Our first example regards programming a mathematical model that predicts the position 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 vertical position \( y \) of the ball at time \( t \). From the model one gets the formula $$ y = v_0 t - 0.5gt^2 , $$ where \( v_0 \) is the initial upwards velocity and \( g \) is the acceleration of gravity, for which \( 9.81\hbox{ ms}^{-2} \) is a reasonable value (even if it depends on things like location on the earth). With this formula at hand, and when \( v_0 \) is known, you may plug in a value for time and get out the corresponding height.
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):
% 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.
A computer program is plain text, as here in the file ball.m, which
contains instructions 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.
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.
%
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 realize 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
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 \( v_0 \)
in mathematics. The next two lines
g = 9.81; % Acceleration of gravity
t = 0.6; % Time
v0, g, t) and their values. These variables are then
used by Matlab when it reads the next line, the actual "formula",
y = v0*t - 0.5*g*t^2 % Vertical position
* 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
expression). Having read the line, Matlab performs the mathematics
on the right-hand 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.
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 argument, 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.
When Matlab interprets some code in a file, it is concerned with every character in the file, exactly as it was typed in. This makes it troublesome to write the code into a file with word processors like, e.g., Microsoft Word, since such a program will insert extra characters, invisible to us, with information on how to format the text (e.g., the font size and type). Such extra information is necessary for the text to be nicely formatted for the human eye. Matlab, however, will be much annoyed by the extra characters in the file inserted by a word processor. Therefore, it is fundamental that you write your program in a text editor where what you type on the keyboard is exactly the characters that appear in the file and that Matlab will later read. There are many text editors around. Some are stand-alone programs like Emacs, Vim, Gedit, Notepad++, and TextWrangler. Many prefer to use the text editor that comes with the graphical Matlab environment.
Reading only does not teach you computer programming: you have to
program yourself and practice heavily before you master mathematical
problem solving via programming. Therefore, it is crucial at this stage that you
write and run a Matlab program. We just went through the program ball.m
above, so let us next write and run that code.
But first a warning: there are many things that must come
together in the right way for ball.m to run correctly on
your computer. There might
be problems with your Matlab installation, with your writing of the
program (it is very easy to introduce errors!),
or with the location of the file, just to mention some of the
most common difficulties for beginners. Fortunately, such problems are
solvable, and if you do not understand how to fix the
problem, ask somebody. Typically, once you are beyond these common
start-up problems, you can move on to learn
programming and how programs can do a lot of otherwise complicated
mathematics for you.
The term Matlab refers to both the software package Matlab from MathWorks Inc., and the programming language Matlab. Matlab programs can either be run in the commercial Matlab software package, or they can be run in the free GNU Octave software, usually just called Octave. We first describe how to operate the Matlab software and then Octave.
The first step is to generate a directory in which you will place your
future Matlab code. Do this in a terminal window (Terminal on Mac,
Power Shell or Command Prompt on Windows, or (e.g.) gnome-terminal on
Linux). Write mkdir mycode to create a directory with name mycode.
Then move into that directory by writing cd mycode.
Start Matlab and try out the following.
ball.m. Do this by choosing
File/New/Script from the menu in the Command window. In the editor
window that pops up, simply write the code lines there as they were
given above for ball.m. Now save this with the name ball.m in the
right directory, i.e. myCode, via Save As from the File menu.
The program is now ready for use!Octave users must write the program in a plain text editor such as Gedit on Linux computers; TextWrangler on Mac, or Notepad++ on Windows. Popular, but more advanced text editors, primarily Emacs and Vim, are also available for these platforms.
ball.m by launching a text editor and
write each line exactly as they are listed in the ball.m program.
Save the file as ball.m in the mycode directory.octave. The Octave program is started and
gives you a prompt octave:1>, which indicates that you can
give Octave commands. Type run ball.m and press enter. Octave will
now run the program.
A program such as ball.m, i.e., code stored in a file with the
extension .m, is usually referred to as an m-file.