The trapezoidal rule for computing Cauchy principal value integral on circle (Q1666928)

Summary: The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel \(\text{cot}((x - s) / 2)\) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.