http://www.maths.cam.ac.uk/undergrad/catam/MATLAB/manual/booklet.pdf
Contents
1 Introduction 1
1.1 Suggestions, comments and corrections . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Other documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Using Windows XP on the Mathematics PWF 2
2.1 Logging in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Windows basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 The start menu and task bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Window elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.5 Files and folders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5.1 Backing up your files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.7 Logging out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.8 Further documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Introduction to MATLAB 7
3.1 Starting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Programming in MATLAB 11
4.1 A simple program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.1 Programming tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1.2 My program is running out-of-control or not responding . . . . . . . . 13
4.2 Improving the output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Reducing typing and a noddy guide to functions . . . . . . . . . . . . . . . . . 15
4.3.1 Script files (a.k.a. M-Files) . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Help! 21
i
6 Vectors and matrices 22
6.1 Creating matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Manipulating matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7 A few more functions 26
7.1 Scalar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.2 Vector functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.3 Matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8 Program ow control 28
8.1 The if-else control structure . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.2 The while control structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.3 The switch-case control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.3.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9 Elementary graph plotting 35
9.1 The plot command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.2 Other 2D graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9.3 Multiple figures and plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9.4 Saving your figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
9.4.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.5 3D graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10 Random number generation 41
11 This and that 41
11.1 Programming style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
11.2 Some terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ii
12 Acknowledgments 42
A A generalised printsquares code 43
1 IntroductionThis guide is intended to help you to learn how to program with MATLAB whether you are
new to programming, or whether you have programmed before but would like to know more
about MATLAB.
You should skim over or skip sections containing material that is already familiar to you.
However, you should realise that writing programs is the only way to learn a programming
language — only by typing in and running your own programs will you learn to translate
mathematics into computer algorithms and thence into computer programs. Although it might
be tedious to type in long programs, instead of just loading them in, you can learn a lot in the
process of typing in, running and changing the programs in this guide.
If you are looking for a quick start to MATLAB, you may also skip sections which appear on
a grey background. Such sections provide more advanced material on MATLAB and program-
ming in general, and may be more useful on a second pass through the tutorial.
Section §2 covers material suited to those who are new to Windows, and mentions functionality
that more advanced users may already be familiar with.
The remaining sections cover learning to use MATLAB. The early sections concentrate on
programming techniques by deliberately using examples that are mathematically very simple.
You are encouraged to modify the example programs and to write your own programs.
• MATLAB has its own built-in help and do cumentation.
• The MathWorks provide an introduction Getting Started with MATLAB. You can access
this by ‘left-clicking’ on the Getting Started link at the top of a MATLAB ‘Com-
mand Window’. Alternatively there is an on-line version available at1
http://www.mathworks.com/access/helpdesk/help/techdoc/learn_matlab/learn_matlab.shtml
1 These links work at the time of writing. Unfortunately The MathWorks have an annoying habit of breaking
their links.
1
A printable version is available fromhttp://www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/getstart.pdf
• The MathWorks also provide links to a whole a raft of other tutorials
http://www.mathworks.co.uk/academia/student_center/tutorials/launchpad.html
In addition their MATLAB documentation page gives more details on maths, graphics,
object-oriented programming etc.; see
http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.shtml
• There is also a plethora of books on MATLAB; for instance MATLAB Guide by D.J.
Higham & N.J. Higham (SIAM, 2nd Ed. 2005, ISBN 0-89871-578-4).
3 Introduction to MATLAB
We recommend that you use MATLAB for the Computational Projects. However, you are
not required to use MATLAB, and if you choose you could program in Mathematica, Maple,
Scilab, C, C++, C#, Python, or any other language.
One of the advantages of MATLAB is that it has an ‘environment’ which includes an editor
and a debugger. However, even if you decide to use MATLAB you need not use either of
these (e.g. you could use the Emacs editor instead of the integrated editor).
MATLAB is available free of charge from the Faculty of Mathematics for installation on your
own personal computer running Windows, Mac OS or Linux.9 MATLAB is also pre-installed
on the computers in the CATAM PWF room in the CMS, and on other PWF computers
(including those at a number of Colleges and that in the Betty and Gordon Moore Library);
for a list of University and College PWF sites see
http://www.cam.ac.uk/localusersonly/cs/pwf/facilities/manclus.html
9 You can collect a copy of MATLAB from the CMS Reception. You will need to bring along your University
identity card, or other from of picture identity. While MATLAB is free to you, it is not free to the University;
hence please only collect the MATLAB software if you intend to take Part IB of the Mathematical Tripos.
7
3.1 Starting MATLABIf you are sitting in front of a PWF Windows computer, you can start MATLAB from the
start menu as described in §2.3. If you have installed MATLAB on your own machine there
may be an icon on the desktop from which you can start MATLAB; alternatively there will be
an entry for MATLAB in your ‘start’ or ‘finder’ menu. After a short while a window should
open with three or four panes.10
1. One pane is labelled ‘Command Window’. This is the pane in which you will type
MATLAB commands.
2. One pane is labelled ‘Current Directory’, and lists the files in that directory.
3. One pane is labelled ‘Workspace’, and lists the variables that you have defined. Displayed
variables may be viewed, manipulated, saved, and cleared.
4. One pane is labelled ‘Command History’. This pane lists your previous commands. Inter
alia you can execute a previous command by double-clicking on it.
Henceforth, unless stated otherwise, you should type the MATLAB commands in this guide
into the ‘Command Window’.
3.2 The basics
MATLAB includes all the operations and functions you would find on a calculator. It will
attempt to evaluate mathematical expressions that you input. Into the ‘Command Window’
type
>> 2+2
and press the return key. MATLAB should print the line
ans = 4
Note that all inputs in MATLAB are terminated by hitting the return key. It is assumed you
will do this from now on. Type
>> cos(pi)
MATLAB returns
ans = -1
1 0 Please note the ‘should’. Depending on how MATLAB has been configured you may end up with one,
two, three or four panes.
8
as ‘pi’ is the built-in expression for p. Now type
>> pi^2;
In this instance, MATLAB evaluates p2 but the semi-colon at the end of the line causes the
output to be suppressed. Typing
>> pi^2
gives the expected output
ans = 9.8696
Note that this is not the actual precision to which MATLAB has calculated the answer, it is
only the output precision.
Of course, MATLAB is much more than a calculator. The first step to programming in
MATLAB is learning how to define and use variables. Variables are identified by a name
(often a mnemonic name) beginning with a letter. They are assigned values using the symbol
= (often read as ‘set equal to’.) The assignment statement takes the form
‘variable’ = ‘value’
where ‘value’ can be a number or algebraic expression, and the algebraic expression can include
other variables.
Next, into the ‘Command Window’ type
>> a = 2
MATLAB returns
a = 2
Now type
>> a = 3;
Although the output is suppressed by the semi-colon, the value of variable a has changed.
Confirm this by typing
>> a*(a+1)
You should obtain
ans = 12
It is important to remember that = assigns values to variables. For example, a=sqrt(a) is not
an equation nor a recursive definition; it simply assigns to variable ‘a’ the square ro ot of the
current value of a.
9
Unlike many programming languages, MATLAB does not require variables to be defined beforethey are used. MATLAB is of course aware of variable types, e.g. integer, real, string, array,
logical, but variables are not forced into type-specific roles and they may change their type
during the course of a program. MATLAB does not observe any naming conventions for
variable types.
It is, however, good practice to initialize large arrays with null values before they are used.
This allows MATLAB to set aside su cient memory before the program begins.
The uidity of variable type gives MATLAB great exibility. For instance, as we will see in the
next section, if v is a vector-valued variable then v(1) is its first element. However, v(1.0)
also returns the first element, as does v(pi>3). The latter case works because pi>3 returns
logical 1 and this is then used as an array index. On the other hand, v(1.2) continues to
make no sense.
