matlab tutorial - doe.carleton.ca

MATLAB Tutorial

Ehsan Ghias–Begloo Department of Electronics Carleton University [email protected] http://www.doe.carleton.ca/~egbegloo/

Updated September 2013

Intro to Matlab >> % ----------- %{

This is wrap comment; same as /* … */ in C++ or Java

%} %% creates a subsection

MATLAB codes can be saved in “.m” files as scripts & run

accordingly

© 2013 Ehsan Ghias-Begloo 1

Matlab Command Lines start with “>>”

“%” defines comment

Declaring Variables >> a = 1; >> a = 1; b = 2

b = 2

>> global kilo, meg, gig;


% “;” prevents Matlab from publishing the result of the line

%{ Multiple statements can be in a single line, separated by “,” or “;” }%

% Global variable accessible within all files

Numeric Formats 1.23456789e6 = 1,234,567.89 Output number format:

>> format **** short: 0.5000 short e: 5.0000e–001 long: 0.500000000000000 rational: 1/2 hex: 3fe0000000000000 …


“%” Short fixed point

“%” Long fixed point

Real vs. Imaginary >> a = 1i + 2

a = 2.0000 + 1.0000i

>> a = 1*i + 2 a = 2.0000 + 1.0000i

>> b = i + 2 b = 2.0000 + 1.0000i

>> i = 1; c = i + 2 c = 3


Result with variable name

Both “i” & “j” can either represent an imaginary unit or a variable!

% “a” is an imaginary number

%} “b” is an imaginary number since “i” hasn't been defined as a variable; hence it's an imaginary unit }%

%{ “c” is a real number, since “i” has already been defined as a variable, NOT an imaginary unit }%

Defining Vectors 1. >> a = [3 1 7 -21 5 6] or a = [3, 1, 7, -21, 5, 6]

a = 3 1 7 -21 5 6

2. >> b = 1:3:28 b = 1 4 7 10 13 16 19 22 25 28

3. >> c = linspace(1,28,10) c = 1 4 7 10 13 16 19 22 25 28

4. >> l = logspace(4,8,8) l = 1.0e+008 * 0.0001 0.0004 0.0014 0.0052 0.0193 0.0720 0.2683 1.0000


Same vectors; i.e. b = c

They all create row vectors

Same as c, but in log scale

Vector Elements >> b(5)

ans = 13

>> c(4:7) 10 13 16 19

>> a'

3 1 7 -21 5 6

>> min(a), max(a) -21 7


“ans” will appear when name is not specified

% Accessing the 5th element of the vector

% Accessing the 4th to 7th elements of the vector

% Transpose the row vector to a column vector

% Minimum and maximum values in the vector

Vectors have their index or subscript begin at 1

Vector Operations >> b + c

2 8 14 20 26 32 38 44 50 56

>> a + b ??? Error using ==> plus Matrix dimensions must agree

>> -3 * b -3 -12 -21 -30 -39 -48 -57 -66 -75 -84

>> c / 2 0.5000 2.0000 3.5000 5.0000 6.5000 8.0000 9.5000 11.0000 12.5000 14.0000

>> c ^ 2 Error using ^ Inputs must be a scalar and a square matrix. To compute elementwise POWER, use POWER (.^) instead.

>> c .^ 2 1 16 49 100 169 256 361 484 625 784


One of the most common errors in Matlab

Errors in Matlab are shown this way

% Addition

% Multiplication by a constant

% Division by a constant

% Power to a constant

Proper way

Defining Matrices 1. >> A = [1 2 3; 3 4 5; 6 7 8]

1 2 3 3 4 5 6 7 8

2. >> B = [[4 7 1]' [9 3 6]' [2 5 8]'] 4 9 2 7 3 5 1 6 8

3. >> D = []

4. >> D = [1, D, 2, 3] 1 2 3

5. >> D = [D, a] 1 2 3 3 1 7 -21 5 6


% “;” adds a row

% Creates an empty matrix; i.e. 0x0

% Matrix concatenation

Defining Matrices 5. >> ones(3)

1 1 1 1 1 1 1 1 1

6. >> zeros(2,5) 0 0 0 0 0 0 0 0 0 0

7. >> rand(3,4) 0.6551 0.4984 0.5853 0.2551 0.1626 0.9597 0.2238 0.5060 0.1190 0.3404 0.7513 0.6991

