Introduction to MATLAB Math 375 Getting started Arithmetics Matrices Plotting Functions Program Control Introduction to MATLAB Math 375 Natalia Vladimirova (many ideas, examples, and excersises are borrowed from Profs. Monika Nitsche, Richard Allen, and Stephen Lau) January 24, 2010
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Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
Introduction to MATLAB
Math 375Natalia Vladimirova
(many ideas, examples, and excersises are borrowed fromProfs. Monika Nitsche, Richard Allen, and Stephen Lau)
January 24, 2010
Introduction toMATLAB
Math 375
Getting started
Starting MATLAB
Desktop Tools
Getting Help
Saving your work
Arithmetics
Matrices
Plotting
Functions
Program Control
Starting MATLABI Under windows
Click on the ”Start” menu buttonClick on the ”MATLAB” menu entry
I Under LinuxUse SSH to log onto linux.unm.eduType matlab
Introduction toMATLAB
Math 375
Getting started
Starting MATLAB
Desktop Tools
Getting Help
Saving your work
Arithmetics
Matrices
Plotting
Functions
Program Control
Desktop ToolsI Command Window
type commands
I Workspaceview program variablesdouble click on a variable to see it in the Array Editor
I Command Historyview past commands
I Current Directoryview and select files in the current directorythe current directory can be changed in the toolbar
I To switch to the default windows arrangement, go to menuDesktop > Desktop Layout > Default
Introduction toMATLAB
Math 375
Getting started
Starting MATLAB
Desktop Tools
Getting Help
Saving your work
Arithmetics
Matrices
Plotting
Functions
Program Control
Getting HelpI Using the Help Browser
I From command window
>> help
>> help plot
I Running demos
>> demos
>> help demos
I On the webhttp://www.mathworks.com
Introduction toMATLAB
Math 375
Getting started
Starting MATLAB
Desktop Tools
Getting Help
Saving your work
Arithmetics
Matrices
Plotting
Functions
Program Control
Saving your workI in class, create your directory (folder) in T: drive
I MATLAB files reside on the current directory or the searchpath
>> addpath T:\myfolder
>> addpath ~/matlab
I you can save a whole session using
>> diary "diary_Jan23.txt"
>> diary off
>> diary on
I bring your flash card (floppy, CD-RW) to save class work
I or upload your files to your UNM storage (?)
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
VariablesI creating a variable (MATLAB is case sensitive);
semicolon suppresses output
>> x = 5;
>> x = 5
x =
5
I checking variables
>> whos
I deleting a variable
>> clear x
I deleting all variables
>> clear
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
Build-in constants
Imaginary unit i = j =√−1 and π
>> clear
>> pi
pi = 3.1416
>> i
i = 0 + 1i
>> j
j = 0 + 1i
Build-in constants can be overridden
>> i = 8
i = 8
>> clear
>> i
i = 0 + 1i
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
Arithmetic operators
+ addition
− subtraction
* multiplication
/ division
ˆ power
‘ complex conjugate and transpose
.* element-by-element mult
./ element-by-element div
.̂ element-by-element power
.‘ transpose
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
Relational and logical operators
== equal
∼= not equal
< less than
<= less than or equal
> greater than
>= greater than or equal
& AND
| OR
∼ NOT
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
Mathematical functionsI Elementary functions (sin, cos, sqrt, abs, exp, log10, round)
>> help elfun
I Advanced functions (bessel, beta, gamma, erf)
>> help specfun
>> help elmat
I Example
>> x=2.3; a=sin(x), b=cos(x)
a = 0.7457
b = -0.6663
>> c = ( sin(x) )^2 + ( cos(x) )^2
c = 1.0000
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Variables
Build-in constants
Arithmetic operators
Logical operators
Math functions
Scripts
Matrices
Plotting
Functions
Program Control
MATLAB scriptsI Example of MATLAB script file (myscript.m)
myscript.m
% This is a comment.
% convert cartesian coordinares (x,y)
% to polar coordinates (r,theta)
x = 4;
y = 3;
r = sqrt(x^2 + y^2)
theta = atan(y/x)
% continuation character is ‘‘...’’
