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Universitt Bern Kurt Schmidheiny/Manuel Wlti

Doing Economics with the Computer Summer Term 2002

11.04.2002 matlabguide.doc
Introduction into Matlab, Version 6.0.0
1 Introduction 3
2 Matlab basics 4
2.1 Prelimaries 4
2.2 Declaring variables 5
2.3 Basic manipulations of matrices 11
2.4 Matrix and array operations 13
2.5 Relational and logical operators 15
2.6 Building special matrices 17
2.7 More built-in functions 18
2.8 Getting Help 22
3 Graphics 24
3.1 Histogram 24
3.2 Plotting series versus their index 24
3.3 Scatterplots (2-D plots) 25
3.4 Adding titles, labels and grids 27
3.5 Creating subplots 28
3.6 3-D plots 29
3.7 Save graphs 30
4 Scripts and functions 31
4.1 Script M-files 31
4.2 Function M-files 32
5 Controlling the flow 36
5.1 The FOR-loop 36
5.2 The WHILE-loop 37
5.3 The IF-statement 38
Introduction intoMatlab, Version 6.0.0 2
5.4 How to break a FOR or WHILE-loop 39
6 Importing and exporting data 40
6.1 The diary command 40
6.2 Save workspace variables on disk 40
6.3 Retrieving data and results 41
6.4 Request user input 42
7 References 43
Introduction into Matlab, Version 6.0.0 3
1 Introduction
Matlab is a high level programming language for technical computing.
1
Its
basic data element is amatrix (the name Matlab therefore stands for matrix
laboratory). Since many models and solution procedures in economics are
naturally stated as a series of matrix operations, Matlab together with
programming languages of the same class is widely used in economics.
Within this course, we shall meet the programming language of Matlab and
learn how to use it, step by step. Matlab offers a huge amount of so-called
operators and functions. This manual introduces just a selection of them.
However, the Matlab Help library offers a complete overview on all existing
elements in Matlab. The software bundle also contains a so called Help desk
that is useable as an Internet page.
1
Comparable high level programming languages would be GAMS, GAUSS and
Mathematica. Lower level languages are Basic, Fortran, C, C++ and Java.
Introduction intoMatlab, Version 6.0.0 4
2 Matlab basics
2.1 Prelimaries
Do start the software as usual (double-click the respective icon). The Matlab
command window (some time called Matlab prompt) will be opened. For the
time being, we will work in this window only. You will, however, later on get
to know the Matlab editor/debugger as well.
As you work in the command window, Matlab remembers the commands you
enter as well as the values of any variables you create. These commands and
variables are said to reside in the Matlab workspace.
Some times it is useful to delete variables from the workspace. Just tape
clear
and all variables defined so far are gone.
As in the example above, Matlab code is written in Courier. There is a
command after every . If you want to compute a command, press Enter
(Return key) and the result will appear in the command window. After a
semicolon (;) at the end of a command line, Matlab will not display the
executed command:
2+3;

in contrast to
2+3
5
Introduction into Matlab, Version 6.0.0 5
Preceding commands can be re-displayed by pressing the up-cursor.
When Matlab displays numerical results, it follows several rules. By default,
Matlab displays a real number with approximately four digits to the right of
the decimal point. If the significant digits in the result are outside this range,
Matlab displays the result in scientific notation, similar to scientific
calculators. If you prefer to see numerical results up to 15 places behind the
dot, use the command
format long
In order to put back the default setting type
format short
To get the current working directoryor folder displayed, type
cd
cd directory-name sets the current directory to the one specified. cd ..
moves to the directory above the current one. The command dir provides a
list of all files in the current directory (or folder).
To interrupt a running execution, type Ctrl+C.
2.2 Declaring variables
In Matlab each variable is a matrix (or rectangular data array). Since a matrix
contains m rows and n columns, it is said to be of dimension m-by-n. A m-by-
1 or 1-by-n matrix is called avector. A scalar, finally, is a 1-by-1 matrix.
Introduction intoMatlab, Version 6.0.0 6
2.2.1 Building a matrix
In order to build a matrix you have several options. One way to do it is to
declare a matrix
as you would write it by hand
A=[1 2 3
4 5 6]
A =
1 2 3
4 5 6
by separating rows with ;
B=[4 5 6; 7 8 9]
or element by element
C(1,1)=3;
C(1,2)=4;
C(1,3)=5;
C(2,1)=6;
C(2,2)=7;
C(2,3)=8;
You can give amatrix whatever name you want. Numbers may be included.
