MATLAB/Octave编程入门：数值模拟实践指南

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The First Few Steps

1.1 What Is a Program? And What Is Programming?

Today, most people are experienced with computer programs, typically programs

such as Word, Excel, PowerPoint, Internet Explorer, and Photoshop. The interaction

with such programs is usually quite simple and intuitive: you click on buttons, pull

down menus and select operations, drag visual elements into locations, and so forth.

The possible operations you can do in these programs can be combined in seemingly

an inﬁnite number of ways, only limited by your creativity and imagination.

Nevertheless, programs often make us frustrated when they cannot do what we

wish. One typical situation might be the following. Say you have some measure-

ments from a device, and the data are stored in a ﬁle with a speciﬁc format. You

may want to analyze these data in Excel and make some graphics out of it. How-

ever, assume there is no menu in Excel that allows you to import data in this speciﬁc

1© The Author(s) 2016

S. Linge, H.P. Langtangen, Pro gramming for Computations – MA TLAB/Octave,

Texts in Computational Science and Engineering 14, DOI 10.1007/978-3-319-32452-4_1

21TheFirstFewSteps

format. Excel can work with many different data formats, but not this one. You start

searching for alternatives to Excel that can do the same and read this type of data

ﬁles. Maybe you cannot ﬁnd any ready-made program directly applicable. You

have reached the point where knowing how to write programs on your own would

be of great help to you! With some programming skills, you may write your own

little program which can translate one data format to another. With that little piece

of tailored code, your data may be read and analyzed, perhaps in Excel, or perhaps

by a new program tailored to the computations that the measurement data demand.

The real power o f computers can only be utilized if you can program them.

By programming you can get the computer to do (most often!) exactly what you

want. Programming consists of writing a set of instructions in a very specialized

language that has adopted words and expressions from English. Such languages

are known as programming or computer languages. The set of instructions is given

to a program which can translate the meaning of the instructions into real actions

inside the computer.

The purpose of this book is to teach you to write such instructions dedicated to

solve mathematical and engineering problems by fundamental numerical methods.

There are numerous computer languages for different purposes. Within the en-

gineering area, the most widely used computer languages are Python, MATLAB,

Octave, Fortran, C, C++, and to some extent Maple, and Mathematica. How you

write the instructions (i.e. the syntax) differs between the languages. Let us use an

analogy.

Assume you are an international kind of person, having friends abroad in Eng-

land, Russia and China. They want to try your favorite cake. What can you do?

Well, you may write down the recipe in those three languages and send them over.

Now, if you have been able to think correctly when writing down the recipe, and

you have written the explanations according to the rules in each language, each of

your friends will produce the same cake. Your recipe is the “computer program”,

while English, Russian and Chinese represent the “computer languages” with their

own rules of how to write things. The end product, though, is still the same cake.

Note that you may unintentionally introduce errors in your “recipe”. Depending on

the error, this may cause “baking execution” to stop, or perhaps produce the wrong

cake. In your computer program, the errors you introduce are called bugs (yes,

small insects! . . . for historical reasons), and the process of ﬁxing them is called

debugging. When you try to run your program that contains errors, you usually

get warnings or error messages. However, the response you get depends on the er-

ror and the programming language. You may even get no response, but simply the

wrong “cake”. Note that the rules of a p rogramming language have to be followed

very strictly. This differs from languages like English etc., where the meaning might

be understood even with spelling errors and “slang” included.

This book comes in two versions, one that is based on Python, and one based on

Matlab. Both Python and Matlab represent excellent programming environments

for scientiﬁc and engineering tasks. The version you are reading now, is the Matlab

version.

Readers who want to expand their scientiﬁc programming skills beyond the

introductory level of the present exposition, are encouraged to study the book

A Primer on Scientiﬁc Programming with Python [9]. This comprehensive book

is as suitable f or beginners as for professional p rogrammers, and teaches the art

1.2 A Matlab Program with Variables 3

of programming through a huge collection o f dedicated examples. This book is

considered the primary reference, and a natural extension, of the programming

matters in the present book.

Some computer science terms

Note that, quite often, the terms script and scripting are used as synonyms for

program and programming, respectively.

The inventor of the Perl programming language, Larry Wall, tried to explain

the difference between script and program in a humorous way (from perl.com

Suppose you went back to Ada Lovelace

and asked her the difference between

a script and a program. She’d probably look at you funny, then say something

like: Well, a script is what you give the actors, but a program is what you give

the audience. That Ada w as one sharp lady . . . Since her time, we seem to

have gotten a bit more confused about what we mean when we say scripting. It

confuses even me, and I’m supposed to be one of the experts.

There are many other widely used computer science terms to pick up. Writing

a program (or script or code) is often expressed as implementing the program.

Executing a program means running the program. An algorithm is a recipe for

how to construct a program. A bug is an error in a program, and the art of

tracking down and removing bugs is called debugging (see, e.g., Wikipedia

Simulating or simulation refers to using a program to mimic processes in the

real world, often through solving differential equations that govern the physics

of the processes.

