Derivative Corrections to the Trapezoidal Rule

TL;DR

This paper explores advanced modifications to the trapezoidal rule that incorporate derivative information, providing error bounds and alternative methods for periodic and real-line integrals of analytic functions.

Contribution

It introduces derivative-based extensions to the trapezoidal rule with new error bounds and discusses different approaches for integrals of analytic functions.

Findings

Derivative corrections improve accuracy for periodic integrals.

Error bounds are established without using derivatives.

Alternative methods for incorporating derivatives are analyzed.

Abstract

Extensions to the trapezoidal rule using derivative information are studied for periodic integrands and integrals along the entire real line. Integrands which are analytic within a half plane or within a strip containing the path of integration are considered. Derivative-free error bounds are obtained. Alternative approaches to including derivative information are discussed.