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Manuscript Title: The computer calculation of Lie point symmetries of large systems of differential equations.
Authors: B. Champagne, W. Hereman, P. Winternitz
Program title: SYMMGRP.MAX
Catalogue identifier: ACBI_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 66(1991)319
Programming language: Macsyma.
Computer: VAX 11/750.
Operating system: VMS 3.7 OR HIGHER.
RAM: 500K words
Word size: 32
Keywords: Computer algebra, General purpose, Symmetry group, Equations differential, Lie algebras, Symbolic computation, Macsyma.
Classification: 4.2, 5.

Nature of problem:
The symmetry group of a given system of differential equations modelling a physical phenomenon may be used to achieve several goals. These include the classification of the solutions of the system, the generation of new solutions from known ones, the simplification of the system by the method of symmetry reduction, to name a few. In the case where particular symmetries must be present, the symmetry group can be used to determine the validity of the modelling differential equations.

Solution method:
The construction of the symmetry groups of differential equations is based on an adaptation of the method. This procedure is translated into a MACSYMA program that performs the most elaborate part of the job, namely the construction of a complete list of determining equations which is free of redundant factors, repetitions and trivial differential consequences.

For complicated systems of differential equations involving derivatives of high order, time limits and available computer memory may cause restrictions. Further limitations are discussed in Section 3.5 of the Long Write-Up.

Unusual features:
The flexibility of this program and the possibility of using it in a partly interactive mode, allow one to find the symmetry groups of essentially arbitrarily large systems of equations. This is the main justification for presenting a new symbolic manipulation program in a field where several programs already exist. Furthermore, this program has been in use (at the Universite de Montreal and elsewhere) for over five years, it has been tested on hundreds of systems of equations and has thus been comprehensively debugged.

Running time:
Given a system of m differential equations of order k with q unknowns and p independent variables, running time is an increasing function of m, k, q and p. Typical running times (CPU) for an example are given in Section 4.