The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor's formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at $n+1$ equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.
翻译:本文的目的是在Rn中得出经改进的第一阶扩展公式,目标是获得与通常泰勒公式相比的最佳减量剩余部分。对于一个特定函数,我们得出的公式是通过引入第一个衍生物的线性组合获得的,该公式的计算方法是以相等的间距点计算,以n+1美元计算。我们说明该公式如何适用于两个重要的应用:内插错误和有限元素误差估计。在这两种情况下,我们说明在什么条件下可以大大改进错误,即如何利用经改进的扩展来减少误差估计的上限。