Introduction to MATLAB

Zen and the Art of
MatLab
DAMIAN GORDON

Introduction to MatLab
 MatLab is an interactive, matrix-based system for
numeric computation and visualisation
 MATrix LABoratory
 Used in image processing, image synthesis,
engineering simulation, etc.

References
 “Mastering MatLab” Duane
Hanselman, Bruce Littlefield
 “The MatLab Primer”
http://www.fi.uib.no/Fysisk/Teori/KU
RS/WRK/mat/mat.html
 “The MatLab FAQ”
http://www.isr.umd.edu/~austin/en
ce202.d/matlab-faq.html

MATLAB Command
Window
To get started, type one of these: helpwin, helpdesk, or demo.
For product information, type tour or visit www.mathworks.com.
»
» help
HELP topics:

Creating Variables
>> varname = 12
varname =
12
>> SS = 56; N = 4; Tot_Num = SS + N
Tot_Num =
60

Operations
+ Addition
- Subtraction
* Multiplication
^ Power
Division
/ Division
• Add vars
• Subtract vars
Multiplication
• Raise to the power
• Divide vars (A div B)
• Divide vars (B div A)

Creating Complex
Numbers
>> 3 + 2i
>> sqrt(9) + sin(0.5)*j
ans =
3.0000 + 0.4794i
Num = sqrt(9) + sin(0.5)*j
real(Num)
imag(Num)

Entering Matrices (1)
>> A = [1 2 3; 4 5 6; 7 8 9]
OR
>> A = [
1 2 3
4 5 6
7 8 9 ]

 To create an NxM zero-filled matrix
>> zeros(N,M)
 To create a NxN zero-filled matrix
>> zeros(N)
 To create an NxM one-filled matrix
>> ones(N,M)
 To create a NxN one-filled matrix
>> ones(N)

 To create an NxM randomly-filled
matrix (which is uniformly
distributed)
>> rand(N,M)
 To create an NxM randomly-filled
matrix (which is normally
distributed)
>> randn(N,M)

Complex Matrices
 To enter a complex matrix, you may do it in one
of two ways :
>> A = [1 2; 3 4] + i*[5 6;7 8]
OR
>> A = [1+5i 2+6i; 3+7i 4+8i]

MATLAB Command Window
» who
Your variables are:
a b c
» whos
Name Size Bytes Class
a 8x8 512 double array
b 9x9 648 double array
c 9x9 648 double array
Grand total is 226 elements using 1808 bytes

Matrix Addition
» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» A + B
ans =
4 4 4
6 6 6
8 8 8

Matrix Subtraction
» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» B - A
ans =
2 2 2
2 2 2
2 2 2

Matrix Multiplication
» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» A * B
ans =
12 12 12
24 24 24
36 36 36

Matrix - Power
» A ^ 2
ans =
6 6 6
12 12 12
18 18 18
» A ^ 3
ans =
36 36 36
72 72 72
108 108 108

Matrix Transpose
A =
1 1 1
2 2 2
3 3 3
» A'
ans =
1 2 3
1 2 3
1 2 3

Matrix Division
Left Division
x = AB (is A*x=B)
>> A = rand(4)
>> B = rand(4)
>> C = A B
=> A * C = B
Right Division /
x=A/B (is x*A=B)
>> A = rand(4)
>> B = rand(4)
>> C = A / B
=> C * A = B

Matrix Operations
+ Addition
- Subtraction
* Multiplication
^ Power
‘ Conjugate Transpose
Left Division
/ Right Division
• Add matrices
• Subtract matrices
Matrix Multiplication
• Raise to the power
• Get transpose
• x = AB (is A*x=B)
• x=A/B (is x*A=B)

MatLab Operators
DAMIAN GORDON

Magic Matrix
MAGIC Magic square.
MAGIC(N) is an N-by-N matrix constructed from the
integers
1 through N^2 with equal row, column, and diagonal sums.
Produces valid magic squares for N = 1,3,4,5,...

