Error bounds for Gauss-Kronrod quadrature formulae

Abstract

The Gauss-Kronrod quadrature formula Q 2 n + 1 G K Q_{2n + 1}^{GK} , is used for a practical estimate of the error R n G R_n^G of an approximate integration using the Gaussian quadrature formula Q n G Q_n^G . Studying an often-used theoretical quality measure, for Q 2 n + 1 G K Q_{2n + 1}^{GK} we prove best presently known bounds for the error constants \[ c s ( R 2 n + 1 G K ) = sup ‖ f ( s ) ‖ ∞ ≤ 1 | R 2 n + 1 G K [ f ] | {c_s}(R_{2n + 1}^{GK}) = \sup \limits _{{{\left \| {{f^{(s)}}} \right \|}_\infty } \leq 1} |R_{2n + 1}^{GK}[f]| \] in the case s = 3 n + 2 + κ , κ = ⌊ n + 1 2 ⌋ − ⌊ n 2 ⌋ s = 3n + 2 + \kappa ,\kappa = \left \lfloor {\frac {{n + 1}}{2}} \right \rfloor - \left \lfloor {\frac {n}{2}} \right \rfloor . A comparison with the Gaussian quadrature formula Q 2 n + 1 G Q_{2n + 1}^G shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.