One drawback of not defining variable types is that when type-specificity is required, you
will need to ensure the correct variable type is being used. If you make a mistake, however,
MATLAB will generally alert you to this at runtime.
3.3 Vectors
MATLAB has been designed to reference and manipulate vectors extremely e ciently. It has
an extensive library of operations and functions that can be applied to vectors taken as a whole
or on an element-by-element basis. Utilizing these built-in features, will allow you to write
streamlined and speedy code.
There is a fuller discussion of vectors and matrices in section 6, but it will be helpful to have
a small taster now.
Into the ‘Command Window’ type
>> x = [-1 0 1 2]
The variable x is row vector with 4 elements. You can ask for its third element by typing
>> x(3)
Note that row (and column) indices in MATLAB start at 1.
A faster way to define x is by typing
>> x = -1:1:2
The right hand side of the assignment statement is interpreted as ‘start at -1 then increment
by 1 until 2 is exceeded’. In fact, MATLAB assumes an increment of 1 unless otherwise stated.
Therefore, x = -1:2 is even more succinct.
Now type
10
>> y = exp(x)Note that the exp function acts on the vector x element-wise: it exponentiates each element
of x. It follows that y will be the same length as x.
On the other hand, MATLAB operations tend not to act element-by-element. For instance, y
= x^2 requires x to be a scalar or a square matrix, and MATLAB will return an error if this
is not the case. To force a MATLAB operation to act element-wise, one inserts a . before the
operator. To see this, type
>> y = x.^2
3.4 Plots
The plot command is an example of MATLAB’s use of vectors. The plot command takes
vectors x and y of the same length and plots the points (x(1),y(1)), (x(2),y(2)), ...
By default, plot also connects the points with straight line segments.
We will look at plotting in detail in §9. In the meantime, using x and y defined above, type
>> plot(x,y)
MATLAB should open a graphics window and display a very segmented-looking parabolic
curve. To make the curve more smooth, type
>> x = -1:0.1:2;
>> y = x.^2;
>> plot(x,y)
Note that it is important to redefine y in the second line, otherwise plot will fail attempting
to plot x of length 31 against y of length 4.
4 Programming in MATLAB
MATLAB is a high-level computer programming language and, like other ‘high-level’ pro-
gramming languages, a MATLAB program is essentially a sequence of statements. However,
unlike languages such as C and C++, MATLAB programs are not compiled before they are
executed.11
1 1 This is not strictly true. MATLAB do es have a ‘just-in-time’ compiler that is invoked if code is executed
from a M-File (as described in §4.3.1).
11
4.1 A simple programAs an introduction we will write a simple program to write out a table of the squares of the
first 10 natural numbers. To do this, we will intro duce the concept of ‘loops’ (a concept that
applies to many other ‘high-level’ programming languages).
Into the ‘Command Window’ type
>> Ilow = 1;
>> Ihigh = 10;
>> for I = Ilow : 1 : Ihigh
Isquare = I*I
end
In the above:
The variables Ilow and Ihigh. The variables Ilow and Ihigh are assigned values 1 and 10
respectively by the assignment statements:
>> Ilow = 1;
>> Ihigh = 10;
As noted in the previous section, because a semi-colon has been added at the end of the
line the values of Ilow and Ihigh are not printed out when they are assigned.
The for loop. The syntax of the for loop is as follows. The lo op starts with the word for.
The next statement is the loop counter condition I=Ilow:1:Ihigh,12 where I is called
the loop counter. The loop counter condition tells the computer to execute the loop once
for each value of I from Ilow to Ihigh, adding 1 to I after each lo op, i.e. it tells the
computer to execute the loop once with I = 1 = Ilow, once with I = 2 = Ilow + 1, . . . ,
once with I = 10 = Ihigh.13 The for loop ends with end, and the statements between
the loop counter condition and end are referred to as the body of the for loop.
Hence, on reaching the for loop, the computer sets I = Ilow then it checks if I Ihigh
and, if true, it executes the statements between the for statement and the end statement.
Next, I is incremented by 1, the computer re-checks if I Ihigh and, if true, it re-
executes the body of the for loop. This continues until I > Ihigh, at which point the
loop ends and execution continues with the first statement, if any, after the loop.
Note that Ilow:1:Ihigh is in the form of a MATLAB vector. In MATLAB, a loop
counter is simply assigned the elements of a vector sequentially.14 Furthermore, the
expression Ilow:Ihigh would also work in the loop counter condition since, as we saw
earlier, an increment of 1 is assumed by default.
The body of the for loop. The mathematical work of the program is carried out in the state-
ments that form the body of the loop. The assignment statement
1 2 Or, with spaces, I = Ilow : 1 : Ihigh. Whether you include spaces for clarity is a personal choice.
1 3 Since the for lo op increment is one, the :1 in Ilow:1:Ihigh is optional. Note that integer increments
other than one, including negative integers, are allowed; but an increment of zero is not wise.
1 4 This means the loop counter does not have to be an integer.
12
Isquare = I*Icomputes the square of I, assigns that value to the variable Isquare, and prints it
out (since there is no final semi-colon). The ‘*’ is an ‘operator’ that means ‘multiply’.
Operators include:
Operator Operation
+ add
- subtract
* multiply
^ raise to the power
/ divide
For further information on arithmetic operators enter help arith, or helpwin arith
within the MATLAB ‘Command Window’.
4.1.1 Programming tips
Although MATLAB do es not care whether or not you leave blank lines or blank space between
variables and operators, it is a considerable help when checking the logic of your program if
you get into the habit of indenting the statements in loops (and spacing out at least some of
the elements in statements). Decide on your policy for indentation and the use of blanks, and
try to stick to it. You will find that when you type in a program within many editors, the
editor can automatically indent for you.
It is rather common programming practice to use the variable names i and j for lo op counters.
In the above example we have not followed this practice, because in MATLAB i and j are
predefined to be the principal square root of -1. However, if you overwrite this prior definition
and you subsequently want to use i and/or j as the imaginary unit, you can reset them by
clearing the current value[s] with clear i and/or clear j.
4.1.2 My program is running out-of-control or not responding
Sometimes you will make a mistake in your programming, and your program will run out-of-
control or not respond. Hitting Ctrl+C in the ‘Command Window’ should restore normality.
4.2 Improving the output
The output from our program is not very readable! Matters can be improved slightly by asking
MATLAB to produce compact formatting. Try
>> format compact
before executing the for lo op again:
13
>> for i=Ilow:Ihigh, Isquare = I^2, endwhere we have reduced the for loop to one line (at the expense of readability) by use of
commas.15
However even after opting for a compact format, the output is still a little like drinking from
a fire hydrant (especially if Ihigh was much greater than 10). Ideally we would like not to
produce two lines of output when one line would do. Instead try the following (note the use
of ’;’ after ’Isquare = I*I’):
>> for I=Ilow:Ihigh
Isquare = I*I;
disp(Isquare)
end
or the more compact
>> for I=Ilow:Ihigh
disp(I*I)
end
This is better, since the use of the display command disp has reduced the output for each
value of I to one line. However the output is not as informative as it might be, since we do
not know the significance of the numbers printed. So try one last change:
>> for I = Ilow:Ihigh
disp([’I = ’ num2str(I) ’, I*I = ’ num2str(I*I)])
end
To see what is happening here, note that [’I = ’ num2str(I) ’, I*I= ’ num2str(I*I)]
is a MATLAB vector with 4 elements (separated by spaces). The disp command puts the
elements of the vector on the same line of output. The elements of the vector, however, are
string variables not numerical variables. Single quotes are used to enclose the value of a string
variable. Therefore, regardless of the value of the numerical variable I, ’I = ’ is the literal
text given in quotes. To insert a numerical variable into text as we have to do here, we use the
built-in MATLAB function num2str, which converts a numerical variable into a string variable.
