Axiom computer algebra system
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| Axiom | |
|---|---|
| Developed by | Independent group of people |
| OS | Cross-Platform |
| Genre | Computer Algebra System |
| License | modified BSD License |
| Website | Axiom Home Page |
Axiom is a free general purpose computer algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed, mathematically (mostly) correct type hierarchy.
Contents |
[edit] History
Axiom has been in development since 1973, originally as Scratchpad by researchers at IBM under the direction of Richard Dimick Jenks. In the 1990s it was sold to NAG and given its current name. In 2001 it was withdrawn from the market and released to Tim Daly, the project lead developer, under the Modified BSD License.
[edit] Philosophy
The Axiom project focuses on the "30 Year Horizon". The primary philosophy is that Axiom needs to develop several fundamental features in order to be useful to the next generation of computational mathematicians. It needs to:
- Be well documented. In order for the next generation to use the system the algorithms need strong documentation. Knuth's literate programming technology is used throughout the source code.
- Proven correct. Axiom plans to use proof technology to prove the correctness of the algorithms (such as Coq and ACL2).
- Mathematically sound. Axiom has a type hierarchy that is strongly based on standard mathematical category theory.
[edit] Documentation
- Jenks, R.D. and Sutor, R. "Axiom, The Scientific Computation System"
- Daly, T. "Axiom Volume 1: Tutorial"
[edit] Related Projects
The Axiom project forked twice in the summer of 2007, (see the articles on FriCAS and OpenAxiom)
[edit] Design
In Axiom, all objects have a type. Examples of types are mathematical structures (such as rings, fields, polynomials) as well as data structures from computer science (e.g., lists, trees, hash tables).
A function can take a type as argument, and its return value can also be a type. For example, Fraction is a function, that takes an IntegralDomain as argument, and returns the field of fractions of its argument. As another example, the ring of 
matrices with rational entries would be constructed as SquareMatrix(4, Fraction Integer). Of course, when working in this domain, 1 is interpreted as the identity matrix and A^-1 would give the inverse of the matrix A, if it exists.
Several operations can have the same name, and the types of both the arguments and the result are used to determine which operation is applied, somewhat similar to OOP.
The extension language Axiom comes with is called SPAD. All the mathematical knowledge of Axiom is written in this language. The interpreter accepts roughly the same language.
SPAD was further developed under the name A# and later Aldor. The latter can still be used as an alternative extension language. It is, however, distributed under a different license.
[edit] Features
Within the interpreter environment, Axiom uses type inference and a heuristic algorithm to make explicit type annotations mostly unnecessary.
It features 'HyperDoc', an interactive browser-like help system, and can display two and three dimensional graphics, also providing interactive features like rotation and lighting.
 HyperDoc displaying the available operations for a domain |
 Axiom displaying a surface |
Axiom is currently the only CAS with an (almost) complete implementation of the Risch algorithm for elementary integration, which was done by Manuel Bronstein and Barry Trager.
[edit] User Interface
There exists a specialised interaction mode for Emacs, as well as a plugin for the TeXmacs editor.
