AI-powered coding agents increase productivity for many developers. But do these agents produce good-quality code? Some say this doesn’t matter, we are heading toward dark software factories with source code never inspected, and maybe we should eliminate source code altogether—“source code is the new assembly code”. Others have a different view. Humans sometimes need to […]

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When you’re interested in studying a family of functions, it can be useful to look at a differential equation that the functions solve. This is a theme I’ve written about several times, most recently here and here, but also three years ago here. Orthogonal polynomials are mathematically elegant as well as very useful in applications [1]. […]

The post Junk solutions first appeared on John D. Cook.

When I hear the term ultraspherical I think of something extremely spherical. For example, a baseball is spherical, but a billiard ball is more spherical. Maybe a highly polished billiard ball is ultraspherical. Using this line of thought, the term ultraspherical polynomial is inexplicable. This is an example of the arcane terminology I wrote about recently. In […]

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The previous post gave several examples of three-term recurrence relations for special functions. These relations can be computationally useful, but they have to be applied carefully. Several years ago I wrote a post on stable and unstable recurrences. In that post I show that the stability of the recurrence relation for Bessel functions produces depends […]

The post Numerical (in)stability of recurrece relations first appeared on John D. Cook.

There many examples of families of functions where each function can be computed as a linear combination of the two previous terms where a and b are functions of x but not on n. This is called a three-term recurrence formula. It’s amazing how often you can run into three-term recurrence formulas. There are theorems that give conditions […]

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Probability density function must integrate to 1, and so if you know a density function up to a constant, the constant is determined. When you’re looking at a probability density f(x) for the first time, it helps to ignore the normalizing constant. Concentrate on the part of the function involving x and know that the normalizing […]

The post The von Mises-Fisher distribution first appeared on John D. Cook.

Special functions often have arcane names that not very helpful without some context. The previous post goes into some reasons for this. This post will expand on a point at the end of the post about “modified” functions. Things are given their names for a reason. Discovering that reason helps you understand their motivation and […]

The post What exactly is modified about a modified Bessel function? first appeared on John D. Cook.

Special functions are special because they’re useful. They can also be shrouded in arcane terminology. These two facts are related. The more widely useful a function is, the more likely it is that the function will be discovered independently multiple times. Independent discoveries lead to varying definitions and notations. For example, there are two widely […]

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Suppose you wanted to find the distance between Earth and Mars over time. To first approximation, both planets orbit the sun in elliptic orbits in the same plane. If you wanted to be more accurate, you’d need to take into account the fact that the orbit of Mars is tilted about 1.85° relative to the […]

The post The difference orbit inclination makes first appeared on John D. Cook.

There’s a pizza shop near my home with a sign out front that says “Coming Soon.” When I drove by it this morning I thought about how you would model the time until an event happens that is “coming soon.” Suppose I look at the sign one day and guess how many days until the […]

The post Coming soon first appeared on John D. Cook.

A few weeks ago I wrote about the number system used in labeling spreadsheet columns. Labels run from A through Z, then AA through AZ, etc. This looks a lot like base 26, but it’s not quite the same. It has no analog of zero. If Z were like zero, Y would be followed by […]

The post How would you know whether an ancient culture had zero? first appeared on John D. Cook.

I like AI-generated ASCII diagrams. Because nobody would ask AI to generate ASCII diagrams, and so, it’s congruous. I like incongruity [1]. Aside from the incongruity of using a gazillion-parameter neural network to make 1970’s style ASCII art, ASCII diagrams have some uses. They’re absolutely tiny compared to image files. But more importantly they can […]

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A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area. For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, […]

The post Big little hexagon first appeared on John D. Cook.

The imbalance conjecture is now a theorem. James Alexander Schreib and Yousof Yavari posted a proof last week. What does the conjecture theorem say? Start with a graph G and for every edge, calculate the absolute value of the difference of the degree of each end. Then the theorem says there exists another graph H whose vertices […]

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Suppose you have a planet in an elliptical orbit around a star. The math is identical for any light object orbiting a heavy object, such as a moon or satellite orbiting a planet, but we’ll call the heavy object a star and the light object a planet. The center of the star is not quite […]

The post Mean distance to the sun first appeared on John D. Cook.

