John D. Cook

Bernstein’s Matrix Mathematics is impressive. It’s over 1500 pages and weighs 5.3 pounds (2.4 kg). It’s a reference book, not the kind of book you just sit down to read. (Actually, I have sat down to read parts of it.) I’d used a library copy of the first edition, and so when Princeton University Press […]

The Moore-Penrose pseudoinverse of a matrix is a way of coming up with something like an inverse for a matrix that doesn’t have an inverse. If a matrix does have an inverse, then the pseudoinverse is in fact the inverse. The Moore-Penrose pseudoinverse is also called a generalized inverse for this reason: it’s not just […]

One of my complaints about math writing is that definitions are hardly ever subtractive, even if that’s how people think of them. For example, a monoid is a group except without inverses. But that’s not how you’ll see it defined. Instead you’ll read that it’s a set with an associative binary operation and an identity […]

A probability distribution is called “fat tailed” if its probability density goes to zero slowly. Slowly relative to what? That is often implicit and left up to context, but generally speaking the exponential distribution is the dividing line. Probability densities that decay faster than the exponential distribution are called “thin” or “light,” and densities that […]

The Duffing equation is an ordinary differential equation describing a nonlinear damped driven oscillator. If the parameter μ were zero, this would be a damped driven linear oscillator. It’s the nonlinear x³ term that makes things nonlinear and interesting. Using an analog computer in 1961, Youshisuke Ueda discovered that this system was chaotic. It was […]

The first post in this series looked at a possible formula for the shape of an egg, how to fit the parameters of the formula, and the curvature of the shape at each end of the egg. The second post looked at the volume. This post looks at the surface area. If you rotate the […]

The previous post looked at an equation to fit the shape of an egg. In two dimensions we had In this post, we’ll rotate that curve around the x-axis to find the volume. Then we’ll see how it compares to that of an ellipsoid. If we rotate the graph of a function f(x) around the x-axis with x ranging […]

How would you fit an equation to the shape of an egg? This site suggests an equation of the form Note that if k = 0 we get an ellipse. The larger the parameter k is, the more asymmetric the shape is about the y-axis. Let’s try that out in Mathematica: ContourPlot[ x^2/16 + y^2 (1 + 0.1 […]

I said something about Perl 6 the other day, and someone replied asking whether anyone actually uses Perl 6. My first thought was I bet more people use Perl 6 than Haskell, and it’s well known that people use Haskell. I looked at the TIOBE Index to see whether that’s true. I won’t argue how […]

Researchers have discovered that for some problems, deep neural networks (DNNs) can get by with low precision weights. Using fewer bits to represent weights means that more weights can fit in memory at once. This, as well as embedded systems, has renewed interest in low-precision floating point. Microsoft mentioned its proprietary floating point formats ms-fp8 and […]

The IEEE standard 754-2008 defines several sizes of floating point numbers—half precision (binary16), single precision (binary32), double precision (binary64), quadruple precision (binary128), etc.—each with its own specification. Posit numbers, on the other hand, can be defined for any number of bits. However, the IEEE specifications share common patterns so that you could consistently define theoretical […]

Categorical data analysis could mean a couple different things. One is analyzing data that falls into unordered categories (e.g. red, green, and blue) rather than numerical values (e..g. height in centimeters). Another is using category theory to assist with the analysis of data. Here “category” means something more sophisticated than a list of items you […]

This post will introduce posit numbers, explain the interpretation of their bits, and discuss their dynamic range and precision. Posit numbers are a new way to represent real numbers for computers, an alternative to the standard IEEE floating point formats. The primary advantage of posits is the ability to get more precision or dynamic range out […]

I recently ran into a tweet saying that if ** denotes exponentiation then // should denote logarithm. With this notation, for example, if we say 3**4 == 81 we would also say 81 // 3 == 4. This runs counter to convention since // has come to be a comment marker or a notation for integer […]

Motivating example: planet spacing My previous post showed that planets are roughly evenly distributed on a log scale, not just in our solar system but also in extrasolar planetary systems. I hadn’t seen this before I stumbled on it by making some plots. I didn’t think it was an original discovery—I assume someone did this […]

The previous post was about Kepler’s observation that the planets were spaced out around the sun the same way that nested regular solids would be. Kepler only knew of six planets, which was very convenient because there are only five regular solids. In fact, Kepler thought there could only be six planets because there are only […]

