W6 MATLAB Fundamentals

8/2/2019 W6 MATLAB Fundamentals

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Centre for Computer Technology

ICT114Mathematics for

Computing

Week 6

Fundamentals of MATLAB


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March 20, 2012 Copyright Box Hill Institute

Objectives

Review Week5

MATLAB workspace

Basic Operations Matrix and Array Operations

Control statements

Script files

M-files


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Queue Characteristics

Queueing systems are characterized byfive components

Arrival pattern of customers

Service pattern

Number of servers

The capacity of the facility to hold customers The order in which customers are served


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Kendalls notation

Kendalls notation for specifying a queuecharacteristics is v/ w/ x/ y/ z

v indicates arrival patternw denotes service pattern

x denotes the number of servers

y represents system capacity (number ofcustomers)

z designates queue discipline


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Some notations for v/ w/ z

Queue characteristic Symbol Meaning

Interarrival timeor

Service time

D

M

Ek (k = 1, 2, 3..)

G

Deterministic

Exponentially distributed

Erlang type distributed

Any other distribution

Queue discipline

FIFO

LIFOSIRO

PRI

GD

First in first out

Last in first outService in random order

Priority ordering

Any other ordering


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Example

Write an algorithm to determine astudents final grade and indicate whether

it is passing or failing. The final grade iscalculated as the average of four marks.

(Algorithms and Flowcharts)


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Example - Pseudocode

Input a set of 4 marks

Calculate their average by summing and dividingby 4

if average is below 50

Print FAIL

else

Print PASS



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Example - Algorithm

Step 1: Input M1,M2,M3,M4

Step 2: GRADE (M1+M2+M3+M4)/4Step 3: if (GRADE < 50) then

Print FAILelse

Print PASS

endif



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Flowchart - Example

PRINTPASS

Step 1: Input M1,M2,M3,M4Step 2: GRADE (M1+M2+M3+M4)/4Step 3: if (GRADE


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Centre for Computer Technology

Fundamentals of MATLAB


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Introduction

MATLAB stands for Matrix Laboratory It is a Numerical Computation software for

Engineering and Scientific calculations Developed by John Little and Cleve Moler

of MathWorks, Inc. The basic Data Type is a Matrix, which

doesnt require dimensioning MATLAB can be used to generate a

variety of Graphs


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MATLAB Variables

Legal variable names

Begin with one of az or AZ

Have remaining characters chosen fromaz, AZ, 09, or _

Maximum length of 63 characters (ver 6.5)

Should not be the name of a built-invariable, built-in-function, or user-definedfunction


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Basic Operations

The basic data type is a numerical rectangularmatrix with real or complex numbers

MATLAB statements are of the form

variable =expression

Expressions typed manually (typed by user) areinterpreted immediately and evaluated

If the statements ends with a semicolon ; , it isevaluated, but the display is suppressed


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March 20, 2012Copyright Box Hill Institute

Basic Operations The variables are case sensitive.

casesen off turns this off

exit or quit, are the commands to exit

MATLAB help, is for online help

helpfunction name, is to obtain informationon a specific function

lookfor, searches the summary informationfor each function

clc clears the command window


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Matrices and Vectors

All Matlab variables are matrices

A Matlab vector is a matrix with one

row orone column

A Matlab scalar is a matrix with onerow andone column


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Basic Operations

A matrix1 2 3

A= 2 3 4 is represented by

3 4 5

A = [1 2 3; 2 3 4; 3 4 5];or

A = [ 1 2 32 3 43 4 5 ];


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Basic Operations A row vector

B = [ 6 9 12 15 18] is represented byB = [ 6 9 12 15 18] or B = [6, 9, 12, 15, 18]

A column vectorC = 6 C = [ 69 912 is represented by 1215 1518 18]

orC = [6;9;12;15;18 ]


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Matrix Operations

The basic matrix operations are addition(+),subtraction(-), multiplication(*) and conjugatetranspose ()

There are two forms of matrix division, the leftinverse operator (\) and the right inverseoperator (/)

size(A) is used to obtain the size of a matrix

A, the result will be a row matrix with twovalues. The is the number of rows and thesecond the number of columns of A


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Matrix Operations The sum and difference of matrices with same

dimensions can be determined For two matrices E and F with same dimensions

E = [7 2 3; 4 3 6; 8 1 5];

F = [1 4 2; 6 7 5; 1 9 1];

