GNU Octave  An Introduction
Originally published in the LinuxGazette.net, December 2004, Issue 109.This is the first of a series of articles in which I will introduce GNU Octave and demonstrate some of its many features. GNU Octave is a highlevel language for numerical computations. I use it every day in my PhD research which involves manipulating large vectors and matrices. It is very similar in syntax and function to a commercial application called Matlab. The biggest difference between the two is that Octave is released under the GNU General Public License, which means it can be freely distributed and/or modified, while a singleuser academic license for the basic Matlab currently costs US$700.
I have convinced a few of my colleagues to give Octave a try instead of Matlab. In every case, once that person stops looking for the differences between the two and decides to give Octave a real chance, they begin to embrace its usefulness, its features and its free availability. They realise that they can install a copy of Octave onto every one of their simulation servers, their laptops and their home computers without having to purchase costly new licenses for each one.
Installing and Running Octave
The source code for Octave can be downloaded from http://www.octave.org/download.html. This site also contains information on where to get Octave in binary form for Apple's OS X and Windows. Most Linux distributions include Octave as standard and if it is not already installed on your system it should simply be a matter of installing the Octave package from your installation CDs or the Internet.Starting the Octave interpreter under Linux is as simple as
typing the `octave
' command:
$ octave GNU Octave, version 2.1.50 (i686pclinuxgnu). Copyright (C) 1996,1997,1998,1999,2000,2001,2002,2003 John W. Eaton. This is free software; see the source code for copying conditions. There is ABSOLUTELY NO WARRANTY; not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE. For details, type `warranty'. Please contribute if you find this software useful. For more information, visit http://www.octave.org/helpwanted.html Report bugs to <bugoctave@bevo.che.wisc.edu>. octave:1>
Documentation
A 380 page manual is included with the Octave source code in HTML, DVI and PS format. This manual is also available online at Octave's home page. For those of you who installed via binary packages, you should be able to access the manual via the `info
' command:$ info octave
If you are unfamiliar with 'info', then try using KDE's interface to info by typing `
info:octave
' into Konqueror's location
bar.
In this article I only intend to touch on the very basics of Octave to demonstrate just how easy it is to pick up and use effectively. I would strongly recommend, at the very least, skimming through the available documentation to get a fuller flavor of what Octave has to offer.
First Steps by Example
Let's look at the problem of solving a system of n linear equations in n unknowns:x + 3y  2z = 3 3x  4y + 3z = 28 5x  5y + 4z = 7Such a system of linear equations can be written as the single matrix equation
Ax =
b
, where A
is the coefficient
matrix, b
is the column vector
containing the righthand side of the linear equations and
x
is the column vector representing
the solution. If you've forgotten your linear algebra then don't
worry  this will all become at lot clearer as we use Octave to
solve this for us:
octave:1> A = [ 1, 3, 2; 3, 4, 3; 5, 5, 4 ] A = 1 3 2 3 4 3 5 5 4 octave:2> b = [ 3; 28; 7 ] b = 3 28 7 octave:3> A \ b ans = 5.0000 2.0000 7.0000 octave:4>You will notice that each line of the interpreter is numbered sequentially; I will use these line numbers when referring to particular commands. On line 1 I defined
A
as a
3x3
matrix containing the coefficients of the linear
system above (a coefficient is the number to the left of the
unknown variables x
, y
and
z
). The rows are delimited with a semicolon and the
individual elements on each row are delimited by a comma. Each of
these is recommended but optional: a space is all that is needed to
delimit elements in a row and the return key could have been used
instead of semicolons. I defined the column vector
b
on line 2 in the same way.
Line 3 computes the solution of the linear system using the
`left division' operator which, for the mathematicians among you,
is conceptually equivalent to
A^{1}b
. By solution, I mean
that x = 5
, y = 2
and z = 7
will satisfy all three equations of the linear system.
Plotting the solution to a problem in mathematics is often the
key to fully understanding that problem. Octave has a number of
functions for plotting two and threedimensional graphs which use
Gnuplot to handle the actual graphics themselves. As a simple
example, let's plot the sin( x )
:
octave:9> x = [ pi:0.01:pi ]; octave:10> plot( x, sin(x) ) octave:11>Which produces:
Let's examine line 9 above in more detail:
pi
is one of many constants built in to Octave for convenience and evaluates as 3.1415... Octave has a range operator of the form
begin:step:end
such that[ 1:1:5 ]
is the vector[ 1 2 3 4 5 ]
. The step is optional and if it is omitted then a step size of1
is assumed:1:1:5
is the same as1:5
.  You might have noticed that line 9 above produced no output to
the screen like similar commands did before. This is because we
ended the command with a semicolon which suppresses the output. In
the case above,
[ pi:0.01:pi ]
creates a vector of length 629 which we really don't want to print to the screen!
Data Types, Simple Arithmetic and Standard Functions
Octave's builtin data types are real and complex scalars and matrices, character strings and a data structure type. All of the standard arithmetic functions are available for scalars and matrices:a + b

Addition (Subtraction). If both operands are matrices then the number of rows and columns must both agree. If one operand is a scalar and the other is a matrix, then that scalar will be added (subtracted) to (from) every element of the matrix. 
a .+ b

Componentwise addition (subtraction) (also known as elementbyelement addition). 
x * y

Multiplication. If both operands are matrices then
the number of columns of x must agree with the number
of rows or y . 
x .* y

Componentwise multiplication. 
x / y

Right division. Conceptually equivalent to (
(y^{T})^{1} * x^{T}
)^{T} 
x ./ y

Componentwise right division 
x \ y

Left division. Conceptually equivalent to
x^{1} * y 
x .\ y

Componentwise left division. 
x ^ y

Power operator. See the manual for definitions
when x and/or y is a matrix. 
x .** y

Componentwise power operation. 
x

Negation 
x'

Complex conjugate transpose. 
x.'

