Error Bounds for Gauss-Kronrod Quadrature Formulae

The Gauss-Kronrod quadrature formula Q2n + 1GK, is used for a practical estimate of the error RₙG of an approximate integration using the Gaussian quadrature formula QₙG. Studying an often-used theoretical quality measure, for Q2n + 1GK we prove best presently known bounds for the error constants \[ {c_s}(R2n + 1GK) = _{{{ \| {{f⁽ˢ⁾}} \|}_∞ } ≤ 1} |R2n + 1GK[f]|\] in the case s = 3n + 2 + κ ,κ = n + 12 - n/2. A comparison with the Gaussian quadrature formula Q2n + 1G shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.