INTRODUCTION TO
MATLAB
Victoria Lapuerta
Ana Laverón
Index
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Introducción
Basic elements of Matlab’s
desktop
MATLAB editor
Numbers and operations
Vectors and matrices
Operations with vectors and
matrices
Functions for vectors and
matrices
Data Input and output
Data structures and cell
matrices
Polynomials
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2D and 3D graphics (I)
2D and 3D graphics (II)
2D and 3D graphics (III)
Creating movies
Matlab files
Functions for functions
Programming
Numerical analysis
Exercices
Introduction

What is Matlab? MATrix LABoratory.

MATLAB is a numerical computing environment and programming
language (initially written in C). MATLAB allows easy matrix
manipulation, plotting of functions and data, implementation of
algorithms, creation of user interfaces, and interfacing with programs
in other languages.

MATLAB makes mathematical operations with vectors y matrices.
As a particular case, it can also work with scalar numbers, both reals
and complexes.

It has packages with specialized functions.
Basic elements of Matlab’s desktop

Command Windows: Where all commands and programs are run.
Write the command or program name and hit Enter.

Command History: Shows the last commands run on the
Command Windows. A command can be recovered clicking twice

Current directory: Shows the directory where work will be done.

Workspace: To see the variables in use and their dimensions (if
working with matrices)
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Help (can also be called from within the comand windows)

Matlab Editor: All Matlab files must end in the .m extension.
Basic elements of Matlab’s desktop
Current
directory
Command
Windows
Command
History
Basic elements of Matlab’s desktop
Some comments about the command window

Commands can be retrieved with arrow up / arrow down keys ↓↑

Moving around the command line is possible with left / right arrow
keys → ←. Go to the beginning of the line with Inicio (Home) and
to the end with Fin (End). Esc deletes the whole line.

A program can be stopped with Ctrl+c
Matlab editor
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There can not be empty spaces in the name of the Matlab files
Use “main_” for the name of the main programs, for example:
main_curvature
Write “;” at the end of a line If you don’t want that the intermediate
calculus is written in the window while the program is running
Write “%” at the beginning of a line to write a comment in the
program
Write “…” at the end of a line if you are writing a very long statement
and you want to continue in the next line
Matlab editor
Debugger
Set/Clear breakingpoint: Sets or clears a break point in the
line the cursor is placed.
Clear all breakingpoints: Deletes all breaking points.
Step: Executes the current line of the program.
Step in: Executes the current line of the program, if the line
calls to a function, steps into the function.
Step out: Returns from a function you stepped in to its calling
function without executing the remaining lines individually.
Continue: Continues executing code until the next breaking
point
Quit debugging: Stops the debugger
Numbers and operations
Numerical Data:

Variables are defined with the assignment operator “=“. MATLAB is dynamically typed,
meaning that variables can be assigned without declaring their type, and that their
type can change. There is no need to define variables as integers, reals, etc, as in
other languages
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Integers: a=2
Reals: x=-35.2
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Maximum 19 significant figures

2.23e-3=2.23*10-3
Precision and formats: By defect, it uses a short format defect, but other formats
can be used:
>> format long (14 significant figures)
>> format short (5 significant figures)
>> format short e (exponential notation)
>> format long e (exponential notation)
>> format rat (rational approximation)
See in File menu: Preferences → Command Windows
Numbers and operations
Preferences (in File menu)
Numbers and operations
Numerical data:
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They’re case sensitive: x=5, X=7
Information about the variables used and their dimensions (if they’re
matrices): Workspace. Also typing
>> who
>> whos (gives more information)
To delete a variable (or several), run:
>> clear variable1 variable2

To delete all the variables, run:

Characteristic constants: pi=, NaN (not a number, 0/0), Inf=.

>> clear
Complex numbers: i=sqrt(-1) (only i or j can be used), z=2+i*4, z=2+4i
 Careful not to use ‘i’ or “j” afterwards as a counter for a loop when
working with complex numbers.
Numbers and operations
Basic Arithmetic Operations:

Addition: +, Substraction -

Multiplication: *, Division: /

Power: ^

Priority Order: Power, division and multiplication, and lastly addition
and substraction. Use () to change the priority.

