A Review of Error Estimation in Adaptive Quadrature

TL;DR

This paper reviews various error estimation methods in adaptive quadrature, highlighting their differences, common issues, and introducing a new general error estimation technique to improve accuracy.

Contribution

It provides a comprehensive review of existing techniques and proposes a novel, more general error estimation method for adaptive quadrature routines.

Findings

Existing methods have common shortcomings.

The new technique addresses key limitations.

The review clarifies differences among error estimators.

Abstract

The most critical component of any adaptive numerical quadrature routine is the estimation of the integration error. Since the publication of the first algorithms in the 1960s, many error estimation schemes have been presented, evaluated and discussed. This paper presents a review of existing error estimation techniques and discusses their differences and their common features. Some common shortcomings of these algorithms are discussed and a new general error estimation technique is presented.