Journal of Inequalities and Applications 
Volume 5 (2000), Issue 2, Pages 167-190
doi:10.1155/S1025583400000096

Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals

Kai Diethelm and Peter Köhler

 Institut für Angewandte Mathematik, Technische Universität Braunschweig, Pockelsstr. 14, Braunschweig D-38106, Germany
Received 30 March 1999; Revised 10 May 1999

Abstract

We consider quadrature formulae for Cauchy principal value integrals Iw,ζ[f]=∫abf(x)x−ζw(x)dx,  a<ζ<b. The quadrature formulae considered here are so-called modified formulae, which are obtained by first subtracting the singularity, and then applying some standard quadrature formula Qn. The aim of this paper is to determine the asymptotic behaviour of the constants ki,n in error estimates of the form |Rnmod[f;ζ]|<ki,n(ζ)||f(i)||∞ for fixed i and n→∞, where Rnmod[f;ζ] is the quadrature error. This is done for quadrature formulae Qn for which the Peano kernels Ki,n of fixed order i behave in a certain regular way, including, e.g., many interpolatory quadrature formulae as Gauss–Legendre and Clenshaw–Curtis formulae, as well as compound quadrature formulae. It turns out that essentially all the interpolatory formulae behave in a very similar way.