In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the r th derivative is positive within precision $$\varepsilon >0$$ ε > 0 are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses the minimal number of function evaluations. It is also shown how to adapt the quadratures to the case where the assumption that the r th derivative has a constant sign is not fulfilled. The theoretical results are illustrated and confirmed with numerical tests.

中文翻译：

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关于自动积分的最优自适应正交

在本文中，研究了自动集成问题。构造计算 r 次可微函数的积分的求积函数，假设 r 阶导数在精度 $$\varepsilon >0$$ ε > 0 内为正。对这些正交进行了严格的分析。事实证明，本文提出的网格选择程序是最优的，即它使用最少数量的函数评估。还展示了如何使求积适用于不满足 r 阶导数具有恒定符号的假设的情况。理论结果通过数值试验得到说明和证实。