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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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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(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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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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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

with radius 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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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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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.

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