1 introduction to matlab-2 laboratoire mathématiques, informatique et applications

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Introduction to Matlab-2

Laboratoire Mathématiques, Informatique et Applications

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Desktop Tools (Matlab v6)

Command Window– type commands

Workspace– view program variables– clear to clear – double click on a variable to see it in the Array Editor

Command History– view past commands– save a whole session using diary

Launch Pad– access tools, demos and documentation

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Matlab Files (.m)

Use predefined functions or write your own functions

Reside on the current directory or the search path– add with File/Set Path

Use the Editor/Debugger to edit, run

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Matrices

a vector x = [1 2 5 1]

x = 1 2 5 1

a matrix x = [1 2 3; 5 1 4; 3 2 -1]

x = 1 2 3 5 1 4 3 2 -1

transpose y = x’ y = 1

2 5

1

5

Matrices

x(i,j) subscription

whole row

whole column

y=x(2,3)

y =

4

y=x(3,:)

y =

3 2 -1

y=x(:,2)

y =

2

1

2

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Operators (arithmetic)

+addition- subtraction* multiplication/ division^power‘ complex

conjugate transpose

.* element-by-element mult

./ element-by-element div

.^ element-by-element power

.‘ transpose

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Operators (relational, logical)

== equal~= not equal< less than<= less than or equal> greater than>= greater than or

equal

& AND| OR~ NOT

1

pi 3.14159265…

j imaginary unit,

i same as j

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Generating Vectors from functions

zeros(M,N) MxN matrix of zeros

ones(M,N) MxN matrix of ones

rand(M,N) MxN matrix of uniformly distributed random numbers on (0,1)

x = zeros(1,3)x =

0 0 0

x = ones(1,3)x =

1 1 1

x = rand(1,3)x = 0.9501 0.2311 0.6068

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Operators

[ ] concatenation

( ) subscription

x = [ zeros(1,3) ones(1,2) ]x = 0 0 0 1 1

x = [ 1 3 5 7 9]x = 1 3 5 7 9

y = x(2)y = 3y = x(2:4)y = 3 5 7

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Matlab Graphics

x = 0:pi/100:2*pi;y = sin(x);plot(x,y);xlabel('x = 0:2\pi');ylabel('Sine of x');title('Plot of the Sine

Function');

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Multiple Graphs

t =0:pi/100:2*pi;y1=sin(t);y2=sin(t+pi/2);plot(t,y1,t,y2);grid on;

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Multiple Plots

t =0:pi/100:2*pi;y1=sin(t);y2=sin(t+pi/2);subplot(2,2,1);plot(t,y1);subplot(2,2,2);plot(t,y2);

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If Statements

IF expression statements ELSEIF expression statements ELSE statements END

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Example clear all; close all; n=10; x=1:n; y=zeros([1,n]); for k=1:n

y(k) = 2*k; end

plot(x,y);

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Homework

Generate a tree structure for the system equation y=sin(ax) + bx knowing that x is the system input and y is the system output. X and y are arrays of n values. Assume n = 100;

Generate a random sequence from -2 to 2 to be used as the system input x using the randn command.

Plot the input versus the output of the above system.

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Solution

clear all;close all;

x=2-4*rand(1,100);a=2;b=3;

y=sin(a*x) + b*x;

subplot(2,1,1); plot(x); grid;title('system input');subplot(2,1,2); plot(y,'r'); grid;title('system output');

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Graph Functions (summary)

plot linear plot grid add grid lines xlabel add X-axis label ylabel add Y-axis label title add graph title subplot divide figure window figure create new figure window pause wait for user response

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Math Functions

Elementary functions (sin, cos, sqrt, abs, exp, log10, round)– type help elfun

Advanced functions (bessel, beta, gamma, erf)– type help specfun– type help elmat

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Scripts

Matlab editor Use scripts to execute a series of

Matlab commands

Matlab Desktop

Press to create new m-file in the matlab editor

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Functions

function f=myfunction(x,y)f=x+y;

save it in myfunction.m call it with y=myfunction(x,y)

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Functions Programming in Matlab. Users can write functions which can be called from the

command line. Functions can accept input variable(s)/matrice(s) and

will output variable(s)/matrice(s). Functions will not manipulate variable(s)/matrice(s) in

the Matlab Workspace. In Matlab functions closely resemble scripts and can be

written in the Matlab editor. Matlab functions have the function keyword.

