A unified generalization of some quadrature rules and error bounds

TL;DR

This paper introduces a parameterized framework that unifies various quadrature rules and their error bounds, providing new inequalities and extending existing results for numerical integration.

Contribution

It offers a unified generalization of quadrature rules and derives new sharp error bounds for both odd and even cases, enhancing the theoretical understanding.

Findings

Unified framework for quadrature rules and error bounds

New sharp error inequalities for odd and even cases

Extension of existing results to new quadrature rules

Abstract

By introducing a parameter, we give a unified generalization of some quadrature rules, which not only unify the recent results about error bounds for generalized mid-point, trapezoid and Simpson's rules, but also give some new error bounds for other quadrature rules as special cases. Especially, two sharp error inequalities are derived when n is an odd and an even integer, respectively.