John D. Cook

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 [...]The post What is the quality of software that AI writes? first appeared on John D. Cook.

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 hereand 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 ultrasphericalI 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 polynomialis inexplicable. This is an example of the arcane terminology I wrote about recently. In [...]The post Ultraspherical first appeared on John D. Cook.

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 recurrence 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 wherea andb are functions ofx 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 [...]The post Three-term recurrences first appeared on John D. Cook.

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 densityf(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 reasons. Discovering those reasons may help 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 [...]The post Why special function terminology is arcane first appeared on John D. Cook.

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 [...]The post AI-generated ASCII diagrams first appeared on John D. Cook.

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 graphG with no edge between two nodes of the same degree. Then for every edge, calculate the absolute value of the difference of the degree of each end. [...]The post The imbalance theorem first appeared on John D. Cook.

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 matrixH0 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 Alpoge 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 [...]The post How NASA's Mariner 9 probe encoded images first appeared on John D. Cook.

A Hadamard matrix is an orthogonal matrix whose entries are all either 1 or - 1. For example 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 above into more examples. If H is a [...]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. Manual cryptography If you told someone about RSA encryption, for [...]The post Manually unbreakable cryptography first appeared on John D. Cook.

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 [...]The post DNA and Bessel functions first appeared on John D. Cook.

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 [...]The post A simple range reduction method first appeared on John D. Cook.

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 aTM on my phone. And when I type 's on my phone, it becomes 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 to calculate functions near the center of [...]The post How not to calculate cosine first appeared on John D. Cook.

In a footnote to the previous post, I said that Python's math library can calculate the logarithm of extremely large numbers but not the cosine. This post will expand on that comment. In this post I'll use n = 200! as my example rather than 1000! because this value of N is larger than the [...]The post cos(200!) first appeared on John D. Cook.

The previous post pointed out that the following code such as the following unexpectedly works. >>> from math import log, factorial >>> log(factorial(1000)) 5912.128178488163 If you don't find this unexpected, note that if you replace math.log with numpy.log the code will fail [1]. Functions like natural logarithm operate on real numbers. Real numbers are represented [...]The post Calculating log(1000!) first appeared on John D. Cook.

Last week I wrote a post on hiding cryptographic keys in decks of cards. I wrote some code for that post that shouldn't work, but before fixing I noticed that it in fact did work. The code computes logarithms for integers larger than the largest representable float. For example, the largest float is on the [...]The post The code that didn't break first appeared on John D. Cook.

A few days ago I wrote a post on counting rooted trees. That post looked at the sequence c(n) which counts the number of rooted trees with n nodes. Here one node is distinguished as the root, but the nodes below the root are not distinguished from each other; all that matters is how the [...]The post Enumerating trees and circles first appeared on John D. Cook.

After writing the previous post about metallic ratios, I thought about the analogy to alchemy and the attempt to make precious metals out of base metals. When can you make one metallic ratio out of another? Can you make the golden ratio out of the lead ratio? Before we can make gold out of lead, [...]The post Mathematical alchemy first appeared on John D. Cook.

The golden ratio is the first and best known of the metallic ratios. I've written about the silver ratio a few times, most recently here. And I've mentioned the bronze ratio a couple times. The metallic ratios after bronze don't have standard names. The nth metallic ratio M(n) is the number whose continued fraction representation [...]The post Ratio of metallic ratios first appeared on John D. Cook.

Yesterday I wrote that a lot of the special functions that pop up in mathematical physics are solutions to second order linear differential equations with polynomial coefficients. More generally, holonomic functions are defined to be those functions that are the solutions to linear differential equations, of any order, with polynomial coefficients. Most special functions are [...]The post Holonomic functions first appeared on John D. Cook.

In this post I mentioned two series which I denotedt(n) andc(n). The former is the number of unlabeled rooted trees withn nodes. The latter is the cumulative sum of the former, i.e. The sequencec(n) is also the number of constraints on an n-step Runge-Kutta method; that's how I became interested in it. Now thet(n) sequence [...]The post Estimating a cumulative sum first appeared on John D. Cook.

Second order linear differential equations with polynomial coefficients form their own area of study. This seems like a narrow class of equations, but it's very important in applications. This class of equations seems like a mathematically natural topic, but why is it so important in applications? I did a PhD in differential equations without ever [...]The post Why polynomial coefficients? first appeared on John D. Cook.

Combinatorial problems can be interesting for their own sake, but they are more interesting when there is a connection to a problem outside combinatorics, and the more unexpected the connection the better. Counting the number of unlabeled rooted trees [1] with n nodes is a pure mathematics problem. Designing numerical methods for solving differential equations [...]The post Counting rooted trees first appeared on John D. Cook.

The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of RK the number of stages s also matches the order of the error p. But in order to achieve error on the order ofp >= 5, you need more than p [...]The post Runge-Kutta order versus stages first appeared on John D. Cook.

I was digging into the Runge-Kutta method for solving differential equations and a line from [1] piqued my curiosity. These calculations, which are not reproduced in Kutta's paper (they are however in Huen (1900)), are very tedious. The calculations are a set of eight constraints that the parameters of a fourth-order Runge-Kutta method must satisfy. [...]The post Solving the RK4 design equations first appeared on John D. Cook.

A couple years ago I wrote about how to compute the inverse of factorial. I used that code in writing the previous post because the post required solving the equation log2(n!) >= b given b. That is, given a number of bits b, find the smallest value of nsuch that n! >= 2b. What the [...]The post Inverse factorial improved first appeared on John D. Cook.

The previous post looked at the idea of storing a cryptographic key in the order of a deck of cards. A deck of 52 cards can store 225 bits of data because log2(52!) = 225. Here x isx rounded down to the nearest integer. If we want to store bigger keys, we're going to need [...]The post Cryptographic Keys and Decks of Cards first appeared on John D. Cook.

The latest issue of Paged Out! has an article by Stephen Hewitt An off-line backup of your cryptographic key using playing cards." The idea is to use a deck of 52 to store a 128-bit cryptographic key. To erase the key, shuffle the deck. Hewitt gives his algorithm for embedding a key, one that can [...]The post Hiding data in permutations first appeared on John D. Cook.

My post from yesterday on permutation roots ends with a Mathematica code for finding the probability that a permutation of n elements has a kth root. This is done by finding the coefficient of xn in the generating function I wanted to say more about this, and look at implementing the same code in SymPy. [...]The post Counting permutations with roots first appeared on John D. Cook.

It's well known that you can convert the base 16 (hex) representation of an integer to the base 2 (binary) representation by simply converting each digit from hex to binary. For example, CAFEhex = 1100 1010 1111 1110two I imagine it's less well known that you can do the same thing with floating point numbers. [...]The post Printing floating point numbers in binary first appeared on John D. Cook.

Let be a permutation on n elements. If there is a permutation such that applying twice has the same effect on the list of elements as applying once, we say = ^2 and is a square root of . If we let our n elements be the integers 0 [...]The post Permutation roots first appeared on John D. Cook.

The function expq(x) is defined by taking the power series for exp(x) and keeping only the terms whose index is a multiple of q. For example, exp2(x) keeps only the even-numbered terms in the exponential power series and so equals cosh(x). In general, The first sum uses Iverson's bracket notation: a Boolean expression in brackets [...]The post exp_q first appeared on John D. Cook.

I was working with a wide spreadsheet from a client the other day and I had to convert between Excel column labels and column numbers. I had never paid attention to how Excel labels columns and implicitly thought it was base 26 using letters rather than digits. But then I realized that's not right. Excel [...]The post Excel column numbering first appeared on John D. Cook.