Remember that the filename of a function will be its calling function name.

Don’t overload any built-in functions by using the same filename for your functions or scripts!

Functions can be opened for editing using the open command. Many built-in Matlab functions can also be viewed using this command.

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>> I=iterate(5)I = 1 4 9 16 25

Functions (continued)

outputinputfunction name

for statement block

function keyword

help lines for function

Access the comments of your Matlab functions >> help iterate Make sure you save changes to the

m-file before you call the function!

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>> [i j]=sort2(2,4)i = 4j = 2>>

Functions (continued)

Functions can have many outputs contained in a matrix

Remember to use the Matlab help command for syntax>> help if

if statement block

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More flow control

Method is linear>>

i =

4

i =

16

i =

256

While statement block Switch statement block

Without ; to print output

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Debugging

Set breakpoints to stop the execution of code>> [i j]=sort2(2,4)K>> K>> whos Name Size Bytes Class a 1x1 8 double array b 1x1 8 double arrayGrand total is 2 elements using 16 bytesK>> aa = 2K>> returni =

4j = 2

Click mouse on the left of the line of code to create a breakpoint

local function workspace

exit debug mode

Debug menus

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Visualisation - plotting data

>> figure % create new figure>> t=0:pi/12:8*pi;>> y=cos(t);>> plot(t,y,‘b.-')

Investigate the function >> y=A*cos(w*t+phi);for different values of phi (eg: 0, pi/4, pi/3, pi/2), w (eg: 1, 2, 3, 4) and A (eg: 1, 0.5, 2). Use the hold on Matlab command to display your plots in the same figure. Remember to type hold off to go back to normal plotting mode. Try using different plot styles (help plot) A = amplitude

phi = phasew = angular frequency = 2*pi*frequency

Plot style

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Flow Control

A=3; B=2;if A > B

'greater'elseif A < B

'less'else

'equal'end

for x = 1:10r(x) = x;

end

• if statement• switch statement

• for loops• while loops

• continue statement• break statement

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Miscellaneous

Loading data from a file– load myfile.dat

Definition – x = [1 2 5 1];

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Getting Help

Using the Help Browser (.html, .pdf)– View getstart.pdf, graphg.pdf, using_ml.pdf

Type – help – help function, e.g. help plot

Running demos – type demos– type help demos

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>> help stem

STEM Discrete sequence or "stem" plot. STEM(Y) plots the data sequence Y as stems from the x

axisterminated with circles for the data value. STEM(X,Y) plots the data sequence Y at the values specified in

X. STEM(...,'filled') produces a stem plot with filled markers. STEM(...,'LINESPEC') uses the linetype specified for the stems

and markers. See PLOT for possibilities. H = STEM(...) returns a vector of line handles. See also PLOT, BAR, STAIRS.

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Random Numbers

x=rand(100,1);stem(x);

hist(x,100)

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Coin Tosses

Simulate the outcomes of 100 fair coin tosses

x=rand(100,1);p=sum(x<0.5)/100

p = 0.5400

Simulate the outcomes of 1000 fair coin tosses

x=rand(1000,1);p=sum(x<0.5)/1000

p = 0.5110

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Coin Tosses

Simulate the outcomes of 1000 biased coin tosses with p[Head]=0.4

x=rand(1000,1);p=sum(x<0.4)/1000

p = 0.4160

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Sum of Two Dies

Simulate 10000 observations of the sum of two fair dies

.

.

.

.

.

. . . . .

.

. . . .. .

... .

.

..

.

.

...

.. .. .. .

