Generalized composite integration rules in the presence of a singularity

Abstract

A composite integration rule results from applying a basic integration rule exact for constants and with all abscissas in the integration interval to each ofn equal subintervals of the interval of integration. A sequence of such composite rules converges to the interval for all Riemann-integrable functions and for functions with an endpoint singularity which are dominated by a monotonic improperly integrable function. These rules are generalized to allow a partition into unequal subintervals or different basic rules in each subinterval. In each of these situations, restrictions must be placed on the basic rule or rules or on the partition to ensure convergence in the singular case. Such restrictions also ensure the convergence of a modified rule in the case of an interior singularity.