Complete MAT LAB Exercise in attached file.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

⋆ 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

and Command History windows. 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.

⋆ 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 ,.

⋆ 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 =

be obtained by typing

π

5.

A list of elementary math functions can

>> help elfun

⋆ 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.

⋆ To get a list of other groups of MATLAB programs already available enter help:

>> help

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⃝2011

Stefania Tracogna, SoMSS, ASU

MATLAB sessions: Laboratory 1

⋆ Another way to obtain help is through the desktop Help menu, Help > Product Help.

⋆ 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

⋆ The quantities Inf (∞) and NaN (Not a Number) also appear frequently. Compare

>> c=1/0

c =

Inf

with

>> d=0/0

d =

NaN

Plotting with MATLAB

⋆ 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

x2 − sin(πx) + ex

y = f (x) =

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

Columns 8 through 14

0.7000

0.8000

0.9000

1.0000

1.1000

Columns 15 through 21

1.4000

1.5000

1.6000

1.7000

1.8000

0.5000

0.6000

1.2000

1.3000

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.

⋆ 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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⃝2011

Stefania Tracogna, SoMSS, ASU

MATLAB sessions: Laboratory 1

(This format was used for all previous outputs).

⋆ 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

Columns 7 through 12

-3.0777

-5.6491 -11.3888 -29.6059

Columns 13 through 18

26.7395

20.5610

17.4156

15.4634

Columns 19 through 21

12.3468

11.7832

11.3891

-1.1679

-1.7974

Inf

45.2318

14.1068

13.1042

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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⃝2011

Stefania Tracogna, SoMSS, ASU

MATLAB sessions: Laboratory 1

Figure L1a: A Figure window

Scripts and Functions

⋆ 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 select the MATLAB desktop File menu File > New > Script. In the MATLAB text

editor window enter the commands as you would in the Command window. To save the ﬁle use

the menu File > Save or File > Save As…, or the shortcut 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.

To create a new function select the MATLAB desktop File menu File > New > Function. 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.

⋆ Examples of script/function:

1. script

myplot.m

x=0:.01:2;

y=(x.^2-sin(pi.*x)+exp(x))./(x-1);

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⃝2011

Stefania Tracogna, SoMSS, ASU

% x-values

% y-values

MATLAB sessions: Laboratory 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’);

%

%

%

%

plot in red with wider line

set range and add grid

add title

add labels

%

%

%

%

%

%

x-values

evaluate myfunction at x

plot in red

set range and add grid

add title

add labels

2. script+function (two separate ﬁles)

myplot2.m (driver script)

x=0:.01:2;

y=myfunction(x);

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’);

myfunction.m (function)

function y=myfunction(x)

y=(x.^2-sin(pi.*x)+exp(x))./(x-1);

% defines function

% 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.

⋆ 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.

⋆ 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).

⋆ 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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⃝2011

Stefania Tracogna, SoMSS, ASU

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.

⋆ 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)

ans =

3

% evaluate the function f at t = 2 and y = 1

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)

⋆ 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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⃝2011

Stefania Tracogna, SoMSS, ASU

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 diary ﬁle 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 diary ﬁle 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 class time 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

with radius r, i.e. satisfy the equation x2 + y 2 = r2 . Create a row vector for θ with the values

5π

0, π4 , π2 , 3π

4 , π, and 4 .

Take r = 2 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 = x2 + y 2 .

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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⃝2011

Stefania Tracogna, SoMSS, ASU

MATLAB sessions: Laboratory 1

2. Use the linspace command or the colon operator : to create a vector t with 91 elements:

et/10 sin(t)

(make sure you use “;” to suppress

1, 1.1, 1.2, . . . , 10 and deﬁne the function y =

t2 + 1

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 ≤ 20.

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.

2

4

4. Plot y = cos x in red with a solid line and z = 1 − x2 + x24 in blue with a dashed line for 0 ≤ 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.

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

y(x) =

dy

= x + 2 is

dx

x2

+ 2x + C

2

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 ≤ 4, with initial conditions y(0) = −1, 0, 1.

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 = ?? ;

y1 = f(??);

y2 = f(??);

y3 = f(??);

plot(??)

title(??)

legend(??)

end

%

%

%

%

%

%

% define the vector x in the interval [0,4]

compute the solution with C = -1

compute the solution with C = 0

compute the solution with C = 1

plot the three solutions with different line-styles

add a title

add a legend

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 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.

Add the title ’Solutions to dy/dx = x + 2’.

Add a legend on the top left corner of the plot with the list of C values used for each graph.

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⃝2011

Stefania Tracogna, SoMSS, ASU

MATLAB sessions: Laboratory 1

(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.

yex

as an inline or anonymous function (see page 6).

x+1

Evaluate the function at x = 2 and y = −1.

6. (a) Enter the function f (x, y) = x3 +

(b) Type clear f to clear the value of the function from part (a). Now write a function M-ﬁle

yex

. Save the ﬁle as f.m (include the M-ﬁle in your report).

for the function f (x, y) = x3 +

x+1

Evaluate the function at x = 2 and y = −1 by entering f(2,-1) in the command window.

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⃝2011

Stefania Tracogna, SoMSS, ASU

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