Mathematica
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225915578
2008-07-16T00:16:39Z
CyberSkull
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[[Category:Apple Design Award winners]]
{{otheruses2|Mathematica}}
{{Infobox Software
| name = Mathematica
| logo = [[Image:Spikey_v6-small.png|100px]]
| screenshot =
| caption =
| developer = [[Wolfram Research]]
| released = {{initial release year|1988}}
| frequently_updated = yes<!-- Release version update? Don't edit this page, just click on the version number! -->
| programming language =
| platform = [[Cross-platform]] [http://www.wolfram.com/products/mathematica/platforms/ (list)]
| language = multilingual
| genre = [[Computer algebra system|Computer algebra]], [[List of numerical analysis software|numerical computations]], [[Information visualization]], [[List_of_statistical_packages|statistics]], [[Graphical user interface| user interface creation]]
| license = [[Proprietary software|Proprietary]]
| website = [http://www.wolfram.com/products/mathematica/index.html Mathematica homepage]
}}
'''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]].<ref>[http://www.businessweek.com/magazine/content/05_40/b3953024.htm Stephen Wolfram: Simple Solutions; The iconoclastic physicist's Mathematica software nails complex puzzles], BusinessWeek, October 3, 2005.</ref>
== Overview ==
[[Image:Simple-manip-mathematica-png.png|thumb|325px|Create a plot and manipulation dialog with a line of code]]
Mathematica is split into two parts, the [[kernel (computer science)|"kernel"]] and the "front end". The kernel interprets expressions (Mathematica code) and returns result expressions. The Front End provides a document centered [[GUI]].
=== Features ===
Some features of Mathematica include<ref>[http://reference.wolfram.com Mathematica documentation]</ref>:
* Pattern matching on symbolic data is the core of the language
* Notebook interface allows easy access to previous calculations
* Libraries of elementary and [[Special functions|special]] mathematical functions
* 2D and 3D data and function [[Visualization (graphic)|visualization]] tools
* Matrix and data manipulation tools including support for [[sparse array]]s
* Solvers for systems of equations, [[Ordinary differential equation|ODEs]], [[Partial differential equation|PDEs]], [[Recurrence relation|recurrence relations]] and integrals
* Multivariate [[statistics]] libraries
* Constrained and unconstrained local and global [[Optimization (mathematics)|optimization]].
* A programming language supporting [[Procedural programming|procedural]], [[Functional programming|functional]] and object oriented constructs
* A toolkit for adding [[user interface]]s to calculations and applications
* Import and export filters for data, images, video, sound, [[CAD]], [[GIS]], document and biomedical formats.
* A database containing economic, scientific, mathematical, and other information in a format that is easy to access within Mathematica
* Support for [[complex number]], [[Arbitrary precision arithmetic|arbitrary precision]] and symbolic computation for all functions
* Technical word processing
== Front ends ==
The Mathematica Front End provides an interface which allows the creation and editing of Notebook documents which can contain program code with [[prettyprint]]ing, formatted text together with results including typeset mathematics, graphics, GUI components, tables and sounds. All contents and formatting can be generated algorithmically or interactively edited and most standard word processing capabilities are supported but only one level of "undo" is supported.
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 [[Mathematica#Connections_with_other_applications|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 (software)|Eclipse]] based [[Integrated development environment|IDE]], introduced in 2006, that provides project based code development tools for Mathematica, including revision management, debugging, profiling, and testing.<ref>[http://www.macworld.com/news/2006/06/21/workbench/index.php MacWorld review of Wolfram Workbench]</ref>
* [[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.<ref>[http://robotics.caltech.edu/~radford/jmath/ JMath website]</ref>
* MASH makes it possible to run self contained Mathematica programs (with arguments) from the UNIX command line.<ref>[http://ai.eecs.umich.edu/people/dreeves/mash/ MASH website]</ref>
== High-performance computing ==
In recent years the capabilities for [[High-performance computing]] have been extended with the introduction of packed arrays (version 4, 1999) <ref>[http://goliath.ecnext.com/premium/0199/0199-1526706.html Math software packs new power; new programs automate such tedious processes as solving nonlinear differential equations and converting units] by Agnes Shanley, ''Chemical Engineering'', March 1, 2002.</ref>, sparse matrices (version 5, 2003)<ref>[http://www.accessmylibrary.com/coms2/summary_0286-9587712_ITM Mathematica 5.1: additional features make software well-suited for operations research professionals] by ManMohan S. Sodhi, ''OR/MS Today'', December 1, 2004.</ref>, and by adopting the free [[GNU Multi-Precision Library]] to evaluate high-precision arithmetic.
