_Matlab+Introduction+%28ang%29.pdf

An Introduction to Matlab

Version 2.3

David F. Griths

Department of Mathematics

The University

Dundee DD1 4HN

With additional material by Ulf Carlsson

Department of Vehicle Engineering

KTH, Stockholm, Sweden

This introduction may be distributed provided that it is not be altered in any way and that its source

is properly and completely specied.

Contents

15 Examples in Plotting

1 MATLAB

16 Matrices|Two{Dimensional Arrays 13

16.1 Size of a matrix . . . . . . . . . . . . 14

16.2 Transpose of a matrix . . . . . . . . 14

16.3 Special Matrices . . . . . . . . . . . 14

16.4 The Identity Matrix . . . . . . . . . 14

16.5 Diagonal Matrices . . . . . . . . . . 15

16.6 Building Matrices . . . . . . . . . . . 15

16.7 Tabulating Functions . . . . . . . . . 15

16.8 Extracting Bits of Matrices . . . . . 16

16.9 Dot product of matrices ( .* ) . . . . 16

16.10Matrix{vector products . . . . . . . 16

16.11Matrix{Matrix Products . . . . . . . 17

16.12Sparse Matrices . . . . . . . . . . . . 17

2 Starting Up 2

2.1 Windows Systems . . . . . . . . . . 2

2.2 Unix Systems . . . . . . . . . . . . . 2

2.3 Command Line Help . . . . . . . . . 2

2.4 Demos . . . . . . . . . . . . . . . . . 3

3 Matlab as a Calculator

4 Numbers & Formats

5 Variables 3

5.1 Variable Names . . . . . . . . . . . . 3

6 Suppressing output

17 Systems of Linear Equations

17.1 Overdetermined system of linear equa-

tions . . . . . . . . . . . . . . . . . . 18

7 Built{In Functions 4

7.1 Trigonometric Functions . . . . . . . 4

7.2 Other Elementary Functions . . . . . 4

18 Characters, Strings and Text

8 Vectors 4

8.1 The Colon Notation . . . . . . . . . 5

8.2 Extracting Bits of a Vector . . . . . 5

8.3 Column Vectors . . . . . . . . . . . . 5

8.4 Transposing . . . . . . . . . . . . . . 5

19 Loops

20 Logicals 21

20.1 While Loops . . . . . . . . . . . . . . 22

20.2 if...then...else...end . . . . . . 23

9 Keeping a record

21 Function m{les 23

21.1 Examples of functions . . . . . . . . 24

10 Plotting Elementary Functions 6

10.1 Plotting|Titles & Labels . . . . . . 7

10.2 Grids . . . . . . . . . . . . . . . . . . 7

10.3 Line Styles & Colours . . . . . . . . 7

10.4 Multi{plots . . . . . . . . . . . . . . 7

10.5 Hold . . . . . . . . . . . . . . . . . . 7

10.6 Hard Copy . . . . . . . . . . . . . . 8

10.7 Subplot . . . . . . . . . . . . . . . . 8

10.8 Zooming . . . . . . . . . . . . . . . . 8

10.9 Formatted text on Plots . . . . . . . 8

10.10Controlling Axes . . . . . . . . . . . 9

22 Further Built{in Functions 25

22.1 Rounding Numbers . . . . . . . . . . 25

22.2 The sum Function . . . . . . . . . . . 25

22.3 max & min . . . . . . . . . . . . . . . 26

22.4 Random Numbers . . . . . . . . . . 26

22.5 find for vectors . . . . . . . . . . . . 27

22.6 find for matrices . . . . . . . . . . . 27

23 Plotting Surfaces

24 Timing

11 Keyboard Accelerators

25 On{line Documentation

12 Copying to and from Word and other

applications 10

12.1 Window Systems . . . . . . . . . . . 10

12.2 Unix Systems . . . . . . . . . . . . . 10

26 Reading and Writing Data Files 29

26.1 Formatted Files . . . . . . . . . . . . 30

26.2 Unformatted Files . . . . . . . . . . 30

13 Script Files

27 Graphic User Interfaces

14 Products, Division & Powers of Vec-

tors 11

14.1 Scalar Product ( * ) . . . . . . . . . . 11

14.2 Dot Product ( .* ) . . . . . . . . . . . 11

14.3 Dot Division of Arrays ( ./ ) . . . . . 12

14.4 Dot Power of Arrays ( .^ ) . . . . . . 12

28 Command Summary

1 MATLAB

from the separate Help window found under

the Help menu or

Matlab is an interactive system for doing nu-

merical computations.

from the Matlab helpdesk stored on disk or

on a CD-ROM.

