A truncated Clenshaw-Curtis formula approximates integrals over a semi-infinite interval

Abstract

The Clenshaw-Curtis (C-C) rule is a quadrature formula for integrals on the finite interval [− 1,1] and known to be efficient for smooth integrands and suitable for constructing an automatic method owing to nice features, i.e., the C-C rule family is nested and the accuracy is easy to check. In this paper, for an integral on a semi-infinite interval $[0,\infty )$ with an integrand decaying exponentially as $x\to \infty $ , we propose a truncated formula of the C-C rule and present an automatic method to approximate the integral to the accuracy of double precision and its Matlab code. We reduce the interval $[0,\infty )$ to a finite interval [0,a], choosing a so that the ignored integral on $[a,\infty )$ is sufficiently small. To approximate the integral Ia on [0,a], we consider a wider interval [0,2a]. By a change of variables, we transform nodes of the C-C rule on [− 1,1] to those on [0,2a]. To approximate Ia, our formula uses the nodes belonging to [0,a]. Similarly, a truncated formula of the Gauss-Legendre (GLe) rule is available. For an analytic function f(z) on $[0,\infty )$ , we give an error analysis for our method. Using numerical examples, we compare our formula with the truncated GLe, Gauss-Laguerre and double exponential formulae in performance. Numerical examples show that our formula, as well as the truncated GLe formula, is efficient, particularly, for semi-infinitely oscillatory integrals.