绝对值函数的一类新的光滑近似

A new smoothed spproximation of absolute value functions

绝对值函数或最大值函数在工程应用以及理论分析中有着至关重要的作用.然而,其不可微性对理论分析造成了巨大困难,所以构造简单的光滑化近似便成了重要的研究课题.在比较和分析了一些光滑逼近函数的性质后,文中利用正弦函数构造和证明了绝对值函数的一类新的光滑近似函数,通过引进的有界常数对常用的几类光滑近似作定量比较,发现新构造的光滑近似有一个数量级的提升,最后在相同参数下给出了这些光滑近似函数的逼近效果.

Absolute value functions or maximum value functions in engineering applications as well as theoretical analysis have a vital role.However,its non-differentiability poses great difficulties for the theoretical analysis,so the construction of simple and smoothing approximations becomes an important research topic.After comparing and analyzing the properties of some smoothed approximation functions,we first use the sine function to construct and then prove a new smoothing approximation of the absolute value function inthis paper.Secondly,through the introduction of bounded constants to make a quantitative comparison of several types of smoothed approximations commonly used,we show that this newly constructed smoothed approximation has an order of magnitude improvement than others.Finally,the approximation effects of these smoothing approximation functions for the same parameters are given graphically.