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Algebra Packages  
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Commercial/ Proprietary
Free/ Open Source  
I think I was supposed to provide pointers to free alternatives to Mathematica (and Maple?) after yesterday. Here are the major general purpose ones, I think (apart from `Mx calc' within the one true editor, obviously):  
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http://www.axiomdeveloper.org/ http://www.openaxiom.org/ and http://fricas.sourceforge.net/ . Axiom is packaged in Debian. Not for RedHat as far as I can tell.  
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> >  Gentoo lAvailable versions: ~200711!s ~200803!s ~200805!s  
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http://maxima.sf.net/ Installed on lv3.nwgrid.ac.uk. Packaged for Debian and RedHat.  
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> >  Available Gentoo versions: 5.14.0 ~5.17.1r1 5.18.1 ~5.19.2 {X clisp cmucl ecl emacs gcl latex linguas_es linguas_pt linguas_pt_BR nls sbcl tk unicode xemacs}
Homepage: http://maxima.sourceforge.net/
 
 
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http://www.texmacs.org/ Packaged for Debian, at least.  
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There are various other lesscapable general purpose ones (e.g. Jacal), and more specialpurpose ones (e.g. PARI, GAP) that might be useful to mathematicians, for instance. I don't know how useful the lists at http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems and http://www.sai.msu.su/sal/A/1/ are.
As far as I know, much criticism of Mathematica in one of the most penetrating reviews I've ever seen is still relevant, but I don't know which: http://www.cs.berkeley.edu/~fateman/papers/mma.pdf.  
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> >  I should also have said that I'm not aware that any of these have been parallelized, though you can probably get Maxima to use the Common Lisp MPI package fairly easily. Also, they may not be suitable for serious numerical array work; the Axiom link to the NAG library hasn't yet been replaced, and the LAPACK in Maxima is done by translation of SLATEC to Lisp, not an interface to a system library.
Other Notes Maxima is a GPLlicensed computer algebra system that is similar to commercial products such as Mathematica and Maple. It can be used to do integration, differentiation, matrix mathematics, solve differential and linear equations, factor and expand polynomial expressions, and plot the results of functions and data in both two and three dimensions. It is available in source format and as a Windows executable or RPM package for Fedora Core 2, and can also be accessed online. Documentation can be found on Maxima's Web site. Octave complements symbolic algebra programs like Maxima by performing numerical calculations. Octave is both an environment for performing numerical calculations and an interpreted programming language. In Linux systems, running octave from the command line opens up an Octave shell where Octave commands can be entered and evaluated. The default Octave package is able to perform matrix and vector mathematics, numerically solve differential equations, and plot data utilizing Gnuplot. Its language is mostly compatible with the language used in the commercial product Matlab and therefore can be used interchangeably with Matlab in most situations. Octave should be able to run most Matlab addon packages. Additionally, users can create there own addon packages to add functionality to Octave. Among the packages in The GNU Octave Repository are a variety of addons for specialized fields. Octave is available in the base packages of many Linux distributions, including Fedora Core and Red Hat Enterprise Linux, and for Windows. Information on Octave commands can be found in Octave's manual online or purchased for $30.
General Maths

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Algebra PackagesI think I was supposed to provide pointers to free alternatives to Mathematica (and Maple?) after yesterday. Here are the major general purpose ones, I think (apart from `Mx calc' within the one true editor, obviously):
There are various other lesscapable general purpose ones (e.g. Jacal), and more specialpurpose ones (e.g. PARI, GAP) that might be useful to mathematicians, for instance. I don't know how useful the lists at http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems and http://www.sai.msu.su/sal/A/1/ are. As far as I know, much criticism of Mathematica in one of the most penetrating reviews I've ever seen is still relevant, but I don't know which: http://www.cs.berkeley.edu/~fateman/papers/mma.pdf.  DaveLove  04 Dec 2009 