One of the great ironies in mathematics instruction is that typical instructional activities are bound to textbooks. Students expect mathematical structures, concepts, and techniques to begin and end in their texts when, in fact, they spend most of their days surrounded by great mathematical problems. In the fall, students learn about vector fields in calculus, but are never asked to give a thought to the leaves swirling outside the classroom window. The mathematical sciences are the sciences of abstraction, but observations of the world around us can inspire us to think about mathematical objects and problems in new and exciting ways. In this issue of SIAM Review, our Education section features two articles, the first of which is a perfect example of finding interesting mathematical problems in sports. In this case, the sport is rugby and asking a very simple question about the best approach to earning extra points reveals a very elaborate answer. The problem evolves into an expression that needs to be solved numerically, giving us a nice segue into our second paper, which explores fixed point iterations and Newton’s methods for solving nonlinear problems; so by coincidence our two Education papers follow a natural sequence.
In “Optimal Ball Placement in Rugby Conversions,” author Pedro Freitas takes a careful look at a simple question arising in rugby. If the words “ruck,” “scrum,” and “lineout” do not mean anything to you, here is a layman’s explanation: A rugby player after scoring a try can attempt to earn extra points by placing the ball along a certain line determined by the field position from which the try was scored. From this position, the player scores extra points by kicking the ball through the upright goal posts. The question is simple: Where is the best place for the player to put the ball along the line? It’s a geometric optimization problem, and Prof. Freitas is not the first person to make a concerted effort to definitively resolve the issue. Like all good modeling problems, posing this one is a bit like peeling an onion. If you don’t go too deeply into it, you can make rapid progress without shedding any tears. In this article, Prof. Freitas takes the problem to a deeper level than has been considered in the past by posing the question in terms of the full three-dimensional geometry: The optimal placement is that which results in the goal presenting the greatest solid angle to the player. The assumption is that the greatest solid angle permits the player to commit the greatest error and still score. The resulting system requires a numerical solution, leading us to the second paper, which, by happy coincidence, focuses on methods for solving nonlinear problems.
In “Fixed Point and Newton’s Methods for Solving a Nonlinear Equation: From Linear to High-Order Convergence,” authors François Dubeau and Calvin Gnang revisit fixed point iterations and Newton’s methods for solving nonlinear problems. These topics are familiar fare in a numerical analysis course. Less common in these courses is a clear, concise presentation of Schröder’s processes for boosting the convergence rates for fixed point and Newton’s methods for simple roots. The authors demonstrate the connection between the two processes and provide a number of nice examples while identifying several directions for further study.
The arrival of these two particular articles together in the publication queue was purely coincidental, but they illustrate the deep connections between (1) reality, (2) mathematical structures as models for reality, and finally (3) insights into reality from analysis and numerical solutions. The first paper is suitable for a modeling seminar or capstone course. The second paper is perfect for an advanced treatment of undergraduate numerical methods or a graduate course in the same. Taken together, these articles may help students learn to take inspiration from everyday problems. After all, a simple question can lead to the most remarkable discoveries.
Read the papers! (Requires subscription or SIAM membership)
Optimal Ball Placement in Rugby Conversions
SIAM Review, 56(4), 673-690.
Fixed Point and Newton’s Methods for Solving a Nonlinear Equation: From Linear to High-Order Convergence
SIAM Review, 56(4), 691-708.