(Matlab is sensitive with respect to the upper and lower case.)
If a variable has previously been assigned a value, the new value overrides the
predecessor.
Suppose you want to look at variable C. All you have to do is typing either
C
C =
3 4 5
6 7 8
Introduction into Matlab, Version 6.0.0 7
or
disp(C)
3 4 5
6 7 8
In general, when you enter C, Matlab does the following: It checks to see if
C is a variable in the Matlab workspace; if not it checks to see if C is a built-
in-function; if not it checks if an M-file named C.m exists in the current
directory; if not it checks to see if C.m exists anywhere on the Matlab search
path, by searching the path in the order in which it is specified. Finally, if
Matlab cant find C.m anywhere on the Matlab search path either, an error
message will appear in the command window.
2.2.2 Declaring an empty matrix
In Matlab, there is no need to declare a variable before assigning a value to it
but it is recommended.
D=[]
D =
[]
2.2.3 Building a matrix out of several submatrices
You can build matrices out of several submatrices. Suppose you have
submatrices A to D.
A=[1 2; 3 4];
B=[5 6 7; 8 9 10];
C=[3 4; 5 6];
D=[1 2 3; 4 5 6];
Introduction intoMatlab, Version 6.0.0 8
In order to build E, which is given by
]
]
]

D C
B A
E , you type:
E=[A B; C D]
E =
1 2 5 6 7
3 4 8 9 10
3 4 1 2 3
5 6 4 5 6
2.2.4 Declaring a string matrix
A variable in Matlab is one of two types: numeric or string. A string matrix is
like any other, except the elements in it are interpreted as ASCII numbers. To
create a string variable, we enclose a string of characters in apostrophes.
Since a string variable is in fact a row vector of numbers, it is possible to
create a list of strings by creating a matrix in which each row is a separate
string. Note: As with all standard matrices, the rows must be of the same
length.
x=['ab';'cd']
x =
ab
cd
x = ['ab' 'cd']
x =
abcd
2.2.5 Declaring a vector
A vector is simply a 1-by-n or m-by-1 matrix.
f=[1 2 3]
Introduction into Matlab, Version 6.0.0 9
f =
1 2 3
Matlab creates a vector always as a row (or 1-by-n) vector. However, it is
very easy to create a column (or n-by-1) vector out of it. All you have to do is
adding aapostrophe (transposition of a matrix - compare matrix operations
further below).
f=[1 2 3]'
f =
1
2
3
You can declare a vector by indicating an initial value, the step and afinal
value.
g=[1:0.5:5]
g =
Columns 1 through 5
1.0000 1.5000 2.0000 2.5000 3.0000
Columns 6 through 9
3.5000 4.0000 4.5000 5.0000
No step means step 1.
x=[3:5]'
x =
3
4
5
Introduction intoMatlab, Version 6.0.0 10
2.2.6 Declaring a scalar (or 1-by-1 matrix)
A scalar is given by a 1-by-1 matrix. E.g., consider the following single
number:
a=2
2.2.7 Managing the Matlab workspace
To see what variable names are in the Matlab workspace, issue the command
who:
who
A B C
For more detailed information, use the command whos:
whos
Name Size Bytes Class
A 2x3 48 double array
B 2x3 48 double array
C 2x3 48 double array
Grand total is 18 elements using 144 bytes
or check the workspace window.
As shown earlier, the command
clear
deletes variables from the Matlab workspace.
Introduction into Matlab, Version 6.0.0 11
2.3 Basic manipulations of matrices
2.3.1 Using partions of matrices
One of the most basic operation is to extract some elements of a matrix
(called a partition of matrices). Consider
A =
1 2 3 4 5
2 3 4 5 1
3 4 5 1 2
4 5 1 2 3
5 1 2 3 4
In paragraph 2.2.1 Building a matrix weve learnt, that e.g. the element a
23
of
Matrix A is called A(2,3) in Matlab. Lets now isolate the central matrix
B =
2 3
3 4
4 5
5 1
In order to do this we type
B=A(1:4,2:3)
B =
2 3
3 4
4 5
5 1
Introduction intoMatlab, Version 6.0.0 12
Suppose we just want to select columns 1 and 3, but take all the lines.
