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Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments. It may also be used as a batch-oriented language.
SciPy is a collection of mathematical routines that builds on the NumPy array extension. It collects a large number of general-purpose scientific and engineering routines. With NumPy + SciPy, you can use Python much as you would use Octave, Scilab, Matlab, or IDL
The GNU Scientific Library (GSL) is a numerical library for C and C++ programmers.
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.
The ATLAS (Automatically Tuned Linear Algebra Software) project is an ongoing research effort focusing on applying empirical techniques in order to provide portable performance. It provides C and Fortran77 interfaces to a portably efficient BLAS implementation, as well as a few routines from LAPACK.
O2scl is an object-oriented library for numerics in C++ useful for solving, minimizing, differentiating, integrating, interpolating, fitting, and more. Classes operate on generic member functions and generic vector types. Includes classes from GSL and CERNLIB.
XMDS is a code generator that integrates equations. You write them down in human readable form in a XML file, and it goes away and writes and compiles a C++ program that integrates those equations as fast as it can possibly be done in your architecture.
JMathLib is a system of mathematical functions designed to be used in evaluating complex expressions and display the results graphically. It can be used either interactively or to interpret script files. It works like Matlab, Octave, Freemat and Scilab.