Mathematica
Developed by	Wolfram Research
Initial release	1988 (19–20 years ago)
Stable release	6.0.2  (February 25th 2008) [+/−]
Preview release	6.0.2  (n/a) [+/−]
Platform	Cross-platform (list)
Available in	multilingual
Genre	Computer algebra, numerical computations, Information visualization, statistics, user interface creation
License	Proprietary
Website	Mathematica homepage

Mathematica is a computer program used mainly in scientific, engineering and mathematical fields. It was originally conceived by Stephen Wolfram and developed by a team of mathematicians and programmers that he assembled and led. It is sold by Wolfram Research of Champaign, Illinois.^[1] Look up mathematica in Wiktionary, the free dictionary. ... Image File history File links No higher resolution available. ... For other uses, see Software developer (disambiguation). ... Wolfram Research is part of the Wolfram Group which consists of four companies: Wolfram Research Inc. ... Year 1988 (MCMLXXXVIII) was a leap year starting on Friday (link displays 1988 Gregorian calendar). ... A software release refers to the creation and availability of a new version of a computer software product. ... February 25 is the 56th day of the year in the Gregorian Calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance to the Gregorian calendar. ... A software release refers to the creation and availability of a new version of a computer software product. ... In computing, a platform describes some sort of framework, either in hardware or software, which allows software to run. ... A cross-platform (or platform independent) programming language, software application or hardware device works on more than one system platform (e. ... See Language (journal) for the linguistics journal. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Listed here are a number of computer programs used for performing numerical calculations: Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. ... Information visualization is a complex research area. ... This is an incomplete list of software that is designed for the explicit purpose of performing statistical analyses. ... GUI redirects here. ... A software license is a legal agreement which may take the form of a proprietary or gratuitous license as well as a memorandum of contract between a producer and a user of computer software. ... Proprietary software is software with restrictions on copying and modifying as enforced by the proprietor. ... A website (alternatively, Web site or web site) is a collection of Web pages, images, videos or other digital assets that is hosted on one or several Web server(s), usually accessible via the Internet, cell phone or a LAN. A Web page is a document, typically written in HTML... Stephen Wolfram (born August 29, 1959 in London) is a physicist known for his work in theoretical particle physics, cellular automata, complexity theory, and computer algebra, and is the creator of the computer program Mathematica. ... Wolfram Research is part of the Wolfram Group which consists of four companies: Wolfram Research Inc. ... Champaign is a city in Champaign County, Illinois, in the United States. ...

It provides cross-platform support for tasks such as symbolic or numerical calculations, arbitrary precision arithmetic, data processing, and plotting. Mathematica has a programming language which supports functional and procedural programming styles.

Overview

Create a plot and manipulation dialog with a line of code

Mathematica is split into two parts, the "kernel" and the "front end". The kernel interprets expressions (Mathematica code) and returns result expressions. In Mathematica, dialogs, graphics, math, and general code are all represented by expressions. In the sense that objects in Mathematica are expressions and that the codes required to manipulate them are also expressions, Mathematica is a unified system. The front end provides an interface called a notebook for viewing, creating, manipulating, and mixing programs, graphics, maths, text, and dialogs. It also provides a debugger, a notebook presentation mode, and the interfaces to devices such as keyboards, mice, and gamepads. 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, although this is not common. A kernel connects the application software to the hardware of a computer. ... For other uses, see Mathematica (disambiguation). ...

Features

Some features of Mathematica include^[2]:

• 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 database containing economic, scientific, mathematical, and other information in a format that is easy to access within Mathematica
• Support for complex number, arbitrary precision and symbolic computation for all functions
• Tools for transforming and pattern matching abstract symbolic data models
• Technical word processing

In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... It has been suggested that Interactive visualization be merged into this article or section. ... A sparse array in computing is an array where most of the elements have the same value (called the default value -- usually 0) and only a few elements have a non-default value. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... This article is about the field of statistics. ... In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... This article is about the computer programming paradigm. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ... The user interface is the part of a system exposed to users. ... CAD is a TLA that may stand for: Cadiz Railroad (AAR reporting mark CAD) Canadian dollar – ISO 4217-code Capital Adequacy Directive Card Acceptance Device Children of the Anachronistic Dynasty Computer-aided design Computer-aided detection (medical) Computer-aided diagnosis (medical) Computer-assisted dispatch Computer-assisted drafting Coronary artery disease... A geographic information system (GIS) is a system for managing data that has a spatial specialized form of an information system. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers and rational numbers with an arbitrary number of digits of precision, limited only by the available memory of the host system. ...