The sum H = E + F The matrix H will appear as

H =

8 6 510 10 119 10 6


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Matrix Operations

For the matrices E and F the differenceG = E F

The difference will appear as

G =6 -2 1-2 -4 17 -8 4

A scalar matrix may also be added or subtractedfrom a matrix in a similar manner


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Matrix Operations

For two matrices X (n x m) and Y (i x j) Matrix Multiplication is defined if the innermost operands

of the two operands are the same, i.e. in the above casem=i

For the matrices E and F (both 3x3), the productQ = E*F will appear asQ =

22 69 27

28 91 2919 84 26 The matrix can be multiplied by a scalar in a similar

fashion


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Matrix Operations Matrix division can be either left inverse operator (\) or

right inverse operator (/) For two matrices x and y

x/y is equivalent to x div yx\y is equivalent to y div x

If a*b = c and a is a nonsingular matrix, thena\c is equivalent to b= inv(a)*c

a/c is equivalent to b = c*inv(a)

Where inv is the MATLAB function to obtain the inverseof a matrix


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Syntax

A = ones(nrows,ncols)

A = zeros(nrows,ncols)

A = eye(n)

A = eye(nrows,ncols)

A = diag(v) v= diag(A)


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March 20, 2012

Copyright Box Hill Institute

Example


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March 20, 2012

Copyright Box Hill Institute


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Subscript Notation

If A is a matrix, A(i,j)selects the element in the ith rowand jth column.

Subscript notation can be used on the right hand side ofan expression to refer to a matrix element.

>> A = [1 2 3; 4 5 6; 7 8 9];>> b = A(3,2)

b = 8

>> c = A(1,1)c = 1


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Subscript Notation

Subscript notation is also used to assignmatrix elements

>> A(1,1) = c/bA =0.2500 2.0000 3.0000

4.0000 5.0000 6.00007.0000 8.0000 9.0000


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Array Operations

Array operations are the element byelement arithmetic operation

The normal matrix operations(*,\,/,^)preceded by (.) represents an arrayoperation

^ represents raising to the power Addition and Subtraction operations are

the same for a matrix and an array


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Array Operations For two matrices with the same dimensions

A1=[2 7 6;8 9 10]; and B1=[6 4 3;2 3 4]; C1=A1.*B1 will result in C1 =

12 28 18

16 27 40 For two matrices with the same dimensions

R1=[7 3 5]; and S1=[2 3 4]; Q1=R1.^S1 will result in Q1=

49 9 625

Q2=R1.^2 will result in Q2=49 9 25

Q3=2.^S1 will result in Q3=4 8 16


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Array Operations

For two matrices with the same dimensionsA1=[2 7 6;8 9 10]; and B1=[6 4 3;2 3 4];

D1=A1./B1 will result in

D1= 0.3333 1.7500 2.00004.0000 3.0000 2.5000

E1=A1.\B1 will result inE1=

3.0000 0.5714 0.50000.2500 0.3333 0.4000


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Complex Numbers

Numbers of the form z=3+4i are calledcomplex numbers

In MATLAB the above number isrepresented as z=3+4*I

There are no spaces around the + sign.

This is because, if there is a spaceMATLAB considers it as two differentnumbers


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Complex Numbers

produces the conjugate transpose

For a matrix x=1+2*i 2+1*i

3-1*i 4+5*i xt=x will result in xt= 1-2*i 3+1*i

2-1*i 4-5*i

. produces the unconjugate transpose xut=x. will result in xt= 1+2*i 3-1*i

2+1*i 4+5*i


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Generating Vectors, Iterations

A very important operator in MATLAB is thecolon symbol (:)

It can be used- to create vectors and matrices

-to specify submatrices and vectors- to perform iterations

Syntaxv_variable = i_val: inc/dec: f_val

v_variable is the vector variablei_val is the initial valuef_val is the final valueinc/dec is the increment or decrement


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Generating Vectors - Examples

v1= 1:8 will generatev1=

1 2 3 4 5 6 7 8 v2= 4:-0.5:1 will generate

v2=4.0 3.5 3.0 2.5 2.0 1.5 1.0

v3= [(5:-1:0);(8:0.2:9)] will generatev3=

5.0 4.0 3.0 2.0 1.0 0.08.0 8.2 8.4 8.6 8.8 9.0


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Generating Vectors

Vectors can be generated by the MATLABfucntions linspaceand logspace

linspacegenerates linearly evenly spacedvectors

logspacegenerates logarithmically evenlyspaced vectors Syntax

linspace(i_val,f_val,num)

logspace(i_val,f_val,num)i_val is the initial valuef_val is the final valuenum is the number of elements in the vector


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Control Statementsfor Loops

The for loop allows a statement or a group ofstatements to repeat a fixed number of times

