1

Computer Applications Lab

Lab 1

Introduction to Matlab

Chapter 1

Sections 1,2,3,5

Dr. Iyad Jafar

Adapted from the publisher slides

What is Matlab ?

• A Computer programming language and software environment to manage data interactively

• Originally developed in 1970s for applications involving matrices, linear algebra and numerical analysis

• Maintained and sold by the MathWorks, Inc.

• Functionality: manage variables, import/export data, calculations, generate plots, and more ……

• Toolboxes such as image processing, statistics, control, financial analysis, and more …

2

2

Matlab Desktop

3

Toolbar

Menus

Current Directory

Command Window

Command History

Workspace

Matlab as interactive calculator

>> 8/10

ans=

0.8000

>> 5*ans

ans=

4

>> r=8/10

r =

0.8000

>> r

r =

0.8000

>> s=20*r

s =

16 4

NOTES

• ans is a built-in special variable to store the

result of the last computation. Can be reused by

typing ans

• variables defined by the assignment operator ‘=‘

are stored in memory and can be used again using

their names

3

Matlab as interactive calculator

• Matlab retains your previous keystrokes.

• Use the up-arrow key to scroll back through the commands.

• Press the key once to see the previous entry, and so on.

• Use the down-arrow key to scroll forward. Edit a line using the left-and right-arrow keys the Backspace key, and the Delete key.

• Press the Enter key to execute the command.

5

Scalar Arithmetic Operations

6

Symbol Operation

Mathematical

Syntax

Matlab Syntax

^ Exponentiation ab a^b

* Multiplication ab a*b

/ Forward Division a/b a/b

\ Backward Division a\b a\b

+ Addition a+b a+b

- Subtraction a-b a-b

4

Order of Precedence of

Arithmetic Operators

7

Note: if

precedence is

equal, evaluation is

performed from left

to right.

Parentheses

Exponentiation

Multiplication & division

Addition and subtraction

Examples of Precedence

>> 8 + 3*5

ans=

23

>> 8 + (3*5)

ans=

23

>> (8 + 3)*5

ans=

55

>> 4^2-12- 8/4*2

ans=

0

>> 4^2-12- 8/(4*2)

ans=

3 8

5

Examples of Precedence

(continued) >> 3*4^2 + 5

ans=

53

>> (3*4)^2 + 5

ans=

149

>> 27^(1/3) + 32^(0.2)

ans=

5

>> 27^(1/3) + 32^0.2

ans=

5

>> 27^1/3 + 32^0.2

ans=

11 9

The Assignment Operator ‘=’

• Typing x = 3 assigns the value 3 to the variable x.

• We can then type x = x + 2. This assigns the value 3 + 2 = 5 to x. But in algebra this implies that 0 = 2.

• In algebra we can write x + 2 = 20, but in Matlab we cannot.

• In Matlab the left side of the assignment operator ‘=‘ must be a single variable.

• The right side must be a computable value.

10

6

Variables in Matlab

No declaration or dimension statements are required to

define variables in Matlab.

To define a new variable, simply write the variable name

followed by the assignment operator and the value to be

stored in the variable.

num_students = 25

If the variable already exists, Matlab changes its contents and,

if necessary, allocates new storage.

To view the value of the variable, type its name on the

command prompt and hit Enter.

Variable names consist of a letter, followed by any number

of letters, digits, or underscores. 11

Special Variables and Constants

12

Variable Description

ans Temporary variable containing

the most recent answer

i and j The imaginary unit √−1

inf Infinity

NaN Indicates an undefined numerical result

pi The number π

7

Commonly Used Mathematical

Functions

13

NOTES

• Trigonometric functions in Matlab use radian measure

• cos2(x) is written (cos(x))^2 in Matlab

Function Matlab Syntax

ex exp(x)

√x sqrt(x)

ln x log(x)

log10 x log10(x)

cos x cos(x)

sin x sin(x)

tan x tan(x)

cos−1 x acos(x)

sin −1 x asin(x)

tan −1 x atan(x)

|x| abs(x)

Complex Number Operations

• The number c1= 1 –2i is entered as follows:

c1 = 1-2i

• An asterisk is not needed between i or j

and a number, although it is required with

a variable, such as c2 = 5 - i*c1.

• Be careful ! The expressions y = 7/2*i and

x = 7/2i give two different results

y = (7/2)i = 3.5i

x = 7/(2i) = –3.5i

14

8

Commands for managing the

work session

15

Command Operation

clc Clears the command window

clear Clears all the variables from the workspace

clear v1 v2 Clears variables v1 and v2 only

exist(‘var’) Check if the variable or function exist

quit Exits Matlab session

who List the names of the variables defined in the work space

whos List the names of the variables defined in the work space

and their details

Semicolon (;) Suppresses the display of results in the command window

Ellipsis (…) Line continuation

Numeric Display Formats

16

Format Operation Example

short Four decimal digits (the default

format);

13.6745

long 16 decimal digits; 17.27484029463547

short e Five digits (four decimals) plus

exponent 6.3792e+03

long e 16 digits (15 decimals) plus

exponent; 6.379243784781294e–04

9

Command Resolving in Matlab

What happens when you type problem1 in the

command prompt ?

1. Matlab first checks to see if problem1 is a variable and if so,

displays its value.

2. If not, Matlab then checks to see if problem1 is one of its

own commands, and executes it if it is.

3. If not, Matlab then looks in the current directory for a file

named problem1.m and executes problem1if it finds it.

4. If not, Matlab then searches the directories in its search

path, in order, for problem1.m and then executes it if found.

17

The Help Navigator

Available through the help menu

Contents

◦ Contents: contents listing tab

◦ Index : a global index tab,

◦ Search: a search tab having a find function and

full text search features

◦ Demos: a bookmarking tab to start built-in

demonstrations.

18

10

Help Functions

• help funcname: Displays in the Command window a description of the specified function funcname.

• lookfor topic: Displays in the Command window a brief description for all functions whose description includes the specified key word topic.

• doc funcname: Opens the Help Browser to the reference page for the specified function funcname, providing a description, additional remarks, and examples.

19

1

Computer Applications Lab

Lab 2

Arrays in Matlab

Chapter 2

Sections 1,2,6,7

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Introduction

Arrays in Matlab

Vectors and arrays

◦ Creation

◦ Addressing

◦ Functions

Cell and structure arrays

2

2

Introduction

Matlab has the capability of handling arrays of numbers and many data types

Array manipulation in Matlab is much simpler when compared to other programming languages (C = A + B is done without writing loops)

This makes Matlab the choice for many engineering application that require processing of data sets.

3

Frequently Used Arrays in Matlab

Numeric:

◦ Single and double precision

◦ Signed integers; Int8, int16, in32

◦ Unsigned integers; uint8, uint16, uint32

Character: array of strings

Logical: contains ‘1’ or ‘0’ that corresponds to

‘True’ or ‘False’

Cell and Structure arrays

◦ Data structures that allows the storage of different data

types such as strings and numbers

4

3

Vectors in Matlab

Vectors are special case of arrays,

with one of its dimensions being 1.

The vector p can be specified by

three components: x, y, and z, and

is written as

p = xi + yj + zk

In Matlab, vector p can be written

as

p= [x, y, z]

Matlab can use vectors having

more than three elements.

5

NOTE

• We can define vectors in

Matlab with more than

three elements

• scalar values in Matlab

are treated as a vector of

size 1x1

Creating Vectors in Matlab 1. To create a row vector, separate the elements by commas or spaces. For

example,

>> p = [3,7,9]

p =

3 7 9

2. You can create a column vector by using the transpose notation (').

>> p = [3,7,9]'

p =

3

7

9

3. You can also create a column vector by separating the elements by

semicolons. For example

>> g = [3;7;9]

6

4

Creating Vectors in Matlab

4. You can create larger vectors by appending one vector to

another. For example, to create the row vector u whose first

three columns contain the values of r = [2,4,20] and whose

fourth, fifth, and sixth columns contain the values of w = [9,-

6,3], you type u = [r,w]. The result is the vector u = [2,4,20,9,-

6,3].

7

NOTE

• In order to append vectors they should be either

row vectors or column vectors. Otherwise,

transposition is required before appending.

• Example >> u = [1,3,4] ;

>> r = [6;8;9] ;

>> z = [u,r’]

z =

[1,3,4,6,8,9]

Creating Vectors in Matlab

5. The colon operator (:) easily generates a large vector of regularly spaced elements.

Syntax: ◦ x = [m:q:n] to create a vector x of values with a spacing q.

◦ The first value is m. The last value is n if m–n is an integer multiple of q. If not, the last value is less than n.

◦ Number of elements = ((n-m)/q )+ 1

Example:

typing x = [0:2:8]creates the vector x = [0,2,4,6,8]

typing x = [0:2:7]creates the vector x = [0,2,4,6].

Default increment: ◦ If the increment q is omitted, it is presumed to be 1.

◦ Thus typing y = [-3:2]produces the vector y = [-3,-2,-1,0,1,2].

8

5

Creating Vectors in Matlab

6. The linspace command also creates a linearly spaced row vector, but instead you specify the number of elements rather than the increment.

Syntax:

◦ linspace(x1,x2,n) where x1 and x2 are the lower and upper limits and n is the number of points.

◦ Increment = (x2-x1) / (n-1)

Example:

◦ linspace(5,8,31)is equivalent to [5:0.1:8].

Default spacing: If n is omitted, the spacing is 1.

9

Magnitude and Length of a

Vector

• The length(v) command gives the number

of elements in the vector.

• The magnitude of a vector x having

elements x1, x2, …, xn is a scalar, given by

and is the same as the vector's

geometric length.

10

2 2 21 2 n x x + ... x

6

Matrices

A matrix has multiple rows and columns. For

example, the matrix

has four rows and three columns.

Vectors are special cases of matrices having one

row or one column.

11

Creating Matrices in Matlab

1. For small matrices, type the elements such that

◦ Elements in the same row are separated by

commas or spaces

◦ Rows are separated by semicolons

Example

>>A = [2,4,10;16,3,7];

Creates the matrix

12

7

Creating Matrices in Matlab

13

Addressing Array and Vector

Elements

To access a single element use Matrix A(row number , column number)

Vectors v(element position)

or

Row vectors v (1, column number)

Column vectors v(row number ,1)

14

Row =2 , column = 3

Row =1 , column = 1

8

Addressing Array and Vector

Elements

To access a group of elements, rows, columns, or

subarrays of arrays use the colon symbol

Examples

15

Array Addressing : Example

16

9

Additional Array Functions

17

Exercise: open the Matlab help browser and read the documentation of the

following array-related functions: find(A), max(A) , min(A), cat(n,A,B,C)

Function Description

size(A) Returns a row vector [m n] containing

the size of the mxn array A

sort(A) Sorts each column of the array A in

ascending order and returns an array

of same size as A

sum(A) Sums the elements in each column of

the array A and returns a row vector

containing the sums

inv(A) Computes the inverse of array A

diag(A) Returns the elements along the main

diagonal of A

fliplr(A) Flips array A about it central column

flipud(A) Flips array A about it central row

Examples

18

10

Multidimensional Arrays

Matlab supports multidimensional arrays.

Examples

◦ Three dimensional arrays have the dimension

m x n x q

◦ four dimensional arrays have the dimension

m x n x q x r

The first two dimensions are the row and the

column.

Higher dimensions are called pages.

