纳皮尔对数的数学证明及精度分析

Proof of Napier’s Log and Analysis of Precision

对数函数在自动驾驶系统中有着广泛应用,如自动驾驶感知系统所使用的深度学习或卷积神经网络常会利用对数函数来设计损失函数,故研究对数发明的历史对掌握对数的概念和应用具有重要意义。该文阐述了纳皮尔对数的定义及其3张表,分析了前人的两类证明方法,提出了新的基于指数函数构造的证明方法。同时,该文还分析了纳皮尔的计算方法,与对照方法相比,给出了纳皮尔对精度范围的优化结果,通过MPRF库进行了计算,结果表明纳皮尔的方法的计算结果更接近真实值。

In the automatic driving systems,logarithm function has been widely used.For example,logarithm function is often used to design loss function in deep learning or convolutional neural network,which serves as the basis for the automatic driving perception system.Therefore,studying the history of invention of logarithm is of great significance to master the concept and application.This paper studies the definition of Napier’s logarithm and his three tables,analyzes two kinds of proof methods of predecessors,and puts forward new proof methods based on the exponential function.Meanwhile,this paper also analyzes Napier’s calculation method.Compared with other alternative methods,the optimization results of Napier’s interval approximation are given.The calculation by MPRF library shows that Napier’s method is more convergent to the true value.