Syntax forindex=i_val:inc:f_valstatement group y

endi_valis initial value, incis increment, f_valis final value

eg. sum = 0;for i = 1 : 100sum = sum + i^2

endsum


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Nested for loops

General formforindex1=i_val1 : inc1 : f_val1

forindex2=i_val2 : inc2 : f_val2

statement group yend

end

i_val1, i_val2 areinitial valuesf_val1,f_val2 arefinal values

inc1, inc2 are respectiveincrements


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Control Statementsifstatements

if statements use logical operations to performsteps

Syntaxiflogical expression1

statement group1

endthe statement group1 will execute if logical expression1 is true(1)else execute the statement following the end statement

eg. if x


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Nested if statements

General formiflogical expression1

statement group1

iflogical expression2statement group2

endstatement group3

endstatement group4


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Control Statements Relational Operators

When a relational operator is used to compare betweenpairs of corresponding elements, the resulting matrixconsists of 1s(true) and 0s(false)

Operator Function

=

= =

~ =

Less than

Less than or equal

Greater than

Greater than or equal

Equal

Not equal


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Control Statementsif-else

statements if-else statement executes one set of statements

if a logical expression is true and a different setof statements if the logical statement is false

Syntax

iflogical expression1

statement group1

elsestatement group2

end


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Control Statements - if-elseif-else

statement With if-elseif-else,group of statements are executed if

other logical expressions are false Syntax

iflogical expression1statement group1

elseiflogical expression2statement group2

elseiflogical expression3statement group3else statement group3

end


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Control Statements - while loop

while loop allows to execute a group ofstatements as long as a specified

condition is satisfied Syntax

whileexpression1

statement group1end

statement group2


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Plotting

Two dimensional plots are created with theplot function

Syntax:plot(x,y)plot(xdata,ydata,symbol)

plot(x1,y1,x2,y2,...)plot(x1,y1,symbol1,x2,y2,symbol2,...)


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Displayfprintf

fprintf writes both to the screen (and to a file)eg:

x = 0 : 5 : 10;

fprintf( The value of x is %2.2f\n , x)

MATLAB displays

The value of x is 0.00The value of x is 5.00

The value of x is 10.00



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Displaydisplay

display(x) prints the value of a variable or

expression x

eg:

x = 0 : 5 : 10;

display(x)

MATLAB displays 0.00 5.00 10.00



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MATLAB Programming

Two different kinds of MATLAB programs

-- MATLAB Scripts-- MATLAB Functions

Both are basically MATLAB files with .m

extension. Also called as m-files


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m-files

To create an m-file

click on FileNew select M-File from the menu

The MATLAB Editor/Debugger screen is presented.The code is typed / edited here

After the code is entered in the Editor/Debugger screen

click on FileSave As (e.g., firstgraph.m ) Save

Make sure that the file is saved in the directory that is inMATLAB's search path


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Script m-files

Script Files

Not really programs

No input/output parameters

Script variables are part of workspace

Useful for tasks that never change Useful as a tool for documenting homework


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Script files - Example

Enter statements for a script file sineplot.m1. Choose New. . . from File menu2. Enter lines listed below

x=linspace(0,2*pi);y=sin(x);plot(x,y)

3. Choose Save. . . from File menu

Save as sineplot.m 4. Run it

>> sineplot


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Script files - disadvantages

Create and change variables in theworkspace

Give no warning that workspace variableshave changed

Can cause bugs that are hard to track

down The algorithm cannot be used for different

set of data


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Function m-files

Use input and output parameters tocommunicate with other functions and thecommand window

Use local variables that exist only while thefunction is executing. These are distinct fromvariables of the same name in the workspace orin other functions

Input parameters allow the same calculationprocedure to be applied to different data

Functions can call other functions


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Syntax

function [outArgs] = funcName(inArgs)

outArgsare enclosed in [ ]

outArgsis a comma-separated list of variable names

[ ] is optional if there is only one parameter functions with no outArgsare legal

inArgsare enclosed in ( )

inArgsis a comma-separated list of variable names functions with no inArgsare legal


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Script files - example

twosum.m - two inputs, no output

function twosum(x,y)% twosum Add two matrices -- comment% and print the result -- commentx+y

>> twosum(2,2)

ans =4

Please note that % is used to write comments


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Script files - example

addmult.m two inputs, two outputs

function [s,p] = addmult(x,y)

% Compute sum and product of two matrices

s = x+y

p = x*y

>> addmult(3,2)


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Summary

MATLAB statements are of the form

variable = expression

All Matlab variables are matrices Matrix and Array operations

Control Statements

Script files Function m-files


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Reference

Gerald W. Recktenwald, NumericalMethods with MATLAB, Implementationand Application, Prentice Hall

John O. Attia, Electronics and CircuitAnalysis using MATLAB, 2nd ed, BocaRaton, FL, CRC Press 2004

Stephen J. Chapman, Essentials ofMATLAB Programming, Thomson Nelson

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