Doug Hensley, Texas A&M University
I well remember my first days with Mathematica. I was working on a paper about the Hausdorff dimension of continued fraction Cantor sets. The basic idea was to represent functions by power series and linear operators by their matrix with respect to the coefficients of the power series. But I wanted to demonstrate that the idea worked in practice, and for that, I had to have an algorithm and a machine on which to execute it.
Along came the NeXT computer, which came bundled with Mathematica. The machine was expensive and weak by today’s standards, but it was awesome by the standards of its own day and the coding was a breeze in Mathematica compared to what it would have been in some other language. (I never got around to learning C; I never needed to!)
I worked that machine pretty hard, and remember it clacking away in the night in our bedroom/my home office as its optical drive was wont to do, grinding out coefficients for a matrix and then hitting a starting vector with the matrix enough times to ascertain whether the leading eigenvalue was more, or less, than 1.
The machine eventually gave out, and better became available for cheaper anyhow. Mathematica never gave out, and better became available for the price of an upgraded license. I’ve got something in the works now, again using Mathematica heavily, for the Hurwitz complex continued fraction algorithm. The invariant density for this has not, so far as I know, been computed to any substantial accuracy, and the plan is to fix that.
This picture shows a Monte Carlo experimentally generated picture of the density.
I no longer have the code for that 1990s paper in software form, but it’s published as part of “A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets,” Journal of Number Theory, 58(1), 1996 pp.9–45. [The paper had a convoluted road to publication.]