matlab notes_pwr.pdf

Alexandria University Mathematics Faculty of Engineering February 2013

Lab Notes

MATLAB

Session (1)

Eng. Sara Hassan Kamel

MATLAB Notes By: Eng. Sara Hassan Kamel

Introduction

MATLAB (Matrix Laboratory) is a high performance language for technical

computing.

MATLAB Environment:

Command window

This is where MATLAB

commands are written

and executed

Workspace Tab: (click to view)

This is where the variables are shown

Command History:

All commands previously executed in the

command window appear here.

When typing commands in the command

window you may use the “up↑” and

“down↓” arrows to navigate through

previous commands and modify them or re-

execute them.


MATLAB is case sensitive, that is two variables A and a correspond to

different variables.

Variable names must begin with a letter NOT a number

o e.g.: a A a_1 A1 fmax

If you assign a different value to an existing variable, its current value will be

overwritten. If you will need the current value, you must use a new variable to

store your data.

To view a previously written command use the “up” key and navigate using

“up” and “down” keys.

To clear the command window use >> clc

To clear the workspace use >> clear all

To close figure windows >> close all

Scalars, Vectors, Arrays (Matrices) and Dimensioning:

Arrays (Matrices):

An array (or a matrix) is a data unit that consists of many elements. For example, a

4-by-3 matrix consists of 4 rows by 3 columns of numbers:

908

632

096

541

A

Vectors:

A vector is a special case of an array; it is an array that consists of either one column

OR one row only.

6431x Row Vector

8

5

2

y Column Vector

Scalar:

A scalar is an array of one element only (a 1-by-1 matrix). It can simply be regarded

as a number.


Defining Variables in MATLAB:

4321

542

101

321

10

x

A

a

5

4

3

y

To define these variables use the following commands:

>> a=10;

>> A=[1 2 3;1 0 1;-2 4 5];

>> x=[1 2 3 4]; OR x=[1,2,3,4];

>> y=[3;4;5];

NOTES:

If we remove the semicolon “ ; ” from the end of the command line the output

is displayed in the command window:

Use square brackets to define matrices, row vectors and column vectors.

Use a comma “ , ” or space to move between elements in the SAME ROW.

Use a semicolon to move to the NEXT ROW.

Variables are saved temporarily in your workspace as shown:

Predefined Constants:

Among the predefined constants in MATLAB are the values of π written as pi and

the value of 1i written as i or j


Basic Operators:

+ Addition - Subtraction .* Element-by-element Multiplication * Matrix Multiplication ./ Element-by-element Division / A/B means 1AB .^ Element-by-element exponentiation ^ A^3 means matrix multiplication as follows: AAA

Element-by-Element Multiplication:

Matrices must be of the SAME SIZE

Multiplying A.*B means multiplying every element in A by the corresponding element

in B (i.e. the element in the same location) to produce a matrix of the SAME SIZE as

A and B.

For example:

403

201210

2*29*01*3

4*53*410*1*

291

4310

203

541

BA

BA

Matrix Multiplication:

Matrices must satisfy the following condition The number of columns of the first

matrix must be equal to the number of rows of the second matrix (i.e. their “inner

dimensions” agree) and the resulting matrix has same number of rows as the first

matrix and the same number of columns as the second matrix.

Meaning that

pmpnnm CBA *

For example:

13

8

8

4

7

4

3*32*2

3*22*1

0*34*2

0*24*1

1*32*2

1*22*1*

3

2

0

4

1

2

32

21

BA

BA


Built-in Functions:

sin( ) cos( ) tan( ) angle in radians: e.g.: sin(π)

sind( ) cosd( ) tand( ) angle in degrees: e.g.: sind(180)

asin( ) acos( ) atan( ) Output: angle in radians

asind( ) acosd( ) atand( ) Output: angle in degrees

cosh( ) sinh( ) tanh( )

asinh( ) acosh( ) atanh( )

eye( ) Identity matrix

sqrt( ) Square Root

exp( ) Exponential xe

log( ) ln(x)

log10( ) log(x)

real( ) Real part of a complex number

imag( ) Imaginary part of a complex number

abs( ) Magnitude or absolute value

angle( ) Angle of a complex number

max( ) Maximum valued element in a matrix

find( ) gives the indices of the elements that satisfy the argument

The functions mentioned here are only a few of the many functions defined in

MATLAB. You may not use all the function mentioned here during the lab sessions

but they are here for future reference. If you wish to know more about any of these

functions (or a function that is not listed above) you may use MATLAB help.


NOTES:

With scalars, it doesn’t matter whether you use matrix or element-by-element

operations.

To add or subtract matrices they must be of the SAME SIZE

To multiply matrices:

Element-by-Element (A .*B): SAME SIZE

Matrix Multiplication (A*B): No. of Columns of A = No. of Rows of B

Applying some built-in function such as sin, cos, tan, exp,… to a matrix means

applying the function to every element in the matrix:

)4sin()sin(

)2/sin()1sin()sin(

4

2/1

pi

piA

pi

piA

The result of the following commands may confuse you:

o >> sin(0) = 0

o >> sin(pi) = 1.2246e-016

o >> tan(pi/2) = 1.6331e+016

The number 1.2246e-16 means 1.22*10 –16 which is a very small number

almost equal to zero which is the result you might expect when executing

sin(pi). Similarly 1.6331e+16 = 1.6331*10 16 is a very large number.

WHY?? Remember that the value of “pi” is stored in MATLAB is only an

approximation of the actual value “π”.