19

Multidimensional Arrays

Create multidimensional arrays by defining the

pages then appending

Example

>> A = [1 , 2; 3 , 4] ;

>> A(:,:,2) = [5 , 6; 7 , 8];

Creates a three dimensional array with two

pages.

20

11

Cell Arrays A cell array is an array in which each element is a bin, or cell,

which can contain an array.

Information of different data types and dimensions can be stored in cell arrays.

• Creation ◦ You can create a cell array by using the cell function

myCell = cell(m,n)

creates empty cell array C with mxn cells

◦ Using the assignment operator and the curly brackets to define its elements.

myCell(1,1) = {‘CA LAB’} % creates the cell array my cell and initializes element (1,1)

myCell(1,2) = {0907311} % assign value to element (1,2)

21

Cell Arrays Creation

22

12

23

Accessing Cell Arrays

Cell indexing (Using paranthesis) Extracts cell(s) and places them in a new cell array Speed = C(3,4) places the contents of cell (3,4) of the array

C in the new variable Speed D = C(1:3,2:5) this places the contents of the cells in rows

1 to 3, columns 2 to 5 in the new cell array D. The new cell array will have three rows, fours columns, and 12 arrays.

Arithmetic operations are not allowed on cells data. They have to be converted to numeric type first (use the double function), or use Content indexing to extract the content.

Content indexing (Using curly brackets) Extracts the content of the cell and places it in a variable of

the same type speed = C {3,4} assigns ’30 mph’ to variable speed of type

string

24

13

Structure Arrays

Structures arrays are a class of arrays that allows you to store dissimilar arrays together. The elements in the structure are accessed using named fields.

Example

◦ In a student database (e.g. name, social security number, email address, test scores). There are four fields (4 field names): 3 string and 1 vector containing numerical elements.

◦ A structure consists of all this information for a single student and a structure array is an array of such structures for different students.

Creation

◦ Using assignment statements

◦ Using the struct function

◦ Structure arrays use the dot notation (.) to specify and to access the fields.

25

Example >> student.name='John Smith';

>> student.ssn='123-45-6789';

>> student.email='smithj@myschool.edu';

>> student.tests=[67,75,84];

>> student

student =

name: 'John Smith‘

ssn: '123-45-6789‘

email: 'smithj@myschool.edu‘

tests: [67 75 84]

>> size(student)

ans=1 1

26

14

Example (cont’d)

>> student(2).name='Mary Jones';

>> student(2).ssn='987-65-4321';

>>student(2).email='jonesm@myschool.edu

';

>> student(2).tests=[84,78,93];

>> student

student = 1x2 struct array with fields:

name

ssn

email

tests

27

Example (cont’d)

The previous structure array can be constructed using the struct function

>> student(1) =struct(‘name’, ’John Smith’, ’SSN’, ’123-45-6789’,

’email’,

’smithj@myschool.edu’,’tests’,[67,7

5,84])

>> student(2) =struct(‘name’, Mary

Jones, ’SSN’, ’987-65-4321’,

’email’, ’

jonesm@myschool.edu’,’tests’,[84,78

,93])

28

15

Structure Functions

29

Function Description

names = fieldnames(S) Returns the field names associated

with structure array S as names; a cell

array of strings

F = getfield(S,’field’) Returns the contents of the field ‘field’

in the structure array S.

isfield(S,’field’) Returns 1 if ‘field’ is the name of a

field in the stucture array S, and 0

otherwise

S = rmfield(S,’field’) Removes the field ‘field’ from the

structure array S

S = setfield(S,’field’,V) Sets the contents of the field ‘field’ to

the value V in the structure array S

S = struct(‘f1’,’v1’,’f2’,’v2’,…) Creates a structure array with the

fields ‘f1,’f2’, … having the values

‘v1’,’v2’, …

Accessing Structure Arrays Type a period after the structure array name followed by

the field name to access the contents of the field

student(2).name returns the value ‘Mary Jones’

You can assign the result to a variable in the usual way

name2=student(2).name

assigns the value ‘Mary Jones’ to the variable name2

To add more information to an existing database

◦ student(1).phone=‘555=1653’

◦ This adds a phone number field to the first student’s

◦ All other structures in the array will also have the phone number

field, but will contain an empty array until you give them values.

To delete a field from every structure in the array, use the

rmfield function.

new_struc = rmfield(array,’field’)

30

1

Computer Applications Lab

Lab 3

Arithmetic Operations on

Arrays

Chapter 2

Sections 3,4,5

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Element-by-element operations

Matrix operations

Special Matrices

Polynomial operations using arrays

2

2

Element-by-Element Operations

It is used to perform the desired mathematical operation between corresponding elements of arrays or by applying the operation between a scalar and each element of the array

3

Element-by-Element Operations

scalar-array operations ◦ Addition B = A + 4

◦ Subtraction B = A – 4

◦ Multiplication B = A * 4

◦ Division B = A / 4

◦ The operation is repeated for each element in the array separately.

Example

4

3

Element-by-Element Operations

Array-array operations

◦ The mathematical operation is repeated between

corresponding elements in the arrays

◦ Arrays should be of equal dimensions

Example

5

Note

Subtraction is

performed in similar

way

Element-by-Element Operations

Array-array operations

◦ For multiplication (division) use .* (./) to multiply

(divide) corresponding elements of the two arrays

Example

6

Note

Division is performed

in similar way

4

Element-by-Element Operations

7

Element-by-Element Operations

8

5

Element-by-Element Operations

Array-array operations

◦ For exponentiation, use .^ to raise each element in the

array to specified base

Example

9

Element-by-Element

Operations

Can be used to evaluate functions at a set of points

Example

evaluate f(x,y) = xy^2 + 8x at (1,2), (3,1), (5,9)

>> x = [1,3,5];

>> y = [2,1,9];

>> f = x.*(y.^2) + 8*x

ans =

12 27 445

10

6

Matrix Operations

We have discussed addition and subtraction earlier

Multiplication

11

Matrix Operations

12

• In MATLAB, use * to perform multiplication of matrices

7

Matrix Operations

Matrix division is more challenging topic than

matrix multiplication.

Matrix division uses both right and left division

operators, / and \, for various applications

One application is solving linear algebraic equations 6x+ 12y+ 4z= 70

7x–2y+ 3z= 5

2x+ 8y–9z= 64

>> A = [6,12,4;7,-2,3;2,8,-9];

>> B = [70;5;64];

>> Solution = A\B

Solution = 3 5 -2

The solution is x= 3, y= 5,and z= –2.

13

Matrix Operations

Solving the previous set of linear

equations can be done by matrix

inversion.

>> A = [6,12,4;7,-2,3;2,8,-9];

>> B = [70;5;64];

>> solution = inv(A) * B

solution =

3.0000

5.0000

-2.0000

14

8

Matrix Operations

Matrix exponentiation

◦ An is equivalent to multiplying the

matrix by itself n times

◦ A should be a square matrix

◦ AB , where B is a matrix, is not defined

15

Special Matrices

Identity matrix I

◦ eye(n) , eye(m,n) , eye(m,n)

All-ones matrix ◦ ones(n) , ones(m,n) , ones (size(A))

All-zeros matrix ◦ zeros(n), zeros(m,n), zeros(size(A))

16

9

Polynomial Operations Using Arrays

The polynomial

can be represented in MATLAB by

Use the function roots(A) to find the polynomial roots

>> c = roots ([1,5,6])

c = -2 -3

Use the function poly(c) to find the coefficients of the

polynomial from its roots.

>> c = [-1,2]

>> ploy(c)

ans = 1 -1 -2

This means that the answer is x2 – x – 2

17

n n 1 n 21 2 3 n 1f(x) = ax + a x + a x + ... + a

1 2 n 1A = [a ,a , ... , a ]

Polynomial Operations Using Arrays

Multiplication

Use the function conv(a,b) to compute the product of the two polynomials described by the coefficient arrays a and b. The two polynomials need not be the same degree. The result is the coefficient array of the product polynomial.

Division

Use the function [q,r] = deconv(num,den) to compute the result of dividing a numerator polynomial, whose coefficient array is num, by a denominator polynomial represented by the coefficient array den. The quotient polynomial is given by the coefficient array q, and the remainder polynomial is given by the coefficient array r.

18

10

Polynomial Operations Using Arrays

Example: if a = 9x3 – 5x2 + 3x + 7 and b = 6x2 + 2, then find a x b and a / b.

>> a = [9,-5,3,7] ; b = [6,0,2] ;

>> product = conv(a,b)

product =

54 -30 36 32 6 14

(This is equivalent to 54x5 – 30x4 + 36x3 + 32x2 + 6x + 14)

>> [Q,R] = deconv(a,b)

Q =

1.5000 -0.8333

R =

0 0 0 8.6667

19

Polynomial Operations Using Arrays

Evaluation of polynomials

The function polyval(a,x) evaluates a polynomial at specified values of its independent variable x, which can be a matrix or a vector. The polynomial’s coefficients of descending powers are stored in the array a. The result is the same size as x.

>> a = [9,-5,3,7];

>> x = [-2,3,4,5];

>> f = polyval(a,x)

f = -91 214 515 1022

20

1

Computer Applications Lab

Lab 4

Functions and Script Files

Chapter 3

Sections 1,2,3,4

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Passing function arguments

Common mathematical functions in

Matlab

Script files

User-defined functions

Importing and Exporting Data

2

2

Passing Function Arguments

3

Passing Function Arguments

4

3

Common Mathematical

Functions

5

Common Mathematical

Functions

>> x = 3 + 4j

>> abs(x)

ans = 5

>> angle(x)

ans = 53.1301

6

>> conj(x)

ans = 3 -4j

>> imag(x)

ans = 4

>> real(x)

ans = 3

4

Common Mathematical

Functions

7

>> x = [5,7,15] ;

>> y = sqrt(x) ;

y =

2.2361 2.6358 3.8730

>> ceil(y)

ans = 3,3,4

>> fix(y)

ans = 2,2,3

>> floor(y)

ans = 2,2,3

>> round(y)

ans = 2,3,4

>> sign(y)

Ans = 1 1 1

Common Mathematical

Functions

8

5

Common Mathematical

Functions

9

Common Mathematical

Functions

10

6

Script Files - Introduction

Up to this point, we have used Matlab in the interactive mode (commands are entered in the Command window) .

The interactive mode is desirable when we have few commands.

For larger problems we usually use a Script file.

Basically, the script file contains the same Matlab commands that we type on the command prompt (we don’t have to retype everything if we want to execute/change the commands again.

11

Script Files - Creation

Use the Matlab editor/debugger to create your script

file. Available from File>New>M-file or by clicking the

new M-file shortcut .

Enter your commands as desired

Use

◦ ‘;’ to suppress the result of commands

◦ ‘%’ to insert comments; inline or full-line comments

Script files are saved in Matlab with .m extension.

Run your script file by typing fileName.m on the

command prompt or click on the run shortcut in

the editor .

12

7

Script Files - Example

Problem

The speed v of a falling object dropped with no initial

velocity is given as a function of time t by v= gt.

Plot v as a function of t for 0 ≤ t≤ tf, where tf is the

final time entered by the user.

13

Script Files - Example

% Comments section

% Program falling speed.m:

% Plots speed of a falling object.

