This page provides a simple list of software which provides (non-trivial) algorithms for numerical calculations, graph theory (including advanced graph layout), number theory or higher algebra in some form. Listing a software or project does not mean an endorsement or that the software has even been reviewed.
The list is currently limited to OSS. There is no limit regarding programming languages, license or API design. Computer algebra systems (CAS) are included. Function graphing, charts and other visualization as well as very specialized topics like neural network and high energy physics are currently not included. Library for basic calculations only, even for high precision integers and fractions or for vectors and matrices, aren't included either. Maybe this will be added later.
Java libraries
[http://www.spaceroots.org/software/mantissa/index.html Mantissa] - linear algebra, least squares, curve fitting, ordinary differential equations integrators, polynomials, random numbers, basic statistical analysis, optimization
[http://sourceforge.net/projects/hartmath HartMath]
[http://hoschek.home.cern.ch/hoschek/colt/ Colt] (Is this the correct Colt home page?)
http://jgrapht.sourceforge.net/
http://openjgraph.sourceforge.net/
C/C++
[http://math-atlas.sourceforge.net ATLAS] - highly optimized BLAS library
Gnu Scientific Library (gsl?)
[http://www.boost.org/ Boost]
[http://pari.math.u-bordeaux.fr/ pari/pari-gp]
[http://www.ginac.de/ GiNAC]
[http://www.math.uni-augsburg.de/opt/goblin.html Goblin] - graph theory
http://pigale.sourceforge.net/
Python
Numerical Python (URLs TBD)
Transcendentals
CAS
[http://sourceforge.net/projects/maxima Maxima]
[http://sourceforge.net/projects/yacas Yacas]
[http://www.r-project.org/ R]
http://sourceforge.net/projects/jscl-meditor
[http://savannah.gnu.org/projects/pspp PSPP] - statiistical analysis