8. >> A' 1 3 6 2 4 7 3 5 8


“ones”, “zeros”, and “rand” can generate square or non-square matrices

% Creates a 3x3 square matrix composed of 1s

% Creates a 2x5 matrix composed of 0s

% Matrix transpose

% Creates a 3x2 matrix composed random nubmbers

Matrix Functions >> inv(A)

1.0e+015 * -2.7022 4.5036 -1.8014 5.4043 -9.0072 3.6029 -2.7022 4.5036 -1.8014

>> inv(a) ??? Error using ==> inv Matrix must be square.

>> size(a), size(A) 1 6

3 3

>> length(A) 3


% Generates the inverse matrix

%{ “inv” is only applicable to square matrices; i.e. cannot use “inv” for vectors }%

% Give size of vectors & matrices

% “size” generates a 1x2 vector defining rows x cols

% Same as max(size(A))

Matrix Extras >> A(2,:)

3 4 5

>> A(:,2) 2 4 7

>> A(2,3) 5

>> A(3,2) 7

>> A(2,2:3) 4 5


% Returns A’s entire 2nd row

% Returns A’s entire 2nd column

% Element selection – row 3, col 2

% Element selection – 2nd row, cols 2 to 3

% Element selection – row 2, col 3

Matrix Extras >> whos

Name Size Bytes Class Attributes A 3x3 72 double B 3x3 72 double a 1x6 48 double b 1x10 80 double c 1x10 80 double

>> sum(A(2,:))

12


% Generates a list of variables in the workspace

% Sum of elements in 2nd row

Matrix Operations >> cos(A)

0.5403 -0.4161 -0.9900 -0.9900 -0.6536 0.2837 0.9602 0.7539 -0.1455

>> log10(B) 0.6021 0.9542 0.3010 0.8451 0.4771 0.6990 0 0.7782 0.9031


An entire matrix can be passed to a function or operation at once; hence, no element by element process is required.

Matrix Operations >> A + B

5 11 5 10 7 10 7 13 16

>> A * B 21 33 36 45 69 66 81 123 111

>> A * a ??? Error using ==> mtimes Inner matrix dimensions must agree.

>> b * c ??? Error using ==> mtimes Inner matrix dimensions must agree.


% Element by element addition

% Matrix multiplication: A x B

Matrix sizes should match: [m x n] x [n o] = [m x o]

One of the most common errors in Matlab

Matrix Multiplication >> c' * b

1 4 7 10 13 16 19 22 25 28 4 16 28 40 52 64 76 88 100 112 …

>> b * c' 2845

>> M1 = A * B 21 33 36 45 69 66 81 123 111

>> B * A 43 58 73 46 61 76 67 82 97


% 1x10 x 10x1 => 1x1

% 10x1 x 1x10 => 10x10

AxB ≠ BxA

% M1(m,n) = A(m,:) x B(:,n)

A .* B ≠ A * B

Matrix Multiplication >> M2 = A .* B

4 18 6 21 12 25 6 42 64

>> M3 = cross(A, B) -39 3 0 23 51 -8 -5 -30 5

>> M4 = dot(A,B) 31 72 95


% Cross product

% Dot product: M4(m) = A(m,:) x B(m,:)

%{ Element by element multiplication: M2(m,n) = A(m,n) x B(m,n) }%

A .* B = B .* A

Matrix Division b / c

1.0000

b ./ c 1 16 49 100 169 256 361 484 625 784

A ./ B 0.2500 0.2222 1.5000 0.4286 1.3333 1.0000 6.0000 1.1667 1.0000


% Matrix element division – creates a single number

% Element by element division – creates a vector

Matrix Division d = [1 2 3]' A \ d

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.469447e-018. 1.0e+015 * 0.9007 -1.8014 0.9007

B \ d 0.0222 0.0222 0.3556


% Matrix division: Am = b → m = A\v

Mesh >> u = [1 2 3]; v = [4 5 6]; >> u .* v

4 10 18

>> [x, y] = meshgrid(u, v) 1 2 3 1 2 3 1 2 3

4 4 4 5 5 5 6 6 6

>> x .* y 4 8 12 5 10 15 6 12 18


%{ 1-to-1 relationship for each element of u to only one element of v & vice versa }%

