Chapter 2Chapter 2

MATLAB MATLAB FundamentalsFundamentals

MATLABMATLAB

MatMatrix rix LabLaboratoryoratory

Problem-Solving MethodologyProblem-Solving Methodology

State the problem clearly Describe the Input/Output (I/O) Work the problem by hand Algorithm - Numerical Method Develop a MATLAB Solution Debugging and Testing Documentation

Why MATLAB?Why MATLAB? Industry standard software application Wealth of built-in functions and libraries Toolboxes (add-on software modules) – image and

signal processing, control systems design, fuzzy logic, etc.

Has own structured programming language Ease of application and testing (pre- and post-

processing without lots of programming and formatting)

Platform independent

MATLABMATLAB MATLAB is a numerical analysis system

Can write “programs”, but they are not formally compiled

Should still use structured programming

Should still use comments

Comments are indicated by “%” at the beginning of the line

Comments!!!Comments!!!

Program DocumentationProgram Documentation

You must include comments in the computer programs you turn in -- otherwise we will have great difficulty knowing what you are doing

For example, here is some cryptic code without comment

for j=0:2

k=(2-j)*(1+3*j)/2

end

What does it do?What does it do?

Put a comment inPut a comment in

% turns (0,1,2) into (1,2,0)

MATLAB WindowsMATLAB Windows Command Window

-- enter commands and data

-- print results Graphics Window

-- display plots and graphs Edit Window

-- create and modify m-files

Managing MATLAB EnvironmentManaging MATLAB Environment who or whos -- See the current runtime environment clear -- remove all variables from memory clc -- clear the command window clf -- clear the graphics window save -- save the workspace environment load -- restore workspace from a disk file abort -- CTRL-C help -- help “command”

Really good “help” command

MATLAB SyntaxMATLAB Syntax No complicated rules

Perhaps the most important thing to remember is semicolons (;) at the end of a line to suppress output

Type “more on” to keep text from leaving screen too fast

diary “filename” saves a text record of session

diary off turns it off

MATLABMATLAB MATLAB’s basic component is a Vector or Matrix

Even single value variables (Scalars)

All operations are optimized for vector use

Loops run slower in MATLAB than in Fortran (not a vector operation)

“size” command gives size of the matrix

Scalars, Vectors, MatricesScalars, Vectors, Matrices MATLAB treat variables as “matrices” Matrix (m n) - a set of numbers arranged in rows (m) and columns

(n) Scalar: 1 1 matrix Row Vector: 1 n matrix Column Vector: m 1 matrix

227150

592342

5231

217

32

025

21732025 275

..

..D

.

.

.

'BC

...B.A

>> pians = 3.1416

>> size(pi)ans = 1 1

>> a=[1 2 3; 4 5 6]a = 1 2 3 4 5 6

>> size(a)ans = 2 3

a=14163.pi

654

321a

Complex variablesComplex variables MATLAB handles complex arithmetic automatically No need to compute real and imaginary parts

separately The unit imaginary number i = is preassigned1

» x=5+2*ix = 5.0000 + 2.0000i» y=5*x+3y = 28.0000 +10.0000i

» x=3+5-0.2x = 7.8000» y=3*x^2+5y = 187.5200» z=x*sqrt(y)z = 106.8116» A=[1 2 3; 4 5 6]A = 1 2 3 4 5 6» b=[3;2;5]b = 3 2 5» C=A*bC = 22 52

» who

Your variables are:

A b y C x z

» whos Name Size Bytes Class

A 2x3 48 double array C 2x1 16 double array b 3x1 24 double array x 1x1 8 double array y 1x1 8 double array z 1x1 8 double array

» save

Saving to: matlab.mat

» save matrix1

MATLAB ExampleMATLAB Example

default filename

filename “matrix1”

Data typesData types

All numbers are double precision

Text is stored as arrays of characters

You don’t have to declare the type of data (defined when running)

MATLAB is case-sensitive!!!MATLAB is case-sensitive!!!

