**MATLAB** (*matrix laboratory*) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, C#, Java, Fortran and Python. The heart of MATLAB is the MATLAB language, a matrix-based language allowing the most natural expression of computational mathematics.

Although MATLAB is intended primarily for numerical computing, an optional toolbox uses the MuPAD symbolic engine, allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems.

As of 2018, MATLAB has more than 3 million users worldwide. MATLAB users come from various backgrounds of engineering, science, and economics.

*MATLAB combines a desktop environment tuned for iterative analysis and design processes with a programming language that expresses matrix and array mathematics directly*.

## What can you do with MATLAB?

Using MATLAB, you can:

- Analyze data
- Develop algorithms
- Create models and applications

The language, apps, and built-in math functions enable you to quickly explore multiple approaches to arrive at a solution. MATLAB lets you take your ideas from research to production by deploying to enterprise applications and embedded devices, as well as integrating with Simulink^{®} and Model-Based Design.

## Who uses MATLAB?

Millions of engineers and scientists in industry and academia use MATLAB. You can use MATLAB for a range of applications, including deep learning and machine learning, signal processing and communications, image and video processing, control systems, test and measurement, computational finance, and computational biology.

## History

Cleve Moler, the chairman of the computer science department at the University of New Mexico, started developing MATLAB in the late 1970s.^{[} He designed it to give his students access to LINPACK and EISPACK without them having to learn Fortran. It soon spread to other universities and found a strong audience within the applied mathematics community. Jack Little, an engineer, was exposed to it during a visit Moler made to Stanford University in 1983. Recognizing its commercial potential, he joined with Moler and Steve Bangert. They rewrote MATLAB in C and founded MathWorks in 1984 to continue its development. These rewritten libraries were known as JACKPAC.^{[} In 2000, MATLAB was rewritten to use a newer set of libraries for matrix manipulation, LAPACK.^{[}

MATLAB was first adopted by researchers and practitioners in control engineering, Little’s specialty, but quickly spread to many other domains. It is now also used in education, in particular the teaching of linear algebra and numerical analysis, and is popular amongst scientists involved in image processing.^{[}

## Syntax

The MATLAB application is built around the MATLAB scripting language. Common usage of the MATLAB application involves using the Command Window as an interactive mathematical shell or executing text files containing MATLAB code.^{[}

### Variables

Variables are defined using the assignment operator, `=`

. MATLAB is a weakly typed programming language because types are implicitly converted. It is an inferred typed language because variables can be assigned without declaring their type, except if they are to be treated as symbolic objects, and that their type can change. Values can come from constants, from computation involving values of other variables, or from the output of a function. For example:

>>x = 17 x = 17>>x = 'hat' x = hat>>x = [3*4, pi/2] x = 12.0000 1.5708>>y = 3*sin(x) y = -1.6097 3.0000

### Vectors and matrices

A simple array is defined using the colon syntax: *initial*`:`

*increment*`:`

*terminator*. For instance:

>> array = 1:2:9 array = 1 3 5 7 9

defines a variable named `array`

(or assigns a new value to an existing variable with the name `array`

) which is an array consisting of the values 1, 3, 5, 7, and 9. That is, the array starts at 1 (the *initial* value), increments with each step from the previous value by 2 (the *increment* value), and stops once it reaches (or to avoid exceeding) 9 (the *terminator* value).

>> array = 1:3:9 array = 1 4 7

the *increment* value can actually be left out of this syntax (along with one of the colons), to use a default value of 1.

>> ari = 1:5 ari = 1 2 3 4 5

assigns to the variable named `ari`

an array with the values 1, 2, 3, 4, and 5, since the default value of 1 is used as the incrementer.

Indexing is one-based,^{[} which is the usual convention for matrices in mathematics, although not for some programming languages such as C, C++, and Java.

Matrices can be defined by separating the elements of a row with blank space or comma and using a semicolon to terminate each row. The list of elements should be surrounded by square brackets: []. Parentheses: () are used to access elements and subarrays (they are also used to denote a function argument list).

>> A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1] A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 >> A(2,3) ans = 11

Sets of indices can be specified by expressions such as “2:4”, which evaluates to [2, 3, 4]. For example, a submatrix taken from rows 2 through 4 and columns 3 through 4 can be written as:

>> A(2:4,3:4) ans = 11 8 7 12 14 1

A square identity matrix of size *n* can be generated using the function *eye*, and matrices of any size with zeros or ones can be generated with the functions *zeros* and *ones*, respectively.

>> eye(3,3) ans = 1 0 0 0 1 0 0 0 1 >> zeros(2,3) ans = 0 0 0 0 0 0 >> ones(2,3) ans = 1 1 1 1 1 1

Transposing a vector or a matrix is done either by the function *transpose* or by adding prime after a dot to the matrix. Without the dot MATLAB will perform conjugate transpose.

>> A = [1 ; 2], B = A.', C = transpose(A) A = 1 2 B = 1 2 C = 1 2 >> D = [0 3 ; 1 5], D.' D = 0 3 1 5 ans = 0 1 3 5

Most MATLAB functions can accept matrices and will apply themselves to each element. For example, `mod(2*J,n)`

will multiply every element in “J” by 2, and then reduce each element modulo “n”. MATLAB does include standard “for” and “while” loops, but (as in other similar applications such as R), using the vectorized notation often produces code that is faster to execute. This code, excerpted from the function *magic.m*, creates a magic square *M* for odd values of *n* (MATLAB function `meshgrid`

is used here to generate square matrices I and J containing 1:n).