MATLAB incorporates many approaches to handling input and output (often based on the
syntax of the C programming language). For instance, the following uses fprintf (which is
MATLAB’s version of C’s printf) to improve the readability of the output:
>> for I = Ilow:Ihigh
fprintf(’I = %2g, I*I = %3g\n’,I,I*I)
end
1 5 Note that format loose returns you to the default formatting.
14
The fprintf statement. Thefprintf(’I = %2g, I*I = %3g\n’,I,I*I)
component of the above code displays the values of the variables I and I*I on the same
line. The string of characters ’I = %2g, I*I = %3g\n’, i.e. the characters up to the
first comma not inside quotes, tells MATLAB what characters to print, where to print
[any] numbers, and in what ‘format’ to print these numbers. For instance the "%2g" tells
MATLAB to print out a number with a width of 2 characters (specifying the width of a
field ensures the numbers are printed out in neat columns), and the "\n" tells MATLAB
to move to a new line. The next two arguments (separated by commas) specify the two
numbers that are to be output.
There are many more refinements to the fprintf statement; e.g. "%s" tells MATLAB
to print a string of characters. However, this is probably not the stage at which to delve
into the many options available with fprintf, so we will not. However, having been
warned, if you are interested you can learn more about fprintf by entering
>> help fprintf
(or helpwin fprintf), or for some slightly more detailed documentation
>> doc fprintf
4.3 Reducing typing and a noddy guide to functions
As noted earlier, the suppliers of MATLAB, The MathWorks, tinker with the graphical inter-
face. As a result in what follows there are di erences between the version of MATLAB on the
PWF (version 7.8 or R2009a) and the version of MATLAB that the Faculty of Mathematics
is distributing for your personal computers (version 7.10 or R2010a). The convention adopted
below is to follow version R2010a (with fo otnotes noting some of the di erences with R2009a).
There are even bigger di erences with earlier versions of MATLAB.
4.3.1 Script files (a.k.a. M-Files)
Unless you have discovered the wonders of the arrow keys16, backspace and delete within the
‘Command Window’, in the last section you will have typed the same code in a number of
times. Really, what you would like to be able to do is to save your program somewhere so that
you can re-run it after minor changes. This section explains how to do this.
We are going to store your programs in a file called a script file. However, so that your
MATLAB programs do not get muddled up with other files, we will put the MATLAB programs
in a separate folder. To create a such a folder move the mouse so that the pointer sits in an
unmarked, i.e. white, part of the ‘Current Directory’ pane. Right-click, and select New folder.
1 6 I.e. up-arrow, left-arrow, right-arrow and down-arrow.
15
When a new folder appears in the pane, type in a name (e.g. MATLAB) and hit the return key.Now double click on your new folder to open that folder (it should be empty).
Next click on File on the top line of the MATLAB window, followed by New and Script17 A
new ‘Editor’ window should open containing a cursor on line 1. Type in the following code:18
Ilow = 1;
Ihigh = 10;
for I = Ilow:Ihigh
disp([’I = ’ num2str(I) ’, I*I = ’ num2str(I*I)])
end
Or, if you prefer to use fprintf,
Ilow=1;
Ihigh=10;
for I=Ilow:Ihigh
fprintf(’I = %2g, I*I = %3g\n’,I,I*I)
end
Once you have typed in the above code it is necessary to save it in a file before it can be run. To
do this click on File on the top line of the ‘Editor’ window, followed by Save As.... A new
window should appear with a default File name of something like Untitled.m or untitled.m
or UntitledN.m for some integer N; this is rather uninformative. To change the file name click
in the box next to File name and change the entry to, say, listsquares.m. Then click Save
to accept this name. If you lo ok in the ‘Current Directory’ pane a file listsquares.m should
now have appeared.
Having saved your code, return to the ‘Command Window’ and enter
>> listsquares
The result should be the output
I = 1, I*I = 1
I = 2, I*I = 4
I = 3, I*I = 9
I = 4, I*I = 16
I = 5, I*I = 25
I = 6, I*I = 36
I = 7, I*I = 49
I = 8, I*I = 64
I = 9, I*I = 81
I = 10, I*I = 100
1 7 Or Blank M-File if you are running version R2009a, or Blank M-File if you are running version R2008b.
1 8 Or cut-and-paste the code from the ‘Command Window’.
16
Or, in the case of using fprintf,I = 1, I*I = 1
I = 2, I*I = 4
I = 3, I*I = 9
I = 4, I*I = 16
I = 5, I*I = 25
I = 6, I*I = 36
I = 7, I*I = 49
I = 8, I*I = 64
I = 9, I*I = 81
I = 10, I*I = 100
If you have made a mistake then you will need to return to the ‘Editor’ window and mo dify
your co de. (If you have closed the ‘Editor’ window either double click on listsquares.m in
the ‘Current Directory’ pane, or enter edit listsquares in the ‘Command Window’.) Once
you have made your corrections save the code by clicking on File on the top line of the ‘Editor’
window, followed by Save (there is no need to use Save As... since the file has already been
created).
Suppose now we wish to list the squares from 11 to 20. Rather than typing in the commands
again we can edit listsquares.m to read:
Ilow=11;
Ihigh=20;
for I=Ilow:Ihigh
disp([’I = ’ num2str(I) ’, I*I = ’ num2str(I*I)])
end
Or
Ilow=11;
Ihigh=20;
for I=Ilow:Ihigh
fprintf(’I = %2g, I*I = %3g\n’,I,I*I)
end
Once you have made the changes save the code by clicking on File on the top line of the
‘Editor’ window, followed by Save. Then in the ‘Command Window’ enter
>> listsquares
and the result should be the output
17
I = 11, I*I = 121I = 12, I*I = 144
I = 13, I*I = 169
I = 14, I*I = 196
I = 15, I*I = 225
I = 16, I*I = 256
I = 17, I*I = 289
I = 18, I*I = 324
I = 19, I*I = 361
I = 20, I*I = 400
4.3.2 Functions
If we wish to change Ilow and Ihigh regularly then there is a better way than repeatedly
editing listsquares.m, which is to embed the co de in a function.
If the version of MATLAB you are running is 7.7 or higher19 the easiest way to do this is to
click on File on the top line of the MATLAB window, followed by New and Function.20 A
new ‘Editor’ window should open containing:
function [ output_args ] = Untitled2( input_args )
%UNTITLED2 Summary of this function goes here
% Detailed explanation goes here
end
Functions. The above code is the bare bones of a function. MATLAB functions are like
mathematical functions or mappings: they take zero or more input arguments, and
pro duce zero or more output arguments. The name of the above function is that given
after the = sign on the first line, i.e. it has the name Untitled2.21
Lines containing %. Anything on a line after a % is interpreted as a comment, and is nearly
always ignored by MATLAB.22 Inter alia you can use comment lines to describe what a
function and/or program is meant to be doing. Comments can appear anywhere in a line
and are used to make the program clearer for reading by humans. Even if you wrote the
program yourself, you will still find it easier to understand and debug if you comment it.
end. The line at the end of the function consisting of end indicates the end of the function.
In many cases it is optional.
1 9 Enter ver to discover the version of MATLAB that you are running.
2 0 Or Function M-File or Function M-File. If the version of MATLAB you are running is 7.6 or lower
then read on without opening the editor.
2 1 Depending on what you have done in your MATLAB session it may be called UntitledN for some integer
N.
2 2 One of the exceptions to ‘nearly always ignored’ is the first time that we encounter comment lines. If the
second line of a function called function name is a comment line, then that line, and any others immediately
following it that are comment lines, are output in response to the command help function name.
