×

zbMATH — the first resource for mathematics

Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs. (English) Zbl 1256.65074
Summary: 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 ordinary differential equation (ODE) introduced recently by M. Kontsevich [private communication]. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F50 Computational methods for sparse matrices
65F05 Direct numerical methods for linear systems and matrix inversion
68W30 Symbolic computation and algebraic computation
65Y15 Packaged methods for numerical algorithms
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Kontsevich, M., private communication.
[2] 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).
[3] 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. · Zbl 0908.35124
[4] Mikhailov, A.V. and Sokolov, V.V., Integrable ODEs on Associative Algebras, Comm. in Math Phys., 2000, vol. 211, pp. 231–251. · Zbl 0956.37040
[5] Efimovskaya, O.V., Integrable Cubic ODEs on Associative Algebras, Fundamentalnaya i Prikladnaya Matematika, 2002, vol. 8, no. 3, pp. 705–720. · Zbl 1034.34046
[6] 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).
[7] Olver, P.J., Applications of Lie Groups to Differential Equations, in Graduate Texts in Mathematics, New York: Springer-Verlag, 1993, vol. 107, 2nd ed. · Zbl 0785.58003
[8] REDUCE–A Portable General-Purpose Computer Algebra System, free download site: http://reducealgebra.sourceforge.net , 2009.
[9] The program streamsolve.red for Solving Linear Algebraic Systems in REDUCE (2007), http://lie.math.brocku.ca/papers/TsWo2007/ .
[10] The program LSSS.red for Solving Linear Algebraic Selection Systems in REDUCE (2011), http://lie.math.brocku.ca/papers/LSSS/ .
[11] Neun, W., The Computer Algebra System Reduce with an Extension Allowing 20M Identifiers: http://www.zib.de/Symbolik/reduce/twentyM.zip .
[12] 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). · Zbl 1153.37432
[13] Wolf, T., Applications of CRACK in the Classification of Integrable Systems, CRM Proceedings and Lecture Notes, 2004, vol. 37, pp. 283–300. · Zbl 1073.37081
[14] 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 .
[15] 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).
[16] Project LinBox: Exact Computational Linear Algebra, http://www.linalg.org/ .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.