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MATLAB (an abbreviation of "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 2020, MATLAB has more than 4 million users worldwide.[20] 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.[21] 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.[22] In 2000, MATLAB was rewritten to use a newer set of libraries for matrix manipulation, LAPACK.[23]

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.[21]

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.[24]

Variables are defined using the assignment operator, =. MATLAB is a weakly typed programming language because types are implicitly converted.[25] 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,[26] 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 =

>> x = 'hat'
x =

>> 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).

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,[27] 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

>> A(2,3)
ans =

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 dot-prime after the matrix (without the dot, prime will perform conjugate transpose for complex arrays):

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


MATLAB supports structure data types.[28] 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[29] (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.[30]
Function handles

MATLAB supports elements of lambda calculus by introducing function handles,[31] or function references, which are implemented either in .m files or anonymous[32]/nested functions.[33]
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.[34] 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).[35]

Method call behavior is different between value and reference classes. For example, a call to a 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
function greet(obj)

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

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

Graphics and graphical user interface programming

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 = sin(x);

produces the following figure of the sine function:

Matlab plot sin.svg

MATLAB supports three-dimensional graphics as well:

[X,Y] = meshgrid(-10:0.25:10,-10:0.25:10);
f = sinc(sqrt((X/pi).^2+(Y/pi).^2));
axis([-10 10 -10 10 -0.3 1])
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));
axis([-10 10 -10 10 -0.3 1])
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:
MATLAB mesh sinc3D.svg MATLAB surf sinc3D.svg

MATLAB supports developing graphical user interface (GUI) applications.[36] UIs can be generated either programmatically or using visual design environments such as GUIDE and App Designer.[37][38]
Interfacing with other languages

MATLAB can call functions and subroutines written in the programming languages C or Fortran.[39] 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.[40][41] Since 2014 increasing two-way interfacing with Python was being added.[42][43]

Libraries written in Perl, Java, ActiveX or .NET can be directly called from MATLAB,[44][45] 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[46] which is sold separately by MathWorks, or using an undocumented mechanism called JMI (Java-to-MATLAB Interface),[47][48] (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.[49]

As alternatives to the MuPAD based Symbolic Math Toolbox available from MathWorks, MATLAB can be connected to Maple or Mathematica.[50][51]

Libraries also exist to import and export MathML.[52]
China ban

In 2020 Chinese state media reported that MATLAB had withdrawn services from two Chinese universities as a result of US sanctions, and said this will be responded to by increased use of open-source alternatives and by developing domestic alternatives.[53]
See also: list of numerical analysis software and comparison of numerical analysis software

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

Wolfram Mathematica

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

GNU Octave

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:

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:

NumPy/SciPy/matplotlib for Python
Perl Data Language for Perl
SciLua/Torch for Lua
SciRuby for Ruby
IT++ for C++
ILNumerics for .NET
Numeric.js for JavaScript

Release history

Version[54] Release name Number Bundled JVM Year Release date Notes
MATLAB 1.0 1984
MATLAB 2 1986
MATLAB 3 1987
MATLAB 3.5 1990 Ran on DOS but needed at least a 386 processor; version 3.5m needed math coprocessor
MATLAB 4 1992 Ran on Windows 3.1x and Macintosh
MATLAB 4.2c 1994 Ran on Windows 3.1x, needed a math coprocessor
MATLAB 5.0 Volume 8 1996 December 1996 Unified releases across all platforms
MATLAB 5.1 Volume 9 1997 May 1997
MATLAB 5.1.1 R9.1
MATLAB 5.2 R10 1998 March 1998 Last version working on classic Macs
MATLAB 5.2.1 R10.1
MATLAB 5.3 R11 1999 January 1999
MATLAB 5.3.1 R11.1 November 1999
MATLAB 6.0 R12 12 1.1.8 2000 November 2000 First release with bundled Java virtual machine (JVM)
MATLAB 6.1 R12.1 1.3.0 2001 June 2001 Last release for Windows 95
MATLAB 6.5 R13 13 1.3.1 2002 July 2002
MATLAB 6.5.1 R13SP1 2003
MATLAB 6.5.2 R13SP2 Last release for Windows 98, Windows ME, IBM/AIX, Alpha/TRU64, and SGI/IRIX[55]
MATLAB 7 R14 14 1.4.2 2004 June 2004 Introduced anonymous and nested functions[56]

Re-introduced for Mac (under Mac OS X)