Example: main_number_operations.m. Try the Debugger
Numbers and operations
Matlab Functions:

exp(x), log(x) (base e), log2(x) (base 2), log10(x) (base 10),
sqrt(x)

Trigonometric functions: sin(x), cos(x), tan(x), asin(x), acos(x),
atan(x), atan2(x) (entre –pi y pi)

Hyperbolic functions: sinh(x), cosh(x), tanh(x), asinh(x), acosh(x),
atanh(x)
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Other functions: abs(x) (absolute value), int(x) (integer part ),
round(x) (rounds to the closest integer), sign(x) (sign function)

Functions for complex numbers: real(z) (real part), imag(z)
(imaginary part), abs(z) (modulus), angle(z) (angle), conj(z)
(conjugated)
Example: main_number_operations.m
Vectors and matrices
Defining vectors:

Row vectors; elements separated by spaces or comas
>> v =[2 3 4]

Column vectors: elements separated by semicolon (;)
>> w =[2;3;4;7;9;8]

Length of a vector w: length(w)

Generating row vectors:
 Specifying the increment h between the elements v=a:h:b
 Specifying the dimension n: linspace(a,b,n) (by default n=100)
 Elements logarithmically spaced logspace(a,b,n) (n points
logarithmically spaced between 10a y 10b. By default n=50)
Example: main_matrix_operations.m
Vectors and matrices
Defining matrices:

It’s not needed to define their size before hand (a size can be
defined and changed afterwards).

Matrices are defined by rows; the elements of one row are
separated by spaces or comas. Rows are separated by semicolon
(;).
» M=[3 4 5; 6 7 8; 1 -1 0]

Empty matrix: M=[ ];

Information about an element: M(1,3), a row M(2,:), a column M(:,3).

Changing the value of an element: M(2,3)=1;

Deleting a column: M(:,1)=[ ], a row: M(2,:)=[ ];

Example: main_matrix_operations.m
Vectors and matrices
Defining matrices:
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Generating de matrices:
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Generating a matrix full of zeros, zeros(n,m)
Generating a matrix full of ones, ones(n,m)
Initializing an identity matrix eye(n,m)
Generating a matrix with random elements rand(n,m)
Adding matrices: [X Y] columns, [X; Y] rows
Example: main_matrix_operations.m
Operations with vectors and matrices
Operating vectors and matrices with scalars:
v: vector, k: scalar:
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v+k addition
v-k sustraction
v*k product
v/k divides each element of v by k
k./v divides k by each element of v
v.^k powers each element of v to the k-power
k.^v powers k to each element of v
Example: main_matrix_operations.m
Operations with vectors and matrices
Operating vectors and matrices
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+ addition
– subtraction
* matrix product
.* product element by element
^ power
.^ power element by element
\ left-division
/ right-division
./ y .\ right and left division element by element
Transposed matrix: B=A’ (in complex numbers, it returns the
conjugated transposed, to get only the trasposed: B=A.’)
Example: main_matrix_operations.m
Functions for vectors and matrices
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sum(v) adds the elements of a vector
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prod(v) product of the elements of a vector
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dot(v,w) vectors dot product
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cross(v,w) cross product
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mean(v) (gives the average)
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diff(v) (vector whose elements are the differenceof the elements of v)
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[y,k]=max(v) maximum value of the elements of a vector (k gives the
position), min(v) (minimum value). The maximum value of a matrix M is
obtained with max(max(M)) and the minimum with min(min(v))

Some of these operations applied to matrices, give the result by
columns.
Functions for vectors and matrices
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[n,m]=size(M) gives the number of rows and columns
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Inverted matrix: B=inv(M), rank: rank(M)
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diag(M): gives the diagonal of a matrix. sum(diag(M)) sums the
elements of the diagonal of M. diag(M,k) gives the k-th diagonal.
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norm(M) norm of a matrix (maximum value of the absolute values of
the elements of M)
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flipud(M) reorders the matrix, making it symmetrical over an
horizontal axis. fliplr(M) ) reorders the matrix, making it symmetrical
over a vertical axis.
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[V, landa]=eig(M) gives a diagonal matrix landa with the eigen
values, and another V whose columns are the eigenvectors of M
Example: main_matrix_operations.m
Data input and output

