**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.

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.

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.

The MATLAB application is built around the MATLAB programming 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 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*sinx y = -1.6097 3.0000

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 increment.

Indexing is one-based, which is the usual convention for matrices in mathematics, unlike zero-based indexing commonly used in other 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 >> A2,3 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:

>> A2:4,3:4 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.

>> eye3,3 ans = 1 0 0 0 1 0 0 0 1 >> zeros2,3 ans = 0 0 0 0 0 0 >> ones2,3 ans = 1 1 1 1 1 1

Transposing a vector or a matrix is done either by the function `transpose`

or by adding dot-prime after the matrix without the dot, prime will perform conjugate transpose for complex arrays:

>> A = [1 ; 2], B = A.', C = transposeA 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 functions accept arrays as input and operate element-wise on each element. For example, `mod2*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 is encouraged and is often faster to execute. The following 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] = meshgrid1:n; A = modI + J - n + 3 / 2, n; B = modI + 2 * J - 2, n; M = n * A + B + 1;

MATLAB supports 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..

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. Variables and functions are case sensitive.

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

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, otherwise value class methods must return a new instance if it needs to modify the object.

An example of a simple class is provided below.

classdef Hello methods function greetobj disp'Hello!' end end end

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

>> x = Hello; >> x.greet; Hello!

MATLAB 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 = sinx; plotx,y

produces the following figure of the sine function:

MATLAB supports three-dimensional graphics as well:

[X,Y] = meshgrid-10:0.25:10,-10:0.25:10; f = sincsqrtX/pi.^2+Y/pi.^2; meshX,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 = sincsqrtX/pi.^2+Y/pi.^2; surfX,Y,f; axis[-10 10 -10 10 -0.3 1] xlabel'{\bfx}' ylabel'{\bfy}' zlabel'{\bfsinc} {\bfR}'

MATLAB supports developing graphical user interface GUI applications. UIs can be generated either programmatically or using visual design environments such as *GUIDE* and *App Designer*.

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 Interface that 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.

There are a number of competitors to MATLAB. Some notable examples include:

There are also free open source alternatives to MATLAB, in particular:

which are somewhat compatible with the MATLAB language. GNU Octave is unique from the others in that it aims to be drop-in compatible with MATLAB syntax-wise see MATLAB Compatibility of GNU Octave.

Among other languages that treat arrays as basic entities array programming languages are:

There are also libraries to add similar functionality to existing languages, such as:

Re-introduced for Mac under Mac OS X

The number or release number is the version reported by Concurrent License Manager program FLEXlm.

For a complete list of changes of both MATLAB and official toolboxes, consult the MATLAB release notes.