Exit Criteria for Newton–Cotes Quadrature Rules

Abstract

A common procedure for deciding when to exit from a quadrature routine is to repeatedly double the number of nodes and stop when the difference between successive approximations is less than a preassigned tolerance. This is shown to be a valid procedure for any Newton–Cotes compound rule when the appropriate higher derivative of the integrand has constant sign. This result is shown to be sharp in the sense that an exit rule based on a weaker inequality will fail for some function in the above class. Conditions are also given under which this exit criterion is asymptotically valid.