% Created on March 1, 2004 by W. Palm

%

% Input Variable:

% tf= final time (in seconds)

%

% Output Variables:

% t = array of times at which speed is

% computed (in seconds)

% v = array of speeds (meters/second)

14

8

Script Files - Example

% Parameter Value:

g = 9.81;

% Acceleration in SI units

% Input section:

tf = input(’Enter final time in seconds:’);

% Calculation section:

dt= tf/500;

% Create an array of 501 time values.

t = [0:dt:tf];

% Compute speed values.

v = g*t;

% Output section:

Plot(t,v),xlabel(’t(s)’),ylabel(’vm/s)’)

15

Notes on Script File Names

1. The name of a script file must begin with a letter,

and may include digits and the underscore

character, up to 31 characters.

2. Do not give a script file the same name as a

variable.

3. Do not give a script file the same name as a

Matlab command or function. You can check to

see if a command, function or file name already

exists by using the exist command.

16

9

Some Input/output Commands

17

Command Description

disp(A) Displays the contents, but not the

name, of the array A.

disp(’text’) Displays the text string enclosed within

quotes.

x =

input(’text’)

Displays the text in quotes, waits for

user input from the keyboard, and

stores the value in x.

x =

input(’text’,’s’)

Displays the text in quotes, waits for

user input from the keyboard, and

stores the input as a string in x.

User-defined Functions

Another type of m-files.

It is simply a script file that implements some operation that is not available in Matlab.

It uses Matlab commands and accepts user arguments.

To define your function, begin the script file with

function [output variables] = name(input variables)

Note:

◦ the output variables are enclosed in square brackets.

◦ the input variables must be enclosed with parentheses.

◦ The function name should be the same as the file name in which it is saved (with the .m extension).

18

10

User-defined Functions - Example Example

function z = fun(x,y)

u = 3*x;

z = u + 6*y.^2;

Call this function with its output argument:

>> z = fun(3,7)

z = 303

The function uses x = 3 and y = 7 to compute z.

Note

• the use of a semicolon at the end of the lines prevents the values

of u and z from being displayed.

• the use of the array exponentiation operator (.^). This enables the

function to accept y as an array.

• the variables defined inside the function are local to the function

and are cleared when the function returns

19

Example of Function Definition

Lines

1. One input, one output: function names in red letters

function [area_square] = square(side) square.m

2. Brackets are optional for one input, one output:

function area_square = square(side) square.m

3. Three inputs, one output:

function [volume_box] = box(height,width,length) box.m

4. One input, two outputs:

function [area_circle,circumf] = circle(radius) circle.m

5. No named output:

function sqplot(side) sqplot.m

20

11

Functions- Local Variables

The names of the input variables given in the

function definition line are local to that

function.

This means that other variable names can be

used when you call the function.

All variables inside a function are erased after

the function finishes executing, except when

the same variable names appear in the output

variable list used in the function call.

21

Functions – Global Variables

The global keyword declares certain variables

global, and therefore their values are available to

the basic workspace and to other functions that

declare these variables global.

The syntax to declare the variables a, x, and q is

global a x q

Any assignment to those variables, in any function

or in the base workspace, is available to all the

other functions declaring them global.

22

12

Subfunctions

Primary : the first function in an m-file and has the same name as the file name. used to call the function from the command line or from other m-file.

Subfunctions : functions defined in the same file as the primary function and they are available to it only (except with function handles). Used to break large tasks into smaller oners

23

Subfunctions - Example

% function to compute circle parameters

% Inputs: radius of the circle

% outputs:

% 1) area: area of circle

% 2) cricum: circumference of the circle

% the function contains two subfunctions; computeArea

and

% computeCircum to calculate the area and the

circumference

% the given radius

24

13

% main function

function [area,circum] = circleParameters(radius)

area = computeArea(radius) ;

circum = computeCircum(radius) ;

% function to compute the area

function a = computeArea(r)

a = pi*r^2;

% function to compute circumference

function c = computeCircum(r)

c = 2*pi*r;

25

Subfunctions – Example Cont’d

Functions Handle @

It is used to call functions indirectly.

The handle then can be used as an

argument to other functions or as a new

function that accepts arguments

For Matlab functions

fHanldle = @functionName

For anonymous functions

fHandle = @ (argList) anonymus function

26

14

Functions Handle @

Example 1 : creating handle for a Matlab function >> fhandle = @humps;

>> x = fminbnd(fhandle, 0.3, 1)

x =

0.6370

Example 2: creating and using a handle to anonymous function >> sqr = @(x) x.^2;

>> a = sqr(5)

a =

25

>> quad(sqr, 0, 1)

ans =

0.3333

27

fzero function

Use fzero function to find the zero(s) of a function of single variable, denoted by x.

fzero(‘function’,x0)

function is the function expression as a string.

x0 is the initial guess of the solution.

For example, fzero(’cos’,2) returns the value 1.5708.

The function fzero(‘function’,x0) tries to find a zero of function near x0, if x0 is a scalar. fzero(‘function’,[x1 x2] searches for the zero in the interval [x1,x2]. The value returned by fzero is near a point where function changes sign or NaN if the search fails.

28

15

fminbnd function

The fminbnd function finds the minimum of a function of a single variable, which is denoted by x.

One form of its syntax is fminbnd(’function’, x1, x2)

where function is a string containing the expression to be minimized.

The fminbnd function returns a value of x that minimizes the function in the interval x1 ≤x ≤x2. For example, fminbnd(’cos’,0,4) returns the value 3.1416.

29

fminsearch Function

The fminsearch function finds the minimum of a function of several variables, which is denoted by x.

One form of its syntax is fminsearch(function, x0)

where function is a function handle for the function to be minimized and x0 is a vector for initial point.

Example - find the minimum of f(x,y) = x*exp(-x2 –y2)

>> % define the function handle

>> f = @ (x) x(1)*exp(-(x(1))^2 – (x(2))^2)

>> fminsearch(f,[0,0])

ans = -0.7071 0.000

30

16

Working with Data Files

Matlab has the capability of importing data

created by other application into its

workspace.

Also, it is capable of exporting workspace

variables so that it can be used by other

applications

31

Importing & Exporting

Spreadsheet Files

The command

A = xlsread(’filename’)

imports the Microsoft Excel workbook file filename.xls

into the array A.

The command

[A, B] = xlsread(’filename’)

imports all numeric data into the array A and all text data

into the cell arrayB

The command

[SUCCESS,MESSAGE]=xlswrite(FILE,ARRAY)

writes the ARRAY into and xls file whose name is given by

FILE. 32

17

Spreadsheet Examples >> [num,charc] =

xlsread(‘e1.xls’)

num =

1.3000 2.1000 29.0000

3.5000 9.0300 3.1000

0.1000 2.0330 2.1000

-2.0000 -0.2000 0.3000

22.0000 1.0000 0.5552

14.0000 -123.0000 7.0000

charc =

'x' 'y' 'z'

33

e1.xls

>> a = [2 , 3 ; 4 , 5] ;

>> [succes,message] =

xlswrite(‘e2.xls’,a)

succes =

1

message =

message: ''

identifier: ''

Exam

ple

1

Exam

ple

2

Working with CSV files

CSV stands for Comma-Separated Values

One format that is used to exchange data between applications

Some properties

◦ Each record is a line

◦ Records are separated by a carriage return and a line feed pair.

◦ Record fields are separated by commas.

◦ leading and trailing spaces adjacent to commas are ignored.

34

18

Working with CSV files

Commands to

◦ Read CSV data into matrix X

X = csvread (fileName)

◦ Write matrix M into file fileName

csvwrite(fileName,M)

>> x = csvread('csvExample.csv')

x =

1 3 10 20

-9 2 1 12

35

Working with ASCII Files

Many applications support the ASCII file format.

If the ASCII file contains a header or the data is separated by

commas , you have to remove the header and replace the

commas with spaces (one at least); otherwise Matlab will

produce an error message.

To load the data in filename into matrix S

S = load(‘filename’)

To save ASCII data

save filename A - ASCII

36

1

Computer Applications Lab

Lab 5

Programming in Matlab

Chapter 4

Sections 1,2,3,4

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Program design and development

Relational operators and logical variables

Logical operators and functions

Loops

The switch statement

Debugging Matlab programs

2

2

Program Design and

Development Algorithm: an ordered sequence of precisely defined instructions

that performs some task in a finite amount of time.

Ordered means that the instructions can be numbered, but an

algorithm must have the ability to alter the order of its

instructions using a control structure.

There are three categories of algorithmic operations:

◦ Sequential operations: Instructions executed in order.

◦ Conditional operations: Control structures that first ask a question to be

answered with a true/false answer and then select the next instruction

based on the answer.

◦ Iterative operations (loops): Control structures that repeat the execution of

a block of instructions.

3

Program Design and

Development

Structured Programming

A technique for designing programs in which a hierarchy of

modules is used, each having a single entry and a single exit

point, and in which control is passed downward through the

structure without unconditional branches to higher levels of the

structure (the use of GOTO). In Matlab these modules can be

built-in or user-defined functions.

4

3

Program Design and

Development Structured Programming

Advantages

1. Structured programs are easier to write because the programmer can study the overall problem first and then deal with the details later.

2. Modules (functions) written for one application can be used for other applications (this is called reusable code).

3. Structured programs are easier to debug because each module is designed to perform just one task and thus it can be tested separately from the other modules.

4.Structured programming is effective in a teamwork environment because several people can work on a common program, each person developing one or more modules.

5. Structured programs are easier to understand and modify, especially if meaningful names are chosen for the modules and if the documentation clearly identifies the module’s task.

5

Program Design and

Development Steps for creating computer solutions

1. State the problem concisely.

2. Specify the data to be used by the program. This is the “input”.

3. Specify the information to be generated by the program. This is the “output”.

4. Work through the solution steps by hand or with a calculator; use a simpler set of data if necessary.

5. Write and run the program.

6. Check the output of the program with your hand solution.

7. Run the program with your input data and perform a reality check on the output.

8. If you will use the program as a general tool in the future, test it by running it for a range of reasonable data values; perform a reality check on the results.

6

4

Program Design and

Development

Documentation

1. Proper selection of variable names to reflect the quantities

they represent.

2. Use of comments within the program.

3. Use of structure charts.

4. Use of flowcharts.

5. A verbal description of the program, often in pseudocode.

7

Program Design and

Development

Documenting with Charts

Two types of charts aid in developing structured programs

and in documenting them. These are structure charts and

flowcharts.

A structure chart is a graphical description showing how the

different parts of the program are connected together.

8

5

Program Design and

Development

Documenting with

Charts

Flowcharts are useful for

developing and documenting

programs that contain

conditional statements, because

they can display the various

paths (called “branches”) that a

program can take, depending on

how the conditional statements

are executed.

9

Program Design and

Development

Documenting with Pseudocode

We can document with pseudocode, in which

natural language and mathematical expressions are

used to construct statements that look like

computer statements but without detailed syntax.

Each pseudocode instruction may be numbered, but

should be unambiguous and computable.

10

6

Relational Operators

Six operators that are used for comparison of

variables (scalars or arrays) or expressions

They have equal precedence (evaluated from left to

right)

11

Relational Operators

For example, suppose that x = [6,3,9] and y = [14,2,9].

The following Matlab session shows some examples.

>>z = (x < y)

z = 1 0 0

>>z = (x ~= y)

z = 1 1 0

>>z = (x > 8)

z = 0 0 1

12

Note

Comparison is performed

element-wise

7

Relational Operators

The relational operators can be used for array addressing.