Identity Function
>> eye (4)
ans =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

Upper Triangle Matrix
» a = ones(5)
a =
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
» triu(a)
ans =
1 1 1 1 1
0 1 1 1 1
0 0 1 1 1
0 0 0 1 1
0 0 0 0 1

Lower Triangle Matrix
» a = ones(5)
a =
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
» tril(a)
ans =
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1

Hilbert Matrix
» hilb(4)
ans =
1.0000 0.5000 0.3333 0.2500
0.5000 0.3333 0.2500 0.2000
0.3333 0.2500 0.2000 0.1667
0.2500 0.2000 0.1667 0.1429

Inverse Hilbert Matrix
» invhilb(4)
ans =
16 -120 240 -140
-120 1200 -2700 1680
240 -2700 6480 -4200
-140 1680 -4200 2800

Toeplitz matrix.
TOEPLITZ
TOEPLITZ(C,R) is a non-symmetric Toeplitz matrix
having C as its
first column and R as its first row.
TOEPLITZ(R) is a symmetric (or Hermitian) Toeplitz
matrix.
-> See also HANKEL

Summary of Functions
• magic
• eye(4)
• triu(4)
• tril(4)
• hilb(4)
• invhilb(4)
• toeplitz(4)
- magic matrix
- identity matrix
- upper triangle
- lower triangle
- hilbert matrix
- Inverse Hilbert matrix
- non-symmetric
Toeplitz matrix

Dot Operator
 A = magic(4); b=ones(4);
 A * B
 A.*B
 the dot operator performs element-by-element
operations, for “*”, “” and “/”

Concatenation
 To create a large matrix from a group of smaller
ones
 try
 A = magic(3)
 B = [ A, zeros(3,2) ; zeros(2,3), eye(2)]
 C = [A A+32 ; A+48 A+16]
 Try some of your own !!

Subscripts
 Row i and Column j of matrix A is denoted by A(i,j)
 A = Magic(4)
 try
 A(1,4) + A(2,4) + A(3,4) + A(4,4)
 try
 A(4,5)

The Colon Operator (1)
 This is one MatLab’s most important operators
 1:10 means the vector
 1 2 3 4 5 6 7 8 9 10
 100:-7:50
 100 93 86 79 72 65 58 51
 0:pi/4:pi
 0 0.7854 1.5708 2.3562 3.1416

The Colon Operator (2)
 The first K elements in the jth column is
 A(1:K, j)
 Sum(A(1:4, 4)) is the sum of the 4th column
or
 Sum(A(:, 4)) means the same

Deleting Rows and Columns (1)
• Create a temporary matrix X
• X=A;
• X(:, 2) = []
• Deleting a single element won’t result in a
matrix, so the following will return an error
• X(1,2) = []

Deleting Rows and Columns (2)
• However, using a single subscript, you can
delete
– a single element
– sequence of elements
So X(2:2:10) = []
gives
• x = 16 9 2 7 13 12 1

The ‘FIND’ Command (1)
>> x = -3:3
x =
-3 -2 -1 0 1 2 3
K = find(abs(x) > 1)
K =
1 2 6 7

The ‘FIND’ Command (2)
A = [ 1 2 3 ; 4 5 6 ; 7 8 9]
[i, j] = find (A > 5)
i =
3
3
2
3
j =
1
2
3
3

Special Variables
• ans
• pi
• eps
• flops
• inf
• NaN
• i,j
• why
- default name for results
- pi
- “help eps”
- count floating point ops
- Infinity, e.g. 1/0
- Not a number, e.g. 0/0
- root minus one
- why not ?

