1.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.

Axiom has been in development since 1973 and was sold as a commercial product. It has been released as free software (under a BSD-like license).

Efforts are underway to extend this software to

• Develop a better user interface;
• Make it useful as a teaching tool;
• Develop an algebra server protocol;
• Integrate additional mathematics;
• Rebuild the algebra in a literate programming style;
• Integrate logic programming;
• Develop an Axiom Journal with refereed submissions.

Axiom web page : http://www.nongnu.org/axiom/

Axiom project : http://savannah.nongnu.org/projects/axiom

2.Giac

Giac Is A Computer algebra system. The system has been designed by Bernard Parisse and is under active development.

3.GTybalt

GTybalt is a free computer algebra system which is built on top of GiNaC, CLN and a program to interpret C and C++ commands. gTybalt, which is still in an experimental stage, is maintained by Stefan Weinzierl. Some of the main features of gTybalt are the following:

• Object Oriented: gTybalt allows symbolic calculations within the C++ programming language.
• Efficiency for large scale problems: Solutions developed with gTybalt can be compiled with a C++ compiler and executed independently of gTybalt.
• Short development cycle: gTybalt can interpret C++ and execute C++ scripts. Solutions can be developed quickly for small-scale problems, either interactively or through scripts, and once debugged, the solutions can be compiled and scaled up to large-scale problems.

4.Macaulay 2

Macaulay 2 is a new software system devoted to supporting research in algebraic geometry and commutative algebra. The software is available now in source code for porting, and in compiled form for Linux, SunOS, Solaris, Windows, and a few other unix machines. An interface with TeXmacs is currently being implemented.

5.Maxima

Maxima is not only one of the oldest and best computer algebra systems around, it is also one of the only general purpose systems for which there is a free implementation. The current free Maxima implementation was started by William F. Schelter. It is dedicated to his memory. Here follow some features of Maxima:

• Plotting via netmath over the network.
• Computations over network
• Well tested on a large array of problems.
• Source level Debugger for maxima code
• Documentation available as html, texinfo, info, dvi and postscript. The documentation can be read inside TeXmacs.
• Easy to extend in fundamentally new ways, because you have complete access to source, and access to Common Lisp.
• Portable to many systems.

6.Pari

Pari-gp is a software package for computer-aided number theory. It consists of a C library, libpari (with optional assembler cores for some popular architectures), and of the programmable interactive gp calculator. While you can write your own libpari-based programs, many people just start up a gp session, or have gp execute their scripts. Pari sessions can now be started inside TeXmacs.

Originally developed at Bordeaux by a team led by Henri Cohen, PARI-GP is now maintained by Karim Belabas at the Université Paris-Sud Orsay with the help of many volunteer contributors.

7.Sage

Sage is an open source system for mathematical computations. It is written in Python and provides interfaces for a wide variety of other systems, such as Axiom, Gap, Pari GP, Macaulay 2, Maxima, Octave, and Singular. The system was started by William Stein.

8.Yacas

Yacas is, as it's name suggest, yet another computer algebra system. Things implemented include: arbitrary precision, rational numeric, vector, complex, and matrix computations (including inverses and determinants and solving matrix equations), derivatives, solving, Taylor series, numerical solving (Newtons method), and a lot more non-mathematical algorithms. The language natively supports variables and user-defined functions. There is basic support for univariate polynomials, integrating functions and tensor calculations. Yacas is still under development.