This paper investigates the implementation of Clenshaw–Curtis algorithms on singular and highly oscillatory integrals for efficient evaluation of the finite Fourier-type transform of integrands with endpoint singularities. In these methods, integrands are truncated by orthogonal polynomials and special function series term by term. Then their singularity types are computed using third and fourth-order homogeneous recurrence relations. The first approach reveals its efficiency for low, moderate and very high frequencies, whereas the second one, is more efficient for small values of frequencies. Moreover, all the results were found quite satisfactory. Algorithms and programming code in MATHEMATICA® 9.0 are provided for the implementation of methods for automatic computation on a computer. Lastly, illustrative numerical experiments and comparison of the proposed Clenshaw–Curtis algorithms to the steepest descent method are mentioned in support of our theoretical analysis in the examples section.

本文研究了Clenshaw-Curtis算法在奇异和高度振荡积分上的实现，以有效评估具有端点奇点的被积数的有限傅里叶型变换。在这些方法中，整数被正交多项式和特殊功能序列逐项截断。然后使用三阶和四阶齐次递归关系计算它们的奇异类型。第一种方法显示了其在低，中和非常高频率下的效率，而第二种方法则对较小的频率值更有效。此外，发现所有结果都令人满意。提供了MATHEMATICA®9.0中的算法和编程代码，用于在计算机上实现自动计算的方法。最后，