GNU Octave - A Brief Introduction to Octave

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# A Brief Introduction to Octave

This manual documents how to run, install and port GNU Octave, and how to report bugs.

GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments. It may also be used as a batch-oriented language.

GNU Octave is also freely redistributable software. You may redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. The GPL is included in this manual in section GNU GENERAL PUBLIC LICENSE.

This document corresponds to Octave version 2.0.13.

## Running Octave

On most systems, the way to invoke Octave is with the shell command ``octave'`. Octave displays an initial message and then a prompt indicating it is ready to accept input. You can begin typing Octave commands immediately afterward.

If you get into trouble, you can usually interrupt Octave by typing `Control-C` (usually written `C-c` for short). `C-c` gets its name from the fact that you type it by holding down `CTRL` and then pressing `c`. Doing this will normally return you to Octave's prompt.

To exit Octave, type `quit`, or `exit` at the Octave prompt.

On systems that support job control, you can suspend Octave by sending it a `SIGTSTP` signal, usually by typing `C-z`.

## 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, I recommend that you try these examples to begin learning Octave by using it. Lines marked with ``octave:13>'` are lines you type, ending each with a carriage return. Octave will respond with an answer, or by displaying a graph.

### Creating a Matrix

To create a new matrix and store it in a variable so that it 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. Ending a command with a semicolon tells Octave to not print the result of a 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 any variable, simply type the name of the variable. For example, to display the value stored in the matrix `b`, type the command

    octave:3> b

### Matrix Arithmetic

Octave has a convenient operator notation for performing matrix arithmetic. 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

To form the matrix product type the command

    octave:6> a' * a

### Solving Linear Equations

To solve the set of linear equations `ax = b`, use the left division operator, ``\'`:

    octave:7> a \ b

This is conceptually equivalent to 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.

### Integrating Differential Equations

Octave has built-in functions for solving nonlinear differential equations of the form

For Octave to integrate equations of this form, you must first provide a definition of the function 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:8> 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

    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)

    t = linspace (0, 50, 200)';

it is easy to integrate the set of differential equations:

    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.

### Producing Graphical Output

To display the solution of the previous example graphically, use the command

    plot (t, x)

If you are using the X Window System, Octave will automatically create a separate window to display the plot. If you are using a terminal that supports some other graphics commands, you will need to tell Octave what kind of terminal you have. Type the command

    gset term

to see a list of the supported terminal types. Octave uses `gnuplot` to display graphics, and can display graphics on any terminal that is supported by `gnuplot`.

To capture the output of the plot command in a file rather than sending the output directly to your terminal, you can use a set of commands like this

    gset term postscript
    gset output "foo.ps"
    replot

This will work for other types of output devices as well. Octave's `gset` command is really just piped to the `gnuplot` subprocess, so that once you have a plot on the screen that you like, you should be able to do something like this to create an output file suitable for your graphics printer.

Or, you can eliminate the intermediate file by using commands like this

    gset term postscript
    gset output "|lpr -Pname_of_your_graphics_printer"
    replot

### 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, type `Control-p` (usually written `C-p` for short). `C-p` gets its name from the fact that you type it by holding down `CTRL` and then pressing `p`. 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 in section Command Line Editing.

### Getting Help

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. This name of the function may not always be obvious, but a good place to start is to just type `help`. This will show you all the operators, reserved words, functions, built-in variables, and function files. You can then get more help on anything that is listed 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 exit 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 in section Commands for Getting Help.

## Conventions

This section explains the notational conventions that are used in this manual. You may want to skip this section and refer back to it later.

### Fonts

Examples of Octave code appear in this font or form: `svd (a)`. Names that represent arguments or metasyntactic variables appear in this font or form: first-number. Commands that you type at the shell prompt sometimes appear in this font or form: ``octave --no-init-file'`. Commands that you type at the Octave prompt sometimes appear in this font or form: `foo --bar --baz`. Specific keys on your keyboard appear in this font or form: `ANY`.

### Evaluation Notation

In the examples in this manual, results from expressions that you evaluate are indicated with ``=>'`. For example,

    sqrt (2)
         => 1.4142

You can read this as "`sqrt (2)` evaluates to 1.4142".

