1. Introduction

AXIOM is a powerful computer algebra system which provides a complete environment for anyone needing to manipulate and solve mathematical formulae. Its application is wide-ranging, from pure mathematics research through branches of physics, chemistry, biology and engineering to financial modelling and cryptography.

It was originally developed as a research tool by IBM in collaboration with experts around the world; IBM has a long history of research in symbolic algebra with many significant contributions to the field. The copyright of AXIOM is now assigned to NAG.

2. Symbolic Solvers

Symbolic solvers have revolutionised the way that people think about the computation of mathematical problems. Computers are now able to manipulate formulae as well as data, providing analytical insight and precision in results which was previously unattainable.

The symbolic solver allows the machine and the user to communicate in terms of algebraic formulae, in the language of the scientist. Both input and output can be purely algebraic or in numerical form of specified precision.

3. What can AXIOM do for you?

For the novice, AXIOM can be used as easily as a sophisticated desk calculator. The User Interface provides hypertext and graphics so that familiarisation is straightforward and rapidly achieved. There is an extensive library of mathematical functions and operations so that AXIOM is immediately useful in the educational field for a range of courses in algebra and computational mathematics generally.

The more expert user can perform complex mathematical calculations using the built-in capabilities which are available for use in the fields of calculus, modern algebra and number theory.

The system is designed to be used by scientists and engineers, undergraduates, research and teaching staff, financial analysts, planners and of course, mathematicians.

4. What does AXIOM Provide?

The mathematical consistency and sheer power of the AXIOM system delivers reliable, accurate and usable results. There is an interactive language, an on-line help and documentation facility in hypertext format and a powerful graphics capability for the manipulation of 2D and 3D objects.

Please refer to the end of this document for a list of some of the features provided.

The unique strength of AXIOM is derived from its object oriented approach and its overall structure which is strongly typed and hierarchical. This means that algorithms can be implemented in their most natural setting and that users can develop their own extensions in a robust, consistent environment without recourse to the supplier.

A further advantage is found in the open nature of the software. AXIOM Library source code and definitions are available on-line so that users can see exactly how AXIOM computes its solutions and thus be reassured that they are getting the 'right' answer.

5. What is Different about AXIOM?

AXIOM provides an interactive command-line driven environment like other systems. It employs a very expressive language with concise syntax. Procedural and functional styles of programming are readily accommodated. Although it is a strongly typed language, the interpreter uses powerful type inferencing techniques to minimize the need for type declarations.

Speed cannot always be traded for comfort. In AXIOM, you (the user) have the best of both worlds. User-written functions are automatically compiled for appropriate types at first invocation. The user has the further option of creating an AXIOM Library module. The new functions (and associated documentation) will then be treated by AXIOM exactly as the system supplied functions. The more care you put into writing your module, the more widely usable and efficient it can be. AXIOM was designed around the basic idea of unlimited extensibility without performance or usability degradation.

The AXIOM Library is unique in its design, scope and rigour. The object-oriented hierarchy of datatypes follows closely the development of Modern Algebra. As a result, AXIOM is the ideal prototyping and developing environment for advanced symbolic algorithms.

To help you get started with the AXIOM system, the command-line interface is augmented by extensive on-line help in hypertext format and common commands are introduced by a fill-in-the-blanks form. The AXIOM User Guide is available on-line and contains thousands of examples of AXIOM commands ready to be run at the click of a button. Hypertext links take you across related topics. The AXIOM Reference Guide to the contents of the AXIOM Library is only available in hypertext form (as befits an object-oriented Library). Thousands of operations from hundreds of modules are documented. Furthermore, we provide all the source code for the AXIOM Library.

6. Release 2.0

The next release of AXIOM (Release 2.0) will offer two new powerful facilities.

A new compiler will allow the construction of stand-alone applications linked with the AXIOM Library and other foreign-language libraries (graphics/numerics libraries). This feature is unique among comparable systems and frees you from the constraints of an interactive system.

The second facility is interactive interpreter access to the NAG Fortran Library (possibly residing and executing at a remote fast machine). Tools will be provided to enable similar access to user-specified programs and libraries. This will be the first such integrated and supported link for a comparable application.

7. The Future

NAG and AXIOM contributors will continue to co-operate and collaborate towards further development of AXIOM. The AXIOM Library compiler will be enhanced and the algebraic components of the package will continue to be extended. The software will be made available on an increasing number of hardware platforms and heterogeneous networks.

Symbolic solvers have long been expected to become the new paradigm in technical computing; AXIOM fulfils that promise now and for the future.

8. Features of AXIOM include: