Introduction to Matlab

The purpose of this intro is to show some of Matlab's basic capabilities.

Nir Gavish, 2.07

Contents

Getting help

Matlab's documentation is % accesable by pressing F1 in Matlab or via the net at

http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html

In addition, for specific command type

help sin

 SIN    Sine of argument in radians.
    SIN(X) is the sine of the elements of X.
 
    See also ASIN, SIND.

    Overloaded functions or methods (ones with the same name in other directories)
       help darray/sin.m
       help sym/sin.m

    Reference page in Help browser
       doc sin

or for a graphical reference of the same help

doc  sin

Matlab development enviroment

Matlab includes a full development enviroment which is composed of

edit mfileName

See this movie for more features of Matlab's development enviroment http://www.mathworks.com/products/demos/shipping/matlab/WhatsNew_1DevEnviro_viewlet_swf.html?product=ML

Variable definitions

Matlab variables are defined by assigment. There is no need to declare in advance the variables that we want to use or their type.

% Define the scalar variable x
x=1
% Now a (row) vector
y=[1 2 3]
% and a column vector
z=[1;2;3]
% Finally, we define a 3x3 matrix
A=[1 2 3;4 5 6;7 8 9]
% List of  the variables defined
whos

x =
     1
y =
     1     2     3
z =
     1
     2
     3
A =
     1     2     3
     4     5     6
     7     8     9
  Name      Size            Bytes  Class     Attributes

  A         3x3                72  double              
  ans       1x59              118  char                
  x         1x1                 8  double              
  y         1x3                24  double              
  z         3x1                24  double              

Mathematical operations

The basic mathematical operators of Matlab work with scalar, vector and matrices. Any combination works, as long as it is mathematically possible.

% Adding one to a scalar
result1=x+1
% Multiply a vector by a scalar
result_x_times_y=x*y
% Vector multiplication, same in syntax as any other multiplication
result_y_times_z=y*z
% Notice that vector multiplication is not commutative
result_z_times_y=z*y
% More mathematical operators
result=(x+1)/2-3*x^2

result1 =
     2
result_x_times_y =
     1     2     3
result_y_times_z =
    14
result_z_times_y =
     1     2     3
     2     4     6
     3     6     9
result =
    -2

Term by term operations

As noted, mathematical operators such as multiplication (*), division (/) or power (^) work between vectors or matrices. In many cases, however, we would like to preform element-wise operations between the two operands. For example, raise to the power of two each term of a matrix, as opposed to multiplying the matrix by itself. This is implemented by 'term by term operations' in Matlab - '.*', './', '.^'.

original_matrix=A
% Here we use the classical power operator ^ - which multiplies the matrix by itself
classical_power_operator=A^2
% Now we use the term-by-term power operator .^ (notice the dot) - which multiplies _each term_ of the matrix by itself
term_by_term_power_operator=A.^2

original_matrix =
     1     2     3
     4     5     6
     7     8     9
classical_power_operator =
    30    36    42
    66    81    96
   102   126   150
term_by_term_power_operator =
     1     4     9
    16    25    36
    49    64    81

More complicated vector definitions - the semicolon operator

Clearly, the definition of vectors by explicitly stating its terms is impractical for vectors with more than a few terms. A better approach is to use the semicolon (:) operator which defines a range of values. Notice that for long enough vector it is recommended to suppress the output to the command window by using ';' .

% Define vector [1 2 3 4 5]
x=1:5
% Define spacing different than one
x=1:0.125:5
% Suppress output
x=1:0.125:5;

x =
     1     2     3     4     5
x =
  Columns 1 through 8
    1.0000    1.1250    1.2500    1.3750    1.5000    1.6250    1.7500    1.8750
  Columns 9 through 16
    2.0000    2.1250    2.2500    2.3750    2.5000    2.6250    2.7500    2.8750
  Columns 17 through 24
    3.0000    3.1250    3.2500    3.3750    3.5000    3.6250    3.7500    3.8750
  Columns 25 through 32
    4.0000    4.1250    4.2500    4.3750    4.5000    4.6250    4.7500    4.8750
  Column 33
    5.0000

Vector functions and operators

Here are some Matlab operations and functions defined for vectors. Notice that many of these functions can be implemented by a simple loop in our program. It is, however, significantly faster to use Matlab's vector function than to use loops in Matlab.