B=A(:,[1 3])
B =
1 3
2 4
3 5
4 1
5 2
where (:) means select all.
2.3.2 Making vectors from matrices and reverse
Suppose that we want to obtain the vectorialization of a matrix, that is you
want to obtain vector B from matrix A.
A =
1 2
3 4
5 6
B=A(:)
B =
1
3
5
2
4
6
Suppose we have a vector B. We want to obtain matrix A from B.
A=reshape(B,3,2)
Introduction into Matlab, Version 6.0.0 13
A =
1 2
3 4
5 6
2.4 Matrix and array operations
2.4.1 Transposition of a Matrix
C =
3 4 5
6 7 8
C'
ans =
3 6
4 7
5 8
Note: The ans variable is created automatically when no output argument is
specified. It can be used in subsequent operations.
2.4.2 Basic matrix operators
+ - Addition and subtraction
* Multiplication
/ right division
^ Exponentiation
E.g., consider the vectors a and b and a scalar i:
a=1:5;
b=[6:10]';
i=3;
The vector d and the matrix E are given by:
d=a+i*a
Introduction intoMatlab, Version 6.0.0 14
d =
4 8 12 16 20
E=b*a
E =
6 12 18 24 30
7 14 21 28 35
8 16 24 32 40
9 18 27 36 45
10 20 30 40 50
What, if the matrices involved in the operation are of incompatible sizes? Not
surprisingly, Matlab will complain with a statement such as:
E=a*d
??? Error using ==> *
Matrix dimensions must agree.
2.4.3 Array Operators
To indicate an array (element-by-element) operation, precede a standard
operator with a period (dot). Matlab array operations include multiplication
(.*), division (./) and exponentiation (.^). (Array addition and subtraction
are not needed and in fact are not allowed, since they would simply duplicate
the operations of matrix addition and subtraction.) Thus, the "dot product" of
x and y is
x=[1 2 3];
y=[4 5 6];
x.*y
ans =
4 10 18
You may divide all the elements in one matrix by the corresponding elements
in another, producing a matrix of the same size, as in:
Introduction into Matlab, Version 6.0.0 15
C = A ./ B
In each case, one of the operands may be a scalar. This proves handy when
you wish to raise all the elements in a matrix to a power. For example:
x =
1 2 3
x.^2
ans =
1 4 9
2.5 Relational and logical operators
Matlab relational operators can be used to compare two arrays of the same
size or to compare an array to a scalar. In the second case, the scalar is
compared with all elements of the array, and the result has the same size as
the array.
< > Less (greater) than
<= >= Less (greater) than or equal to
== Equal to
~= Not equal to
Introduction intoMatlab, Version 6.0.0 16
Lets consider the following matrix G. The command G<=2 finds elements of
Gthat are less or equal 2. The resulting matrix contains elements set to logical
true (1) where the relation is true, and elements set to logical false (0) where it
is not.
G=[1 2; 3 4];
T=G<=2
T =
1 1
0 0
Suppose we want to select the elements that are greater or equal to 5 in the
matrixA on page 11 and store them in vector B. We do this in two steps. First
we tell Matlab to create a 5-by-5 matrix which contains 1, if an element in A
is greater or equal than 5 and 0 otherwise. Second we tell Matlab to collect all
the elements of A in a row vector, for which we got a logical true (1) answer
in the first step.
B=A(A>=5)
B =
5
5
5
5
5
Logical operators provide a way to combine or negate relational expressions.
Matlab logical operators include:
& AND
| OR
~ NOT
Introduction into Matlab, Version 6.0.0 17
An example is:
T=~(G<=2)
T =
0 0
1 1
2.6 Building special matrices
2.6.1 Declaring a zero (or Null) matrix
zeros(n) returns an n-by-n matrix of zeros. An error message appears if n is
not a scalar. zeros(m,n) or zeros([m n]) returns an m-by-n matrix of
zeros. Thus,
D=zeros(2,3)
D =
0 0 0
0 0 0
2.6.2 Declaring a ones matrix
E=ones(2,3)
E =
1 1 1
1 1 1
2.6.3 Declaring a identity matrix
F=eye(3)
F =
1 0 0
0 1 0
0 0 1
Introduction intoMatlab, Version 6.0.0 18
2.6.4 Declaring a random matrix
The rand function generates arrays of random numbers whose elements are
uniformly distributed in the interval (0,1).
