On the Petras algorithm for verified integration of piecewise analytic functions

TL;DR

This paper extends the analysis of Petras' verified integration algorithm to a broader class of piecewise analytic functions, providing both upper and lower bounds on its complexity in terms of evaluations needed.

Contribution

It offers new complexity estimates for Petras' algorithm applicable to a wider function class, including both upper and lower bounds.

Findings

Complexity bounds are established for the algorithm.

Examples demonstrate complexity of order | ln(ε)|/ε^{p-1} for p > 1.

The analysis broadens understanding of the algorithm's efficiency.

Abstract

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 $\Theta(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.