1 Introduction

Symbolic calculations, carried out by computer algebra systems, have become an integral part in the daily work of scientists. The advance in algorithms and computer technology has led to remarkable progress in several areas of natural sciences. gTybalt was developed as a tool for certain kind of calculations. The characteristics of these calculations are: First of all, these tend to be “long” calculations, e.g. the system needs to process large amounts of data and efficiency in performance is a priority. Secondly, the algorithms for the solution of the problem are usually developed and implemented by the scientists themselves. This requires support from the computer algebra system for a programming language which allows to implement complex algorithms for abstract mathematical entities. In other words, it requires support of object oriented programming techniques from the system. On the other hand, these calculations usually do not require that the computer algebra system provides sophisticated tools for all branches of mathematics. Thirdly, despite the fact that these calculations process large amounts of data, the time needed for the implementation of the algorithms usually outweights the actual running time of the program. Therefore convenient development tools are also important.

1.1 Main features of gTybalt

gTybalt is a free computer algebra system. It is distributed under the terms and conditions of the GNU General Public Licence.

The main features of gTybalt are:

• Object Oriented: gTybalt allows symbolic calculations within the C++ programming language.
• Efficiency for large scale problems: Solutions developed with gTybalt can be compiled with a C++ compiler and executed independently of gTybalt. This is particular important for computer-extensive problems and a major weakness of commercial computer algebra systems.
• Short development cycle: gTybalt can interpret C++ and execute C++ scripts. Solutions can be developed quickly for small-scale problems, either interactively or through scripts, and once debugged, the solutions can be compiled and scaled up to large-scale problems.
• High quality output: Mathematical formulae are visualized using TeX fonts and can easily be converted to LaTeX on a what-you-see-is-what-you-get basis.

Compared to other computer algebra systems, gTybalt does not try to cover every domain of mathematics. Some desirable algorithms, like symbolic integration are not implemented. However, the modular design of gTybalt allows to incorporate easily new algorithms.

The functionality of a computer algebra system can be divided into different modules, e.g. there will be a module, which displays the output, a second module analysis and interprets the input, a third module does the actual symbolic calculation. Writing a computer algebra system from scratch is a formidable task. Fortunately it is not required, since there are already freely available packages for specific tasks. gTybalt is based on several other packages and provides the necessary communication mechanisms among these packages. gTybalt is therefore a prototype of a “bazaar”-style program and an example of what is possible within the free software community. It should be clearly stated, that without these already existing packages gTybalt would never have been developed and my thanks go to the authors of these packages for sharing their programs with others. In particular gTybalt is build on the following packages:

• The TeXmacs-editor is used to display the output of formulae in high quality mathematical typesetting using TeX fonts.
• gTybalt can also be run from a text window. Then the library eqascii is used to render formulae readable in text mode.
• Any interactive program needs an interpreter for its commands. gTybalt uses the CINT C/C++ interpreter which allows execution of C++ scripts and C++ command line input.
• At the core of any computer algebra system is the module for symbolic and algebraic manipulations. This functionality is provided by the GiNaC-library.
• One aspect of computer algebra systems is arbitrary precision arithmetic. Here GiNaC (and therefore gTybalt) relies on the Class Library for Numbers (CLN library).
• Plotting functions is very helpful to quickly visualize results. The graphic abilities of gTybalt are due to the Root-package.
• The GNU scientific library is used for Monte Carlo integration.
• Optionally gTybalt can be compiled with support for the expansion of transcendental functions. This requires the nestedsums library to be installed.
• Optionally gTybalt can be compiled with support for factorization of polynomials. This requires the NTL library to be installed.

Additional documentation: Since gTybalt incoporates GiNaC, Root and TeXmacs, you are advised to read also the documentation on GiNaC, Root and TeXmacs.

1.2 License

gTybalt is Copyright © 2001-2011 Stefan Weinzierl and Roberta Marani.

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; see the file COPYING. If not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.