Most instructors manage two opposing currents in classroom dynamics. The first is a desire to describe the subject in its broadest sense including all the subtleties and nuances. It gives the students a road map to the field, and if they can appreciate it at the time, it provides structure to a course that might otherwise appear to be a loosely connected series of topics. The second is the desire to ground the subject with concrete examples that students can grasp at the finest level. The examples link prerequisite knowledge with one of the new subjects and provide students with a foundation upon which to construct a deeper understanding. Few courses draw together so many disparate topics as numerical methods for partial differential equations (PDEs). When first exposed to finite differences for numerical PDEs, students will readily notice the regular structures in matrices, but few texts address these structures systematically and connect them directly to the analytical properties of the solution. There are many good examples, but it’s hard to see the broader landscape. In this Education section article, “Functions of Difference Matrices Are Toeplitz Plus Hankel,” authors Gil Strang and Shev MacNamara fill in some of the big picture by presenting some new results and observations about the structure of difference matrices for approximating solutions to PDEs.
A common approach in the analysis of numerical methods for PDEs is to discretize the spatial operator and leave time continuous so that one studies the semidiscrete system. The exact solution to these systems will be functions of the underlying difference matrices, hence the structure of functions of difference matrices is a natural point of interest for students, instructors, and researchers. Toeplitz matrices, having entries that are constant along diagonals, arise naturally in difference matrices. As the authors point out, this is to be expected because of the translation-invariance of the differentiation operator. In an unbounded domain, if initial condition u0 leads to a solution u, a translation of u0 will lead to a solution that is a translation of u. The authors discover that solutions to the semidiscrete systems are Toeplitz plus Hankel. The role of Hankel matrices, matrices that are constant along antidiagonals, is a little more subtle to explain. If one considers the role of boundary conditions for canonical problems like the heat or wave equations, one can see how this might arise. In the case of a single Neumann boundary condition for the heat equation, the classic method of images provides some insight. Shifting the initial condition toward the boundary shifts the image solution toward the boundary outside the domain, which shifts the influence of the image away from the boundary within the domain. Thus, shifting the initial condition has both a Toeplitz part from transition invariance and a Hankel part from the influence of the boundaries. The Toeplitz plus Hankel decomposition paves the way for exact solutions to semidiscrete approximations, allowing better error estimates and a deeper general understanding of spatial discretization.
While it makes sense intuitively that functions of difference matrices should be Toeplitz plus Hankel, the authors enthusiastically explore the topic using a variety of methods and techniques from advanced undergraduate or introductory graduate mathematics including spectral decompositions, Riemann sums, and Fourier analysis. In other words, the destination is interesting, but the voyage is even more fun! This paper would serve as an excellent module for a numerical analysis course or an integrative seminar for undergraduates or graduate students interested in numerical methods.
Read the paper! (Requires subscription or SIAM membership)
Functions of Difference Matrices Are Toeplitz Plus Hankel
SIAM Review, 56(3), 525-546.