R=rand(5,1)
R =
0.6154
0.7919
0.9218
0.7382
0.1763
The randn function generates arrays of random numbers whose elements are
normally distributed with mean 0 and variance 1.
N=randn(2,3)
N =
-0.4326 0.1253 -1.1465
-1.6656 0.2877 1.1909
2.7 More built-in functions
The type of commands used to build special matrices are called built-in
functions. There are a large number of built-in functions in Matlab. Apart
from those mentioned above, the following are particularly useful:
2.7.1 Display text or array
disp(X) displays an array, without printing the array name. If X contains a
text string, the string is displayed.
disp(' Corn Oats Hay')
Corn Oats Hay
Introduction into Matlab, Version 6.0.0 19
2.7.2 Sorting a matrix
sort(A) sorts the elements in ascending order. If A is a matrix, sort(X)
treats the columns of A as vectors, returning sorted columns. E.g.:
A=[1 2; 3 5; 4 3]
A =
1 2
3 5
4 3
sort(A)
ans =
1 2
3 3
4 5
2.7.3 Sizes of each dimension of an array
size(A) returns the sizes of each dimension of matrix A in a vector. [m,n]
= size(A) returns the size of matrix A in variables m and n (recall: in
Matlab arrays are defined as m-by-n matrices). length(A) returns the size of
the longest dimension of A. E.g.:
[m,n]=size(A)
m =
3
n =
2
length(A)
ans =
3
Introduction intoMatlab, Version 6.0.0 20
2.7.4 Sum of elements of a matrix
If A is a vector, sum(A) returns the sum of the elements. If A is a matrix,
sum(A) treats the columns of A as vectors, returning a row vector of the sums
of each column.
B=sum(A)
B =
8 10
If A is a vector, cumsum(A) returns a vector containing the cumulative sum
of the elements of A. If A is a matrix, cumsum(A) returns a matrix the same
size as A containing the cumulative sums for each column of A.
B=cumsum(A)
B =
1 2
4 7
8 10
2.7.5 Smallest (largest) elements of an a rray
If A is a matrix, min(A) treats the columns of A as vectors, returning a row
vector containing the minimum element from each column. If A is a vector,
min(A) returns the smallest element in A. max(A) returns the
maximum elements.
min(A)
ans =
1 2
Introduction into Matlab, Version 6.0.0 21
2.7.6 Descriptive statistics of matrices
mean(X) returns the mean values of the elements along the columns of an
array. std(X) returns the standard deviation using
( )
1
2
2
1
1
1
n
i
i
std x x
n

| `

. ,

where n is the number of elements in the sample.

For matrices where each row is an observation and each column a variable,
cov(x) is the covariance matrix. Likewise, corrcoef(X) returns amatrix
of correlation coefficients calculated from an input matrix whose rows are
observations and whose columns are variables.
2.7.7 Determinant of a matrix
det(A) returns the determinant of the square matrix A.
A=[4 2; 1 3];
B=det(A)
B =
10
2.7.8 Inverse of a matrix
inv(A) returns the inverse of the square matrixA.
C=inv(A)
C =
0.3000 -0.2000
-0.1000 0.4000
In practice, it is seldom necessary to form the explicit inverse of a matrix. A
frequent misuse of inv arises when solving the system of linear equations
Introduction intoMatlab, Version 6.0.0 22
Ax = b. One way to solve this is with x = inv(A)*b. A better way, from
both an execution time and numerical accuracy standpoint, is to use the
matrix division operator x = A\b.
2.7.9 Eigenvectors and eigenvalues of a matrix
The n-by-n (quadratic) matrix A can often be decomposed into the form
1
PDP A , where D is the matrix of eigenvalues and P is the matrix of
eigenvectors. Consider
A =
0.9500 0.0500
0.2500 0.7000
[P,D]=eig(A)
P =
0.7604 -0.1684
0.6495 0.9857
D =
0.9927 0
0 0.6573
Probably you are just interested in the eigenvalues.
eig(A)
ans =
0.9927
0.6573
2.8 Getting Help
Typinghelp topic displays help about that topic if it exists, e.g.
help det
Introduction into Matlab, Version 6.0.0 23
shows information about the use of the determinant function. Another way is
to consult the help desk in the Help Menu.
