Introduction to MATLAB

1

Introduction to MATLAB

INGE3016 Algorithms and Computer Programming With MATLABDr. Marco A. Arochaoct 25, 2007; june 12, 2012

2

Some facts: MATLABMATrix LABoratory Matrix mathematics >1000 functions

3

Some factsAdvantages Ease of use Platform independent Predefined functions Plotting Graphical User

Interface Compiler

Disadvantages Interpreted language Slower Expensive

Matrix & Array OperatorsScalar & Matrix Array

Addition + before .+ now +

Subtraction - before .- now -

Multiplication * .*Division / and \ ./ and .\Exponentiation ^ .^Modulus mod(x,y) mod(x,y)

4

To associate expressions use “( )”, i.e., regular parentheses. Do no use brackets “[ ].”

Do not use “( )” or “.” or “x” to mean multiplication.

5

Examples>> 2+3-2*4 <E>% note the precedence orderans = -3The division operators:>> 6/2 <E> % numerator/denominator

ans = 3>> 2\6 <E> % denominator/numerator

ans = 3

6

Rules to construct variable and file names:

uppercase letters: A-Z (26 of them) lowercase letters: a-z digits: 0-9 underscore _ first must be a letter any length, first 63 are significant (if more than

63 the rest will be ignored)

7

Rules to construct variable and file names:

No blank spaces within a name case sensitive, unless instructed otherwise by the case

function (i.e., case off) This rules must be extended to the construction of

MATLAB file names as file names can become variables

filename.m

Same rules as variable name

8

All variables are arrays The unit of data in MATLAB is the array A MATLAB variable is a region of memory

containing an array Array is a collection of data organized into

rows and columns and known by a single name

9

Common Library Functions & Parameters

Mathematics MATLAB Math & MATLAB

sqrt(x) sin(x)

ln(x) log(x) cos(x)

log10(x) log10(x) tan(x)

ex exp(x) asin(x)

xy x^y acos(x)

|x| abs(x) atan(x)

, 3.14159… pi

x

10

Common MistakePlease code in MATLAB the following

Correct: y=3*x*exp(x^2)-1

Incorrect: y=3*x*e^(x^2)-1Incorrect: y=3*x*exp^(x^2)-1

132

xxey

11

Scientific Notation Math: 1x10-3 Matlab: 1e-3 (correct) Matlab: 1*10^-3 ( accepted by MATLAB but not accepted by the instructor;

student using this approach denotes lack of knowledge of the scientific notation)

Common Mistakes: 1xe-3 10e-3 1e^-3 1e*-3 1*e-3

12

Assignment Statement The Syntax:

Variable Name = Expression

The assignment statement operates assigning the value of the right-hand side expression to the left-hand side variable.

13

Examplea=1;b=2;c=a+b; % c=3

Note: a mistake is to write a + b = c which is wrong as the compiler cannot assign the value of c

to a variable named a+b.

14

Examplea=1;a=a+1; % the new value of a in the LHS is 2

Think of the last statement as:

1;aa valueoldthevaluenewthe

15

Quiz, your first algorithm

Write a piece of program that switches the value of a and ba=1;b=3;……fprintf(‘a=%d, b=%d’, a, b);

Your statements

Output should be:

a = 3, b = 1

16

Quiz, your first algorithm

Write a piece of program that switches the value of a and ba=1;b=3;a=b;b=a;fprintf(‘a=%d, b=%d’, a, b);

Student first approach, is it right?

Output should be:

a = 3, b = 1

17

Quiz, apply your knowledge

Given: a=1; b=3; c=5;

Transfer the values of a c b a c b

18

Multiple Statements

>>x=1; y =2; % or>>x=1, y=2; multiple assignment statements can be

placed on a single line separated by semicolons or commas

19

clear; clc; <ctrl><c>

Commands to manage the work session in the Command Window clear;

remove the values of all variables from memory clear is short for clear all

clc; clears the Command Window but the values of the variable

remains [also: MATLAB edit menu: EditClear Command Window]

<ctrl><c>

stops program execution and return to editing phase

20

semicolon (;) commandIf an assignment statement is typed without the semicolon at the end, the results of the statement are automatically displayed in the command window:

>> x = 5*20 % statement without the colonx = 100

Displaying partial results is an excellent way to quickly check your work (debugging), but it slows down the execution of a program. Recommendation: leave off the semicolon while debugging and use it after the program is bugs-free.

21

The array unitUnder the array unit in MATLAB we can store: Scalar

1x1 array

Vectors: 1-D arrays Column-vector: m x 1 array Row-vector: 1 x n array

Multidimensional arrays m x n arrays Also called matrices One or more rows by one or more columns

22

Scalars, Vectors, Matrices Scalars are array with only one row and one

column a = [3.4] a = 3.4

a is a 1x1 array (a scalar) containing the value 3.4 Brackets are optional in this case and usually we don’t use it

for 1x1 array The only element is a(1) with a value of 3.4

Both are valid

23

Initializing 1-D arrays (vectors) Vectors are one-dimension either row-vectors or column-vectors Values are listed using brackets [ ] Blank spaces or commas can be used to separate values

Row vectors:>> prime=[2 3 5 7 13]prime = 2 3 5 7 13>> prime=[2, 3, 5, 7, 13]prime = 2 3 5 7 13

Two choices for Array assignmentUse either one

Memory

Row vectors:>> prime=[2, 3, 5, 7, 11, 13]prime = 2 3 5 7 11 13

The elements of an array are identified by indices starting with 1 With arrays pay attention to indices and values, they are not the same Each value is stored in memory in the order listed as:

prime(1)=2prime(2)=3prime(3)=5prime(4)=7prime(5)=11prime(6)=13

Array elements’ indices always start in one

indices values

25

Retrieving array elementsEach element can be retrieved one by one or all at once:

>> prime(1) <E>ans = 2>> prime(2) <E>ans = 3…>> prime(6) <E>ans = 13>> prime <E>prime = 2 3 5 7 11 13

26

Manipulating arrays

with individual elements:

>> prime(1)*prime(2)ans = 6>> prime(3)^2ans = 25>> log(prime(1))ans = 0.6931

Or with the whole array:

>> x=prime+1x = 3 4 6 8 14>> z=prime-2z = 0 1 3 5 11>> y=x+zy = 3 5 9 13 25>> m=x.*ym = 9 20 54 104 350

prime(1)=2prime(2)=3prime(3)=5prime(4)=7prime(5)=13After you have initialized prime all values are in memory

and we can perform operations

27

Clearing array values We can clear all the values previously entered

with the clear function>> clear prime

If we try to retrieve the values of prime after using clear prime, an error message is found

>> prime(1)

??? Undefined function or variable 'prime'.

28

Column vectors: Rows can be separated by

semicolon or new lines

>> b = [3; 2; 1] <E>>> b= [3 <E>

2 <E> 1] <E>

b is a 3 x 1 array (or simply a column-vector)

3

2

1

b

Transpose operator(‘)transpose a row-vector into a column-vector and vice verse

>> x=[1 2 3]’ x = 1 2 3

>> x=[1 ;2; 3]’

x = 1 2 3

29

• Transpose a column vector into a row vector• Transpose a row vector into a column vector• MATLAB displays row vectors horizontally and column vectors vertically• In the examples, changes are permanent, i.e., x changed form row to

column and from column to row not just on the printed output.

30

Colon Operator (:)

The colon operator (:) can generate large row vectors of regularly spaced elements.

>> x = [first:incr:last]first = first valuelast = last valueincr = increment, if omitted, the increment is 1

Ex

>> x = [0:2:8]x =

0 2 4 6 8

memory:x(1)=0x(2)=2x(3)=4x(4)=6x(5)=8

31

Colon Operator, examples>> x = [0:2:6]x = 0 2 4 6

>> u =[10:-2:4]u = 10 8 6 4

>> y = [1:1:4]' % combines the colon and transpose operatorsy = 1 2 3 4

Que sería la vida sin “:”

(1) x = [0 2 4 6 8]

(2) j=1;for ii=0:2:8

x(j)=ii;j=j+1;

end% j is the index generator% ii is the values generator

(3)j=1;x(j)=0;for j=2:1:5

x(j)=x(j-1)+2;end% j generates both: % indices and values

32

Tres ejemplos para inicializar x con otros métodos diferentes al operador “colon” y lograr en memoria: x(1)=0 x(2)=2 x(3)=4 x(4)=6 x(5)=8

33

QUIZ Using the Colon operator, calculate the

values of x from 1 up to 3 in increments of 0.1, (Application on the Trapezoidal integration rule)

34

linspace function: The linspace (short for linear space) function

creates a linearly spaced row vector, but instead you specify the number of values rather than the increment.

Syntax: linspace(first, last, pts.)

where: first= first valuelast= last valuepts.= number of points

35

linspace, exampleProblem: Want to produce the following values:

5.0 5.1 5.2 5.3 5.4 5.55.65.7 5.8 5.9 6.0

first value

last value

11 points x=linspace(5, 6, 11)

linspace, effect in memory Could you tell what is the effect in memory for the

linspace(5,6,11) instruction?

