GNU Octave
GNU Octave is software featuring a high-level programming language, primarily intended for numerical computations. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License.

History
The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Real development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 6.4.0 was released on Oct 30, 2021.

The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations.

Development history

Developments
In addition to use on desktops for personal scientific computing, Octave is used in academia and industry. For example, Octave was used on a massive parallel computer at Pittsburgh Supercomputing Center to find vulnerabilities related to guessing social security numbers.

Acceleration with OpenCL or CUDA is also possible with use of GPUs.

Technical details
 Octave is written in C++ using the C++ standard library.
 Octave uses an interpreter to execute the Octave scripting language.
 Octave is extensible using dynamically loadable modules.
 Octave interpreter has an OpenGL-based graphics engine to create plots, graphs and charts and to save or print them. Alternatively, gnuplot can be used for the same purpose.
 Octave includes a Graphical User Interface (GUI) in addition to the traditional Command Line Interface (CLI); see #User interfaces for details.

Octave, the language
The Octave language is an interpreted programming language. It is a structured programming language (similar to C) and supports many common C standard library functions, and also certain UNIX system calls and functions. However, it does not support passing arguments by reference although function arguments are copy-on-write to avoid unnecessary duplication. 

Octave programs consist of a list of function calls or a script. The syntax is matrix-based and provides various functions for matrix operations. It supports various data structures and allows object-oriented programming.

Its syntax is very similar to MATLAB, and careful programming of a script will allow it to run on both Octave and MATLAB.

Because Octave is made available under the GNU General Public License, it may be freely changed, copied and used. The program runs on Microsoft Windows and most Unix and Unix-like operating systems, including Linux, Android, and macOS.

Notable features

Command and variable name completion
Typing a TAB character on the command line causes Octave to attempt to complete variable, function, and file names (similar to Bash's tab completion). Octave uses the text before the cursor as the initial portion of the name to complete.

Command history
When running interactively, Octave saves the commands typed in an internal buffer so that they can be recalled and edited.

Data structures
Octave includes a limited amount of support for organizing data in structures. In this example, we see a structure "x" with elements "a", "b", and "c", (an integer, an array, and a string, respectively):
octave:1> x.a = 1; x.b = [1, 2; 3, 4]; x.c = "string";
octave:2> x.a
ans =  1
octave:3> x.b
ans =

   1   2
   3   4

octave:4> x.c
ans = string
octave:5> x
x =
{
  a =  1
  b =

     1   2
     3   4

  c = string
}

Short-circuit Boolean operators
Octave's && and || logical operators are evaluated in a short-circuit fashion (like the corresponding operators in the C language), in contrast to the element-by-element operators & and |.

Increment and decrement operators

Octave includes the C-like increment and decrement operators ++ and -- in both their prefix and postfix forms.
Octave also does augmented assignment, e.g.  x += 5.

Unwind-protect
Octave supports a limited form of exception handling modelled after the unwind_protect of Lisp. The general form of an unwind_protect block looks like this:
unwind_protect
   body
unwind_protect_cleanup
   cleanup
end_unwind_protect

As a general rule, GNU Octave recognizes as termination of a given block either the keyword end (which is compatible with the MATLAB language) or a more specific keyword end_block. As a consequence, an unwind_protect block can be terminated either with the keyword end_unwind_protect as in the example, or with the more portable keyword end.

The cleanup part of the block is always executed. In case an exception is raised by the body part, cleanup is executed immediately before propagating the exception outside the block unwind_protect.

GNU Octave also supports another form of exception handling (compatible with the MATLAB language):
try
   body
catch
   exception_handling
end

This latter form differs from an unwind_protect block in two ways. First, exception_handling is only executed when an exception is raised by body. Second, after the execution of exception_handling the exception is not propagated outside the block (unless a rethrow( lasterror ) statement is explicitly inserted within the exception_handling code).

Variable-length argument lists
Octave has a mechanism for handling functions that take an unspecified number of arguments without explicit upper limit. To specify a list of zero or more arguments, use the special argument varargin as the last (or only) argument in the list.

function s = plus (varargin)
   if (nargin==0)
      s = 0;
   else
      s = varargin{1} + plus (varargin{2:nargin});
   end
end

Variable-length return lists
A function can be set up to return any number of values by using the special return value varargout. For example:

function varargout = multiassign (data)
   for k=1:nargout
      varargout{k} = data(:,k);
   end
end

C++ integration
It is also possible to execute Octave code directly in a C++ program. For example, here is a code snippet for calling rand([10,1]):
#include <octave/oct.h>
...
ColumnVector NumRands(2);
NumRands(0) = 10;
NumRands(1) = 1;
octave_value_list f_arg, f_ret;
f_arg(0) = octave_value(NumRands);
f_ret = feval("rand", f_arg, 1);
Matrix unis(f_ret(0).matrix_value());

C and C++ code can be integrated into GNU Octave by creating oct files, or using the MATLAB compatible MEX files.

MATLAB compatibility
Octave has been built with MATLAB compatibility in mind, and shares many features with MATLAB:

 Matrices as fundamental data type.
 Built-in support for complex numbers.
 Powerful built-in math functions and extensive function libraries.
 Extensibility in the form of user-defined functions.

