MAT 275 Laboratory 1

20 Mar

MATLAB sessions: Laboratory 1
MAT 275 Laboratory 1
Introduction to MATLAB
MATLAB is a computer software commonly used in both education and industry to solve a wide range
of problems.
This Laboratory provides a brief introduction to MATLAB, and the tools and functions that help
you to work with MATLAB variables and files.
The MATLAB Environment
F To start MATLAB double-click on the MATLAB shortcut icon. The MATLAB desktop will open.
On the left side you will generally find the Current Folder window and on the right the Workspace.
The Command Window is where the MATLAB commands are entered and executed. Note that windows
within the MATLAB desktop can be resized by dragging the separator bar(s).
If you have never used MATLAB before, we suggest you type
demo at the MATLAB prompt. Click
on
Getting Started with MATLAB and run the file.
Basics And Help
Commands are entered in the Command Window.
F Basic operations are +, , *, and /. The sequence
>> a=2; b=3; a+b, a*b
ans =
5
ans =
6
defines variables a and b and assigns values 2 and 3, respectively, then computes the sum a+b and product
ab. Each command ends with , (output is visible) or ; (output is suppressed). The last command on a
line does not require a
,.
F Standard functions can be invoked using their usual mathematical notations. For example
>> theta=pi/5;
>> cos(theta)^2+sin(theta)^2
ans =
1
verifies the trigonometric identity sin2 θ + cos2 θ = 1 for θ = π5 . A list of elementary math functions can
be obtained by typing
>> help elfun
F To obtain a description of the use of a particular function type help followed by the name of the
function. For example
>> help cosh
gives help on the hyperbolic cosine function.
F To get a list of other groups of MATLAB programs already available enter help:
>> help
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MATLAB sessions: Laboratory 1
F Another way to obtain help is through the help button in the toolbar.
F MATLAB is case-sensitive. For example
>> theta=1e-3, Theta=2e-5, ratio=theta/Theta
theta =
1.0000e-003
Theta =
2.0000e-005
ratio =
50
F The quantities Inf (1) and NaN (Not a Number) also appear frequently. Compare
>> c=1/0
c =
Inf
with
>> d=0/0
d =
NaN
Plotting with MATLAB
F To plot a function you have to create two arrays (vectors): one containing the abscissae, the other the
corresponding function values. Both arrays should have the same length. For example, consider plotting
the function
y = f(x) = x2 sin(πx) + ex
x 1
for 0
x 2. First choose a sample of x values in this interval:
>> x=[0,.1,.2,.3,.4,.5,.6,.7,.8,.9,1, …
1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2]
x =
Columns 1 through 7
0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000
Columns 8 through 14
0.7000 0.8000 0.9000 1.0000 1.1000 1.2000 1.3000
Columns 15 through 21
1.4000 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000
Note that an ellipsis was used to continue a command too long to fit in a single line.
Rather than manually entering each entry of the vector
x we can simply use
>> x=0:.1:2
or
>> x=linspace(0,2,21)
Both commands above generate the same output vector x.
F The output for x can be suppressed (by adding ; at the end of the command) or condensed by entering
>> format compact
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MATLAB sessions: Laboratory 1
(This format was used for all previous outputs).
F To evaluate the function f simultaneously at all the values contained in x, type
>> y=(x.^2-sin(pi.*x)+exp(x))./(x-1)
y =
Columns 1 through 6
-1.0000 -0.8957 -0.8420 -0.9012 -1.1679 -1.7974
Columns 7 through 12
-3.0777 -5.6491 -11.3888 -29.6059 Inf 45.2318
Columns 13 through 18
26.7395 20.5610 17.4156 15.4634 14.1068 13.1042
Columns 19 through 21
12.3468 11.7832 11.3891
Note that the function becomes infinite at x = 1 (vertical asymptote). The array y inherits the dimension
of
x, namely 1 (row) by 21 (columns). Note also the use of parentheses.
IMPORTANT REMARK
In the above example *, / and ^ are preceded by a dot . in order for the expression to be evaluated for
each component (entry) of
x. This is necessary to prevent MATLAB from interpreting these symbols
as standard linear algebra symbols operating on arrays. Because the standard
+ and operations on
arrays already work componentwise, a dot is not necessary for
+ and .
The command
>> plot(x,y)
creates a Figure window and shows the function. The figure can be edited and manipulated using the
Figure window menus and buttons. Alternately, properties of the figure can also be defined directly at
the command line:
>> x=0:.01:2;
>> y=(x.^2-sin(pi.*x)+exp(x))./(x-1);
>> plot(x,y,’r-’,’LineWidth’,2);
>> axis([0,2,-10,20]); grid on;
>> title(’f(x)=(x^2-sin(\pi x)+e^x)/(x-1)’);
>> xlabel(’x’); ylabel(’y’);
Remarks:
The number of x-values has been increased for a smoother curve (note that the stepsize is now :01
rather than
:1).
The option ’r-’ plots the curve in red.
’LineWidth’,2 sets the width of the line to 2 points (the default is 0:5).
The range of x and y values has been reset using axis([0,2,-10,20]) (always a good idea in the
presence of vertical asymptotes).
The command grid on adds a grid to the plot.
A title and labels have been added.
The resulting new plot is shown in Fig. L1a. For more options type
help plot in the Command
Window.
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MATLAB sessions: Laboratory 1
Figure L1a: A Figure window
Scripts and Functions
F Files containing MATLAB commands are called m-files and have a .m extension. They are two types:
1. A
script is simply a collection of MATLAB commands gathered in a single file. The value of the
data created in a script is still available in the Command Window after execution. To create a
new script click on the
New Script icon in the upper left corner of the Home toolbar. In the
MATLAB text editor window enter the commands as you would in the Command window. To
save the file click on the save button .
Variable defined in a
script are accessible from the command window.
2. A
function is similar to a script, but can accept and return arguments. Unless otherwise specified
any variable inside a function is local to the function and not available in the command window.
On the upper left corner of the Home toolbar, click on
New and select function from the pulldown
menu. A MATLAB text editor window will open with the following predefined commands
function [ output_args ] = Untitled3( input_args )
%UNTITLED3 Summary of this function goes here
% Detailed explanation goes here
end
The \output args” are the output arguments, while the \input args” are the input arguments. The
lines beginning with % are to be replaced with comments describing what the functions does. The
command(s) defining the function must be inserted after these comments and before
end.
To save the file proceed similarly to the Script M-file.
Use a function when a group of commands needs to be evaluated multiple times.
F Examples of script/function:
1.
script
myplot.m
x=0:.01:2; % x-values
y=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-values
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MATLAB sessions: Laboratory 1