I Save it in your working directoryI Call it from the command window
>> myscript
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Creating vectors
>> x = [1 2 5 3] % separated by spaces
x =
1 2 5 1
>> x = [1, 2, 5, 3] % separated by commas
x =
1 2 5 3
>> y = [6; 7; 4; 8] >> y = x’
y = y =
6 1
7 2
4 5
8 3
Note: A’ = transpose ( A )
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
More ways of creating vectorsI Using increment x1 : ∆x : xn
>> x = 0:0.2:1
x =
0.0 0.2 0.4 0.6 0.8 1.0
I Using linspace (x1, xn, n)
>> x = linspace(0,1,6)
x =
0.0 0.2 0.4 0.6 0.8 1.0
I Note: both end points are included, ∆x = xn−x1
n−1
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Creating matricesI Explicitly
>> A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2 3
4 5 6
7 8 9
I Using MATLAB functions
>> M = 2; N = 3;
>> a = zeros(M, N)
>> b = ones(M, N)
>> c = rand(M, N)
>> I = eye(N)
I scalar is 1× 1 matrix
I vector is 1× N or N × 1 matrix
I you can create N1 × N2 × ...× Nn matrices
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Matrix addition (or subtraction)
element by element C = A + B : Cij = Aij + Bij
(a11 a12
a21 a22
)+
(b11 b12
b21 b22
)=
(a11 + b11 a12 + b12
a21 + b21 a22 + b22
)
>> A = [1 2; 3 4] >> B = [5 6; 7 8]
A = B =
1 2 5 6
3 4 7 8
>> C = A + B
C =
6 8
10 12
Matrices being added must be the same size
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Matrix multiplication
not element by element C = AB: Cij =∑n
k=1 AikBkj
(a11 a12
a21 a22
)(b11 b12
b21 b22
)=
(a11b11 + a12b21 a11b12 + a12b22
a21b11 + a22b21 a21b12 + a22b22
)
>> A = [1 2; 3 4] >> B = [4 3; 2 1]
A = B =
1 2 4 3
3 4 2 1
>> C = A * B >> C = A .* B
C = C =
8 5 4 6
20 13 6 4
For element by element multiplication use .*
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Accessing elements of a matrix
>> A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2 3
4 5 6
7 8 9
I A(i , j) subscription - numbering starts from 1 (Fortran style)
>> A(2,3)
ans = 6
I whole row
>> x = A(2,:)
x =
4 5 6
I whole column
>> y = A(:,3)
y =
3
6
9
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Accessing elements of a matrix (cont.)
I Matrix elements can be accessed by index array.Try the following:
>> v = 10:10:80
>> ind = [3,4,7]
>> v(ind)
>> v(ind) = -1
I ... or by condition:
>> v = 10:10:80
>> v>30
>> v(v>30) = -1
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Matrix operationsI [ ] concatenation
>> x = [ zeros(1,3) ones(1,2) ]
x =
0 0 0 1 1
I ( ) subscription
>> x = [ 1 3 5 7 9];
>> y = x(2)
y =
3
>> y = x(2:4)
y =
3 5 7
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
Finding matrix size
Explain the output of the following command
>> A = [1 2 3; 4 5 6]
>> ndims(A)
>> length(A)
>> size(A)
>> [n m] = size(A)
>> sum(A)
>> sum(A,1)
>> sum(A,2)
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Creating vectors
More ways...
Creating matrices
Addition
Multiplication
Accessing elements
... more
Matrix operations
Matrix size
Exercises
Plotting
Functions
Program Control
What happens when you run the followingcommands? Why?
>> x = 0:1:5
>> y = x*x
>> y = x.*x
>> y = x^2
>> y = x’*x
>> y = x*x’
>> y = 2*x
>> y = 1./x
>> x = 0:0.1:1
>> y = x(4:8)
>> y = x(5:end)
>> y = x([1,5,7,11])
>> A = [1 2 3; 4 5 6; 7 8 9]
>> B = A((2:3),(2:3))
>> C = sin([0, 0.5*pi, pi, 1.5*pi])
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
Simple plotI MATLAB plots arrays. Arrays must be created first!
>> t = 1:1:7
>> plot(t, sin(t))
I More points - smoother line
>> x = 1:0.1:7;
>> y = sin(x);
>> plot(x, y)
I Note: y(x) is plotted, not x(t) and y(t).
I Vectors x and y must be the same length.
