octave: Simple Examples
1.2 Simple Examples
===================
The following chapters describe all of Octave’s features in detail, but
before doing that, it might be helpful to give a sampling of some of its
capabilities.
If you are new to Octave, we recommend that you try these examples to
begin learning Octave by using it. Lines marked like so, ‘octave:13>’,
are lines you type, ending each with a carriage return. Octave will
respond with an answer, or by displaying a graph.
1.2.1 Elementary Calculations
-----------------------------
Octave can easily be used for basic numerical calculations. Octave
knows about arithmetic operations (+,-,*,/), exponentiation (^), natural
logarithms/exponents (log, exp), and the trigonometric functions (sin,
cos, ...). Moreover, Octave calculations work on real or imaginary
numbers (i,j). In addition, some mathematical constants such as the
base of the natural logarithm (e) and the ratio of a circle’s
circumference to its diameter (pi) are pre-defined.
For example, to verify Euler’s Identity,
i*pi
e = -1
type the following which will evaluate to ‘-1’ within the tolerance of
the calculation.
octave:1> exp (i*pi)
1.2.2 Creating a Matrix
-----------------------
Vectors and matrices are the basic building blocks for numerical
analysis. To create a new matrix and store it in a variable so that you
can refer to it later, type the command
octave:1> A = [ 1, 1, 2; 3, 5, 8; 13, 21, 34 ]
Octave will respond by printing the matrix in neatly aligned columns.
Octave uses a comma or space to separate entries in a row, and a
semicolon or carriage return to separate one row from the next. Ending
a command with a semicolon tells Octave not to print the result of the
command. For example,
octave:2> B = rand (3, 2);
will create a 3 row, 2 column matrix with each element set to a random
value between zero and one.
To display the value of a variable, simply type the name of the
variable at the prompt. For example, to display the value stored in the
matrix ‘B’, type the command
octave:3> B
1.2.3 Matrix Arithmetic
-----------------------
Octave uses standard mathematical notation with the advantage over
low-level languages that operators may act on scalars, vector, matrices,
or N-dimensional arrays. For example, to multiply the matrix ‘A’ by a
scalar value, type the command
octave:4> 2 * A
To multiply the two matrices ‘A’ and ‘B’, type the command
octave:5> A * B
and to form the matrix product ‘transpose (A) * A’, type the command
octave:6> A' * A
1.2.4 Solving Systems of Linear Equations
-----------------------------------------
Systems of linear equations are ubiquitous in numerical analysis. To
solve the set of linear equations ‘AX = b’, use the left division
operator, ‘\’:
X = A \ b
This is conceptually equivalent to ‘inv (A) * b’, but avoids computing
the inverse of a matrix directly.
If the coefficient matrix is singular, Octave will print a warning
message and compute a minimum norm solution.
A simple example comes from chemistry and the need to obtain balanced
chemical equations. Consider the burning of hydrogen and oxygen to
produce water.
H2 + O2 --> H2O
The equation above is not accurate. The Law of Conservation of Mass
requires that the number of molecules of each type balance on the left-
and right-hand sides of the equation. Writing the variable overall
reaction with individual equations for hydrogen and oxygen one finds:
x1*H2 + x2*O2 --> H2O
H: 2*x1 + 0*x2 --> 2
O: 0*x1 + 2*x2 --> 1
The solution in Octave is found in just three steps.
octave:1> A = [ 2, 0; 0, 2 ];
octave:2> b = [ 2; 1 ];
octave:3> x = A \ b
1.2.5 Integrating Differential Equations
----------------------------------------
Octave has built-in functions for solving nonlinear differential
equations of the form
dx
-- = f (x, t)
dt
with the initial condition
x(t = t0) = x0
For Octave to integrate equations of this form, you must first provide a
definition of the function ‘f(x,t)’. This is straightforward, and may
be accomplished by entering the function body directly on the command
line. For example, the following commands define the right-hand side
function for an interesting pair of nonlinear differential equations.
Note that while you are entering a function, Octave responds with a
different prompt, to indicate that it is waiting for you to complete
your input.
octave:1> function xdot = f (x, t)
>
> r = 0.25;
> k = 1.4;
> a = 1.5;
> b = 0.16;
> c = 0.9;
> d = 0.8;
>
> xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
> xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);
>
> endfunction
Given the initial condition
octave:2> x0 = [1; 2];
and the set of output times as a column vector (note that the first
output time corresponds to the initial condition given above)
octave:3> t = linspace (0, 50, 200)';
it is easy to integrate the set of differential equations:
octave:4> x = lsode ("f", x0, t);
The function ‘lsode’ uses the Livermore Solver for Ordinary Differential
Equations, described in A. C. Hindmarsh, ‘ODEPACK, a Systematized
Collection of ODE Solvers’, in: Scientific Computing, R. S. Stepleman et
al. (Eds.), North-Holland, Amsterdam, 1983, pages 55–64.
1.2.6 Producing Graphical Output
--------------------------------
To display the solution of the previous example graphically, use the
command
octave:1> plot (t, x)
Octave will automatically create a separate window to display the plot.
To save a plot once it has been displayed on the screen, use the
print command. For example,
print -dpdf foo.pdf
will create a file called ‘foo.pdf’ that contains a rendering of the
current plot in Portable Document Format. The command
help print
explains more options for the ‘print’ command and provides a list of
additional output file formats.
1.2.7 Help and Documentation
----------------------------
Octave has an extensive help facility. The same documentation that is
available in printed form is also available from the Octave prompt,
because both forms of the documentation are created from the same input
file.
In order to get good help you first need to know the name of the
command that you want to use. The name of this function may not always
be obvious, but a good place to start is to type ‘help --list’. This
will show you all the operators, keywords, built-in functions, and
loadable functions available in the current session of Octave. An
alternative is to search the documentation using the ‘lookfor’ function
(described in 
Getting Help).
Once you know the name of the function you wish to use, you can get
more help on the function by simply including the name as an argument to
help. For example,
help plot
will display the help text for the ‘plot’ function.
Octave sends output that is too long to fit on one screen through a
pager like ‘less’ or ‘more’. Type a <RET> to advance one line, a <SPC>
to advance one page, and <q> to quit the pager.
The part of Octave’s help facility that allows you to read the
complete text of the printed manual from within Octave normally uses a
separate program called Info. When you invoke Info you will be put into
a menu driven program that contains the entire Octave manual. Help for
using Info is provided in this manual, Getting Help.
1.2.8 Editing What You Have Typed
---------------------------------
At the Octave prompt, you can recall, edit, and reissue previous
commands using Emacs- or vi-style editing commands. The default
keybindings use Emacs-style commands. For example, to recall the
previous command, press ‘Control-p’ (written ‘C-p’ for short). Doing
this will normally bring back the previous line of input. ‘C-n’ will
bring up the next line of input, ‘C-b’ will move the cursor backward on
the line, ‘C-f’ will move the cursor forward on the line, etc.
A complete description of the command line editing capability is
given in this manual, Command Line Editing.
Info Catalog
octave: Running Octave
octave: Introduction
octave: Conventions