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- Highschooler develops a new integration technique that works on 73% of the common integrals used in Calculus 2 named Maclaurin Integration.
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.
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:pbuk said: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
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?pbuk said: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'm comparing the red line in this graphvela said: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 expect to be similar (after a vertical shift) to some level of approximation.vela said:my orange curve is shifted vertically from his.
Not at all: if my comments could be construed as hateful then please suggest an edit or report them.vela said:You seem to have something against the high school kid. Why so much hate?
Nice idea and impressive work for a high-schooler! Kid has a bright future.jedishrfu said: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/
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 said: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?
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.valenumr said: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.
So yes, it was a bad example - or rather a poorly illustrated example. I have sent the author a constructive note.pbuk said: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?
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?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
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!Since this set of conditions are fairly liberal
valenumr said: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.
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##.pbuk said:I expect to be similar (after a vertical shift) to some level of approximation.
It's just that your comments came across as unnecessarily negative and dismissive. (It could just be me.)pbuk said:Not at all: if my comments could be construed as hateful then please suggest an edit or report them.
This is pretty much what I wrote to him, and he has acknowledged.jasonRF said: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.
ohwilleke said: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.
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.PhDeezNutz said:From what I understand Bucholz High School has very close ties to UF.
At the very least they are in the same town.
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 said: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.
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.WWGD said:UF is not a bad school.
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.ohwilleke said: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.
I concur. Perhaps a powerful mathematical engine such as Mathematica or WolframAlpha could incorporate such an algorithm in their database. Thank you!MevsEinstein said: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!
Thank you! I'm writing my second right now and I agree that it is significantly easier.jedishrfu said: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.
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).jedishrfu said: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.