In some cases, matrix values that are returned by expressions are displayed like this

    [1, 2; 3, 4] == [1, 3; 2, 4]
         => [ 1, 0; 0, 1 ]

and in other cases, they are displayed like this

    eye (3)
         =>  1  0  0
             0  1  0
             0  0  1

in order to clearly show the structure of the result.

Sometimes to help describe one expression, another expression is shown that produces identical results. The exact equivalence of expressions is indicated with ``=='`. For example,

    rot90 ([1, 2; 3, 4], -1)
    ==
    rot90 ([1, 2; 3, 4], 3)
    ==
    rot90 ([1, 2; 3, 4], 7)

### Printing Notation

Many of the examples in this manual print text when they are evaluated. Examples in this manual indicate printed text with ``-|'`. The value that is returned by evaluating the expression (here `1`) is displayed with ``=>'` and follows on a separate line.

    printf ("foo %s\n", "bar")
         -| foo bar
         => 1

### Error Messages

Some examples signal errors. This normally displays an error message on your terminal. Error messages are shown on a line starting with `error:`.

    struct_elements ([1, 2; 3, 4])
    error: struct_elements: wrong type argument `matrix'

### Format of Descriptions

Functions, commands, and variables are described in this manual in a uniform format. The first line of a description contains the name of the item followed by its arguments, if any. The category--function, variable, or whatever--is printed next to the right margin. The description follows on succeeding lines, sometimes with examples.

#### A Sample Function Description

In a function description, the name of the function being described appears first. It is followed on the same line by a list of parameters. The names used for the parameters are also used in the body of the description.

Here is a description of an imaginary function `foo`:

Function: foo (x, y, ...)
  The function `foo` subtracts x from y, then adds the remaining arguments to the result. If y is not supplied, then the number 19 is used by default.

      foo (1, [3, 5], 3, 9)
           => [ 14, 16 ]
      foo (5)
           => 14

  More generally,

      foo (w, x, y, ...)
      ==
      x - w + y + ...

Any parameter whose name contains the name of a type (e.g., integer, integer1 or matrix) is expected to be of that type. Parameters named object may be of any type. Parameters with other sorts of names (e.g., new_file) are discussed specifically in the description of the function. In some sections, features common to parameters of several functions are described at the beginning.

Functions in Octave may be defined in several different ways. The catagory name for functions may include another name that indicates the way that the function is defined. These additional tags include

Built-in Function
  The function described is written in a language like C++, C, or Fortran, and is part of the compiled Octave binary.
Loadable Function
  The function described is written in a language like C++, C, or Fortran. On systems that support dynamic linking of user-supplied functions, it may be automatically linked while Octave is running, but only if it is needed. See section Dynamically Linked Functions.
Function File
  The function described is defined using Octave commands stored in a text file. See section Function Files.
Mapping Function
  The function described works element-by-element for matrix and vector arguments.

#### A Sample Command Description

Command descriptions have a format similar to function descriptions, except that the word `Function' is replaced by `Command. Commands are functions that may called without surrounding their arguments in parentheses. For example, here is the description for Octave's `cd` command:

Command: cd dir
Command: chdir dir
  Change the current working directory to dir. For example, `cd ~/octave` changes the current working directory to ``~/octave'`. If the directory does not exist, an error message is printed and the working directory is not changed.

#### A Sample Variable Description

A variable is a name that can hold a value. Although any variable can be set by the user, built-in variables typically exist specifically so that users can change them to alter the way Octave behaves (built-in variables are also sometimes called user options). Ordinary variables and built-in variables are described using a format like that for functions except that there are no arguments.

Here is a description of the imaginary variable `do_what_i_mean_not_what_i_say`.

Built-in Variable: do_what_i_mean_not_what_i_say
  If the value of this variable is nonzero, Octave will do what you actually wanted, even if you have typed a completely different and meaningless list of commands.

Other variable descriptions have the same format, but `Built-in Variable' is replaced by `Variable', for ordinary variables, or `Constant' for symbolic constants whose values cannot be changed.

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