18
Using the editor modify the function template to read:23function [ Isquares ] = printsquares ( Ilow, Ihigh )
%PRINTSQUARES Function to print the squares of integers
% PRINTSQUARES(Ilow,Ihigh) prints the squares from Ilow to Ihigh in
% steps of one, and returns the answers in the (Ihigh-Ilow+1) x 2
% matrix Isquares
%
Isquares=zeros(Ihigh-Ilow+1,2);
for I=Ilow:Ihigh
Isquares(I,1)=I;
Isquares(I,2)=I*I;
disp([’I = ’ num2str(Isquares(I,1)) ’, I*I = ’ num2str(Isquares(I,2))])
end
%
end
Or replace
disp([’I = ’ num2str(Isquares(I,1)) ’, I*I = ’ num2str(Isquares(I,2))])
with
fprintf(’I = %2g, I*I = %3g\n’,Isquares(I,1),Isquares(I,2))
Compared with listsquares.m we have made a number of changes.
The first line. The first line is the function declaration. The function is called printsquares,
it has two input arguments, namely Ilow and Ihigh, and it has one output argument,
Isquares.
The array Isquares. The output argument, Isquares, is a matrix (or array) into each row
of which will be written a number and its square. Thus, in order to hold all results, the
matrix needs to be size (Ihigh-Ilow+1)×2. The line
Isquares=zeros(Ihigh-Ilow+1,2);
initialises this matrix with zeros by means of setting Isquares equal to an array of
zeros of size (Ihigh-Ilow+1)×2 (enter help zeros and/or doc zeros into the ‘Com-
mand Window’ for information about the zeros function).
The comment lines. The comment lines have been modified to provide [bare-bones] help about
the function.
2 3 The easiest way to do this is to cut-and-paste from listsquares.m. Note that if the version of MATLAB
you are running is 7.6 or lower, then you will need to open a new blank M-file and type in the co de from
scratch.
19
Once you have typed in the above code you need to save your code in a file with the same
name as the function (i.e. printsquares), but with a .m appended. To do this click on File
on the top line of the ‘Editor’ window, followed by Save As.... A new window should appear
with a default File name of printsquares.m. click Save to accept this name. If you look in
the ‘Current Directory’ pane a file printsquares.m should now have appeared.
To test your function return to the ‘Command Window’ and first enter
>> help printsquares
The comment lines at the top of your function should be printed out. Next enter
>> printsquares(1,10);
The result should be the output (or with a few more spaces if you are using fprintf)
I = 1, I*I = 1
I = 2, I*I = 4
I = 3, I*I = 9
I = 4, I*I = 16
I = 5, I*I = 25
I = 6, I*I = 36
I = 7, I*I = 49
I = 8, I*I = 64
I = 9, I*I = 81
I = 10, I*I = 100
If you have made a mistake then, as before, you will need to return to the ‘Editor’ window and
modify your code. Once you have made your corrections save the code by clicking on File on
the top line of the ‘Editor’ window, followed by Save.
The function, as written, can also return the squares in a matrix. To test this enter
>> amatrix=printsquares(1,10);
>> amatrix
Note that we have used amatrix as the name of the matrix in which to store the output from
the function printsquares; we did not have to use the name Isquares. The matrix name
Isquares is said to be only local ly defined within the function.
Next, for your choices of m and n you should check that
>> printsquares(m,n);
produces the results that you expect.
Finally you should try closing the editor window by clicking on File on the top line of the
‘Editor’ window, followed by Close Editor. You can always return to editing a file by double
clicking on the file name in the Current Directory pane.
20
4.4 Exercises
1. Test your function printsquares with m 0 and m>n, and then modify your code to work
with both m 0 and m>n.
Hints.
m 0. In this case you need to ensure that the matrix Isquares does not have
a zero or negative index.
m>n. In this case entering help sign and help abs might suggest a route
forward.
One possible solution to this problem is given in Appendix A on page 43.
2. Write a function, say called printpowers, to display a table listing I, I2, I3 and I4;
Hint: how to edit a file and save it under a new name. If you want to modify an old pro-
gram to produce a new one you do not need to type it all in again. For in-
stance, suppose that you want to create a printpowers function by modifying
your printsquares function. To do this first double click on printsquares.m in
the ‘Current Directory’ pane. An ‘Editor’ window should open up. Make your
modifications within this window including, say, changing the name of the function
from printsquares to printpowers. Once you have done this you will need to
save your changes. As before click on File on the top line of the ‘Editor’ window,
followed by Save As.... A new window should appear with a default File name of
printsquares.m, rather than the printpowers.m desired. click in the box next to
File name and change the entry from printsquares.m to printpowers.m. Then
click Save to accept this name. If you look in the ‘Current Directory’ pane, a file
printpowers.m should now have appeared.
5 Help!
One of the advantages of MATLAB is that there is a plethora of ways of getting help. In §1
on page 1 some links to MathWorks’ own tutorials have been given.
There are several commands in MATLAB to help you get information and find out about your
set-up.
help <function> and helpwin <function>. These provide information about <function>
in the ‘Command Window’ and a new window, respectively.
type <filename>. This displays the contents of the file <filename>. If <filename> is a
built-in MATLAB function, you will be told.
which <function>. This locates functions and files, e.g. which roots tells you whether roots
is a built-in command, a function or doesn’t exist.
lookfor <keyword>. This performs a keyword search, e.g. lookfor empty finds commands
that deal with empty arrays and brie y describes them.
21
helpbrowser and doc. These open a new window to display MATLAB documentation.
doc <function>. This tells you about <function> in a new window.
demo <function>. This command opens the Demos pane in the Help browser, listing demos
for all installed pro ducts, e.g. demo matlab.
who, whos and workspace. who and whos list the current variables (the latter in long form) in
the Command Window. workspace does the same thing in its own pane/window and
provides a GUI (graphical user interface) to manipulate the variables.
why. This command was written either by a Monty Python fan, or by someone who had just
calculated an answer of 42.
path. Prints the current MATLAB path; this is a list of directories. When you type a
<command>, MATLAB lo oks in these directories for the corresponding file.
ver. Tells you which versions of which toolboxes (libraries) are installed.
6 Vectors and matrices
MATLAB has been designed to work e ciently with matrices, including vectors (i.e. matrices
with only one row or one column) and scalars (i.e. a 1 × 1 matrix).
6.1 Creating matrices
There are a number of ways of entering matrices (or arrays).
1. Matrices can be entered explicitly element by element. For instance try the following
commands in the ‘Command Window’:
>> clear
>> rowvec1=[1 2 3 4]
>> colvec1=[4;3;2;1]
>> rowvec2=colvec1’
>> colvec2=rowvec1’
>> rowvec3=-4:1:0
>> colvec3=[0:-1:-4]’
>> A=[1 2 3; 4,5,6; 7 8 9]
>> A’
>> B=[1 sqrt(-1); 1+i 4+3i]
>> C=[’upper line’;’lower line’]
>> C(1,2), C(2,9)
Remarks.
(i) The command clear clears all variables and functions from memory so that you
start with a clean slate.
22
(ii) The command ’ forms the transpose of a matrix.(iii) The construct a:b:c works as in for loops, i.e. it generates a row vector starting
at a in increments24 of b until c is exceeded (c is only included if c-a is a multiple
of b).
(iv) The elements within a row of a matrix may be separated by commas as well as
blanks.
(v) As illustrated by the row vector B, MATLAB understands complex numbers. It was
for this reason that we used I and not i in §4.1. If we had [accidentally] redefined
i it can be reset to the square root of -1 by the command clear i; alternatively
j may be used as the square root of -1 (as long as it has not been redefined).
(vi) Matrices can consist of strings of characters as well as numbers.
(vii) You can refer to the element in row m and column n of a matrix, say C, by C(m,n).
A matrix, or vector, will only accept positive integers as indices, starting from 1.
