Education

Abstract

One of the great things about studying and teaching mathematical sciences is that many themes, methods, and concepts resurface in new areas with increasing complexity and depth. Notions such as algebraic operation, function, and solutions of systems continue to expand and deepen as students move through a standard mathematics curriculum and perhaps on into graduate school. In his poem “Ulysses,” Alfred Lord Tennyson penned the sentiment perfectly. Yet all experience is an arch wherethrough Gleams that untraveled world whose margin fades Forever and forever when I move. John Boyd, this issue's contributor, points out that rootfinding often stops with a few “favorites” such as bisection, Newton's method, or Brent's algorithm. In “Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix,” he presents Chebyshev-proxy rootfinding (CPR), which brings together several disparate mathematical topics to yield a robust scheme for solving a nonlinear holomorphic equation $f(x)=0$ on a closed interval. The term proxy refers to the fact that one replaces the original function with a Chebyshev polynomial approximation $f_N(x)$. Then one finds the roots of Chebyshev polynomial by finding the eigenvalues of its Chebyshev--Frobenius companion matrix. While the algorithm is more complex than something like bisection, specialized mathematical software now makes implementation much more accessible, and, of course, CPR is more accurate and robust. The style of presentation in this article makes it an excellent addition as a module in a course on numerical methods. Before introducing the nuts and bolts of the algorithm, the author surveys the notion of proxy functions as a way of unifying other rootfinders conceptually. As the author carries the reader through the algorithm, he confronts the details that require most of the hard work that goes into any effective algorithm. For instance, naively creating a proxy adaptively can be very expensive, so he points out that there are clever ways to reuse information from a coarse Chebyshev grid when constructing a refined one. In fact, the discussion of the algorithm is only about half the paper. The other half includes additional examples and covers techniques for extending the method to functions that are not smooth or to functions on unbounded domains. In short, this issue's Education feature synthesizes several distinct mathematical concepts and focuses them on rootfinding, which is one of the most fundamental quantitative problems. It offers a great way to liven up an introductory numerical analysis course, or provide depth to a more advanced methods course. In doing so, it provides us all with a way to expand our mathematical horizons.