I To create a graph in logaritmic coordimatesuse loglog, semilogx, or semilogy instead of plot.
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
Label you plots!I Once we created the plot...
>> x = linspace(0, 2*pi, 50);
>> y = sin(x);
>> plot(x, y, ’o-r’) % plot with red circles
I we can add features to it
>> axis([0, 2*pi, -1.1, 1.1])
>> xlabel(’x’); ylabel(’y’)
>> title(’Plot of the sine function’)
>> legend(’y(x) = sin(x)’, ’Location’, ’NorthEast’)
>> grid on
I You can edit figure using menu functions.
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
Saving plots to files
I Save your figure as *.fig - this way you can modify it later.
I Save PDF on letter-size paper. Here “gcf” is the handle tocurrent figure (“get current figure”). The plot position on thepaper is given as the coordinates of upper-left and lower-rightcorners
>> set(gcf, ’papersize’, [8.5, 11], ...
’paperposition’, [1, 1, 6, 4])
>> saveas(gcf,’myplot.pdf’,’pdf’)
I If figure to be inserted into a document, use paper size sameas plot size
>> set(gcf, ’papersize’, [6, 4], ...
’paperposition’, [0, 0, 6, 4])
>> saveas(gcf,’myplot.pdf’,’pdf’)
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
Marker options (type help plot to remind)
b blue . point - solid
g green o circle : dotted
r red x x-mark -. dashdot
c cyan + plus -- dashed
m magenta * star (none) no line
y yellow s square
k black d diamond
w white v triangle (down)
^ triangle (up)
< triangle (left)
> triangle (right)
p pentagram
h hexagram
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
Multiple graphs, panels, and figuresI Multiple graphs on the same plot (also see help hold)
>> x = linspace(0, 2*pi, 100);
>> y1 = cos(3*t) .* exp(-0.5*t);
>> y2 = sin(3*t) .* exp(-0.5*t);
>> plot(t,y1,’x’, t,y2, ’*’)
I To open new figure (new window)
>> figure
>> figure(1) %make figure 1 current figure
I To divide figure to panels
>> subplot(2,2,1)
>> plot(t,y1)
>> subplot(2,2,2)
>> plot(t,y2)
>> subplot(2,2,3)
>> plot(y1,y2)
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Simple plot
Label you plots!
Saving plots
Markers
Multiple Plots
In class exercise
Functions
Program Control
In class exercise
By plotting y(t) below with different values of parameter a find aat which y(t) passes through given “experimental” points as closeas possible
y(t) = e−t2 sin(at)
t = 0.1 0.5 0.9 1.3 1.7 ...
2.1 2.5 2.9 3.3 3.7
y = 0.648 -0.445 0.287 -0.167 0.078 ...
-0.015 -0.028 0.056 -0.072 0.079
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
User defined ...
... more
... more
Function handles
Program Control
User defined functionsI Unlike scripts, functions can take arguments and return values
I Like scripts, MATLAB functions reside in M-files (*.m)
I Name of the file = name of the function
I Example of function file mycomplex.m
mycomplex.m
function z = mycomplex(x,y)
z = x + i*y;
end
here x and y are arguments, z is returned value,and mycomplex is the name of the function
I Call it from command line or from another function
>> c = mycomplex(a,b)
Introduction toMATLAB
Math 375
Getting started
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Matrices
Plotting
Functions
User defined ...
... more
... more
Function handles
Program Control
User defined functions (cont.)I Functions can take matrices as arguments
I Functions can return multiple values
I You can put several functions in a single M-file.
I Example: function file myplot.m
myplot.mfunction myplot
[x,y] = mydata(n);
plot(x,y);
end
function [x,y] = mydata(n)
x = linspace(0, 2*pi, n);
y = sin(x);
end
I Can we call mydata(n) from command line?
Introduction toMATLAB
Math 375
Getting started
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Matrices
Plotting
Functions
User defined ...
... more
... more
Function handles
Program Control
User defined functions (cont.)I Example: function file mycart2pol.m
mycart2pol.m
function[ radius, theta ] = mycart2pol(x, y)
% convert cartesian coordinares (x, y)
% to polar coordinates (r, theta)
theta = atan2(y, x);
radius = sqrt(x.^2 + y.^2);
end
I What happens when you call mycart2pol as following?
