Mathematica

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Mathematica is a widely-used computer algebra system originally conceived by Stephen Wolfram, developed by a team of mathematicians and programmers that he assembled and led, and sold by his company Wolfram Research. Mathematica is also a powerful programming language emulating multiple paradigms on top of term-rewriting.

Overview

Wolfram and his team started to work on the program in 1986 and released the first version in 1988. The current version is 5.2 (released July 12, 2005). It is available on a wide variety of computer systems.

The Mathematica programming language is based on term-rewriting and supports both functional and procedural programming (though functional code is much more efficient in general). It is implemented in an object-oriented variant of C, but the bulk of the extensive code library is actually written in the Mathematica language that can be used to extend the system. Typically, new code is added in the form of Mathematica "packages", i.e., text files written in the Mathematica language.

In the Mathematica system, the core language is interpreted by a kernel that performs the actual computations. The results are usually communicated to one of several front ends. Communication between the kernel and the front end (or any other client, like user-written programs) uses the MathLink protocol, often over a network. It is possible for several front end processes to connect to the same kernel, and for one front end to be connected to several kernels.

Unlike some other computer algebra systems, for example Maxima or Maple, Mathematica tries to apply the currently stored transformation rules as long as possible, looking for a fixed point. For this to be meaningful, absence of side-effects is beneficial (though not enforced), hence the similarity to functional programming. Functions and code are first-class and not opaque. Scoping is dynamic, but there are also some constructs that try to simulate lexical scope (all of these can easily be broken).

History

Wolfram has released the following versions:

• 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)

Examples

The following Mathematica sequence will find the determinant of the 6×6 matrix whose i, j'th entry contains ij with all zero entries replaced as 1.

 In[1]:= Det[Array[Times, {6, 6}, 0] /. 0 -> 1]
 Out[1]= 0

So the determinant of such a matrix is 0.

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}

See List of hello world programs for the prototypical Hello World program.

Multiple paradigms, one language

Mathematica permits multiple programming paradigmatic approaches to programming. Consider a simple 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:

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 outer product operator, Range corresponds to the iota operator. The Outer function permits any function, whether it be named, or anonymous, which are functions specified on the fly by using #n to specify the function argument and appending an &. The above function could be equivalently specified as Outer[GCD[#1, #2] &, Range[5], Range[5]], but Mathematica permits the above shortcut as well.

An iterative approach:

In[6]:= l1 = {}; (* initialize as empty list, since we want a list in the end *)
       For[i = 1, i <= 5, i++,
       l2 = {}; 
          For[j = 1, j <= 5, j++,
             l2 = Append[l2, GCD[i, j]] 
             ];                                      
          l1 = Append[l1, l2]; (* append the sublist, that is, the row *)
       ]; l1
Out[6]=  {{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, common 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[7]:= x^4 + 1
Out[7]= 1+x4

If the FullForm command is used on this expression however:

In[8]:= FullForm[x^4 + 1]
Out[8]= Plus[1, Power[x, 4]]

Nearly all objects in Mathematica have the basic form head[e₁, e₂, ...] (which may be displayed or entered in some other fashion). For example, the head of the above example is Plus, and symbols such as x have form Symbol["x"]. Lists have this structure too, where the head is List.

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

In[9]:= Expand[(Cos[x] + 2 Log[x^11])/13][[2, 1]]
Out[9]= 2/13

The reverse can also occur -- lists can be modified to behave like ordinary expressions:

In[10]:= Map[Apply[Log, #] &, {{2, x}, {3, x}, {4, x}}]
Out[10]= {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 higher-order function found in many functional languages.

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[11]:= Apply[Log, {{2,x}, {3,x}, {4,x}}, 1]
Out[10]= {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.

Front ends

Mathematica consists of two parts - a front end with which users interact and a kernel which performs the computations. The two communicate via the Mathlink protocol, which is discussed in another section. It is possible to use the kernel on one computer and the front end on another, although this is not how most people use Mathematica.