Variable NamesVariable Names Usually, the name is identified with the problem Variable names may consist of up to 31 characters Variable names may be alphabetic, digits, and the

underscore character ( _ ) Variable names must start with a letter

ABC, A1, C56, CVEN_302

day, year, iteration, max

time, velocity, distance, area, density, pressure

Time, TIME, time (case sensitive!!)(case sensitive!!)

Initializing VariablesInitializing Variables Explicitly list the values reads from a data file uses the colon (:) operator reads from the keyboard

A = [1; 3; 5; 10]; B = [1 3 5; -6 4 -1]

C = [2 3 5 1; 0 1 … (continuation)

1 -2; 3 5 1 -3]

E = [A; 1; A]; F = [C(2,3); A]

Matrix ConcatenationMatrix Concatenation 497y ; 321x

497321yxz

321497

497321 xy ;y xv

497

321y ;x u

41

12

10

52

F

410

123

101

521

C

Colon OperatorColon Operator Creating new matrices from an existing matrix

C = [1,2,5; -1,0,1; 3,2,-1; 0,1,4]

F = C(:, 2:3) = [2,5; 0,1; 2,-1; 1,4]

123

101E

410

123

101

521

C

Colon OperatorColon Operator Creating new matrices from an existing matrix

C = [1,2,5; -1,0,1; 3,2,-1; 0,1,4]

E = C(2:3,:) = [-1 0 1; 3 2 -1]

Colon OperatorColon Operator Creating new matrices from an existing matrix

C = [1,2,5; -1,0,1; 3,2,-1; 0,1,4]

G = C(3:4,1:2) = [3,2; 0,1]

10

23G

410

123

101

521

C

Colon OperatorColon Operator Variable_name = a:step:b

time = 0.0:0.5:2.5

time = [0.0, 0.5, 1.0, 1.5, 2.0, 2.5]

Negative increment

values = 10:-1:2

values = [10, 9, 8, 7, 6, 5, 4, 3, 2]

linspace Function linspace Function linspace(x1, x2) gives 100 evenly spaced values

between x1 and x2

x = linspace(x1,x2) linspace(a,b,n) generate n equally spaced points

between a and b

x = linspace(a,b,n)

» linspace(0,2,11)

ans =

Columns 1 through 7

0 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000

Columns 8 through 11

1.4000 1.6000 1.8000 2.0000

logspace Functionlogspace Function logspace(a,b,n) generates a logarithmically

equally spaced row vector

x = logspace(a,b,n) logspace(a,b) generates 50 logarithmically

equally spaced points

x = logspace(a,b)

» logspace(-4,2,7)

ans =

0.0001 0.0010 0.0100 0.1000 1.0000 10.0000 100.0000

Special MatricesSpecial Matrices

1111

1111ones(2,4)

111

111

111

)3(ones

00

00

00

zeros(3,2)

100

010

001

)3(eye

bababa

baa

bba

bababaab

aa

baab

; nsubtractio and Addition

\ \ division Left\

/ ;* ; division and tionMultiplica/ *

Negation

^ tionExponentia^

Form MATLABOperation Symbol

Scalar Arithmetic OperationsScalar Arithmetic Operations

Example: x = (a + b*c)/d^2

count = count + 1

(Matrix inverse)

In order of priority

Order of Precedence of Order of Precedence of Arithmetic OperationsArithmetic Operations

1. Parentheses, starting with the innermost pair

2. Exponentiation, from left to right

3. Multiplication and division with equal precedence, from left to right

4. Addition and subtraction with equal precedence, from left to right

Examples: factor = 1 + b/v + c/v^2

slope = (y2 - y1)/(x2 - x1)

loss = f * length/dia * (1/2 * rho * v^2)

func = 1 + 0.5*(3*x^4 + (x + 2/x)^2)

Order of Precedence of Order of Precedence of Arithmetic OperationsArithmetic Operations

The priority order can be overridden with parentheses

» y = -7.3^2

y =

-53.2900

» y=(-7.3)^2

y =

53.2900

» a=3; b=5; c=2;

