Quadrature rules with (not too many) derivatives

TL;DR

This paper introduces new quadrature formulas that require only endpoint derivatives and provides error bounds based on the function's smoothness, improving efficiency in numerical integration.

Contribution

It presents novel quadrature rules with minimal derivative evaluations at endpoints and derives error bounds for functions with different smoothness conditions.

Findings

Quadrature formulas requiring only endpoint derivatives are identified.

Error bounds are established for functions with various smoothness assumptions.

The methods improve efficiency in numerical integration by reducing derivative evaluations.

Abstract

Quadrature formulas for $\int_a^b f(x) dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to\mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous so that $f^{(n)}\in L^p([a,b])$, and when $f^{(n-1)}$ is merely continuous.