Introduction intoMatlab, Version 6.0.0 24
3 Graphics
Matlab can produce both planar plots and 3-D plots. The first important
instruction is the clf instruction that clears the graphic screen.
3.1 Histogram
The instruction hist(x) draws a 10-bin histogram for the data in vector x.
hist(x,c), where c is a vector, draws a histogram using the bins specified
in c. Here is an example:
c=-2.9:0.2:2.9;
x=rand(5000,1);
hist(x,c) ;
3.2 Plotting series versus their index
2-D graphs essentially rely on the instruction plot. plot(y) plots the data
in vector y versus its index, i.e.,
y=rand(100,1);
clf;
plot(y)
produces the following graph:
Introduction into Matlab, Version 6.0.0 25
plot([Y X]) plots the two column vectors of the matrix [Y X] versus their
index (within the same graph).
3.3 Scatterplots (2-D plots)
The general form of the plot instruction is
plot(x,y,S)
where x and y are vectors or matrices and S is a one, two or three character
string specifying color, marker symbol, or line syle. plot(x,y,S) plots y
against x,if y and x are vectors. If y and x are matrices, the columns of y are
plotted versus the columns of x in the same graph. If only y or x is a matrix,
the vector is plotted versus the rows or columns of the matrix. Note that
Introduction intoMatlab, Version 6.0.0 26
plot(x,[a b],S) and plot(x,a,S,x,b,S), where a and b are vectors,
are alternative ways to create exactly the same scatterplot.
The S string is optional and is made of the following (and more) characters:
Line style:
Solid Dashed Dotted line Dash-dot line
Symbol - -- : -.
Colour:
Yellow Red Green Blue Cyan Magenta White Black
Symbol y r g b C M w k
Marker specifier:
Point Circle Asterisk Cross
Symbol . O * x
Thus,
plot(X,Y,b*)
plots a blue asterisk at each point of the data set.
Here is an example:
x=pi*(-1:.01:1);
y=sin(x);
clf;
plot(x,y,b-);
Introduction into Matlab, Version 6.0.0 27
3.4 Adding titles, labels and grids
You can add titles and labels to your plot, using instructions :
title(atitle) % Obvious
xlabel(alabel) % Label on the x axis
ylabel(alabel) % Label on the y axis
You can even add a grid on your plot, using grid. Here is an example:
x=pi*(-3:.01:3);
y1=sin(x);
y2=cos(x);
clf;
plot(x,y1,y,x,y2,r);
title(A nice example ?);
ylabel(The functions);
xlabel(The values);
grid;
Introduction intoMatlab, Version 6.0.0 28
3.5 Creating subplots
You can also have many graphs on the screen using instruction
subplot(rcn) where r, c, and n denotes respectively the number of rows,
columns and the number of the graph:
x=[-3:.01:3];
y1=exp(-0.5*x.^2)/sqrt(2*pi);
y2=sin(x);
y3=cos(x);
y4=abs(sqrt(x));
clf;
subplot(221); %figure 1 out of 4
plot(x,y1);
ylabel(Y1);
xlabel(X);
title(Gaussian);
subplot(222); %figure 2 out of 4
plot(x,y2);
ylabel(Y2);
xlabel(X);
title(Sin(X));
subplot(223); %figure 3 out of 4
plot(x,y3);
ylabel(Y3);
xlabel(X);
title(Cos(X));
subplot(224); %figure 4 out of 4
plot(x,y4);
ylabel(Y4);
xlabel(X);
title(Abs(Sqrt(X)));
Introduction into Matlab, Version 6.0.0 29
3.6 3-D plots
The plot command from the two-dimensional world can be extended into
three dimensions with plot3. The format is the same as the two-dimensional
plot, except the data are in triples rather than in pairs. The generalized
format of plot3 is plot3(x
1
,y
1
,z
1
,S
1
,x
2
,y
2
,z
2
,S
2
,), where x
n
, y
n
,
and z
n
are vectors or matrices, and S
n
are optional character strings specifying
color, marker symbol, or line syle. Here is an example:
t=linspace(0,10*pi);
plot3(sin(t),cos(t),t,b*);
xlabel(sin(t)),zlabel(t);
Next we consider mesh plots. Matlab defines a mesh surface by the z-
coordinates of points above a rectangular grid in the x-y plane. It forms a plot
Introduction intoMatlab, Version 6.0.0 30
by joining adjacent points with straight lines. The result looks like a fishing
net with the knots at the data points. Mesh plots are very useful for visualizing
large matrices or for plotting functions of two variables.
