Maxima
The official maxima logo.
Developer: Independent group of people
License: GNU GPLv2
Web: http://maxima.sourceforge.net/
About
Various graphical user interfaces are available for Maxima. wxMaxima is a cross platform GUI based on wxWidgets. The GNU TeXmacs mathematical editor program can be used to provide an interactive GUI for Maxima, as can SAGE. Other options include the Imaxima front end as well as an Emacs interaction mode.
Maxima includes a complete programming language with ALGOL-like syntax but Lisp-like semantics, so it can readily be used for teaching programming and computer algebra.
It uses Gnuplot for drawing.
Since Maxima is written in Common Lisp, it is easily accessed programmatically and extended, as the underlying Lisp can be called from Maxima.
Numeric calculations
Maxima is a full-featured CAS that specializes in operations]. It also offers special numerical capabilities such as arbitrary-precision arithmetic: integers and rational numbers which can grow to sizes limited only by machine memory, and [floating point numbers] whose precision can be set arbitrarily large ("bfloats").
For calculations which use floating point and arrays heavily, Maxima offers the possibility of generating code in other programming languages (notably Fortran) which may execute it more efficiently.
Maxima is a general-purpose system, and special-case calculations such as factorization of large numbers, manipulation of extremely large polynomials, etc. are often better done in specialized systems.