Małgorzata Moczurad, Piotr Zgliczyński, Włodzimierz Zwonek
We give an improved lower bound for the error of any quadrature computing $\int_{-1}^1 f(x) dα(x)$ of analytic functions bounded in the neighborhood of $[-1,1]$.
Małgorzata Moczurad, Piotr Zgliczyński
We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples with complexity $Θ(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.