The first step in generating the mesh plot of a function of two variables,
z = f(x,y) , is to generate X and Y matrices consisting of repeated rows and
columns, respectively, over some range of the variables x and y. Matlab
provides the function meshgrid for this purpose. [X,Y]=meshgrid(x,y)
creates a matrix X whose rows are copies of the vector x, and a matrix Y
whose columns are copies of the vector y. This pair of matrices may then be
used to evaluate functions of the two variables using Matlabs array
mathematics features. Here is an example:
x=[-2:.1:2];
y=x;
[X,Y]=meshgrid(x,y);
R=sqrt(-X.^1-Y.^2);
mesh(X,Y,Z);
3.7 Save graphs
To save the resulting graph in a file just use the instruction print. Its general
syntax is given by :
print -options name_of_file
To save the graph in a .jpeg-format you type:
print -djpeg graph.jpeg
To save the graph in a .ps-format you type:
print -dps graph.eps
Introduction into Matlab, Version 6.0.0 31
4 Scripts and functions
Up to now, we were content to input individual commands in the Matlab
command window. For more complex functions or frequently needed
sequences of instructions this is, however, extremely unsatisfying. Matlab
offers us two possibilities of processing or of automating such sequences
more comfortably, script M-files and function M-files.
4.1 Script M-files
Matlab allows you to place Matlab commands in a simple text file, and then
tell Matlab to open the file and evaluate commands exactly as it would if you
had typed them in the Matlab command window. These files are called script
M-files. The term script symbolizes the fact that Matlab simply reads from
script found in the file. The term M-file recognizes the fact that script
filenames must end with the extension .m. Further, the name of a M-file has
to begin with an alphabetical letter
To create a script M-file, choose Newfrom the File menu and select M-file
(alternatively you just type edit in the Matlab prompt and press Enter). This
procedure brings up a text editor window, the Matlab editor/debugger. Lets
type the following sequence of statements in the editor window:
A=[1 2];
B=[3 4];
C=A+B
This file can be saved as the M-file example.m on your disk by choosing Save
from the File menu. Matlab executes the commands in example.m when you
Introduction intoMatlab, Version 6.0.0 32
simply type example in the Matlab command window (provided there is a
file example.m in your working directory or path
2
).
example
C =
4 6
The use of scripts is particularly handy when you are coding problems that
consists of a large number of connected and complex indexing steps. Scripts
can operate on existing data in the workspace, or they can create new data on
which to operate. Although scripts do not return output arguments, any
variables that they create remain in the workspace, to be used in subsequent
computations. Moreover, scripts can produce graphical output using functions
like plot.
4.2 Function M-files
A function M-file is similar to a script file in that it is a text file having a .m
extension. As with script M-files, function M-files are created with a text
editor. A function M-file is different from a script file in that a function
communicates with the Matlab workspace only through their variables passed
to it and though the output variables it created. Intermediate variables within
the function do not appear in, or interact with, the Matlab workspace.
Functions operate on variables within their own workspace, separate from the
workspace you access at the Matlab command window. If we call a function
2
Do modify the default adjustments over File/Set Path/Path or by pressing the button
Path Browser in the command window toolbar. A third way would be to use DOS-
commands: dir lists the files in the current directory; cd prints out the current
directory; cd .. changes to the directory above the current one.
Introduction into Matlab, Version 6.0.0 33
within a code, Matlab will search the work directory for the name of the
invoked M-file, minus its extension.
We have encountered examples of function M-files in paragraph 2, already.
Some built-in functions, like ones, are part of the Matlab core. Although
they are very efficient, the computational details are not readily accessible.
Others, like std, are implemented in M-files. You can see the code and
even modify it if you want.
3
Apart from the function M-files incorporated in
the Matlab package, you can write functions on your own, download them
from other users homepages on the internet or buy them separately in so
called toolboxes.
4.2.1 Defining your own function M-files
We add new functions to Matlab's vocabulary by expressing them in terms of
existing commands and functions. The general syntax for defining afunction
is
Function[output1,..]=<name of function>(input1,..);
statements;
The first line of a function M-file defines the M-file as a function and
specifies its name (its file name without the .m extension). The name of the
M-file and the one of the function should be the same. It also defines its input
and output variables.
