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1) issue MATLAB commands in one,
2) see the results of graphics in another,
3) create and change MATLAB programs in others.
The User's Guide for MATLAB is quite well written and contains
both a Tutorial and a Reference section. Both are quite brief so that
until you learn the basic procedures and the names of a lot of
functions, you will find that you have to spend considerable time
looking for ways to use the language. These notes were written to
help students get the background to understand how to do many of the
basic operations, but you will find many other features in the Study
Guide and a description of most of the functions available to help
you
This chapter will demonstrate the following MATLAB procedures.
2.1 Discussion
2.2 Entering MATLAB and Using help and demo
2.3 Defining Arrays and Arithmetic Operations with them
2.4 Saving and Printing your Work
2.5 Matrix Operations with Arrays
2.6 Element by Element Operations on Arrays
2.7 Plotting Results
2.8 Find Roots of Equations: Using Polynomial Approximations
To get into MATLAB you simply open the main menu in the gray part of
the screen with the right mouse button. Darken the MATLAB bar in the
menu by pointing to it with the right button held down and let go of
the button. If you want your MATLAB session to access a directory of
yours other than your home directory, you can change directories
while in MATLAB by using the cd, command. (See help
cd). Files can then be saved to this directory and be loaded or
referenced from it . Entering MATLAB is much simpler to do than to
describe and should become automatic after a few sessions.
Immediately after opening MATLAB, a MATLAB Command Window, should
appear for you to work in.
You will get a message that includes information like:
To get started, type one of these commands: helpwin, helpdesk, or demo. For information on all of the MathWorks products, type tour.
The demo command shows how to use a variety of Matlab tools
with emphasis on array operations and graphics.
There is one inconsistency in the way a number of things are
presented in MATLAB. Most messages about MATLAB functions (such as
those that you get from HELP) capitalize function names, but MATLAB
is case sensitive in its default state. The computer will only
respond to function names that are typed in lower case. Thus to get
HELP from the helpdesk or to run the DEMO packages, you must type
these as helpdesk, and demo.
>> demo
will allow you to choose maong many MATLAB 5.1 demonstration
programs which construct many interesting and detailed graphs. They
also give a good description of how MATLAB works. You are urged to
try several of these to get some feeling for what can be done with
MATLAB. In particular note how very simple programs can produce
rather sophisticated results.
To get out of MATLAB, type:
>> quit
and all MATLAB windows should disappear.
2.3 Defining Arrays and Arithmetic Operations with
them
MATLAB allows you to define three types of arrays:
1) scalars with a single element,
2) vectors containing ordered lists of elements,
3) matrices with ordered elements in a rectangular arrangement.
The elements stored in the arrays may be numeric or character types. No distinction is made between integer and real numbers. Numeric and character scalars may be defined using commands similar to FORTRAN. The following session will show that MATLAB automatically echoes the values you set unless you end your variable definition with a semicolon.
>> s=1.2
s =
1.2000 <-- s is a numberic vector with one element in it.
>> s=1.2;
>> s
s =
1.2000
>> a='qwerty'
a =
qwerty <-- a is a character string with the characters: qwerty in it
Since the form used to show the value is rather verbose, you will
usually want to end array definitions with a semicolon. If you forget
this and start listing a very long array, simply press CONTROL C to
terminate the listing. You will also find that MATLAB inserts blank
lines to separate commands unless you instruct it to eliminate them
with the command:
>> format compact
As in the second example defining s, you can always see what a
variable's value is by typing its name.
Let's define a few vectors:
>> v=[1 2 3]
v =
1 2 3 <-- v is a numeric vector with three elements
>> vc=['asdf' '123']
vc =
asdf123 <-- vc is a character string with the characters: asdf123 in it
The square brackets must be used in defining numeric vectors and
may be useful in constructing character arrays. A character vector
(or string) consists of any linear arrangement of characters, so
adding more characters takes place by direct concatenation.
Now finally, some matrices:
>> m=[1 2 3
4 5 6]
m =
1 2 3
4 5 6
>> mc=['asdf'
'123 ']
mc =
asdf
123
Suppose you want to see the names of the variables that you have created:
>> who Your variables are: a m n v demos mc s vc
Some of these names are familiar: a, m, mc, v, and
vc, were created in the sessions shown here. The variables
demos, and n, must have been created when the
demo command was tried.