%{ 1-to-1 relationship for each element of u to 3 elements of v & vice versa }%

“meshgrid” replicates the 1x3 grid vectors u & v to produce a full grid. 3x3 x & y contain copies of u & v in each column.

u .* v is the first row of the meshgrid

Mesh

Without meshgrid With meshgrid


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xy

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z = x2 + y2

Finding Roots Nearest zero

>> fzero(@cos, 2) 1.5708

>> fzero(@(x) sin(2*x),2) 1.5708

Polynomial roots >> roots([1,0,2,1])

0.2267 + 1.4677i 0.2267 - 1.4677i -0.4534 + 0.0000i


% pi/2 % nearest zero of cosine function to 2

% roots of x3+2x+1=0

IF Statements %% Dr. Sheldon Cooper's breakfast schedule … % Any code If day == 'Mon' if brkfast == 'Oatmeal'; brkfastFate= ' Consumed'; end elseif day == 'Sat' if brkfast == 'Cereal with Dr. Who'; brkfastFate= ' Consumed'; end … else brkfastFate = 'Dumped in the garbage'; end


IF b

lock

s

% 1st IF statement % Inner IF statement

% Else IF

% Equivalent to default statement

For each IF statement, an “end” statement is required at the end of the block

Switch Cases %% Nelson Muntz's response depending on his mood … % Any code

mood = ['hungry', 'angry', 'Martin Prince', …]; switch mood case 'hungry' response = 'Rob lunch'; case 'angry' response = 'Punch'; case 'Martin Prince' response = 'Give wedgie'; …. otherwise response = 'ha ha'; end


% Case 1

% Case 2

% Case 3

% Default case

% The only “end” statement required, at the end of the block

Sw

itch

blo

ck

Loops 1. For Loop

u = 1:3; C = zeros(3,3); D = []; for i = u'; for j = 1:3; C(i,j) = A(i,j); D = [D, A(i,j)];

end end

1 2 3 3 4 5 6 7 8

2. While Loop % Settlers of Catan game flowchart; whoever reaches 10 wins the game scores = zeros(1,4); while max(scores) < 10 scores = playNextRound(scores); end


% “D” is an empty matrix; i.e. 0x0

% “i” is assigned a vector

% Cij = Aij

% Concatenating D with Aij

% D

% Game is played until a player's score reaches 10

% Play next round & update “scores”

For

blo

cks

Wh

ile b

lock

% C = A

Subroutines 1. M-File

Function definition function [arrivalYr, realAge] = FuturamaAgeCalctr(f, departureYr, age) travelTime = 1000; arrivalYr = f(departureYr) + travelTime; realAge = age + travelTime;


% 1st output

% 2nd output

• Needs to be saved on a separate “.m” file. • Outputs are passed in the form of an array. • Does NOT require “end” statement.

Function Name Inputs Outputs

Subroutines Calling the function

PJFry = [1999, 25]; [PJFry(3), PJFry(4)] = FuturamaAgeCalctr(@abs, PJFry (1), PJFry (2)); % Some string to show the results in a sentence str = sprintf('Phillip J. Fry was frozen in year %d, when he was %d. … He was defrosted in year %d, %d years after his birth.', PJFry(1), … PJFry(2), PJFry(3), PJFry(4)); disp(str)

Phillip J. Fry was frozen in year 1999, when he was 25. He was defrosted in year 2999, 1025 years after his birth.


% Fry's departure year and age

% Results are in the form of an array

% Displaying the string

• File path has to be set to the function location. • Function outputs are in the form of an array.

