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Axiom
Axiom is a general purpose Computer Algebra system. It is useful for research and development of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler.
More informationGAP
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. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. GAP is developed by international cooperation. The system, including source, is distributed freely under the terms of the GNU General Public License. You can study and easily modify or extend GAP for your special use. The current version is GAP 4, the older version GAP 3 is still available.
More informationMuPad
MuPAD is a mathematical expert system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy. For example, the number of significant digits can be chosen freely. Apart from a vast variety of mathematical libraries the system provides tools for high quality visualization of 2- and 3-dimensional objects. On Microsoft Windows, Apple Macintosh and Linux systems, MuPAD offers a flexible notebook concept for creating mathematical documents combining texts, graphics, formulas, computations and mathematical visualizations and animations. On Microsoft Windows MuPAD further supports the technologies OLE, ActiveX Automation, DCOM, RTF and HTML. Thus it offers a natural integration in Office applications like Word or PowerPoint as well as others.
More informationPARI/GP
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.
More informationPolyBoRi
The core of PolyBoRi is a C++ library, which provides high-level data types for Boolean polynomials and monomials, exponent vectors, as well as for the underlying polynomial rings and subsets of the powerset of the Boolean variables. As a unique approach, binary decision diagrams are used as internal storage type for polynomial structures. On top of this C++-library we provide a Python interface. This allows parsing of complex polynomial systems, as well as sophisticated and extendable strategies for Gröbner base computation. PolyBoRi features a powerful reference implementation for Gröbner basis computation.
More informationRisa/Asir
Risa/Asir is a general computer algebra system and also a tool for various computation in mathematics and engineering. The development of Risa/Asir started in 1989 at FUJITSU. Binaries have been freely available since 1994 and now the source code is also free. Currently Kobe distribution is the most active branch of its development. We characterize Risa/Asir as follows: (1) An environment for large scale and efficient polynomial computation. (2) A platform for parallel and distributed computation based on OpenXM protocols.
More informationSage
SAGE is a framework for number theory, algebra, and geometry computation. It is open source and freely available under the terms of the GNU General Public License (GPL). SAGE is a Python library with a customized interpreter. It is written in Python, C++, and C (via Pyrex). Python (http://www.python.org) is an open source object-oriented interpreted language, with a large number of libraries, e.g., for numerical analysis, which are available to users of SAGE. Python can also be accessed in library mode from C/C++ programs. SAGE provides an interface to several important open source libraries, including Cremona’s MWRANK library for computing with elliptic curves, the PARI library (pari.math.u-bordeaux.fr) for number theory, Shoup’s number theory library NTL (http://www.shoup.net/ntl/), SINGULAR (http://www.singular.uni-kl.de) for commutative algebra, GAP (http://www.gap-system.org) for group theory and combinatorics, and maxima (http://maxima.sourceforge.net) for symbolic computation and calculus.
More informationThe Tensor Library
TTL (The Tensor Library) facilitates the translation of tensor expressions into C++ without compromising speed. A Tensor here can be any multidimensional array, including vectors and matrices. Expressions including several indices can be written in C++, removing the need for explicit loops. Expression templates and similar techniques are used to achieve this goal.
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