For example, with x = [6,3,9] and y = [14,2,9], typing

z = x(x<y)

finds all the elements in x that are less than the

corresponding elements in y. The result is z = 6.

The arithmetic operators +, -, *, /, and \ have precedence

over the relational operators. Thus the statement z = 5 > 2

+ 7 is equivalent to z = 5 >(2+7) and returns the result z =

0.

We can use parentheses to change the order of

precedence; for example, z = (5 > 2) + 7evaluates to z = 8.

13

The Logical Class

When the relational operators are used, such as x

= (5 > 2) they create a logical variable, in this case, x.

Prior to Matlab 6.5 logical was an attribute of any

numeric data type. Now logical is a first-class data

type and a Matlab class, and so logical is now

equivalent to other first-class types such as character

and cell arrays.

Logical variables may have only the values 1 (true)

and 0 (false).

14

8

Logical Operators

15

Logical Operators

16

9

Precedence of Different

Operators

17

Logical Functions

18

10

Logical Functions

19

Logical Functions

Use Matlab documentation to check the

following logical functions.

1. isstr

2. isnumeric

3. isvector

4. ishold

5. isequal

6. isscalar

20

11

The Find Function

The Matlab function find computes an array containing the indices of the nonzero elements of the numeric array x.

Example

>> x = [-2, 0, 4];

>> y = find(x)

Y = 1 3

The resulting array y = [1, 3] indicates that the first and third elements of x are nonzero.

[u,v,w] = find(A)

Computes the arrays u and v containing the row and column indices of the nonzero elements of the array A and computes the array w containing the values of the nonzero elements. The array w may be omitted.

21

The find Function

When the find function is used with arrays, the return value is the linear indices of the nonzero elements.

With linear indices, we can refer to array elements with a single value A(k), where k is the element location if the columns of the array are appended to each other.

Example A =

A(2) is equivalent to A(2,1) which is 5

A(5) is equivalent to A(2,2) which is 6

22

2 1 05 6 71 2 3

12

The Find Function

With logical operators

The expression inside the function is evaluated first, then the search for nonzero elements is performed

Consider the session >> x = [5, -3, 0, 0, 8]; y = [2, 4, 0, 5, 7];

>> z = find(x&y)

z = 1 2 5

Note that the find function returns the indices, and not the values.

To extract the values, use array indexing y = x(z)

23

Conditional Statements

The if statement’s basic form is

if logical expression

statements

end

Every if statement must have an accompanying end statement.

The end statement marks the end of the statements that are to be executed if the logical expression is true.

24

13

Conditional Statements

The else statement

The basic structure for the use of the else

statement is

if logical expression

statements group 1

else

statement group 2

end

25

Conditional Statements

26

14

Conditional Statements

Example

x = [4,-9,25];

if x < 0

disp(’Some elements of x are negative.’)

else

y = sqrt(x)

end

Because the test if x < 0 is false, when this program

is run it gives the result

y = [2, +3.000i, 5]

27

Conditional Statements

Example

x = [4,-9,25];

if x >= 0

y = sqrt(x)

else

disp(’Someelements of x are negative.’)

end

When executed, it produces the following message: Some elements of x are negative. The test if x < 0 is false, and the test if x >= 0 also returns a false value because x >= 0 returns the vector [1,0,1].

28

15

Conditional Statements

The elseif Statement

The general form of the if statement is if logical expression 1 statement group 1 elseif logical expression 2 statement group 2 else statement group 3 end

The else and elseif statements may be omitted if not required. However, if both are used, the else statement must come after the elseif statement to take care of all conditions that might be unaccounted for.

29

Conditional Statements

30

16

Conditional Statements

Example, suppose that

The following statements will compute y if x already has a scalar value

if x > 10

y = log(x)

elseif x >= 0

y = sqrt(x)

else

y = exp(x) -1

end

31

exp(x) 1,x 0

y sqrt(x),0 x 10

log(x),x 10

Loops

For Loops.

Used to repeat a set of statements for specific number of times.

The loop variable k is initially assigned the value 5, and x is calculated from x = k^2. Each successive pass through the loop increments k by 10 and calculates x until k exceeds 35. Thus k takes on the values 5, 15, 25, and 35, and x takes on the values 25, 225, 625, and 1225. The program then continues to execute any statements following the end statement.

32

17

33

Note the following rules when using for loops

with the

loop variable expression k = m:s:n

The step value s may be negative.

Example: k = 10:-2:4 produces k = 10, 8, 6, 4.

• If s is omitted, the step value defaults to 1.

• If s is positive, the loop will not be executed if m is greater than n.

• If s is negative, the loop will not be executed if m is less than n.

• If m equals n, the loop will be executed only once.

• If the step value s is not an integer, round-off errors can cause the loop to execute a different number of passes than intended.

34

18

Loops – The While Statement

The while loop is used when the looping process terminates because a specified condition is satisfied, and thus the number of passes is not known in advance.

The structure of a while loop

while logical expression

statements

end

For the while loop to function properly, the following two conditions must occur:

1.The loop variable must have a value before the while statement is executed.

2. The loop variable must be changed somehow by the statements.

35

Loops

While loop

A simple example of a while loop is

x = 5;

while x < 25

disp(x)

x = 2*x -1;

end

The results displayed by the disp statement are 5, 9, and 17.

36

19

Loops

Infinite Loops

You can create an infinite while loop which is a loop that never ends:

x = 8;

while x ~=0

x = x -3;

end

Within the loop the variable x takes on the values 5, 2, -1, -4….., and the condition x~=0 is always satisfied, so the loop never stops.

37

Loop Vectorization

We can often avoid the use of loops and branching and thus create simpler and faster programs by using a logical array as a mask that selects elements of another array. This called loop vectorization.

Example Here is one way to compute the sine of 1001 values ranging from 0 to 10:

i = 0;

for t = 0:.01:10

i = i + 1;

y(i) = sin(t);

End

A vectorized version of the same code is t = 0:.01:10 ;

y = sin(t) ; The second example executes much faster than the first and is the way Matlab is meant to be use

38

20

Loop Vectorization

Example: for the array A = [0, -1, 4; 9, -14, 25; -34, 49, 64], find all the elements that are greater than or equal to 0.

Using loops [m,n] = size(A) ;

C = zeros(m,n) ;

for k = 1:m*n

if A(k) >= 0

C(k) = A(k)

end

end

Using array masking and relational operators to vectorize

the loop C = A ;

C(C<0) = 0 ;

39

The Switch Structure

The switch structure provides an alternative to

using the if, elseif, and else commands. Anything

programmed using switch can also be

programmed using if structures. However, for

some applications the switch structure is more

readable than code using the if structure.

40

21

The Switch Statement

switch input expression

case value1

statement group 1

case value2

statement group 2

.

.

.

otherwise

statement group n

end 41

The Switch Statement

Example

The following switch block displays the point on the compass that corresponds to that angle.

switch angle case 45

disp(’Northeast’)

case 135

disp(’Southeast’)

case 225

disp(’Southwest’)

case 315

disp(’Northwest’)

otherwise

disp(’DirectionUnknown’)

end

42

22

Program Design and

Development

Finding Bugs

Debugging a program is the process of finding and removing the “bugs,” or errors, in a program. Such errors usually fall into one of the following categories.

1.Syntax errors such as omitting a parenthesis or comma, or spelling a command name incorrectly. Matlab usually detects the more obvious errors and displays a message describing the error and its location.

2. Runtime errors; These errors occur due to an incorrect mathematical procedure. They do not necessarily occur every time the program is executed; their occurrence often depends on the particular input data. A common example is division by zero.

43

Program Design and

Development

Locate Errors

1. Always test your program with a simple version of the problem, whose answers can be checked by hand calculations.

2. Display any intermediate calculations by removing semicolons at the end of statements.

3. To test user-defined functions, try commenting out the function line and running the file as a script.

4. Use the debugging features of the Editor/Debugger.

44

23

Matlab Editor/Debugger

Menus: File, Edit, text, Tools, Debug, Desktop, Window, Help

45

The Debug Menu

Breakpoints

Set/Clear Breakpoint

Step

Step In

Step Out

Continue

Go Until Cursor

Exit Debug Mode

Enable/Disable Breakpoint

Stop if Errors/Warnings

46

24

The Debug Menu

Example Two functions that find the values of a vector x that are

greater than the average values of all elements in x

47

% fun1.m function = fun1(x)

avg = sum(x) / length(x);

y= fun2(avg,x)

%fun2.m function fun2(x,avg)

above = length(find(x>avg));

>> % Test the functions

>> x = [1,2,3,4,10];

>> y = fun1(x)

y =

3

ERROR !!! Answer should be 1 !!!

Let’s see how we can debug …..

Strings

A string is a variable that contains characters.

Strings are useful for creating input prompts and messages and for storing and operating on data such as names and addresses.

To create a string variable, enclose the characters in single quotes.

For example, the string variable name is created as follows:

>>name = ’Leslie Student’

name =

Leslie Student

48

25

Strings and the Input Command

The input function, whose syntax is

x = input(’prompt’, ’s’)

displays the string prompt on the screen, waits for input

from the keyboard, and returns the entered value in the

string variable x.

49

Concatenating Strings

Given strings S1, S2, and S3, a string S that

is composed of appending the three stings

can be formulated by S = [S1 S2 S3]

concatenates character

Example >> S1 = ‘CPE’ ;

>> S2 = ‘0907311’;

>> S = [S1 S2]

S =

CPE0907311

50

26

Strings Conversion

Use the num2str() function to convert numeric values in strings.

Use the str2num() function to convert numeric values that are stored as strings to numbers.

Example

>> d = 1.23;

>> class(d)

ans = double

>> f = num2str(d)

f = 1.23

>> class(f)

ans = char

51

1

Computer Applications Lab

Lab 6

Plotting

Chapter 5

Sections 1,2,3,8

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

xy Plotting Functions

Subplots

Special Plot Types

Three-Dimensional Plotting

2

2

xy Plotting The most common types of plots.

Plots the function y = f(x). (other symbols can be used also)

Anatomy of a plot

3

xy Plotting

The command that is used for xy plots in MATLAB is

plot(x,y)

where both x and y are vectors of the same length.

Example

The following MATLAB session plots y = 0.4 √1.8x for 0 ≤x ≤52, where y

represents the height of a rocket after launch, in miles, and x is the horizontal

(downrange) distance in miles.

>>x = [0:0.1:52];

>>y = 0.4*sqrt(1.8*x);

>>plot(x,y)

>>xlable(’Distance(miles)’) % Adds label to x-axis

>>ylable(’Height(miles)’) % Adds label to y-axis

>>title(’Rocket Height as a Function of Downrange Distance’)

% Adds title to the plot.

4

3

xy Plotting

5

xy Plotting

Requirements for a Correct Plot

1. Each axis must be labeled with the name of the quantity

being plotted and its units!

2. Each axis should have regularly spaced tick marks at

convenient intervals—not too sparse, but not too dense—

with a spacing that is easy to interpret and interpolate. For

example, use 0.1, 0.2, and so on, rather than 0.13, 0.26, and so

on.

3. If you are plotting more than one curve or data set, label

each on its plot or use a legend to distinguish them.

4. If you are preparing multiple plots of a similar type or if the

axes‟ labels cannot convey enough information, use a title.