Version 5.2 (2005) added automatic multi-threading when computations are performed on modern multi-core computers.<ref>[http://www.accessmylibrary.com/coms2/summary_0286-12336000_ITM The 21st annual Editors' Choice Awards], Macworld, February 1, 2006.</ref> This release included CPU specific optimized libraries. In addition Mathematica is supported by third party specialist acceleration hardware such as [[ClearSpeed]].<ref>[http://www.thefreelibrary.com/ClearSpeed+Advance(TM)+Accelerator+Boards+Certified+by+Wolfram...-a0147498410 ClearSpeed Advance(TM) Accelerator Boards Certified by Wolfram Research; Math Coprocessors Enable Mathematica Users to Quadruple Performance.]</ref>
In 2002 [[gridMathematica]] was introduced to allow user level [[Parallel_computing|parallel programming]] on heterogeneous clusters and multiprocessor systems and supports grid technology such as the [[Windows_Server_2003|Microsoft Compute Cluster Server]].<ref>[http://www.macworld.com/news/2002/11/20/mathematica/index.php gridMathematica offers parallel computing solution] by Dennis Sellers, MacWorld, November 20, 2002.</ref>
== Development ==
Several solutions are available for deploying applications written in Mathematica:
* 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.<ref>[http://www.gizmag.com/mathematica-player-pro-new-application-delivery-system-for-mathematica/9096/ Mathematica Player Pro - new Application Delivery System for Mathematica] www.gizmag.com</ref>
* 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.
== Connections with other applications ==
Communication with other applications occurs through a protocol called [http://www.wolfram.com/solutions/mathlink/mathlink.html MathLink]. It allows communication between the Mathematica kernel and front-end, and also provides a general interface between the kernel and other applications. Wolfram Research freely distributes a developer kit for linking applications written in the [[C (programming language)|C programming language]] to the Mathematica kernel through ''MathLink''.<ref>[http://www.accessmylibrary.com/coms2/summary_0286-9256461_ITM New Mathematica: faster, leaner, linkable and QuickTime-compatible: MathLink kit allows ties to other apps. (Wolfram Research Inc. ships Mathematica 2.1, new QuickTime-compatible version of Mathematica software)] by Daniel Todd, MacWeek, June 15, 1992.</ref> Two other components of Mathematica, whose underlying protocol is ''MathLink'', allow developers to establish communication between the kernel and a .NET or [[Java (programming language)|Java]] program: ''.NET/Link''.<ref>[http://www.wolfram.com/solutions/mathlink/netlink/ .NET/Link]: .NET/Link is a toolkit that integrates Mathematica and the Microsoft .NET Framework.</ref> and ''J/Link''<ref>[http://www.macworld.com/2002/11/reviews/mathematica/ Mathematica 4.2: Feature-Rich Math Program Integrates with the Web, Adds Full Java Support] by Charles Seiter, ''Macworld'', November 1, 2002.</ref>
Using ''.NET/Link'', a .NET program can ask Mathematica to perform computations; likewise, a Mathematica program can load .NET [[class (computer science)|classes]], manipulate .NET objects and perform method calls. This makes it possible to build .NET [[graphical user interface]]s 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]].<ref>[http://www.databasejournal.com/news/article.php/3453911 Mathematica 5.1 Available ], Database Journal, Jan 3, 2005.</ref>.
