Wolfram Mathematica - Mathematica Tutorial

Wolfram Mathematica (usually termed Mathematica) is a mathematical symbolic computation program, sometimes termed a computer algebra system or program, used in many scientific, engineering, mathematical, and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in Mathematica.

Features

Features of Wolfram Mathematica include:

• Libraries of mathematical elementary functions and special functions
• Support for complex number, arbitrary precision arithmetic, interval arithmetic, and symbolic computation
• Matrix and data manipulation tools including support for sparse arrays
• 2D and 3D data, function and geo visualization and animation tools
• Solvers for systems of equations, diophantine equations, ordinary differential equations (ODEs), partial differential equations (PDEs), differential algebraic equations (DAEs), delay differential equations (DDEs), stochastic differential equations (SDEs), and recurrence relations
• Finite element analysis including 2D and 3D adaptive mesh generation
• Numeric and symbolic tools for discrete and continuous calculus including continuous and discrete integral transforms
• Constrained and unconstrained local and global optimization
• Multivariate statistics libraries including fitting, hypothesis testing, and probability and expectation calculations on over 160 distributions.
• Support for censored data, temporal data, time series, and unit based data
• Calculations and simulations on random processes and queues
• Supervised and unsupervised machine learning tools for data, images and sounds including artificial neural networks
• Tools for text mining including regular expressions and semantic analysis
• Data mining tools such as cluster analysis, sequence alignment and pattern matching
• Computational geometry in 2D, 3D and higher dimensions
• Libraries for signal processing including wavelet analysis on sounds, images and data
• Linear and non-linear control system libraries
• Tools for 2D and 3D image processing and morphological image processing including image recognition
• Tools for visualizing and analysing directed and undirected graphs
• Tools for combinatoric problems
• Number theory function library
• Tools for financial calculations including bonds, annuities, derivatives, options etc.
• Group theory and symbolic tensor functions
• Import and export filters for data, images, video, sound, computer-aided design (CAD), geographic information systems (GIS), document and biomedical formats
• Database collection for mathematical, scientific, and socio-economic information and access to Wolfram Alpha data and computations
• Technical word processing including formula editor and automated report generator
• Programming language supporting procedural, functional, and object-oriented constructs
• Toolkit for adding user interfaces to calculations and applications
• Tools to connect to dynamic-link library (DLL), Structured Query Language (SQL), Java, .NET, C++, Fortran, CUDA, OpenCL, and Hypertext Transfer Protocol (HTTP) based systems
• Tools for parallel programming
• Using both "free-form linguistic input" (a natural language user interface) and Wolfram Language in notebook when connected to the Internet

The Notebook interface

Wolfram Mathematica is split into two parts, the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions.

The front end, designed by Theodore Gray in 1988, provides a GUI, which allows the creation and editing of Notebook documents containing program code with prettyprinting, formatted text together with results including typeset mathematics, graphics, GUI components, tables, and sounds. All content and formatting can be generated algorithmically or edited interactively. Standard word processing capabilities are supported, including real-time multi-lingual spell-checking.

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 in a slideshow environment for presentations. Notebooks and their contents are represented as Mathematica expressions that can be created, modified or analyzed by Mathematica programs or converted to other formats.

The front end includes development tools such as a debugger, input completion, and automatic syntax highlighting.

Among the alternative front ends is the Wolfram Workbench, an Eclipse based integrated development environment (IDE), introduced in 2006. It provides project-based code development tools for Mathematica, including revision management, debugging, profiling, and testing. There is a plugin for IntelliJ IDEA based IDEs to work with Wolfram Language code which in addition to syntax highlighting can analyse and auto-complete local variables and defined functions. The Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU readline and MASH which runs self-contained Mathematica programs (with arguments) from the UNIX command line.

High-performance computing

In recent years, the capabilities for high-performance computing have been extended with the introduction of packed arrays (version 4, 1999) and sparse matrices (version 5, 2003), and by adopting the GNU Multi-Precision Library to evaluate high-precision arithmetic.

Version 5.2 (2005) added automatic multi-threading when computations are performed on 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 in 2008 parallel computing technology was included in all Mathematica licenses including support for grid technology such as Windows HPC Server 2008, Microsoft Compute Cluster Server and Sun Grid.

Support for CUDA and OpenCL GPU hardware was added in 2010. Also, since version 8 it can generate C code, which is automatically compiled by a system C compiler, such as GCC or Microsoft Visual Studio.

Deployment

There are several ways to deploy applications written in Wolfram 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.
• A free-of-charge version, Wolfram CDF Player, is provided for running Mathematica programs that have been saved in the Computable Document Format (CDF). It can also view standard Mathematica files, but not run them. It includes plugins for common web browsers on Windows and Macintosh.
• 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. Due to bandwidth limitations interactive 3D graphics is not fully supported within a web browser.
• Wolfram Language code can be converted to C code or to an automatically generated DLL.
• Wolfram Language code can be run on a Wolfram cloud service as a web-app or as an API

Connections with other applications

Communication with other applications occurs through a protocol called Wolfram Symbolic Transfer Protocol (WSTP). It allows communication between the Wolfram 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 programming language C to the Mathematica kernel through WSTP. Using J/Link., a Java program can ask Mathematica to perform computations; likewise, a Mathematica program can load Java classes, manipulate Java objects and perform method calls. Similar functionality is achieved with .NET /Link, but with .NET programs instead of Java programs. Other languages that connect to Mathematica include Haskell, AppleScript, Racket, Visual Basic, Python and Clojure.

Links are available to many mathematical software packages including OpenOffice.org Calc, Microsoft Excel, MATLAB, R, SageMath (which can also pull up Mathematica), Singular, Wolfram SystemModeler, and Origin. Mathematical equations can be exchanged with other computational or typesetting software via MathML.

Communication with Structured Query Language (SQL) databases is achieved through built-in support for Java Database Connectivity (JDBC). Mathematica can also install web services from a Web Services Description Language (WSDL) description. It can access HDFS data via Hadoop.

Mathematica can capture real-time data via a link to LabVIEW, from financial data feeds and directly from hardware devices via GPIB (IEEE 488), USB and serial interfaces. It automatically detects and reads from HID devices.

Computable data

Wolfram Mathematica includes collections of curated data provided for use in computations. Mathematica is also integrated with Wolfram Alpha, an online service which provides additional data, some of which is kept updated in real time. Some of the data sets include astronomical, chemical, geopolitical, language, biomedical and weather data, in addition to mathematical data (such as knots and polyhedra).

Reception

BYTE in 1989 listed Mathematica as among the "Distinction" winners of the BYTE Awards, stating that it "is another breakthrough Macintosh application ... it could enable you to absorb the algebra and calculus that seemed impossible to comprehend from a textbook".

Version history

Wolfram Mathematica built on the ideas in Cole and Wolfram's earlier Symbolic Manipulation Program (SMP). The name of the program "Mathematica" was suggested to Stephen Wolfram by Apple cofounder Steve Jobs although Wolfram had thought about it earlier and rejected it.

Wolfram Research has released the following versions of Mathematica: