On the Petras algorithm for verified integration of piecewise analytic functions

We consider the algorithm for verified integration of piecewise analytic functions presented in Petras' paper (Petras, 2002). The analysis of the algorithm contained in that paper is limited to a narrow class of functions and gives upper bounds only. We present an estimate of the cost (measured by the number of evaluations of the integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples of functions with cost Θ(∣lnε∣/εp−1), for any p>1, where ε is the desired accuracy of the computed integral.