Mathematica can also install [[Web_services|web services]] from a [[Web Services Description Language|WSDL]] description<ref>[http://www.w3.org/Math/Documents/Notes/services.xml Mathematical Web Services: W3C Note 1 August 2003]</ref><ref>[http://reference.wolfram.com/mathematica/WebServices/tutorial/Introduction.html Introduction to Web Services], Mathematica Web Services Tutorial</ref>.
== Language examples ==
The following Mathematica sequence will find the [[determinant]] of the 6×6 [[matrix (mathematics)|matrix]] whose ''i'', ''j''<nowiki>'</nowiki>th entry contains ''ij''.
<small>In[1]:=</small> Det[Array[Times, {6, 6}]]
<small>Out[1]=</small> 0
So the determinant of such a matrix is 0 (i.e. it is [[Invertible Matrix|singular]]).
The following numerically calculates the root of the equation ''e''<sup>''x''</sup> = ''x''<sup>2</sup> + 2, starting at the point ''x'' = -1.
<small>In[2]:=</small> FindRoot[Exp[x] == x^2 + 2, {x, -1}]
<small>Out[2]=</small> {x -> 1.3190736768573652}
Mathematica can do integral and differential [[calculus]], in particular evaluation of integrals in terms of [[special functions]]. For example:
<small>In[3]:=</small> Integrate[x/Sin[x], x]//OutputForm
<small>Out[3]=</small> x (Log[1 - E<sup>I x</sup>] - Log[1 + E<sup>I x</sup>]) + I (PolyLog[2, -E<sup>I x</sup>] - PolyLog[2, E<sup>I x</sup>])
Here, <tt>E</tt> and <tt>I</tt> are the fundamental constants [[e (mathematical constant)|e]] and [[imaginary unit|i]] respectively, and <tt>PolyLog[s,z]</tt> is the [[Polylogarithm|polylogarithm function]] <math>\mathrm{Li}_s\left(z\right)</math>.
Symbolic sums can be calculated. For example:
<small>In[4]:=</small> Sum[z^k/k^s, {k, 1, Infinity}]
<small>Out[4]=</small> PolyLog[s, z]
=== Multiple programming paradigms ===
Mathematica permits multiple approaches to programming. Consider this example: we want a table of values of gcd(''x'', ''y'') for 1 ≤ ''x'' ≤ 5, 1 ≤ ''y'' ≤ 5.
The most concise approach is to use one of the many specialized functions:
<small>In[3]:=</small> Array[GCD, {5, 5}]
<small>Out[3]=</small> {{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:
<small>In[4]:=</small> Table[GCD[x, y], {x, 1, 5}, {y, 1, 5}]
<small>Out[4]=</small> {{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 (programming language) | APL]]-style approach:
<small>In[5]:=</small> Outer[GCD, Range[5], Range[5]]
<small>Out[5]=</small> {{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}}
<tt>Outer</tt> corresponds to the generalized outer product operator, <tt>Range</tt> corresponds to the iota operator. <tt>Outer</tt> admits general functions, whether they be named, or ''anonymous''. Anonymous functions are specified by using <tt>#''n''</tt> to as the function argument and appending an &. The above function could be equivalently specified as <tt>Outer[GCD[#1, #2] &, Range[5], Range[5]]</tt>.
An approach using loops:
<small>In[6]:=</small> 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}]
<small>In[7]:=</small> l1
<small>Out[7]=</small> {{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}}
Observe that this solution is considerably larger than the previous ones.
