Notes on the convergence of trapezoidal - rule quadrature

Numerical quadrature is another name for numerical integration, which refers to the approximation of an integral ́ f (x)dx of some function f (x) by a discrete summation ∑wi f (xi) over points xi with some weights wi. There are many methods of numerical quadrature corresponding to different choices of points xi and weights wi, from Euler integration to sophisticated methods such as Gaussian quadrature, with varying degrees of accuracy for various types of functions f (x). In this note, we examine the accuracy of one of the simplest methods: the trapezoidal rule with uniformly spaced points. In particular, we discuss how the convergence rate of this method is determined by the smoothness properties of f (x)—and, in practice, usually by the smoothness at the end­ points. (This behavior is the basis of a more sophisticated method, Clenshaw-Curtis quadrature, which is essentially trapezoidal integration plus a coordinate transforma­ tion to remove the endpoint problem.) For simplicity, without loss of generality, we can take the integral to be for x ∈ [0,2π], i.e. the integral ˆ 2π I = f (x)dx, 0