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Mathematica is a computer program used widely in scientific, engineering and mathematical fields. It is mainly known as a computer algebra system, but it has various other features for technical computing. It was originally conceived by Stephen Wolfram and developed by a team of mathematicians and programmers that he assembled and led. It is developed by Wolfram Research of Champaign, Illinois.

Mathematica is split into two parts, the "kernel" and the "front end". The kernel interprets expressions (Mathematica code) and returns result expressions. The Front End provides a document-centered GUI.

- Pattern matching on symbolic data is the core of the language
- Notebook interface allows easy access to previous calculations
- Libraries of elementary and special mathematical functions
- 2D and 3D data and function visualization tools
- Matrix and data manipulation tools including support for sparse arrays
- Solvers for systems of equations, ODEs, PDEs, recurrence relations and integrals
- Multivariate statistics libraries
- Constrained and unconstrained local and global optimization.
- A programming language supporting procedural, functional and object oriented constructs
- A toolkit for adding user interfaces to calculations and applications
- Import and export filters for data, images, video, sound, CAD, GIS, document and biomedical formats
- A collection of databases of mathematical, scientific, and socio-economic information
- Support for complex number, arbitrary precision and symbolic computation for all functions
- Technical word processing including formula editing

Documents can be structured using a hierarchy of cells, which allow for outlining and sectioning of a document and support automatic numbering index creation. Documents can be presented a slide show environment for presentations. Starting with version 3.0 of the software, notebooks and their contents are represented as Mathematica expressions that can be created, modified or analysed by Mathematica programs. This allows conversion to other formats such as TeX or XML.

The Mathematica Front End includes development tools such as a debugger, input completion and automatic syntax coloring.

The kernel and the front end communicate via the MathLink protocol. It is possible to use the kernel on one computer and the front end on another.

The standard Mathematica front end is used by default, but alternative front ends are available:

- The Wolfram Workbench is an Eclipse based IDE, introduced in 2006, that provides project based code development tools for Mathematica, including revision management, debugging, profiling, and testing.
- Microsoft Excel can be used as a front end to Mathematica.
- GUIKit allows the construction of custom interfaces to Mathematica using the JAVA Swing libraries.
- Mathematica includes a command line front end.
- JMath is a third party front end based on GNU readline that runs on UNIX-like operating systems.
- MASH makes it possible to run self contained Mathematica programs (with arguments) from the UNIX command line.

Version 5.2 (2005) added automatic multi-threading when computations are performed on modern multi-core computers. This release included CPU specific optimized libraries. In addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed.

In 2002 gridMathematica was introduced to allow user level parallel programming on heterogeneous clusters and multiprocessor systems and supports grid technology such as the Microsoft Compute Cluster Server.

- Mathematica Player Pro is a runtime version of Mathematica that will run any Mathematica application but does not allow editing or creation of the code.
- Mathematica Player is a free interactive player is provided for running Mathematica programs that have been digitally signed for non-commercial use via a Wolfram Research web service, or published on the Wolfram Demonstrations Project website. It can also view unsigned Mathematica files, but not run them.
- webMathematica allows a web browser to act as a front end to a remote Mathematica server. It is designed to allow a user written application to be remotely accessed via a browser on any platform. It may not be used to give full access to Mathematica. The current version of webMathematica (2.3) does not support Mathematica 6.

Using .NET/Link, a .NET program can ask Mathematica to perform computations; likewise, a Mathematica program can load .NET classes, manipulate .NET objects and perform method calls. This makes it possible to build .NET graphical user interfaces from within Mathematica. Similar functionality is achieved with J/Link, but with Java programs instead of .NET programs.

Communication with SQL databases is achieved through built-in support for JDBC..

Mathematica can also install web services from a WSDL description.

Out[1]= 0

So the determinant of such a matrix is 0 (i.e. it is singular).

The following numerically calculates the root of the equation e^{x} = x^{2} + 2, starting at the point x = -1.

In[2]:= FindRoot[Exp[x] == x^2 + 2, {x, -1}] Out[2]= {x -> 1.3190736768573652}

Mathematica can do integral and differential calculus, in particular evaluation of integrals in terms of special functions. For example:

In[3]:= Integrate[x/Sin[x], x]//OutputForm

Out[3]= x (Log[1 - E^{I x}] - Log[1 + E^{I x}]) + I (PolyLog[2, -E^{I x}] - PolyLog[2, E^{I x}])

Here, `E` and `I` are the fundamental constants e and i respectively, and `PolyLog[s,z]` is the polylogarithm function $mathrm\{Li\}\_sleft(zright)$.

