John D. Cook

The previous post linked to a post I wrote a few years ago about the Hilbert transform and Fourier series. That post says that if the Fourier series of a function is then the Fourier series of its Hilbert transform is When I looked back at that post I thought about how if you thought […] The post Hilbert transform as an infinite matrix first appeared on John D. Cook .

The previous post announced some notes I wrote up based on an article by Henry Baker implementing functions of a complex variable in terms of functions of a real variable. That is, it finds functions u(x, y) and v(x, y) such that f(x + iy) = u(x, y) + i v(x, y) where x, y, u, and v are […] The post Real and imaginary parts first appeared on John D. Cook .

A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations You can do something analogous for all the elementary functions, though some of the equations are quite a bit more complicated than the ones above. See the […] The post Building complex functions out of real parts first appeared on John D. Cook .

“You can’t always get what you want. But sometimes you get what you need.” — The Rolling Stones Circular functions and hyperbolic functions aren’t invertible, but we invert them anyway. These functions map many points in the domain to each point in the range, and we invert them by mapping a point in the range […] The post Couth and uncouth function pairs first appeared on John D. Cook .

The difference between a circular function and a hyperbolic function is a rotation or two. For example, cosh(z) = cos(iz). You can read that as saying that to find the hyperbolic cosine of z, first you rotate z a quarter turn to the left (i.e. multiply by i) and then take the cosine. For another example, […] The post Circular and hyperbolic functions differ by rotations first appeared on John D. …

How should we define √(z² − 1)? Well, you could square z, subtract 1, and take the square root. What else would you do?! The question turns out to be more subtle than it looks. When x is a non-negative real number, √x is defined to be the non-negative real number whose square is x. When x is […] The post Square root of x² − 1 first appeared on John D. Cook .

The previous post derived the identity and said in a footnote that the identity holds at least for x > 1 and y > 1. That’s true, but let’s see why the footnote is necessary. Let’s have Mathematica plot The plot will be 0 where the identity above holds. The plot is indeed flat for x > 1 […] The post Closer look at an identity first appeared on John D. Cook .

Markov numbers are integer solutions to x² + y² + z² = 3xyz. The Wikipedia article on Markov numbers mentions that Don Zagier studied Markov numbers by looking the approximating equation x² + y² + z² = 3xyz + 4/9 which is equivalent to f(x) + f(y) = f(z) where f(t) is defined as arccosh(3t/2). It wasn’t clear to me why the […] The post Approximating Markov’s equation first appeared on John D. Coo…

I’ve written a couple posts lately about reverse engineering the internal state of a random number generator, first Mersenne Twister then lehmer64. This post will look at xorshift128, implemented below. import random # Seed the generator state a: int = random.getrandbits(32) b: int = random.getrandbits(32) c: int = random.getrandbits(32) d: int = random.getrandbits(32) MASK = […] The post Recover…

If you look very closely at my previous post, you’ll notice that I initialize a 128-bit integer with a 64-bit value. The 128-bit unsigned integer represents the internal state of a random number generator. Why not initialize it to a 128-bit value? I was trying to keep the code simple. A surprising feature of C […] The post Initialize and print 128-bit integers in C first appeared on John D. Cook .

A couple days ago I wrote about hacking the Mersenne Twister. I explained how to recover the random number generator’s internal state from a stream of 640 outputs. This post will do something similar with the lehmer64 random number generator. This generator is very simple to implement. Daniel Lemire found it to be “the fastest […] The post Hacking the lehmer64 RNG first appeared on John D. Cook .

This morning I wrote a post about the probability that a random matrix over a finite field is invertible. If the field has q elements and the matrix has dimensions n × n then the probability is In that post I made observation that p(q, n) converges very quickly as a function of n [1]. […] The post Euler function first appeared on John D. Cook .

What is the inverse of shifting a sequence to the right? Shifting it to the left, obviously. But wait a minute. Suppose you have a sequence of eight bits abcdefgh and you shift it to the right. You get 0abcdefg. If you shift this sequence to the left you get abcdefg0 You can’t recover the […] The post Inverse shift first appeared on John D. Cook .

The two latest posts have involved invertible matrices with 0 and 1 entries. If you fill an n × n matrix with 0s and 1s randomly, how likely is it to be invertible? What kind of inverse? There are a couple ways to find the probability that a binary matrix is invertible, depending on what […] The post Probability that a random binary matrix is invertible first appeared on John D. Cook .

The previous post looked at the tempering step of the Mersenne Twister, formulating a sequence of bit operations as multiplication by a matrix mod 2. This post will look at the components more closely. The theorems of linear algebra generally hold independent of the field of scalars. Typically the field is ℝ or ℂ, but […] The post The linear algebra of bit twiddling first appeared on John D. Cook…

The Mersenne Twister (MT) is a random number generator with good statistical properties but bad cryptographic properties. In buzzwords, it’s a PRNG but not a CSPRNG. This post will show how the internal state of a MT generator can be recovered from its output. We’ll do this using linear algebra. The bit twiddling approach is […] The post Reverse engineering Mersenne Twister with Linear Algebra fi…

Curvature is conceptually simple but usually difficult to calculate. For a level set curve f(x, y) = c, such as in the previous couple posts, the equation for curvature is Even when f has a fairly simple expression, the expression for κ can be complicated. If we define then the level set of f(x, y) = c is […] The post Calculating curvature first appeared on John D. Cook .

The previous post constructed a triangular analog of the squircle, the unit circle in the p-norm where p is typically around 4. The case p = 2 is a Euclidean circle and the limit as p → ∞ is a Euclidean square. The previous post introduced three functions Li(x, y) such the level set of each function forms a […] The post Smoothed polygons first appeared on John D. Cook .

TimF left a comment on my guitar pick post saying the image was a “squircle-ish analog for an isosceles triangle.” That made me wonder what a more direct analog of the squircle might be for a triangle. A squircle is not exactly a square with rounded corners. The sides are continuously curved, but curved most […] The post Triangular analog of the squircle first appeared on John D. Cook .

I try to maintain a consistent work environment across computers that I use. The computers differ for important reasons, but I’d rather they not differ for unimportant reasons. Unified keys One thing I do is remap keys so that the same key does the same thing everywhere, to the extent that’s practical. This requires remapping […] The post Unified config files first appeared on John D. Cook .