John D. Cook

Yesterday I wrote about Householder’s higher-order generalizations of Newton’s root finding method. For n at least 2, define and iterate Hn to find a root of f(x). When n = 2, this is Newton’s method. In yesterday’s post I used Mathematica to find expressions for H3 and H4, then used Mathematica’s FortranForm[] function to export […]

Are there infinitely many positive integers n such that tan(n) > n? David P. Bellamy and Felix Lazebnik [1] asked in 1998 whether there were infinitely many solutions to |tan(n)| > n and tan(n) > n/4. In both cases the answer is yes. But at least as recently as 2014 the problem at the top […]

There are variations on Newton’s root finding method that use higher derivatives and converge faster. Alston Householder developed a sequence of such methods of the form where the superscripts in parentheses indicate derivatives. When n = 2, Householder’s method reduces to Newton’s method. Once Newton’s method is close enough to a root, the error is […]

The utility numfmt, part of Gnu Coreutils, formats numbers. The main uses are grouping digits and converting to and from unit suffixes like k for kilo and M for mega. This is somewhat useful for individual invocations, but like most command line utilities, the real value is using it as part of pipeline. The --grouping […]

I wanted to stress test the bc calculator a little and so I calculated π to 10,000 digits a couple different ways. First I ran time bc -l <<< "scale=10000;4*a(1)" which calculates π as 4 arctan(1). This took 2 minutes and 38 seconds. I imagine bc is using some sort of power series to compute […]

The previous post looked at an algorithm for generating De Bruijn sequences B(k, n) where k is a prime number. These are optimal sequences that contain every possible consecutive sequence of n symbols from an alphabet of size k. As noted near the end of the post, the case k = 2 is especially important […]

Last week I wrote about a way to use De Bruijn cycles. Today I’ll write about a way to generate De Bruijn cycles. De Bruijn cycles Start with an alphabet of k symbols. B(k, n) is the set of cycles of that contain every sequence of n symbols, and is as short as possible. Since […]

Google announced today that it has demonstrated “quantum supremacy,” i.e. that they have solved a problem on a quantum computer that could not be solved on a classical computer. Google says Our machine performed the target computation in 200 seconds, and from measurements in our experiment we determined that it would take the world’s fastest […]

As I mentioned in my computational survivalist post, I’m working on a project where I have a dedicated computer with little more than basic Unix tools, ported to Windows. It’s given me new appreciation for how the standard Unix tools fit together; I’ve had to rely on them for tasks I’d usually do a different […]

If you look at a sine wave, its curvature is sorta like its values. When sine is zero, the curve is essentially a straight line, and the curvature is zero. When sine is at its maximum, the function bends the most and so the curvature is at a maximum. This makes sense analytically. The second […]

Suppose you have a keypad that will unlock a door as soon as it sees a specified sequence of four digits. There’s no “enter” key to mark the end of a four-digit sequence, so the four digits could come at any time, though they have to be sequential. So, for example, if the pass code […]

My previous post contained the image below. The point of the post was that the numbers that came up in yet another post made the fractal image above when you wrote them in binary. Here’s the trick I used to make that image. To make the pattern of 0’s and 1’s easier to see, I […]

As mentioned in the previous post, the Gauss-Wantzel theorem says you can construct a regular n-gon with a straight edge and compass if and only if n has the form 2k F where k is a non-negative integer and F is a product of distinct Fermat primes. Let’s look at the binary representation of these […]

If you know the sine of any angle, you can find its cosine from the Pythagorean theorem. And if you know the sine of an angle you can find the sine of any multiple of that angle using the identity for the sine of a sum. You can find that sin 30° = 1/2 from […]

Suppose you want to sent pictures from Jupiter back to Earth. A lot could happen as a bit travels across the solar system, and so you need some way of correcting errors, or at least detecting errors. The simplest thing to do would be to transmit photos twice. If a bit is received the same […]

Yesterday I wrote a post looking at the frequency of Koine Greek letters and the corresponding entropy. David Littleboy asked what an analogous calculation would look like for a language like Japanese. This post answers that question. First of all, information theory defines the Shannon entropy of an “alphabet” to be bits where pi is […]

Kenneth G. Libbrecht has posted a 523-page book on snow to arXiv.