3
In order to check out whether a built-in function is an M-file or not, you can use your
Windows Explorer Tools/Find/Files or Folders and look for the respective name of the
function plus its extension, e.g. std.m. If it is indeed an M-file you can open it.
Introduction intoMatlab, Version 6.0.0 34
Next, there is a sequence of comment lines with the text displayed in response
to the help command:
help <name of function>.
Finally the remainder of the M-file contains Matlab commands that create the
output variables.
Lets consider an example: We would like to build a function, that gives us
the mean and the standard deviation of a vector. (Suppose we dont know that
there is an respective built-in function.)
function [mx,stx]=stat(x);
%
% function [mx,stx]=stat(x)
% Computes the mean and standard deviation of a
% vector x
% x: a vector (column or row)
% mx: mean of vector
% stx: standard deviation of vector
%
lx=length(x);
mx=sum(x)/lx;
stx=(sum((x-mx).^2)/(lx-1))^(1/2);
The name of the function is stat. The input vector is called x. Output
consists of two respective variables, mx and stx. After every line that starts
with a % there is your text of the online help. What follows next are some
familiar Matlab statements. First we define the scalar lx as the length of the
input vector. Then we define the scalars mx and stx.
You now want to include and use your new function. For this purpose you
change to the command window. Suppose, your input is a 1-by-100 random
vector.
Introduction into Matlab, Version 6.0.0 35
x=randn(1,100);
[mx,stx]=stat(x)
mx =
0.0407
stx =
0.8797
You get the variables mx and stx as an output.
(e.g., an algorithm or an estimation procedure etc.) from the internet. You will
probably find it convenient to create your own library of tools that you will
use often.
4.2.3 Toolboxes
Finally, function M-files can be part of a so-called toolbox. A toolbox is a
collection of function M-files that extend the capability of Matlab. In addition
to the toolboxes installed by default
4
toolboxes to specific topics (like
statistics or optimization) can be bought separately. For an overview visit
http://www.cx.unibe.ch/matlab/fulldocset.html.
4
In your Matlab folder you will find a folder called toolbox. By default it contains three
toolboxes: local, matlaband tour.
Introduction intoMatlab, Version 6.0.0 36
5 Controlling the flow
The structure of the sequences of instructions used so far was rather
straightforward; commands were executed one after the other, running from
top to bottom. Like any computer programming language and programmable
calculators Matlab offers, however, features that allow you to control the flow
of command execution. If you have used these features before, this section
will be familiar to you. On the other hand, if controlling the flow is new to
you, this material may seem complicated at first; if this is so, take it slow.
Controlling the flow is extremely powerful, since it lets past computations
influence future operations. With the three following sets of instruction it is
possible to cope with almost all the problems we shall encounter in
computation. Because they often encompass numerous Matlab commands,
they frequently appear in M-files, rather than being typed directly at the
Matlab command window.
5.1 The FOR-loop
The most common use of a FOR-loop arises when a set of statements is to be
repeated a fixed number of times n. The general form of a FOR-loop is:
For variable = expression;
statements;
end;
The variable expression is thereby a row vector of the length n, that is
processed element-by-element (in many cases it is the row vector (1:n)).
Statements stands for a sequence of statements to the program. The
Introduction into Matlab, Version 6.0.0 37
columns of the expression are stored one at a time in the variable while the
following statements, up to the end, are executed. The individual instructions
are generally separated by semicolons, in order to avoid a constant output
from section results in the command window.
A simple example of such a loop is:
for i=1:10;
x(i)=i;
end;
disp(x')
5.2 The WHILE-loop
In contrast to the FOR-loop, with which the loop will pass through a fixed
number n, WHILE-loops are executed, until an abort condition is fulfilled.
Note: You have to care that the abort condition is achieved by the program in
finite time in each case, in order to avoid a continuous loop.
The general syntax for WHILE loop is the following:
while condition;
statements;
end;
So while the condition is satisfied, the statements will be executed.
