# zbMATH — the first resource for mathematics

Error bounds for Gauss-Kronrod quadrature formulae. (English) Zbl 0790.41020
Summary: The Gauss-Konrod quadrature formula $$Q^{GK}_{2n+1}$$ is used for a practical estimate of the error $$R^ G_ n$$ of an approximate integration using the Gaussian quadrature formula $$Q^ G_ n$$. Studying an often-used theoretical quality measure, for $$Q^{GK}_{2n+1}$$ we prove best presently known bounds for the error constants $c_ s(R^{GK}_{2n+1})=\sup_{\| f^{(s)}\|_ \infty\leq 1} | R^{GK}_{2n+1}[f]|$ in the case $$s= 3n+ 2+ \kappa$$, $$\kappa=\Bigl\lfloor{n+1\over 2}\Bigr\rfloor-\Bigl\lfloor{n\over 2}\Bigr\rfloor$$. A comparison with the Gaussian quadrature formula $$Q^ G_{2n+1}$$ shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.