his decision, closeness to home, scholarship, other school friends attending the same university, relatives nearby, likes the sports teams or area...

 

[PhDeezNutz]

Honestly, I'm a little surprised he's going to U of Florida. Nothing wrong with it, but he would probably have no problem getting into CalTech, MIT or Princeton, and would benefit from having more brilliant peers at schools like those.

From what I understand Bucholz High School has very close ties to UF.

At the very least they are in the same town.

 

[MevsEinstein]

From what I understand Bucholz High School has very close ties to UF.

At the very least they are in the same town.

My dad went to my state for free because he got a better job. Glenn can get a free ticket to a powerful university since that university will need bright people like him.

 

[WWGD]

My dad went to my state for free because he got a better job. Glenn can get a free ticket to a powerful university since that university will need bright people like him.

UF is not a bad school. Maybe not top 10, but not bad. Not so sure undergrad school needs to be top 10. If he blazes through, maybe he can transfer somewhere better.

 

[MevsEinstein]

UF is not a bad school.

It's actually one of the best universities on Earth (99th to be specific). I was just saying that Glenn can act like Napoleon and get himself to Harvard or something.

 

[glennbruda]

Hey, that's me haha! Thank you everybody for the constructive criticism and kind words, I really appreciate it. If I had a chance to rewrite the paper now, I would make a lot of changes, but I suppose that is what revisions are for.

Honestly, I'm a little surprised he's going to U of Florida. Nothing wrong with it, but he would probably have no problem getting into CalTech, MIT or Princeton, and would benefit from having more brilliant peers at schools like those.

Both of my parents are UF alums, I already know a few professors there from the process of writing this paper, I've been a Gator all my life (sports aren't the most important to me but it is a plus), and the cost of those colleges is extremely high. It just made sense for me to go to UF; perhaps I'll go to one of those colleges for grad school.

Maclaurin Integration's formula is too big and lengthy, but computer scientists can make an algorithm based on it to find the integrals of functions that can't be integrated by other techniques.

Nice job Glenn Bruda!

I concur. Perhaps a powerful mathematical engine such as Mathematica or WolframAlpha could incorporate such an algorithm in their database. Thank you!

 

[jedishrfu]

Welcome to PF!

Its impressive for one so young to have a published math paper. It seems the first paper is always the toughest one.

Have you taken the MAA tests or the Putnam yet?

Often young math talent take them as part of their education into challenging problems with twists and turns.

 

[glennbruda]

Welcome to PF!

Its impressive for one so young to have a published math paper. It seems the first paper is always the toughest one.

Have you taken the MAA tests or the Putnam yet?

Often young math talent take them as part of their education into challenging problems with twists and turns.

Thank you! I'm writing my second right now and I agree that it is significantly easier.

I have not; I never was a big fan of competition in math in general. I find that such a discipline is more elegant when people work together rather than against each other. Additionally, I don't like the time pressure aspect of such competitions usually. Perhaps I'll try the Putnam competition though, having a limited amount of very challenging questions given a lot of time seems palatable. Thanks for the suggestions.

 

[jedishrfu]

Yes so true on the competitions. They can never delve deeply into some problem and instead rely on the student noticing some insight above and beyond what they learned in traditional math courses. I took the MAA once and was lucky to get a single problem right.

My friend on the other was quite talented and became an MAA champion and was on a team that competed internationally with England and Russia. They failed miserably mostly due to the test structure. The US MAA was a multiple choice affair where you could strike out some non-answers and focus on the reduced set and make an educated guess. The international test was fill in the blanks for which the English and Russian teams routinely tested on.

The English in particular have the Tripos tests which students study for like crazy often using experienced tutor coaches. These tests dictated where in the hierarchy of academia you stood. The math tripos was the oldest of these tests.

https://en.wikipedia.org/wiki/Mathematical_Tripos

I think it was GH Hardy who lobbied for changes in these tests as he felt that they had held England back a hundred years or more in mathematics as compared to Europe. Basically the English were focused on applied math (tripos type problems) and Europe was into rigorous proofs and pure math.

https://en.wikipedia.org/wiki/G._H._Hardy

 

[ergospherical]

The English in particular have the Tripos tests which students study for like crazy often using experienced tutor coaches. These tests dictated where in the hierarchy of academia you stood. The math tripos was the oldest of these tests.

Tripos isn't competition maths but rather the name given to end-of-year exams at undergrad level. Nonetheless there are still a few oddities (if you top the maths tripos then you're designated the senior wrangler).