systems and libraries, most interesting

• Albert.
• GAP [open source]
  • mailing list
  • manual
  • Peter Webb's computations of (co)homology of simplical complexes and groups
  • HAP and HAPprime - packages for computations of various (co)homologies, including those of Lie algebras and groups, and for topological data analysis
  • Using Table Automorphisms for Constructing Character Tables in GAP by Thomas Breuer
  • Wedderga package - Wedderburn decomposition of group algebras.
  • Simplicial Homology GAP package by Jean-Guillaume Dumas, includes computation of the Smith normal form of a sparse matrix.
  • Circle - package for dealing with adjoing groups of associative rings (a*b = ab + a + b)
  • grape package contains routines for computing automorphisms of graphs.
  • small groups by Malka Schaps.
  • kbmag - package for dealing with finitely presented groups via rewriting systems; computes growth function.
  • homological algebra package
• PARI/GP. Number theory, linear algebra over number fields, etc. Tricks using PARI by Anton Mellit. PARI basics. Scripts by Max Alekseyev.
  To compute reverse of power series: serreverse(-x + x^2 - 1/6*x^3 + O(x^4));
  composition of powers series S and P: subst (S, x, P);
  to find roots of a polynomial: polroots (1 + x + x^2 + x^3); to factorize polynomial: factor (1 + x + x^2 + x^3)
• Sage [open source; partially funded by Microsoft and Google]. Python-based, incorporates GAP, PARI, Singular, Octave and virtually everything else.

systems and libraries, other (in alphabetical order)

• Alg. Scheme package by Stephen F. Siegel. Lie algebras, free Lie algebras, homology of groups.
• Alg [Windows binary]. Calculations in small finite universal algebras, by Jaroslav Jezek.
• Anick. C-based. for calculating Anick resolutions.
• Axiom. wiki. Seemingly supports nonassociative rings. Site is very messy - it is diffuclt to locate no-nonsense downloads and installation instructions.
  • Guessing formulas for sequences package.


• Cadabra - manipulations with tensors.
• CategoryTheory.
• CoCoA. CoCoA at work MSRI presentation by Lorenzo Robbiano.
• com.perisic.ring - Java classes for multivariate and univariate polynomials over the rings by Marc Conrad.
• Coq - automatic theorem proving
• CoRoPa (Computational Rough Paths), uncluding libalgebra - C++ library for computations in free Lie and associative algebras, including Campbell-Baker-Hausdorff formula: [sourceforge] [nongnu.org]
• CUDD - essentially, C++ libraries for dealing with boolean functions.


• Derive. Commercial? Symbolic integration?
• DoCon - written in Haskell.


• Felix. free. "Special computer algebra system for the computation in commutative and non-commutative rings and modules. The central method is Buchberger's algorithm and its generalizations to non-commutative rings, in particular to free k-algebras and algebras of solvable type. Among the implemented applications there are syzygy computations and basic ideal operations. Felix provides a complete programming language which in standard mode is interpreted but also on-line compiler and linker are included".
• Fermat. Polynomials and matrix computation.
• FORM - "large scale symbolic computations". Parallel version. Communication with external resources. Example of computation of Gerstenhaber bracket.


• GDCT - "Java application for storage and display of finitely presented categories and functors among them".
• Giac/Xcas.
• GiNaC. screenshot.
• Givaro: C++ library for integer, rational (wrapper around gmp), modular and polynomial arithmetics. Used by Sage for modular arithmetics. Documentation is absent.
• GMP library. Icelandic GNU mirror.
• Group Explorer.


• HOL - automated proof system for higher order logic.


• JSci - a "science" API for Java (including Lie algebras, numerical linear algebra, etc.)


• LambdaTensor. open source. LISP-based. symbolic and numeric higher-rank sparse tensor multilinear algebra. computations in classical and Kac-Moody Lie algebras and Lie groups.
• LiDIA - C++ library for computational Number Theory, including linear algebra over Z and finite fields. patch to build LiDIA 2.20 with gcc 4.3.3.
• LinBox - C++ library for exact linear algebra, including sparse matrices and Lanczos method. Very unmature.


• Macauley2 [open source]. Algebraic geometry, commutative algebra, and homological algebra.
• Magnus. Combinatorial group theory.
• Magma [commercial]. Magma 2.12 (~137M). Explicitly supports Lie algebras and (co)homological calculations. Implements inner automorphisms of simple Lie algebras. (free) online interface. Code samples. Brauer-Manin obstruction.
• Maple [commercial]. "free trial" version. Maple 8 for Linux (~60M).
  • Maple applications (especially algebra section (including Lie algebra and commutators packages)).
  • John Stembridge's packages for symmetric functions, characters of symmetric group, root systems, etc.
  • CREP - "Designed to deal with categories whose morphism spaces are finite-dimensional over a field k. The main example of a category with this property is the category of finite-dimensional representations of an associative unital k-algebra".
• Math::Algebra::Symbols and Math::Symbolic CPAN modules.
• Mathematica [commercial; produced by Wolfram]
  • wolframalpha (sort of online interface)
  • doc
  • Mathematica 5.2 for windows
  • Combinatorica package for Young tableaux.
• Maxima. Seems to be, together with Axiom, the major open source alternative to Mathematica and Maple. It seems that the difference between Maxima and Axiom is that Maxima tries to do the things right away, while Axiom tries to do the things right. So Axiom may be lagging behind Maxima in capabilities, but it could be more superior in concept (and in a long run?). A 10 minute tutorial by Antonio Cangiano. example of computation of combinatorial identities
• mpir - "angry fork" from gmp. Rational and integer arithmetics. Used by Sage.
• MuPAD (commercial, part of MATLAB).


• nauty - program and C library for computing automorphisms of graphs.


• PolyBoRi - polynomials over boolean rings, including Gröbner bases computations.
• Prover9 and Mace4 - automatic theorem prover. Proof that in an associative ring x^3 = x implies commutativity. People occasionally use it to prove apparently something meaningful.


• REDUCE [open source]: [reduce-algebra.com] [uni-koeln.de]. QSUM: a package for summation of q-hypergeometric series.


• SALSA - polynomial systems and Gröbner bases.
• SIMATH. online calculator.
• Singular - polynomial (aka commutative algebra) computations. Examples. Plural - "noncommutative Singular" (supports only a very particular PBW-type of noncommutative rings, and not general noncommutative rings).
• Symmetrica [open source]. manual. A set of routines written in C: representation of symmetric group, tableaux, Schur polynomials, etc.


• The Universal Algebra Calculator by Ralph S. Freese and Emil W. Kiss - computations in universal algebras [elte.hu] [hawaii.edu]
• TPTP - The TPTP Problem Library for Automated Theorem Proving.


• Yacas. manual including a lot of theoretical background.

everything else

• RISC - could provide remote access to symbolic computation facilities.
• Computational Category Theory Project
• http://wwwmaths.anu.edu.au/research.groups/algebra/soft.html - (C) programs for different computations on groups, including Schur multiplier.
• Computations related to algebraic topology by Christian Nassau.
• Notes on Symbolic computations by Larry A. Lambe.
• CHomP (Computational Homology Project) including software for computing homology of simplicial complexes accompanying the book Computational Homology by Kaczynski, Mischaikow and Mrozek.
• Lie algebras manipulation in Haskell.
• computing Betti numbers for smooth complete intersections by Donu Arapura.
• Implementation of permutations in the standard C++ library.
• permutations in combinations in awk
• The Stony Brook Algorithm Repository