Maxima – A general purpose CAS (Computer Algebra System) is a program that’s able to perform symbolic manipulation for the resolution of common problems. As a matter of fact, modern CAS covers an extremely wide range of functionalities
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Scilab – Matlab is the standard for numerical computing, but there are a few clones and valid alternatives that are entirely free. Scilab is the closest that you can get to Matlab without spending a penny. It’s very compatible with Matlab (albeit not 100%) but it’s really flexible, powerful and comes with a Matlab converter and Scicos which is a block diagram modeler and simulator.
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Scilab – Matlab is the standard for numerical computing, but there are a few clones and valid alternatives that are entirely free. Scilab is the closest that you can get to Matlab without spending a penny. It’s very compatible with Matlab (albeit not 100%) but it’s really flexible, powerful and comes with a Matlab converter and Scicos which is a block diagram modeler and simulator.
R – For statistical computing and analysis in the Open Source world, it doesn’t get any better than R. It is a programming language and environment that enables you to do pretty much anything that the commercial software (S-Plus) does. It is so widely adopted that it can be considered a standard in the field.
lpsolve is another one that I like for linear (integer) programming. LaTeX is also a great tool – a typesetting system; it includes features designed for the production of technical and scientific documentation (many will already be familiar with it but if you are not, check it out).
Related: Statistics for Experimenters Second Edition – Online Mathematics Textbooks – Annals of Mathematics – Open Source for LEGO Mindstorms
For several–specific–numerical mathematical routines, there is also the following URL:
http://www.akiti.ca/Mathfxns.html
The routines posted off this page are based on good-quality code, mostly from the NETLIB repository of algorithms. Very handy (quick, easy-to-use, and free!) for when numbers are required, but a user might not want to install and start up a whole software package.
— Robert