Octave

High level language for numerical computations

If you're scientific or mathematical work involves highly complex linear and non linear problem solving, then you may find that Octave provides the power you've been looking for.

Octave is a high-level language developed by the open source community which functions simply by providing a command line interface for solving linear and nonlinear numeric problems. For those that are familiar with Matlab, they will have few problems picking up Octave because it works in conjunction with the former language.

The program basically features several very powerful tools for solving the kind of linear algebra equations that would send shivers down the spine of even the most experienced mathematician. According to the developers, it also includes functions for manipulating polynomials and integrating differential and differential-algebraic equations if you know what those are in the first place! The one bonus for those familiar with other languages is that Octave can also be extended or used in conjunction with modules written in C++, C, Fortran and others.

This is an extremely complex language that only those familiar with Matlab or Fortran will pick up quickly. Anyone else should be prepared to put a lot of hours in although the problem solving skills it will provide you with seem virtually limitless.