1.2 A Matlab Program with Variables

Our ﬁrst example regards programming a mathematical model that predicts the po-

sition of a ball thrown up in the air. From Newton’s 2nd law, and by assuming

negligible air resistance, one can derive a mathematical model that predicts the ver-

tical position y of the ball at time t. From the model one gets the formula

y D v

t  0:5gt

;

where v

is the initial upwards velocity and g is the acceleration of gravity, for

which 9:81 ms

2

is a reasonable value (even if it depends on things like location

on the earth). With this formula at hand, and when v

is known , you may plug in

a value for time and get out the corresponding height.

1.2.1 The Program

Let us next look at a Matlab program for evaluating this simple formula. Assume

the program is contained as text in a ﬁle named ball.m. The text looks as follows

(ﬁle ball.m):

http://www.perl.com/pub/2007/12/06/soto-11.html

http://en.wikipedia.org/wiki/Ada_Lovelace

http://en.wikipedia.org/wiki/Software_bug#Etymology

41TheFirstFewSteps

% Program for computing the height of a ball in vertical motion

v0 =5; % Initial velocity

g = 9.81; % Acceleration of gravity

t = 0.6; % Time

y = v0*t - 0.5*g*t^2 % Vertical position

Computer programs and parts of programs are typeset with a blue background

in this book. A slightly darker top and bottom bar, as above, indicates that the code

is a complete program that can be run as it stands. Without the bars, the code is just

a snippet and will (normally) need additional lines to run properly.

1.2.2 Dissection of the Program

A computer program is plain text, as here in the ﬁle ball.m, which contains in-

structions to the computer. Humans can read the code and understand what the

program is capable of doing, but the program itself does not trigger any actions on

a computer before another program, the Matlab interpreter, reads the program text

and translates this text into speciﬁc actions.

You must learn to play the role of a computer

Although Matlab is responsible for reading and understanding your program, it is

of fundamental importance that you fully understand the program yourself. You

have to know the implication of every instruction in the program and be able to

ﬁgure out the consequences of the instructions. In other words, you must be able

to play the role of a computer. The reason for this strong demand of knowledge is

that errors unavoidably, and quite often, will be committed in the program text,

and to track down these errors, you have to simulate what the computer does

with the program. Next, we shall explain all the text in ball.m in full detail.

When you run your program in Matlab, it will interpret the text in your ﬁle line

by line, from the top, reading each line from left to right. The ﬁrst line it reads is

% Program for computing the height of a ball in vertical motion.

This line is what we call a comment. That is, the line is not meant for Matlab to read

and execute, but rather for a human that reads the code and tries to understand what

is going on. Therefore, one rule in Matlab says that whenever Matlab encounters

the sign % it takes the rest of the line as a comment. Matlab then simply skips

reading the rest of the line and jumps to the next line. In the code, you see several

such comments and probably r ealize that they make it easier for you to understand

(or guess) what is meant with the code. In simple cases, comments are probably not

much needed, but will soon be justiﬁed as the level of complexity steps up.

The next line read by Matlab is

v0 =5; % Initial velocity

1.2 A Matlab Program with Variables 5

According to its rules, Matlab will now create a variable with the name v0 and

set (the value of) that variable equal to 5. We say that 5 is assigned to v0.This

means that whenever Matlab reads v0 hereafter, it plugs in 5 instead of the name

v0, since it knows that v0 has the value 5. You may think of v0 as a variable v

mathematics. The next two lines

g = 9.81; % Acceleration of gravity

t = 0.6; % Time

are of the same kind, so having read them too, Matlab knows of three variables (v0,

g, t) and their values. These variables are then used b y Matlab when it reads the

next line, the actual “formula”,

y = v0*t - 0.5*g*t^2 % Vertical position

Again, according to its rules, Matlab interprets * as multiplication, - as minus and

^ as exponent (let us also add here that, not surprisingly, + and / would have been

understood as addition and division, if such signs had been present in the expres-

sion). Having read the line, Matlab performs the mathematics on the right-han d

side, and then assigns the result (in this case the number 1.2342) to the variable

name y. Also, since the ﬁnal line has no semi-colon, Matlab understands that we

also want the result printed to screen. When ball.m is run, the number 1.2342

appears on the screen.

Note that leaving out a semi-colon provides an easy way to print things to screen

in general. Simply writing, e.g.,

v0 in the program above, i.e. without the semi-

colon, will make the content of v0 be printed to screen.

In the code above, you see several blank lines too. These are simply skipped by

Matlab and you may use as many as you want to make a nice and readable layout

of the code.

1.2.3 WhyNotJustUseaPocketCalculator?

Certainly, ﬁnding the answer as done by the program above could easily have been

done with a pocket calculator. No objections to that and no programming would

have been needed. However, what if you would like to have the position of the ball

for every milli-second of the ﬂight? All that punching on the calculator would have

taken you something like four hours! If you know how to program, however, you

could modify the code above slightly, using a minute or two of writing, and easily

get all the positions computed in one go within a second. A much stronger argu-

ment, however, is that mathematical models from real life are often complicated and

comprehensive. The pocket calculator cannot cope with such problems, even not

the programmable ones, because their computational power and their programming

tools are far too weak compared to what a real computer can offer.

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