Transpose. 
There are many standard functions builtin to Octave and these include the scalar functions:
sin()

asin()

log()

abs()

cos()

acos()

log2()

sqrt()

tan()

atan()

log10()

sign()

round()

floor()

ceil()

mod()

the vector functions:
max()

sum()

median()

any()

min()

prod()

mean()

all()

sort()

var()

std()

and the matrix functions:
eig()

 eigenvalues and eigenvectors 
inv()

 inverse 
poly()

 characteristic polynomial 
det()

 determinant 
size()

 return the size of a matrix 
norm(,p)

 compute the pnorm of a matrix 
rank()

 the rank of a matrix 
Strings can be declared with either single or double quotes:
> fname = "Barry";
Strings can be concatenated
using the same notation as matrix definitions:
> sname = "O'Donovan";
> [ fname, " ", sname ]
ans = Barry O'Donovan
There are many string functions available as standard, including
functions for converting strings to numbers and viceversa. There
are also a number of functions for printing strings to the screen
such as disp()
and printf()
, and for
reading data from the user such as input()
.
The Octave Environment
In all the cases above where we had an assignment command such asA = ...
, the variable A
is created or
overwritten with the information on the righthand side of the
assignment operator (=
). Variable names are case
sensitive and made up of letters, digits and underscores but must
begin with a letter or underscore. Variables remain in the
interpreter's environment until you either exit the interpreter or
clear the variable:> clear A
deletes the variable
A
, while:> clear
deletes all variables currently stored. The
who
command can be used to list all variables
currently stored in the environment.
We would often like to save the current environment to disk as a
backup or to come back to it later and continue on from where we
left off. We can use the following two commands for this:
> save
filename
to save all of the currently defined variables to filename
and:
> load
filename
to load them again at a later point.
Loops and Conditional Statements
Just like any other programming language, Octave has its loop and conditional constructs. The following example demonstrates how to generate the first 10 values of Fibonacci's sequence using a for loop:octave:11> fib = [ 0, 1 ]; octave:12> for i = 3:10 > fib = [ fib, fib( i2 ) + fib( i1 ) ]; > endfor octave:13> fib fib = 0 1 1 2 3 5 8 13 21 34 octave:14>Fibonacci's sequence is described by
F_{k} =
F_{k1} + F_{k2}
with F_{0} =
0
and F_{1} = 1
. It is often used to
describe the population growth of rabbits: suppose that a newly
born pair of rabbits produce no offspring in the first month of
their lives and one new pair on each subsequent month. Starting
with F_{1} = 1
pairs in the first month,
F_{k}
is the number of pairs in the
kth month assuming that none of the rabbits die.
Fibonacci's sequence occurs naturally in a variety of places and it
is one of those rare occurrences in mathematics where a simple
formula can be truly fascinating.
Notice that in the above code:
 we redefine the vector
fib
using itself and one new value; and  we can access individual elements of a vector by specifying the element number in parenthesis.
The following example evaluates the randomness of Octave's
rand()
function and demonstrates it's conditional
statements:
octave:14> a = b = c = d = 0; octave:15> for i = 1:100000 > r = rand(1); > if ( r < 0.25 ) > a++; > elseif ( r < 0.5 ) > b++; > elseif ( r < 0.75 ) > c++; > else > d++; > endif > endfor octave:16> a,b,c,d a = 25115 b = 24870 c = 25045 d = 24970 octave:17>Line 14 sets the scalar variables
a
, b
,
c
and d
to zero. We then generate 100,000
random numbers between 0 and 1 and increase a
by one
if it falls between 0 and 0.25, b
if it falls between
0.25 and 0.5, and so forth. Once the loop completes, we would
expect the values of a
, b
, c
and d
to be approximately 25,000 if
rand()
generates truly random numbers, which, as can
be seen above, it does.
A Brief Overview of the Features of Octave
Octave was originally written and is still maintained by John W. Eaton who made the first public release in 1993. Since then many other people have contributed to it as they found it lacked features they needed. As it stands, Octave comes with many builtin functions grouped into related packages.Matrix manipulation is at the heart of Octave and it includes all the operators you would expect for matrix arithmetic including addition, subtraction, multiplication (matrix and componentwise), division, transposition, etc. It also has a number of functions for generating common matrices including:
eye()
 the identity matrix;ones()
andzeros()
 a matrix of all ones or zeros;hankel()
 the famous Hankel matrix; andhilb()
andinvhilb()
 the Hilbert and inverse Hilbert() matrix.
The groups of specialised functions include:
 input and output;
 plotting;
 matrix manipulation;
 linear algebra;
 nonlinear equations;
 differential equations;
 optimisation;
 statistics;
 financial functions;
 sets;
 polynomial manipulations;
 control theory;
 signal processing;
 image processing; and
 audio processing.
Some of these are complete and some only contain a few functions. Each is added by various people when and as needed. Over the next couple of months we will look at creating new functions with Octave as well as writing new functions in C++. The Octave developers welcome new additions and hopefully by the end of this series you might be writing and contributing your own Octave functions.
Final Words
Hopefully this article will have demonstrated just how easy it is to pick up the basics of Octave. For any teachers or lecturers trying to teach their students matrices and/or linear algebra, why not introduce Octave into your course as a teaching tool? And for the lecturers or students of university departments such as maths, mathematical physics, physics, engineering, computer science, etc.  it's often difficult to have to come up with new and exciting final year projects every year. Why not have a student implement some mathematical functionality that Octave lacks from your own area of research that might be of interest to others?Next month: Writing new Octave functions and writing Octave scripts that can be executed from the command line.
Copyright © 2004, Barry O'Donovan. Released under the Open Publication license.