R is a language and environment for statistical computing and graphics. R provides a wide variety of statistical (linear and nonlinear modelling, classical statistical tests, time-series analysis, classification, clustering, ...) and graphical techniques, and is highly extensible. The S language is often the vehicle of choice for research in statistical methodology, and R provides an Open Source route to participation in that activity.
Python is a high level programming language that aims to combine remarkable power with very clear syntax. Anaconda is a free cross-platform Python distribution from Continuum Analytics . It comes built in with various scientific Python packages such as NumPy, SciPy, Pandas, Matplotlib, Numba, etc.
From the GNU website: "The GNU Scientific Library (GSL) is a numerical library for C and C++ programmers. It is free software under the GNU General Public License. The library provides a wide range of mathematical routines such as random number generators, special functions and least-squares fitting. There are over 1000 functions in total with an extensive test suite."
This software was discontinued on MSI systems as of January 8, 2016. AMPL/KNITRO is a large, sparse, linear and nonlinear programming solver and a general purpose solver for nonlinear optimization. It is designed to handle large problems and is effective for solving linear, quadratic, and nonlinear smooth optimization problems, both convex and nonconvex. SNOPT is highly effective for problems with a nonlinear objective function and large numbers of sparse linear constraints.
OpenFOAM is an open source software package which is designed to scale well on large distributed memory systems. While originally written for CFD, with it's open source strategy and a large user base OpenFOAM has grown to be a general purpose finite element toolbox.