Would the letters in an ancient Greek text carry more or less information than in modern English? To address this question, I downloaded a copy of the Greek New Testament from Project Gutenberg and ran the word frequency script from my previous post. This lead to the follow table of letters and percent frequency. α […]

Once in a while I need to know what characters are in a file and how often each appears. One reason I might do this is to look for statistical anomalies. Another reason might be to see whether a file has any characters it’s not supposed to have, which is often the case. A few […]

A few days ago I wrote about computational survivalists, people who prepare to be able to work on computers with only software that is available everywhere. Of course nothing is available everywhere, and so each person interprets “everywhere” to mean computers they anticipate using. If you need to edit a text file on a Windows […]

The idea behind differential privacy is that it doesn’t make much difference whether your data is in a data set or not. How much difference your participation makes is made precise in terms of probability statements. The exact definition doesn’t for this post, but it matters that there is an exact definition. Someone designing a […]

Queueing theory is the study of waiting in line. That may not sound very interesting, but the subject is full of surprises. For example, when a server is near capacity, adding a second server can cut backlog not just in half but by an order of magnitude or more. More on that here. In this […]

There are a couple different ways to combine random variables into a new random variable: means and mixtures. To take the mean of X and Y you average their values. To take the mixture of X and Y you average their densities. The former makes the tails thinner. The latter makes the tails thicker. When […]

Primes usually look odd. They’re literally odd [1], but they typically don’t look like they have a pattern, because a pattern would often imply a way to factor the number. However, 12345678910987654321 is prime! I posted this on Twitter, and someone with the handle lagomoof replied that the analogous numbers are true for bases 2, […]

Some programmers and systems engineers try to do everything they can with basic command line tools on the grounds that someday they may be in an environment where that’s all they have. I think of this as a sort of computational survivalism. I’m not much of a computational survivalist, but I’ve come to appreciate such […]

How many days are in a year? 365. How many times does the earth rotate on its axis in a year? 366. If you think a day is the time it takes for earth to rotate once around its axis, you’re approximately right, but off by about four minutes. What we typically mean by “day” […]

In the previous post, I said that Lissajous curves are the result of plotting a curve whose x and y coordinates come from (undamped) harmonic oscillators. If we add a little bit of dampening, multiplying our cosine terms by negative exponentials, the resulting curve is called a harmonograph. Here’s a bit of Mathematica code to […]

Suppose that over time the x and y coordinates of a point are both given by a harmonic oscillator, i.e. x(t) = cos(nx t + φx) y(t) = cos(ny t + φy) Then the resulting path is called a Lissajous curve. If you add a z coordinate also given by harmonic oscillator z(t) = cos(nz […]

For an ellipsoid with equation the Gaussian curvature at each point is given by Now suppose a ≥ b ≥ c > 0. Otherwise relabel the coordinate axes so that this is the case. Then the largest curvature occurs at (±a, 0, 0), and the smallest curvature occurs at (0, 0, ±c). You could prove […]

Take a calculator and enter any number. Then press the cosine key over and over. Eventually the numbers will stop changing. You will either see 0.99984774 or 0.73908513, depending on whether your calculator was in degree mode or radian mode. This is an example of a fixed point, a point that doesn’t change when you […]

Suppose you have a polynomial p(x) and in interval [a, b] and you want to know how many distinct real roots the polynomial has in the interval. You can answer this question using Sturm’s algorithm. Let p0(x) = p(x) and letp1(x) be its derivative p‘(x). Then define a series of polynomials for i ≥ 1 […]

Tie a knot in a rope and join the ends together. At each point in the rope, compute the curvature, i.e. how much the rope bends, and integrate this over the length of the rope. The Fary-Milnor theorem says the result must be greater than 4π. This post will illustrate this theorem by computing numerically […]

Julian Havil writes what I think of as serious recreational mathematics. His books are recreational in the sense that they tell a story rather than cover a subject. They are lighter reading than a text book, but require more advanced mathematics than books by Martin Gardner. Havil’s latest book is Curves for the Mathematically Curious. […]

I didn’t realize until recently that there’s a connection between the control key on a computer keyboard and controlling a mechanical device. Both uses of the word control are related via ASCII control characters as I discovered by reading the blog post Four Column ASCII. Computers work with bits in groups of eight, and there […]

