Gauss-Kronrod integration rules for Cauchy principal value integrals

Abstract

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n, and when it does, several significant figures are lost in its use.