It is possible to create multi-dimensional arrays, as in this example where a 3 × 4 × 2
matrix is generated element by element:
>> for i1=1:1:3
for i2=1:1:4
for i3=1:1:2
multi(i1,i2,i3)=i1+i2*i3;
end
end
end
>> multi
2. Matrices can be also be formed using one of a number of built-in functions. For instance
try the following commands in the ‘Command Window’:
>> a1=zeros(2)
>> a2=zeros(2,3)
>> b1=ones(2)
>> b2=ones(3,2)
>> c1=eye(2)
>> c2=eye(2,3)
>> c3=eye(3,2)
>> d1=diag(colvec1)
>> d2=diag(rowvec1,1)
>> d3=diag(rowvec2,-2)
>> e1=rand(3)
>> e2=rand(2,3)
>> f=hilb(4)
>> g=magic(5)
Remark. Use the help command to find out more information about these functions,
e.g. help diag.
2 4 Or decrements if b is negative.
23
3. Matrices can also be read in from a file. Open up an empty script file as described atthe start of §4.3.1 (although we are not going to write a script into it), and enter the
following
-3 5 5 6
24 3 -5 0
Next save this as a file tempmat.dat (note the .dat replacing the .m).25 Then in the
‘Command Window’ execute
>> load tempmat.dat
>> tempmat
Remark. You have created a 2 × 4 matrix called tempmat.
As well as loading matrices from a file, it is also possible to save matrices to a file. For
instance try
>> tranmat=tempmat’;
>> save tempmat.dat tranmat -ascii -double
If you double click on tempmat.dat in the Current Directory pane you will see that the
original 2 × 4 matrix has been overwritten with its 4 × 2 transpose.
There are certain restrictions on the matrices than can be written in human-readable, or
ASCII, form using the save command. Complex matrices and large multi-dimensional
matrices (such as multi) have to be saved in binary ‘MAT-file’ form. To illustrate this
try the following:
>> whos
>> save tempvar tempmat tranmat multi
>> clear
>> whos
>> load tempvar
>> whos
Remark. If you wish to tidy up your files you can delete tempmat.dat and tempvar.mat
by right clicking on them, selecting Delete and then clicking on Yes in the pop-up
window.
6.2 Manipulating matrices
We have already seen that we can refer to an element of a matrix by its indices. Rows, columns
and sub-matrices can also be selected from a matrix and manipulated. For example:
2 5 The extension .dat could be anything, but it’s best to avoid .m and .mat.
24
>> D=[ 1 3 5; 2 4 6];>> E=D(2,:) % row 2
>> G=D(:,1) % column 1
>> H=D(:,2:3) % columns 2 and 3
>> J=D(:,[1,3]) % columns 1 and 3
>> K=[G H] % the orginal matrix D
>> disp(D-K) % check
We can do arithmetic with matrices, where the normal matrix rules apply. Hence the addition
and subtraction operators, + and -, only make sense when acting on matrices that have the
same numbers of rows and columns. For example try
>> A=hilb(5)+ones(5)
>> B=hilb(5)-ones(4)
The operator * performs normal matrix multiplication, so the number of columns of the matrix
to the left of * must equal the number of rows of the matrix to the right. We have noted
previously that . before an operator tells MATLAB to carry out the operation elementwise.
Therefore, .* performs element-by-element multiplication for matrices that have the same
number of rows and columns. To illustrate this try
>> A=hilb(5)
>> B=eye(5)
>> disp(A*B)
>> disp(A.*B)
Similarly ^ raises a matrix to a power, while .^ raises each element to a given power. For
illustration try
>> A=[1 2 3; 0 1 2; 0 0 1]
>> disp(A^3)
>> disp(A.^3)
>> disp(3.^A)
‘Division’ needs a little more care. A\B is the matrix division of A into B, which is roughly the
same as A-1B, except it is computed in a di erent way using Gaussian elimination. A/B is the
matrix division of B into A, which is roughly the same as A*B-1, except it is computed using
A/B = (B’\A’)’. If the inverse of a matrix is needed then it can be calculated using the inv
function. To illustrate this try26
>> A=rand(5)
>> B=rand(5) % B will not be the same as A: why?
>> inv(A)
2 6 Note that whereas I was used as a scalar in §4.1, in this example it is defined to be a 5×5 matrix. Hence
the advisable use of clear I.
25
>> inv(B)>> C=A\B-inv(A)*B
>> norm(C)
>> D=A/B-A*inv(B)
>> norm(D)
>> clear I; I=eye(5)
>> norm(A\I-inv(A))
Similarly there are element-by-element operators A.\B and A./B. As an illustration try
>> A=hilb(3)
>> B=ones(3)
>> I=eye(3)
>> A.\B
>> A./B
>> A.\I
>> A./I
7 A few more functions
We have already encountered a number of functions, e.g. disp, fprintf, zeros, ones, eye,
diag, rand, hilb, magic, inv and norm. MATLAB has hundreds of other predefined functions
to perform many mathematical, graphical and other operations. Further, many more functions
are provided in the optional ‘toolboxes’ (enter ver to see which toolboxes are available to you),
and even more functions are freely available on the web. Below we list a few more mathematical
functions that [primarily] act on scalars, vectors and matrices.
7.1 Scalar functions
The following functions essentially operate on scalars, but operate element-by-element on ma-
trices:
Function Description Function Description
cos cos (angle in radians) acos inverse cos
sin sin (angle in radians) asin inverse sin
tan tan (angle in radians) atan inverse tan
exp exponential atan2 2-argument form of inverse tan
cosh hyperbolic cos acosh inverse hyperbolic cos
sinh hyperbolic sin asinh inverse hyperbolic sin
tanh hyperbolic tan atanh inverse hyperbolic tan
log natural log rem remainder after integer division
log10 base 10 log fix round towards 0
abs absolute value floor round down to the nearest integer
sign sign (either -1 or +1) ceil round up to the nearest integer
sqrt square root round round to the nearest integer
26
7.2 Vector functionsThere are other MATLAB functions that operate on row or column vectors; most also act on
matrices column-by-column to pro duce a row vector containing the results of each column. A
few of these functions are
Function Description
all true if all elements of a vector are nonzero
any true if any element of a vector is a nonzero number or is logical 1
max largest element
mean mean value
median median value
min smallest element
prod product of elements
sum sum of elements
It follows that the largest entry in a matrix A is given by max(max(A)) rather than max(A).
Alternatively, max(A(:)) can be used since A(:) reshapes matrix A as a single-column vector.
7.2.1 Exercise
Using the function max find the largest entry of the matrix multi defined on page 23.
7.3 Matrix functions
There are yet other MATLAB functions that perform common operations on matrices; the
table below lists a small subset.
27
Function Descriptioncond Condition number with respect to inversion
det Matrix determinant
diag Create or extract diagonals
eig Eigenvalues and eigenvectors
expm Matrix exponential
full Convert a sparse matrix to a full matrix
inv Matrix inverse
lu LU matrix factorization
norm Vector and matrix norms
null Null space
poly Characteristic polynomial
rank Rank of matrix
rref Reduced row echelon form
size Size of matrix
sparse Convert a matrix to sparse form
svd Singular value decomposition
trace Sum of diagonal elements
tril Lower triangular part of matrix
triu Upper triangular part of matrix
8 Program ow control
The for construct is one of several control structures available for programming in MATLAB.
In this section we will describe some of the other control structures.
8.1 The if-else control structure
The if–else control structure allows you to execute di erent sets of instructions depending on
whether a test condition, or expression, is true or false. The general form of the statement is
if <expression>
<statements>
elseif <expression>
<statements>
else
<statements>
end
The else and elseif parts are optional, and multiple elseif clauses are allowed.
As an example, consider the following function that uses real arithmetic27 to solve a quadratic
equation, and gives real or complex solutions as appropriate.
2 7 As noted earlier MATLAB supports complex arithmetic, but as an example we cho ose to restrict ourselves
to real arithmetic.
