1.Axiom
1.The Axiom
system
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 builtin compiler.
Axiom has been in development since 1973 and was sold as a commercial
product. It has been released as free software (under a BSDlike
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
1.Giac
2.The Giac
system
Giac Is A Computer algebra system.
The system has been designed by Bernard Parisse
and is under active development. It is a generalpurpose computation
kernel written in C++ with the following features.

Algebra

Arbitrary precision integers, integer and polynomial arithmetic,
Gröbner bases, simplification, equation solver, partial
fraction decomposition, linear algebra (vectors, matrices, row
reduction to echelon form, eigenvalues and eigenvectors),
permutations, combinatorial analysis, computing in and

Calculus

Derivatives, integration, limits, series expansion, vector
calculus, calculus of variations, implicit differentiation,
curve interpolation, differential equations, special functions

2D and 3D plotting

Function graphs, surfaces, plotting implicit curves, plotting
the solution of a system of inequalities, polar and parametric
plots, scatter and line plots, bar plots, pie charts, histograms

2D and 3D geometry

Points, segments, lines, triangles, polygons, circles, conics,
parametric curves, curve intersection, tangents, planes,
surfaces

Probability and statistics

Probability distributions, random variables, efficient sampling,
maximumlikelihood fitting, statistical tests, kernel density
estimation

Signal processing and audio tools

Convolution, (auto)correlation, continuous/fast Fourier
transform, Fourier series, filtering, windowing, loading,
creating and saving audio clips, audio playback, resampling,
noise removal, waveform plotting, spectrum plotting

Optimization

Mixed integer linear programming, finding local and global
extrema, nonlinear programming, transportation problem

Graph theory

Creating (random) graphs and digraphs, operations on graphs,
modifying graphs, importing and exporting graphs to dot file format, examining properties of graphs,
traversing graphs, vertex and edge coloring, graph isomorphism,
graph drawing

Programming

Functions, local variables, conditionals, loops, choice of
syntax (Clike, Python, Maple, Mupad, TI89)

Syntax compatibility with other systems

Maple compatibility, MuPAD compatibility, Python compatibility,
MathML export (content and presentation), LaTeX export
Giac output is rendered in TeXmacs from its
Scheme tree representation, which results in semantically correct
formulas which can be copied back to Giac.
Interactive plotting and mathematical input are supported as well.
A note on session plots.
Any interactive Giac figure can be
automatically embedded into the TeXmacs document by pressing
“OK”. However, the PostScript conversion requires eps2eps script to be
available in PATH. Linux users simply need to install the
Ghostscript package. Windows users need to install either MiKTeX or
Ghostscript. If the latter is installed, then the paths to bin and lib folders in the Ghostscript
installation directory (e.g. C:\Program
Files\gs\gs9.53.3\bin and also lib) must be added to the PATH
variable, which can be edited by typing “env”' in the
Windows start menu.
1.GTybalt
3.The GTybalt system
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 smallscale problems, either interactively or through
scripts, and once debugged, the solutions can be compiled and
scaled up to largescale problems.
1.Macaulay 2
4.The Macaulay
2 system
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.
1.Maxima
5.The Maxima
system
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.
1.Pari
6.The Pari
system
Parigp is a software package for
computeraided 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, PARIGP is now maintained
by Karim Belabas at the Université
ParisSud Orsay with the help of many volunteer contributors.
1.Reduce
7.Reduce
computer algebra system
Reduce is one of the oldest
computer algebra systems. It is powerful, stable and highly efficient.
Its TeXmacs interface is described in the arXiv article.
1.Sage
8.
The Sage system
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.
1.Yacas
9.The Yacas
system
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 nonmathematical algorithms. The language
natively supports variables and userdefined functions. There is basic
support for univariate polynomials, integrating functions and tensor
calculations. Yacas is still under development.
© 1999–2003 by Joris van der Hoeven