Matlab
offers extensive facilities for displaying vectors and matrices as graphs. In
this section we will describe a few of the most important graphics functions and
provide examples.
5.1 Creating a Plot
The
plot function can be used in different ways, depending on the input argument(s).
If y is a vector, plot(y) produces a piecewise linear graph of the elements of y
versus the index of the elements of y. If we specify two vectors as arguments,
plot(x,y) produces a graph of y versus x.
For
example, here is how to plot the cosine function from 0 to 2p.
» theta=0:pi/100:2*pi;
» y=sin(theta);
» plot(theta,y) % Check the
plot on your screen
You can
easily create multiple graphs with a single call to plot function by using
multiple x-y pairs. For example, here is how to plot sine and cosine functions
together over 0 to 2p.
» theta=0:pi/100:2*pi;
» y1=sin(theta);
» y2=cos(theta);
» plot(t,y1,t,y2)
title
command adds a title text of your choice, to the top of the current axis as your
graph’s title. The title text must be enclosed in single quotes.
» title('sine(t) and cosine(t)')
xlabel('text') and ylabel('text') commands add label texts beside the x and
y-axes respectively, on the current axis. If you need to use a single quote sign
' in the title or label texts you should put it twice.
» xlabel('Angle ''Theta''')
» ylabel('Amplitude')
The
hold on command holds the current plot and so that you can overlay several
plots. The hold off returns to the default mode whereby plot commands erase the
previous plots before making new plots.
The
grid on command adds grid lines to the current axes and the grid off command
takes them off.
Both
hold and grid commands without any arguments toggle the hold and grid states
respectively. To explore these functions, issue the following lines one by one
and see what happens.
» x=1:20;
» plot(x,x.^2)
» hold on
» plot(x,2*x)
» grid on
The
subplot function allows you to display multiple plots in the same figure window.
Typing subplot(m,n,p) breaks the figure window into an m-by-n matrix of small
subplots and selects the pth subplot for the current plot. The plots are
numbered along first the top row of the figure window, then the second row, and
so on. For example, we can make two plots in two different subregions of the
same figure window as follows.
» theta=0:pi/100:2*pi;
» subplot(2,1,1);plot(sin(t))
» subplot(2,1,2);plot(2*sin(t)+cos(t))
In this section of the
tutorial, we will show how “for” and the “while” loops are used in Matlab.
First, we discuss the “for loop” with examples for row operations on matrices
and for Euler's Method to approximate an ordinary differential equation (ODE).
Following the “for loop”, we will see the use of “while loop” in an example.
For
loops allow us to repeat a group of commands. If you want to repeat some action
in a predetermined way, you can use a “for loop”. As a “for loop” will have
Matlab loop around some statements, and you must tell Matlab where to start and
where to end. All of loop structures in Matlab start with a keyword such as
“for” or “while” and they all end with the keyword “end”. Here is the format of
the “for loops”.
for loopvar = expression,
statement, ..., statement
end
In the
“for” statement or loop, Matlab will loop through for each possible value of the
variable loopvar. For example, here is a simple loop that Matlab will go around
3 times.
» for j=1:3,
j
end
j =
1
j =
2
j =
3
As
another example, let us define a vector and later change its entries in a for
loop.
» v=1:3:10;
v =
1 4 7 10
» for j=1:4,
v(j)=j;
end
» v
v =
1 2 3 4
NOTE: Since Matlab is
an interpreted language, for loops get executed very slowly. Therefore, if speed
is a constraint on your Matlab code, try to avoid using “for loops” where
possible.
Yet
another example is below, in which we perform an operation on the rows of a
matrix repeatedly. We want to start at the second row of the matrix and subtract
the previous row from it and then repeat this operation on the following rows. A
simple “for loop” will do the job for us.