28
function solvequad( a, b, c )
%SOLVEQUAD Solves a quadratic equation using real arithmetic
%
% SOLVEQUAD(a,b,c) solves the quadratic equation a*x*x+b*x+c=0
%
if abs(a*c) <= eps*b*b
disp(’a*c is too small compared with b*b’)
return
end
%
disc = b*b-4*a*c;
if disc >= 0
disc = sqrt(disc)/(2*a);
r = -b/(2*a) + disc;
fprintf(’Solution 1 is %14.4g\n’,r)
r = -b/(2*a) - disc;
fprintf(’Solution 2 is %14.4g\n’,r)
else
re = -b/(2*a);
im = sqrt(-disc)/(2*a);
fprintf(’Solution 1 is %14.4g + %14.4g i\n’,re,im)
fprintf(’Solution 2 is %14.4g - %14.4g i\n’,re,im)
end
end
The first if control structure. abs(a*c) calculates the absolute value of a*c (i.e. |a*c|), while
eps is a MATLAB positive constant that is set to the spacing between two successive
oating point numbers as approximated in the computer. If abs(a*c) is too small
compared with b*b then rounding errors can be important.
The first if control structure tests whether abs(a*c) is less than or equal to eps*b*b.
If this test is true then an informative message is printed out using disp, and control
is ‘returned’ to the program which called the function (i.e. execution of the function is
terminated).
The second if control structure. Immediately before this control structure the variable disc
is set equal to the discriminant b*b-4*a*c. The if–else control structure then tests the
value of disc to decide whether to calculate real or complex solutions.
disc greater than or equal to zero. If disc is greater than or equal to zero, then the lines
disc = sqrt(disc)/(2*a);
r = -b/(2*a) + disc;
fprintf(’Solution 1 is %14.4g\n’,r)
r = -b/(2*a) - disc;
fprintf(’Solution 2 is %14.4g\n’,r)
are executed. The expression sqrt(disc) computes the square root of disc (see
§7 for more details on functions). Then following some more arithmetic, the real
29
roots are written to the display. The format string ‘%14.4g’ in fprintf tells thecomputer to output the real variable in a field fourteen characters (or positions)
wide with four decimal places of accuracy.
disc less than zero. If disc is negative then the lines
re = -b/(2*a);
im = sqrt(-disc)/(2*a);
fprintf(’Solution 1 is %14.4g + %14.4g i\n’,re,im)
fprintf(’Solution 2 is %14.4g - %14.4g i\n’,re,im)
are executed instead, and the complex roots are written to the display.
Logical expressions. The logical expressions used in conditional statements are often construc-
ted using the following relational operators:
Operator Operation
== equal to (=)
~= not equal to (=)
> greater than (>)
< less than (<)
>= greater than or equal to ( )
<= less than or equal to ( )
Conditional statements can also be constructed with the following logical operators (see
also help relop and page 34):
Operator Operation
&& Short-circuit logical AND
|| Short-circuit logical OR
& Element-wise logical AND
| Element-wise logical OR
~ Logical NOT
xor Logical EXCLUSIVE OR
any True if any element of vector is nonzero
all True if all elements of vector are nonzero
Further help. Further details of the if–else statement can be found by entering
>> help if
or
>> doc if
8.1.1 Exercises
(i) Type the above function into a file solvequad.m. Run the function for various choices of
a, b and c, e.g. solvequad(1,1,1).
30
(ii) Mo dify the above code to return the two solutions as output arguments. Compare your
answer with the output of the MATLAB command roots, which you can find out more
about using help roots.
Hints.
(a) The MATLAB command roots accepts a vector argument of the form [a b c];
e.g. try
>> a=1; b=1; c=1;
>> solvequad(a,b,c)
>> roots([a b c])
(b) To learn more about vectors, matrices and arrays use MATLAB’s extended do cu-
mentation. To do this click on ‘Help’ on the top line of the MATLAB window, then
click on ‘Product Help’. In the new window that opens there is a search box towards
the top left-hand corner. Enter ‘creating and concatenating matrices’ into this
search box, and then press ‘return’ to start the search. Documentation will appear
in the main pane.
(iii) Check if the two roots are the same and, if so, only output one of them. Note that
because the computer only stores real numbers approximately (with a relative error of
eps), you will find that checking if a real number is equal to zero will not work — you
will need to use a test that checks if a real number is ‘close’ to zero.
8.2 The while control structure
A while lo op repeats statements an indefinite number of times while an expression is true.
The general form of a while statement is:
while <expression>
<statements>
end
• The break statement can be used to terminate the lo op prematurely, i.e. to force a break
to the statement immediately following the end.
• The continue statement can be used to pass control to the next iteration of a for or
while loop in which it appears, skipping any remaining statements in the body of the
for or while loop.
A simple example which writes out the factors of a natural number is the program:
31
function printfactors( intnum )I = int32(intnum);
J = 1;
while J <= I
k=I/J;
if k*J == I
fprintf(’%4d\n’,J)
end
J=J+1;
end
fprintf(’ are factors of %d\n’,I)
end
The int32 function. The statement
I = int32(intnum);
rounds the [ oating point] number intnum to the nearest signed 32-bit integer. The value
of a signed 32-bit integer can be in the range -231 to (231 - 1).28
The arithmetic operations +, -, *, ˆ and / are defined for 32-bit integers, where the
divide operator / rounds to the nearest integer. Arithmetic operations involving both
oating-point numbers and 32-bit integers result in 32-bit integers. Hence the statement
k=I/J;
results in a 32-bit integer (since I is a 32-bit integer).
The while loop. The loop starts with the word while. Then comes the loop test condition
J <= I, which is true if J is less than or equal to I and false otherwise. Then comes a
set of [indented] statements, namely
k=I/J;
if k*J == I
fprintf(’%4d\n’,J)
end
J=J+1;
If the loop test condition is true these statements are executed. The the loop test condi-
tion is then re-evaluated and if true the statements are executed again. This sequence of
‘test and re-execution’ is repeated until the loop test condition becomes false; then the
loop ends and execution continues with the first statement after the loop.
A single equals sign (=) means ‘set equal to’. The statement J=J+1 means ‘set the value of J
to its old value plus 1’; it does not mean that J is equal to J+1.
The factor test. The statement
if k*J == I
2 8 Within MATLAB you can find the range using intmin(’int32’) and intmax(’int32’). Values outside
the range -23 1 to (23 1 - 1) are said to ‘saturate on over ow’, namely they are mapped to -23 1 or (23 1 - 1).
32
tests whether k multiplied by J is equal to I. This is true if J is a factor of I, in whichcase J is printed.
8.2.1 Exercise
Change the printsquares function in §4.3 to use a while loop in place of the for lo op. There
are advantages and disadvantages in using each of these control structures (e.g. for lo ops are
particularly useful when manipulating array and string variables). Try to think of an example
that would be simpler to write using a while loop than a for loop.
8.3 The switch-case control
The if–else pair is often the best way of controlling the possible branching in the execution of
a program. Sometimes though switch-case control structure, which switches among several
cases based on an expression, contributes to neater-looking code. The general form of the
switch statement is:
switch <switch_expression>
case <case_expresion>,
<statements>
case {<case_expression1>, <case_expression2>,...}
<statements>
...
otherwise,
<statements>
end
The statements that follow the first case in which the switch expression matches with the
case expression are executed. When the case expression is a succession of alternative ex-
pressions (within braces and separated by commas), as in the second case above, the statements
that follow the case are executed if the switch expression matches any of the alternatives.
If none of the case expressions match with the switch expression then the otherwise
clause is executed (if present). Only one case is ever executed and execution resumes with the
statement after the end statement.29
A simple example, which prints out a di erent message according to an input choice, is the
program:
2 9 Unlike C, the switch statement does not ‘fall through’ from one case to the next, and hence breaks are
unnecessary.