Saving to files and recovering data:

save –mat file_name matrix1_name, matrix2_name

load –mat file_name matrix1_name, matrix2_name
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save file_name matrix1_name –ascii (saves 8 figures after the
decimal point)
save file_name matrix1_name –ascii –double (saves 16 figures
after the decimal point)
Example: main_matrix_operations.m
Data structures and cell matrices

Matlab allows store variables with a tree structure:
>> structure1.data1= value of data 1
>> structure1.data2=value of data 2
>> structure1.data3.subdata31=value of subdata 31
>> structure1.data3.subdata32=value of subdata 32
estructura1
dato1
dato2
dato3
subdato31

subdato32
Matlab allows store different variables in cell matrices:
>> cell_matrix1= {data1 data2; data3 data4}
Example: main_data_structure.m
data1
data2
data3
data4
Polynomials

Polynomials are written in Matlab as a row vector whose dimension
is n+1, n being the degree of the polynomial.
Example: x3+2x-7 is written:
>> pol1= 1 0 2 -7 

Obtaining the roots: roots (returns a column vector, even though
pol1 is a row vector)
>>roots_data=roots(pol1)
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A polynomial can be reconstructed from its roots, using the
command poly
>> p=poly(roots_data) (returns a row vector)

If the input for poly is a matrix, the output is the characteristic
polynomian of the matrix
Example: main_polynomials.m
Polynomials
Matlab functions for polynomials

Calculate the value of a polynomial p in a given point x: polyval
>>y=polyval(p,x)
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Multiplying and dividing polynomials: conv(p,q) y deconv(p,q)
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Calculate the derivative polynomial: polyder(p)
2D and 3D Graphics (I)
Basic 2D and 3D Graphic Functions

2D: plot() creates a graphic from vectors, with linear scales on both
axes,
>> plot(X,Y,’option’) (option: allows chosing color and stroke
of the curve)
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hold on: allows to draw more graphics on the same figure
(deactivate with hold off)
grid activates a grid on the drawing. Writing grid again deactivates it.
2D: loglog() logarithmic scale on both axes, semilogx():
logarithmic scale on the abcises axis, and linear on the ordinates
axis, semilogy(): linear scale on abscises and logarithmic on
ordinates.
Example: main_graphics.m, and see in Demos: Graphics
2D and 3D Graphics (I)
Basic 2D and 3D graphic functions
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2D: subplot(n,m,k) divides a drawing window in n horizontal parts
and m vertical parts, where k is the activated partition.
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2D: polar(angle,r) to draw in polars
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2D: fill(x,y,’option’) draws a closed curve and fills it with the color
indicated in ‘option’
3D: plot3 is similar to the 2D plot.
» plot3(X,Y,Z, ’option’)
2D and 3D Graphics(I)
Basic 2D and 3D graphic functions
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2D: ezplot(f) simplified graphic functions. By default [–2π ≤ x ≤ 2π]

ezplot(f,[a,b]) plots f in a different interval

f can be an implicit function f(x,y)=0.
>> ezplot(f); % plots f(x,y)=0 in -2*pi<x<2*pi and -2*pi<y<2*pi
>> ezplot(f, [a,b]); % plots f(x,y)=0 in a<x<b and a<y<b
>> ezplot(f, [xmin,xmax,ymin,ymax]);

ezplot can plot parametric functions x(t), y(t):
>> ezplot('sin(t)','cos(t)'); % plots with 0<t<2*pi
>> ezplot('sin(t)','cos(t)', [t1,t2]); % plots with t1<t<t2
2D: ezpolar(f) to plot in polar coordinates
3D: ezplot3 the same idea as ezplot but 3D
Example: main_graphics.m
2D and 3D Graphics (I)
Selecting the axes scale

axis([x0 x1 y0 y1]) (2D), axis([x0 x1 y0 y1 z0 z1]) (3D)
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axis auto: returns to the default scale
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axis off: deactivates the axes labels. Axes, labels and grid disappear, axis
on: activates it again.
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axis equal: same scale factor for both axes

axis square: encloses the area delimited by the axes in a square.

