John D. Cook

The Greek letter β (beta) and the German letter ß (eszett) look similar, especially in some fonts. Now suppose an OCR program sees some character that could be a beta or could be an eszett. It could calculate some kind of distance between between the pixel pattern of the character and the pixel patterns of […] The post Bayesian OCR first appeared on John D. Cook .

The idea of using computers to assist with proofs is not new. The first major computer-assisted proof was published in 1976, the proof of the four color theorem by Kenneth Appel and Wolfgang Haken. The authors reduced the proof of the four color theorem to verifying calculations on 1,834 configurations, each checked by a computer […] The post A 50-year-old computer-assisted proof first appeared o…

A rising tide may lift all boats, but the AI tide lifts some boats much more than others. By all accounts, the best programmers have had the biggest productivity boost from AI. And top tier mathematicians are using AI to settle long-standing mathematical conjectures. AI is a powerful tool, but tools don’t come to life […] The post AI is an intelligence multiplier first appeared on John D. Cook .

Yesterday OpenAI announced a proof that settled a long-standing question about the Navier-Stokes equations from fluid dynamics. The announcement has created a lot of buzz, as one would expect. But there’s an aspect of OpenAI’s work that I haven’t seen anyone talk about: they posted a Lean 4 formal proof at the same time as […] The post The part of Navier-Stokes no one is talking about first appea…

There are rumors that a long-standing math problem, one of the Millennium Prize problems, has been solved. The problem concerns technical properties of solutions to the Navier-Stokes equations [1], an equation that describe the dynamics of fluid flow. Popular accounts of the problem are often oversimplified and misleading. Some reports will speak of the problem […] The post Navier-Stokes in the n…

The Online Etymological Dictionary gives the following etymology for grok: grok (v.) “understand empathically,” 1961, an arbitrary formation by U.S. science fiction writer Robert A. Heinlein (1907-1988) in his book “Stranger in a Strange Land.” In the book it is a transliteration of a Martian word and is said to mean etymologically “to drink.” It attained […] The post Ngram error rate first appea…

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A […] The post Proof of the rank-trace theorem first appeared on …

Suppose you want to know the rank of an n × n matrix A, the number of linearly independent rows of A, or equivalently the number of linearly independent columns. There are at least three difficulties. Difficulties in computing rank First of all, rank is not a continuous function of a matrix. Since rank is an […] The post Computing a lower bound on matrix rank first appeared on John D. Cook .

Eric Lu announced on X today that he has factored RSA-260, a number N with 260 digits (862 bits) that is the product of two large primes [1]. RSA numbers are challenge problems posed to gauge the security of RSA encryption, which rests on the difficulty of factoring large numbers [2]. The naming scheme is […] The post New RSA number factored first appeared on John D. Cook .

I just found out Brian Beckman and I got a patent on work we did for GSI Technology [1]. Nearly all the work I do is under an NDA, so I don’t often get a chance to talk about my projects. This work is public now that it’s in a patent; I suppose it has […] The post Patented application of linear algebra first appeared on John D. Cook .

I watched a few videos this morning, looking for ideas of what I could use AI to do. In one video, someone had an agent monitor tech news sites every 30 minutes to notify him of a variety of developments. No doubt that’s less effort than visiting a bunch of sites every half hour, but […] The post Making the unnecessary easier first appeared on John D. Cook .

This post provides a couple examples to go along with two earlier posts. The pattern we’re illustrating is families of polynomials pn(x) that each satisfy a differential equation and a three-term recurrence. The differential equations have a second solution qn(x) that is the larger solution with respect to x but the smaller solution with respect to n. In both […] The post Second solutions first a…

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 q…

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 […] The post Ultraspherical first a…

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 re…

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 […] The post Three-term recurrences first appeared on John D.…

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 Joh…

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 functio…