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Seven sins of numerical linear algebra
In numerical linear algebra we are concerned with solving linear algebra problems accurately and efficiently and understanding the sensitivity of the problems to perturbations. We describe seven si…
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The latest blog posts to take off on Hacker News. 2,391 posts.
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Simulating Wind on Procedural Terrain Using Lattice Boltzmann Method
Note: The full source code for this article can be found here (summarized version). Individual sections contain references to specific code snippets below.
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On AlphaTensor’s new matrix multiplication algorithms
Two acquaintances independently asked about this today, so it seems worth a write-up: recently (as of this writing), DeepMind published a new paper about a new practical fast matrix multiplication …
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Knuth's Art of Computer Programming, V 4B, has gone into print
(I'm still looking for the catchiest way to appreciate the number MMXXVI. Exercise 7.2.1.6–122 gives 1234+.5*6+789 and -1+2345*6/7+8+9 and 1+(2/3)*45*67.8-9 ; other suggestions are welcome. The most elegant that I've seen so far is a beauty that was suggested independently by Arnaud Lefebvre and by Evangelos…
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What Are the Odds?
An unusual lottery result made the news recently: on October 1, 2022, the PCSO Grand Lotto in the Philippines, which draws six numbers from $latex {1}&fg=000000$ to $latex {55}&fg=000000$ a…
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An Algorithm for Polygon Intersections
In this post we'll work our way towards an algorithm that can compute convex polygon intersections. We'll also explore a method for intersections between axis-aligned rectangles, a function that can determine the intersection of two line segments, as well as a point in polygon test.
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Explained from scratch: private information retrieval and homomorphic encryption
This is a from-scratch explanation of private information retrieval built using homomorphic encryption. I try to assume only a general math / computer science background. To simplify things, my explanations may not always match academic definitions. You can check out this paper or code for more.
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Turns are better than radians
Switching away from radians makes code simpler, faster, and more precise.
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Using GPT-3 to pathfind in random graphs
I wondered how GPT-3 might do on the following problem: given a partially-connected graph, find a path between two nodes, or determine that no path exists. I wrote some code to automate generating …
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Number theory explained from first principles
A lot of modern cryptography builds on insights from number theory, which has been studied for centuries.
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Death by hockey sticks
You may be familiar with the term “hockey stick curve,” used describe a trend that has been flat/stable for a very long time, but shoots up at the end of the series in dramatic fashion, resembling the shape of a hockey stick. Hockey can be a violent sport, and it’s easy to get hurt by even one well-aimed swing…
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Parallel Curves of Cubic Béziers
Blog of Raph Levien.
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Ramanujan Factorial Approximation (2012)
Ramanujan came up with an approximation for factorial that resembles Stirling's famous approximation but is much more accurate. As with Stirling's approximation, the relative error in Ramanujan's approximation decreases as n gets larger. Typically these approximations are not useful for small values of n. For n = 5…
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Someone’s Been Messing with My Subnormals
TL;DR: After noticing an annoying warning, I went on an absurd yak shave, and discovered that because of a tiny handful of Python packages b...
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Busy Beaver Updates: Now Even Busier
Way back in the covid-filled summer of 2020, I wrote a survey article about the ridiculously-rapidly-growing Busy Beaver function. My survey then expanded to nearly twice its original length, with …
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Converting a fisheye image into a panoramic, spherical or perspective projection
The source code implementing the projections below is only available on request for a small fee. It includes a demo application and an invitation to convert an image of your choice to verify the code does what you seek. For more information please contact the author .
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Why do arrays start at 0?
It's not the reason you think. No, it's not that reason either.
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Visual Sum of Cubes
This article discusses a ‘visual’ derivation of the formula for 1³+2³+…+n³.
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The future of interactive theorem proving?
This is a guest post, written by Zhangir Azerbayev. Zhangir is an undergraduate at Yale, majoring in computer science and mathematics. He completed this work while visiting Carnegie Mellon’s …
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Deconstructing Bézier Curves
Graphists, animators, game programmers, font designers, and other graphics professionals and enthusiasts are often working with Bézier curves. They're popular, extensively documented, and used pretty much everywhere. That being said, I find them being explained almost exclusively in 2 or 3 dimensions, which can be a…
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Efficient simulation through linear algebra
I spent a lot of time working on a project in which physically plausible simulation of soft materials with pressure chambers was a key part, and in doing so, we managed to improve a part of our simulation by a significant amount. I was very happy with how this small part of the whole system turned out, and I've been…
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Noether's Theorem in a Nutshell (2020)
Noether's theorem is an amazing result which lets physicists get conserved quantities from symmetries of the laws of nature. Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, and so on.
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Let's Talk SkipList
the forgotten data structure (not really)
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Juris Hartmanis 1928–2022
A sure foundation for Computational Complexity source—wonderful 2015 CACM interview Juris Hartmanis passed away this morning. He was a professor in Cornell’s computer science department…
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Shor, I’ll do it (2007)
I’ve been talking a lot recently about how quantum algorithms don’t work. But last week JR Minkel, an editor at Scientific American, asked me to write a brief essay about how quantum al…
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The Fourth Operation: What Comes After Exponentiation
Tetration -- repeated exponentiation a fractional number of times.
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Knuth changes his mind on Bernoulli number B_1
(I thank Yoshiyuki Kotani for this identity. We'll be able to use the same idea again in 2024 and 2027!)