To chose the labels on the axes:
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set(gca, ‘XTick’,-pi:pi/2,pi) %gca:get current axis
set(gca, ‘XTicklabel’,({‘-pi’,’-pi/2’,0,’pi/2’,’pi’})
2D and 3D Graphics (I)
Functions to add titles to the graphic

title('title') adds a title to the drawing. To include in the text the
value of a numerical variable, it has to be transformed with:

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int2str(n) converts the value of the integer n to a character
num2str(x) converts the value of a real or complex variable x to
a character. Example: title(num2str(x))
xlabel(‘text’) adds a label to the abscises axis. With xlabel off it
disapppears. Same with ylabel(‘text’) or zlabel(‘text’)
text(x,y,'text') places 'text‘ on the specific coordinates x and y. If x
and y are vectors, the text is placed on each pair of elements.
gtext('text') places text with help of the mouse.
2D and 3D Graphics (I)
Matlab functions for 2D and 3D graphics

Printing graphics: Print (File button on the graphic window)
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Saving graphics: Save (File button on the graphic window): A .fig
file is created, it can be re-edited and saved again.

Exporting graphics: Export (File button on the graphic window)
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figure(n): calls to a new figure or to a figure already done.

close all deletes all figures, close(figure(n)) deletes one figure in
particular.
2D and 3D Graphics (II)
Drawing surfaces
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Creating a grid from two vectors [X, Y]=meshgrid(x,y)
Drawing of the grid builton a surface Z(X,Y):
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Graphic of the surface Z(X,Y):
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mesh(X,Y,Z),
meshc(X,Y,Z) (also draws the level lines on the z=0 surface)
surf(X,Y,Z),
surfc(X,Y,Z)
pcolor(Z) draws the projection with colored shadows on the flat
surface (the color range is related to the variations of Z)
contour(X,Y,Z,v) and contour3(X,Y,Z,v) generate contour lines of
a surface for the values given in v. To label the lines, first
cs=contour(Z) (to know the contour values) and then clabel(cs) or
directly clabel(cs,v)
contourf(X,Y,Z,v): to fill with colours the space between contour
lines
ezsurf (f). Plots a 3D graphic of f(x,y). By default:–2 < x, y < 2.
Example: main_surface_graphics.m and see in Demos: Graphics
2D and 3D Graphics (II)
Drawing surfaces
 Different ways of drawing colored polygons:
 shading flat: shadows with flat color for each polygon.
 shading interp: shadows with interpolated colors between
corners of each polygon.
 shading faceted: flat shadowing with superposed black lines
(default option)

hidden off (deactivates the option of hidden lines), hidden on
(activates it)

Manipulating graphics:
 view(azimut, elev), view([xd,yd,zd])
 rotate(h,d,a) or rotate(h,d,a,o), ‘h’ is the object, ‘d’ is a vector
that gives the direction, ‘a’ an angle and ‘o’ the rotation origin
 In graphic window: View (camera toolbar)
2D and 3D Graphics (III)
Coordinate System Transformation
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[ang,rad]=cart2pol(x,y), from cartesians to polars
[ang,rad,z]=cart2pol(x,y,z), from cartesians to cylindric
[x,y]=pol2cart(ang,rad), from polars to cartesians
[x,y,z]=pol2cart(ang,rad,z), from cylindric to cartesians
[angx,angz,rad]=cart2sph(x,y,z), from cartesians to spheric
[x,y,z]=aph2cart(angx,angz,rad), from espheric to cartesians
Creating movies

A movie is made out of several images or frames

getframe is used to save all those images. It returns a column
vector with the required information for reproducing the image that
has been represented, for example with the plot function. These
vectors are stored in a matrix, M.

movie(M,n,fps) represents n times the movie stored in M at a fps
frames per second speed
X=0:0.01:2*pi;
for j=1:10
plot(x,sin(j*x)/2)
M(j)=getframe;
end
movie(M,4,6)

Example: main_movie.m
Matlab Files

Program files: Scripts
They are built with a series of commands. The main file will be named
main_name.m

Function files
To create your own functions. They are called from within the scripts.