The ‘PLOT’ Command (1)
>> X = linspace(0, 2*pi, 30);
>> Y = sin(X);
>> plot(X,Y)
>> Z = cos(X);
>> plot(X,Y,X,Z);

The ‘PLOT’ Command (2)
>> W = [Y ; Z]
>> plot (X,W)
Rotate by 90 degrees
>> plot(W,X)

PLOT Options
>> plot(X,Y,’g:’)
>> plot(X,Y,’r-’)
>> plot(X,Y,’ko’)
>> plot(X,Y,’g:’,X,Z,’r-’,X,Y,’wo’,X,Z,’c+’);

PLOT Options
>> grid on
>> grid off
>> xlabel(‘this is the x axis’);
>> ylabel(‘this is the y axis’);
>> title(‘Title of Graph’);
>> text(2.5, 0.7, ’sin(x)’);
>> legend(‘sin(x)’, ‘cos(x)’)

SUBPLOT Command
 Subplot(m,n,p)
 creates a m-by-n matrix in and plots in the pth plane.
subplot(2,2,1)
plot(X,Y)
subplot(2,2,2)
plot(X,Z)
subplot(2,2,3)
plot( X,Y,X,Z)
subplot(2,2,4)
plot(W,X)

Specialised matrices
• compan
• gallery
• hadamard
• hankel
• pascal
• rosser
• vander
• wilkinson
• Companion matrix
• Higham test matrices
• Hadamard matrix
• Hankel matrix
• Pascal matrix.
• Classic symmetric
eigenvalue test problem
• Vandermonde matrix
• Wilkinson's eigenvalue
test matrix

Polynomials
Polynomials are represented as
row vectors with its coefficients in
descending order, e.g.
X4
- 12X3
+ 0X2
+25X + 116
p = [1 -12 0 25 116]

Polynomials
The roots of a polynomial are found as follows
r = roots(p)
roots are represented as a column vector

Polynomials
Generating a polynomial from its roots
polyans = poly(r)
includes imaginary bits due to rounding
mypolyans = real(polyans)

Polynomial Addition/Sub
 a = [1 2 3 4]
 b = [1 4 9 16]
 c = a + b
 d = b - a

Polynomial Addition/Sub
 What if two polynomials of different order ?
X3
+ 2X2
+3X + 4
X6
+ 6X5
+ 20X4
- 52X3
+ 81X2
+96X + 84
 a = [1 2 3 4]
 e = [1 6 20 52 81 96 84]
 f = e + [0 0 0 a] or f = e + [zeros(1,3) a]

Polynomial Multiplication
 a = [1 2 3 4]
 b = [1 4 9 16]
Perform the convolution of two arrays !
 g = conv(a,b)
g =
1 6 20 50 75 84 64

Polynomial Division
 a = [1 2 3 4]
 g = [1 6 20 50 75 84 64]
Perform the deconvolution of two arrays !
[q,r] = deconv(g, a)
q = {quotient}
1 4 9 16
r = {remainder}
0 0 0 0 0 0 0 0

Polynomial Differentiation
 f = [1 6 20 48 69 72 44]
 h = polyder(f)
h =
6 30 80 144 138 72

Polynomial Evaluation
x = linspace(-1,3)
p = [1 4 -7 -10]
v = polyval(p,x)
plot (x,v), title(‘Graph of P’)

Rational Polynomials
 Rational polynomials, seen in Fourier, Laplace and
Z transforms
 We represent them by their numerator and
denominator polynomials
 we can use residue to perform a partial fraction
expansion
 We can use polyder with two inputs to
differentiate rational polynomials

Data Analysis Functions (1)
 corrcoef(x)
 cov(x)
 cplxpair(x)
 cross(x,y)
 cumprod(x)
 cumsum(x)
 del2(A)
 diff(x)
 dot(x,y)
 gradient(Z, dx, dy)
 Correlation coefficients
 Covariance matrix
 complex conjugate
pairs
 vector cross product
 cumulative prod of
cols
 cumulative sum of cols
 five-point discrete
Laplacian
 diff between elements
 vector dot product
 approximate gradient