6

4

xy Plotting Requirements for a Correct Plot

5 . If you are plotting measured data, plot each data point with a

symbol such as a circle, square, or cross (use the same symbol

for every point in the same data set). If there are many data

points, plot them using the dot symbol.

6. Sometimes data symbols are connected by lines to help the

viewer visualize the data, especially if there are few data

points. However, connecting the data points, especially with a

solid line, might be interpreted to imply knowledge of what

occurs between the data points. Thus you should be careful to

prevent such misinterpretation.

7. If you are plotting points generated by evaluating a function

(as opposed to measured data), do not use a symbol to plot the

points. Instead, be sure to generate many points, and connect the

points with solid lines.

7

xy Plotting The Figure Window

It has different menus to save, export, and edit

generated figures.

For example, we can

◦ save the figure to be used later on (it has .fig extension).

◦ print the plot

◦ we can insert text boxes, arrows, shapes, and legends.

◦ zoom in and out and rotate plot

◦ clear the figure

◦ copy the plot to be inserted in other documents

◦ export the figure in other data formats using File>Export

Setup

8

5

xy Plotting

More Commands grid command

The grid command displays gridlines at the tick marks corresponding to the tick labels.

Type grid on to add gridlines; type grid off to stop plotting gridlines. When used by

itself, grid toggles this feature on or off, but you might want to use grid on and grid off

to be sure.

axis command

Is used to override the MATLAB selections for the axis limits. The basic syntax is

axis([xmin xmax ymin ymax]). This command sets the scaling for the x-and y-axes

to the minimum and maximum values indicated. Note that, unlike an array, this command

does not use commas to separate the values.

figure command

Use this command to open a new figure window to be used for new plots. Using the plot

command more than once update the same figure window that was used by the first plot

command, unless the hold command is used

9

xy Plotting

10

6

xy Plotting Some Variations of the plot Command

If y is a vector and we use plot(y), MATLAB plots the

elements of y versus their indices.

If y is a vector of complex numbers, the use of plot(y) plots

the imaginary part of the elements versus the real part.

In other words, plot(y) in case y contains complex values is

equivalent to plot(real(y),imag(x)).

Example

>> z = 0.1 + 0.9i;

>> n = [0:0.01:10];

>> plot(z.^n),

xlabel(‘Real’),ylabel(‘imaginary’);

11

xy Plotting

The plot command with complex numbers

12

7

xy Plotting The function plot command fplot

It is a smart command that takes the function to be plotted

and determine the required number of points to show most

of the features of the function

Syntax

fplot(‘string’,[xmin xmax ymin ymax])

[x,y] = fplot(‘string’,[xmin xmax ymin ymax])

Example

compare the result of using fplot(‘cos(tan(x)) –

tan(sin(x))’,[1 2]) with

>> x = [1:0.01:2] ;

>> y = cos(tan(x)) – tan(sin(x));

>> plot(x,y)

13

xy Plotting

Using fplot command

14

8

xy Plotting

15

Using plot command

xy Plotting

Plotting Polynomials

The polyval(coeff,x) function in MATLAB can be used to

evaluate a polynomial whose coefficients are given in coeff

evaluated at values specified by x

Example

>> x = [-6:0.01:6];

>> y = [3,2,-100,2,-7,90];

>> plot(x,polyval(y,x))

>> xlable(‘x)

>> ylable(‘p’)

16

-6 -4 -2 0 2 4 6-3000

-2000

-1000

0

1000

2000

3000

4000

5000

p

x

9

xy Plotting

Subplots

You can use the subplot command to obtain several smaller “subplots”

in the same figure.

Syntax

subplot(m,n,p)

This command divides the Figure window into an array of rectangular

panes with m rows and n columns.

The variable p tells MATLAB to place the output of the plot

command following the subplot command into the pth pane.

For example, subplot(3,2,5)creates an array of six panes, three panes

deep and two panes across, and directs the next plot to appear in the

fifth pane (in the bottom-left corner).

17

xy Plotting

Subplots

Example

x = [0:0.01:5];

y = exp(-1.2*x).*sin(10*x+5);

subplot(1,2,1)

plot(x,y),axis([0 5 -1 1])

x = [-6:0.01:6];

y = abs(x.^3-100);

subplot(1,2,2)

plot(x,y),axis([-6 6 0 350])

18

10

xy Plotting - Example

19

xy Plotting Data Markers and Line Types

We can change the data markers, line styles, and colors for our plots by using a string in a

plot command

For example, plot(x,y,‟b-.o‟) plots y versus x using a blue dash-dotted line with circles as data

markers.

More data markers, line styles, and colors are listed below

20

11

xy Plotting

Controlling Tick-Mark Spacing and Labels

The set command is a powerful command for controlling objects

One of these objects is the axes in the plot.

We can change the tick mark spacing and use labels instead of numbers

as tick marks.

Syntax

Set(gca,’XTick’,[xmin:dx:xmax],’YTick’,[ymin:dy:ymax])

◦ gca is a function that returns a handle for the current axes

Example

set(gca,’XTick’,[0:0.1:2],’YTick’,[0:0,1:1])

21

xy Plotting

Controlling Tick-Mark Spacing and Labels

We can assign labels to tick marks instead of numbers.

Example >> x = [1:6];

>> y = [13,5,7,14,10,12]

>> set(gca,‟Xticklabel‟,[„Jan‟;‟Feb‟;‟Mar‟;Apr‟;‟May‟;‟Jun‟])

>> set(gca,‟XTick„,[1:6]), axis([1 6 0 15]) , xlabel(„Month‟) …

,ylabel(„Monthly Sales $1000‟), title(„Printer Sales for January to June, 1997‟)

22

12

xy Plotting

Labeling Curves and Data When we plot more than one curve on the same plot, we

have to distinguish them with ◦ Legends

legend(‘string1’,’string2’,…)

◦ By a label next to each curve

gtext(‘string’) (it requires user interaction to specify the location of the text by the mouse)

or use

text(x,y,’string’) where x and y specify the location of the text

Example

23

xy Plotting

24

13

xy Plotting

The hold Command

We can plot y versus x and v versus u on the same figure using the plot

command by

plot(x,y,u,v)

If we use the plot command once more, the figure gets updated with the

new plot and the old plot is erased.

To preserve earlier plots, use the hold on command.

This command will allow the new plot to be superimposed over the old

plot.

Use hold off to disable the hold command.

25

xy Plotting

Annotating Plots

We already saw the title(‘string’) that is used to insert a title for the plot.

We can insert mathematical symbols in the title, axes labels, and text

labels by using the backslash character \ followed by the symbol name.

Superscripts and subscripts are denoted by ^ and _ , respectively. To set

multiple characters as superscripts or subscripts, enclose them by {}.

Example : the command

>> title(‘Ae^{-t/\tau} sin(\omega t)’)

produces

in the figure title.

26

t/Ae sin( t)

14

xy Plotting Hints for Improving Plots

1. Start scales from zero whenever possible. This technique prevents a false

impression of the magnitudes of any variations shown on the plot.

2. Use sensible tick-mark spacing. If the quantities are months, choose a

spacing of 12 because 1/10 of a year is not a convenient division. Space

tick marks as close as is useful, but no closer. If the data is given monthly

over a range of 24 months, 48 tick marks might be too dense, and also

unnecessary.

3. Minimize the number of zeros in the data being plotted. For example, use

a scale in millions of dollars when appropriate, instead of a scale in dollars

with six zeros after every number

4. Determine the minimum and maximum data values for each axis before

plotting the data. Then set the axis limits to cover the entire data range

plus an additional amount to allow convenient tick-mark spacing to be

selected.

27

xy Plotting Hints for Improving Plots

5. Use a different line type for each curve when several are plotted on a

single plot and they cross each other; for example, use a solid line, a

dashed line, and combinations of lines and symbols. Beware of using

colors to distinguish plots if you are going to make black-and-white

printouts and photocopies.

6. Do not put many curves on one plot, particularly if they will be close to

each other or cross one another at several points.

7. Use the same scale limits and tick spacing on each plot if you need to

compare information on more than one plot

28

15

xy Plotting Special Types of Plots

Exercise: use the help browser to check the stairs

command.

29

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

Stem Plots: use stem(x,y) Bar Plots: use bar(x,y)

1 4 5 8 90

1

2

3

4

5

6

7

8

9

10

x

y

3-D Plotting Plotting of Curves

Use the command plot3(x,y,z) to plot three-

dimensional curves.

We can use the grid command and the attributes for

selecting the line style, color and data markers.

Example

30

16

3-D Plotting

Surface Mesh Plots

The function z = f(x,y) represents a surface

when plotted on xyz axes, and the mesh

function provides the means to generate a

surface plot. Before you can use this function,

you must generate a grid of points on the xy-

plane, and then evaluate the function f(x,y) at

these points. The meshgrid function

generates the grid.

31

3-D Plotting

Surface Mesh Plots

Syntax

[X,Y] = meshgrid(x,y)

Where x = [xmin:xspacing:xmax] and y = [ymin:yspacing:ymax],

This computes the Cartesian product between x and y.

The function [X,Y] = meshgrid(x) is the equivalent to [X,Y] =

meshgrid(x,x) and can be used if x and y have the same minimum

values, the same maximum values, and the same spacing. Using this

form, you can type [X,Y] = meshgrid(min:spacing:max), where min and

max specify the minimum and maximum values of both x and y and

spacing is the desired spacing of the x and y values.

Once the grid is obtained, we can plot the three dimensional surface

using the mesh(x,y,z) function. As always, the grid, label, and text

functions can be used with the mesh function.

32

17

3-D Plotting

Surface Mesh Plots

Example

33

3-D Plotting Contour Plots

Contour plot contain lines that indicate constant elevation (values).

Contour plots can be generated in MATLAB by the contour(x,y,z) command.

We need first to use the meshgrid.

34

18

3-D Plotting

35

3-D Plotting Examples

36

mesh(x,y,z) meshc(x,y,z)

meshz(x,y,z) waterfall(x,y,z)

1

Computer Applications Lab

Lab 7

Designing GUI with Matlab

Dr. Iyad Jafar

Adapted from Mathworks

Outline

What is Graphical User Interface (GUI) ?

GUI in Matlab

How does GUI work ?

Structure of GUI in Matlab

The GUIDE Tool

Structure of GUI M-File

A Simple GUI Example

Adding Menus

2

2

What is GUI ? A graphical user interface (GUI) is a graphical display that

contains devices, or components, that enable a user to

perform interactive tasks.

To perform these tasks, the user of the GUI does not have

to create a script or type commands at the command line.

Often, the user does not have to know the details of the

task at hand.

3

How Does a GUI Work?

Each component, and the GUI itself, is associated with one or more user-written routines known as callbacks. The execution of each callback is triggered by a particular user action such as a button push, mouse click, selection of a menu item, or the cursor passing over a component. You, as the creator of the GUI, provide these callbacks.

This kind of programming is often referred to as event-driven programming.

The writer of a callback has no control over the sequence of events that leads to its execution or, when the callback does execute.

4

3

Structure of GUI in Matlab

Any GUI in MATALB is associated with two files

◦ Figure file with extension .fig that contains all the components or controls to be used in the GUI application. The FIG-file is a binary file and you cannot modify it except by changing the layout in GUIDE.

◦ M-file that contains the code for the GUI initialization and the callbacks/functions for different components in the GUI layout and other hidden functions of the GUI.

