Numerical integration in arbitrary-precision ball arithmetic

TL;DR

This paper introduces a rigorous, efficient implementation of arbitrary-precision numerical integration in the Arb library, utilizing the Petras algorithm for rapid convergence with guaranteed error bounds.

Contribution

It presents a novel, general implementation of complex-analytic numerical integration with rigorous error bounds, outperforming existing non-rigorous methods.

Findings

Achieves rapid convergence for complex integrals

Provides rigorous error bounds without derivative evaluations

Outperforms existing non-rigorous software

Abstract

We present an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library. Rapid convergence is ensured for piecewise complex analytic integrals by use of the Petras algorithm, which combines adaptive bisection with adaptive Gaussian quadrature where error bounds are determined via complex magnitudes without evaluating derivatives. The code is general, easy to use, and efficient, often outperforming existing non-rigorous software.