Johann Kepler discovered in 1596 that the ratios of the orbits of the six planets known in his day were the same as the ratios between nested Platonic solids. Kepler was understandably quite impressed with this discovery and called it the Mysterium Cosmographicum. I heard of this in a course in the history of astronomy […]

I was looking at my daughter’s statistics homework recently, and there were a pair of questions about testing the level of lead in drinking water. One question concerned testing whether the water was safe, and the other concerned testing whether the water was unsafe. There’s something bizarre, even embarrassing, about this. You want to do […]

Curvature is tedious to calculate by hand because it involves calculating first and second order derivatives. Of course other applications require derivatives too, but curvature is the example we’ll look at in this post. Computing derivatives It would be nice to write programs that only explicitly implement the original function and let software take care […]

The generalized normal distribution adds an extra parameter β to the normal (Gaussian) distribution. The probability density function for the generalized normal distribution is Here the location parameter μ is the mean, but the scaling factor σ is not the standard deviation unless β = 2. For small values of the shape parameter β, the […]

One attempt at rationalizing astrology is to say that the gravitational effects of celestial bodies impact our bodies. To get an idea how hard the stars and planets pull on us, let’s compare their gravitational attraction to that of cows at various distances. Newton’s law of gravity says that the gravitational attraction between two objects […]

Every positive integer can be written as the sum of three palindromes, numbers that remain the same when their digits are reverse. For example, 389 = 11 + 55 + 323. This holds not just for base 10 but for any base b ≥ 5. The result and algorithms for finding the palindromes was published […]

Take any positive integer n and sum the squares of its digits. If you repeat this operation, eventually you’ll either end at 1 or cycle between the eight values 4, 16, 37, 58, 89, 145, 42, and 20. For example, pick n = 389. Then 3² + 8² + 9² = 9 + 64 + […]

Our adventure starts with the following ordinary differential equation: Analytic solution We can solve this equation in closed-form, depending on your definition of closed-form, by multiplying by an integrating factor. The left side factors to and so The indefinite integral above cannot be evaluated in elementary terms, though it can be evaluated in terms of […]

Ratios of gamma functions come up often in applications. If the two gamma function arguments differ by an integer, then it’s easy to calculate their ratio exactly by using (repeatedly if necessary) the fact at Γ(x + 1) = x Γ(x). If the arguments differ by 1/2, there is no closed formula, but the there […]

Differential privacy adds Laplace-distributed random noise to data to protect individual privacy. (More on that here.) Although it’s simple to generate Laplacian random values, the Laplace distribution is not always one of the built-in options for random number generation libraries. The Laplace distribution with scale β has density The Laplace distribution is also called the double […]

I heard somewhere that Pluto receives more sunlight than you might think, enough to read by, and that sunlight on Pluto is much brighter than moonlight on Earth. I forget where I heard that, but I’ve done a back-of-the-envelope calculation to confirm that it’s true. Pluto is about 40 AU from the sun, i.e. forty times […]

Suppose you know a number is between 30 and 42. You want to guess the number while minimizing how wrong you could be in the worst case. Then you’d guess the midpoint of the two ends, which gives you 36. But suppose you want to minimize the worst case relative error? If you chose 36, […]

The previous post looked at how much information is contained in zip codes. This post will look at how much information is contained in someone’s age, birthday, and birth date. Combining zip code with birthdate will demonstrate the plausibility of Latanya Sweeney’s famous result [1] that 87% of the US population can be identified based […]

If you know someone’s US zip code, how much do you know about them? We can use entropy to measure the amount of information in bits. To quantify the amount of information in a zip code, we need to know how many zip codes there are, and how evenly people are divided into zip codes. […]

Dark debt is a form of technical debt that is invisible until it causes failures. The term was coined in the STELLA conference and codified in the conference report. Dark debt is found in complex systems and the anomalies it generates are complex system failures. Dark debt is not recognizable at the time of creation. … It […]

Given two functions f and g, the product rule tells you how to take the first derivative of their product, and the chain rule tells you how to take the first derivative of their composition. What if you want to take higher order derivatives? You could repeatedly apply basic calculus rules, but there are formulas […]

Pick random numbers uniformly between 0 and 1, adding them as you go, and stop when you get a result bigger than 1. How many numbers would you expect to add together on average? You need at least two samples, and often two are enough, but you might get any number, and those larger numbers […]