» s1 = a-b*c

s1 =

-7

» s2=(a-b)*c

s2 =

-4

Exponentiation has higher priority than negation

Multiplication has higher priority than subtraction

Array OperationsArray Operations An array operation is performed

element-by-element

B(5);* A(5) C(5)B(4);* A(4) C(4)B(3);* A(3) C(3)B(2);* A(2) C(2)B(1);* A(1) C(1)

MATLAB: C = A.*B;

] ,[]^ ,^[] ,[].^ ,[

] ,[]^ ,^[] ,[.^

] ,[]^ ,^[].^ ,[.^tionexponentia Array.^

].,.[]\ ,\[] ,[\]. ,[division left Array\.

].,.[]/ ,/[] ,/[]. ,[division right Array./

] ,[] ,[*]. ,[tionmultiplica Array*.

] ,[] ,[] ,[nsubtractio Array-

] ,[] ,[] ,[addition Array

] ,[] ,[nsubtractio array-Scalar-

] ,[] ,[addition array-Scalar

ExampleFormOperationSymbol

812543254235

24395323523

8643234324BA

71430667257835873A.\B

4001375057835873A./B

1863263A.*B

343864BA

9123864BA

32638bA

97364bA

Element-by-Element OperationsElement-by-Element Operations

But a*b gives an error (undefined) because dimensions are incorrect. Need to use .*

Vector and Matrix operationsVector and Matrix operations

11

7

3

ba

6

4

2

b

5

3

1

a

30

12

2

6*5

4*3

2*1

b.*a

Vectorized Matrix OperationsVectorized Matrix Operations

9243813B).^3(F

15122783.^AE

5.06.175.02B/.AD

240122B.*AC

2541B

1832A

Array Operations for Array Operations for m m n Matrices n Matrices

510515

2015105

2015105

5.*AB

1213

4321

4321

1 - 21 3-4;:-1:1 -4;:1A

18127

642781

642781

3.^AC

Matrix TransposeMatrix Transpose

4)2(3)1)(2()3)(4(

2-

1

3

324'y*x

639

426

8412

213

3

2

4

y'*x

2-

1

3

y' ;

3

2

4

'x

213y ; 324x

Built-in FunctionsBuilt-in Functions All the standard operators +, , *, /, ^

Sqrt( ), abs( ), sin( ), cos( ), exp( ), tanh( ), acos( ), log( ), log10( ), etc.

These operators are vectorized

exp(4)

exp(5)

exp(3)

exp(a) ;

sin(4)

sin(5)

sin(3)

)asin( ;

4

5

3

a

Built-in FunctionsBuilt-in Functions Certain functions, such as exponential and square

root, have matrix definition also Use “help expm” and “help sqrtm” for details

>> A = [1 3 5; 2 4 6; -3 2 -1]A = 1 3 5 2 4 6 -3 2 -1>> B = sqrt(A)B = 1.0000 1.7321 2.2361 1.4142 2.0000 2.4495 0 + 1.7321i 1.4142 0 + 1.0000i>> C = sqrtm(A)C = 2.1045 + 0.0000i 0.1536 - 0.0000i 1.8023 + 0.0000i 1.7141 - 0.0000i 1.1473 + 0.0000i 1.7446 + 0.0000i -2.0484 + 0.0000i 1.3874 + 0.0000i 0.5210 - 0.0000i

(element by element)

(C*C = A)

MATLAB GraphicsMATLAB Graphics One of the best things about MATLAB is

interactive graphics “plot” is the one you will be using most often Many other 3D plotting functions -- plot3,

mesh, surfc, etc. Use “help plot” for plotting options To get a new figure, use “figure” logarithmic plots available using semilogx,

semilogy and loglog

plot(x,y) defaults to a blue line

plot(x,y,’ro’) uses red circles

plot(x,y,’m*’) uses magenta asterisks

If you want to put two plots on the same graph, use “hold on”

plot(a,b,’r:’) (red dotted line)hold onplot(a,c,’ko’) (black circles)

