A major strength of MATLAB is its ability to handle collections of numbers (or other entities), called arrays, as if they were a single variable. If A and B are arrays with the same dimensions, MATLAB can add them by simply typing the typing the command:
C = A + B.
In most other programming languages, this operation would require several statements or commands.
This capability makes MATLAB a natural choice for solving engineering and data analysis problems.
MATLAB allows you to work with several types of arrays:
In the following sections, you will learn how to:
2.1 Defining Arrays
MATLAB allows you to define several types of arrays:
The elements stored in the arrays may be numeric or character types. No distinction is made between integer and real numbers. 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 numeric 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 echoing of the input 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 mc s v vc
>>
These variables and their values also appear in the Workspace window, while the Command History captures the full record of this session (see illustration below).
We will now introduce a simple way to define a vector of numbers of the form:
a, a + c, a + 2c, . . . b-c, b
where c is a constant increment. Consider for example the vector u=[ 0.0 0.5 1.0 1.5 2.0]. To put it in the previous form, we need to choose
a=0, c=0.5, b=2To define this vector, issue the following MATLAB command:
>> u=0:.5:2
t =
0 0.5000 1.0000 1.5000 2.0000
>> 2*u
ans =
0 1 2 3 4
>> f=sin(u)
f =
0 0.4794 0.8415 0.9975 0.9093
To keep track of the elements of an array, we use the row and column numbers (or array indices) of each element in an array. With the colon operator, you can select individual elements, rows, columns or "subarrays" of arrays. Consider, for example, the vector:
>> v = [1 2 3 4 5 6];
To display all the elements of the vector, issue the command:
>> x = v(:)
x =
1
2
3
4
5
6
To create a subvector with the 3rd, 4th and 5th elements of the vector, issue the command:
>> z= v(3:5)
z =
3 4 5
Consider now the two-dimensional array:
M =
1 2 3
4 5 6
7 8 9
To display all the elements of the third column of M, issue the command:
>> M(:,3)
ans =
3
6
9
To display all the elements in the 2nd and third columns, issue the command:
>> M(:,2:3)
ans =
2 3
5 6
8 9
Finally, issue the following command to create a subarray containing all the elements on the 2nd and 3rd column that also belong to the 1st and 2nd row:
>> w=M(1:2,2:3)
w =
2 3
5 6
You can edit the elements of an array, find the array in the Workspace window and double-click on it. The Array Editor will open in a separate window (see ilustartion below) and you can interactively change the values of any array element. Close the editor to accept the new vlues.
2.2 Performing Array Operations
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.
The transpose of a matrix is found
with the single quote symbol . Matrix multiplication is done with the * operator. Addition and subtraction are done with the usual operators as shown below:
>> 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
2.3 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:
|
First Two Digits |
No. of Plots |
Orientation |
|
|
|
side by side |
|
|
|
above and below |
|
|
|
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: .
2.5 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
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 chbe301/chbe303 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.
Continue on to Chapter 3
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