Symbolic sums can be calculated. For example:

In[4]:= Sum[z^k/k^s, {k, 1, Infinity}]

Out[4]= PolyLog[s, z]

The most concise approach is to use one of the many specialized functions:

In[3]:= Array[GCD, {5, 5}] Out[3]= {{1, 1, 1, 1, 1}, {1, 2, 1, 2, 1}, {1, 1, 3, 1, 1}, {1, 2, 1, 4, 1}, {1, 1, 1, 1, 5}}

There are at least three other approaches to this: In[4]:= Table[GCD[x, y], {x, 1, 5}, {y, 1, 5}] Out[4]= {{1, 1, 1, 1, 1}, {1, 2, 1, 2, 1}, {1, 1, 3, 1, 1}, {1, 2, 1, 4, 1}, {1, 1, 1, 1, 5}}

An APL-style approach:

In[5]:= Outer[GCD, Range[5], Range[5]]Out[5]= {{1, 1, 1, 1, 1}, {1, 2, 1, 2, 1}, {1, 1, 3, 1, 1}, {1, 2, 1, 4, 1}, {1, 1, 1, 1, 5}}

An approach using loops: In[6]:= l1 = {}; (* initialize as empty list, since we want a list in the end *) Do[l2 = {}; Do[l2 = Append[l2, GCD[i, j]], {j, 1, 5}];

l1 = Append[l1, l2], (* append the sublist, that is, the row *){i, 1, 5}]

In[7]:= l1Out[7]= {{1, 1, 1, 1, 1}, {1, 2, 1, 2, 1}, {1, 1, 3, 1, 1}, {1, 2, 1, 4, 1}, {1, 1, 1, 1, 5}} Note that this solution is considerably larger than the previous ones.

In[8]:= x^4 + 1

Out[8]= 1+x^{4}

If the `FullForm` command is used on this expression however:

In[9]:= FullForm[x^4 + 1]

Out[9]= Plus[1, Power[x, 4]]

All objects in Mathematica, except atomic ones such as symbols, numbers, and strings, have the basic form head[e_{1}, e_{2}, ...] (which may be displayed or entered in some other fashion). For example, the head of the above expression is `Plus`. Lists have this structure too, where the head is `List`, and the elements are e_{1}, e_{2}, etc. The concept of head is defined for atomic objects as well (`Symbol` for symbols, `Integer` for whole numbers, etc.), but they have no extractable subparts.

The principle permits ordinary expressions unrelated to lists to be operated on with list operators:

In[10]:= Expand[(Cos[x] + 2 Log[x^11])/13][[2, 1]]

Out[10]= 2/13The reverse can also occur -- lists can be modified to behave like ordinary expressions: In[11]:= Map[Apply[Log, #] &, {{2, x}, {3, x}, {4, x}}] Out[11]= {Log[x]/Log[2], Log[x]/Log[3], Log[x]/Log[4]} where the

Because of this equivalence between a regular mathematical object represented in Mathematica to that of a simple list structure, some built-in Mathematica functions permit threading, where functions map themselves over lists without much further specification. Indeed, `Apply` threads itself over lists when invoked as
In[12]:= Apply[Log, {{2,x}, {3,x}, {4,x}}, 1]
Out[12]= {Log[x]/Log[2], Log[x]/Log[3], Log[x]/Log[4]}
where the third argument being a 1 specifies that `Apply` replaces the heads of its argument only at the first level in the list, which is what we want, and is equivalent to the above example.

When writing a large piece of code ("Module"), variables references are by default global rather than local and must be declared at the top rather than inline.

A regular single-user license for Mathematica 6.0 includes one year of service which includes updates, technical support, a home use license, a webMathematica Amateur license, and a Wolfram Workbench license. Discounts are available for government, charity, educational, pre-college, school, student and retiree use and depend on geographical region. Educational site licenses allow use by students at home. A license manager similar to Flexlm is available to provide efficient sharing of licenses within a group.

Wolfram Research has released the following versions of Mathematica:

- Mathematica 1.0 (1988)
- Mathematica 1.2 (1989)
- Mathematica 2.0 (1991)
- Mathematica 2.1 (1992)
- Mathematica 2.2 (1993)
- Mathematica 3.0 (1996)
- Mathematica 4.0 (1999)
- Mathematica 4.1 (2000)
- Mathematica 4.2 (2002)
- Mathematica 5.0 (2003)
- Mathematica 5.1 (2004)
- Mathematica 5.2 (2005)
- Mathematica 6.0 (2007)

- Comparison of computer algebra systems
- Comparison of numerical analysis software
- Comparison of statistical packages
- IMTEK Mathematica Supplement, an open source Mathematica add-on for Finite Element Simulation
- List of numerical analysis software
- List of statistical packages
- List of graphing software
- Mathematical software
- MathML
- MathModelica System Designer, modeling and simulation environment for Mathematica
- Theodore Gray
- Publicon

- Wolfram Mathematica Documentation Center
- comp.soft-sys.math.mathematica discussion group
- Google Directories entry containing a wide variety of useful links and tutorials
- Mathematica-users, a wiki for users of Mathematica
- Mathematica Photo Gallery, examples of art using Mathematica
- Will it rot my students' brains if they use Mathematica? by Theodore W. Gray and Jerry Glynn, excerpted from The Beginner's Guide to Mathematica V4, published by Cambridge University Press.
- " Levels Of A Mathematica Expression" by Enrique Zeleny, The Wolfram Demonstrations Project.

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Last updated on Friday October 10, 2008 at 15:21:17 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 10, 2008 at 15:21:17 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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