5

Creating GUI Using GUIDE

GUIs in Matlab can be created either

programmatically or by using the GUIDE

tool.

Type guide on the command prompt to

start the GUIDE tool.

With this tool, you can use the mouse to

add different components and set their

properties (size, text, callbacks)

6

4

7

8

Use This Tool... To...

Layout Editor Select components from the component palette, at the left side of the Layout

Editor, and arrange them in the layout area. See Adding Components to the GUI for

more information.

Figure Resize Tab Set the size at which the GUI is initially displayed when you run it. See Setting the

GUI Size for more information.

Menu Editor Create menus and context, i.e., pop-up, menus. See Creating Menus for more

information.

Align Objects Align and distribute groups of components. Grids and rulers also enable you to

align components on a grid with an optional snap-to-grid capability. See Aligning

Components for more information.

Tab Order Editor Set the tab and stacking order of the components in your layout. See Setting Tab

Order for more information.

Toolbar Editor Create Toolbars containing predefined and custom push buttons and toggle buttons.

See Creating Toolbars for more information.

Icon Editor Create and modify icons for tools in a toolbar. See Creating Toolbars for more

information.

Property Inspector Set the properties of the components in your layout. It provides a list of all the

properties you can set and displays their current values.

Object Browser Display a hierarchical list of the objects in the GUI. See Viewing the Object

Hierarchy for more information.

Run Save and run the current GUI. See Saving and Running a GUIDE GUI for more

information.

M-File Editor Display, in your default editor, the M-file associated with the GUI. See GUI Files: An

Overview for more information.

Position Readouts Continuously display the mouse cursor position and the positions of selected

objects

5

Structure of GUI M-file

The GUI M-file that GUIDE generates is a function file. It has the same name as the GUI figure file.

It contains the callbacks for the GUI components which are subfunctions of the main function.

When GUIDE generates an M-file, it automatically includes templates for the most commonly used callbacks for each component.

The M-file also contains initialization code, as well as an opening function callback and an output function callback.

You must add code to the component callbacks for your GUI to work as you want.

9

Structure of GUI M-file Example, the GUI opening function % --- Executes just before simple_gui is made visible.

function simple_gui_OpeningFcn(hObject, eventdata, handles, varargin)

% This function has no output args, see OutputFcn.

% hObject handle to figure

% eventdata reserved - to be defined in a future version of Matlab

% handles structure with handles and user data (see GUIDATA)

% varargin command line arguments to simple_gui (see VARARGIN)

% Choose default command line output for simple_gui

handles.output = hObject;

handles.x = 7

% Update handles structure

guidata(hObject, handles);

The handles argument is an important structure that contains the data of the GUI. This structure is available for all components in the GUI. It can be used to store any data.

10

6

Structure of GUI M-file

Example, callback function for a pushbutton % --- Executes on button press in pushbutton1.

function pushbutton1_Callback(hObject, eventdata, handles)

% hObject handle to pushbutton1 (see GCBO)

% eventdata reserved - to be defined in a future version

of Matlab

% handles structure with handles and user data (see

GUIDATA)

handles.x = sqrt(5)

% Update handles structure

guidata(hObject, handles);

Each time the handles data is modified, the function guidata has to be used to save the new data to the handles structure.

11

The Get function

The get function is extensively used in programming GUIs. It is used to get properties and values of the GUI component.

Syntax

get(h,'PropertyName')

h is a handle to the component and propertyName is the name of the property that we want to get for the component specified by the handle.

Components handles can specified using the handles.tagValue, where tagValues is the vaule of the tag property for the component.

12

7

The Set Function

The set function is the opposite of the get

function.

It is used to change the value of certain

property of a component.

Syntax set(h,'PropertyName',PropertyValue,...)

The properties differ from component to

component.

13

Components Properties

Each GUI component has its own properties that can be changed by the property inspector or by the set function.

Some of these properties are

◦ ‘String’, ‘fontWeight’, ‘fontSize’, ‘fontName’, ‘fontAngle’, which are used to format and specify the component label.

◦ ‘value’ property depends on component type.

◦ ‘Tag’ is the name of the component that is used to get and set its properties in the code.

◦ ‘Enable’ is used to disable or enable the component.

◦ ‘Visible’ controls if the component is visible or not.

◦ ‘Resize’ enable or disable resizing fo components or the GUI figure.

14

8

Example

We want to create a GUI that plots a sine or a cosine function over a range that is specified by the user. When the GUI is started, the default plot is a sine wave over [-π,π] with 100 points. The user should have the option to show or hide gridline

Required components ◦ Axes to plot the function.

◦ Popmenu to allow the user to choose which function to plot.

◦ Two text boxes to specify the minimum and the maximum of the range.

◦ One text box to specify the number of points to be used in the plot

◦ A push button to update the plot based on the user input.

◦ A check box to show or hide gridlines

15

Example – Starting GUIDE

Start the GUIDE tool by typing guide on the command prompt.

The guide quick start tool is started.

Select blank GUI then click OK.

16

9

Example – Adding Components

Use the GUI preferences from the File menu to display the names of the components.

17

Example – Adding Components Using the mouse, drag and drop the required

components in the figure

18

10

Example – Adding Components Use the property editor to change the labels of different

components. Change the property ‘String’ for each component.

19

Example - Saving

Save your GUI with the name guiExample by clicking on save from the File menu.

Matlab will create two files

◦ guiExample.fig

◦ guiExample.m

Matlab will open the editor to show the contents of the guiExample.m file.

The m-file contains the common callbacks for different components in the GUI.

20

11

Example - Programming the GUI

Initialization part

We want the GUI to generate a default plot

when it is first started. This code is added to

the opening function of the GUI.

In the M-file editor, click on show function

icon . . A menu will appear listing all

functions in the M-file.

Select guiExample_OpeningFcn. The editor

goes to the line where this function starts.

Add the following code to opening function.

21

Example - Programming the GUI % Set the values for the text boxes to default values

% access the handles for the text boxes from the handles

structure

set(handles.xMin,'string',num2str(-pi))

set(handles.xMax,'string',num2str(pi))

set(handles.nPoints,'string',num2str(100))

% define default range, points, and function to be

plotted

xMin = -pi , xMax = pi , nPoints = 100 ;

% define default range, points, and function to be plotted

x = xMin: (xMax-xMin)/(100-1):xMax;

y = sin(x) ;

plot(x,y), xlabel(‘x’),ylabel(‘y’);

% Update handles structure

guidata(hObject, handles);

22

12

Example - Programming the

GUI

Programming the Push Button

The callback function of the push button should perform the following tasks:

◦ Read the minimum and maximum of the plotting range from xMin and xMax textboxes.

◦ Read the number of points to be used from nPoints textbox.

◦ Read the function to be plotted from the popmenu.

◦ Plot the required function based no user selection.

Select the plotButton callback function from the function icon . . Insert the following code in the function.

23

Example - Programming the

GUI

Programming the Push Button % when this key is pressed, we have to read the values from the text

boxes

% and the function to be plotted, the plot.

xMin = str2num(get(handles.xMin,'string'));

xMax = str2num(get(handles.xMax,'string'));

nPoints = str2num(get(handles.nPoints,'string'));

func = get(handles.funcMenu,'value')

% generate x vector and plot the function based on the value of the popmenu

x = xMin : (xMax-xMin)/(nPoints-1) : xMax ;

if func == 1

plot(x,sin(x));

title('Sin(x)'), xlabel('x'),ylabel('y');

else

plot(x,cos(x));

title('cos(x)'), xlabel('x'),ylabel('y');

end

%clear selection of the grid checkbox

set(handles.gridChkBox,'value',0)

24

13

Example - Programming the

GUI

Programming the Checkbox

When this checkbox is on, gridlines should appear on the axes of the GUI. When it is not selected, no gridlines are shown.

To program this, just add the grid command in the body of the checkBox callback function (the tag name is gridChkBox).

Each time the checkbox is selected or deselected, the function is executed and the use of the grid command toggles the gridlines.

25

Final GUI

26

14

Notes

To run your GUI, type its name on the command prompt. Or click on the run icon in GUIDE.

To open your GUI, start the guide tool then click on the open existing GUI tab and select your file.

To close your GUI from your code, use the close command.

If you have more than one axes in your GUI and you want activate one of them for plotting, use axes(h) where h is a handle to axes. Example axes(handles.axes1) set axes1 in the GUI as the current axes.

27

Adding Menus We can add menus to our GUI similar to those found in

most GUI application. Let’s add a File menu to the

previous example. The menu should contain two items:

close, to close the GUI, and reset to clear the axes and

the text boxes.

This can be done using the menu editor.

Click on the menu editor icon to start the menu

editor.

28

15

Adding Menus

Click on the new menu icon to insert a

new menu. Click on the new menu and

update its properties as shown.

29

Adding Menus

Click on the new menu item icon and add

two items with labels Reset and Close, and tags

restItem and closeItem.

30

16

Adding Menus

Now if you run you GUI by clicking on the

run icon in GUIDE, you will see the

created menu.

31

Adding Menus

We need now to define the callbacks for the menus items.

In the editor, select resetItem_callback from the funtion list and add the following line of code

% clear current axes1

cla(handles.axes1)

set(handles.xMin,'string','')

set(handles.xMax,'string','')

set(handles.nPoints,'string','')

set(handles.gridChkBox,'value',0)

Similarly, add the close command to the callback function of closeItem to close the GUI.

32

17

The uigetfile Command

The uigetfile command opens the standard box for retrieving files. It can be used with other data importing functions to open files interactively.

Syntax [FileName,PathName,FilterIndex] = uigetfile(FilterSpec)

The returned values are strings

Example:

>> [fName,fPath,fInex] = uigetfile(‘*.csv’) ;

>> % Matlab will open the dialog box

>> class(fName)

ans =

char

33

The uigetfile Command

If the Cancel button is pressed, an empty string is returned.

34

18

The msgbox Command

We can generate message windows by

using the msgbox command.

Syntax msgbox(Message,Title,Icon)

Message is the message to be displayed as

string.

Title is the title of the window as a string.

Icon is the icon to be displayed on the

message. It can be ’error’, ‘help’, or ‘warn’.

35

The msgbox Command

Example msgbox(‘Division by Zero’,’My

Message’,’Warn’)

36

1

Computer Applications Lab

Lab 9 Probability, Statistics, Interpolation,

and Calculus

Chapter 7

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Statistics and Probability

Histograms

Normal and Uniform Distributions

Random Number Generation

Interpolation

2

2

Common Statistical Functions MATALB provides several functions to compute statistical properties of

any set of numerical data

For the first three commands, if x is a matrix, Matlab returns a raw vector

in which each element is the result of applying these function to its

columns. cumsum returns a matrix that contains cumulative sums of the

columns.

3

Function Description

mean (x) Computes the average value for the elements of vector x.

median(x) Computes the median value for the elements in vector x.

std(x) Computes the standard deviation of the elements in vector

x.

cumsum(x) Returns a vector the same size of x containing the

cumulative sum of the elements in x.

Examples

>> x =[ones(1,5)*3,ones(1,4),ones(1,6)*(2)];

>> mean(x)

ans = 31

>> median(x)

ans = 2

>> std(x)

ans = 0.7988

>> cumsum(x)

ans = 3 6 9 12 15 16 17

18 19 21 23 25 27 29

31

4

3

Histograms Given a set of data values stored in vector x, the histogram of x

is a display of statistical information that uses rectangles to

show the frequency of data items in successive numerical

intervals of equal size.

