Indeed, there are plenty of open source solutions that can solve your particular problem, and many problems don't need any sort of symbolic tools. There's also Octave with a nice collection of packages on Octave-Forge, there's PARI (included in Sage), and there's LAPACK, and I'm sure there are others I'm either forgetting or haven't heard of.
I too can attest to the extensive abilities of the Sage package. It's extremely powerful and easy to use given that it's based in Python. Even if you don't want to download it, give the notebook a try: http://www.sagenb.org/.