Optimal error estimates for corrected trapezoidal rules

TL;DR

This paper derives optimal error estimates for corrected trapezoidal rules under various smoothness assumptions on the function, including cases where derivatives are integrable or distributions, and provides formulas for making the rules exact for cubic polynomials.

Contribution

It introduces a modified trapezoidal rule with an optimally chosen correction term and establishes sharp error bounds under broad conditions on the second derivative.

Findings

Optimal correction coefficient minimizes error estimates.

Error bounds are sharp for the given assumptions.

Formulas are provided for exactness on cubic polynomials.

Abstract

Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term $k[f'(a)-f'(b)]$. The coefficient $k$ in the quadrature formula is found that minimizes the error estimates. It is shown that when $f'$ is merely assumed to be continuous then the optimal rule is the trapezoidal rule itself. In this case error estimates are in terms of the Alexiewicz norm. This includes the case when $f"$ is integrable in the Henstock--Kurzweil sense or as a distribution. All error estimates are shown to be sharp for the given assumptions on $f"$. It is shown how to make these formulas exact for all cubic polynomials $f$. Composite formulas are computed for uniform partitions.