Compute

Core Algorithms

With the world's largest collection of algorithms in a single system, each able to operate across the widest applicable scope of numeric, symbolic, or graphical input, Mathematica provides broad coverage for mathematical computations and equation solving in every field.

• Uniform expression model: Mathematica handles many different kinds of concepts: mathematical formulas, lists, and graphics, to name just a few. Although they often look very different, Mathematica represents all of these in one uniform way—as expressions.
  
• Equation solving: Mathematica's numeric and symbolic equation solving capabilities, all automatically selected through a small number of powerful functions, include algebraic, differential, recurrence, and functional equations and inequalities, as well as linear systems.
  
• Graphs and networks: Mathematica includes a large suite of fundamental graph operations and algorithms, including finding paths, cycles, cliques, and more. Create families of special graphs, generate random graphs, or construct graphs interactively. Import and export to standard graph and matrix formats.
  

• Linear algebra: Symbolic matrices, numerical matrices of any precision, dense and sparse matrices, and matrices with millions of entries: Mathematica handles them all, seamlessly switching among large numbers of optimized algorithms.
  
• Discrete calculus: Mathematica delivers a comprehensive system for discrete calculus, covering symbolic operations, difference equations, generating functions, sequences, and numerical discrete calculus.
  
• Polynomial algebra: Mathematica supports all aspects of polynomial algebra, including factoring and decomposition, structural operations, polynomial division, and more. Carefully tuned strategies automatically select optimal algorithms, allowing large-scale polynomial algebra.
  
• Number theory: A complete library of functions covering multiplicative, analytic, additive, and algebraic number theory, including factoring, primes, congruences, and modular arithmetic, makes Mathematica the ideal platform for number theoretic experiment, discovery, and proof.
  

• Mathematical constants and data: Built-in datasets of finite groups, graphs, knots, lattices, polyhedra, and more are all suitable for direct integration into calculations. Computations can also use mathematical constants to any precision, and millions of digits of constants like π and e can be calculated in seconds.
  
• Probability and statistics: Mathematica's broad coverage of statistics and data analysis means more statistical distributions than any other system, distributions that can be defined directly from data, support for classical statistics, large-scale data analysis, statistical model analysis, exploratory data analysis, symbolic manipulation and numeric analysis, charting, and more.
  

• Calculus and analysis: Covering differentiation, integration, series, Fourier analysis, integral transforms, differential operators, and more, Mathematica's powerful capabilities span the breadth of symbolic and numeric calculus.
  
• Computational systems: Mathematica made possible Stephen Wolfram's exploration of the computational universe and the emerging field of Wolfram Science (NKS). Whether for modeling, algorithm discovery, or basic NKS, Mathematica has immediate built-in capabilities for the systematic study of a broad range of computational systems.
  

• Logic and Boolean algebra: Incorporating state-of-the-art quantifier elimination, satisfiability, and equational-logic theorem proving, Mathematica provides a powerful framework for investigations based on Boolean algebra.
  
• Special functions: Mathematica has the broadest and deepest coverage of special functions, all of which support arbitrary-precision evaluation for complex values of parameters; arbitrary series expansion even at branch points; and an immense web of exact relations, transformations, and simplifications.
  

Numerical Computing

Mathematica uses the power of symbolic computation to make numerical computing faster and more accurate. Automatic algorithm selection and the ability to use any calculation precision enhance Mathematica's capabilities in linear algebra, quadrature, local and global optimization, differential equation solving, and many more areas.

• Task-oriented automatic solvers: Task-based Mathematica functions solve problems by automatically selecting the appropriate numerical method, even switching in mid-calculation. With hundreds of methods to choose from, this optimized algorithm selection improves speed and reliability over manual specification.
  
• Results at any precision: Any number precision or number size can be used across all functions, allowing answers accurate to almost any number of digits. Internally, higher-precision calculations are often used automatically.
  
• Linear algebra and sparse arrays: Improve speed and memory use with robust linear algebra on dense matrices using industry-standard, high-performance libraries; sparse arrays of any dimension; and numerical linear algebra on arbitrary precision and mixed symbolic-numeric matrices.
  

• Integration and summation: Compute single and multidimensional numerical integrals and numerical sums and products of sequences. Many integration methods including globally adaptive subdivision, Gaussian and Clenshaw-Curtis quadrature rules, and specialized high-dimensional and oscillatory rules.
  

• Numerical equation solving: Numerical root-finding of functions and systems of simultaneous equations is built into Mathematica. Methods include Newton, Secant, and Brent as well as specialized methods for efficient numerical solutions of systems of polynomial equations.
  
• Symbolically enhanced numeric computing: With behind-the-scenes symbolic calculations, Mathematica optimizes performance of numerical computations for time and accuracy—and makes previously unsolvable calculations directly computable. Examples include the intelligent handling of piecewise functions, discontinuities, and automatic expression transformation ahead of numerical sampling.
  
• Unique numerical precision tracking: Mathematica automatically tracks and communicates how many digits of the result are accurate, giving almost complete protection from numerical errors, be they round-off errors or from badly conditioned systems.
  
• Differential equations: Numerically solve delay and differential-algebraic equations, partial differential equations, and systems of nonlinear differential equations of any order. Mathematica's built-in methods include implicit and explicit Runge-Kutta and multistep methods, specialized methods for stiff equations, method of lines, and many more.
  

