Summer of Math Exposition

Good video on a niche root-finding algorithm. I felt a bit out of the loop after you listed off a number of obscure theorems in the introduction, but I got caught up throughout the video. Good introduction, explanation, and segue with V(x). Solid entry overall.

Nice that you listed cons for other methods (Lehmer-Schur), but it would be nice to see some pros of Wilf’s method before you dove into the explanation.

First and foremost - you really need to work on your annunciation. Play it for people who don’t know what you’re talking about. Watch how many times they say “What did he say?”. The topic seems interesting, but you need to motivate it better and to create interest. I didn’t even realize you were talking about complex functions til you were well within the video. You also tend to jump around a lot citing different methods and theorems, that makes it really hard to follow.

Felt a bit lost at the start without the knowledge of the Lehmer-Schur video.

I was not familiar with the topic, and I couldn’t follow.

I don’t think this video was well-suited to this contest. It seems to be more oriented toward a specialist audience, or perhaps viewers of your channel who have already studied many approaches to root-finding and are already well established with the concepts. Specifically by the rubric, you did not really present a motivation for why we should want to do this, and the clairty is rather weak. As a specific suggestion for improvement, I think a reorganization of the video would make the concepts much easier to follow, especially covering the rectangle test before the search section - because it was very hard to understand the search section without already having a clear mental picture of what the rectangle test does. Then within the rectangle test section, if I followed you correctly, the Sturm sequence is an improvement of the simpler-and-more-straightforward technique from Fourier, so, it would be better then to cover the Fourier technique first, and then introduce the Sturm technique as an elaboration. Finally, I like your use of an example, but you ommitted a few key details, and also went too quickly through some important formulas, that made it difficult to follow the full argument. Better to go more slowly through all the steps.

Very hard to understand as very little is explained.

I remember you from last year! Glad to see you back. I’m going to comment in dialogue with the feedback you received last year. From everyone, mostly because, amusingly, i don’t remember which of the feedback was mine.

• You showed a lot more restraint with the formulas you displayed on screen, which I agree does make this a lot more inviting than the Lehmer-Schur video.
• Still, motivation continues to be the weakest point. You have convinced me that root-finding is hard, but a brief reminder of why it’s important would go a long way. Ideally, with an example that illustrates the benefit of being able to find complex and repeated roots, since this is a claimed benefit of this algorithm over others.
• In both this video and the last one, there’s a bit of cookbook-style approach to these algorithms, which is to say, that you tell us how without telling us why. For instance, in this video it’s not clear what kind of work p-bar is doing for us. Now, I assume this is because Wilf’s argument is rather complicated; but if that is indeed the case, it would still be appreciated to hear you say it. (I do think in this video you do a better job hinting at this with the screenshots of Wilf’s paper, but it’s still too subtle for my taste. If there is something I’m not meant to be equipped to understand with the information given in the video, I want to be reassured of this.)
• Unfortunately I think the “how” is also not quite up to par; when you say to expand p(x) around Q_k I assume this means substituting x-Q_k. That would be equivalent to the k=3 case in the second bullet point, but I’m not clear what I’m supposed to do with the k=1,2,4 substitutions, and what relationship these have to p-bar.
• I do think you did a better job identifying the critical moments in your argument and slowing down. In general I feel that the script seems much more aware of its audience here

The visualizations were clean, though the text in certain places would benefit from being larger. This video is hard to follow at the high-school level due to the amount of terms that are mentioned without explanation of what they are.

At 1:45 I’m confused as to what the Cauchy bound is. Is it the equation on the following bullet point, or some other concept/equation?

At 1:52 Is there an explanation as to why this circle should contain all the roots?

The graph at 2:59 has axis labels that are too small to read

At 3:54 using $i$ as an index while also dealing with imaginary numbers is a bit confusing.

At 7:14 it’s hard to follow what is trying to be said, especially without knowing what the Cauchy index is.