Abstract
The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.
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References
Kontsevich, M., private communication.
Kontsevich, M., Noncommutative Identities, Opening Talk at the Arbeitstagung 2011 of the Max Planck Institute Bonn/Germany http://www.mpim-bonn.mpg.de/webfm-send/146 (2011).
Olver, P.J. and Sokolov, V.V., Integrable Evolution Equations on Associative Algebras, Comm. in Math. Phys., 1998, vol. 193, no. 2, pp. 245–268.
Mikhailov, A.V. and Sokolov, V.V., Integrable ODEs on Associative Algebras, Comm. in Math Phys., 2000, vol. 211, pp. 231–251.
Efimovskaya, O.V., Integrable Cubic ODEs on Associative Algebras, Fundamentalnaya i Prikladnaya Matematika, 2002, vol. 8, no. 3, pp. 705–720.
Wolf, T. and Efimovskaya, O., On Integrability of the Kontsevich Non-Abelian ODE System, accepted for publication in Lett. in Math. Phys., 9 pages, DOI: 10.1007/s11005-011-0527-4 and http://lie.math.brocku.ca/twolf/papers/EfWNew11.pdf (2011).
Olver, P.J., Applications of Lie Groups to Differential Equations, in Graduate Texts in Mathematics, New York: Springer-Verlag, 1993, vol. 107, 2nd ed.
REDUCE—A Portable General-Purpose Computer Algebra System, free download site: http://reducealgebra.sourceforge.net, 2009.
The program streamsolve.red for Solving Linear Algebraic Systems in REDUCE (2007), http://lie.math.brocku.ca/papers/TsWo2007/.
The program LSSS.red for Solving Linear Algebraic Selection Systems in REDUCE (2011), http://lie.math.brocku.ca/papers/LSSS/.
Neun, W., The Computer Algebra System Reduce with an Extension Allowing 20M Identifiers: http://www.zib.de/Symbolik/reduce/twentyM.zip.
Tsarev, S.P. and Wolf, T., Classification of 3-Dimensional Integrable Scalar Discrete Equations, Lett. in Math. Phys., DOI: 10.1007/s11005-008-0230-2, also arXiv: 0706.2464 (2008).
Wolf, T., Applications of CRACK in the Classification of Integrable Systems, CRM Proceedings and Lecture Notes, 2004, vol. 37, pp. 283–300.
Gonnet, G.H. and Monagan, M.B., Solving Systems of Algebraic Equations or the Interface between Software and Mathematics, Research report CS-89-13, University of Waterloo (1989), http://www.cs.uwaterloo.ca/research/tr/1989/CS-89-13.pdf.
Pearce, R., Solving Sparse Linear Systems in Maple, http://www.mapleprimes.com/posts/41191-Solving-Sparse-Linear-Systems-In-Maple, source code at: http://www.cecm.sfu.ca/~rpearcea/sge/sge.mpl (2007).
Project LinBox: Exact Computational Linear Algebra, http://www.linalg.org/.
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Wolf, T., Schrüfer, E. & Webster, K. Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs. Program Comput Soft 38, 73–83 (2012). https://doi.org/10.1134/S0361768812020065
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DOI: https://doi.org/10.1134/S0361768812020065