We will show some of the more important
properties of arithmetic operations with arrays, in the following
session: First, a short way to get a vector of numbers of the
form:
a, a + 1, a + 2, . . . b - a, b
then some operations on that vector to show how vector functions make life easy in MATLAB. Set t as a vector with elements at a regular spacing. Choosing:
a=0, interval=0.5, b=2
>> t=0:.5:2
t =
0 0.5000 1.0000 1.5000 2.0000
>> 2*t
ans =
0 1 2 3 4
>> f=sin(t)
f =
0 0.4794 0.8415 0.9975 0.9093
2.4 Saving and Printing your Work
If you run out of time to finish a problem in MATLAB, you should save
the workspace by typing:
>> save
which produces a file called matlab.mat., This may then be retrieved by:
>> load
when you can continue work on the problem. If you have several problems that you want separate workspaces for, simply give them different names as in:
>> save prob1
which can be retrieved by:
>> load prob1
If you want to start over with a fresh workspace, type:
>> clear
If you want to get rid of only a few variables in your active
workspace, give the names of the variables to be deleted after the
clear, command. You can also save only a few of the variables in a
named workspace by listing the names of those variables after the
workspace name in the save command.
Note that when you use the save, command, you create a binary file
that can not be listed or edited or otherwise used in another
environment. If you have a long numeric array that you want to save
so that it can be used in another environment, save that variable
with the command:
>> save name.dat name -ascii
This will create a file called name.dat, with the data that was in
the MATLAB variable name, stored so that it can be listed, edited or
used elsewhere as any other ASCII file. It can also be loaded back
into MATLAB to create the numeric array with the same data in it or
with new data if the file has been edited.
Here is an example where the variable xm, is saved, then the
resulting file is edited to add more data. In MATLAB the array listed
as:
xm =
12 15 17 19
2 -3 -5 -15
100 10000 1000000 100000000
It was saved by:
>> save xm.dat xm -ascii
to produce the file xm.dat, that lists as:
wsname% cat xm.dat 1.2000000e+01 1.5000000e+01 1.7000000e+01 1.9000000e+01 2.0000000e+00 -3.0000000e+00 -5.0000000e+00 -1.5000000e+01 1.0000000e+02 1.0000000e+04 1.0000000e+06 1.0000000e+08
Suppose we edit it to add another row:
wsname% cat xm.dat 1.2000000e+01 1.5000000e+01 1.7000000e+01 1.9000000e+01 2.0000000e+00 -3.0000000e+00 -5.0000000e+00 -1.5000000e+01 1.0000000e+02 1.0000000e+04 1.0000000e+06 1.0000000e+08 400 10.5 -18 0
Then in MATLAB we can get this new version of the variable by:
>> clear >> load xm.dat >> format short e >> xm xm = 1.2000e+01 1.5000e+01 1.7000e+01 1.9000e+01 2.0000e+00 -3.0000e+00 -5.0000e+00 -1.5000e+01 1.0000e+02 1.0000e+04 1.0000e+06 1.0000e+08 4.0000e+02 1.0500e+01 -1.8000e+01 0
If you want to save a copy of your session in a file for someone to study or to print, you can do so by starting the session with the diary, command. This is shown in the following example session:
>> diary mat1.t >> t=0:.1:6; >> plot(t,sin(t)^2) ??? Error using ==> ^ Matrix must be square. >> plot(t,sin(t).^2) >> quit 183 flops
Note the error in the squaring operation. The omitted period is
one of the most common errors in MATLAB. We wanted an element by
element operation on an array and must specify that.
The file mat1.t, will then list exactly like the session just shown
with only the line involving diary missing. The graphics window will
show your curve with axes marked at integer or simple fractions. Try
the commands shown in the example session to see how simple the use
of diary, and plot, is. If you save several sessions in the same
file, new ones are appended. Of course, the graphics output is not
saved. The recording in the file may be turned on and off with the
commands:
>> diary off
any commands that you do not want saved
>> diary on
The diary command rarely produces a file that is suitable for
submission as a solution to assignments. In nearly all cases such
files include rough output with numerous errors and statements out of
order so that a grader can not follow what was done in completing the
assignment. You should always plan to edit such files to make it
clear by adding comments, reordering the output and deleting errors
so that the results are easier to follow. In particular, you should
show in the file the problem that is being worked and clearly
designate the final answers to the problem. You may find it easier to
make assignment solutions by copying the results in a Matlab or Maple
session to a file that you are editing as you complete the
assignment. When assignments are ready for submission, they should be
stored in a file in your ceng301/ceng303 directory and clearly
labeled with a name that tells what is in the file. For example,
assignment 1 solution might be called solution1. When the file is
ready to be graded, your TA should be sent an e-mail message telling
her/him that the file is now ready for grading.
