牛顿-柯特斯公式的简单推导及实际误差
A Simple Derivation of Newton-Cotes Formulas with Realistic Errors
AI总结：
本文通过未定系数法和牛顿多项式基，推导出基于等距节点的插值型求积规则，并引入实际误差修正项，为经典牛顿-柯特斯公式提供了一种计算替代方案。
摘要：
为了近似积分 $I(f)=\int_a^b f(x) dx$，其中 $f$ 是足够光滑的函数，使用给定的 $n$（$n\geq 2$）个等距点面板开发了求积规则的模型。这些模型源于未定系数法，使用多项式的牛顿基。尽管最终结果在代数上等价于众所周知的闭型牛顿-柯特斯规则，但获得的算法并非经典算法。在基本模型中，采用最简单的求积规则 $Q_n$（即所谓的左矩形规则），并构造修正项 $\tilde E_n$，使得最终规则 $S_n=Q_n+\tilde E_n$ 是插值型的。修正项 $\tilde E_n$ 依赖于数据的差商，可视为 $Q_n$ 的“实际修正”，因为 $\tilde E_n$ 应接近 $Q_n$ 真实误差的大小，且符号正确。对规则 $S_n$ 的理论误差分析以及差商的一些经典性质建议在给定面板中引入一个或两个新点。当 $n$ 为偶数时引入一个点，否则引入两个点。在这两种情况下，该方法能够为“扩展或修正”规则 $S_n$ 计算实际误差 $\bar E_{S_n}$。相应的输出 $(Q_n, \tilde E_n, S_n, \bar E_{S_n})$ 包含关于近似 $Q_n$ 和 $S_n$ 质量的可靠信息，前提是满足涉及函数 $f$ 导数比率的某些条件。这些简单规则很容易转化为复合规则。数值例子表明，这些求积规则可作为经典牛顿-柯特斯公式的计算替代方案。
英文摘要：
In order to approximate the integral $I(f)=\int_a^b f(x) dx$, where $f$ is a sufficiently smooth function, models for quadrature rules are developed using a given {\it panel} of $n (n\geq 2)$ equally spaced points. These models arise from the undetermined coefficients method, using a Newton's basis for polynomials. Although part of the final product is algebraically equivalent to the well known closed Newton-Cotes rules, the algorithms obtained are not the classical ones. In the basic model the most simple quadrature rule $Q_n$ is adopted (the so-called left rectangle rule) and a correction $\tilde E_n$ is constructed, so that the final rule $S_n=Q_n+\tilde E_n$ is interpolatory. The correction $\tilde E_n$, depending on the divided differences of the data, might be considered a {\em realistic correction} for $Q_n$, in the sense that $\tilde E_n$ should be close to the magnitude of the true error of $Q_n$, having also the correct sign. The analysis of the theoretical error of the rule $S_n$ as well as some classical properties for divided differences suggest the inclusion of one or two new points in the given panel. When $n$ is even it is included one point and two points otherwise. In both cases this approach enables the computation of a {\em realistic error} $\bar E_{S_n}$ for the {\it extended or corrected} rule $S_n$. The respective output $(Q_n,\tilde E_n, S_n, \bar E_{S_n})$ contains reliable information on the quality of the approximations $Q_n$ and $S_n$, provided certain conditions involving ratios for the derivatives of the function $f$ are fulfilled. These simple rules are easily converted into {\it composite} ones. Numerical examples are presented showing that these quadrature rules are useful as a computational alternative to the classical Newton-Cotes formulas.
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