The package CRACK for solving large overdetermined systems

Abstract

The program CRACK is a computer algebra package written inREDUCE for the solution of over-determined systems of algebraic,ordinary or partial differential equations with at most polynomialnon-linearity. It is available as part of version 3.8 of the REDUCEsystem (dated April 2004) and the latest development version can bedownloaded from http://lie.math.brocku.ca/crack/. The purpose of this poster is to accompany a softwaredemonstration of CRACK at ISSAC 2005. The poster is supposed togive a graphical overview of CRACK, emphasizing features which arespecial to the package. Those are • a rich interface, with visualization aids forinspecting large systems, including - for each equation its properties, history and investigationsthat have already been done with the equation, - the occurrence of all derivatives of selected functions in anyequation, - a statistical overview of the system (number of equations andfunctions in dependence of number of independent variables), - a matrix display of occurrences of unknown constants/functionsin all equations, - a count of the total number of appearances of eachunknown, - the determination of not under-determined subsystems, - a listing of all sub- and sub-sub-.. cases investigated sofar, with their assumptions, number of steps and number ofsolutions, - graphical displays of size related measures of the computationdone so far; • a discussion of possibilities to trade interactivelyor automatically the speed of the solution process versus safety(avoidance of expression swell): - only length-reducing versus complete Gröbner basiscomputation steps. - substitutions in shorter equations only, i.e. only in asub-system versus substitutions in the complete system, - growth bounded substitutions versus general substitutions, - case splittings (induced by factorizations, substitutions withpotentially vanishing coefficients or adhoc case distinctions)versus Gröbner basis steps, - an investment in the length reduction of equations to reachsparse systems with multiple benefits; • algorithmic extensions which include - the ability to collect and apply syzygies which result as aby-product in the process of computing a differentialGröbner basis to integrate linear PDEs. - the treatment of inequalities: their usage, active collectionand derivation, and their constant update in an ongoing reductionprocess based on newly derived equations. - the capability added by Winfried Neun (ZIB Berlin) to run in atruly parallel mode on a beowulf cluster, recently also ported to64bit AMD processors. - post-computation procedures, especially the possibility tomerge solutions of parametric linear algebraic systems and toautomate the production of web-pages for solutions that arefound. - the ability to separate expressions with respect toindependent variables which occur polynomially but with variableexponents, leading to automatically investigated case distinctionsof exponents being equal or not; • safety enhancing measures as - the ability to backup and re-load the whole session, - the automatic storing of the complete keyboard input during asession with the opportunity to feed this stored input into a newsession, - the possibility to impose time restrictions of notoriouslyslow sub-steps, like factorizations and sometimes the computationand reduction of S-polynomials in a Gröbner basiscomputations, - a method to interrupt an ongoing automatic computation andchange it to interactive mode The poster in landscape format will display the above fourtopics in boxes. For some of the sub-items above a screen outputwill give a visual impression, like the matrix indicatingoccurrences of unknowns in equations. In the following publicationsthe solution of large overdetermined systems was a crucialingredient: • the solution of large bi-linear algebraic systems andautomatic merging of obtained solutions: - Wolf, T., Efimovskaya, O. V.: Classification of integrablequadratic Hamiltonians on e(3), Regular and Chaotic Dynamics, vol8, no 2 (2003), p 155--162. - Sokolov, V. V., Wolf, T.: Classification of integrablepolynomial vector evolution equations, J. Phys. A: Math. Gen. 34(2001) 11139--11148. - Tsuchida, T. and Wolf, T.: Classification of integrablecoupled systems with one scalar and one vector unknown, preprintnlin.SI/0412003 (2004) 60 pages, to appear in J. Phys. A: Math.Gen. - Sokolov, V. V. and Wolf, T.: Integrable quadratic Hamiltonianson so(4) and so(3, 1),preprint (2004) 16 pages, nlin.SI/0405066. - Kiselev, A. and Wolf, T.: New recursive chains of N=1homogeneous superequations, to appear in proceedings of "Symmetryin Nonlinear Mathematical Physics", Kyiv 2005. • the solution of extensive overdeterminedODE/PDE-systems: - Anco, S. and Wolf, T.: Some symmetry classifications ofhyperbolic vector evolution equations, nlin.SI/0412015, JNMP,Volume 12, Supplement 1 (2005), p 13--31.