2.5 Matrix Operations with Arrays
There are two kinds of operations with arrays called: Matrix
operations and Array operations. The first set includes taking the
transpose of an array, ``matrix" multiply, adding and subtracting
arrays of the same shape, etc. The transpose of a matrix is found
with the single quote symbol . Matrix multiply is done with a * and
addition and subtraction use their normal symbols as seen in the
following:
>> m'
ans =
1 4
2 5
3 6
>> m*v'
ans =
14
32
>> m1=[6 5 4
7 4 1];
>> m+m1
ans =
7 7 7
11 9 7
Finally matrix division is used to solve sets of linear equations. To solve the set of equations:
5x1 + 3x2 - x3 = 5 2x1 - 7x2 - 3x3 = 0 x1 + 5x2 + 6x3 = -7
we need to define a 3 by 3 matrix with the coefficients of the unknowns in it and divide a column vector with the right hand side coefficients by this matrix. Unfortunately you have to get used to the fact that there are two divide symbols in MATLAB.
A\B gives the solution to: A*X = B
A/B gives the solution to: X*A = B
In our case we need to use the first form so:
>> a=[5 3 -1
2 -7 -3
1 5 6];
>> a\[5 0 -7]' <-- Note the quote or prime
to transpose the vector.
ans =
0.1168
0.8426
-1.8883
We can confirm our solution by using matrix multiply.
>> a*ans
ans =
5.0000
0.0000
-7.0000
If we forget how many elements there are in a vector, the command length, will tell us. The size, command will tell how many rows and columns there are in a defined matrix. These are shown in the following session:
>> v=[1 3 5];
>> size(v)
ans =
1 3
>> length(v)
ans =
3
2.6 Element by Element Operations on Arrays
Array operations in MATLAB require some care. Essentially there are
two types of operations involving arrays.
1) Element by element operations
2) Vector-Matrix operations.
Confusing these can lead to real difficulties. The next session shows the formation and addition of two vectors:
>> a=[1 2 3];
>> b=[2 5 8];
>> a+b
ans =
3 7 11
Now suppose we try to multiply them:
>> a*b ??? Error using ==> * Inner matrix dimensions must agree.
What happened? The multiplication symbol by itself when applied to vectors or matrices is interpreted to mean matrix multiply. We could have done this by transposing the second vector:
>> a*b'
ans =
36
This gave (1*2 + 2*5 + 3*8) = 36, the usual scalar product of two
vectors.
If we want the element by element product, we must put a period
before the multiplication symbol:
>> a.*b
ans =
2 10 24
The same procedure must be followed in doing division and exponentiation. For example, if you want the square of each element in a vector you must use:
>> a.^2
ans =
1 4 9
Forgetting the period will lead to:
>> a^2 ??? Error using ==> ^ Matrix must be square.
The fact that MATLAB gives vector and array results with most of
its built-in functions is one of its main features. The fact that
sin(t), with t, a vector gives the value of sin of each
element makes it trivial to look at a plot of the function.
2.7 Plotting Results
The following set of commands produces a plot of sin(t) and
cos(t) for t from 0 to 4pi.
>> t=0:pi/16:4*pi;
>> plot(t,[sin(t);cos(t)])
>> title('sin(t) and cos(t)')
>> xlabel('t')
>> ylabel('sin and cos')
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The five lines in this sequence do the following:
1) Sets up a vector t with the elements: 0, pi/16, pi/8, .... 4pi.
2) Creates a matrix with its first row: sin(0),sin(/16),sin(/8)...sin(4). and its second row: cos(0),cos(/16),cos(/8)...cos(4). and plots the two curves.
3) Puts a title at the top of the plot.
4) Labels the x axis.
5) Labels the y axis.
If you want to save your figure to use in a later Matlab session or for someone else to see in such a session use the Save As command in the file menu on the plot figure. This will save the figure with the name you specify and an appendage of .fig. You may then retreive the file with the open command in another Matlab session.
Two three dimensional plotting routines are available. Both are
demonstrated in the EXPO package. The mesh, command constructs
a three dimensional plot of the values in a matrix vs the indices
that specify positions in the matrix. The contour, command
gives a contour plot of a matrix interpreted in the same way.
You can create up to four subplots in the same window. The first two
digits in the argument of subplot, specify how many plots and their
orientation as:
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First Two Digits |
No. of Plots |
Orientation |
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side by side |
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above and below |
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in the Quadrants |
>> t=0:.1:6; >> plot(t,sin(t)) >> grid
Try that and you can see that there is a root between 3 and 4:
Figure 2.2 Looking for a root
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We could redefine t to span the interval (3,4) and plot that result to home in on the root, but will investigate using a polynomial fit of the values for our function:
>> t=3:.25:4;
>> c=polyfit(t,sin(t),3)
c =
0.1537 -1.4420 3.5107 -1.5619
>> roots(c)
ans =
5.6727
3.1415
0.5704
The polyfit, function finds a polynomial that fits the data given it in the least squares sense. In our case, it found that:
p(t) = 0.1537t^3 - 1.4420t^2 + 3.5107t - 1.5619
closely approximates sin(t) over the interval (3,4). The program
roots finds all roots of a polynomial. We can recognize that
the procedure found an answer very close to the one that we know is
correct: .
Continue on to Chapter 3
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