>> mycart2pol(1,1)
>> r = mycart2pol(1,1)
>> [r th] = mycart2pol(1,1)
>> [r th] = mycart2pol([-1 -2 -3], [0 0 0])
>> help mycart2pol
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
User defined ...
... more
... more
Function handles
Program Control
Function handlesI There are a number of useful MATLAB functions which take
function handle as an argumentI Example: to numerically evaluate integral∫ 2
1
x3dx =1
4
(24 − 14
)= 3.75
using MATLAB function quad we first create file cube.m
cube.mfunction y = cube(x)
y = x.^3;
end
I and then pass the function handle (@cube) and integrationlimits (1,2) to quad
>> quad(@cube, 1, 2)
I or create inline function, defined by the string
>> f = inline(’x.^3’);
>> quad(f, 1, 2)
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
For loopI The for loop executes a statement or group of statements a
predetermined number of times.
for x = x1 : ∆x : xnstatement
end
I The increment ∆x can be omitted; default ∆x = 1
for x = x1 : xnstatement
end
I The index of a for loop can be an array.
v=v1 : ∆v : vnfor x = v
statementend
I For loops can be nested
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
For loop: examples
>> format compact
>> for n = 1:5
disp(n)
end
>> for n = 0.5:0.1:0.8
disp(n)
end
>> v = [2, 4, 8, 16];
>> for n = v
disp(n)
end
>> for j = 1:3
for i = 1:3
A(i,j) = i+j;
end
end
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
For loop: examples
Create vector [1 2 3 4 5] in a “classical” way
>> v = zeros(1,5) % allocate space
>> for i=1:5 % cycle through the elements
v(i)=i % assign value to each element
end
Note: allocation is optional (but it makes the code run faster).Try the following:
>> clear v
>> for i=1:5
v(i)=i
end
>> v = zeros(1,5)
>> a = v(8)
>> v(8) = 8
Introduction toMATLAB
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Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
While loop
The while loop executes a statement or group of statementsrepeatedly as long as the controlling expression is true
while conditionstatement
end
>> x=1;
>> while x > 0.01
x = x/2
end
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
If statement
If evaluates a logical expression and executes a group ofstatements based on the value of the expression.
if conditionstatement
end
Example: find the maximum element in an array:
>> v=rand(1,6)
>> vmax = -1;
>> for i=1:length(v)
if v(i) > vmax
vmax = v(i);
end
end
>> disp(vmax)
But, of coarse, it is easier just to do
>> vmax = max(max(v))
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
If-else statement
if-else can take care for both branches of the fork...
if condition1statement1
elsestatement2
end
... or multiple branches
if condition1statement1
elseif condition2statement2
elseif condition3statement3
elsestatement4
end
Introduction toMATLAB
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Getting started
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Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
If-else example
Script rps.m simulates a rock-paper-scissors player
rpm.s
% simulate a rock-paper-scissors player
x = rand(1) % get random number between 0 to 1
x = x*3 % rescale x so that 0 < x < 3
if x<1
disp(’rock!’)
elseif x<2
disp(’paper!’)
else
disp(’scissors!’)
end
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
Other program control statements
continue passes control to the next iteration of the loop
break terminates the execution of a loop
return exits the currently running function
switch executes statements based on the value of a variable orexpression
try, catch error control statements
Note: there is no goto statement in MATLAB
Introduction toMATLAB
Math 375
Getting started
Arithmetics
Matrices
Plotting
Functions
Program Control
For loop
For loop: examples
For loop: examples
While loop
If statement
If-else statement
If-else example
Other statements
Simple text I/O
Simple text input and outputI C-style output to the screen
>> fprintf(’square root of %d is %6.3f\n’, 3, sqrt(3))
square root of 3 is 1.732
I The same output can be directed to a file
>> fid=fopen(’mydata.txt’, ’wt’);
>> fprintf(fid, ’%s\n’, ’% k sqrt(k)’);
>> for k=1:3
fprintf(fid, ’%d %6.3f\n’, k, sqrt(k));
end
>> fclose(fid);
I An easy way to load text data
>> load ’mydata.txt’
>> disp(mydata)
1.0000 1.0000
2.0000 1.4140
3.0000 1.7320
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