Subroutines 2. feval [PJF(1) PJF(2)] = feval('FuturamaAgeCalctr', @abs, 1999, 25)

2999 1025


Another way of Calling an external function

Function Name Inputs

Subroutines 3. inline % Function definition

f = inline('1000 + (x - y)', 'x', 'y'); % Calling the function

f(1999, 1974) 1025

4. @ % Function definition

f = @(x, y) (1000 + (x - y)); % Calling the function

f(1999, 1974) 1025


Internal function, only accessible within the “.m” file

% Fry's real age in arrival

% Fry's departure and birth year

Similar to inline

2D Plots >> x = -pi:pi/1000:pi; y = sin(x); >> plot(x, y,'r--')

Plot Options

grid on / off title ('Title') xlabel ('X-axis label') ylabel ('Y-axis label') axis ([Xmin Xmax Ymin Ymax]) legend ('A1', 'A2', …, 'Location', 'East')


-4 -3 -2 -1 0 1 2 3 4-1

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y = Sin(x) vs. x

xy

= si

n(x)

Sin(x)

Histogram A histogram shows the distribution of data values >> mark = 0:12; >> marks = 12*rand(1,100); >> hist(marks, mark)


0 1 2 3 4 5 6 7 8 9 10 11 120

2

4

6

8

10

12

Mark (F, D-, D, D+, ..., A+)

# of

Stu

dent

s

Distribution of Marks for 100 Students

Other 2D Plots semilogx semilogy loglog scatter stem contour


Same as 'plot', with a log scale X-axis

Same as 'plot', with a log scale Y-axis

Same as 'plot', with both axis in log scale

Discrete equivalent of 'plot'

Plot options can be used in any plot types!

Same as 'scatter', with stems connected to x-axis

Contour equivalent of 'plot'

Multiple Plots Shared Y-axis

o >> plot(x, sin(x), 'g--', x, cos(x), 'b:')

o >> plot(x, sin(x), 'g--‘) >> hold on >> plot(x, cos(x), 'b:') >> hold on

Double Y-axis >> plotyy(x, sin(x), x, cos(x), 'function')


-4 -3 -2 -1 0 1 2 3 4-1

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plot, semilogx, semilogy, loglog, stem, ..

Alternative methods

Subplots Creates several [sub]plots on the same window

>> subplot(1,2,1) >> plot(A, d, 'ms--') >> subplot(1,2,2) >> plot(B, d, 'b^:')


% 1st plot of a 1x2 window

% 2nd plot of a 1x2 window

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Sin(x) vs. x

x

sin(

x)

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Cos(x) vs. x

x

Cos

(x)

3D Plots >> t = -10*pi : pi/100 : 10*pi; >> x = t.*cos(t); y = t.*sin(t);

>> plot3(x, y, t);


-40-30

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(t, t*cos(t), t*sin(t))

t * cos(t)

t * s

in(t)

Other 3D Plots mesh contour3 surf scatter3


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z = y2 – x2

Need meshgrid inputs

Mathematical Functions abs(x) Absolute value

sqrt(x) Square root

exp(x) Exponential; ex

log(x) Natural log

log10(x) Base 10 log

log2(x) Base 2 log

real(x) Complex real part

imag(x) Complex imaginary part

conj(x) Complex conjugate

round(x) Rounds to the nearest integer

…


Trigonometric Functions sin(x) Sine

cos(x) Cosine

tan(x) Tangent

asin(x) Inverse sine

acos(x) Inverse cosine

atan(x) Inverse tangent

sinh(x) Hyperbolic sine

cosh(x) Hyperbolic cosine

tanh(x) Hyperbolic tangent

…


Trigonometric functions expect their argument to be in radian!

Logical & Relational Operators & AND

| OR

~ NOT

< Less than

> Greater than

<= Less than or equal to

>= Greater than or equal to

== Equal to

~= Not equal to


Displaying Strings sprintf fprintf int2str num2str disp …


Print

Saving figure or model to disk as image or MATLAB file

>> print -dpsc myPlot.ps Postscript

>> print -dwinc Windows printer

>> print –dbitmap Windows bitmap clipboard

>> saveas(gcf, 'myPlot', 'pdf') Save file


File types: pdf, jpeg, gif, bmp, fig, etc.

Helpful Commands help plot Displays help about “plot” in Command Window helpwin plot Displays help about “plot” in Help Window lookfor plot Searches help for “plot”

clear Clears variables & functions from memory

hold on / off Holds current graph

pause Pauses code & wait for user response, i.e. key

pause(t) Pauses for t seconds & resume afterwards

figure Creates a new figure window

close Closes a figure/window

clc Clears the command window

…


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