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

We consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass ofC^k(a, b]\Z), whereZis 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 thanεare bounded byO(ε^−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 thanεwithO(ε^−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 withO(ε^−1/k) arithmetical operations and guaranteed precisionε.