Matrix Concatenation:

Concatenation means “gluing” two or more matrices together to create a larger

matrix. There are basically two types of concatenation: horizontal and vertical.

Horizontal Concatenation:

Matrices must have the same number of ROWS.

The resulting matrix has the same number of rows as the original ones and

the number of columns is equal to the sum of the number of columns of the

concatenated matrices.

Vertical Concatenation:

Matrices must have the same number of COLUMNS.

The resulting matrix has the same number of columns as the original ones

and the number of rows is equal to the sum of the number of rows of the

concatenated matrices.

Vertical Horizontal Both

Horizontal Concatenation:


Vertical Concatenation:

Both:

Question:

If we have A3x4, B3x2 and C4x6, which of them can be concatenated and how?

Answer:

A and B Horizontal Concatenation [A B]

A and C Vertical Concatenation [A ; C]

A and B horizontally then C vertically [A B ; C]

A

C

B


Matrix Indexing:

To read a certain element (or multiple elements) within a matrix or vector:

Given:

>> A =

1 2 0

3 4 -2

5 6 8

>> A(3,2)

6

>> A([1 2],[3 1])

0 1

-2 3

>> A(1,:)

1 2 0

>> A(8)

-2 In this case MATLAB counts the elements column by column.

>> B=[1 2 8 4];

>> B(2)

8

>> B(end)

4


Getting Help with MATLAB:

There are basically two methods to access MATLAB help:

1. By clicking the Help Button in the MATLAB toolbar as shown:

This will open a new window:

Use the search bar to type what you’re looking for.

2. Inline Help:

If you know the name of the function you’re looking for, for example “plot”, you can

type the following command in the command window:

>> help plot

This displays a brief explanation of how to use the function and its syntax. To access

the full help document (the reference page in the Help Browser) go to the last line

displayed on the screen and click on doc plot.

Type here!

Alexandria University Mathematics Faculty of Engineering March 2013

Lab Notes

MATLAB

Session (2)

Plots Using MATLAB



Two-Dimensional Plots:

In order to plot any line or curve, you need a set of points, namely, the

coordinates (x, y) of each point. Once you’ve marked these points on the x-y

plane, you join them together. This is the way MATLAB works. You give it the

set of points (data) in the form of two vectors equal in length, then MATLAB

plots the points and joins them together using straight lines.

Example:

Given the following data:

x 0 1 2 3 4

y 0 1 4 9 16

To plot these we will view them as follows:

x 0 1 2 3 4

y 0 1 4 9 16

Now we plot the points:

(0,0) (1,1) (2,4) (3,9) (4,16)

(0,0)

(1,1)

(2,4)

(3,9)

(4,16)

Notice how the points are

joined by straight lines even

though clearly these points

are most likely to represent

the curve y=x2


The simplest way to plot a function is by executing the following commands:

>> x=[0 1 2 3 4];

>> y=[0 1 4 9 16];

>> plot(x, y);

This leads us to the following question: What if we need to plot the function

y=x2 from x=0 to x=100 using 101 points? Here we will use the colon operator

or the built-in function linspace.

The Colon Operator:

Command Output Vector

>> x=0:100; Produces a vector of 101 points from 0 to 100 with a unit step (step=1)

>> x=0:2:10; x = 0 2 4 6 8 10

>> x=0:3:10; x = 0 3 6 9

>> x=10:-1:1; x = 10 9 8 7 6 5 4 3 2 1

>> x=10:-2:1; x = 10 8 6 4 2

x = start:step:end;

NOTES:

The default step size is 1. So if no step size is specified – as in the first

example – then it will be assumed 1.

If you wish to create a sequence of descending numbers use a negative

step. Remember here that you must specify the step size even if it’s -1

(MATLAB will not assume a step size of -1 if you write a starting number

that is greater than the end of the sequence; instead, you’ll get an

empty vector.


The linspace function:

This function is used to generate linearly space vectors. Use it when you know

how many points you want in the vector rather than the step (as well as, of

course, the starting and end point of the vector).

linspace(start, end, NumberOfPoints)

Example:

x=linspace(0,10,11); is the same as x=0:10;

x=linspace(0,10,12); gives

0, 0.9091, 1.8182, 2.7273, 3.6364, 4.5455, 5.4545, 6.3636,

7.2727, 8.1818, 9.0909, 10.0000

length function:

To find the length (number of points) in a vector use this function.

>> length(x)


2D Plots:

To plot a function f(x,y) you need to define both x and y.

Example: To plot x+y=1 from x= –5 to x=5

>> x=-5:5;

>> y=1-x;

>> plot(x,y)

To plot 2xy

>> y=x.^2; Here we use element-by-element multiplication

>> plot(x,y)

Here the curve is not smooth; that’s

because we used x=-5:5 (i.e. with a

step equal to 1) so MATLAB computes

the value of y at (-5,-4, -3.....2,3,4,5)

only and joins them with straight lines.

To get a smoother curve, use a

smaller step:

>> x=-5:0.1:5;


Editing the plot:

You can add a grid, title, labels and legends to the plot. You can also change the

colour of the plot and use markers each point on the plot. You can plot more than

one function in the same figure.

Quick Note: Strings in MATLAB:

A string is a sequence of characters (letters, numbers and/or symbols). In some of

the functions used here you will need to enter a string input such as the title of the

plot. In that case, for MATLAB to recognize your input as a string you need to put it

between single quotation marks. (e.g. 'MATLAB Session (2): First Plot').

plot Function:

plot(x,y,'r--o',x2,y2,'g-*',....)