Suppose you want to test whether something you’re doing is having any effect. You take a few measurements and you compute the average. The average is different than what it would be if what you’re doing had no effect, but is the difference significant? That is, how likely is it that you might see the […]

California’s new privacy law takes effect January 1, 2020, less than 100 days from now. The bill was written in a hurry in order to prevent a similar measuring from appearing on a ballot initiative. The thought was that the state legislature would pass something quickly then clean it up later with amendments. Six amendments […]

The GDPR‘s right-to-be-forgotten has been in the news this week. This post will look at a couple news stories and how they relate. Forgetting about a stabbing On Monday the New York Times ran a story about an Italian news site that folded as a result of resisting requests to hide a story about a […]

Using program exceptions as a learning tool: When I’m learning something new, I sometimes find myself practicing EDD (exception driven development). I try to evaluate some code, get an exception or error message, and then Google the error message to figure out what the heck happened. From Mastering Clojure Macros

If something is outside your grasp, it’s hard to know just how far outside it is. Many times I’ve intended to sit down and understand something thoroughly, and I’ve put it off for years. Maybe it’s a programming language that I just use a few features of, or a book I keep seeing references to. […]

Typesetting zodiac symbols in LaTeX is admittedly an unusual thing to do. LaTeX is mostly used for scientific publication, and zodiac symbols are commonly associated with astrology. But occasionally zodiac symbols are used in more respectable contexts. The wasysym package for LaTeX includes miscellaneous symbols, including zodiac symbols. Here are the symbols, their LaTeX commands, […]

I read in Wikipedia this morning that there’s a pattern to the parity of flight numbers. Among airline flight numbers, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights. I never noticed this. I could see how it might be a useful convention. It would mean that […]

I was reading Rupert Miller’s book Beyond ANOVA when I ran across this line: I never use the Kolmogorov-Smirnov test (or one of its cousins) or the χ² test as a preliminary test of normality. … I have a feeling they are more likely to detect irregularities in the middle of the distribution than in […]

There are tools that I’ve used occasionally for many years that I’ve just started to appreciate lately. “Oh, that’s why they did that.” When you see something that looks poorly designed, don’t just exclaim “Why would anyone do that?!” but ask sincerely “Why would someone do that?” There’s probably a good reason, or at least […]

Mersenne primes are prime numbers of the form 2p – 1. It turns out that if 2p – 1 is a prime, so is p; the requirement that p is prime is a theorem, not part of the definition. So far 51 Mersenne primes have discovered [1]. Maybe that’s all there are, but it is […]

Here is a 12-minute video from Minute Physics, in collaboration with the US Census Bureau, giving an overview of differential privacy and how the 2020 census will use it to protect privacy. Related posts Scaling up differential privacy: lessons from the US Census Protecting privacy while keeping detailed date information Comparing differential privacy to Safe […]

The Collatz conjecture asks whether the following procedure always terminates at 1. Take any positive integer n. If it’s odd, multiply it by 3 and add 1. Otherwise, divide it by 2. For obvious reasons the Collatz conjecture is also known as the 3n + 1 conjecture. It has been computationally verified that the Collatz […]

One of the things I liked about Perl was string interpolation. If you use a variable name in a string, the variable will expand to its value. For example, if you a variable $x which equals 42, then the string "The answer is $x." will expand to “The answer is 42.” Perl requires variables to […]

In my previous post on VIN numbers, I commented that if a check sum has to be one of 11 characters, it cannot detect all possible changes to a string from an alphabet of 33 characters. The number of possible check sum characters must be at least as large as the number of possible characters […]

A VIN (vehicle identification number) is a string of 17 characters that uniquely identifies a car or motorcycle. These numbers are used around the world and have three standardized formats: one for North America, one for the EU, and one for the rest of the world. Letters that resemble digits The characters used in a […]

The Collatz conjecture is for computer science what until recently Fermat’s last theorem was for mathematics: a famous unsolved problem that is very simple to state. The Collatz conjecture, also known as the 3n+1 problem, asks whether the following function terminates for all positive integer arguments n. def collatz(n): if n == 1: return 1 […]