MATLAB编程与数值方法实战指南

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"《在MATLAB中的编程和数值方法》是由S.R. Otto和J.P. Denier合著的一本书，旨在引导读者了解如何在MATLAB环境下进行编程和应用数值方法。书中包含111个插图，适用于科学和工程学本科课程的学生，通过MATLAB这个强大的工具，使学生能够快速构建原型，测试和理解数值计算的方法。" MATLAB是一种广泛应用于科学计算的交互式环境，它以矩阵和数组操作为基础，简化了编程和数据分析的过程。在MATLAB中编程，可以帮助用户快速实现复杂算法，进行数据可视化，并解决各种数学问题。书中详细讲解了MATLAB的基础语法、函数使用以及与其他编程语言的交互。数值方法是解决数学问题的一种实用手段，尤其在处理无法解析求解的问题时。这些方法包括但不限于：线性代数中的求解系统、插值和拟合、数值微积分、非线性方程的求解、常微分方程的数值解法（如欧拉方法、龙格-库塔方法）、优化算法等。MATLAB提供了丰富的内置函数和工具箱，支持这些数值方法的实现。在本书中，作者可能详细讨论了如何在MATLAB中应用这些数值方法，例如： 1. 矩阵和向量操作：MATLAB的基石，用于构建和操作数据。 2. 数值求解线性方程组：使用高斯消元、LU分解或QR分解等方法。 3. 插值和拟合：如拉格朗日插值、样条插值，以及最小二乘法拟合数据。 4. 数值积分：如梯形法则、辛普森法则和高斯积分。 5. 非线性方程求解：如牛顿迭代法、二分法等。 6. 常微分方程数值解：如四阶龙格-库塔法和其他高级方法。 7. 优化问题：包括无约束优化、有约束优化和多变量优化算法。此外，书中可能会涉及错误控制和精度分析，帮助读者理解数值计算中的稳定性问题。通过实例和练习，学生可以深入理解和掌握这些概念，提升在实际问题中的应用能力。 MATLAB还提供了如Simulink这样的图形化建模工具，使得系统仿真和控制设计变得更加直观。书中可能也会介绍Simulink在模拟控制系统和信号处理中的应用。《在MATLAB中的编程和数值方法》是一本全面的指南，涵盖了MATLAB编程基础和数值计算的核心内容，对于想要在科学和工程领域使用MATLAB的人来说，是一本不可多得的学习资料。通过这本书，读者可以掌握用MATLAB进行数值计算的技能，为进一步的科研工作打下坚实的基础。

2 1. Simple Calculations with MATLAB

We shall start by introducing MATLAB commands which can be typed at

the MATLAB prompt; these will ultimately form part of our vocabulary of

MATLAB commands. MATLAB already has an extensive vocabulary: however

we will learn that we can expand this set. As the name MATLAB (MATrix

LABoratory) suggests, most of the commands work with matrices and these

will be discussed in due course. We shall start with scalar operations, for which

MATLAB acts like a very powerful calculator.

1.2 Scalar Quantities and Variables

We will begin with the basic ideas of equations and variables. Try entering the

commands as they are given. Consider the following two commands:

>>a=3

>>b=4;

These two commands are entered on separate lines; the MATLAB prompt is

denoted by >> (which does not need to be typed), as distinguished from the

standard greater than sign >. The command on the ﬁrst line sets the variable

a to be equal to three (3) and that on the second line sets the variable b to

be equal to four (4). The two commands also diﬀer because the second one

ends with a semicolon. This instructs MATLAB to execute the command but

suppress any output; whereas above we can see that the value of a has been

set to 3. These commands can be read as

set a equal to 3

set b equal to 4 (and suppress output)

Reading the commands in this way it should be clear that it is not possible

to have a command of the form 7=x(set 7 equal to x), whereas we could

have x = 7 (set x equal to 7). These variables can now be used again, for

instance

be accessed using the commands demo or tour. There is also a good help facility

which, unsurprisingly, can be accessed by typing help followed by the command in

question. There is also a facility to use a web browser (helpdesk or helpbrowser).

Here, you would type a=3, and then press RETURN, and then type b=4;and

press RETURN again. The spaces are included purely for clarity.

1.2 Scalar Quantities and Variables 5

with (whether it be a ﬁle containing MATLAB code, or data etc). We will

see more on this later in the section on script ﬁles.

2. Variable names in MATLAB are case sensitive, so that a and A are two

diﬀerent objects.

3. It is good programming practise to employ meaningful variable names.

In our initial examples we have only used very simple (but appropriate)

names: however as the examples become more complex our variable names

will be more informative.

4. Variables names should not coincide with a predeﬁned MATLAB command

or with any user-deﬁned subroutines. To see whether a variable name is al-

ready in use we can use the command type variable

name, but it may

be better to use the command which variable

name (this will tell you

whether the name variable

name corresponds to an existing code or in-

trinsic function.

1.2.2 Precedence: The Order in Which Calculations Are

Performed

This represents one of the most common sources of errors and it is often the

most diﬃcult to detect. Before proceeding we brieﬂy comment on the question

of precedence, or the order in which commands are executed. Consider the

mathematical expression a(b + c) which you might read as “a times b plus c”

which would appear to translate to the MATLAB command a*b+c. Hopefully

you can see that this actually is equal to ab+c. The correct MATLAB command

for a(b + c)isa*(b+c). The brackets have been used to force MATLAB to ﬁrst

evaluate the expression (b+c) and then to multiply the result by a. We should

avoid falling into the trap of assuming that commands are performed from left-

to-right, for instance c+a*b is equal to c + ab (not (c + a)b as if the addition

was performed ﬁrst).

At this point we should pause brieﬂy and make sure the ideas of brackets are

ﬁrmly in place. Brackets should always appear in pairs and the mathematics

contained within brackets (or equivalently MATLAB) will be evaluated ﬁrst.

Hopefully this concept is familiar to you: however it is worth reiterating, since

one of the most common problems in using MATLAB occurs due to either

unbalanced or incorrectly placed brackets. For example the commands (3+4/5)

and (3+4)/5 are obviously diﬀerent, the former being 3

and the latter being

3+4

The most critical use of brackets, which circumvents another popular source

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