NOTES:

To plot several plots on one figure there are two methods:

Use one plot function as indicated above

After the first plot, type >> hold on then type the next plot commands.

When you’re done and you don’t want the following commands to affect the

previous plot type >> hold off

The following functions take one string input:

o title( )

o xlabel( )

o ylabel( )

The function legend adds a small box at the top right corner of the figure that shows

a sample of each line or curve in the plot described by a text label that you specify.

For example, if you have three curves, you may write:

>> legend('curve 1','curve 2','curve 3')

Line Color

Line Style

Marker Type

1st Curve 2nd Curve . . .


Example:

>> x=0:10;

>> y1=x.^2;

>> y2=1-exp(-2*x);

>> y3=x;

>> plot(x,y1,'b--',x,y,'r*',x,y3,'g')

>> grid on

>> title('Lab 2')

>> xlabel('x')

>> ylabel('y')

>> legend('y=x^2','y=1-e^-^2^x','y=x')


Using LaTex Interpreter:

To use Greek letters, subscripts, superscripts and style of the text in a string such as

the title we use the LaTex Interpreter tool.

^ Superscript (e.g. x2)

_ Subscript (e.g. Pav)

\alpha α

\pi π

\int Integral ∫ \delta ∆

\theta θ

The full list is available in MATLAB help “Text Properties” OR “Annotation

Textbox Properties”.

Markers, Colours & Line Types:

Syntax Marker Type

No Marker (default)

o Circular

s Square

p Pentagonal

h Hexagonal

d Diamond

v Triangular (down)

< Triangular (left)

> Triangular (right)

x Cross

You may also use the following:

hold on To make several superimposed plots with several “plot”

commands.

xlim([a b]) To limit the x-axis from a to b only.

ylim([a b]) To limit the y-axis from a to b only.

Syntax Colour

r red

g green

b blue (default)

k black

m magenta

y yellow

c cyan

w white

Syntax Line Type

- Solid (default)

- - Dashed

: Dotted

-. Dash-dotted


More plotting commands:

fplot(function,limits) function is a string with a variable x

e.g. (‘sin(x)’ OR ‘x^2’).

o Example:

>> fplot('exp(x)',[-2 2])

stem( )

o >> x=0:10;

o >> y=x.^3;

o >> stem(x,y)

stairs( )

o >> stairs (x,y)


Subplots:

>> subplot(m,n,p)

It divides the figure into an m-by-n matrix, and plots the figure in the following

command in the pth element, where the elements are counted along the top row then

the second row and so on...

>> subplot(4,1,1)

>> subplot(4,2,3)

>> subplot(4,2,4)

>> subplot(4,4,9)

>> subplot(4,4,10)

>> subplot(4,4,13)

>> subplot(4,4,14)

>> subplot(2,2,4)

After every subplot command write the plotting command for the function you

want to plot in that specified space.


Using the figure command:

Executing the following command opens an empty figure ready for you to plot:

>> figure

This can be useful if you want to plot several functions each in a separate window.

But BE CAREFUL:

If you have several figures open, and you open a new figure to plot, then you go

back and click on one of the previous plots, the new plot command will be executed

in the “current or active figure” – in other words – the one you last clicked on!!

Tips:

All vectors used to plot a 2D or a 3D figure must all be of the SAME LENGTH.

If you want to plot a point, use scalars instead of vectors in the plotting

function (e.g. plot(3,5, 'o')). If you want to plot several points, put them

in a vector, and when specifying the line color, type and marker, just write the

color and the marker like this for example: plot(x,y,'ro'). This could also

be useful if you want to use markers of a different color!

You can some properties of the resulting plot by using the plot editor in the figure:

(Click the arrow to start editing).


Three-Dimensional Plots (3D plots):

There are two main types of 3D plots:

1. Line Plots

2. Surface Plots

As you should know, you can draw a line or a curve in a 3-dimensional space

(i.e. with 3 coordinates; x, y and z). You can also draw a plane or a surface. That

is how the types of 3D plots are classified in MATLAB. The following part will

give you more details about how the program deals with each type.

Line Plot figure (1.1)

Surface Plot figure (1.2)


3D Line Plots:

As you have already learned in 2D plots, MATLAB plots a set of points given to

the plot function in the form of two equal-length vectors and joins them with

straight lines. Similarly, we deal with line plots as a set of points (this time,

each has 3 coordinates) that we join together by straight lines.

To plot these we will view them as follows:

x 0 1 2 3 4

y 0 1 4 9 16

z 0 1 2 3 4

Now we plot the points using the function plot3 for line plots:

plot3(x,y,z)

(0,0,0) (1,1,1) (2,4,2) (3,9,3) (4,16,4)

(0,0,0)

(1,1,1)

(2,4,2)

(3,9,3)

(4,16,4)

plot3(0:4,[0:4].^2,0:4)

OR x=0:4; y=x.^2; z=x; plot3(x,y,z)


Parametric Representation: Say we have a direct relation between the variables x and y: y=x

2

We can use an extra variable - a “parameter”- to rewrite the equation as: x=t

y=t2

This is just a simple example, where it may appear that using the direct relation is much more convenient, but in other cases, the direct relation is more complicated and using the parametric form is a lot easier.

