Error bounds for Romberg quadrature (Q1806009)

An important question in the theory of numerical integration rules is the relative quality of different types of integration rules. The authors derive new error bounds for the Romberg rules and compare these bounds with previously obtained bounds for Gauss rules. The bounds are derived for two large classes of integrands, integrands with bounded derivatives, and integrands with bounded total variation of derivatives. The derivations are quite technical and the main tool used is the Peano form for the error in an integration rule. The analysis explicitly shows the superiority of the Gaussian rules, except for integrands with limited smoothness.