Core Algorithms

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.

Built-in symbolic tensors and vector analysis

Mathematica provides integrated support for symbolic array objects, from simple vectors to arrays of any rank, dimensions, and symmetry. Tensor algebra operations allow the construction of polynomials of symbolic arrays that can be simplified into a standard form. Mathematica has a structured array type that stores only symmetry and independent components, leading to substantial memory saving. Differential operators for vector analysis can handle explicit arrays of any type and rank and interpret them in various orthogonal coordinate systems.

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.

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, the ability to associate weights with data points, hypothesis tests, support for classical statistics, large-scale data analysis, statistical model analysis, exploratory data analysis, symbolic manipulation and numeric analysis, charting, and more.

Time series and stochastic differential equations

Mathematica has sophisticated functionality for time series and stochastic differential equation (SDE) random processes, and includes a full suite of scalar and vector time series models such as MA, AR, and ARMA, as well as several extensions. Time series models can be easily simulated, estimated from data, and used to generate forecasts. SDE processes can be specified using both parametric and general Ito or Stratonovich processes. They can be easily simulated numerically, and many of their properties can be computed symbolically.

Random processes

Using a symbolic representation of a process, Mathematica makes it easy to simulate its behavior, estimate parameters from data, and compute state probabilities at different times. There is additional functionality for special classes of random processes such as Markov chains, queues, time series, and stochastic differential equations.

Graphs and networks

Mathematica includes a complete and rich set of graph and network analysis functionality including network flows, social network analysis, and more. Create families of special graphs, generate random graphs, or construct graphs interactively. Import and export to standard graph and matrix formats.

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.

Systemwide units support

Mathematica has a unit system containing thousands of different units—all integrated with Wolfram|Alpha's sophisticated unit interpretation system. This creates an advanced unit system that combines the flexibility of free-form linguistics with the computational power of numerical and symbolic algorithms. The units framework integrates seamlessly with visualization, numeric, and symbolic functions.

Social network analysis

High-level functions for community detection, cohesive groups, and centrality and similarity measures, as well as access to social networks from a variety of sources—including directly from social media sites such as Facebook, LinkedIn, and Twitter—make network analysis easier and more flexible than ever before.

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.

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.

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.

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.

Discrete calculus

Mathematica delivers a comprehensive system for discrete calculus, covering symbolic operations, difference equations, generating functions, sequences, and numerical discrete calculus.

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.

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.