Octave treats incompatibility with MATLAB as a bug; therefore, it could be considered a software clone, which does not infringe software copyright as per Lotus v. Borland court case.

MATLAB scripts from the MathWorks' FileExchange repository in principle are compatible with Octave. However, while they are often provided and uploaded by users under an Octave compatible and proper open source BSD license, the FileExchange Terms of use prohibit any usage beside MathWorks' proprietary MATLAB.

Syntax compatibility
There are a few purposeful, albeit minor, syntax additions:

 Comment lines can be prefixed with the # character as well as the % character;
 Various C-based operators ++, --, +=, *=, /= are supported;
 Elements can be referenced without creating a new variable by cascaded indexing, e.g. [1:10](3);
 Strings can be defined with the double-quote " character as well as the single-quote ' character;
 When the variable type is single (a single-precision floating-point number), Octave calculates the "mean" in the single-domain (MATLAB in double-domain) which is faster but gives less accurate results;
 Blocks can also be terminated with more specific Control structure keywords, i.e., endif, endfor, endwhile, etc.;
 Functions can be defined within scripts and at the Octave prompt;
 Presence of a do-until loop (similar to do-while in C).

Function compatibility
Many, but not all, of the numerous MATLAB functions are available in GNU Octave, some of them accessible through packages in Octave Forge. The functions available as part of either core Octave or Forge packages are listed online.

A list of unavailable functions is included in the Octave function  __unimplemented.m__. Unimplemented functions are also listed under many Octave Forge packages in the Octave Wiki.

When an unimplemented function is called the following error message is shown:
  octave:1> guide
  warning: the 'guide' function is not yet implemented in Octave

  Please read <http://www.octave.org/missing.html> to learn how you can contribute missing functionality.
  error: 'guide' undefined near line 1 column 1

User interfaces
Octave comes with an official graphical user interface (GUI) and an integrated development environment (IDE) based on Qt. It has been available since Octave 3.8, and has become the default interface (over the command line interface) with the release of Octave 4.0. 
It was well-received by EDN contributor, who said "[Octave] now has a very workable GUI."

Several 3rd-party graphical front-ends have also been developed, like ToolboX for coding education.

GUI applications 
With Octave code, the user can create GUI applications GUI Development (GNU Octave (version 6.3.0)). Below are some examples:

Button, edit control, checkbox# create figure and panel on it
f = figure;
# create a button (default style)
b1 = uicontrol (f, "string", "A Button", "position",[10 10 150 40]);
# create an edit control
e1 = uicontrol (f, "style", "edit", "string", "editable text", "position",[10 60 300 40]);
# create a checkbox
c1 = uicontrol (f, "style", "checkbox", "string", "a checkbox", "position",[10 120 150 40]);Textboxprompt = {"Width", "Height", "Depth"};
defaults = {"1.10", "2.20", "3.30"};
rowscols = [1,10; 2,20; 3,30];
dims = inputdlg (prompt, "Enter Box Dimensions", rowscols, defaults);Listbox with message boxes.my_options = {"An item", "another", "yet another"};
[sel, ok] = listdlg ("ListString", my_options, "SelectionMode", "Multiple");
if (ok == 1)
  msgbox ("You selected:");
  for i = 1:numel (sel)
    msgbox (sprintf ("\t%s", my_options{sel(i)}));
  endfor
else
  msgbox ("You cancelled.");
endifRadiobuttons# create figure and panel on it
f = figure;
# create a button group
gp = uibuttongroup (f, "Position", [ 0 0.5 1 1])
# create a buttons in the group
b1 = uicontrol (gp, "style", "radiobutton", "string", "Choice 1", "Position", [ 10 150 100 50 ]);
b2 = uicontrol (gp, "style", "radiobutton", "string", "Choice 2", "Position", [ 10 50 100 30 ]);
# create a button not in the group
b3 = uicontrol (f, "style", "radiobutton","string", "Not in the group","Position", [ 10 50 100 50 ]);