R 1-D array (row-vector) with 11 elements:

... linspace(5,6,11) is equivalent to x =[5:0.1:6]

36

5.0 5.1 5.2 5.3 5.9 6.0

37

Compare colon operator with linspace function

a= first valueb= last valuen= number of points -1 (=panels number, integration rule)

h=(b-a)/n = increment

x=[a:h:b] % orx=linspace(a,b,n+1)

Equivalent instructionsBoth produce n+1 elements of xBoth produce row-vectors

Compare colon operator with linspace functionx=[2,4,6,8,10,12,14,16,18,20];

a) Colon Operator x=[2:2:20];

b) linspace function x=linspace(2,20,10);

38

x=[a:h:b]x=linspace(a,b,n+1)

h=(b-a)/n

39

Data Types & Variable Declaration

Most common data types (default): double char

double means double precision 15-16 significant-digit variables automatically created whenever a numerical value

is assigned to a variable name.

40

char variables & strings Variables of type char consist of :

scalars (one character) or arrays (many characters),

(char arrays are most commonly called strings) Example:

cheo=‘x’ % one character pepe = ‘This is a character string’ % many characters

pepe is a 1x26 character array.

Strings are character arrays containing more than one character

41

Variable initialization

Three common ways to initialize a variable in MATLAB:

Assign data to the variable in an assignment statement

Input data into the variable from the keyboard (input function)

Read data from an external file

42

Initializing Variables with Keyboard input function>> x = input('Enter an input value = ') <E>

Output in the Command Window:>> Enter an input value = 5 <E>

x = 5Effect in Memory? x(1)=5

user writes 5 and <E>

43

Initializing Variables with Keyboard input function>>x = input('Enter an input value = ')

Output:>>Enter an input value = [1,2,3] <E> x = 1 2 3

Effect in Memory?x(1)=1; x(2)=2; x(3)=3

user writes [1,2,3] and <E>

44

Initializing Variables with Keyboard Input>>x = input('Enter an input value = ')<E>

Output in the Command Window:>> Enter an input value = [1 2; 3 4] <E>

x = 1 2 3 4

Effect in memory: 2x2 array with elements

x(1,1)=1 x(1,2)=2x(2,1)=3 x(2,2)=4

45

Initializing Variables with Keyboard Input>> x = input('Enter an input value = ')<E>

Output in the Command Window:>>Enter an input value = [1;2;3;4] <E>x = 1 2 3 4

Effect in memory: column vector with elements:

x(1)=1x(2)=2x(3)=3x(4)=4 Also could be a 4x1

2-D array with elements:

x(1,1)=1x(2,1)=2x(3,1)=3x(4,1)=4

46

Initializing Variables with Keyboard Input

>>x = input('Enter an input value = ') <E>

Output in the Command Window:>>Enter an input value = ‘Albert Einstein’<E>x =

Albert Einstein

Effect in memory: 15 elements with character values:

x(1)=’A’x(2)=’l’x(3)=’b’x(4)=’e’x(5)=’r’x(6)=’t’x(7)=’ ‘ (i.e., nada)x(8)=’E’x(9)=’i’x(10)=’n’x(11)=’s’x(12)=’t’x(13)=’e’x(14)=‘i’x(15)=’n’

You write this input

47

Initializing Variables with Keyboard Input The data type of the variable is decided

during the execution by typing a particular value: an scalar or an array within brackets, or a string within quotes and then <enter>

48

Initializing Variables with Keyboard Input ‘s’ (meaning string) as a second input parameter,

then the data returned is a character string. Note the difference in syntax with previous instruction:

>>x = input('Enter an input value = ', 's')

Output in the Command Window:>>Enter an input value = Albert Einstein <E> ( NO

quotes)x =

Albert Einstein

49

2D Arrays (Matrices)

Matrices are arrays with two or more dimensions Their size is specified by the number of rows

and columns (rows first) Number of elements =row x column Reference an array, two forms:

a(2,1) address an individual element, by identifying the row and column

a without parenthesis address the whole array

2D Arrays (matrices)

50

1 2 3

5 64

a(1,1) a(1,2) a(1,3)

a(2,1) a(2,2) a(2,3)

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

Two syntax methods to initialize:

a is a 2 x 3 array

Rows can be separated by semicolon or new lines

Example of operations:

