We introduce a new quadrature rule based on Chebyshev’s and Simpson’s rules. The corresponding composite rule induces the adaptive method of approximate integration. We propose a stopping criterion for this method and we prove that if it is satisfied for a function which is either 3-convex or 3-concave, then the integral is approximated with the prescribed tolerance. Nevertheless, we give an example of a function which does satisfy our criterion, but the approximation error exceeds the assumed tolerance. The numerical experiments (performed by a computer program created by the author) show that integration of 3-convex functions with our method requires considerably fewer steps than the adaptive Simpson’s method with a classical stopping criterion. As a tool in our investigations we present a certain inequality of Hermite–Hadamard type.

中文翻译：

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关于某种自适应的近似积分方法及其停止准则

我们介绍了一种基于切比雪夫和辛普森规则的新正交规则。相应的合成规则引发了近似积分的自适应方法。我们为该方法提出了一个停止准则，并且证明了如果对于3凸或3凹函数满足要求，则积分将以规定的公差近似。不过，我们给出了一个满足我们标准的函数示例，但是逼近误差超过了假定的容差。数值实验（由作者创建的计算机程序执行）表明，将3-凸函数与我们的方法集成所需的步骤比具有经典停止准则的自适应辛普森方法所需的步骤少得多。作为我们调查的一种工具，我们提出了Hermite–Hadamard类型的某些不等式。