On the error and its control in a two-parameter generalised Newton-Cotes rule

A quadrature formula containing two free (phase) parameters k, k′, and recently written by Van Daele et al. [13], is rederived using an extension of Lagrangeʹs identity. By using this method, a closed-form expression is determined for the local error term E[f] and the relevant Peano kernel is given explicitly. Sufficient conditions are established, under which this kernel remains definite, thus allowing a particularly simple expression to be deduced for E[f] which reduces to the classical result for Booleʹs rule in the limit k,k′ → 0. work on optimisation of global error is extended to include the new rule. It is shown that this error can be reduced by a factor O(h2) on a certain curve in the phase space. A further reduction by a factor O(h2) is sometimes possible by choosing the phase parameters on the intersection of this curve and another such. When these curves intersect only at complex values, the reduction is still achieved. A closed-form expression for the global error is also derived under these circumstances and this is seen to be asymptotically O(h10) as h → 0. nd limiting form of the five-point formula is found to reduce to the generalised Booleʹs rule written by Vanden Berghe et al. [14] and a third limiting form is also written. This third rule, which is a special case of a one-parameter family of generalised Booleʹs rules, is seen to perform better than the other two in the two examples studied. l numerical examples are given, with extensive diagrams, to illustrate all uses of the techniques proposed.