=== Common structures and manipulations ===
One guiding principle in Mathematica is a unified structure behind almost all objects representable in Mathematica. For example, the expression <math>x^4+1</math> if entered will be represented as if it were written:
<small>In[8]:=</small> x^4 + 1
<small>Out[8]=</small> 1+x<sup>4</sup>
If the <tt>FullForm</tt> command is used on this expression however:
<small>In[9]:=</small> FullForm[x^4 + 1]
<small>Out[9]=</small> Plus[1, Power[x, 4]]
All objects in Mathematica, except atomic ones such as symbols, numbers, and strings, have the basic form ''head''[''e''<sub>1</sub>, ''e''<sub>2</sub>, ''...''] (which may be displayed or entered in some other fashion). For example, the head of the above expression is <tt>Plus</tt>. Lists have this structure too, where the head is <tt>List</tt>, and the elements are ''e''<sub>1</sub>, ''e''<sub>2</sub>, etc. The concept of ''head'' is defined for atomic objects as well (<tt>Symbol</tt> for symbols, <tt>Integer</tt> 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:
<small>In[10]:=</small> Expand[(Cos[x] + 2 Log[x^11])/13]<nowiki>[[2, 1]]</nowiki>
<small>Out[10]=</small> 2/13
The reverse can also occur -- lists can be modified to behave like ordinary expressions:
<small>In[11]:=</small> Map[Apply[Log, #] &, {{2, x}, {3, x}, {4, x}}]
<small>Out[11]=</small> {Log[x]/Log[2], Log[x]/Log[3], Log[x]/Log[4]}
where the <tt>Apply</tt> function changes the head of its second argument to that of the first, and <tt>Map</tt> behaves like the [[higher-order function|map function]] found in many functional languages. Note that <tt>Log[b,x]</tt> is the base <tt>b</tt> logarithm, which is [[Logarithm#Change of base|converted]] to <tt>Log[x]/Log[b]</tt> on input.
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, <tt>Apply</tt> threads itself over lists when invoked as
<small>In[12]:=</small> Apply[Log, {{2,x}, {3,x}, {4,x}}, 1]
<small>Out[12]=</small> {Log[x]/Log[2], Log[x]/Log[3], Log[x]/Log[4]}
where the third argument being a 1 specifies that <tt>Apply</tt> 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.
== Licensing ==
Mathematica is [[proprietary software]] protected by both [[trade secret]] and [[copyright]] law.<ref>[http://www.wolfram.com/terms/MathematicaLicenseAgreement.pdf Wolfram Mathematica License Agreement]</ref>
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.<ref>[http://www.wolfram.com/products/webmathematica/versions/amateurorder.html webMathematica terms]</ref> 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.
== Platform Availability ==
Mathematica 6 is supported on various versions of [[Microsoft Windows]], Apple's [[Mac OS X]], [[Linux]], Sun's [[Solaris_operating_system | Solaris]] (UltraSPARC and AMD x86 processors only), HP's [[HP-UX]] and IBM's [[AIX]] platforms. All platforms are supported with 64-bit implementations.<ref>[http://www.wolfram.com/products/mathematica/platforms/ Supported platforms list]</ref>
== Version history ==
Mathematica built on the ideas in Cole and Wolfram's earlier [[Symbolic Manipulation Program]] (SMP).<ref>[http://www.accessmylibrary.com/coms2/summary_0286-11287695_ITM Math, the universe, and Stephen: the author of Mathematica created a whirlwind of scientific controversy this year when, after more than 10 years of research, he published his treatise on the ability of simple structures to create unpredictable complex patterns. (2002 Scientist Of The Year).(Stephen Wolfram)] by Tim Studt, R&D, November 1 , 2002.</ref><ref>[http://query.nytimes.com/gst/fullpage.html?res=940DE5DA173AF937A15755C0A96E948260 A Top Scientist's Latest: Math Software] by Andrew Pollack, ''The New York Times'', June 24, 19988.</ref>
[[Wolfram Research]] has released the following versions of Mathematica:
<div style="-moz-column-count:2; column-count:2;">
