摘  要在研究经典物理数学过程中，学者们通常研究适定性问题。什么叫作适定问题？满足接下来的三个要求的问题被叫做适定问题：第一，解是存在的；第二，解是唯一的；第三，解连续地依附定解条件。只要同时符合这三个条件，那么这个问题就是适定问题。值得注意的是，要是只不满足条件3，则它被称为Hadamard意义下的不适定问题。86527

数值微分很明显等于在Hadamard意义下的不适定问题。丈量过程中发生细小误差都很有可能造成数值成果的误差变大。我们是有一些办法来解决这些问题的。在本篇论文中，主要讨论的就是能够有效解决数值微分问题的几种方法，并且做出了详细的方法阐述。任何一个方法的最终目的都是希望得出一个较为精确的数值结果。

在本篇论文中介绍的几个基本的方法，如：差分型数值微分、插值型数值微分、数值积分求数值微分、Richardson外推法。最后通过对每一种方法的计算，来证实运用这些方法计算出的数值都可能达到预期计算结果的精度的。

毕业论文关键词：差分型数值微分；插值型数值微分；数值积分求数值微分；外推法求数值微分

Abstract In classical mathematics and physics, well post problems usually are researche -d by people。 Well posed problems were satisfied with the following three requireme -nts: (1) Existence of solution; (2) Uniqueness of solution; (3) Continuity of solutions。 

The ill-posed problem is that it does not meet one of these conditions。 In particular, if it was dissatisfied with the conditions (3), then it is called the ill-posed problems in the sense of Hadamard。

Numerical differentiation is a classical ill-posed problem in the sense of hadamard。  The large errors in numerical results may be caused by small errors in the measurement。 Some methods have been used to solve the problems。 This paper mainly discussed the issue of numerical differentiation that can be solved effectively by several methods。 And each method is corresponding  to a detailed explanation。 The accurate numerical result can be found by every method which can reduce the error in numerical value。

Several basic methods in this paper such as difference-type numerical differenti -ation, interpolation-type numerical differentiation, numerical integration for numerical differentiation and Richardson extrapolation for numerical differentiation are discussed。 Through calculation of each method, the desired accuracy can be achieved。

Keywords：numerical differentiation; numerical integration; numerical differentiation; extrapolation 

目   录

 第一章 绪   论 1

1。1 数值微分的研究背景 1

1。2 数值微分的研究现状 1

1。3 本文的主要工作 2

第二章 数值微分的基本方法 3

2。1 预备知识 3

2。2差商型数值微分 4

2。2。1差商型数值微分定义 4

2。2。2 差商型数值微分实例 5

2。3 插值型数值微分 6

2。3。1插值型数值微分定义 6

2。3。2 插值型数值微分实例 8

2。4 数值积分求数值微分 9

2。4。1 数值积分法定义 9

2。4。2 数值积分法实例