Front ends

The default Mathematica front end features layout and graphics capabilities. It allows for the formatting of mathematical expressions, performs prettyprinting (automatic indentation) and includes development tools such as a debugger, input completion and automatic syntax coloring. It also provides Mathematica documents in a form called a notebook where user input (both text and Mathematica input) as well as results sent by the kernel (including graphics, sound and interactive interfaces) are placed in a hierarchy of cells. This allows for outlining and sectioning of a document. All notebook contents can be placed in-line within text regions or within input. Starting with version 3.0 of the software, notebooks are represented as expressions that can be created, modified or analysed by the kernel. To prettyprint (or pretty-print) is to present an object to a human reader, so that it is easier to perceive the objects structure, or, less commonly, to simply make it more attractive. ...

The Mathematica Player is a front end provided by the manufacturer free of charge, which can read notebooks and perform other functions of the full front end. It includes a Mathematica kernel so that calculations can be updated in response to interactive elements. It does not, however, allow documents to be edited or created.

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

A screenshot of a PDA accessing a Mathematica kernel using WITM is shown.

• 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.^[3]
• 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.^[4]
• MASH makes it possible to run self contained Mathematica programs (with arguments) from the UNIX command line.^[5]
• WITM, which is an acronym for Web Interface to Mathematica, is a free web browser front which accepts arbitrary input.^{[citation needed]} It is intended to allow the licensed user remote access to a Mathematica kernel. Standard Mathematica licenses disallow the use of tools like WITM for making Mathematica available on the web or for sharing with multiple users. An extended license is available as part of webMathematica which allows public access to the Mathematica kernel, though not for arbitrary input.^[6]

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... User with Treo (PDA with smartphone functionality) Personal digital assistants (PDAs) are handheld computers, but have become much more versatile over the years. ... Eclipse is an open-source software framework written primarily in Java. ... An integrated development environment (IDE), also known as integrated design environment and integrated debugging environment, is a programming environment that has been packaged as an application program,that assists computer programmers in developing software. ... GNU readline is a software library created and maintained by the GNU project. ...

Connections with other applications

Communication with other applications occurs through a protocol called 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 distributes freely a developer kit for linking applications written in the C programming language to the Mathematica kernel through MathLink.^[7] Two other components of Mathematica, whose underlying protocol is MathLink, allow developers to establish communication between the kernel and a .NET or Java program: .NET/Link.^[8] and J/Link^[9] C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... Java language redirects here. ...

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. In object-oriented programming, a class is a programming language construct used to group related fields and methods. ... GUI redirects here. ...

Communication with SQL databases is achieved through built-in support for JDBC.^[10]. SQL (IPA: or ) is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ... Java Database Connectivity, or JDBC, is an API for the Java programming language that defines how a client may access a database. ...

Mathematica can also install web services from a WSDL description^[11][12]. A web service is a collection of protocols and standards used for exchanging data between applications. ... WSDL redirects here. ...

High-performance computing

In the first versions of Mathematica, performance for large numerical computations lagged far behind specialized software,^{[citation needed]} but recent versions saw great improvement in this area with the introduction of packed arrays (version 4, 1999) ^[13], sparse matrices (version 5 (2003)^[14], and by adopting the free GNU Multi-Precision Library to evaluate high-precision arithmetic. The GNU Multiple-Precision Library, also known as GMP, is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. ...

Version 5.2 (2005) added automatic multi-threading when computations are performed on modern multi-core computers.^[15] This release included CPU specific optimized libraries. In addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed.^[16] ClearSpeed is a company that develops processors for supercomputers. ...

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.^[17] gridMathematica is a product sold by Wolfram Research which is based on its main product Mathematica. ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ... Windows Server 2003 is a server operating system produced by Microsoft. ...

Deployment

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.^[18]
• 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.

Language examples

The following Mathematica sequence will find the determinant of the 6×6 matrix whose i, j'th entry contains ij. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

 In[1]:= Det[Array[Times, {6, 6}]] Out[1]= 0 

So the determinant of such a matrix is 0 (i.e. it is singular). In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

The following numerically calculates the root of the equation e^x = x² + 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: For other uses, see Calculus (disambiguation). ... In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ...

 In[3]:= Integrate[x/Sin[x], x]//OutputForm Out[3]= x (Log[1 - EI x] - Log[1 + EI x]) + I (PolyLog[2, -EI x] - PolyLog[2, EI x]) 

Here, E and I are the fundamental constants e and i respectively, and PolyLog[s,z] is the polylogarithm function . e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... The polylogarithm (also known as de Jonquières function) is a special function Lis(z) that is defined by the sum It is in general not an elementary function, unlike the related logarithm function. ...

