Error inequalities for an optimal 3-point quadrature formula of closed type

TL;DR

This paper studies the problem of designing an optimal 3-point closed-type quadrature rule by optimizing the associated kernel to minimize a natural error bound. The authors derive the optimal rule on $[0,1]$ with nodes at $0$, $\tfrac12$, and $1$ and weights $\frac{\sqrt{2}}{8}$, $1-\frac{\sqrt{2}}{4}$, and $\frac{\sqrt{2}}{8}$, achieving a sharp error bound $\left|\int_{0}^{1} f(t)dt - \left[ \frac{\sqrt{2}}{8}f(0) + \left(1-\frac{\sqrt{2}}{4}\right) f\left(\tfrac{1}{2}\right) + \frac{\sqrt{2}}{8}f(1)\right] \right| \le \frac{2-\sqrt{2}}{48} \|f''\|_{\infty}$. Simpson’s rule arises as a special case with a larger error constant. The paper then provides a comprehensive suite of Ostrowski-like error inequalities for the quadrature error under various smoothness assumptions (bounded $f'$, $f'\in L^2$, $f''\in L^1$ or $L^2$), with sharp constants and constructions demonstrating optimality. Finally, the results are extended to composite numerical integration on $[a,b]$, giving explicit global error bounds and alternative estimates in terms of $n$, derivative norms, and related functionals, thereby offering practical guidance for implementing the optimal rule in numerical integration.

Abstract

An optimal 3-point quadrature formula of closed type is derived. Various error inequalities are established. Applications in numerical integration are also given.

Theorems & Definitions (22)

• Theorem 1
• Remark 2
• Theorem 3
• Lemma 4
• Theorem 5
• Remark 6
• Theorem 7
• Theorem 8
• Remark 9
• Example 10