REDUCE is a system for doing scalar, vector and matrix algebra by computer, which also supports arbitrary precision numerical approximation and interfaces to gnuplot to provide graphics. It can be used interactively for simple calculations (as illustrated in the screenshot above) but also provides a full programming language, with a syntax similar to other modern programming languages.
Established in 1969 the CPC Program Library now contains more than 2200 programs in computational physics and chemistry. Papers describing the programs are published in the Computer Physics Communications Journal and are available online via Science Direct.
The ATLAS (Automatically Tuned Linear Algebra Software) project is an ongoing research effort focusing on applying empirical techniques in order to provide portable performance. At present, it provides C and Fortran77 interfaces to a portably efficient BLAS implementation, as well as a few routines from LAPACK.
SciPy is open-source software for mathematics, science, and engineering. The SciPy library depends on NumPy, which provides convenient and fast N-dimensional array manipulation. The SciPy library is built to work with NumPy arrays, and provides many user-friendly and efficient numerical routines such as routines for numerical integration and optimization. Together, they run on all popular operating systems, are quick to install, and are free of charge. If you need to manipulate numbers on a computer and display or publish the results, give SciPy a try!
Sage is a free, open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common, Python-based interface. The project goal is to create a viable free, open source alternative to Magma, Maple, Mathematica and Matlab.