A numerical analyst called Cleve Moler wrote

the rst version of Matlab in the 1970s. It

has since evolved into a successful commercial

software package.

Another useful facility is to use the ' lookfor keyword '

command, which searches the help les for the key-

word. See Exercise 16.1 (page 17) for an example

of its use.

Matlab relieves you of a lot of the mundane

tasks associated with solving problems nu-

merically. This allows you to spend more time

thinking, and encourages you to experiment.

2.2 Unix Systems

You should have a directory reserved for sav-

ing les associated with Matlab. Create such

a directory ( mkdir ) if you do not have one.

Change into this directory ( cd ).

Matlab makes use of highly respected algo-

rithms and hence you can be condent about

your results.

Powerful operations can be performed using

just one or two commands.

Start up a new xterm window (do xterm & in

the existing xterm window).

You can build up your own set of functions

for a particular application.

Launch Matlab in one of the xterm windows

with the command

Excellent graphics facilities are available, and

the pictures can be inserted into L A T E X and

Word documents.

matlab

After a short pause, the logo will be shown

followed by a window containing the Matlab

interface. Should you wish to run Matlab in

an xterm window, use the command

These notes provide only a brief glimpse of the

power and exibility of the Matlab system. For a

more comprehensive view we recommend the book

matlab -nojvm

Matlab Guide

D.J. Higham & N.J. Higham

SIAM Philadelphia, 2000, ISBN: 0-89871-469-9.

and, following dislpay of the logo, the Matlab

prompt >> will appear.

Type quit at any time to exit from Mat-

lab.

2 Starting Up

2.1 Windows Systems

2.3 Command Line Help

Help is available from the command line prompt.

Type help help for \help" (which gives a brief syn-

opsis of the help system), help for a list of topics.

The rst few lines of this read

On Windows systems MATLAB is started by double-

clicking the MATLAB icon on the desktop or by

selecting MATLAB from the start menu.

The starting procedure takes the user to the Com-

mand window where the Command line is indicated

with ' >> '. Used in the calculator mode all Matlab

commands are entered to the command line from

the keyboard.

Matlab can be used in a number of dierent ways or

modes; as an advanced calculator in the calculator

mode, in a high level programming language mode

and as a subroutine called from a C-program. More

information on the rst two of these modes is given

below.

Help and information on Matlab commands can be

found in several ways,

HELP topics:

matlab/general - General purpose commands.

matlab/ops - Operators and special char...

matlab/lang - Programming language const...

matlab/elmat - Elementary matrices and ma...

matlab/elfun - Elementary math functions.

matlab/specfun - Specialized math functions.

(truncated lines are shown with . . . ). Then to ob-

tain help on \Elementary math functions", for instance,

type

from the command line by using the ' help

topic ' command (see below),

>> help elfun

This gives rather a lot of information so, in order to see

the information one screenful at a time, rst issue the

command more on , i.e.,

Command Example of Output

>>format short 31.4162 (4{decimal places)

>>format short e 3.1416e+01

>>format long e 3.141592653589793e+01

>>format short 31.4162 (4{decimal places)

>>format bank 31.42 (2{decimal places)

>> more on

>> help elfun

Hit any key to progress to the next page of information.

2.4 Demos

Demonstrations are invaluable since they give an indi-

cation of Matlabs capabilities. A comprehensive set are

available by typing the command

format|how Matlab prints numbers|is controlled by

the \format" command. Type help format for full list.

Should you wish to switch back to the default format

then format will suce.

The command

format compact

is also useful in that it suppresses blank lines in the

output thus allowing more information to be displayed.

>> demo

( Warning: this will clear the values of all current vari-

ables.)

3 Matlab as a Calculator

5 Variables

The basic arithmetic operators are + - * / ^ and these

are used in conjunction with brackets: ( ) . The symbol

^ is used to get exponents (powers): 2^4=16 .

You should type in commands shown following

the prompt: >> .

>> 3-2^4

ans =

-13

>> ans*5

ans =

-65

>> 2 + 3/4*5

ans =

5.7500

The result of the rst calculation is labelled \ ans " by

Matlab and is used in the second calculation where its

value is changed.

We can use our own names to store numbers:

Is this calculation 2 + 3/(4*5) or 2 + (3/4)*5 ? Mat-

lab works according to the priorities:

1. quantities in brackets,

2. powers 2 + 3^2 ) 2 + 9 = 11 ,

3. * / , working left to right ( 3*4/5=12/5 ),

4. + - , working left to right ( 3+4-5=7-5 ),

Thus, the earlier calculation was for 2 + (3/4)*5 by

priority 3.