Inverting a power series As a student, one of the things that seemed curious to me about power series was that a function might have a simple power series, but its inverse could be much more complicated. For example, the coefficients in the series for arctangent have a very simple pattern but the coefficients in […]

Here’s an identity relating Fibonacci numbers and binomial coefficients. You can think of it as a finite sum or an infinite sum: binomial coefficients are zero when the numerator is an integer and the denominator is a larger integer. See the general definition of binomial coefficients. Source: Sam E. Ganis. Notes on the Fibonacci Sequence. The […]

This evening, after I got off a phone call discussing a project with a colleague, I thought “Huh, I guess you could call that project management.” I worked as a project manager earlier in my career, but what I’m doing now feels completely different and much more pleasant. Strip away the bureaucracy and politics, and […]

There’s now an “Animate” link on the exponential sum pages that lets you watch the curves being drawn. Sometimes these are surprising. The plot of the partial sums might bounce all over in the process of filling in a very symmetric plot. Here’s an example of that.

Probability is full of theorems that say that probability density approximates another as some parameter becomes large. All the dashed lines in the diagram below indicate a relationship like this. You can find details of what everything in the diagram means here. How can you quantify these approximations? One way is to use Kullback-Leibler […]

I’ve run into potential polynomials a lot lately. Most sources I’ve seen are either unclear on how they are defined or use a notation that doesn’t sit well in my brain. Also, they’ll either give an implicit or explicit definition but not both. Both formulations are important. The implicit formulation suggests how potential polynomials arise […]

Bell polynomials come up naturally in a variety of contexts: combinatorics, analysis, statistics, etc. Unfortunately, the variations on Bell polynomials are confusingly named. Analogy with differential equations There are Bell polynomials of one variable and Bell polynomials of several variables. The latter are sometimes called “partial” Bell polynomials. In differential equations, “ordinary” means univariate (ODEs) […]

If you visit this blog once in a while, here are a few ways to hear from me more regularly. Subscription You can subscribe to the blog via RSS or email. I often use SVG images because they look great on a variety of devices, but most email clients won’t display that format. If you […]

A “squircle” is a sort of compromise between a square and circle, but one that differs from a square with rounded corners. It’s a shape you’ll see, for example, in some of Apple’s designs. A straight line has zero curvature, and a circle with radius r has curvature 1/r. So in a rounded square the […]

Suppose you have a differential equation of the form If the function f(x) is constant, the differential equation is easy to solve in closed form. But when it is not constant, it is very unlikely that closed form solutions exist. But there may be useful closed form approximate solutions. There is no linear term in […]

Yesterday I said on Twitter “Time to see whether practice agrees with theory, moving from LaTeX to Python. Wish me luck.” I got a lot of responses to that because it describes the experience of a lot of people. Someone asked if I’d blog about this. The content is confidential, but I’ll talk about the […]

Sir Michael Atiyah recommends Hermann Weyl’s book The Classical Groups for its clarity and beautiful prose. From my interview with Atiyah: Hermann Weyl is my great model. He used to write beautiful literature. Reading it was a joy because he put a lot of thought into it. Hermann Weyl wrote a book called The Classical Groups, […]

Consider the following Taylor series for sin(θ/7) and the following two functions based on the series, one takes only the first non-zero term def short_series(x): return 0.14285714*x and a second that three non-zero terms. def long_series(x): return 0.1425714*x - 4.85908649e-04*x**3 + 4.9582515e-07*x**5 Which is more accurate? Let’s make a couple plots plot to see. First […]

Here’s an apparent paradox. You’ll hear that Monte Carlo methods are independent of dimension, and that they scale poorly with dimension. How can both statements be true? The most obvious way to compute multiple integrals is to use product methods, analogous to the way you learn to compute multiple integrals by hand. Unfortunately the amount […]

Spheres and balls are examples of common words that take on a technical meaning in math, as I wrote about here. Recall the the unit sphere in n dimensions is the set of points with distance 1 from the origin. The unit ball is the set of points of distance less than or equal to 1 from the […]

For a random variable X, the kth moment of X is the expected value of Xk. For any random variable X with 0 mean, or negative mean, there’s an inequality that bounds the 3rd moment, m3 in terms of the 4th moment, m4: The following example shows that this bound is the best possible. Define u […]

Here’s a useful LaTeX command that I learned about recently: \underbrace. It does what it sounds like it does. It puts a brace under its argument. I used this a few days ago in the post on the new prime record when I wanted to show that the record prime is written in hexadecimal as […]