(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

(3,2) (4,2) (5,2) (6,2)

(3,3) (4,3) (5,3) (6,3)

(2,2)(1,2)

(2,3)(1,3)

(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)

(3,5) (4,5) (5,5) (6,5)(2,5)(1,5)

(3,6) (4,6) (5,6) (6,6)(2,6)(1,6)

1 2 3 4 5 6

1

2

3

4

5

6

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Sum of Two Dies

for i=2:12z(i)=sum(y==i)/10000

endbar(z)

Boolean Operations

» x=[0 1 1]; y=[1 1 1];» and(x,y)» ans =» 0 1 1» or(x,y)» ans =» 1 1 1

» x=[0 1 1]; y=[1 1 1];» and(x,y)» ans =» 0 1 1» or(x,y)» ans =» 1 1 1

Boolean Operators

= = equal to

> greater than

< less than

~ not

& and

| or

isempty()

isfinite(), etc. . . .

any()

all()

1 = TRUE0 = FALSE

»bool_ops

Solving Simultaneous Equationsusing “Left Division”

If [A] is square matrix (m = n):

For overdetermined system (m>n): using Least squares regression “curve fit” of data

Warning if rank deficient (dependent columns) - solution not unique

For undetermined system (m<n): using QR factorization with column pivoting

Never unique

[A]{x} = {b} {x} = ?

{x} = [A]-1{b}» x = inv(A)*b;

» x = A\b;

» x = inv(A)*b;

» x = A\b;

Error if singularWarning if nearly singular

[A] = mxn{x} = nx1{b} = mx1

» x = A\b;» x = A\b;

» x = A\b;» x = A\b;

Solve this set of simultaneous equations:

Example: Solving Equations

» A = [-1 1 2; 3 -1 1;-1 3 4];

» b = [2;6;4];

» x = inv(A)*b

x =

1.0000

-1.0000

2.0000

» x = A\b

x =

1.0000

-1.0000

2.0000

» A = [-1 1 2; 3 -1 1;-1 3 4];

» b = [2;6;4];

» x = inv(A)*b

x =

1.0000

-1.0000

2.0000

» x = A\b

x =

1.0000

-1.0000

2.0000

-x1 + x2 + 2x3 = 2

3x1 - x2 + x3 = 6

-x1 + 3x2 + 4x3 = 4

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Return to our example

clear all;close all;n=20;x=rand(1,n);x1= [0 x(1:n-1)];x2= [0 0 x(1:n-2)];a=2;b=5;y= a*x1 + b*x2;figure; plot(y,'r'); grid;title('system output');

String Arrays

Created using single quote delimiter (')

Each character is a separate matrix element (16 bits of memory per character)

Indexing same as for numeric arrays

» str = 'Hi there,'

str =

Hi there,

» str2 = 'Isn''t MATLAB great?'

str2 =

Isn't MATLAB great?

» str = 'Hi there,'

str =

Hi there,

» str2 = 'Isn''t MATLAB great?'

str2 =

Isn't MATLAB great?

1x9 vectorstr = H i t h e r e ,

» str ='Hi there,';

» str1='Everyone!';

» new_str=[str, ' ', str1]

new_str =Hi there, Everyone! » str2 = 'Isn''t MATLAB great?';

» new_str2=[new_str; str2]new_str2 =Hi there, Everyone!Isn't MATLAB great?

» str ='Hi there,';

» str1='Everyone!';

» new_str=[str, ' ', str1]

new_str =Hi there, Everyone! » str2 = 'Isn''t MATLAB great?';

» new_str2=[new_str; str2]new_str2 =Hi there, Everyone!Isn't MATLAB great?

1x19 vector

1x9 vectors

String Array Concatenation

Using [ ] operator:Each row must be same length

Row separator:semicolon (;)

Column separator:space / comma (,)For strings of different

length:•STRVCAT•STR2MAT

» new_str3 = strvcat(str, str2)new_str3 =Hi there, Isn't MATLAB great?

» new_str3 = strvcat(str, str2)new_str3 =Hi there, Isn't MATLAB great?

2x19 matrix

2x19 matrix(zero padded)

1x19 vectors

Working with String Arrays

String Comparisons– STRCMP - compare whole strings– STRNCMP - compare first ‘N’ characters– FINDSTR - finds substring within a larger string

Converting between numeric & string arrays:– NUM2STR - convert from numeric to string array– STR2NUM - convert from string to numeric array

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End of Lecture