Plotting CommandsPlotting Commands

Free-Falling Bungee JumperFree-Falling Bungee Jumper Use built-in functions sqrt & tanh

>> g = 9.81; m = 75.2; cd = 0.24;

>> t = 0:1:20t =

Columns 1 through 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Columns 16 through 21

15 16 17 18 19 20

>> v=sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t)v =

Columns 1 through 9

0 9.7089 18.8400 26.9454 33.7794 39.2956 43.5937 46.8514 49.2692

Columns 10 through 18

51.0358 52.3119 53.2262 53.8772 54.3389 54.6653 54.8956 55.0579 55.1720

Columns 19 through 21

55.2523 55.3087 55.3484

Plot the velocity versus time curve>> plot(t,v); grid on

>> title('Free Falling Bungee Jumper')

>> xlabel('time t (second)'); ylabel('velocity v (m/s)')

>> print –djpeg bungee.jpg

How to change line thickness, line color, font size, symbols,

marker size, etc.?

Color, Symbols, and Line TypesColor, Symbols, and Line Types Use “help plot” to find available Specifiers

b blue . point - solid

g green o circle : dotted

r red x x-mark -. dashdot

c cyan + plus -- dashed

m magenta * star

y yellow s square

k black d diamond

v triangle (down)

^ triangle (up)

< triangle (left)

> triangle (right)

p pentagram

h hexagram

Colors Symbols Line Types

» x=0:0.1:5;» y=2*x.^3-12*x.^2+8*x-6;» H=plot(x,y,'b',x,y,'r*');» set(H,'LineWidth',3,'MarkerSize',12)» xlabel('x'); ylabel('y');» title('f(x)=2x^3-12x^2+8x-6');» print -djpeg075 poly.jpg

element-by-element operations x.^n

Adjust line thickness, font size, marker size, etc.

» x=0:0.1:10;» y=sin(2.*pi*x)+cos(pi*x);» H1=plot(x,y,'m'); set(H1,'LineWidth',3); hold on;» H2=plot(x,y,'bO'); set(H2,'LineWidth',3,'MarkerSize',10); hold off;» xlabel('x'); ylabel('y');» title('y = sin(2\pix)+cos(\pix)');» print -djpeg075 function.jpg

» x=0:0.1:3; y1=exp(-x); y2=sqrt(x);» H=plot(x,y1,'b-',x,y2,'r--');» set(H,'LineWidth',3)» xlabel('x'); ylabel('y'); title('MATLAB Plots');» H1=text(1.8,0.2,'exp(-x)'); set(H1,'FontSize',18);» H2=text(1.8,1.3,'sqrt(x)'); set(H2,'FontSize',18);

plot (x, y) plot(x1, y1, x2, y2)

plot (x, y, ‘color symbol line style’)

» x = linspace(0, 2*pi);» y = sin (2.*x);» z = cos (0.5*x);» plot (x, y)» plot (x, y, x, z)» figure (2)» plot (x, y, 'r o -'); grid on» hold on» plot (x, z, 'b * :')

(red, circle, solid line)

(blue, star, dotted line)

figure or figure (#) : open a figure

Plotting CommandsPlotting Commands

» xlabel (Time)» ylabel (Temperature)» title (Temperature Record : 1900 - 2000)» text (17, 120, Record High )» text (85, -40, Record Low )» axis ([0 100 -50 140])» hold off

xlabel ( label ) ylabel ( label )title ( title of the plot )text ( x_location, y_location, text )axis ( [ x_min x_max y_min y_max ] )

- text string

Graphics CommandsGraphics Commands Axis, Labels, and Title

CVEN 302-501CVEN 302-501

Homework No. 1Homework No. 1

Chapter 2 Problems 2.2 (15), 2.4 (15), 2.7 (20),

2.10 (20), 2.11 (20).

Due Friday. 09/05/2008 at the Due Friday. 09/05/2008 at the beginning of the periodbeginning of the period