Example:

We want to plot a helix. A helix is a 3D spiral shape whose equations (in

parametric form) are:

tz

ty

tx

sin

cos

So we execute the following commands:

t=-10*pi:pi/100:10*pi; x=cos(t); y=sin(t); z=t; plot3(x,y,z) grid on


Notice that the function plot3 has the same options as plot, meaning that you

can change the color, line type and marker type as follows:

plot3(x,y,z,'r--o')

This will change the line to a red, dashed line with circular markers. You can also add a grid, title, legend, x, y and z labels.

Commands:

t=-2*pi:pi/100:2*pi; x=cos(t); y=sin(t); z=t; plot3(x,y,z,'r--o',z,y,x,'b-*') grid on title('Helix') xlabel('x-axis') ylabel('y-axis') zlabel('z-axis') legend('Helix1','Helix2')

Note: The figure here has been rotated to get a better view of the plots. The default

view may look different when running the commands in MATLAB. Rotation will be

explained later in these notes.


Notice the difference between the following 3D plots of a helix. Look at the

ranges of t in each case.

t=-pi:pi/100:pi; plot3(cos(t),sin(t),t)

t=-pi:pi/100:pi; plot3(cos(3*t),sin(3*t),3*t)

t=-5*pi:pi/100:5*pi; plot3(cos(t),sin(t),t)

t=-5*pi:pi/100:5*pi; plot3(cos(t/2),sin(t/2),t/2)


Property Editor:

Instead of using functions like “title”, “xlabel” and editing the colors of the line

using strings like 'r--o' you can do all this editing using the Property Editor.

Click Edit Figure Properties

This opens the property editor beneath the figure, there you can manually

change the different parts of the figure:

1. The background of the figure (default: grey)

2. The axes background (default: white)

3. The line/curve (default: blue, solid, no marker)


The Figure Background:

Notice that the background is selected if the black squares appear as in the

figure. Here you can change the grey background from the “Figure Color” as

indicated.

The “Colormap” will be used later on in surface plots. However, here in line

plots it has no function.

Selection


The Figure Axes:

To select the axes (the yellow part here) click on them. The selection square

markers will show the selected part. Now you can change the color, the grid,

the limits and the labels.


The Line or Curve:

Here you can change the line type, color and thickness as well as the marker

type, facecolor and outline color.

Here’s an example: (Zooming in on a modified line):


You can also use Insert and choose any of the options shown.

For some options, like adding or removing the grid, you can right click on the

axes (or the line, whichever part you wish to modify) and do what you want

using the menu that appears.

It is important to note that all these options are available for 2D plots as well.


3D Surface Plots:

In order to draw a surface, you first need to create a grid in the x-y plane, and

then, at every point in this grid, specify the height (z-coordinate) of the surface

by using z as a function of x and y.

Notice that when dealing with lines we use vectors of the same length. We

take the 1st element in x with the 1st element in y and in z to create the 1st

point, then the 2nd element in x, y and z to create the 2nd point and so on…

To create a grid we want the 1st element in x with ALL the elements of y, then

the 2nd element of x with ALL the element of y and so on…

This is what a grid

looks like.

You don’t actually

see this grid, you

visualize it.

After providing

MATLAB with the

“height” of the surface

at each point this is

how it looks like.


Example:

Plot the function yxz sincos for x and y ranging from 0 to 10.

Commands:

x=0:0.2:10; y=x; [x,y]=meshgrid(x,y); z=cos(x).*sin(y); mesh(x,y,z) figure surf(x,y,z) figure surf(x,y,z) shading interp

This command creates the “grid” i.e.

instead of having two vectors x and y

(each 1x51) we get two matrices (51x51).

Look at the workspace before and after

executing this to see the difference.

mesh(x,y,z) plots a wireframe

surface.

Notice that the gridlines of the surface are

colored while the spaces (or squares) in

between are white.

The highest points on the surface are red

while the lowest points are dark blue.

surf(x,y,z) plots a continuous,

colored surface.

Notice that the gridlines of the surface are

black while the spaces in between are

colored according to the height of the

surface points.


Additional options for 3D surface plots in the Plot Editor:

Colormap: Changes the gradient colors of the surface and mesh plots.

Rotate: (Works for all 3D line and surface plots)

1. Click the “Rotate 3D” button.

2. Go to the plot in the figure.

3. Click and drag to rotate.

4. If you wish to return to the original view, right click (with the rotation

cursor active) and choose “Reset to Original View”.

Executing the command:

>> shading interp

After plotting using “surf” removes

the black gridlines and creates a

smooth colored surface.

This command, however, has no

effect on mesh plots.

Alexandria University Mathematics 6 Faculty of Engineering March 2013

Lab Notes

MATLAB

Session (3)



Using “input” and “disp” functions

Input Function (input):

user_entry = input('prompt')

user_entry = input('prompt', 's')

Use “input” function when you want to ask the user to enter a certain value then you can save it in a certain variable.

Example:

>> x = input(‘Enter the value of x: ‘)

The previous command displays the following in the command window:

Enter the value of x:

Now the user should type in the value of x, say 10. So type 10 and click enter. This way the value 10 is assigned to the variable x and it appears in the workspace.

Example:

>> word = input(‘Enter password: ’, ‘s’)

The previous command has an extra input ‘s’ which means that whatever the

user types in will be treated as a string not a numerical value. So if you type in

lab3 then lab3 will be treated as a string and stored in the variable word.

Display Function (disp):

Use this function to display a certain string or array to the user in the

command window. Here whatever you type in as an input to the function

disp will not be stored anywhere, it will just be displayed.