 plot(x,y,’r-’,’LineWidth’,2); % plot in red with wider line axis([0,2,-10,20]); grid on; % set range and add grid title(’f(x)=(x^2-sin(\pi x)+e^x)/(x-1)’); % add title xlabel(’x’); ylabel(’y’); % add labels

2. script+function (two separate files)

 myplot2.m (driver script) x=0:.01:2; % x-values y=myfunction(x); % evaluate myfunction at x plot(x,y,’r-’,’LineWidth’,2); % plot in red axis([0,2,-10,20]); grid on; % set range and add grid title(’f(x)=(x^2-sin(\pi x)+e^x)/(x-1)’); % add title xlabel(’x’); ylabel(’y’); % add labels myfunction.m (function) function y=myfunction(x) % defines function y=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-values

3. function+function (one single file)

 myplot1.m (driver script converted to function + function) function myplot1 x=0:.01:2; % x-values y=myfunction(x); % evaluate myfunction at x plot(x,y,’r-’,’LineWidth’,2); % plot in red axis([0,2,-10,20]); grid on; % set range and add grid title(’f(x)=(x^2-sin(\pi x)+e^x)/(x-1)’); % add title xlabel(’x’); ylabel(’y’); % add labels %—————————————– function y=myfunction(x) % defines function y=(x.^2-sin(pi.*x)+exp(x))./(x-1); % y-values

