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 I^a 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 I^a, 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.

Clenshaw-Curtis（CC）规则是有限区间[− 1,1]上积分的一个正交公式，已知对光滑被积数有效，并且由于具有良好的功能而适合构造自动方法，即CC规则族嵌套，精度易于检查。在本文中，对于半无限区间\（[0，\ infty）\）上的积分，并且被积数呈指数衰减为\（x \ to \ infty \）的情况，我们提出了CC规则的截断公式，并给出一种将积分近似为双精度精度的自动方法及其Matlab代码。我们将间隔\（[0，\ infty）\）减小为有限的间隔[0，a ]，选择a使得在\（[a，\ infty）\）足够小。为了近似[0，a ]上的积分I ^a，我们考虑了更宽的区间[0,2 a ]。通过变量的变化，我们将CC规则在[− 1,1]上的节点转换为[0,2 a ]上的节点。为了近似I ^a，我们的公式使用属于[0，a ]的节点。类似地，高斯-勒格德（GLe）规则的截断公式可用。对于\（[0，\ infty）\）上的解析函数f（z ），我们对该方法进行了错误分析。使用数值示例，我们将公式与截短的GLe，Gauss-Laguerre和双指数公式的性能进行比较。数值例子表明，我们的公式以及截短的GLe公式都是有效的，特别是对于半无限振荡积分。