Example:

>> disp(' Corn Oats Hay')

>> disp(rand(5,3))


Results in:

Corn Oats Hay

0.2113 0.8474 0.2749

0.0820 0.4524 0.8807

0.7599 0.8075 0.6538

0.0087 0.4832 0.4899

0.8096 0.6135 0.7741

MATLAB m-files:

Instead of writing commands and executing them one by one in the command

window we can create a file, write all commands and then run the file. There

are two types of m-files:

SCRIPTS:

No inputs or outputs defined

Consist of a sequence of instructions

You can choose any filename to save the script

FUNCTIONS:

Can have inputs and outputs

Must start with the following line: function [op1,op2,...]=function_name(ip1,ip2,....)

The filename must be the same as the name specified in the previous

(first) line i.e. in this example it should be function_name.m

NEW m-file


Note that both scripts and functions have the extension “.m”, but the content

of each type is different as mentioned earlier.

Example:

Write a function and a script that count the number of elements of a given

matrix. Test your program for the matrix:

402

001

654

321

Now to do that you can use the function numel( )which directly gives the

number of elements in the given matrix. Another solution would be to use the

function size which had 2 outputs: the number of rows and the number of

columns, then multiply the two results.

SCRIPT:

A=input('Enter the input matrix: '); x=numel(A); disp(['The number of elements in the given matrix is ', num2str(x)])

Or use this: disp('The number of elements in the given matrix is:') disp(x)

Where the function “num2str” converts x from a numerical value to a string. We use this function because the input to disp must be only one input; either a string or a numerical value/array. So we construct a vector whose elements are all strings, the first part is the string ‘The of elements in the given matrix is ’ and the other is the number in x converted to string. To run the script and execute the commands in the script, there are two ways:

1. Type the filename in the command window and click enter >> lab3_script1

2. Click the “Run” button at the top of the m-file window:


FUNCTION:

Here we shall try using size instead of numel function [x]=lab3_fn1(A) [r c]=size(A); x=r*c;

Now the function lab3_fn1 can be used the exact same way you use numel or any other built-in function, as long as the function is saved in your current directory. >> B=[1 2 3;4 5 6;1 0 0;2 0 4]

B =

1 2 3

4 5 6

1 0 0

2 0 4

>> numel(B)

ans =

12

>> lab3_fn1(B)

ans =

12


Write a function that replaces a specific column of a given matrix by some

other column vector. Test your program on the matrix in the previous problem.

Try replacing the second column by the vector v = [ 1 1 0 0]t.

function [A_new]=lab3_fn2(A,x,n_col) A_new=A; A_new(:,n_col)=x;

The last line means that we replace all rows in the column number n_col in

the matrix A by the vector x (which must be a column vector whose size is

suitable for this replacement to occur.

>> lab3_fn2(B,[1;1;0;0],2)

ans =

1 1 3

4 1 6

1 0 0

2 0 4

Write a script file that takes as input a matrix and displays a menu of elementary row operations (interchanging rows, addition of a multiple of a row to another, multiplying a row by a non-zero constant) together with an exit option. Display the resulting matrix after performing the desired row operation.

We will use the switch function:

Syntax:

switch variable

case value1

....

case value2

....

:

:

otherwise

....

end


ANSWER:

A=input('Enter the matrix: '); disp('Which of these row operations do you want to execute?') disp('(1)Interchanging Rows') disp('(2)Multiplying a row by a Scalar') disp('(3)Adding Two Rows') x1=input('Enter 1,2 or 3 for the required row operation: '); switch x1 case 1 disp('Which rows do you want to interchange?') r1=input('Number of 1st row: '); r2=input('Number of 2nd row: '); temp=A(r1,:); A(r1,:)=A(r2,:); A(r2,:)=temp; disp(A) case 2 c=input('Scalar: '); r=input('Number of row: '); x=c*A(r,:); A(r,:)=x; disp(A) case 3 r1=input('First row: '); r2=input('Second row (in which the addition operation

is executed: '); x=A(r1,:)+A(r2,:); A(r2,:)=x; disp(A) end

Write a function to plot a circle. (Hint: use the parametric representation of

the circle).

A circle of radius “a” is written in parametric form as:

X=a*cos(t)

Y=a*sin(t)

NOTE:

A function may not have numerical outputs, such as this one, so we omit the

output part from the first line and we just write

function fn_name(ip1,..)


So the function will be written as follows:

function plotcircle(r) t=0:0.01:2*pi; x=r*cos(t); y=r*sin(t); plot(x,y) axis square

Write a script and a function according to the description:

1) Write a program to plot the function Acos(wt). The user should be able to

specify the amplitude A, the frequency w, the range of t (starting point, end

point and step size separately) as well as the line color, type and marker

type.

Solution:

Script:

ts=input('Enter the starting point of the range: '); tf=input('Enter the end point of the range: '); tstep=input('Enter the step: '); A=input('Enter the Amplitude of the Sine Wave: '); w=input('Enter the value of w in Acos(wt): '); spec=input('Specify the line color, type and marker: ','s'); t=ts:tstep:tf; x=A*sin(w*t); plot(t,x,spec) title('Sine Function') xlabel('x') ylabel('y') grid on

This makes the y-axis

appear the same length

as the x-axis so the

circle doesn’t appear

oval.


Function: (the filename is plot2Dsin.m)

function plot2Dsin(ts,tf,tstep,A,w,spec) t=ts:tstep:tf; x=A*sin(w*t); plot(t,x,spec) title('Sine Function') xlabel('x') ylabel('y') grid on

Exercises: 1- Write a function that solves a 33 system of linear equations using

Cramer’s rule. Your program should test whether the input matrix is singular in which case displays an error message. Make use of the function you created in problem 2. Test your program using the following system of linear equations:

x – y + z = 3, x + 3y + z = –1, x – 3z = –2.

