On the Complexity of Self-Validating Numerical Integration and Approximation of Functions with Singularities

Abstract We consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass of C k ([ a ,  b ]\ Z ), where Z is a finite subset of [ a ,  b ]. Using only function values or values of derivatives, we usually cannot guarantee that the costs for obtaining an error less than e are bounded by O ( e −1/ k ) and we may have much higher costs. The situation changes if we also allow “realistic” estimates of ranges of functions or derivatives on intervals as observations. A very simple algorithm now yields an error less than e with O ( e −1/ k )-costs and an analogous result is also obtained for uniform approximation with piecewise polynomials. In a practical implementation, estimation of ranges may be done efficiently with interval arithmetic and automatic differentiation. 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 functions from a wide class with O ( e −1/ k ) arithmetical operations and guaranteed precision e .