» A=[[1 2 3 4]' [3 2 1 4]' [2
1 3 4]']
A =
1 3 2
2 2 1
3 1 3
4 4 4
» for j=2:4,
A(j,:) = A(j,:) -
A(j-1,:); % Suppressing output
end
» A
A =
1 3 2
1 -1 -1
2 2 4
2 2 0
Another
example where loops come in handy is the approximation of differential
equations. The following example approximates the solution of differential
equation
using Euler's method over an
interval from 0 to 2. (Actually, this is an initial value problem.) Euler's
method simply uses the following approximation to solve such a differential
equation.
First,
we define the step size, h, then we find the grid points, and do an
approximation using a for loop. The approximation is simply a vector, y, in
which the entry y(j) is the approximation to function y(t) at x(j).
» h=0.1;
» x=0:h:2;
» y=0*x; %
Initialization of y (Filled with zeros)
» y(1)=1; % Initial
value
» size(x)
ans =
1 21
» for i=2:21,
y(i) = y(i-1) +
h*(x(i-1)^2 - y(i-1)^2);
end
» plot(x,y)
» plot(x,y,'go')
» plot(x,y,'go',x,y)
The
“while loop” has the following format.
while expression
statement, ..., statement
end
As long
as the expression returns a result whose real part has all non-zero elements,
the statements are executed. The expression is usually includes relational
operators such as ==, <, >, <=, >=, or ~=. For information about these
operators, type help relop at the Matlab prompt.
To
illustrate the use of while loops, we will find an approximate solution for the
following differential equation
by again using Euler’s
method.
» h = 0.001; % A small
step size
» x = 0:h:2;
» y = 0*x;
» y(1) = 1;
» i = 1;
» size(x)
ans =
1 2001
» max(size(x))
ans =
2001
» while(i<max(size(x)))
y(i+1) = y(i) + h*(x(i)-abs(y(i)));
i = i +
1;
end
» plot(x,y,'go')
» plot(x,y)
In this
section, we will introduce you the basic operations for creating script or batch
files in Matlab. Once you formulate specific tasks that you want to carry out in
Matlab, you can group them in a file to identify them as a routine that you can
use whenever you need. These Matlab script files are commonly referred as
M-files and they must have “.m” as their extension to be properly identified by
Matlab.
Suppose, you have developed a set of instructions to plot the cosine function
over a certain interval, but later you have realized that you may use the same
set of instructions to plot the cosine function over different intervals. Let us
see how we can create a simple mfile to implement this idea. First, you will
need to create the file. You can use any text editor to create the file but we
prefer to use a very functional Editor/Debugger that comes with Matlab. This
editor allows you to create m-files, to do file manipulations and to debug your
programs. You can start this editor in two ways. At Matlab command prompt, issue
edit command or on the Matlab toolbar, just click on the “new file icon”. If you
want edit an already existing m-file, either use edit filename or click on the
“open file” icon on the Matlab toolbar and select your file.
Once
the editor appears on the screen, you can either type or copy and paste some
commands from Matlab command window into your m-file.
In our
sample m-file below, we first created an angle vector th, having values from 0
to x, and then we took the cosine of that vector and plotted it. After we were
done with typing/editing, we saved this file as “mycos.m”, under a directory,
let us say, C:\Myfiles.
% file: mycos.m
%
% This Matlab file will plot cos(th) function
from 0 to x
% To run this file you must specify the final
value, x
th = 0:pi/100:x;
y = cos(th);
plot(th,y);title('cosine(x)')
To have
Matlab execute commands in our m-file, we simply type its name cosx at Matlab
prompt. However, we must make sure that Matlab can find our m-file in advance.
We can do this by either switching to the directory where tour m-file resides,
by using the change directory, cd, command, or by adding the directory of our
file to Matlab’s search path. Both of these techniques are shown below.
» cd C:\Myfiles
» mycos
» path(path,'C:\Myfiles')
» mycos
If you
call the m-file without first defining the variable x, you will get an error
message. You must first specify all of the variables that are not defined in the
file itself.
» x=pi;mycos
Writing m-files is not the
only way to prepare programs that Matlab can run, we can also write Matlab
functions. Most of the time, functions are more advantageous over m-files. When
we pass a variable to a function as an input argument, its content and will not
change. Functions return their results or outputs into variables of our choice.
