从计算的角度看分析

Discussing Mathematics Analysis Based on View of Computations

数学分析建立在极限基础之上,围绕极限的存在性分析与判定方法,研究各种类型的极限的存在性分析、判定与计算。在实际计算中,更关注如何找到(或近似找到)存在的极限对象,分析中大量内容在讨论计算:以ε-N语言描述的极限概念体现误差与算法终止步的关系,有关计算的可行性或符号演算基本限制在初等函数类,初等函数类由基本初等函数通过四则运算和复合运算递归生成,其导函数可以实现符号演算;Newton-Leibniz公式表明部分初等函数的定积分的可通过符号演算实现;Taylor展开式和Fourier展开式分别给出了解析函数和可积函数的标准化表示和近似计算,这样的标准化表示其目的是解决计算问题。

It is known that the mathematics analysis is founded on the basis of limit theory, and it mainly analy- zes and decides existence conditions of existence, methods of decision of limits, computation of all kinds of limits. In the practice applications, we focus mainly on finding the objective or its approximation. A majority of contents in the mathematics analysis is to discuss computations. In the view of algorithms, the limit concept described by forms presents the relation between errors and stop-steps in algorithms. The research for computability and sym- bols deduction of limit is often restricted to the class of elementary functions, which is generated by addition, subtraction, multiplication, division and composite operations from basic elementary functions. The derivatives of elementary functions can be computed by symbol deduction, and some integral can be computed by symbol de- duction based on Newton-Leibniz formation. The Taylor expansions and Fourier expansions of functions give two standard representations of functions. Based on such expansions we realize approximate computations of functions.