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Table of contents

1 Starting Octave 2 Entering commands 2.1 Examples of Operators 3 Plotting 3.1 Exercise 3.2 More on commands 3.3 Example 4 Exercises 4.1 Warning 5 Challenge 6 Script files

The aim of this tutorial is to give you a quick introduction to basic Octave and to show that you know a lot of it already. If you should ever get stuck or need more information on an Octave function or command, type

help command

at the Octave prompt. command is the name of the Octave command or function on which to find help. Be warned though that the descriptions can sometimes be a bit technical and filled with jargon.

Starting Octave

Type octave in a terminal window to get started. You should see the following.

GNU Octave, version 2.1.69 (i386-pc-linux-gnu).
Copyright (C) 2005 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'.

Additional information about Octave is available at http://www.octave.org.

Please contribute if you find this software useful.
For more information, visit http://www.octave.org/help-wanted.html

Report bugs to <bug@octave.org> (but first, please read
http://www.octave.org/bugs.html to learn how to write a helpful report).

octave:1> 

Entering commands

The last line above is known as the Octave prompt and, much like the prompt in Linux, this is where you type Octave commands. To do simple arithmetic, use + (addition), - (subtraction), * (multiplication), / (division), and ^ (exponentiation). Many mathematical functions are also available and have obvious names, e.g. sin, cos, log, abs (absolute value).

Here are some examples showing the input typed at the prompt and the output returned by Octave.

2 + 3
octave:1> 2 + 3 ans = 5	octave:2> log(100)/log(10) ans = 2	octave:3> floor((1+tan(1.2)) / 1.2) ans = 2	octave:4> sqrt(3^2 + 4^2) ans = 5

Some things to note:

• Octave requires parentheses around the input of a function (so, log(10) is fine, but (log 10) is not).
• Any spacing before and after arithmetic operators is optional, but allowed.
• Not all Octave functions have obvious names (e.g. sqrt above). Don't panic for now. You will get to know them as we go along.

Examples of Operators

These are all Operators in octave. the operator is stated first followed by its description.

• a + b or 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 or a .- b
• Component-wise addition (subtraction) (also known as element-by-element 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
• Component-wise multiplication.
• x / y
• Right division. Conceptually equivalent to ( (yT)-1 * xT )T
• x ./ y
• Component-wise right division
• x \ y
• Left division. Conceptually equivalent to x-1 * y
• x .\ y
• Component-wise left division.
• x ^ y or x ** y
• Power operator. See the manual for definitions when x and/or y is a matrix.
• x .** y
• Component-wise power operation.
• -x
• Negation
• x'
• Complex conjugate transpose.
• x.'
• Transpose.

Plotting

You are going to plot the following pictures using Octave:

Figure 1 contains a plot of x vs sin x and is generated with the following commands. (It's a bit boring but illustrates the basic functionality.)

x = linspace(0, 2*pi, 100);
y = sin(x);
plot(x, y);

The command that actually generates the plot is, of course, plot(x, y). Before executing this commmand, we need to set up the variables, x and y. The plot function simply takes takes two vectors of equal length as input, interprets the values in the first as x-coordinates and the second as y-coordinates and draws a line connecting these coordinates.

The first command above, x = linspace(0, 2*pi,100), uses the linspace function to make a vector of linearly spaced values. The first value in the vector is 0, the final value is 2π and the vector contains 100 values. This vector is assigned to the variable named x.

The second command computes the sin of each value in the vector variable, x, and stores the resulting vector in the variable y.

(As an aside: the name of a variable can be any sequence of letters, digits and underscores that does not start with a digit. There is no maximum length for variable names, and the case of alphabetical characters is important, i.e. a and A are two different variable names.)

Exercise

Plot the function for . (This is Figure 2).

More on commands

The following commands and functions are useful for setting up variables for plotting 2D graphs.

• linspace creates a vector of evenly (linearly) spaced values.