[J,I] = meshgrid(1:n); A = mod(I + J - (n + 3) / 2, n); B = mod(I + 2 * J - 2, n); M = n * A + B + 1;

### Structures

MATLAB has structure data types.^{[} Since all variables in MATLAB are arrays, a more adequate name is “structure array”, where each element of the array has the same field names. In addition, MATLAB supports dynamic field names^{[} (field look-ups by name, field manipulations, etc.). Unfortunately, MATLAB JIT does not support MATLAB structures, therefore just a simple bundling of various variables into a structure will come at a cost.^{[}

### Functions

When creating a MATLAB function, the name of the file should match the name of the first function in the file. Valid function names begin with an alphabetic character, and can contain letters, numbers, or underscores. Functions are often case sensitive.

### Function handles

MATLAB supports elements of lambda calculus by introducing function handles, or function references, which are implemented either in .m files or anonymous/nested functions.^{[}

### Classes and object-oriented programming

MATLAB supports object-oriented programming including classes, inheritance, virtual dispatch, packages, pass-by-value semantics, and pass-by-reference semantics. However, the syntax and calling conventions are significantly different from other languages. MATLAB has value classes and reference classes, depending on whether the class has *handle* as a super-class (for reference classes) or not (for value classes).^{[}

Method call behavior is different between value and reference classes. For example, a call to a method

object.method();

can alter any member of *object* only if *object* is an instance of a reference class.

An example of a simple class is provided below.

classdefhellomethodsfunctiongreet(this) disp('Hello!')endendend

When put into a file named hello.m, this can be executed with the following commands:

>>x = hello;>>x.greet(); Hello!

## Graphics and graphical user interface programming

MATLAB supports developing applications with graphical user interface (GUI) features. MATLAB includes GUIDE (GUI development environment) for graphically designing GUIs.^{[} It also has tightly integrated graph-plotting features. For example, the function *plot* can be used to produce a graph from two vectors *x* and *y*. The code:

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

produces the following figure of the sine function:

A MATLAB program can produce three-dimensional graphics using the functions *surf*, *plot3* or *mesh*.

[X,Y] = meshgrid(-10:0.25:10,-10:0.25:10); f = sinc(sqrt((X/pi).^2+(Y/pi).^2)); mesh(X,Y,f); axis([-10 10 -10 10 -0.3 1]) xlabel(‘{\bfx}’) ylabel(‘{\bfy}’) zlabel(‘{\bfsinc} ({\bfR})’) hidden off | [X,Y] = meshgrid(-10:0.25:10,-10:0.25:10); f = sinc(sqrt((X/pi).^2+(Y/pi).^2)); surf(X,Y,f); axis([-10 10 -10 10 -0.3 1]) xlabel(‘{\bfx}’) ylabel(‘{\bfy}’) zlabel(‘{\bfsinc} ({\bfR})’) |

This code produces a wireframe 3D plot of the two-dimensional unnormalized sinc function: | This code produces a surface 3D plot of the two-dimensional unnormalized sinc function: |

In MATLAB, graphical user interfaces can be programmed with the GUI design environment (GUIDE) tool.^{[}

## Interfacing with other languages

MATLAB can call functions and subroutines written in the programming languages C or Fortran. A wrapper function is created allowing MATLAB data types to be passed and returned. MEX files (MATLAB executables) are the dynamically loadable object files created by compiling such functions. Since 2014 increasing two-way interfacing with Python was being added.

Libraries written in Perl, Java, ActiveX or .NET can be directly called from MATLAB,^{[} and many MATLAB libraries (for example XML or SQL support) are implemented as wrappers around Java or ActiveX libraries. Calling MATLAB from Java is more complicated, but can be done with a MATLAB toolbox^{[} which is sold separately by MathWorks, or using an undocumented mechanism called JMI (Java-to-MATLAB Interface), (which should not be confused with the unrelated Java Metadata Interfacethat is also called JMI). Official MATLAB API for Java was added in 2016.^{[}

As alternatives to the MuPAD based Symbolic Math Toolbox available from MathWorks, MATLAB can be connected to Maple or Mathematica.

Libraries also exist to import and export MathML.

## License

MATLAB is a proprietary product of MathWorks, so users are subject to vendor lock-in. Although MATLAB Builder products can deploy MATLAB functions as library files which can be used with .NET or Java^{[} application building environment, future development will still be tied to the MATLAB language.

Each toolbox is purchased separately. If an evaluation license is requested, the MathWorks sales department requires detailed information about the project for which MATLAB is to be evaluated. If granted (which it often is), the evaluation license is valid for two to four weeks. A student version of MATLAB is available as is a home-use license for MATLAB, Simulink, and a subset of Mathwork’s Toolboxes at substantially reduced prices.

It has been reported that European Union (EU) competition regulators are investigating whether MathWorks refused to sell licenses to a competitor. The regulators dropped the investigation after the complainant withdrew its accusation and no evidence of wrongdoing was found.^{[}

## Alternatives

See also: list of numerical analysis software and comparison of numerical analysis software

MATLAB has a number of competitors.^{[} Commercial competitors include Mathematica, TK Solver, Maple, and IDL. There are also free open source alternatives to MATLAB, in particular GNU Octave, Scilab, FreeMat, and SageMath, which are intended to be mostly compatible with the MATLAB language; the Julia programming language also initially used MATLAB-like syntax. Among other languages that treat arrays as basic entities (array programming languages) are APL, Fortran 90 and higher, S-Lang, as well as the statistical languages R and S. There are also libraries to add similar functionality to existing languages, such as IT++ for C++, Perl Data Language for Perl, ILNumerics for .NET, NumPy/SciPy/matplotlib for Python, SciLua/Torch for Lua, SciRuby for Ruby, and Numeric.js for JavaScript.

GNU Octave is unique from other alternatives because it treats incompatibility with MATLAB as a bug (see MATLAB Compatibility of GNU Octave), therefore, making GNU Octave a superset of the MATLAB language.