On the Complexity of Validated Numerical Integration for Integrands with Singularities

We consider the complexity of numerical integration of bounded functions in C k ? a; b]nZ , where Z varies over all nite subsets of a; b]. Using only function values or values of derivatives, we usually can not guarantee that the costs for obtaining an "-approximation are bounded by O(" ?1=k) and we may obtain much higher costs. The situation changes if we also allow estimates of ranges of functions or derivatives on intervals as observations. In a practical implementation, estimation of ranges may be done eeciently with interval arithmetic and automatic diierentiation. The cost for each such evaluation (also of ranges of derivatives) is bounded by a constant times the cost for a function evaluation. The mentioned techniques reduce the class of integrands, but still allow numerical integration of a wide class of functions to which conventional integration algorithms may applied reasonably. A very simple algorithm now yields an "-approximation with O(" ?1=k)-costs for the remaining class of functions.