2- Write a function that computes the mean, variance and standard deviation of a given set of data points. Test your program using the following data set: 69x,93x,75x,90x 4321 . Display helpful

messages. Compare your results with those obtained using the corresponding MATLAB built-in functions. (Hint: the estimates for the

mean, variance and standard deviation are given by

n

1iix

n

1m ,

)1n/()mx(vn

1i

2i

and vs , respectively)

HINT: Cramer’s Rule:

Consider a system of linear equations represented in matrix multiplication form as follows:

Ax=b Where A is invertible.

Then the theorem states that:


where Ai is the matrix formed by replacing the ith column of A by the column vector b.

Example:

2x + y + z = 3 x – y – z = 0 x + 2y + z = 0

Coefficient Matrix is D:

Evaluating each determinant, we get:

Cramer's Rule says that x = Dx ÷ D, y = Dy ÷ D, and z = Dz ÷ D. That is:

x = 3/3 = 1, y = –6/3 = –2, and z = 9/3 = 3


PART (II)

Introduction:

In this session, you will learn how to use some programming functions in

MATLAB. These functions include “for”, “if”, “switch” and others. Such

functions are best explained through examples. This session focuses on “for”.

for:

Use “for” to create a loop, i.e. when you want a certain command or a set of

commands to be executed several times.

for variable = initval:step:endval

statement

...

statement

end

Where the “variable” is the counter, you can give it any name, it’s just a

variable. For-loops are best explained through examples.

Example (1):

Use a for-loop to sum the numbers from 1 to 5.

Answer:

s=0; for n=1:5 s=s+n; end disp(s)

Now we shall explain how this works:

The idea is to create a variable, give it an initial value and use the loop to add

the numbers inside this variable one by one:


Snew = Sold + n

n Old value of s New value of s (s=s+n)

1 0 1+0=1 2 1 2+1=3

3 3 3+3=6 4 6 4+6=10

5 10 5+10=15

Notice that n is a vector (1:5 = [1 2 3 4 5]) so when we enter the loop for the

first time, n takes the value of the first element, and so n=1.

In the second iteration (step), n takes the following element’s value - which is 2

- and so on… (see the table above).

Example (2):

Write a program (script) to sum up the first 10 even numbers (excluding zero).

Answer:

s=0; for n=0:2:20 s=s+n; end disp(s)

OR

s=0; for n=1:10 s=s+2*n; end disp(s)

Both solutions are correct. However, if you don’t know that the first 20 even

numbers are from 2 to 20 (like for example if the question asked you to sum up


the first 57 or 396 even numbers!), then the second solution is the more

appropriate one.

For the second solution:

Snew = Sold + 2n

n Old value of s New value of s (s=s+2n)

1 0 2*1+0=2 2 2 2*2+2=6

3 6 2*3+6=12 4 12 2*4+12=20

5 20 2*5+20=30

6 30 2*6+30=42 7 42 2*7+42=56

8 56 2*8+56=72 9 72 2*9+72=90

10 90 2*10+90=110

Note that each time we add an even number; because when n takes the values

1, 2, 3, 4 … 10, 2n takes the values 2, 4, 6, 8 ...20.

Example (3):

Write a function m-file to calculate the factorial of a given integer. Use the

function name “fact”.

Answer:

function f=fact(n) f=1; for i=1:n f=f*i; end

e.g.: The factorial of 5 5! = 5 x 4 x 3 x 2 x 1 = 1 x 2 x 3 x 4 x 5

In general n!= n(n-1)(n-2) … 1


Here we initialize f to 1, because any number multiplied by 1 stays as it is.

The loop here keeps the result in f, each iteration it multiplies the previous

value of f to the following integer in the sequence from 1 to n:

To try out the function, do not click “Run”. Instead you go to the command

window and try the function for any value of n – say 5:

>> fact(5)

n f (Old Value) f (New Value) f=f*n

1 1 1*1=1

2 1 1*2=2 3 2 2*3=6

4 6 6*4=24 5 24 24*5=120

The result:

ans = 120

>> fact(0)

In this case we have f=1, then in the loop n=1:0 (so this means a vector whose

starting point is 1 and end point is 0 with a step equal to 1, and 1+1=2 > 0.

Hence n is an empty matrix) and the for-loop is not executed at all, therefore f

remains 1.

ans =1

Indeed: 0!=1


NOTE:

MATLAB actually has built-in functions to do some of the examples introduced

here, but still the examples show you how to create a for-loop and trace the

code.

sum(x) This function sums up the elements of x

factorial(x) This function computes the factorial

Example (3):

Create the vector *1 2 4 8 16 …. 128+

Answer:

A=[]; for i=0:7 A=[A 2î]; end disp(A)

This is what we do:

1. First we create an empty matrix A (or else when we reach the step inside

the loop A will be undefined).

2. A=[A 2î] will concatenate the new A and the old A to create the vector

step by step.

To see how it works, we’ll remove the semicolon and display the value of i:

A=[]; for i=0:7 disp('i=') disp(i) A=[A 2î] end


Output:

i= 0 A = 1 i= 1 A = 1 2 i= 2 A = 1 2 4 i= 3 A = 1 2 4 8 i= 4 A = 1 2 4 8 16 i= 5 A = 1 2 4 8 16 32 i= 6 A = 1 2 4 8 16 32 64 i= 7 A = 1 2 4 8 16 32 64 128


Example (4):

Write a script to create a 9x8 matrix whose rows are the same as the vector A

in example (3). You’ll need to regenerate A, don’t use a previously created

vector.