The first line is executable and starts with the word function as showed:
function [output_arg1, output_arg2]=function_name(input_arg1,
input_arg2, …, parameters)


The file must be saved as function_name.m
Example: main_plot_sine.m. Use “Step in” in Debugger to enter this
function
Matlab Files


MATLAB allows the definition of functions directly from mathematic
expressions using function inline
By default 'x‘, ‘y’, … are the inputs, althought it is possible to define
them explicitly when we call the function inline.
>> function_1 = inline('cos(x)+2*sin(2*x)');
>> function_2 = inline('cos(x)+2*sin(2*y)‘,’x’,’y’);

Input and Output commands:


input: allows entering data: a=input(‘Type the value of a’);
disp: shows a text on screen: disp(‘The algorythm did not
converge’)
Functions for functions
Function references

They asign a function to a variable. The operator @ is used.
>> @reference_name=function_name

To evaluate a reference the function feval is used as follows
[r1, r2, r3, ...] = feval(reference_name, arg1, arg2, arg3, ...)

Function references are very useful to give a function to other functions. Functions
that execute other functions are called functions for functions.

Function references are variables of MATLAB, so they can be stored in matrices, for
example.
Example: main_functions_for_functions.m
Function for functions

fzero(@name_function,x0): Calculates the zero of a function
closest to the value of the variable x0

fminsearch(@name_function,x0): calculates the relative minimun
of a function closest to x0

fminbnd(@name_function,a,b): calculates a minimun of the
function in the interval [a,b]
Example: main_functions_for_functions.m
Programming
Loops
for k=n1:incre:n2
end
for k=vector_column
end
while
end
Example: main_loops
Programming
Conditional control structures

Logical operators:
 >, <, >=,<=,== (equal)
 | (or), &(and)
 ~ (no), ~= (not equal)
if
end
if
else
end
Example: main_conditional
if
elseif
else
end
Programming
Structures of control condicionated: switch

switch is similar to a sequence of if...elseif
switch_expresion=case_expr3 %example
switch switch_expresion
case case_expr1,
actions1
case {case_expr2, case_expr3,case_expr4,...}
actions2
otherwise, % option by default
actions3
end
Example: main_conditional
Programming
Interpolation

1D:
 A polynomial is defined (example, n=2, ax^2+bx+c),to
interpolate: p=polyfit(x,y,n). To obtain the interpolation on
certain values ’xi’: yi=polyval(p,xi).
 yi = interp1(x,y,xi,method). Methods: ‘linear’ (linear
interpolation), ’cubic’ (cubic), ’spline (cubic spline )

2D:
 matrix_Z=interp2(X,Y,Z,matrix_X,matriz_Y,method). Methods:
’bilinear’ (linear interpolation), ’bicubic’ (cubic)
Numerical Analysis
Integration

1D: quad, quadl: integrate a function in an interval [a,b]
quad(@name_function,a,b)

2D: dblquad: integrate a function in an interval
[xmin,xmax]x[ymin,ymax]
dblquad(@sin,xmin,xmax,ymin,ymax)
Numerical Analysis
Solving differential ecuations

Solving initial value problems for ordinate differential ecuations
(ODEs)

[T,Y]=solver(@F,tspan,Y0)




solver: algorythm to solve ODEs, ode45, ode23, ode113,
ode15s,ode23s.
F: function that has the differentical ecuations in matrix form
Tspan: times vector [t0 tfinal] for the integration.
Y0: column vector with the initial conditions in t0
Exercise I
Represent the functions:
y1= sin(3 π x)/ex
y2=cos(3π x)/ex
with x between 0 and 3 π,obtaining only one figure like:
Exercise II
a)
Solve the equation system:
3x+2y-z=1
5x+y+3z=-2
3y-4z=3
b)
Be A the coefficients matrix of the previous system. Obtain the
maximum eigenvalue of A and its associated eigenvector as the
program output.
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INTRODUCTION TO MATLAB