MATLAB 7.0.1 R14SP1 October 2004
MATLAB 7.0.4 R14SP2 1.5.0 2005 March 7, 2005 Support for memory-mapped files[57]
MATLAB 7.1 R14SP3 1.5.0 September 1, 2005 First 64-bit version available for Windows XP 64-bit
MATLAB 7.2 R2006a 15 1.5.0 2006 March 1, 2006
MATLAB 7.3 R2006b 16 1.5.0 September 1, 2006 HDF5-based MAT-file support
MATLAB 7.4 R2007a 17 1.5.0_07 2007 March 1, 2007 New bsxfun function to apply element-by-element binary operation with singleton expansion enabled[58]
MATLAB 7.5 R2007b 18 1.6.0 September 1, 2007 Last release for Windows 2000 and PowerPC Mac; License Server support for Windows Vista;[59] new internal format for P-code
MATLAB 7.6 R2008a 19 1.6.0 2008 March 1, 2008 Major enhancements to object-oriented programming abilities with a new class definition syntax,[60] and ability to manage namespaces with packages[61]
MATLAB 7.7 R2008b 20 1.6.0_04 October 9, 2008 Last release for processors w/o SSE2. New Map data structure:[62] upgrades to random number generators[63]
MATLAB 7.8 R2009a 21 1.6.0_04 2009 March 6, 2009 First release for Microsoft 32-bit & 64-bit Windows 7, new external interface to .NET Framework[64]
MATLAB 7.9 R2009b 22 1.6.0_12 September 4, 2009 First release for Intel 64-bit Mac, and last for Solaris SPARC; new use for the tilde operator (~) to ignore arguments in function calls[65][66]
MATLAB 7.9.1 R2009bSP1 1.6.0_12 2010 April 1, 2010 bug fixes.
MATLAB 7.10 R2010a 23 1.6.0_12 March 5, 2010 Last release for Intel 32-bit Mac
MATLAB 7.11 R2010b 24 1.6.0_17 September 3, 2010 Add support for enumerations[67]
MATLAB 7.11.1 R2010bSP1 1.6.0_17 2011 March 17, 2011 bug fixes and updates
MATLAB 7.11.2 R2010bSP2 1.6.0_17 April 5, 2012[68] bug fixes
MATLAB 7.12 R2011a 25 1.6.0_17 April 8, 2011 New rng function to control random number generation[69][70][71]
MATLAB 7.13 R2011b 26 1.6.0_17 September 1, 2011 Access-change parts of variables directly in MAT-files, without loading into memory;[72] increased maximum local workers with Parallel Computing Toolbox from 8 to 12[73]
MATLAB 7.14 R2012a 27 1.6.0_17 2012 March 1, 2012 Last version with 32-bit Linux support.[74]
MATLAB 8 R2012b 28 1.6.0_17 September 11, 2012 First release with Toolstrip interface;[75] MATLAB Apps.[76] redesigned documentation system
MATLAB 8.1 R2013a 29 1.6.0_17 2013 March 7, 2013 New unit testing framework[77]
MATLAB 8.2 R2013b 30 1.7.0_11 September 6, 2013[78] Built in Java Runtime Environment (JRE) updated to version 7;[79] New table data type[80]
MATLAB 8.3 R2014a 31 1.7.0_11 2014 March 7, 2014[81] Simplified compiler setup for building MEX-files; USB Webcams support in core MATLAB; number of local workers no longer limited to 12 with Parallel Computing Toolbox
MATLAB 8.4 R2014b 32 1.7.0_11 October 3, 2014 New class-based graphics engine (a.k.a. HG2);[82] tabbing function in GUI;[83] improved user toolbox packaging and help files;[84] new objects for time-date manipulations;[85] Git-Subversion integration in IDE;[86] big data abilities with MapReduce (scalable to Hadoop);[87] new py package for using Python from inside MATLAB,[88] new engine interface to call MATLAB from Python;[89] several new and improved functions: webread (RESTful web services with JSON/XML support), tcpclient (socket-based connections), histcounts, histogram, animatedline, and others
MATLAB 8.5 R2015a 33 1.7.0_60 2015 March 5, 2015
MATLAB 8.5 R2015aSP1 1.7.0_60 October 14, 2015 Last release supporting Windows XP and Windows Vista
MATLAB 8.6 R2015b 34 1.7.0_60 September 3, 2015 New MATLAB execution engine (a.k.a. LXE);[90] graph and digraph classes to work with graphs and networks;[91] MinGW-w64 as supported compiler on Windows;[92] Last version with 32-bit support
MATLAB 9.0 R2016a 35 1.7.0_60 2016 March 3, 2016 Live Scripts: interactive documents that combine text, code, and output (in the style of Literate programming);[93] App Designer: a new development environment for building apps (with new kind of UI figures, axes, and components);[94] pause execution of running programs using a Pause Button
MATLAB 9.1 R2016b 36 1.7.0_60 September 15, 2016 define local functions in scripts;[95] automatic expansion of dimensions (previously provided via explicit call to bsxfun); tall arrays for Big data;[96] new string type;[97] new functions to encode/decode JSON;[98] official MATLAB Engine API for Java[49]
MATLAB 9.2 R2017a 37 1.7.0_60 2017 March 9, 2017 MATLAB Online: cloud-based MATLAB desktop accessed in a web browser;[99] double-quoted strings; new memoize function for Memoization; expanded object properties validation;[100] mocking framework for unit testing;[101] MEX targets 64-bit by default; new heatmap function for creating heatmap charts[102]
MATLAB 9.3 R2017b 38 1.8.0_121 September 21, 2017
MATLAB 9.4 R2018a 39 1.8.0_144 2018 March 15, 2018[103]
MATLAB 9.5 R2018b 40 1.8.0_152 September 12, 2018
MATLAB 9.6 R2019a 41 1.8.0_181 2019 March 20, 2019 MATLAB Projects.
MATLAB 9.7 R2019b 42 1.8.0_202 September 11, 2019 Introduction of 'arguments' block for input validation, and enabling of dot indexing into function outputs
MATLAB 9.8 R2020a 2020 March 19, 2020 Removal of Mupad

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.[104]
See also

Comparison of numerical analysis software
List of numerical analysis software


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Gilat, Amos (2004). MATLAB: An Introduction with Applications 2nd Edition. John Wiley & Sons. ISBN 978-0-471-69420-5.
Quarteroni, Alfio; Saleri, Fausto (2006). Scientific Computing with MATLAB and Octave. Springer. ISBN 978-3-540-32612-0.
Ferreira, A.J.M. (2009). MATLAB Codes for Finite Element Analysis. Springer. ISBN 978-1-4020-9199-5.
Lynch, Stephen (2004). Dynamical Systems with Applications using MATLAB. Birkhäuser. ISBN 978-0-8176-4321-8.

External links

Official website
MATLAB at Curlie

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia



Hellenica World - Scientific Library

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