GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects.

Geomview is an interactive 3D viewing program for Unix. Geomview lets you view and manipulate three-dimensional objects: you use the mouse to rotate, translate, zoom in and out, and so on. Geomview can be used as a standalone viewer for static objects or as a display engine for other programs which produce dynamically changing geometry. Geomview can display objects described in a variety of file formats. Geomview comes with a wide selection of example objects, and you can create your own objects too. Geomview runs on most Unix platforms, including GNU/Linux.

The program kaleido may be used, without any further programming, to compute the metrical properties of the polyhedra, such as angles and radii. and their combinatorial properties, such as the Euler characteristic and the covering density. Furthermore, the program is capable of generating Cartesian coordinates for the vertices and faces, which are then used to display a rotating wire-frame images of the polyhedra, with depth simulated by edge brightness, and to generate a pic file which can be included in any TeX or troff manuscript. The computational features are available on any machine with a decent C compiler. The graphic features are currently available for Unix machines with X Windows or LucasFilm graphics, UNIX V/386 machines, and MS-DOS machines, but may be extended quite easily to other graphic environments. The source code is carefully broken into small logical units, so it may be used by an experienced programmer in any environment which requires a precise computation of polyhedra, such as a computer modeling software.

KANT is a software package for mathematicians interested in algebraic number theory. For those KANT is a tool for sophisticated computations in number fields, in global function fields, and in local fields.

LAPACK is written in Fortran 90 and provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. The associated matrix factorizations (LU, Cholesky, QR, SVD, Schur, generalized Schur) are also provided, as are related computations such as reordering of the Schur factorizations and estimating condition numbers. Dense and banded matrices are handled, but not general sparse matrices. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision.

Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, sets, lists, vectors, matrices and tensors. Maxima yields high precision numerical results by using exact fractions, arbitrary-precision integers and variable-precision floating-point numbers. Maxima can plot functions and data in two and three dimensions.

GNU Octave is a high-level interpreted language, primarily intended for numerical computations. It provides capabilities for the numerical solution of linear and nonlinear problems, and for performing other numerical experiments. It also provides extensive graphics capabilities for data visualization and manipulation. Octave is normally used through its interactive command line interface, but it can also be used to write non-interactive programs. The Octave language is quite similar to Matlab so that most programs are easily portable.

PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves…), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions. PARI is also available as a C library to allow for faster computations.

REDUCE is an interactive system for general algebraic computations of interest to mathematicians, scientists and engineers. Capabilities include:

- expansion and ordering of polynomials and rational functions;
- substitutions and pattern matching in a wide variety of forms;
- automatic and user controlled simplification of expressions;
- calculations with symbolic matrices;
- arbitrary precision integer and real arithmetic;
- facilities for defining new functions and extending program syntax;
- analytic differentiation and integration;
- factorization of polynomials;
- facilities for the solution of a variety of algebraic equations;
- facilities for the output of expressions in a variety of formats;
- facilities for generating optimized numerical programs from symbolic input;
- calculations with a wide variety of special functions;
- Dirac matrix calculations of interest to high energy physicists.

SageMath is built out of nearly 100 open-source packages and features a unified interface. SageMath can be used to study elementary and advanced, pure and applied mathematics. This includes a huge range of mathematics, including basic algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, exact linear algebra and much more. It combines various software packages and seamlessly integrates their functionality into a common experience. It is well-suited for education and research.

SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more.

Scilab is free and open source software for numerical computation providing a powerful computing environment for engineering and scientific applications. Scilab is released as open source under the CeCILL license (GPL compatible), and is available for download free of charge. Scilab is available under GNU/Linux, Mac OS X and Windows XP/Vista/7/8. Scilab includes hundreds of mathematical functions. It has a high level programming language allowing access to advanced data structures, 2-D and 3-D graphical functions.

Tela (the TEnsor LAnguage) is a numerical computing environment mainly targeted for numerical simulation pre- and postprocessing work.