>>a(1,2)*a(1,3)ans

6

>>a.*aans 1 4 9 16 25 36

51

Initialization Example The number of elements in every row of an

array must be the same:a= [1 2 3; 4 5] % produces error

a= 123 not allowed 45

[ ] is an empty array

52

Relational Operators

six operators:<, less than<=, less than or equal to>, greater than>=, greater than or equal to==, equal to~=, not equal to

two results: false (zero, 0) true (one, 1; in general any number different than

zero)

6 operators

53

a>b, example:

conditionalexpressions

true ≠0 (usually =1)

false=0

54

Example with scalars Let

a=5; b=3 ; c=6 ; d=8;

Find the value of a>b c>d

55

Relational Operators with Arrays comparison occurs on an element-by-element basis arrays being compared must have the same dimensions resulting array has the same dimensions as the operators

>> x=[6,3,9];>> y=[14,2,9]; % three conditions:>> z=(x<y) % x(1)<y(1)6<14z = % x(2)<y(2)3<2 1 0 0 % x(3)<y(3)9<9

• 3 results• set is a logical

array

Relational Operators with Arrays>> x=[6,3,9];>> y=[14,2,9];

>> z=(x>y)z = 0 1 0>> z=(x~=y)z = 1 1 0>> z=(x==y)z = 0 0 1

56

results are logical arrays

57

Relational Operators with Arrays

All the elements of an array can be compared to an scalar. Example:

>> x=[6,3,9];

>> z=(x>8)z = 0 0 1

58

Relational Operators with Arrays PrecedenceThe arithmetic operators +, -, *, /, and \ have precedence over the

relational operators

Exercise:Suppose that x =[-5, -4, 7, 8, 9] and y=[-3, -4, 9, 8, 7]. What is the

result of the following operations? Use MATLAB to check your answers.

z = (x < y)z = (x > y)z = (x ~= y)z = (x == y)z = (x > 8)z = 5 > 2 + 7z = 5 > (2+7)z = (5>2) +7

59

Logic OperatorsOperator Operation

& Logical AND

&& Logical AND with shortcut evaluation

| Logical OR

|| Logical OR with shortcut evaluation

xor() Logical Exclusive OR

~ Logical NOT

60

& True TableCond1 & Cond2 Result

T & T T

T & F

F & T

F & F

Expressions containing ‘and’ are true only if both conditions are true

Both scalar and array values

61

&& True TableCond1 Operator Cond2 Result

T && T T

T && F

F && T

F && F

Expressions containing ‘and’ are true only if both conditions are true

Scalar values ONLY, With shortcut evaluationIn 3rd and 4rd rows Cond2 is not evaluated

62

| True TableCond1 Operator Cond2 Result

F | F F

T | F

F | T

T | T

Statements containing ‘or’ are false when both conditions are false.

Both scalar and array values

63

|| True TableCond1 Operator Cond2 Result

F || F F

T || F

F || T

T || T

Statements containing ‘or’ are false when both conditions are false.

Scalar values only, Shortcut evaluation

64

xor() True TableCond1 Cond2 xor(cond1,cond2)

ResultF F F

T F T

F T T

T T F

Statements containing ‘xor’ are false when both conditions are true or false.

Scalar values only

65

~ True Table~ Cond result

~ T

~ F

66


a=5; b=3; c=6; d=8;

Find the value of (a>b) & (c>d) ((5>3) and (6>8))

67


a=5; b=3; c=6; d=8;

Find the value of (a>b) | (c>d) ((5>3) or (6>8))

68


a=5; b=3; c=6; d=8; Find the value of

~( (a>b) && (c>d)) ~ ((5>3) and (6>8))

69

a<x<b MATLAB, two choices: x>a & x<b (common programming syntax) a<x<b

70

Logical Expressions:a=1, b=2, c=3, d=4

if b>a & d >bfprintf(‘Hello World’);

elsefprintf(‘sorry I am busy’);

end

a=1, b=2, c=3, d=4

if d>b >afprintf(‘Hello World’);

elsefprintf(‘sorry I am busy’);

end

71

Example with arrays

>> x=[6,3,9];>> y=[14,2,9];>> z=~xz = 0 0 0>> z=~x>yz = 0 0 0>> z=~(x>y)z = 1 0 1

72

Example with arrays Consider the following:x = [1, 2, 3, 4, 5];y = [-2, 0, 2, 4, 6];z = [8, 8, 8, 8, 8];

The statement: z>x & z>y (reads z is greater than x and z is greater than y) returns ans = 1 1 1 1 1 because z is greater than both x and y for every element. This

result should be interpreted to mean that the condition is true for all the elements.