Also, a function may run much faster than an mfile doing the same job.
The
commands and functions that will form a new function must be put in a file whose
name defines the new function, with a filename extension of “.m”. At the top of
the file there must be a line that contains the syntax definition for the new
function. For example, let us write our “cosine plotter program” in function
form.
function mycosfun(x)
% file: mycosfun.m
% This Matlab function will plot cos(th) from
0 to x
% Input vars : x, the final value
% Output vars: none
th = 0:pi/100:x;
% th is a local variable, it will get lost
after
% the functions returns
y = cos(th);
plot(th,y);title(‘cosine(x)’)
8.1 Help Utilities of
Matlab
Matlab offers extensive help
through several facilities. These are:
·
lookfor
This utility lets you search
all m-files in Matlab’s search path for a keyword, so that you can locate Matlab
function(s) that you can use for a specific task. Here is an example.
» lookfor cosine
ACOS Inverse cosine.
ACOSH Inverse hyperbolic
cosine.
COS Cosine.
COSH Hyperbolic cosine.
TFFUNC time and frequency
domain versions of a cosine modulated Gaussian pulse.
·
help
This
utility lets you reach the on line help service of Matlab.
help,
by itself, lists all primary help topics. Each primary topic corresponds to a
directory name on Matlab’s search path or Matlabpath (for more information about
Matlabpath, type help path at the Matlab prompt.)
help
topic gives help on the specified topic. The topic can be a command/function
name or a directory name. If the topic is a command/function name, help displays
information about that command/function, like its syntax, behavior, examples of
usage, etc. If the topic is a directory name, help displays the table of
contents for the specified directory.
» help cos
COS Cosine.
COS(X) is the cosine of
the elements of X.
NOTE: In the online
help, keywords are capitalized to make them stand out. Always type commands in
lowercase in order Matlab since all command and function names are actually in
lowercase.
For
more information about the help utility, type help help at Matlab prompt.
·
The Matlab
Help Desk
This is an online help
system that you can browse through using your regular web browser. The Matlab
Help Desk provides access to a wide range of help and reference information
stored on a disk or CD-ROM in your local system. The Help Desk can be started by
selecting Help Desk option under the “Help” menu, or by typing helpdesk at the
prompt.
At the
end of tutorial, we are presenting you with some website links and books that
your may find useful.
8.2 WWW Sites on Matlab
1.
Matlab’s official web site:
http://www.mathworks.com
2.
Useful Matlab M-Files and Toolboxes:
http://www.mathtools.net
3.
An extensive Matlab FAQ by Mathworks:
http://www.mathworks.com/FAQ/faq.html
4.
University of Florida Matlab Primer:
http://www.cis.yale.edu/secf/software/Matlab/Matlab-primer.html
5.
City University of Hong Kong, A Beginner's Guide to Matlab:
http://www.image.cityu.edu.hk/Matlab/Matlab-b.pdf
6.
Useful Matlab links and materials:
http://www.phys.ualberta.ca/~kbeaty/Matlab_help/weblinks.htm
7.
Matlab Links on the web by Malgorzata S. (Gosha) Zywno:
http://www.ryerson.ca/~ele749/general/Matlablinks.html
8.3 Books about How to
Use Matlab
All from Prentice-Hall,
http://www.prenhall.com/divisions/esm/catalog.html
1.
Mastering Matlab, A Comprehensive Tutorial and Reference, by D. Hanselman
and B. Littlefield, 1996.
2.
The Student Edition of Matlab, by The MATHWORKS, Inc., 1996.
3.
Engineering Problem Solving With Matlab, by D. Etter, 1997.
4.
Matrices and Matlab: A Tutorial, by M. Marcus, 1993.
5.
Matlab Project Book For Linear Algebra, R.L. Smith, 1997.
6.
ATLAST Computer Exercises for Linear Algebra, S. Leon, et.al, 1996.
