The Error Estimates of Kronrod Extension for Gauss-Radau and Gauss-Lobatto Quadrature with the Four Chebyshev Weights

Abstract

In this paper, we consider the Kronrod extension for the Gauss-Radau and Gauss-Lobatto quadrature consisting of any one of the four Chebyshev weights. The main purpose is to effectively estimate the error of these quadrature formulas. This estimate needs a calculation of the maximum of the modulus of the kernel. We compute explicitly the kernel function and determine the locations on the ellipses where a maximum modulus of the kernel is attained. Based on this, we derive effective error bounds of the Kronrod extensions if the integrand is an analytic function inside of a region bounded by a confocal ellipse that contains the interval of integration.