%The technique of converting a loop in a Matlab program to vector operations is called _'vectorization'_ and is fundamental in preformance improvement in Matlab.

% Transpose of a vector\matrix
y_transpose=y'
% note that B' is Hermitian conjugate, and B.' is transpose
B = [1 2 ; 2*i i]
B'
B.'

y_transpose =
     1
     2
     3
B =
   1.0000             2.0000          
        0 + 2.0000i        0 + 1.0000i
ans =
   1.0000                  0 - 2.0000i
   2.0000                  0 - 1.0000i
ans =
   1.0000                  0 + 2.0000i
   2.0000                  0 + 1.0000i

%Accessing a specific term of a vector (first term is indexed one, not zero)
y2=y(2)
% A partial list of vector functions
% sum(y)  = sum all the values of the vector y
res_sum =  sum(y)
 % prod(y) = multiply all the values of the vector y
res_prod =  prod(y)
% diff(y) = [y(2)-y(1),y(3)-y(2), ..., y(n)-y(n-1)]
res_diff =  diff(y)

y2 =
     2
res_sum =
     6
res_prod =
     6
res_diff =
     1     1

Matlab ("continuous") functions

Numerically, we cannot represent a general continuous function (x,f(x)) because it requires handling infinite data (for each point in the range, we need to keep f(x)). Instead, we represent a function by its values at a finite number of data points (x_i,f(x_i)), where the series of points {x_i} is typically referred to as the sampling points or the grid points. Accordingly, the "continuous" functions in Matlab accepts a vector of point {x_i} and return a vector of values {f(x_i)}. We note that opposed to the numerical approach is the symbolic approach, which is the approach you know from all the basic math classes.

% define the grid {1,1.1,1.2...4.9,5} using the semicolon operator
x=1:0.1:5;
% f(x) = x^2/(4+x), notice the use of *term-by-term operators*
f1  =  x.^2./(4+x);
% sqrt(x) = x^(1/2)
f2 =  sqrt(x + x.^3);
% Note: MATLAB doesn't define the constant `e'.  Use exp(1) to get e.
f3  =  exp(x);
% Note: in MATLAB log() means ln() (i.e., log in base e).
f4  =  log(x+4);
% `pi' is a matlab constant.  Note: sin, cos , etc. are in radians (NOT in degrees!)
f5  =  cos(pi)*tan(x);
% abs(x) := |x|
f6  =  abs(f5);
% sign(x) gives -1 for x<0, 0 for x=0, and +1 for x>0
f7  =  sign(f5);

Plotting graphs

Matlab is well-known for its plotting capabilites and for their simplicity of use. We now go over the most basic plot command and its features - plot(x,y) which plots the data points {x_i,f(x_i)}

% Define the grid and the "continuous" function sin
x=0:0.01:2*pi;
y=sin(x);
% Plot the points (x,sin(x))
plot(x,y,'.');

As you see, the plot command displays a graph of (x,sin(x)) which looks continuous. However, a closer look at the data shows it is not really continuous.

The default command connects every two points, making the graph look continuous.

% Close the last graph to clean up the prior setting of double axes
close all;
% Plot using the default option
plot(x,y);

In order to make sense and be useful, graphics can typically be zoomed, and should always have descriptions of the X-axis and Y-axis, and a title:

plot(x,y);
% Set the axis boundaries. Note: The data should not touch the axis, therefore the y axis is set to be -1.05 to 1.05.
axis([0 2*pi -1.05 1.05])
% Add a label for the x-axis
xlabel('x');
% Add a label for the y-axis
ylabel('sin(x)');
% Add a title for the y-axis
title('sine function graphic')