In case 2 myfunction.m can be used in any other m-file (just as other predefined MATLAB functions).
In case 3
myfunction.m can be used by any other function in the same m-file (myplot1.m) only. Use 3
when dealing with a single project and 2 when a function is used by several projects.
F Note that the function myplot1 does not have explicit input or output arguments, however we cannot
use a script since the construct
script+function in one single file is not allowed.
F It is convenient to add descriptive comments into the script file. Anything appearing after % on any
given line is understood as a comment (in green in the MATLAB text editor).
F To execute a script simply enter its name (without the .m extension) in the Command Window (or
click on the SAVE & RUN button ).
The function
myfunction can also be used independently if implemented in a separate file myfunction.m:
>> x=2; y=myfunction(x)
y =
11.3891
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MATLAB sessions: Laboratory 1
A script can be called from another script or function (in which case it is local to that function).
If any modification is made, the script or function can be re-executed by simply retyping the script
or function name in the Command Window (or use the up-arrow on the keyboard to browse through
past commands).
IMPORTANT REMARK
By default MATLAB saves files in the Current Folder. To change directory use the Current Directory
box on top of the MATLAB desktop.
F A function file can contain a lot more than a simple evaluation of a function f(x) or f(t; y). But in
simple cases
f(x) or f(t; y) can simply be defined using the inline syntax.
For instance, if we want to define the function
f(t; y) = t2 y, we can write the function file f.m
containing
function dydt = f(t,y)
dydt = t^2-y;
and, in the command window, we can evaluate the function at different values:
>> f(2,1) % evaluate the function f at t = 2 and y = 1
ans =
3
or we can define the function directly on the command line with the inline command:
>> f = inline(’t^2-y’,’t’,’y’)
f =
Inline function:
f(t,y) = t^2-y
>> f(2,1) % evaluate the function f at t = 2 and y = 1
ans =
3
However, an inline function is only available where it is used and not to other functions. It is not recommended when the function implemented is too complicated or involves too many statements.
Alternatively, the function can be entered as an
anonymous function
>> f = @(t,y)(t^2-y)
F CAUTION!
The names of script or function M-files must begin with a letter. The rest of the characters may
include digits and the underscore character. You may not use periods in the name other than the
last one in ’.m’ and the name cannot contain blank spaces.
Avoid name clashes with built-in functions. It is a good idea to first check if a function or a script
file of the proposed name already exists. You can do this with the command
exist(’name’),
which returns zero if nothing with name
name exists.
NEVER name a script file or function file the same as the name of the variable it computes. When
MATLAB looks for a name, it first searches the list of variables in the workspace. If a variable of
the same name as the script file exists, MATLAB will never be able to access the script file.
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MATLAB sessions: Laboratory 1
Exercises
Instructions:
You will need to record the results of your MATLAB session to generate your lab report. Create a
directory (folder) on your computer to save your MATLAB work in. Then use the Current Directory
field in the desktop toolbar to change the directory to this folder. Now type
diary lab1 yourname.txt
followed by the Enter key. Now each computation you make in MATLAB will be save in your directory
in a text file named lab1 yourname.txt. When you have finished your MATLAB session you can turn
off the recording by typing
diary off at the MATLAB prompt. You can then edit this file using your
favorite text editor (e.g. MS Word).
Lab Write-up: Now that your diary file is open, enter the command format compact (so that when
you print out your diary file it will not have unnecessary blank lines), and the comment line
% MAT 275 MATLAB Assignment # 1
Include labels to mark the beginning of your work on each part of each question, so that your edited lab
write-up has the format
% Exercise 1
. .
% Exercise 2
Final Editing of Lab Write-up: After you have worked through all the parts of the lab assignment
you will need to edit your diary file.
Remove all typing errors.
Unless otherwise specified, your write-up should contain the MATLAB input commands,
the corresponding output, and the answers to the questions that you have written.
If the exercise asks you to write an M-file, copy and paste the file into your lab write-up in the
appropriate position (after the problem number and before the output generated by the file).
If the exercise asks for a graph, copy the figure and paste it into your lab write-up in the appropriate position. Crop and resize the figure so that it does not take too much space. Use \;”
to suppress the output from the vectors used to generate the graph. Make sure you use enough
points for your graphs so that the resulting curves are nice and smooth.
Clearly separate all exercises. The exercises’ numbers should be in a larger format and in boldface.
Preview the document before printing and remove unnecessary page breaks and blank spaces.
Put your name and course number on each page.
Important: An unedited diary file without comments submitted as a lab write-up is not
acceptable.
1. All points with coordinates x = r cos(θ) and y = r sin(θ), where r is a constant, lie on a circle
r, i.e. satisfy the equation x2 + y2 = r2. Create a row vector for θ with the values
0
; π3 ; π2 ; 23π ; π; and 43π.
Take
r = 3 and compute the row vectors x and y. Now check that x and y indeed satisfy the
equation of a circle, by computing the radius
r = px2 + y2.
Hint: To calculate
r you will need the array operator .^ for squaring x and y. Of course, you
could also compute
x2 by x.*x.
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MATLAB sessions: Laboratory 1
2. Use the linspace command or the colon operator : to create a vector t with 46 elements:
1
; 1:2; 1:4; : : : ; 10 and define the function y = et=20 cos(t)
t2 + 4 (make sure you use \;” to suppress
the output for both
t and y).
(a) Plot the function
y in black and include a title with the expression for y.
(b) Make the same plot as in part (a), but rather than displaying the graph as a curve, show the
unconnected data points. To display the data points with small circles, use
plot(t,y,’o’).
Now combine the two plots with the command
plot(t,y,’o-’) to show the line through the
data points as well as the distinct data points.
3. Use the command
plot3(x,y,z) to plot the circular helix x(t) = sin t; y(t) = cos t; z(t) = t
0 t 30.
Add a grid to the plot using the command
grid on.
NOTE: Use semicolon to suppress the output when you define the vectors
t, x, y and z. Make sure
you use enough points for your graph so that the resulting curve is nice and smooth.
4. Plot
y = sin x in red with a solid line and the Taylor polynomial approximation z = x x63 + 120 x5
in blue with a dashed line for π x π on the same plot.
Hint: Use
plot(x,y,’r’,x,z,’–’).
Add a grid to the plot using the command
grid on.
Type
axis tight to set the axis limits to the range of the data.
NOTE: Use semicolon to suppress the output when you define the vectors
x, y and z. Make sure
you use enough points for your graph so that the resulting curves are nice and smooth.
5. The general solution to the differential equation
dy
dx
= x2 2x is
y(x) = x3
3
x2 + C with y(0) = C:
The goal of this exercise is to write a function file to plot the solutions to the differential equation
in the interval 0
x 5, with initial conditions y(0) = 2; 0; 2.
The
function file should have the structure function+function (similarly to the M-file myplot1.m
Example 3, page 5). The function that defines y(x) must be included in the same file (note that
the function defining
y(x) will have two input arguments: x and C).
Your M-file should have the following structure (fill in all the
?? with the appropriate commands):
function ex5
x = ?? ; % define the vector x in the interval [0,5]
y1 = f(??); % compute the solution with C = -2
y2 = f(??); % compute the solution with C = 0
y3 = f(??); % compute the solution with C = 2

