标题： 牛顿-柯特斯公式的简单推导及现实误差 显示英文标题

标题： A Simple Derivation of Newton-Cotes Formulas with Realistic Errors

摘要： 为了近似积分$I(f)=\int_a^b f(x) dx$，其中$f$是一个足够光滑的函数，使用给定的{\it 面板}个$n (n\geq 2)$等距点来开发求积规则的模型。 这些模型来源于不定系数法，使用牛顿多项式基。 尽管最终产品的一部分在代数上等价于众所周知的闭型牛顿-柯特斯公式，但得到的算法并不是经典的。 在基本模型中采用最简单的求积规则$Q_n$（所谓的左矩形法则）并构造一个修正项$\tilde E_n$，使得最终规则$S_n=Q_n+\tilde E_n$是插值性的。 修正 $\tilde E_n$，根据数据的差商，可以视为 {\em 现实校正}对于 $Q_n$的一种，从意义上说 $\tilde E_n$应该接近 $Q_n$的真实误差的大小，并且符号也正确。 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的面板中加入一个或两个新点。 当 $n$为偶数时加入一个点，否则加入两个点。 在两种情况下，这种方法能够计算 {\em 实际误差} $\bar E_{S_n}$ 对于 {\it 扩展或修正} 规则 $S_n$。 各自输出$(Q_n,\tilde E_n, S_n, \bar E_{S_n})$包含关于近似值质量的可靠信息$Q_n$和$S_n$，前提是满足涉及函数$f$导数比值的某些条件。 这些简单规则很容易转换为{\it 复合}的规则。 给出了数值例子，表明这些求积规则作为经典牛顿-科特斯公式的计算替代方法是有用的。 显示英文摘要

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.