as mentioned in the last post, am zooming in on the “power iteration” algorithm. it is explained as, “if you want to find the dominant eigenvector, use the power iteration”. in my case, found it by discovering that “if you use the power iteration, it will lead to the dominant eigenvector”. kind of subtle right? maybe reminds me of that old saying “all paths lead to rome”. and then ofc, the classic, “rome wasnt built in a day”.
here is the code that compares the (normalized) current state vector to the dominant eigenvector, which apparently ruby organizes it as the leftmost column of the left eigen-decomposition matrix. it uses/ selects the 95th/100th density iteration which tends to lead to a longer trajectory. am in good company, as wikipedia notes the power iteration is used at the core of Google pagerank algorithm! 😀 😎 💡 ⭐ ❗ ❤
hi all, last month collatz installment made some progress, but unf have been a bit tied up with work, where a fiscal year transition/ boundary tends to lead to some crunch-like dynamics, leaving less time for one of my favorite side projects, namely this one, bummer/ ouch.
but, here is a small trickle/ dribble of some newer ideas, mainly benefitting from google searches and maybe a little algebra.
❗ 💡 😀 😎 ⭐ last month was quite a tour de force against collatz, the culmination of many months or even years of hard work and creative ideas, and many different approaches all combined (pyramid-like) to lead to very solid results verging on a “candidate solution”. another theme that was pursued earlier here are “adversarial algorithms” which has been used with great success eg by Google/ Deepmind against Go. the basic theme is “two algorithms competing against each other” so to speak. along these lines there is one final idea to try against the prior Collatz “solution”.
starting these blogs out, sometimes dont really know where they will lead ahead of time, but its a new month and a new pov. so the (“inherently a priori“) title is typically either “whats happening at the moment” or “some general/ larger theme intended to be pursued,” in this case the former. (now pondering that, am intending to revisualize the last collatz experimental angle in particular but on other hand, nearly entire theme of this overall research prj/ program in general could be said to be “revisualization”!)
at 1st was thinking maybe not all the major extra effort for polished/ pretty visualization was worth it at the moment, but couldnt resist, just wanted to see it, and was curious/ wondering about a few additional statistics. there was some real payoff by saving all the intermediate data and coming up with a major refactoring of the visualization code, ie decoupling visualizing and generation phases, and did a fairly massive rewrite without having to rerun the very expensive generation code. all easier imagined/ said than done! took quite awhile/ substantial effort.
the title is semi ironic, because it seems there is never a vacation with a hard problem. only a hiatus? did go on a trip last wk and had a great time, didnt think about math much at all! which is a good thing! work/ life balance and all that! although for some, math is life! would go into more juicy detail for all my loyal readers (like last years epic saga) but alas, havent heard from any of you in ages so not sure you really exist 😳 😥
just picture me in tattered/ dusty/ dirty clothes on the side of a busy cyberhighway, weathered/ sunburned/ wrinkled/ unshaven skin, sitting in the blazing hot sun with a cardboard sign scrawled with marker, million-mile staring-into-distance plaintively…
will write for comments! 😐 😳 🙄