33
function switchcase
%SWITCHCASE A simple example function using the switch statement
%
option=0;
while option ~= 4
disp(’1: Euler’’s Method’);
disp(’2: Leap Frog Method’);
disp(’3: Runge Kutta Method’);
disp(’4: Quit’);
option=[];
while ~isa(option,’numeric’) || isempty(option)
option=input(’Please enter your choice: ’);
end
switch(option)
case 1,
fprintf(’Attempting Euler\n\n’)
% Code to solve ODE using Euler
case 2,
fprintf(’Attempting Leap frog\n\n’)
% Code to solve ODE using LF
case 3,
fprintf(’Attempting Runge Kutta\n\n’)
% Code to solve ODE using RK
case 4,
fprintf(’Exiting function\n\n’)
return
otherwise,
fprintf(’Please choose an integer between 1 and 4\n\n’)
end
end
end
In addition to demonstrating the use of switch, this function also illustrates a number of other
points.
The outer while loop. The outer while loop continues execution until the variable, option,
is found to be equal to ‘4’.
The empty matrix and isempty. [] is the empty matrix. A variable can be tested to see if
it is equal to the empty matrix using the function isempty, which returns true is the
argument is the empty matrix. For more information use help isempty.
The function isa. The function isa(object,’classname’) returns true if object is an in-
stance of ’classname’. In the example program isa is used to test whether or not the
variable option is an integer or a oating point number. The not operator ~ in front of
the call to isa negates the test. For more information use help isa.
34
The logical OR operator ||. The operator || is used between two expressions as in A || B.
The expression A || B is true if A and/or B is true; whereas the expression A && B is
true only if both A and B are true. For more information use help relop.
The function input. The function input(’string’) outputs a character string and then waits
for input from the keyboard. The input can be any MATLAB expression, which is
evaluated, using the variables in the current workspace. If the user presses the return
key without entering anything, input returns an empty matrix. For more information
use help input.
’’ and \n. To output a ’ in a display string we have to enter ’’. As before, to output a
newline in a display string we use \n. For more information use help fprintf.
8.3.1 Exercise
Type in and run the code. Try and break it, e.g. in response to the prompt try 1.2, text, ’a
string’, int32(3), etc.
9 Elementary graph plotting
MATLAB can be used to pro duce a wide variety of plots and curves, including 2D plots, 3D
plots and 3D surface plots.
9.1 The plot command
The plot command creates a linear plot; if x and y are vectors of the same length, the
command plot(x,y) opens a graphics window and draws a plot of the elements of x on the
abscissa versus the elements of y on the ordinate. For instance
>> x=-pi:pi/10:pi
>> y=sin(x)
>> plot(x,y)
plots the sine function between -p and p. Here pi is p, x is a row vector that consists of
real values between -p and p in steps of p/10, while sin(x) calculates the sine of [all] the
elements of x. A smoother graph can be obtained by reducing the step, or mesh, size between
points on the abscissa:
>> x=-pi:pi/1000:pi; y=sin(x); plot(x,y)
However, it is not always possible to plot a smooth graph if you choose the wrong function:
>> x=linspace(-pi, pi, 1000); y=sin(pi*pi./x); plot(x,y)
>> axis([-pi pi -1.5 1.5])
35
Remarks.• The linspace command creates a vector of 1000 elements equally spaced between -p
and p inclusive.30
• Note the use of the element-by-element right array divide, i.e. ‘./’ rather than right
matrix divide, i.e. ‘/’.31
• Once you have a graph it’s easy to add titles. You can either use the menus on the
graphics window (e.g. click on Insert, or enter the following commands into the ‘Com-
mand Window’:
>> title(’The sine function’)
>> xlabel(’radians’)
>> ylabel(’sine’)
• You can plot more than one function on a graph. However, the following does not work
since the first graph is erased.
>> w=cos(x); plot(x,w)
It is first necessary to enter ‘hold on’ to hold the current plot and all axis properties so
that subsequent graphing commands add to the existing graph. Try
>> x=linspace(-pi, pi, 2001); y=sin(x); w=cos(x);
>> plot(x,y)
>> hold on
>> plot(x,w)
>> xlabel(’radians’), title(’The sine and cosine functions’)
• The trouble now is that it is di cult to tell the graphs apart since they are in the same
colour and line style. One way forward is to plot the two lines at one, since then the lines
are automatically drawn in di erent colours. It is then possible to distinguish between
the graphs using the legend command:
>> hold off
>> plot(x,y,x,w)
>> xlabel(’radians’), title(’The sine and cosine functions’)
>> legend(’sine’,’cosine’);
Note. It was necessary to issue a hold off instruction so that the figure could be re-
drawn.
• The plot command has many options to control colour, markers and line style: see
help plot and/or doc plot. Hence to plot the sine graph in a green dashed line and
the cosine graph in a red dash-dot line one can use
3 0 Why choose an even number of elements between -p and p for this particular function?
3 1 A./B denotes element-by-element division. A and B must have the same dimensions unless one is a scalar.
A scalar can be divided with anything.
36
>> plot(x,y,’g--’,x,w,’r-.’)>> xlabel(’radians’), title(’The sine and cosine functions’)
>> legend(’sine’,’cosine’);
• It is possible to add or remove a grid using grid on or grid off respectively.
• By means of the menu options on the top line of the Figure window it is possible to
change many features of graphs. Alternatively you can use the axis command to change
many axis features: range, aspect ratio, etc.
9.1.1 Exercise
Plot the sine and cosine functions with a large number of points (e.g. 2001). Then try
>> hold on;
>> v=tan(x);
>> plot(x,v)
>> title(’The sine, cosine and tangent functions’)
>> xlabel(’radians’); legend(’sine’,’cosine’, ’tan’);
What has happened is that the ordinate has been automatically rescaled. Using help axis,
or otherwise, choose axes to obtain a sensible plot.
9.2 Other 2D graphs
There are many other 2D graphical tools available in MATLAB, e.g.
Function Description Function Description
loglog Log-log scale plot semilogx Semi-log scale plot
semilogy Semi-log scale plot polar Polar co ordinate plot
fplot Plot function barh Horizontal bar graph
bar Bar graph comet Comet-like tra jectory
compass Compass plot ezplot Easy to use function plotter
errorbar Error bar plot ezpolar Easy to use polar coordinate plotter
feather Feather plot fill Filled 2-D polygons
area Filled area plot stem Discrete sequence or ‘stem’ plot
hist Histogram rose Angle histogram plot
pareto Pareto chart pie Pie chart
scatter Scatter plot plotmatrix Scatter plot matrix
stairs Stairstep plot
To find out more use help graph2d or help specgraph (or alternatively doc graph2d or
doc specgraph). The MATLAB graphics tutorial is also helpful: browse to
http://www.mathworks.com/access/helpdesk/help/techdoc/learn_matlab/learn_matlab.shtml
37
and click on Graphics.One of the specialized tools provided is the command fplot, which plots the graph of a
function. For instance we could have plotted the sine function with
>> clf
>> fplot(@sin,[-pi pi])
Remarks.
(i) clf has been used to clear the current figure.
(ii) @sin is a function handle to the function sin. See help function handle to learn more.
(iii) The command fplot(@(x)sin(x),[-pi pi]) is an equivalent alternative.
(iv) fplot has a number of options (see help fplot). Like plot it is possible to plot more
than one graph at a time, e.g. try
>> fplot(@(x)[sin(x),cos(x),tan(x)], 2*pi*[-1 1 -1 1])
9.2.1 Exercise
Work out what the following commands plot:
>> clf, clear
>> f = @(x,n)abs(exp(-1i*x*(0:n-1))*ones(n,1));
>> fplot(@(x)f(x,10),[0 2*pi])
Remarks.
(i) Here, in addition to clearing the current figure, we have also cleared all variables and
functions from memory.
(ii) Note the use of i as the square root of -1: see help i.
(iii) f is an anonymous function: see doc function handle.