Usage: linspace(start, stop, length). The length parameter is optional and specifies the number of values in the returned vector. If you leave out this parameter, the vector will contain 100 elements with start as the first value and stop as the last.

• plot plots a 2-dimensional graph.

Usage: plot(x, y) where x and y are vectors of equal length.

• figure creates a new plotting window.

This is useful for when you want to plot multiple graphs in separate windows rather than replacing your previous graph or plotting on the same axes.

• hold on and hold off sets whether you want successive plots to be drawn together on the same axes or to replace the previous plot.

Example

We are going to plot Figures 3 and 4. Figure 3 contains the 3 trigonometric functions

• cos2x,
• sin4x, and
• 2sinx

on one set of axes. Figure 4 contains the sum of these 3 functions.

Firstly, we use linspace to set up a vector of x-values.

octave:1> x = linspace(0, 2*pi);

Then, we compute the y-values of the 3 functions.

octave:2> a = cos(2*x);
octave:3> b = sin(4*x);
octave:4> c = 2*sin(x);

The following plots the first graph.

octave:5> figure;
octave:6> plot(x, a);
octave:7> hold on;
octave:8> plot(x, b);
octave:9> plot(x, c);

We use line 5 (figure) to tell Octave that we want to plot on a new set of axes. It is good practice to use figure before plotting any new graph. This prevents your accidentally replacing a previous plot with the new one.

Note that on line 7, hold on is used to tell Octave that we don't want to replace the first plot (from line 6) with subsequent ones. Octave will plot everything after hold on on the same axes, until the hold off command is issued.

Finally, we plot the second graph.

octave:10> figure;
octave:11> hold off;
octave:12> plot(x, a+b+c);

Line 10 creates a new graph window and line 11 tells Octave that any subsequent plots should simply replace previous ones. Line 12 generates the plot of the sum of the 3 trigonometric functions.

Exercises

• Plot the absolute value function for .
• Plot a circle of radius 1, centered at the origin. (This is not so easy.)
• Plot your favourite function(s)

Warning

If you try (or have tried) to plot something like x² or , you will run into trouble. The following error messages are common. In the case of x^2:

error: for A^b, A must be square

In the case of sin(x)*cos(x):

error: operator *: nonconformant arguments (op1 is 1x100, op2 is 1x100)

This error occurs whenever you try multiply or divide two vector variables because '*', '/', and "^" are vector operations not element by element scalar operations (remember that x and y are vectors). Octave has corresponding element by element scalar operators: place a '.' before the operator (e.g. x./2 or x.^2 or x.*2) for these operations (see vectors and matrices for more information).

Challenge

Since Octave is a numerical (and not symbolic) mathematics package, it does make numerical errors and does not handle some operations well. To confirm this, make a plot of tan x, for x between -π and π. What is wrong with the resulting picture?

Your task is to generate the (much better looking) graph below using what you have learned so far and the axis function. axis can be used to adjust which part of the plot is actually displayed on screen. Use the command help axis to determine how this function works.

It might take some thinking to get the asymptote lines at right. You can use help plot to find out how to plot dotted lines.

Script files

It is useful to be able to save Octave commands and rerun them later on. You might want to save your work or create code that can be reused (by yourself or somebody else). Such files are known as Octave script files. They should be saved with a .m extension so that Octave can recognise them. (The .m extension is used because MATLAB calls its script files M-files and Octave is based on MATLAB.)

To run an existing script in Octave, you have to be in the same directory as the script file and type in the name of the file without the .m in Octave. For example, if I have a script called myscript.m in an octave directory, the following two commands will execute the script.

chdir('~/octave'); % This changes to the octave directory
myscript;

Note that the chdir('~/octave') command is necessary only if you are not already inside that directory when running Octave.

In the following section you will be shown a number of new statements that you can use to make your Octave code much more powerful. A number of example script files are provided and you should save these into a directory for later use. A good idea is to create a directory called octave in your home directory and store all your Octave files in there.

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