The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product. Starting with a Hadamard matrix H0 and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by Hn+1 = G ⊗ Hn for  […]

The post Proportion of 1s in a Hadamard matrix first appeared on John D. Cook.

Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the Mariner 9 probe to Mars encoded each 6-bit pixel to a 32-bit codeword in such a way that the original pixel could be recovered if no more than 7 bits were […]

The post Probability of correcting errors first appeared on John D. Cook.

Hadamard matrices are in the news following the recent announcement of a newly discovered Hadamard matrix. I’ve written three posts on Hadamard matrices recently, one as a sort of introduction and two on applications: the error correcting code used in the Mariner 9 probe and constructing sphere packings. A Hadamard matrix is an orthogonal matrix […]

The post Compressing a Hadamard matrix first appeared on John D. Cook.

Yesterday Levent Alpöge announced that he and his colleagues had discovered a new Hadamard matrix using Claude AI. That motivated a post I wrote this morning on how to construct Hadamard matrices. I mentioned in that post that these matrices arise in applications. This evening I gave an example, describing how NASA used a Hadamard […]

The post Hadamard Codes and Sphere Packing first appeared on John D. Cook.

NASA set Mariner 9 to photograph Mars in 1971. The images had to be encoded for transmission using an error-correcting code, otherwise they would be significantly corrupted when they were received on Earth. The images were encoded for transmission using a code based on Hadamard matrices, specifically a (32, 6, 16) Hadamard code. This means […]

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A Hadamard matrix is an orthogonal matrix whose entires are all either 1 or − 1. For example \begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix} is a Hadamard matrix of order 2. True to Stigler’s law of eponymy, James Joseph Sylvester investigated Hadamard matrices before Jacques Hadamard. Sylvester saw how to bootstrap the example […]

The post Constructing Hadamard matrices first appeared on John D. Cook.

Suppose you’ve never worked at the command line and you’re reading a book about the bash shell. You read that !$ is a shortcut to refer to the last word of the previous command. That little fact will almost certainly not stick in your head for a couple reasons. First, you probably see no need […]

The post Cryptic but consistent first appeared on John D. Cook.

I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I poked around the data a little. The top names were not at all what I expected, but then again this is limited […]

The post Dogs and fat tails first appeared on John D. Cook.

Suppose you were able to go back in time, to an era before computers, and give someone contemporary cryptography. Encryption methods that are essentially unbreakable now would certainly be unbreakable then. But there’s a catch: not only do attackers not have computers, neither do users. If you told someone about RSA encryption, for example, you’d […]

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The following extraordinary paragraph comes from Knuth’s TAOCP Volume 4A, right before the last set of exercises. Many of the exercises below ask a modern reader to find and/or to correct errors in the literature of bygone days. The point is not to gloat over how smart we are in the 21st century; the point […]

The post Learning from historical mistakes first appeared on John D. Cook.

In science and engineering classes, you might describe a system using Newton’s laws and end up with a differential equation. You then solve the differential equation, analytically or numerically, to see how the solutions behave. You might also do the opposite, especially in a mathematics class: look at what differential equation a set of functions […]

The post Inverse differential equations first appeared on John D. Cook.

I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand […]

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At the end of my post on how not to calculate cosine I said that the first step in calculating cosine, particularly cosine of a large number, would be to do range reduction. This post will present a simple range reduction method by Cody and Waite that is adequate for moderately large arguments. If you […]

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I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ' on my laptop, it becomes ’ on my phone. And when I type 's on my phone, it becomes 痴 on my laptop. Apparently the phone turns the apostrophe (U+0027) […]

The post Corrupted apostrophes first appeared on John D. Cook.

Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don’t. I worked on the implementation of trig functions in hardware, and I can assure you we didn’t just use power series. Power series are an excellent way calculate functions near the center of the […]

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