9.3 Multiple figures and plots
It is possible to have more than one Figure window open at any one time by using the figure
command (see help figure). Suppose figure 1 is the current figure, then the command fig-
ure(2) (or simply figure) will open a second figure (if necessary) and make it the current figure.
The command figure(1) will then expose figure 1 and make it again the current figure. The
command gcf will return the number of the current figure. Try the following
38
>> figure(1); fplot(@sin,[-pi pi]); hold on>> figure(2); fplot(@cos,[-pi pi])
>> figure(1); fplot(@cos,[-pi pi])
Remark. Figure n can be cleared or closed with clf(n) or close(n), respectively.
It is also possible to have many graphs in each figure by using the subplot command to create
axes in tiled positions (see help subplot). So for example if you want 3 rows each containing
2 graphs, and you want to plot sin(x), cos(x), sin2(x), cos2(x), sin3(x) and cos3(x) in the six
graphs, you enter
>> subplot(3,2,1); fplot(@sin,[-pi pi],’b-’)
>> subplot(3,2,2); fplot(@cos,[-pi pi],’g-.’)
>> subplot(3,2,3); fplot(@(x)sin(x)^2,[-pi pi],’r--’)
>> subplot(3,2,4); fplot(@(x)cos(x)^2,[-pi pi],’c-’)
>> subplot(3,2,5); fplot(@(x)sin(x)^3,[-pi pi],’m-.’)
>> subplot(3,2,6); fplot(@(x)cos(x)^3,[-pi pi],’k--’)
When it comes to writing up your projects, it is often clearer to include more than one graph
on a page. If appropriate you are advised to do so.32
9.4 Saving your figures
It is possible to save your figures to a file (or to send them to a printer) using the print
command. The format in which the figure is saved is specified by an option:
Option Format
-dpdf Portable do cument format
-dps PostScript for black and white printers
-dpsc PostScript for color printers
-deps Encapsulated PostScript
-depsc Encapsulated Color PostScript
-dpng Portable Network Graphic
-djpeg<nn> JPEG image, quality level of nn; quality level defaults
to 75 if nn is omitted.
For later inclusion into a LaTeX document print -depsc <filename> (colour encapsulated
postscript) or print -dpdf <filename> (portable document format) might be best. For Word
documents, try print -depsc -tiff <filename>, which adds a TIFF preview.
3 2 Submitting reams of output with only one graph per page may not be the best way to endear yourself to
the person marking your pro jects.
39
9.4.1 ExerciseSave one your graphs in pdf format in a file and open it using an external viewer (e.g.
acroread). Then issue the command orient tall, save the graph again, and view it again
(use help orient or doc orient to discover the reason for the change).
9.5 3D graphs
There are many 3D graphical tools available in MATLAB. You can find out more using
help graph3d and help specgraph (or doc graph3d and doc specgraph).
As an example, the following code draws the surface
z = x2 + y2 ,
over the region -1 x 1, -1 y 1:
>> n=41
>> x=linspace(-1, 1, n);
>> y=linspace(-1, 1, n);
>> [X,Y]=meshgrid(x,y);
>> Z=X.^2-Y.^2;
>> surf(X,Y,Z)
Remarks.
(i) The larger the value of n the smoother the surface.33
(ii) The meshgrid command creates two 2D matrices, each with as many elements as there are
grid-points. The X matrix has the x-coordinates and the Y matrix the y -coordinates. The
Z matrix contains the ‘height’ of the [single-valued] function defined over the rectangular
grid.
(iii) In calculating Z note the use of .^, to square each element of the matrices X and Y, rather
than the use of ^ (which would have squared the matrices not the elements).
(iv) The colouring of the surface is proportional to surface height. A key to the colours can
be added with
>> colorbar
The type of shading can be modified with
>> shading interp
(v) The command mesh(X,Y,Z) could have been used to draw a wireframe mesh instead of
a surface.
3 3 With earlier versions of MATLAB on the PWF there was a ‘bug’ with the result that before setiing n 16
it was necessary to issue the command opengl neverselect.
40
10 Random number generation
The simplest way to obtain a pseudo-random number uniformly distributed in (0, 1) in MAT-
LAB is with the command rand. Calling this with 2 arguments, rand(m,n), returns an m × n
matrix of uniformly distributed pseudo-random numbers in (0, 1). These are random in the
sense that there should be no measurable correlations between successive numbers, but are
pseudo-random in the sense that a deterministic algorithm generates these numbers.
Introducing random numbers into any code adds an interesting wrinkle when trying to debug
that co de. Imagine wanting to check that the code produces the same results before and after
making some small change. However, simply calling rand introduces an unrepro ducible element
to the code which precludes this kind of check. We can fix this by using the deterministic nature
of pseudo-random numbers. We use a random number stream initiated with a user-given seed.
That is, we assign a variable which we pass to rand which defines an initial condition for the
pseudo-random number generator and keeps track of how many numbers have been generated.
>> seed = 1234; % determines starting point in series of random numbers
>> rns = RandStream(’mt19937ar’, ’seed’, seed);
>> randomarray = rand(rns,1,10); % array of 10 random numbers
Remarks.
(i) You can change seed when you want to use a di erent string of pseudo-random numbers.
(ii) Using doc RandStream you can read more in the MATLAB help pages about the gener-
ation algorithms; mt19937ar seems like a reasonable default.
(iii) You can pass variables of type RandStream to functions you write in order to ensure the
whole code uses a single random number stream.
(iv) You can define separate random number streams for use in di erent parts of the code.
11 This and that
11.1 Programming style
As has already been mentioned, when writing your programs it’s goo d (some would say essen-
tial) to include informative comments, and it helps if they are readable (e.g. by sensible use of
indentation).
What constitutes the best programming style is debatable, and to some extent is a theology.
However, before you start to program in earnest it is probably sensible to read at least one
article. As of June 2010 the article in Wikipedia is grouped around the headings
1 Elements of good style
1.1 Code appearance
41
1.1.1 Indentation1.1.2 Vertical alignment
1.1.3 Spaces
1.1.4 Tabs
1.2 Naming, logic, and higher techniques
1.2.1 Appropriate variable names
1.2.2 Boolean values in decision structures
1.2.3 Left-hand comparisons
1.2.4 Looping and control structures
1.2.5 Lists
and is not bad: see
http://en.wikipedia.org/wiki/Programming_style
11.2 Some terminology
Inf. The IEEE (Institute of Electrical & Electronics Engineers) representation of positive
infinity.
NaN. The IEEE representation of ‘Not-a-Number’, used when an operation has an undefined
result (e.g. 0/0). All logical operations involving NaN return false, except for =. If you
try to plot points with the value NaN nothing appears.
3 4 See http://www.eng.cam.ac.uk/help/tpl/programs/matlab.html.
42
A A generalised printsquares codeThere is more than one way to skin this cat, but one answer is35
function [ Isquares ] = printsquares ( Ilow, Ihigh )
%PRINTSQUARES Function to print the squares of numbers
%
% PRINTSQUARES(Ilow,Ihigh) prints the squares from Ilow to Ihigh in
% steps of one
%
Ilow=round(Ilow); Ihigh=round(Ihigh);
%
% Find how many squares to print
range=abs(Ihigh-Ilow)+1;
%
% Set up a matrix of the correct size to store the results
Isquares=zeros(range,2);
%
% Ensure that the matrix indicies are all strictly positive
for I=1:range
Isquares(I,1)=Ilow+sign(Ihigh-Ilow)*(I-1);
Isquares(I,2)=Isquares(I,1)*Isquares(I,1);
fprintf(’I = %2g, I*I = %3g\n’,Isquares(I,1),Isquares(I,2))
end
%
end
Note that we have forced Ilow and Ihigh to be integers by using round (for more information
enter help round).
3 5 This code, or an approximation to it, can be downloaded from
http://www.maths.cam.ac.uk/undergrad/catam/MATLAB/manual/printsquares.m.
43
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