Symbolic sums can be calculated. For example:

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

 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: APL (for A Programming Language) is an array programming language based on a notation invented in 1957 by Kenneth E. Iverson while at Harvard University. ...

 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}} 

Outer corresponds to the generalized outer product operator, Range corresponds to the iota operator. Outer admits general functions, whether they be named, or anonymous. Anonymous functions are specified by using #n to as the function argument and appending an &. The above function could be equivalently specified as Outer[GCD[#1, #2] &, Range[5], Range[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]:= l1 Out[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}} 

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 x⁴ + 1 if entered will be represented as if it were written:

 In[8]:= x^4 + 1 Out[8]= 1+x4 

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[e₁, e₂, ...] (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 e₁, e₂, 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]= 2/13 

The 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 Apply function changes the head of its second argument to that of the first, and Map behaves like the map function found in many functional languages. Note that Log[b,x] is the base b logarithm, which is converted to Log[x]/Log[b] on input. In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... Look up logarithm in Wiktionary, the free dictionary. ...

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.

Licensing

Mathematica is proprietary software protected by both trade secret and copyright law.^[19] Proprietary software is software with restrictions on copying and modifying as enforced by the proprietor. ... A trade secret is a formula, practice, process, design, instrument, pattern, or compilation of information which is not generally known or reasonably ascertainable, by which a business can obtain an economic advantage over competitors or customers. ... Not to be confused with copywriting. ...

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.^[20] 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. For other uses, see Mathematica (disambiguation). ... FLEXlm is a popular software license manager. ...

Platform Availability

Mathematica 6 is supported on various versions of Microsoft Windows, Apple's Mac OS X, Linux, Sun's Solaris (UltraSPARC and AMD x86 processors only), HP's HP-UX and IBM's AIX platforms. All platforms are supported with 64-bit implementations.^[21] Windows redirects here. ... Mac OS X (pronounced ) is a line of graphical operating systems developed, marketed, and sold by Apple Inc. ... This article is about operating systems that use the Linux kernel. ... Solaris is a computer operating system developed by Sun Microsystems. ... HP-UX (Hewlett Packard UniX) is Hewlett-Packards proprietary implementation of the Unix operating system, based on System V (initially System III). ... AIX or Aix may be: Aix, a genus of two species of dabbling ducks, the Wood Duck (Aix sponsa) and the Mandarin Duck (Aix galericulata) AIX operating system Athens Internet Exchange, (AIX) a European IXP a place name: Aix-la-Chapelle, or Aachen, a city in Germany in France: Aix...

Version history

Mathematica built on the ideas in Cole and Wolfram's earlier Symbolic Manipulation Program (SMP).^[22][23] Symbolic Manipulation Program, usually called SMP, was a computer algebra system designed by Chris A. Cole and Stephen Wolfram at Caltech circa 1979 and initially developed in the Caltech physics department under Wolframs leadership with contributions from Geoffrey C. Fox, Jeffrey M. Greif, Eric D. Mjolsness, Larry J. Romans...

Wolfram Research has released the following versions of Mathematica: Wolfram Research is part of the Wolfram Group which consists of four companies: Wolfram Research Inc. ...

See also

• 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

The following tables provide a comparison of computer algebra systems (CAS). ... The following tables provide a comparison of numerical analysis software. ... The following tables compare general and technical information for a number of statistical analysis packages. ... Description The IMTEK Mathematica Supplement (IMS for short) is an open source Mathematica AddOn that aims to provide a wealth of tutorials and packages for Scientific and Engineering purposes, mainly focussed on finite element simulations but going far beyond. ... Listed here are a number of computer programs used for performing numerical calculations: Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. ... This is an incomplete list of software that is designed for the explicit purpose of performing statistical analyses. ... This is an incomplete list of software that either includes the ability to create one or more infographics from a provided data set, or is provided specifically for information visualization. ... Mathematical software is a type of software genre which has mathematics in common. ... Mathematical Markup Language (MathML) is an application of XML for describing mathematical notations and capturing both its structure and content. ... MathModelica System Designer is a platform for multi-engineering simulation and Model-Based Design for dynamic systems. ... Theodore Gray is was one of the founders of Wolfram Research and and is currently Wolframs Director of User Interface Technology. ...

References

External links

Wikibooks has a book on the topic of

Mathematica

• Wolfram Mathematica Documentation Center
• 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.