MATLAB助力数值计算：经典教材与编程提升

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《数值计算与MATLAB》是一本经典的教育书籍，由Cleve B. Moler撰写，由Society for Industrial and Applied Mathematics (SIAM) 出版。这本书在2004年修订再版，旨在通过MATLAB编程的实际应用，帮助读者深入理解和掌握数值计算方法。MATLAB，作为一款强大的数学软件，是本书的核心工具，作者充分利用其直观易用的特性，将理论知识与实践操作相结合。书中内容涵盖了数值计算的基础概念，如数值解微分方程、线性代数、优化算法、常微分方程求解、插值与拟合、傅立叶变换等，这些都是工程和科学研究中不可或缺的技能。每一章节都配以MATLAB代码示例，读者可以通过实际编写和运行这些代码，提升编程能力和理解数值计算的工作原理。此外，书中的内容还包括误差分析和数值稳定性，这是数值计算中至关重要的部分，因为它关乎计算结果的准确性和可靠性。作者还讨论了如何利用MATLAB的图形用户界面和高级功能进行复杂计算，以及如何优化代码以提高效率。书中特别强调了MATLAB的交互式学习方式，鼓励读者在探索和实验中学习，这种实践驱动的教学方法有助于培养解决问题的能力。同时，书中也提到了版权信息，强调所有内容未经许可不得复制或传输，以尊重知识产权。《数值计算与MATLAB》不仅是学术研究者和研究生的必备参考书，也是工程师、科学家和计算机科学学生提升数值计算技能的实用手册。对于想要在数值分析领域深入探索或者寻求编程技巧提升的人来说，这本书提供了丰富的实践经验和理论支持，是学习和应用数值计算的宝贵资源。

1.1. The Golden Ratio 5

Figure 1.2. f(φ) = 0.

runs the script and creates the picture.

% GOLDRECT Plot the golden rectangle

phi = (1+sqrt(5))/2;

x = [0 phi phi 0 0];

y=[00110];

u=[11];

v=[01];

plot(x,y,’b’,u,v,’b--’)

text(phi/2,1.05,’\phi’)

text((1+phi)/2,-.05,’\phi - 1’)

text(-.05,.5,’1’)

text(.5,-.05,’1’)

axis equal

axis off

set(gcf,’color’,’white’)

The vectors x and y each contain ﬁve elements. Connecting consecutive (x

)

pairs with straight lines produces the outside rectangle. The vectors u and v each contain

two elements. The line connecting (u

) with (u

) separates the rectangle into the

square and the smaller rectangle. The plot command draws these lines—the x − y lines

in solid blue and the u − v line in dashed blue. The next four statements place text at

various points; the string ’\phi’ denotes the Greek letter. The two axis statements

cause the scaling in the x and y directions to be equal and then turn off the display of the

axes. The last statement sets the background color of gcf, which stands for get current

ﬁgure, to white.

8 Chapter 1. Introduction to MATLAB

mathematical texts that, among other things, introduced Europe to the Hindu-Arabic notation

for numbers. Even though his books had to be transcribed by hand, they were widely

circulated. In his best known book, Liber Abaci, published in 1202, he posed the following

problem:

A man put a pair of rabbits in a place surrounded on all sides by a wall. How

many pairs of rabbits can be produced from that pair in a year if it is supposed

that every month each pair begets a new pair which from the second month on

becomes productive?

Today the solution to this problem is known as the Fibonacci sequence,orFibonacci

numbers. There is a small mathematical industry based on Fibonacci numbers. A search of

the Internet for “Fibonacci” will ﬁnd dozens of Web sites and hundreds of pages of material.

There is even a Fibonacci Association that publishes a scholarly journal, the Fibonacci

Quarterly.

If Fibonacci had not speciﬁed a month for the newborn pair to mature, he would not

have a sequence named after him. The number of pairs would simply double each month.

After n months there would be 2

pairs of rabbits. That’s a lot of rabbits, but not distinctive

mathematics.

Let f

denote the number of pairs of rabbits after n months. The key fact is that the

number of rabbits at the end of a month is the number at the beginning of the month plus

the number of births produced by the mature pairs:

= f

n−1

+ f

n−2

The initial conditions are that in the ﬁrst month there is one pair of rabbits and in the second

there are two pairs:

= 1,f

= 2.

The following Matlab function, stored in the M-ﬁle fibonacci.m, produces a

vector containing the ﬁrst n Fibonacci numbers.

function f = fibonacci(n)

% FIBONACCI Fibonacci sequence

% f = FIBONACCI(n) generates the first n Fibonacci numbers.

f = zeros(n,1);

f(1) = 1;

f(2) = 2;

for k = 3:n

f(k) = f(k-1) + f(k-2);

end

With these initial conditions, the answer to Fibonacci’s original question about the size of

the rabbit population after one year is given by

fibonacci(12)

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