This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error estimate and also the trapezoidal quadrature error, in order to assess the gain of accuracy we obtain using this new Taylor-like formula.
翻译:本文专门用一个新的第一顺序泰勒式公式,其中对应的剩余部分与古典泰勒公式中通常的公式相比大幅缩减。为了得出这一新公式,我们引入了相关函数第一个衍生物的线性组合,该函数在需要评估该函数的两个点之间以 n+1 平距点计算。我们表明,对线性组合中的加权进行最佳选择,可以最大限度地减少相应的剩余部分。然后,我们分析Lagrange P1内插误差估计值和诱杀性四极错误,以便评估我们使用这种新的泰勒式公式获得的准确性收益。