* Mathematica 1.0 (1988)<ref>[http://query.nytimes.com/gst/fullpage.html?res=940DE6DB133EF933A05753C1A96E948260 Supercomputer Pictures Solve the Once Insoluble], John Markoff, October 30, 1988.</ref>
* Mathematica 1.2 (1989)<ref>[http://www.accessmylibrary.com/coms2/summary_0286-9205258_ITM Mathematica 1.2 adds new graphics options: upgrade also promises concurrent operations] by Elinor Craig, ''MacWee''k, July 25, 1989.</ref>
* Mathematica 2.0 (1991)<ref>[http://www.accessmylibrary.com/coms2/summary_0286-9227849_ITM Mathematica + 283 functions = Mathematica 2.0] by Raines Cohen, ''MacWeek'', January 15, 1991.</ref>
* Mathematica 2.1 (1992)<ref>[http://www.accessmylibrary.com/coms2/summary_0286-9256461_ITM New Mathematica: faster, leaner, linkable and QuickTime-compatible: MathLink kit allows ties to other apps. (Wolfram Research Inc. ships Mathematica 2.1, new QuickTime-compatible version of Mathematica software)] by Daniel Todd, MacWeek, June 15, 1992.</ref>
* Mathematica 2.2 (1993)<ref>[http://www.highbeam.com/doc/1G1-13185601.html New version of Mathematica], ''Mechanical Engineering'', June 1, 1993.</ref>
* Mathematica 3.0 (1996)<ref>[http://www.businessweek.com/1997/02/b3509205.htm New Mathematica] by Stephen H. Wildstrom, ''BusinessWeek'', June 15, 1997.</ref>
* Mathematica 4.0 (1999)<ref>[http://www.highbeam.com/doc/1G1-61370961.html Mathematica 4.0] by Charles Seiters, ''Macworld'', October 1, 1999.</ref>
* Mathematica 4.1 (2000)
* Mathematica 4.2 (2002)<ref>[http://www.macworld.com/2002/11/reviews/mathematica/ Mathematica 4.2: Feature-Rich Math Program Integrates with the Web, Adds Full Java Support] by Charles Seiter, ''Macworld'', November 1, 2002.</ref>
* Mathematica 5.0 (2003)<ref>[http://www.mywire.com/pubs/PCMagazine/2003/09/02/420220?extID=10051 Mathematica 5.0 Adds Up: Exactly 15 years after Mathematica's initial release, Wolfram Research has released Mathematica ], ''PC Magazine'', September 3, 2003.</ref>
* Mathematica 5.1 (2004)<ref>[http://moreresults.factiva.com/results/index/index.aspx?ref=PCW0000020050115e0c60001i Mathematica 5.1's Web Services Add Up; Mathematica 5.1 delivers improvements over Version 5.0 that are vastly out of proportion for a .1 upgrade.] by Peter Coffee, ''eWeek'', December 6, 2004.</ref>
* Mathematica 5.2 (2005)<ref>[http://www.macworld.co.uk/news/index.cfm?NewsID=12069&Page=1&pagePos=6 Mathematica hits 64-bit], ''MacWorld'' UK, July 13, 2005.</ref>
* Mathematica 6.0 (2007)<ref>[http://www.scientific-computing.com/products/review_details.php?review_id=17 Mathematica 6: Felix Grant finds that version 6 of Wolfram Research's symbolic mathematical software really does live up to its expectations.] Scientific Computing, 2007.</ref>
</div>
== See also ==
{{Wikibooks|Mathematica}}
* [[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]]
== References ==
{{reflist}}
== External links ==
* [http://reference.wolfram.com/ Wolfram Mathematica Documentation Center]
* [http://groups.google.com/group/comp.soft-sys.math.mathematica comp.soft-sys.math.mathematica] discussion group
* [http://www.google.com/Top/Science/Math/Software/Mathematica/ Google Directories] entry containing a wide variety of useful links and tutorials
* [http://www.mathematica-users.org/webMathematica/wiki/wiki.jsp?pageName=Main_Page Mathematica-users], a wiki for users of Mathematica
* [http://www.terrystickels.com/gallery/BeautyofMath/index.html Mathematica Photo Gallery], examples of art using Mathematica
* [http://www.theodoregray.com/BrainRot/ 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.
*"[http://demonstrations.wolfram.com/LevelsOfAMathematicaExpression/ Levels Of A Mathematica Expression]" by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
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