 plot(??) % plot the three solutions with different line-styles title(??) % add a title legend(??) % add a legend

end
function y = f(x,C)
y = ?? % fill-in with the expression for the general solution
end
Plot the graphs in the same window and use different color and/or line-styles for each graph. To
plot the graphs in the same window you can use the command
hold on or use the plot command
similarly to Exercise 4.
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MATLAB sessions: Laboratory 1
Add the title ’Solutions to dy=dx = x2 2x’.
Add a legend on the top left corner of the plot with the list of
C values used for each graph.
(Type
help plot for a list of the different line-styles, and help legend for help on how to add a
legend.) Include both the M-file and the plot in your report.
NOTE: the only output of the function file should be the graph of the three curves. Make sure
you use enough points so that the curves are nice and smooth.
6. (a) Enter the function
f(x; y) = x2 + xey
y + 1
as an
inline or anonymous function (see page 6).
Evaluate the function at
x = 1 and y = 2 by entering f(-1,2) in the command window.
(b) Type
clear f to clear the value of the function from part (a). Now write a function M-file
for the function
f(x; y) = x2 + xey
y + 1
. Save the file as
f.m (include the M-file in your report).
Evaluate the function at
x = 1 and y = 2 by entering f(-1,2) in the command window.
c 2016 Stefania Tracogna, SoMSS, ASU 9