TL;DR
This paper examines the limitations of the standard design principle for quadrature formulas, revealing discrepancies between expected and actual exactness in several well-known methods.
Contribution
It demonstrates how the common assumption of exactness for certain classes of functions does not hold for popular quadrature formulas like Newton-Cotes and Gauss methods.
Findings
Newton-Cotes, Clenshaw-Curtis, Gauss-Legendre, and Gauss-Hermite quadratures do not always behave as predicted.
The standard design principle can be misleading in understanding quadrature accuracy.
Several examples show the failure of the principle in practical cases.
Abstract
The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton-Cotes, Clenshaw-Curtis, Gauss-Legendre, and Gauss-Hermite quadrature. Three further examples are mentioned more briefly.
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