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Adaptive integration of 5-convex and 5-concave functions

Szymon Wąsowicz

TL;DR

This work addresses adaptive numerical integration for $5$-convex or $5$-concave functions in $C^6[a,b]$ by leveraging a three-point Gauss quadrature $\mathcal{G}$ and a four-point Lobatto quadrature $\mathcal{L}$. The authors derive a practical composite scheme $Q_n$ with a rigorous error bound $|\int_a^b f- Q_n|\le\tfrac14|\mathcal{L}_n-\mathcal{G}_n|$, and provide a terminating adaptive algorithm that increases the subdivision until $|\mathcal{L}_n-\mathcal{G}_n|\le 4\varepsilon$. Theoretical results show convergence of the composite quadratures and guarantee termination, while numerical experiments on $f(x)=1/x$ and $f(x)=e^x$ demonstrate substantial efficiency gains over prior $3$-convex methods. Implemented in SageMath, the approach offers a robust and faster alternative for high-order convexity classes relevant in numerical analysis and computational mathematics.

Abstract

An adaptive method connected with 3-point Gauss quadrature and 4-point Lobatto quadrature is introduced and investigated for 5-convex functions.

Adaptive integration of 5-convex and 5-concave functions

TL;DR

This work addresses adaptive numerical integration for -convex or -concave functions in by leveraging a three-point Gauss quadrature and a four-point Lobatto quadrature . The authors derive a practical composite scheme with a rigorous error bound , and provide a terminating adaptive algorithm that increases the subdivision until . Theoretical results show convergence of the composite quadratures and guarantee termination, while numerical experiments on and demonstrate substantial efficiency gains over prior -convex methods. Implemented in SageMath, the approach offers a robust and faster alternative for high-order convexity classes relevant in numerical analysis and computational mathematics.

Abstract

An adaptive method connected with 3-point Gauss quadrature and 4-point Lobatto quadrature is introduced and investigated for 5-convex functions.
Paper Structure (5 sections, 4 theorems, 30 equations)

This paper contains 5 sections, 4 theorems, 30 equations.

Table of Contents

  1. Remarks on adaptive methods of approximate integration
  2. Higher-order convexity
  3. Introduction
  4. The Adaptive Method
  5. Numerical Experiments

Key Result

Proposition 1

If the function $f\in C^6[a,b]$ is $5$-convex, then

Theorems & Definitions (6)

  • Proposition 1
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4