Survey and Review

Abstract

The Survey and Review article in this issue is “The Exponentially Convergent Trapezoidal Rule,” by Nick Trefethen and André Weideman. It deals with a fundamental and classical issue in numerical analysis---approximating an integral. By focusing on one deceptively simple method, the authors are able to combine a lively, historical overview with an insightful, up-to-date coverage of recent results. We learn about the work of Euler, Gauss, Maclaurin, and Poisson, and some of the 20th century pioneers of numerical analysis. We also learn about high-precision computation, treatment of singularities, choice of contours to integrate along, Gauss and Clenshaw--Curtis quadrature, Padé approximation, and a “quadrature formula that doesn't even integrate constants right” (see footnote number 15). The first half of the article looks at the mathematics behind the well-known geometric convergence of the trapezoidal rule. Here the two fundamental routes to a proof are (a) exploiting Taylor series and aliasing, and (b) introducing a function that has simple poles at the summation points in order to apply residue calculus. The second half deals with applications where the trapezoidal rule has proved useful, including contour integration, rational approximation, problems with endpoint singularities, inverse Laplace transforms, integral equations, zero finding, special functions, and the numerical solution of parabolic PDEs. Readers familiar with the work of the authors will not be surprised to learn that this article deftly illustrates the power of complex analysis, even for studying problems that appear to live on the real line. The survey also uses the trapezoidal rule as a means to connect ideas in rational approximation, Laplace transforms, Cauchy integrals, sampling theory, interpolation theory, and matrix functions. As the authors make clear, there are many alternatives to the humble trapezoidal rule, but the method deserves attention because of - its ease of use; - its powerful convergence properties, which can be understood through relatively straightforward analysis; and - its utility in generating explicit approximation formulas. Further, the method is shown to be an excellent vehicle for demonstrating deep connections between a wide range of research topics in applied and computational mathematics.