Histograms are useful in studying data properties and

distribution as they can be used to approximate the probability

function.

Suppose x = [1,1,1,3,4,4,5,5,5,6,6,7,7,7,7,7,7], the histogram of

these values is shown below.

5 0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7Histogram of x

x

Absolu

te F

requency

Histograms

To generate histograms in MATALB, use the ◦ hist(x) function or its variants (discussed later)

◦ bar(x,y), where x is a vector containing the centers of the bars (counts) and y is a vector containing the frequencies of the values in each bar.

Example

>> x = [1,1,1,3,4,4,5,5,5,6,6,7,7,7,7,7,7];

>> hist(x,[0:8]); % here we specified the bin centers. If not provided, Matlab default is 10 bins between the min and max of input data

OR

>> x = [0:8] ; % specify bar centers

>> y = [0,3,0,1,2,3,2,6,0]; % absolute frequencies manually computed

>> bar(x,y);

6

4

Histograms

7

Command Description

bar(x,y) Creates a bar chart of y versus x.

hist(y) Aggregates the data in the vector y into 10 bins evenly spaced

between the minimum and maximum values in y.

hist(y,n) Aggregates the data in the vector y into n bins evenly spaced

between the minimum and maximum values in y.

hist(y,x) Aggregates the data in the vector y into bins whose center

locations are specified by the vector x. The bin widths are the

distances between the centers.

[z,x] = hist(y) Same as hist(y) but returns two vectors z and x that contain

the frequency count and the bin locations.

[z,x] = hist(y,n) Same as hist(y,n) but returns two vectors z and x that contain

the frequency count and the bin locations.

[z,x] = hist(y,x) Same as hist(y,x) but returns two vectors z and x that contain

the frequency count and the bin locations. The returned vector

x is the same as the user-supplied vector x

Probability In any experiment or phenomena, the probability of

an event is a number between 0 and 1 that indicates the likelihood of that event to occur if the experiment is repeated infinitely

Example: Rolling a dice. The probability of obtaining any of the number on the dice is 1/6.

Random variable is often the term used to express the outcome of an experiment. In rolling a dice, the random variable may take any of the values 1,2,3,4,5,6.

Based on the experiment, these random variables could be continuous or discrete.

8

5

Probability - Example

9

Probability – Probability Functions

Random variables are usually associated with

functions or distributions to characterize the

experiment they come from.

Such functions are called probability density functions

(pdf).

If available, these distributions can be used to

calculate the probability of the random variable to

take certain value(s).

The area under pdf is always equal to 1. this is based

on the fact that we will obtain any of the possible

outcomes if we run the experiment.

10

6

Common Distributions

Uniform Distribution

Normal Distribution

11

1 , a x b

pdf(x) b a 0 , otherwise

2

2(x )

21pdf(x) e

2

ba

b - a

_____1

-6 -4 -2 0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Computing Probability of an Event

The probability that a random variable x is less than or equal than some value b is computed from the pdf by

Continuous RV Discrete RV

This is usually referred as the cumulative distribution function CDF.

For continuous random variables, the probability P(x=b) = 0.

For discrete random variables, the probability P(x=xk) = pdf(xk)

12

b

P(x b) pdf(x)dx CDF(x)

k

k ii

P(x x ) pdf(x)

7

Histograms as pdf

For practical experiments, it is common

to use the normalized histogram to

represent the pdf of the experiment.

The normalized histogram is defined as

the absolute frequency histogram divided

by total number of frequencies.

13

Example Suppose that x = [1,1,1,1,1,2,2,3,3,3,3,5,5,5,5,7,7,7,7,7,7,7,8]

The following commands generates the normalized histogram of x with

bin centers at 1 through 8.

>> [y,binLocations] = hist(x,[1:8]) ;

>> normhist = y / sum(y) ; % or could use normhist = y / length(x)

>> label(‘x’)

>> ylabel(‘Normalized Histogram’)

The probability of obtaining a number less than or equal to 3 can be found by

>> cdf = cumsum(normhist)

>> prob = cdf(3)

14 1 2 3 4 5 6 7 8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

Nor

mal

ized

His

togr

am

8

Sums and Differences of

Random Variables

15

It can be proved that the mean of the sum (or difference)

of two independent normally distributed random variables

equals the sum (or difference) of their means, but the

variance is always the sum of the two variances. That is,

if x and y are normally distributed with means x and y,

and variances x and y, and if u x + y and u x y,

then

u x + y

u x y

u2 u

2 x2 + y

2

Random Numbers

Random numbers are extensively used in

simulation and game programming.

Matlab provides several functions to

generate random numbers based on a given

distribution.

16

9

Random Numbers

17

Command Description

rand Generates a single uniformly distributed random number

between 0 and 1.

rand(n) Generates an n x n matrix containing uniformly

distributed random numbers between 0 and 1.

rand(m,n) Generates an m x n matrix containing uniformly

distributed random numbers between 0 and 1.

s = rand(’state’) Returns a 35-element vector s containing the current

state of the uniformly distributed generator.

rand(’state’,s) Sets the state of the uniformly distributed generator to s.

rand(’state’,0) Resets the uniformly distributed generator to its initial

state.

rand(’state’,j) Resets the uniformly distributed generator to state j, for

integer j.

rand(’state’,sum(100*clock)) Resets the uniformly distributed generator to a different

state each time it is executed.

Random Numbers

18

Command Description

randn Generates a single normally distributed random number having

a mean of 0 and a standard deviation of 1.

randn(n) Generates an n ´ n matrix containing normally distributed

random numbers having a mean of 0 and a standard deviation of

1.

randn(m,n) Generates an m ´ n matrix containing normally distributed

random numbers having a mean of 0 and a standard deviation of

1.

s = randn(’state’) Like rand(’state’) but for the normally distributed generator

randn(’state’,s) Like rand(’state’,s) but for the normally distributed generator.

randn(’state’,0) Like rand(’state’,0) but for the normally distributed generator.

randn(’state’,j) Like rand(’state’,j) but for the normally distributed generator.

randn(’state’,sum(100*clo

ck))

Like rand(’state’,sum(100*clock)) but for the normally

distributed generator.

10

Random Numbers

Example: check the operation of randn >> x = randn(1,10000);

>> [y,bins] = hist(x,[-4:0.01:4]));

>> y = y / length(x)

>> Bar(bins,y)

>> axis([-5 5 0 max(y)*1.1])

19 -5 -4 -3 -2 -1 0 1 2 3 4 5

0

1

2

3

4

5

x 10-3

Random Numbers To generate uniformly distributed random

numbers in the interval [a,b], use

y = (b-a) * x + a

Where x is a set of uniformly distributed numbers generated by rand function.

Example : the following session generates 1000 uniformly distributed numbers in the interval [-3,3].

>> x = rand(1,1000) ;

>> y = (3+3) * x – 3 ;

>> [yhist,bins] = hist(y,[-4:0.1:4]);

>> bar(bins,yhist);

20

11

Random Numbers

-5 -4 -3 -2 -1 0 1 2 3 4 50

5

10

15

20

25

30

x

Appro

xim

ate

Uniform

Dis

trib

ution

21

Random Numbers

To generate normally distributed random numbers with a mean µ and standard deviation σ

y = σ * x + µ

Where x is a set of normally distributed numbers generated by randn function.

Example : the following session generates 1000 normally distributed numbers with a mean of 5 and deviation of 1.5

>> x = randn(1,1000) ;

>> y = 1.5* x + 5 ;

>> [yhist,bins] = hist(x,[0:0.1:10];

>> bar(bins,yhist);

22

12

Random Numbers

-2 0 2 4 6 8 10 120

5

10

15

20

25

30

35

x

Appro

xim

ate

Norm

al D

istr

ibution

23

Interpolation

Engineering problems often require the analysis of data pairs; cause and effect, or input output relationships.

It is sometimes required to estimate variable’s value between the given data point. This process is known as interpolation.

Plotting the data greatly aids interpolation and extrapolation.

24

13

Interpolation Linear and cubic spline interpolation

25

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

x

y

Linear

Cubic Spline

Interpolation

26

Function Description

Y_int =

interp1(x,y,x_int)

Used to linearly interpolate a function of one variable: y f (x). Returns a linearly interpolated vector y_int at the

specified value x_int, using data stored in x and y.

Z_int =

interp2(x,y,z,x_int,y

_int)

Used to linearly interpolate a function of two variables: y f (x, y). Returns a linearly interpolated vector z_int at

the specified values x_int and y_int, using data stored

in x, y, and z.

Y_int =

spline(x,y,x_int)

Computes a cubic-spline interpolation where x and y

are vectors containing the data and x_int is a vector

containing the values of the independent variable x at

which we wish to estimate the dependent variable y. The

result Y_int is a vector the same size as x_int containing

the interpolated values of y that correspond to x_int.

14

Interpolation

When using interp1 or spline functions, note that x should be sorted in ascending order and x_int values should be within the range of x.

Example: assume that x = [1,2,3,4,5] and y = [2,5,8,10,6]. Estimate the value of y when x_int = [2.5,3.5].

% using linear

>> y_int = interp1(x,y,x_int)

y_int = 6.5000 9.0000

% using cubic spline

>> y_int = spline(x,y,x_int)

y_int = 6.5000 9.3750

27

1

Computer Applications Lab

Lab 10 Numerical Calculus and Symbolic

Processing

Chapter 8 - Sections 8.1 through 8.3

Chapter 10 - Sections 10.1 through 10.3

Dr. Iyad Jafar

Adapted from the publisher slides

Outline

Numerical Calculus

◦ Integration

◦ differentiation

Symbolic Processing ◦ Defining symbolic expressions

◦ Manipulation of symbolic expressions

◦ Symbolic differentiation, integration, and limits

2

2

Numerical Calculus - Integration

Integration of function f(x) in the interval a ≤ x ≤

b can be represented as the area between f(x) and

the x-axis.

3

Numerical Calculus - Integration

Numerical integration can be approximated by

partitioning the area into rectangles or trapezoidals

then summing up their areas.

4

3

Numerical Calculus - Integration

5

Function Description

trapz(x,y) Uses trapezoidal integration to compute the integral

of y with respect to x, where the array y contains the

function values at the points contained in the array x.

quad(‘func’,a,b,tol) Uses an adaptive Simpson’s rule to compute the

integral of the function ’fun’ with a as the lower

integration limit and b as the upper limit. The

parameter tol is optional. tol indicates the specified

error tolerance.

quad1(‘fun’,a,b,tol) Uses Lobatto quadrature to compute the integral of

the function ’fun’. The rest of the syntax is identical to

quad.

Numerical Calculus - Integration

The quad and quadl functions are more accurate than trapz; however, they are restricted to computing the integrals of functions and cannot be used when the integrand is specified by a set of points. For such cases, use the trapz function.

Example: the integration can be computed in Matlab

6

Function Result

quad(‘sin’,0,pi) 2

quad1(‘sin’,0,pi) 2

x = 0:0.01:pi

trapz(x,sin(x)) 1.9797

0

sin(x)dx 2

4

Numerical Calculus - Differentiation

The derivative of a function can be interpreted

graphically as the slope function.

Three methods can be used to estimate the slope at

a point numerically; backward difference, forward

difference, and central difference.