The default Mathematica front end features extensive layout and graphical capabilities, performs prettyprinting and offers a notebook metaphor - user input (both text and Mathematica input) as well as results sent by the kernel (including graphics and sound) are placed in a hierarchy of cells (as is the case for Maple), which also allows for outlining and sectioning of a document. Starting with version 3.0 of the software, notebooks are represented as expressions that can be manipulated by the kernel.

To allow those without Mathematica licenses to view Mathematica notebooks dedicated reader software was made available. This software, called MathReader can be downloaded free of charge. This does not use a Matematica kernel, so no new results can be computed.

All UNIX/Linux versions of Mathematica, include addtionally a command line front-end which can be used from a Dumb_terminal. An example of this is shown below.

% math
Mathematica 5.2 for Sun Solaris (UltraSPARC)
Copyright 1988-2005 Wolfram Research, Inc.
-- Terminal graphics initialized --

In[1]:= Solve[x^2 + 2 x - 9 == 0, x]

Out[1]= {{x -> -1 - Sqrt[10]}, {x -> -1 + Sqrt[10]}}
Old versions of Mathematica which run on MS-DOS had only a command line front end.

WITM, which is an acronym for Web Interface to Mathematica, is a web browser front end which is compatible with virtually any networked computer which has a web browser. This allows Mathematica to be used on devices like PDA's (e.g. Pocket PC or similar) for which there is no version of Mathematica available from Wolfram Research. A screenshot of an HP iPAQ PDA accessing a Mathematica kernel is shown.
Several other front ends are also available, e.g., JMath or MASH, but the standard Mathematica front end is the most popular.

Connections with other applications

Communication with other applications occur through a protocol called Mathlink. It allows not only communication between the Mathematica kernel and front-end, but also provides a general interface between the kernel and arbitrary applications. Wolfram Research distributes freely a developer kit for linking applications written in the C programming language to the Mathematica kernel through MathLink. Two other components of Mathematica, whose underlying protocol is MathLink, allow developers to establish communication between the kernel and a Java or .NET program: J/Link and .NET/Link.

Using J/Link, a Java program can ask Mathematica to perform computations; also, a Mathematica program can load any Java class, manipulate Java objects and perform method calls, making it possible, for instance, to build Java graphical user interfaces from Mathematica. Similarly, a .NET software can invoke the kernel to perform calculations and send results back, and Mathematica developers can easily have access to .NET's functionality.

Mathematica on the Web

Wolfram Research also makes a program called webMathematica with which Web servers can add "interactive calculations and visualization to a website."

On Sloane's Online Encyclopedia of Integer Sequences, Mathematica and MAPLE are the two most often used CASs for which commands are provided with which to calculate the sequences; both of them have their own database fields on the OEIS.

Criticisms

In breaking with most popular programming languages, Mathematica indexes its lists starting from one. See Zeroth and Numbering from 0 or 1?

Many of the Wolfram Reseach forums, as well as the comp.soft-sys.math.mathematica newsgroup are all moderated, which some argue slows down the speed at which assistance can be obtained from the community.

See also

• Axiom
• IGOR Pro
• Maple
• Maxima
• MathCad
• Yacas
• List of computer algebra systems
• List of numerical analysis software
• IMTEK Mathematica Supplement, an open source Mathematica add-on for Finite Element Simulation

External links

• Wolfram Research
• Mathematica (Wolfram Research)
• The History of Mathematica, an overview of the system's history and development
• Wiki-Mathematica, a Mathematica-users wiki
• WITM, a web browser interface to Mathematica, allowing use on any platform.
• MASH, a UNIX-scripting interface to Mathematica
• IMS, the Open Source IMTEK Mathematica Supplement (IMS)
• Mathematica Photo Gallery, for numerous examples of art using Mathematica
• Solving Font issues in Mathematica EPS graphics: Replacing fonts with Times Roman, Computer Modern, autogenerated LaTeX and PSfrag,simple PSFrag based Mathematica package or Including Mathematica fonts in dvips config