Data Analysis Functions (2)
 histogram(x)
 max(x), max(x,y)
 mean(x)
 median(x)
 min(x), min(x,y)
 prod(x)
 sort(x)
 std(x)
 subspace(A,B)
 sum(x)
 Histogram or bar
chart
 max component
 mean of cols
 median of cols
 minimum component
 product of elems in
col
 sort cols (ascending)
 standard dev of cols
 angle between
subspaces
 sum of elems per col

Symbolic Expressions
 ‘1/(2*x^n)’
 cos(x^2) - sin(x^2)
 M = sym(‘[a , b ; c , d]’)
 f = int(‘x^3 / sqrt(1 - x)’, ‘a’, ‘b’)

Symbolic Expressions
diff(‘cos(x)’)
ans =
-sin(x)
det(M)
ans =
a*d - b * c

Symbolic Functions
 numden(m) - num & denom of polynomial
 symadd(f,g) - add symbolic polynomials
 symsub(f,g) - sub symbolic polynomials
 symmul(f,g) - mult symbolic polynomials
 symdiv(f,g) - div symbolic polynomials
 sympow(f,’3*x’) - raise f^3

Advanced Operations
 f = ‘1/(1+x^2)’
 g = ‘sin(x)’
compose(f,g) % f(g(x))
ans=
1/(1+sin(x)^2)

Advanced Operations
• finverse(x^2)
ans =
x^(1/2)
• symsum(‘(2*n - 1) ^ 2’, 1, ‘n’)
ans =
11/3*n + 8/3-4*(n+1)^2 + 4/3*(n+1)^3

Symbolic Differentiation
f = ‘a*x^3 + x^2 + b*x - c’
diff(f) % by default wrt x
ans =
3*a^2 + 2*x + b
diff(f, ‘a’) % wrt a
ans =
x^3
diff(f,’a’,2) % double diff wrt a
ans =
0

Symbolic Integration
f = sin(s+2*x)
int(f)
ans =
-1/2*cos(s+2*x)
int(f,’s’)
ans =
-cos(s+2*x)
int(f, ‘s’, pi/2,pi)
ans=
-cos(s)

Comments &Punctuation (1)
• All text after a percentage sign (%) is
ignored
>> % this is a comment
• Multiple commands can be placed on one
line separated by commas (,)
>> A = magic(4), B = ones(4), C = eye(4)

Comments &Punctuation (2)
• A semicolon may be also used, either after a
single command or multiple commands
>> A = magic(4); B = ones(4); C = eye(4);
• Ellipses (…) indicate a statement is
continued on the next line
A = B/…
C

SAVE Command (1)
>> save
• Store all the variables in binary format in a file
called matlab.mat
>> save fred
• Store all the variables in binary format in a file
called fred.mat
• >> save a b d fred
• Store the variables a, b and d in fred.mat

SAVE Command (2)
>> save a b d fred -ascii
 Stores the variables a, b and d in a
file called fred.mat in 8-bit ascii
format
>> save a b d fred -ascii -double
 Stores the variables a, b and d in a
file called fred.mat in 16-bit ascii
format

Load Command
• Create a text file called mymatrix.dat with
– 16.0 3.0 2.0 13.0
– 5.0 10.0 11.0 8.0
– 9.0 6.0 7.0 12.0
– 4.0 15.0 14.0 1.0
• “load mymatrix.dat”, create variable
mymatrix

M-Files
• To store your own MatLab commands in a
file, create it as a text file and save it with a
name that ends with “.m”
• So mymatrix.m
A = […
16.0 3.0 2.0 13.0
5.0 10.0 11.0 8.0
9.0 6.0 7.0 12.0
4.0 15.0 14.0 1.0];
• type mymatrix

IF Condition
if condition
{commands}
end
If x = 2
output = ‘x is even’
end

WHILE Loop
while condition
{commands}
end
X = 10;
count = 0;
while x > 2
x = x / 2;
count = count +
1;
end

