Computer Algebra Systems
Short descriptions of computer algebra systems are presented in three sections: major systems, special purpose systems, and packages. However, the separation between special purpose systems and packages is not to be taken too literally. An older survey is
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Computer Algebra Systems
Short descriptions of computer algebra systems are presented in three sections: major systems, special purpose systems, and packages. However, the separation between special purpose systems and packages is not to be taken too literally. An older survey is the paper by Calmet and van Hulzen in [Buchberger et al. 1982]. There is now an excellent new book by Wester [Wester 1999] that covers computer algebra systems. Additional systems are described in Chapter 3. For instance, section 3.1.1.1 refers to the SCHOONSCHIP program. The Nobel Prize in physics was awarded in 1999 to its author, and as the citation of the Nobel Foundation reads [www. nobel.se/announcement-99/physics99.html] "At the end of the 1960s ... [Martinus J. G.] Veltman had developed the Schoonschip computer program which, using symbols, performed algebraic simplifications of the complicated expressions that all quantum field theories result in when quantitative calculations are performed .... With the help of Veltman's computer program [Gerardus] 't Hooft's partial results were now verified and together they worked out a calculation method in detail." The editors 4.1 4.1.1
General Purpose Systems AXIOM
AXIOM, a Computer Algebra System with Abstract Data Types 4.1.1.1 Overview AXIOM is a modern computer algebra system with a modular design which takes advantage of the intrinsic relationships between mathematical objects to allow them to be viewed and described in terms of their mathematical properties via a system of abstract data types. AXIOM is a fully interactive system and includes a command-line interpreter, a hypertext-based help system, graphics and a compiler which allows efficient extensions to the system to be produced by users. The system is based on Cambridge Common Lisp (CCL). There is also an interactive link to the NAG Fortran library for numerical processing. AXIOM is a successor to the Scratchpad system developed by Richard Jenks' group at the IBM T.J. Watson Research Center in New York. In 1991 the technology was licensed to NAG Ltd. and AXIOM release 1.0 appeared. The current version is 2.2 and is available on all Unix platforms. In addition there is a version for Windows-95 and Windows NT which has a different user interface and graphics system. 4.1.1.2 The AXIOM Type System AXIOM recognises the fundamental relationships between mathematical objects and is designed to take advantage of them. Its object-centered design provides efficiency, consistency and robustness through a natural type system which categorises objects according to their J. Grabmeier et al (eds.), Computer Algebra Handbook © Springer-Verlag Berlin Heidelberg 2003
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Chapter 4 Computer Algebra Systems
algebraic properties. Users can add their own extensions to the system with the AXIOM Library Compiler (known as ALDoR). At the top level of the AXIOM type system are Categories, which can be viewed as specifications of algebraic structures in terms of their abstract properties. Some of these will be familiar, such as Set or Ring, while
