What are the pluses and minuses of different software packages? Anything new worth checking out?
I'm especially interested in open source packages.
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If you haven't already, you should try Sage.
Someone should also mention Axiom, which started out as commercial package developed by IBM but after a long and tortuous journey is now open source. As I understand it, it is built on a strongly typed, mathematically oriented and user extensible type hierarchy. So it knows what groups, rings and fields are, and it lets you define your own and do computation in them.
I tend to use Macaulay2 for research related stuff
Some points about it.
I've also heard good things about Sage but haven't tried it.
I suggest you to try Reduce, it is OpenSource and there are RPM packeges. It looks like this:
The only open-source CAS (Computer Algebra System) that I know of is called Maxima, but as I have not used it, I cannot say whether it is any good. I own a copy of Mathematica 7.0, and can say that it is superb. The programming language is easy to learn, and in many ways quite similar to that of the TI-89. The university I attend uses Maple 13, which from my experience is not as strong as Mathematica in the area of purely symbolic manipulation, but is superior in terms of numerical modeling.
For quick calculations, I use Wolfram|Alpha, which is not open source but is a free web service.