FOR Loop
for x=array
{commands}
end
for n = 1:10
x(n) = sin(n);
end
A = zeros(5,5); %
prealloc
for n = 1:5
for m = 5:-1:1
A(n,m) = n^2 + m^2;
end
disp(n)
end

Creating a function
function a = gcd(a,b)
% GCD Greatest common divisor
% gcd(a,b) is the greatest common divisor of
% the integers a and b, not both zero.
a = round(abs(a)); b = round(abs(b));
if a == 0 & b == 0
error('The gcd is not defined when both numbers are zero')
else
while b ~= 0
r = rem(a,b);
a = b; b = r;
end
end

Quick Exercise (!)
 Consider Polynomial Addition again :
how would you write a program that takes in two
polynomials and irrespective of their sizes it adds
the polynomials together ? Given that the
function length(A) returns the length of a vector.
Answers on a postcard to : dgordon@maths.kst.dit.ie
oh, and while you’re here anyhow, if you have a
browser open, please go to the following sites :
http://www.the hungersite.com
http://www.hitsagainsthunger.com

Creating Programs
Title : Program.m
function out = program(inputs)
% PROGRAM
<code>

Know Thyself
 Where am I ?
 pwd
 Get me onto the hard disk
 cd C:
 Where am I now ?
 pwd
 Get me to where I know
 cd ..

Quick Answer (!)
function c = mypoly(a,b)
% MYPOLY Add two polynomials of variable lengths
% mypoly(a,b) add the polynomial A to the polynomial
% B, even if they are of different length
%
% Author: Damian Gordon
% Date : 3/5/2001
% Mod'd : x/x/2001
%
c = [zeros(1,length(b) - length(a)) a] + [zeros(1, length(a) -
length(b)) b];

Recursion
function b = bart(a)
%BART The Bart Simpson program writes on the
blackboard
% program, Bart writes the message a few times
% and then goes home to see the Simpsons
if a == 1
disp('I will not....');
else
disp('I will not skateboard in the halls');
bart(a - 1);
end

Curve Fitting
 What is the best fit ?
 In this case least squares curve fit
 What curve should be used ?
 It depends...

POLYFIT : Curve Fitting
 polyfit(x,y,n) - fit a polynomial
 x,y - data points describing the curve
 n - polynomial order
 n = 1 -- linear regression
 n = 2 -- quadratic regression

Curve Fitting Example
x = [0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1];
y = [-.447 1.978 3.28 6.16 7.08 7.34 7.66
9.56 9.48 9.30 11.2];
polyfit(x,y,n)
 n = 1
p = 10.3185 1.4400
 n = 2
p = -9.8108 20.1293 -0.0317
y = -9.8108x2
+ 20.1293x - 0.0317

Curve Fitting Example
xi = linspace(0,1,100);
z = polyval(p,xi)
plot(x,y,'o',x,y,xi,z,':');

Interpolation - 1D
 t = interp1(x,y,.75)
t =
9.5200
also
 interp1(x,y,.75,’spline’)
 interp1(x,y,.75,’cubic’)

Interpolation - 2D
 interp2(x,y,Z,xi,yi,TYPE)
TYPE =
'nearest' - nearest neighbor interpolation
'linear' - bilinear interpolation
'cubic' - bicubic interpolation
'spline' - spline interpolation

Fourier Functions
 fft
 fft2
 ifft
 ifft2
 filter
 filter2
 fftshift
 Fast fourier
transform
 2-D fft
 Inverse fft
 2-D Inverse fft
 Discrete time
filter
 2-D discrete tf
 shift FFT results so -ve
freqs appear first

Tensors
 See ‘Programs’
 ‘Tensors’
 ‘Tensors.html’

3D Graphics
T = 0:pi/50:10*pi;
plot3(sin(t), cos(t), t);

3D Graphics
title('Helix'), xlabel('sin(t)'),
ylabel('cos(t)'), zlabel('t')
grid

3D Graphics
 Rotate view by elevation and azimuth
view(az, el);
view(-37.5, 60);

Introduction to MATLAB

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