>> x = 3-2^4

x =

-13

>> y = x*5

y =

-65

so that x has the value 13 and y = 65. These can

be used in subsequent calculations. These are examples

of assignment statements: values are assigned to

variables. Each variable must be assigned a value before

it may be used on the right of an assignment statement.

4 Numbers & Formats

Matlab recognizes several dierent kinds of numbers

5.1 Variable Names

Legal names consist of any combination of letters and

digits, starting with a letter. These are allowable:

Type

Examples

Integer

1362;217897

Complex 3:21 4:3i (i = p 1)

Inf

1:234;10:76

NetCost, Left2Pay, x3, X3, z25c5

These are not allowable:

Innity (result of dividing by 0)

NaN

Not a Number, 0=0

Net-Cost, 2pay, %x, @sign

The \ e " notation is used for very large or very small

numbers:

-1.3412e+03 = 1:3412 10 3 = 1341:2

-1.3412e-01 = 1:3412 10 1 = 0:13412

All computations in MATLAB are done in double pre-

cision, which means about 15 signicant gures. The

Use names that reect the values they represent.

Special names: you should avoid using

eps = 2.2204e-16 = 2 54 (The largest number such

that 1 + eps is indistinguishable from 1) and

pi = 3.14159... = .

If you wish to do arithmetic with complex numbers,both

i and j have the value p 1 unless you change them

Real

>> i,j, i=3

ans = 0 + 1.0000i

i = 3

>> x = 9;

>> sqrt(x),exp(x),log(sqrt(x)),log10(x^2+6)

ans =

8.1031e+03

ans =

1.0986

ans =

1.9395

6 Suppressing output

One often does not want to see the result of intermedi-

ate calculations|terminate the assignment statement

or expression with semi{colon

exp(x) denotes the exponential function exp(x) = e x

and the inverse function is log :

>> x=-13; y = 5*x, z = x^2+y

y =

-65

>> format long e, exp(log(9)), log(exp(9))

ans = 9.000000000000002e+00

ans = 9

>> format short

z =

104

the value of x is hidden. Note also we can place several

statements on one line, separated by commas or semi{

colons.

and we see a tiny rounding error in the rst calculation.

log10 gives logs to the base 10. A more complete list

of elementary functions is given in Table 2 on page 32.

Exercise 6.1 In each case nd the value of the expres-

sion in Matlab and explain precisely the order in which

the calculation was performed.

8 Vectors

These come in two avours and we shall rst describe

row vectors: they are lists of numbers separated by ei-

ther commas or spaces. The number of entries is known

as the \length" of the vector and the entries are often

referred to as \elements" or \components" of the vec-

tor.The entries must be enclosed in square brackets.

i) -2^3+9 ii) 2/3*3

iii) 3*2/3 iv) 3*4-5^2*2-3

v) (2/3^2*5)*(3-4^3)^2 vi) 3*(3*4-2*5^2-3)

7 Built{In Functions

7.1 Trigonometric Functions

Those known to Matlab are

sin, cos, tan

and their arguments should be in radians.

e.g. to work out the coordinates of a point on a circle of

radius 5 centred at the origin and having an elevation

30 o = =6 radians:

>> v = [ 1 3, sqrt(5)]

v =

1.0000 3.0000 2.2361

>> length(v)

ans =

Spaces can be vitally important:

>> v2 = [3+ 4 5]

v2 =

>> x = 5*cos(pi/6), y = 5*sin(pi/6)

x =

7 5

>> v3 = [3 +4 5]

v3 =

4.3301

y =

2.5000

3 4 5

The inverse trig functions are called asin, acos, atan

(as opposed to the usual arcsin or sin 1

etc.). The

We can do certain arithmetic operations with vectors

of the same length, such as v and v3 in the previous

section.

result is in radians.

>> acos(x/5), asin(y/5)

ans = 0.5236

>> pi/6

ans = 0.5236

>> v + v3

ans =

4.0000 7.0000 7.2361

>> v4 = 3*v

v4 =

3.0000 9.0000 6.7082

>> v5 = 2*v -3*v3

v5 =

-7.0000 -6.0000 -10.5279

>> v + v2

??? Error using ==> +

Matrix dimensions must agree.

7.2 Other Elementary Functions

These include sqrt, exp, log, log10

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