admin 管理员组

文章数量: 1088108


2024年4月15日发(作者：简单c语言程序代码)

《数值计算方法》实验报告

1

什么是数值计算方法及应用与误差计算

1．什么是数值计算方法及应用

计算数学也叫做数值计算方法或数值分析。主要内容包括代数方程、线性代数方程组、

微分方程的数值解法，函数的数值逼近问题，矩阵特征值的求法，最优化计算问题，概率统

计计算问题等等，还包括解的存在性、唯一性、收敛性和误差分析等理论问题。数值计算方

法,是一种研究并解决数学问题的数值近似解方法,是在计算机上使用的解数学问题的方法，简

称计算方法。在科学研究和工程技术中都要用到各种计算方法。 例如，在航天航空、地质勘

探、汽车制造、桥梁设计、 天气预报和汉字字样设计中都有计算方法的踪影.

Numerical analysis involves the study of methods of computing numerical data. In many

problems this implies producing a sequence of approximations by repeating the procedure again and

again. People who employ numerical methods for solving problems have to worry about the

following issues: the rate of convergence (how long does it take for the method to find the answer),

the accuracy (or even validity) of the answer, and the completeness of the response (do other

solutions, in addition to the one found, exist).Numerical methods provide approximations to the

problems in question. No matter how accurate they are，they do not, in most cases, provide the exact

answer. In some instances working out the exact answer by a different approach may not be

possible or may be too time consuming and it is in these cases where numerical methods are most

often used.

The ever-increasing advances in computer technology has enabled many in science and

engineering to apply numerical methods to simulate physical phenomena. Numerical methods are

often divided into elementary ones such as finding the root of an equation, integrating a function or

solving a linear system of equations to intensive ones like the finite element method. Intensive

methods are often needed for the solution of practical problems and they often require the

systematic application of a range of elementary methods, often thousands or millions of times over.

In the development of numerical methods, simplifications need to be made to progress towards

a solution: for example general functions may need to be approximated by polynomials and

computers cannot generally represent numbers exactly anyway. As a result, numerical methods do

not usually give the exact answer to a given problem, or they can only tend towards a solution

getting closer and closer with each iteration. Numerical methods are generally only useful when

they are implemented on computer using a computer programming language.

In the study of numerical methods, we can make a general distinction between a set of methods

such as solving linear systems of equations , solving matrix eigenvalue problems , interpolation ,

numerical integration and finding the roots or zeros of equations , which can be somewhat

《数值计算方法》实验报告

2

considered as the building blocks for larger that arise in engineering/applied mathematics/physics.

For example the problem of solving ordinary differential equations , optimisation and solving

integral equations . But from the point of view of aplied mathematics or engineering, erhaps the

most significant problems in numerical methods is the solution of partial differential equations by

Finite Difference Methods , Finite Element Methods or Boundary Element Methods .

数值分析是涉及计算数字数据的方法的研究。在许多问题中，这意味着一遍又一遍重复

程序以产生近似值得序列。应用数值方法解决问题的人关心以下问题：收敛率（算法需要多

久来找到答案），答案的准确性（或者有效性），响应的完整性（做其他的解决方案，另外一

个发现，存在的话）。数值方法提供了问题的近似答案。无论它是如何准确，它在大多数情况

下也不会提供真切的答案。在某些情况下用不同的方式寻求确切的答案是不太可能的或也可

能是太耗费时间，这就是数值计算方法最经常使用情况。

计算机技术的不断进步使许多科学和工程应用数值方法来模拟物理现象。数值方法通常

分为初级的，如找到一个方程的根，集成功能型向集约型的，如有限元法求解线性系统的方

程。密集的方法往往是解决实际问题的需要，他们往往需要系统应用一系列的基本方法，通

常是数千或数百万次。

在发展的数值方法中，简化是取得进展的解决方案所必需的，例如：大多函数可能需要

近似多项式和计算机不能准确地表示数字。因此，数值方法通常不会给出给定问题确切的答

案，或者它们在每次迭代中只能越来越密切地趋近解决方案。数值方法一般仅在当它们被电

脑上的一种计算机编程语言实现时起作用。

在研究数值方法中，我们可以得到一些一整套方法的大体特性，如求解线性方程组，求

解矩阵特征值问题，插值，数值积分，发现根部或零方程组的，这些可以在一定程度上被视

为解决工程/应用数学/物理问题的桥梁。例如求解常微分方程，优化和解决的问题，积分方程。

但是，。用于数学或工程的角度来看，数值方法的最重要的问题是用有限差分法，有限元方法，

边界元法解决偏微分方程。

2. 误差计算


本文标签： 数值 问题 方法 计算方法

版权声明：本文标题：实验1_误差计算与什么是数值计算方法及应用 内容由网友自发贡献，该文观点仅代表作者本人， 转载请联系作者并注明出处：http://roclinux.cn/p/1713192537a623271.html， 本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如发现本站有涉嫌抄袭侵权/违法违规的内容，一经查实，本站将立刻删除。