7

Matlab provides the diff function for computing derivative estimates. It computes the backward difference.

Syntax

d = diff(x)

where x is a vector of values and the result is a vector d containing the differences between adjacent elements in x.

if x has n elements, then d is n-1 elements.

Example:

>> x = [5,7,12,-20];

>> y = diff(x)

y = 2 5 -32

8

Numerical Calculus - Differentiation

5

Example: compare the true derivative of sin(x) with backward and central derivatives.

x = [0:pi/50:2*pi] ;

n = length(x);

y = sin(x);

% true derivative

td = cos(x) ;

% numerical derivative backward difference

nd = diff(y) ./ diff(x) ;

% numerical derivative using central difference

ncd = (y(3:n)-y(1:n-2))./ (x(3:n)-x(1:n-2)) ;

plot(x,td,'b'), ;hold on

plot(x(2:n),nd,'ro'), plot(x(3:n),ncd,'g*') ;

xlabel('x'), ylabel('Derivative of SIN(x)')

legend('COS(x)','Backward Derivative','Central Derivative')

9

Numerical Calculus - Differentiation

10

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Derivative o

f S

IN(x

)

COS(x)

Backward Derivative

Central Derivative

Numerical Calculus - Differentiation

6

11

Command Description

d = diff(x)

Returns a vector d containing the differences between adjacent

elements in the vector x.

b = polyder(p)

Returns a vector b containing the coefficients of the derivative

of the polynomial represented by the vector p.

b =

polyder(p1,p2)

Returns a vector b containing the coefficients of the polynomial

that is the derivative of the product of the polynomials represented by p1 and p2.

[num, den] =

polyder(p2,p1)

Returns the vectors num and den containing the coefficients of

the numerator and denominator polynomials of the derivative of the quotient p2/p1, where p1 and p2 are polynomials.

Numerical Calculus - Differentiation

Example

F(x) = x2 + 2x + 1 , G(x) = x3 + 5x2 – x + 8, find the derivative of F * G and F/G

>> F = [1,2,1], G = [1,5,-1,8];

% d/dx(F*G) >> d = polyder(F,G)

d = 5 28 30 22 15

% d/dx(F/G) >> [num,denum] = polyder(F,G)

num = -1 -4 -14 6 17

denum = 1 10 23 6 81 -16

64

12

Numerical Calculus - Differentiation

7

Symbolic Processing

Refers to the way that computers

perform operations on mathematical

expressions the way that humans it with

pencil and paper.

Matlab contains many functions to define

symbols, symbolic expressions,

simplification, and solution to symbolic

expressions.

13

Symbolic Processing – Defining Symbols

To define a symbol in Matlab use the sym() or the syms functions. We can create symbolic constants or symbolic variables.

Examples

1. x = sym(‘x’) % creates the symbolic variable with name x

2. syms x y u v % creates four symbolic variables x,y,u, and v

3. fraction = sym(‘1/3’) % create symbolic constant fraction with value 1/3

When mathematical operations are used with symbols, the result is symbolic.

Example >> x = sym(‘sqrt(2)’);

>> 2 * x

ans = 2*2^(1/2)

>> class(ans)

ans = sym

14

8

Symbolic Processing – Symbolic

Expressions We can use symbols to define symbolic expressions. We can use the

operators + - / ^ and the built-in functions.

Example

>> syms x y ;

>> r = sqrt(x^2+y^2)

r = (x^2+y^2)^(1/2)

If we later assign x and y to 3 and 5, typing r will not result in the evaluation

of the expression.

We can use the vector and matrix notation with symbols.

Example

>> n = 3 , syms x, A = x.^((0:n)’*(0:n))

A = [ 1, 1, 1, 1]

[ 1, x, x^2, x^3]

[ 1, x^2, x^4, x^6]

[ 1, x^3, x^6, x^9]

>> class(A)

ans = sym

15

Symbolic Processing The function findsym(E,n) returns the n symbolic variables in

expression E that are closest to x (default symbol in Matlab), with

the tie breaker going to the variable closer to z. >>syms b x1 y

>>findsym(6*b+y)

ans =

b,y

>>findsym(6*b+y,1) %Find the one variable closest to

x

ans =

y

>>findsym(6*b+y+x1,1) %Find the one variable closest

to x

ans =

x1

>>findsym(6*b+y*i) %i is not symbolic

ans =

b, y 16

9

Manipulating Symbolic Expressions

With symbolic expressions in Matlab, we can expand, collect, simplify, and

create new expressions from old expressions using the mathematical

operators.

The function collect(E) collects coefficients of like powers in the

expression E. If there is more than one variable, you can use the optional

form collect(E,v), which collects all the coefficients with the same

power of v.

>>syms x y

>>E = (x-5)^2+(y-3)^2;

>>collect(E) % will collect x powers

ans =

x^2-10*x+25+(y-3)^2

>>collect(E,y) % will collect y powers

ans =

y^2-6*y+(x-5)^2+9

17

Manipulating Symbolic Expressions

The expand and simplify functions. >>syms x y

>>expand((x+y)^2) % applies algebra rules

ans =

x^2+2*x*y+y^2

>>expand(sin(x+y)) % applies trig identities

ans =

sin(x)*cos(y)+cos(x)*sin(y)

>>simplify(6*((sin(x))^2+(cos(x))^2))

% applies another trig identity

ans = 6

18

10

Manipulating Symbolic Expressions

The factor function finds the factors of the expression.

>>syms x y

>>factor(x^2-1)

ans =

(x-1)*(x+1)

The function subs(E,old,new) substitutes new for old

in the expression E, where old can be a symbolic variable or

expression and new can be a symbolic variable, expression, or

matrix, or a numeric value or matrix. For example,

>>syms x y

>>E = x^2+6*x+7;

>>F = subs(E,x,y)

F =

y^2+6*y+7

19

Manipulating Symbolic Expressions

The function poly2sym(p) converts the

coefficient vector p to a symbolic polynomial.

The function sym2poly(E) converts the

expression E to a polynomial coefficient vector.

>> ploy2sym([2,4,5])

ans = 2*x^2 + 4*x + 5

>> ploy2sym([2,4,5],y)

ans = 2*y^2 + 4*y + 5

20

>> sys x ;

>> sym2poly(9*x^2+4)

ans =

9 0 4

11

Manipulating Symbolic Expressions

The function [num,den] = numden(E) returns two expressions num and den for the numerator and denominator for expression E.

>> syms x

>> E1 = x^2 + 5 ;

>> E2 = 1 / (x+6) ;

>> [num,den] = numden(E1+E2)

num = x^3 + 6*x^2 + 5*x + 31

den = x + 6

>> pretty(num) % provides mathematical friendly

expression

3 2

x + 6 x + 5 x + 31

21

Evaluating Symbolic Expressions Use the subs and double functions to evaluate an

expression numerically. Use subs(E,old,new) to replace old with a numeric value new in the expression E. The result is of class double. For example,

% evaluate symbolic

expressions

>>syms x

>>E = x^2+6*x+7;

>>G = subs(E,x,2)

G =

23

>>class(G)

ans =

double

22

% evaluate constant

expressions

>> sqt2 = sym(‘sqrt(2));

>> y = 6 * sqrt2

y = 6 * 2^(1/2)

>> double(y)

ans = 8.4853

12

Plotting Expressions

The Matlab function ezplot(E) generates a plot of a symbolic expression E, which is a function of one variable. The default range of the independent variable is the interval [-2, 2].

The optional form ezplot(E,[xmin xmax]) generates a plot over the range from xmin to xmax.

The ezplot function accepts E as string also.

The ezplot function does not accept the specification of line styles and data markers.

23

ezplot(x^2-6*x+7)

24

-6 -4 -2 0 2 4 6

0

10

20

30

40

50

60

70

80

90

x

x2-6 x+7

13

The Solve Function

It is used to find the solution of an equation, i.e. the zeros of the expression.

>> eq1 = ‘x+5’

>> solve(eq1)

ans = -5

>> solve(‘exp(2*x)+3*exp(x)=54’)

ans =

[log(-9)]

[log(6) ]

>> eq3 = ’x^2+9*y^4=0’;

>>solve(eq3) %Note that x is presumed to be the unknown variable

ans =

[ 3*i*y^2]

[-3*i*y^2]

25

The Solve Function We can solve an equation in terms of a symbol

Example:

>> solve (‘b^2+8*c+2*b=0’)

ans = -1/8*b^2-1/4*b

>> solve (‘b^2+8*c+2*b=0’,’b’) % solve for b

ans = [-1+(1-8*c)^(1/2)]

[-1-(1-8*c)^(1/2)]

We can solve more than equation

>> eq1 = ‘6*x+2*y=14’;

>> eq2 = ‘3*x+7*y=31’;

>> solve(eq1,eq2)

ans = x: [1x1 sym] % the answer is a structure

y: [1x1 sym]

>> x = ans.x

x = 1

>> y = ans.y

y = 4 26

14

Evaluating Symbolic Expressions

Example

>> syms w x y z ;

>> M = w - x^2 + log(y) + 5*z ;

>> % evaluate M with y = 5 and z = 1

>> N = subs(M,[y,z],[5 1])

ans = w-x^2+log(5)+5

>> % solve in terms of w

>> G = solve(N,w) % the result is symbolic array

G = (w+log(5)+5)^(1/2)

-(w+log(5)+5)^(1/2)

>> % evaluate the first root if w = 6

>> subs(G(1),w,6)

ans =

3.5510

27

Solving Symbolic Expressions

Example: find the intersection of y = x2 and y = 2x

>> syms x ;

>> y1 = x^2 ;

>> y2 = 2*x ;

>> solve(y1-y2)

ans =

0

2

Example: if x + 6y = a and 2x – 3y = 9, then find the solution for x and y in

terms of the parameter a.

>> syms a x y ;

>> [x,y] = solve('x+6*y = a','2*x-3*y = 9')

x = 1/5*a+18/5

y = 2/15*a-3/5

28

15

Symbolic Differentiation We can use the diff function to perform symbolic

differentiation by passing the expression directly, as a string, or as a symbolic expression.

>> diff(‘sin(x*y)’) % differentiation with respect to

x

ans = cos(x*y)*y

>> syms x y ; % differentiation with respect to y

>> diff(sin(x*y),y)

ans = x*cos(x*y)

>> diff(x*sin(x*y),y,2) % Computes the second

derivative

ans = -x^3*sin(x*y)

29

Symbolic Integration

We can use the int function to perform symbolic integration by passing the expression directly, as a string, or as a symbolic expression.

>> int(2*x) % x is defined as a symbol

ans = x^2

>> int(sin(x*y),y) % integration with respect to y

ans = -1/x*cos(x*y)

>> int(sin(x*y),y,0,1) % integrates with respect

to y over [0,1]

ans = -(-1+cos(x))/x

>> int(sin(x*y),0,1) %integrates with respect to x

over [0,1]

ans = -(-1+cos(y))/y

30

16

Limits

We can use the limit function to find the limits of

expressions.

>> limit(sin(a*x)/x) % limit as x goes to 0

ans = a

>> limit(sin(a*x)/x,3) % limit as x goes to 3

ans = 1/3*sin(3*a)

>> limit(sin(a*x)/x,a,3) % limit as a goes to 3

ans = sin(3*x)/x

>> limit((1/x),x,0,’right’) % limit as x goes to 0

from right

ans = Inf

31