Plotting multiple graphs together

Often we need to plot more than one function, for example - to compare the output of two processes or get an easy look at various measures at once. There are two ways to plot multiple functions:

% add the cos graph to the exisiting plot
% tell Matlab to hold the graph for the next plot
hold
%  plot the additional graph, the additional parameter 'r' is for 'red'
plot(x,cos(x),'r');
% add a legend for the two plots
legend('sin(x)','cos(x)')
% correct the title
title('sin and cos graph')

Current plot held

We now split the page into several axes by the subplot command Its syntax is subplot(row_num,col_num,curr_plot)

% split the screen into 2 rows and 3 columns of axes, set the next plot to be at the first axes
subplot(2,3,1);
% plot a graph of x
plot(x,x);
% Now go to the second axes
subplot(2,3,2);
% plot a graph of x^2
plot(x,x.^2);
% Next, plot a graph of sin in the fifth axes.  Notice, all editing of the graph applies only to the current axes
subplot(2,3,5);
% plot a graph of sin(x)
plot(x,y);
% set axis
axis([0 2*pi -1.05 1.05])
% Add a label for the x-axis
xlabel('x');
% Add a label for the y-axis
ylabel('sin(x)');
% Add a title
title('a sinus graph')

Examples of more sophisticated graphics

The number of line codes needed to produces these graphs is no more than 10-20 lines.

cruller;
logo;figure;
spharm2;

Flow control

Here I just give examples for the most basic flow control commands. For more info see http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_prog/

Conditional control

a=4;
if a==5
    a=a+1
else
    a=3
end

a =
     3

Loop Control

for ix=1:3
    a=a+ix
end

a =
     4
a =
     6
a =
     9

Notice that I name the enumerator index 'ix' and not 'i' or 'j'. This is because 'i' and 'j' are the complex imaginary numbers, e.g.,

i_square=i^2

i_square =
    -1

Saving results

We can save all our results for future reference. Here we discuss three different objects:

In this case, we can only save future output to the command window. The command

diary 'MyCommandWindow'

saves all output to command window into the .txt file 'MyCommandWindow' until this option is turned off by the command

diary off

The following commands save & load the entire workspace into the .mat file 'MyMatFile'

save 'MyMatFile'
load 'MyMatFile'

To save a specific variable, use

save 'x.mat' x
%

saving in ascii format:

x = (-1:0.4:1)'
y = sin(x*pi)
d = [x y]  % double-column
save 'my_sin.dat' -ascii -double d

x =
   -1.0000
   -0.6000
   -0.2000
    0.2000
    0.6000
    1.0000
y =
   -0.0000
   -0.9511
   -0.5878
    0.5878
    0.9511
    0.0000
d =
   -1.0000   -0.0000
   -0.6000   -0.9511
   -0.2000   -0.5878
    0.2000    0.5878
    0.6000    0.9511
    1.0000    0.0000

The following commands save the current figure

% save as jpeg (not optimal for graphs, good compression)
print -djpeg 'myPic.jpeg'
% save as tiff (much better for graphs, more space)
print -dtiff 'myPic.tif'

Cleaning up

Since memory is not erased at the begging and end of a script, it is a good habit to clean up before and after the script run.

% Close all plot windows
close all;
% Erase the command window (You can still see the last command in the 'command history' window)
clc;
% Clear all variables from the workspace.
clear all;

More references on the web

Tutorials on Matlab's site: https://matlabacademy.mathworks.com

Cleve's Moler (free online) book http://www.mathworks.com/moler/chapters.html

Exercise 1

Do this by taking the sum of only 100 terms using Matlab's 'sum' command. Use the Matlab constant 'pi' to calcuate the accuracy of your calculation.

Exercise 2

In this exercise you will write a program that calculates the factorials 1!, 2!, 3! ... 15! in three different ways

Exercise 3

In this exercise you produce a graphical example of the accuary of a Taylor series

Exercise 4

The Legendre polynomials (Pn(x)) are defined by the following recurrance relation

with

% That's it, let's clean up
close all;clc;clear all;

Published with MATLAB® 7.4