• Local and global optimization: Mathematica includes a full range of state-of-the-art optimization techniques, including constrained and unconstrained local optimization using conjugate gradient, interior point, and other methods; global optimization using Nelder-Mead, simulated annealing, and oher methods; linear programming; traveling salesman problems; and more.
  

Application Areas

In addition to being a comprehensive and powerful computational system, Mathematica builds in specialized functionality for many technical areas, from computational biology to wavelet analysis. Every function is tightly integrated with the whole Mathematica system, making it possible to investigate a single area in great detail and explore new ideas at the intersection of different fields.

• Statistics and data analysis: From calculating basic descriptive statistics to developing and visualizing multidimensional nonlinear models, Mathematica streamlines the entire statistical analysis workflow. With hundreds of distributions, automated parameter estimation, hypothesis testing, statistical model analysis, and more, Mathematica provides a comprehensive set of tools for assessing and understanding data.
  
• GPU programming: Mathematica includes built-in support for CUDA and OpenCL environments, making GPU programming broadly accessible for the first time. GPU operations are fully integrated into the Mathematica system, including direct exchange of data between processes and automatic compilation and linking of GPU code.
  
• Control systems: Design and analyze control systems using classical and state-space techniques, develop control solutions for analog and digital systems, simulate models in open- and closed-loop configurations, and interactively evaluate controllers—all in one system.
  
• Wavelet analysis: Mathematica brings a broad spectrum of wavelet analysis tools to your desktop, including support for many wavelet families, and continuous and discrete wavelet transforms. Wavelet transforms work directly with arrays of any dimension as well as sound and image data, providing a symbolic representation of the transform that can easily be visualized or used for further processing.
  

• Parallel computing: On any multicore computer, Mathematica automatically runs multiple parts of a computation concurrently—making parallel computing easy enough for everyday use. Mathematica's parallel infrastructure is set up to allow seamless scaling to networks, clusters, grids, and clouds, while the symbolic character of the Mathematica language provides straightforward support of many programming paradigms and data-sharing models.
  

• Image processing and analysis: With dozens of features for real-time image acquisition, filtering, segmentation, shape analysis, feature detection, and more, Mathematica offers a complete environment and interactive workflow for image processing and analysis.
  

• Financial engineering: Mathematica includes built-in functions for pricing derivatives, including exotic options; for computing bond values and sensitivity measures; and for calculating time value of money. With more than a hundred financial indicators, easy connectivity to databases and web services, high-performance computing with built-in parallel processing, and more, Mathematica has everything you need for computational finance.
  

• Geographical information systems: With built-in geodesy data and comprehensive support for all standard reference ellipsoids, datums, and projections, Mathematica streamlines the process of importing, visualizing, and computing with geospatial data from varied sources—allowing you to apply high-precision geodetic techniques to your GIS applications.
  

Data Sources & Analysis

Mathematica can combine your imported data with Wolfram|Alpha's computable data and immediately analyze it using advanced model and data fitting, signal processing, classification, or statistical methods. Highly customizable data visualization features let you see your results in new ways.

• Hundreds of file formats: Process and analyze data from diverse sources consistently using Mathematica's uniform symbolic representation. Hundreds of file formats are supported for import and export, including spreadsheets; XML; 2D and 3D graphics formats; multimedia formats; document formats, including PDF and HTML; compressed files; and many other data formats for specific application areas.
  
• Database connectivity: Mathematica connects to any standard SQL database, with support for secure connections, result sets, connection pooling, and transactions. It provides a high-level symbolic representation of databases, queries, and results, as well as full support for traditional string-based SQL queries.
  

• Statistical data analysis: With the largest set of built-in distributions of any system, Mathematica offers a full suite of statistical measures and operations, from mean and variance to cumulants and information entropy. Advanced operations on data include automatic hypothesis testing, distribution estimation, kernel density estimation, and more.
  
• Data visualization: Mathematica includes a full repertoire of functions for visualizing structured and unstructured data in 2D and 3D. Built-in functions include contour and density plots; point, line, and surface plots; vector and streamline plots; histograms; and standard statistical charts, such as pie, bar, bubble, and quantile charts.
  

• Cluster analysis: Mathematica's integrated exploratory data analysis features include cluster detection, nearest neighbor searching, a large library of standard distance and similarity measures, data binning and histogram functions, and more.
  
• Direct access to Wolfram|Alpha data: Instantly access Wolfram|Alpha's continuously growing data collection. Compute with more than 10 trillion pieces of data in every field, including science, engineering, finance, socioeconomics, and more. Access data programmatically, or query in plain English.
  

• Language connectivity and web services: Mathematica can natively call and be called by C, .NET, Java, and other languages; automatically generate C code; compile standalone libraries or executables; link to dynamic libraries at run time; and connect to WSDL web services.
  
• Model fitting: Mathematica automates linear and nonlinear model fitting, including logit and probit regression, and provides full fit diagnostics such as confidence intervals, ANOVA tables, and much more.
  
• Signal processing: Mathematica has complete support for signal processing, with algorithms optimized to enable large volumes of data. Features include linear and nonlinear filters, Fourier and wavelet transforms, correlations and convolutions, fast linear algebra, and interpolation and extrapolation.
  
• String analysis: Mathematica provides optimized algorithms for substring detection, replacement, alignment, and pattern matching using regular expressions and generalized symbolic patterns.
  

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