Answer:

Here we will use two for loops, one inside the other:

A=[]; for i=1:9 B=[]; for j=0:7 B=[B 2^j]; end A=[A;B]; end disp(A)

1. First we initialize A as an empty matrix.

2. The first loop repeats 9 times, once for each row.

3. The code within the outer loop is the same as the one in example (3),

including initializing the matrix B as an empty matrix to make it ready for

concatenation.

4. After the inner loop is finished, matrix B contains the vector [ 1 2 4

…128+ and is vertically concatenated with A (which starts out as an

empty matrix, then one row, then two and so on…)

5. The last line displays the matrix A

Note: This example is for you to learn how to trace two loops one embedded

inside the other. However, you can create the vector *1 2 4 … 128+ and then

use it in one loop to create the matrix. This way you’ll use two separate loops.

B=[]; for j=0:7 B=[B 2^j]; end A=[]; for i=1:9 A=[A;B]; end disp(A)


Example (5):

Suppose you want to create a script that enables a user who does not know

how to write a matrix using MATLAB to do so by prompting the user to enter

the elements one-by-one. Also, the user should be able to decide the size of

the matrix.

Answer:

m=input('Enter the number of rows: '); n=input('Enter the number of columns: '); A=[]; for i=1:m for j=1:n x=input('Enter the element: '); A(i,j)=x; end end disp(A)

First, we ask the user for the number of rows and columns, and we use these in

the loops:

The outer loops is for the rows and the inner one for the columns

The total number of iterations (number of times the commands inside the

inner loop is repeated) is the product of m and n:

For example: if m=3 and n=5: Total number of iterations = 3x5 = 15

i=1

j=1

j=2

j=3

j=4

j=5

i=2

j=1

j=2

j=3

j=4

j=5

i=3

j=1

j=2

j=3

j=4

j=5


This fills up the matrix A row by row:

A(1,1) = …

A(1,2) = …

A(1,3) = …

and so on…

Example (6):

Write a script using a for-loop to produce the following plot where the

functions drawn are sin(x), sin(2x), sin(3x) and sin(4x):

x=0:0.1:10; for n=1:4 subplot(2,2,n) plot(x,sin(n*x)) end

Do not forget to define x before the loop (if it’s inside, it

doesn’t matter but there’s no use repeating it every iteration if

it’s the same in all plots).

Alexandria University Mathematics 6 Faculty of Engineering March 2013

Lab Notes

MATLAB

Simulink and GUI



2

I. Simulink

Simulink is a graphical extension to MATLAB for modeling and simulation of

systems. With Simulink, you can build a model for any system and simulate it

using a block diagram.

To open Simulink, click on the indicated icon and wait for the toolboxes to

load:

The Simulink Library will open:

This is where you will find all the blocks you need to build your system.

Simulink Icon

Open New

Model


3

Basically, to simulate a system using Simulink, you need to do the following

steps:

1. Open a new Simulink Model

2. Drag and drop blocks from the Simulink library into the model

3. Adjust the attributes of the blocks

4. Connect the blocks (CTRL + click)

5. Run

Simulink Library:

The library has some basic toolboxes and other specialized toolboxes or

“blocksets” (like Power systems, communications, image processing,

Aerospace … etc.)

For starters, take a look at the following blocksets:

1) Sources: This one contains signal sources which you can use as inputs to

your system; a sine wave, a unit step function, square pulses….

2) Sinks: Contains different blocks to view the output. You can use a scope

(which looks like the screen of an oscilloscope) or a display (if the output is just

a number not a signal). You can also you “To Workspace” or “simout” to

transfer the output value to the workspace as a variable.

The figure below shows the available sinks:

3) Commonly Used Blocks: This is a useful blockset that contains a collection of

some of the most important basic blocks, including some sinks, sources and

mathematical operators.


4

Example:

We start with this simple model as an example of how to use Simulink:

Drag and drop the shown 4 blocks from the Simulink library to the new model.

Sine Wave From “Sources”

Derivative From “Continuous”

Gain From “Commonly Used Blocks” OR “Math Operations”

Scope From “Sinks”

To connect the blocks you can drag a line from each block to the following one

or click on the first block, press CTRL and click on the other block.

When the mouse pointer turns to a “Plus” sign (+) you can start dragging.

After connecting the blocks, the model looks like this:

This is where

you click to

RUN the model


5

To change the parameters of each block, there are two methods you need to

know:

1. Manually, by double-clicking on the block and changing them.

2. Using an m-file to control the blocks with MATLAB commands.

The manual method is easy and straightforward, for example, this is what you

get when clicking on the Sine Wave block:

If you’re using a new block and don’t know how to change the parameters,

click on the HELP button (next to “OK” and “Cancel”).


6

Using an m-file to change the block parameters (attributes):

First, you need to save the model. Having done that, open a new m-file (script)

and use the following functions:

set_param(Name of File/Name of Block, Name of attribute, Value)

The function “set_param” changes the value of a certain parameter. For

example, to change the amplitude of the sine wave to 2 and the gain factor to

0.5 use the following commands (assume the model name is “mysim1”):

set_param('mysim1/Sine Wave','Amplitude','2'); set_param('mysim1/Gain','Gain','0.5');

If you want to run the model using the m-file (instead of clicking the button)

you can use the following command:

sim('mysim1')

NOTE: When you run the file, nothing appears! But this is normal. You need

to double click on the scope in order to view the output after running the

model.


