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.

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.