Consider the following example:
eps = 1;
while (1+eps) > 1
eps = eps/2;
end;
eps = eps*2
Introduction intoMatlab, Version 6.0.0 38
This example shows one way of computing the special Matlab value eps,
which is the smallest number that can be added to 1 such that the result is
greater than 1 using finite precision. (We use uppercase EPS so that the
Mat lab value eps is not overwritten.) In this example, EPS starts at 1. As long
as (1+EPS)>1 is True (nonzero), the commands inside the WHILE loop are
evaluated. Since EPS is continually divided in two, EPS eventually gets so
small that adding EPS to 1 is no longer greater than 1. (Recall that this
happens because a computer uses a fixed number of digits to represent
numbers. Matlab uses 16 digits, so you would expect EPS to be near 10
-16
.) At
this point, (1+EPS)>1 is False (zero) and the WHILE loop terminates.
Finally, EPS is multiplied by 2 because the last division by 2 made it too
small by a factor of two.
5.3 The IF-statement
A third possibility of controlling the flow in Matlab is called the IF-statement.
The IF-statement executes a set of instructions if a condition is satisfied. The
general form is
if condition 1;
% commands if condition 1 is TRUE
statements 1;
elseif condition 2;
% commands if condition 2 is TRUE
statement 2;
else condition;
% commands if condition 1 and 2 are both FALSE
statements 3;
end;
Introduction into Matlab, Version 6.0.0 39
When you have just two conditions you skip the elseif condition and
immediately go to the else condition.
Assume you have a demand function of the following kind
( )

'

<

otherwise
p if
p if
P
P P D 3 2
2
2
5 . 0 1
0
2
The code will be:
P=input('enter a price :'); % displays the message
% '.' on the screen and
% waits for an answer
if P<=2;
D=0; % statement 1
elseif (P>2)&(P<=3);
D=1-0.5*P; % statement 2
else; % otherwise
D=2*P^(-2); % statement 3
end;
disp(D);
5.4 How to break a FOR or WHILE-loop
break terminates the execution of a for FOR- or WHILE-loop.
Introduction intoMatlab, Version 6.0.0 40
6 Importing and exporting data
With Matlab you can do both, export your data or calculated results and
import external data.
6.1 The diary command
The diary command creates a log of keyboard input and the resulting output
(except it does not include graphics). The log is saved as an ASCII text file in
the current directory or folder. For instance lets type the following
statements:
diary result.out;
disp(A);
diary off;
In our directory appears a file named result with the extension .out. We can
open it in any editor. It contains the (string or numeric) matrix A.
Note: The instruction diary does not overwrite an existing file. We have to
delete it first with delete <afile> if we do not want to addinformation on
the existing file but overwrite it.
6.2 Save workspace variables on disk
The Save workspace as menu item in the File menu opens a standard file
dialog box for saving all current variables. Saving variables does not delete
them from the Matlab workspace. Besides, Matlab provides the command
save.
save
Introduction into Matlab, Version 6.0.0 41
stores all variables in Matlab binary format in the file matlab.mat.
save data
saves all variables in Matlab binary format in the file data.mat.
save data capital ascii
saves the variable capital in 8-digit ASCII text format in the file data.txt.
ASCII-formatted files may be edited using any common text editor.
6.3 Retrieving data and results
The load command uses the same syntax, with the obvious difference of
For example assume the data file, data.txt, takes the form:
.....1.0000 ->.....2.0000
.....3.0000 ->.....4.0000
.....5.0000 ->.....6.0000
We make sure that data.txt is in the right directory and then type
Our data are stored in a matrix called data that we will manipulate as any
other matrix.
If our data is stored in an Excel spreadsheet then there is an even lazier way to
import data. We highlight the data to be imported in the Excel spreadsheet,
type copy, swap to the Matlab prompt, define a matrix (e.g. data=), open the
angular brackets and then type paste. Finally we close the brackets. We end
up with a matrix called data containing our data.
Introduction intoMatlab, Version 6.0.0 42
6.4 Request user input
If we want the user to input information from the keyboard we use the
statement input:
n=input(Give a number);
displays the message Give a number on the screen and wait for an answer.
The result will be evaluated in the variable n.
Introduction into Matlab, Version 6.0.0 43
7 References
Einfhrung in Matlab
http://www.stat.uni-muenchen.de/~boehme/matlab-kurs/lecture1.html
Noisser, Robert (1998): Matlab Einfhrung
http://www.iert.tuwien.ac.at/support/matlab_1.htm
Sigmon, Kermit (1992), Matlab Primer 2
nd
edition, Department of
Mathematics, University of Florida.
The Student Edition of MATLAB, Version 5, Users Guide (1997)
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