73

Example: Consider the following: x = [1, 2, 3, 4, 5]; y = [-2, 0, 2, 4, 6]; z = [8, 8, 8, 8, 8];

The statement: x>y | x>z (reads x is greater than y or x is greater than z) returns ans = 1 1 1 0 0 This result should be interpreted to mean that the condition is

true for the first three elements and false for the last two.

74

Control Structures

Three fundamental control structures in computer programming:

Sequential Selection Repetition

75

SEQUENCIAL

In this structure each statement is executed only once, from top-to-bottom sequentially (i.e., the second statement goes after the first, the third after the second, and so on.)

The sequential structure does not require additional statements as we already covered it.

76

Selection Structure: MATLAB has three if structures

if conditionstatements

end

if condition-1stats-1elseif condition-2

stats-2…elseif condition-nstats-nelsestats-n+1end

% elseif is one word

if conditionstatements-1

elsestatements-2

end

Allows program flows to run into one of several choices

77

if Example

x = 5;if x>=5

x = x +1;

endfprintf(‘x=%d’,x); No semicolon

True or False?

What if x = 6?What if x = 1?

indentation

78

if Example

x=3;if x>=5 x = x+1;else x = x-1;endfprintf(‘ x = %d’, x);

one of these two is executed, but not both

79

%ng=numerical grade

ng=input(‘enter ng \n’);

if ng>=90 fprintf(‘student got A’); elseif ng>=80 fprintf(‘student got B’); elseif ng>=70 fprintf(‘student got C’); elseif ng>=60 fprintf (‘student got D’); else fprintf(‘student got F’); end

ExampleMultialternative if

first true condition is executed, the rest ignored

each false condition carries on information to the following conditions

80

Example

% ng = numeric grade */ ng=input(‘enter ng \n’);

if ng>=90 fprintf(‘student got an A’);endif ng<90 & ng>=80 fprintf(‘student got a B’);endif ng<80 & ng>=70 fprintf(‘student got a C’);endif ng<70 & ng>=60 fprintf (‘student got a D’);endif ng<60 fprintf(‘student got a F’);end

independent if statements

all true conditions are executed

The previous solutioncan be implemented w/ independent if statements

Each if-block doesn’t carry information to the next condition, therefore conditions need a range for each grade

81

com

pare

%ng=numerical gradeng=input(‘enter ng \n’);

if ng>=90 fprintf(‘student got A’); elseif ng>=80 fprintf(‘student got B’); elseif ng>=70 fprintf(‘student got C’); elseif ng>=60 fprintf (‘student got D’); else fprintf(‘student got F’); end

% ng = numeric grade ng=input(‘enter ng \n’);

if ng>=90 fprintf(‘student got an A’);endif ng<90 & ng>=80 fprintf(‘student got a B’);endif ng<80 & ng>=70 fprintf(‘student got a C’);endif ng<70 & ng>=60 fprintf (‘student got a D’);endif ng<60 fprintf(‘student got a F’);end

Discuss differences of the two solutions

82

Flow Charts

83

More examples with if statements: The Trapezoidal rule

)2...22()2/1( 1321 nn fffffhI

n = number of panels

84

Example(*), Trapezoidal ruleacc=0;for ii=1:1:n+1 if ii==1 c(ii)=1.0; elseif ii==n+1 c(ii)=1.0; else c(ii)=2.0; end fx(ii)=tanh(x(ii))^2;

term(ii)=c(ii)*fx(ii); acc=acc+term(ii);

end

This assumes that x(ii) has been computed before

(*) This assumes that the Trapezoidal rules was sent as computer project

85

Example Trapezoidal rule, version 2

acc=0;for ii=1:1:n+1 if ii==1| ii==n+1 % note the logical operators c(ii)=1.0; else c(ii)=2.0; end

fx(ii)=tanh(x(ii))^2; term(ii)=c(ii)*fx(ii);

acc=acc+term(ii);end

This assumes that x array has been computed before

86

More examples with if statements: Simpson 1/3 rule

1 2 3 4 1(1/ 3) ( 4 2 4 ... 4 )n nI h f f f f f f

87

Example, Simpson 1/3 ruleacc=0;for ii=1:1:n+1 if ii==1 | ii==n+1 c(ii)=1; elseif mod(ii,2)==0 c(ii)=4; else c(ii)=2; end

t(ii)=c(ii)*f(ii); acc=acc+t(ii);end

Assuming f has been previously computed

Documents

Introduction to MATLAB