as an engineering student with a strong mathemathical background, i dealing some problems like this at university:
as you can see some are just numerical ones, others have to process some kinds of symbols. currently i'm working with java, but i'm not very pleased with it (can't say exactly why, probably a personal taste) and now i'm searching for a programming language, in which i can easily prototype new algorithms, like for example in python, and don't care about low level stuff, but has the ability to speed things up if neccessary, e.g. with concurrent/parallel programming, etc. (writing it in python and rewrite it in C/C++ isn't really a option i prefer...)
to sum it up:
i've looked around at sites, like http://rosettacode.org/ and picked 2 or 3 favorites: Go, Lisp (and maybe Haskell) but other recommandations are welcome
Common Lisp using SBCL is pretty fast if you take the time to make it fast.
Why does this fit what you want?
I would use python together with cython: http://www.cython.org for speeding up your code. For symbolic computations you have http://code.google.com/p/sympy/
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