Packages 
Octave also has packages available for free. Those packages are located at Octave-Forge Octave Forge - Packages. Available packages are:
 bim - Package for solving Diffusion Advection Reaction (DAR) Partial Differential Equations
 bsltl - The BSLTL package is a free collection of OCTAVE/MATLAB routines for working with the biospeckle laser technique
 cgi - Common Gateway Interface for Octave
 communications - Digital Communications, Error Correcting Codes (Channel Code), Source Code functions, Modulation and Galois Fields
 control - Computer-Aided Control System Design (CACSD) Tools for GNU Octave, based on the proven SLICOT Library
 data-smoothing - Algorithms for smoothing noisy data
 database - Interface to SQL databases, currently only postgresql using libpq
 dataframe - Data manipulation toolbox similar to R data
 dicom - Digital communications in medicine (DICOM) file io
 divand - divand performs an n-dimensional variational analysis (interpolation) of arbitrarily located observations
 doctest - The Octave-Forge Doctest package finds specially-formatted blocks of example code within documentation files
 econometrics - Econometrics functions including MLE and GMM based techniques
 fem-fenics - pkg for the resolution of partial differential equations based on fenics
 financial - Monte Carlo simulation, options pricing routines,  financial manipulation, plotting functions and additional date manipulation tools
 fits - The Octave-FITS package provides functions for reading, and writing FITS (Flexible Image Transport System) files
 fpl - Collection of routines to export data produced by Finite Elements or Finite Volume Simulations in formats used by some visualization programs
 fuzzy-logic toolkit - A mostly MATLAB-compatible fuzzy logic toolkit for Octave (fails to install due to long-standing bug)
 ga - Genetic optimization code
 general - General tools for Octave
 generate_html - This package provides functions for generating HTML pages that contain the help texts for a set of functions
 geometry - Library for geometric computing extending MatGeom functions
 gsl - Octave bindings to the GNU Scientific Library
 image - The Octave-forge Image package provides functions for processing images
 image-acquisition - The Octave-forge Image Acquisition package provides functions to capture images from connected devices
 instrument-control - Low level I/O functions for serial, i2c, parallel, tcp, gpib, vxi11, udp and usbtmc interfaces
 interval - The interval package for real-valued interval arithmetic allows one to evaluate functions over subsets of their domain
 io - Input/Output in external formats e.g. Excel
 level-set - Routines for calculating the time-evolution of the level-set equation and extracting geometric information from the level-set function
 linear-algebra - Additional linear algebra code, including general SVD and matrix functions
 lssa - A package implementing tools to compute spectral decompositions of irregularly-spaced time series
 ltfat - The Large Time/Frequency Analysis Toolbox (LTFAT) is a MATLAB/Octave toolbox for working with time-frequency analysis,  wavelets and signal processing
 mapping - Simple mapping and GIS .shp and raster file functions
 mataveid - System identification package for both MATLAB and GNU Octave
 matavecontrol - Control toolbox for both MATLAB and GNU Octave
 miscellaneous - Miscellaneous tools that would fit nowhere else
 mpi - Octave bindings for basic Message Passing Interface (MPI) functions for parallel computing
 msh - Create and manage triangular and tetrahedral meshes for Finite Element or Finite Volume PDE solvers
 mvn - Multivariate normal distribution clustering and utility functions
 nan - A statistics and machine learning toolbox for data with and w/o missing values
 ncarray - Access a single or a collection of NetCDF files as a multi-dimensional array
 netcdf - A MATLAB compatible NetCDF interface for Octave
 nurbs - Collection of routines for the creation, and manipulation of Non-Uniform Rational B-Splines (NURBS), based on the NURBS toolbox by Mark Spink
 ocs - Package for solving DC and transient electrical circuit equations
 octclip - This package allows users to do boolean operations with polygons using the Greiner-Hormann algorithm
 octproj - This package allows users to call functions of PROJ
 optics - Functions covering various aspects of optics
 optim - Non-linear optimization toolkit
 optiminterp - An optimal interpolation toolbox for octave
 parallel - Parallel execution package
 quaternion - Quaternion package for GNU Octave, includes a quaternion class with overloaded operators
 queueing - The queueing package provides functions for queueing networks and Markov chains analysis
 secs1d - A Drift-Diffusion simulator for 1d semiconductor devices
 secs2d - A Drift-Diffusion simulator for 2d semiconductor devices
 secs3d - A Drift-Diffusion simulator for 3d semiconductor devices
 signal - Signal processing tools, including filtering, windowing and display functions
 sockets - Socket functions for networking from within octave
 sparsersb - Interface to the librsb package implementing the RSB sparse matrix format for fast shared-memory sparse matrix computations
 splines - Additional spline functions
 statistics - Additional statistics functions for Octave
 stk - The STK is a (not so) Small Toolbox for Kriging
 strings - Additional functions for manipulation and analysis of strings
 struct - Additional structure manipulation functions
 symbolic - The Octave-Forge Symbolic package adds symbolic calculation features to GNU Octave
 tisean - Port of TISEAN 3
 tsa - Stochastic concepts and maximum entropy methods for time series analysis
 vibes - The VIBes API allows one to easily display results (boxes, pavings)  from interval methods
 video - A wrapper for ffmpeg's libavformat and libavcodec, implementing addframe, avifile, aviinfo and aviread
 vrml - 3D graphics using VRML
 windows - Provides COM interface and additional functionality on Windows
 zeromq - ZeroMQ bindings for GNU Octave

Comparison with other similar software
Other free alternatives to MATLAB include Scilab and FreeMat. Octave is more compatible with MATLAB than Scilab is, and FreeMat has not been updated since June 2013.

See also

List of numerical-analysis software
Comparison of numerical-analysis software
List of statistical packages
List of numerical libraries
 MATLAB

Notes

References

Further reading

External links

 

Array programming languages
Articles with example MATLAB/Octave code
Cross-platform free software
Data analysis software
Data mining and machine learning software
Free educational software
Free mathematics software
Free software programmed in C++
Octave
High-priority free software projects
Numerical analysis software for Linux
Numerical analysis software for MacOS
Numerical analysis software for Windows
Numerical programming languages
Science software that uses Qt
Software that uses Qt