Computational Discovery with Newton Fractals, Bohemian Matrices, & Mandelbrot Polynomials

Abstract

The authors have been using a largely algebraic form of ``computational discovery'' in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. This paper describes what we mean by ``computational discovery,'' what good it does for the students, and some specific techniques that we used.

Figures (3)

• Figure 1: Basins of attraction for Newton's method on a certain degree $8$ polynomial, displayed in equation \ref{['eq:FibMand6']}, as produced by the Fractals:-Newton command in Maple (left) and in by a custom Python program (right). The view in the complex plane is from $-1.8 \le x \le 1.0$ and $-1.4 \le y \le 1.4$.
• Figure 2: Eigenvalue density plot of all $2^{17} = 131072$ ten by ten Bohemian skew-pentadiagonal matrices with population $\{1,i\}$, plotted on $-3.25 \le \Re(\lambda) \le 3.25$ and $-3.25 \le \Im(\lambda) \le 3.25$ in the complex $\lambda$-plane. Hotter colours have higher density. At this time of writing we have no explanation whatever for the distribution pattern visible here.
• Figure 3: Zeros of $z_{11}(c)$, a polynomial of degree $1024$. Each of these zeros is a value of $c$ which leads to a periodic orbit of period $11$, namely $0$, $z_1(c) = c$, $z_2(c)$, $\ldots$, $z_{10}(c)$, $0$. Such points are also called hyperbolic centers in the Mandelbrot set and are actually quite sparse in the Mandelbrot set.