7

Using the SCOPE:

By double clicking on the scope, the output signal will appear in yellow on a

black background. If the output doesn’t fit the window, right click on the figure

and click “Autoscale”.

If you want to view or use the plot OUTSIDE Simulink, click on the “Data

History” tab.

This button takes

you to the scope

parameters


8

Once you have done these settings, when you run the model, the variable

ScopeData appears in the workspace:

Make sure this is

unchecked

Make sure that

“Save Data to

Workspace” is

checked.

The variable name is the name of the variable

that shall appear in the workspace, it should

have two columns, one containing the x-values

and the second containing the y-values.

The format should be “Array” (in order to have

the data in matrix from).


9

This is an example of an m-file to control and run the model. The script also

plots the output using data from the scope:

set_param('mysim1/Sine Wave','Amplitude','3'); set_param('mysim1/Gain','Gain','2'); sim('mysim1') x=ScopeData(:,1); y=ScopeData(:,2); plot(x,y)

Output:

NOTES:

You can search for blocks in the Simulink Library by typing their name in

the search bar.

You can resize, recolor and rename the blocks in the Simulink model as

well as the background. Right click on any of the blocks to view the

options.

You can rotate any block by selecting it and clicking CTRL+R


10

II. Graphical User Interface (GUI) using MATLAB

GUIDE: Graphical User Interface Development Environment

By clicking on the indicated icon, you can access the GUIDE. Use the GUIDE to

create a user-friendly interface, like this for example:

Here’s how to get started: When you click the icon, this appears. Click OK.

GUIDE Icon


11

Example:

The following GUI can plot a variety of functions. You can choose the function

from a drop-down menu. The range of the horizontal axis can be adjusted

using the edit boxes. The radio buttons allow you to choose the color of the

plot (blue or red). The checkbox applies a grid to the figure if checked.

To plot, click the plot button. To clear the figure, click the clear button.

If you want to edit the name or text in any of the GUI elements, double click

the element. The “Property Inspector” appears. Go to the “String” option and

type whatever you want. For the drop-down menu, a new line represents a

new selection.

It is preferable to leave the edit boxes blank. The default text is “Edit Text”.

So go to the property inspector and delete the words “Edit Text” and leave the

string blank. You could also put a default value if you want.

This is where you will

create the interface

These are the elements you can add to your

interface; buttons, axes, edit boxes…etc.

Run the GUI Access the GUI

m-file


12

IMPORTANT: Every element (button, box, menu…) has a Tag listed in the

property inspector. This is the name you will use later to call the element in the

m-file. You can use the default name, or you can rename it to make it easier for

you to remember what the button does when you are writing the m-file that

controls your GUI.

For example, the default tags for the radio buttons are radiobutton1 for the

Red button, and radiobutton2 for the Blue button. You can rename them

redbutton and bluebutton.


13

Now to the important part: Programming your GUI:

1. Save the GUI figure.

2. Once you save it, an m-file appears. If you want to access this m-file later

you can click on the m-file editor button in the figure.

3. The m-file already has lines of code written. DO NOT ERASE THEM!

4. Each GUI element has a section in which you can write your own

commands. For example, if you want to tell the m-file what to do if the

PLOT button is pressed, go to the line that starts with: function pushbutton1_Callback(hObject, eventdata, handles)

Now take a look at how the final m-file: These are the parts of the m-file that were modified: % --- Executes on button press in checkbox1. function checkbox1_Callback(hObject, eventdata, handles) % hObject handle to checkbox1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) s=get(handles.checkbox1,'value'); axes(handles.axes1); if s==1 grid on else grid off end

% --- Executes on button press in pushbutton1. function pushbutton1_Callback(hObject, eventdata, handles) axes(handles.axes1); x0=str2num(get(handles.edit1,'string')); xf=str2num(get(handles.edit2,'string')); x=x0:0.01:xf; ch1=get(handles.popupmenu1,'value'); switch ch1 case 1 y=sin(x); case 2 y=cos(x); case 3 y=exp(x); case 4 y=log(x); end ch2=get(handles.radiobutton1,'value'); if ch2==1 color='r'; else color='b'; end plot(x,y,color)


14

% --- Executes on button press in pushbutton2. function pushbutton2_Callback(hObject, eventdata, handles) % hObject handle to pushbutton2 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) cla set(handles.edit1,'string',''); set(handles.edit2,'string',''); set(handles.popupmenu1,'value',1); set(handles.checkbox1,'value',0); set(handles.radiobutton1,'value',1); set(handles.radiobutton2,'value',0); axes(handles.axes1); grid off

As you can see, the only buttons that need modifications in the m-file are the PLOT, CLEAR and Grid on/off buttons. More about the functions used: set To change or set the value of an element (on/off, change the text…etc.) get To get the data entered by the user from the GUI (checked/unchecked, text, which selection…etc.) cla Clear Axes axes To call the axes; you need to use this before you plot. If you have more than one plotting regions, you need this to specify which one the program should use. str2num Converts a string to a number. The opposite is num2str.

To run the GUI, click the “Run” button in the GUI figure, NOT the m-file!

Also, if you want to open the figure again anytime, you must do so using the GUIDE and clicking the “Existing GUI” tab then searching for your file from there, not just double clicking the figure file. One final note: If you want to send the GUI or transfer it from one place to another, you must send or transfer BOTH the figure AND the m-file.

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