Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

Abstract

In this paper we show how the transformations associated with the reduction to the Smith form of some classes of mul-tivariate polynomial matrices are computed. Using a Maple implementation of a constructive version of the Quillen-Suslin Theorem, we present two algorithms for the reduction to a particular Smith form often associated with the simplification of linear systems of multidimensional equations.

Share and Cite:

M. Boudellioua, "Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple," American Journal of Computational Mathematics, Vol. 2 No. 1, 2012, pp. 21-26. doi: 10.4236/ajcm.2012.21003.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] H. H. Rosenbrock, “State Space and Multivariable Theory,” Nelson-Wiley, London, 1970.
[2] T. Kailath, “Linear Sys-tems,” Prentice-Hall, Englewood Cliffs, 1980.
[3] A. Fabianska and A. Quadrat, “Applications of the Quillen-Suslin Theo-rem to Multidimensional Systems Theory,” Technical Report 6126, INRIA, Sophia Antipolis, 2007.
[4] D. Quillen, “Projective Modules over Polynomial Rings,” Inventiones Mathematicae, Vol. 36, 1976, pp. 167-171. doi:10.1007/BF01390008
[5] A. Suslin, “Projective modules over Polynomial Rings Are Free,” Soviet Mathematics—Doklady, Vol. 17, No. 4, 1976, pp. 1160-1164.
[6] M. Frost and C. Storey, “Equivalence of a Matrix over R[s; z] with Its Smith Form,” International Journal of Control, Vol. 28, No. 5, 1979, pp. 665-671. doi:10.1080/00207177808922487
[7] M. Morf, B. Levy and S. Kung, “New Results in 2-D Systems Theory: Part I: 2-D Polynomial Matrices, Factorization and Coprimeness,” Proceedings of the IEEE, Vol. 65, No. 6, 1977, pp. 861-872. doi:10.1109/PROC.1977.10582
[8] E. Lee and S. Zak, “Smith Forms over R[z1; z2],” IEEE Transactions on Auto-matic Control, Vol. 28, No. 1, 1983, pp. 115-118. doi:10.1109/TAC.1983.1103118
[9] M. Frost and M. S. Boudellioua, “Some Further Results Concerning Matrices with Elements in a Polynomial Ring,” International Journal of Control, Vol. 43, No. 5, 1986, pp. 1543-1555. doi:10.1080/00207178608933558
[10] Z. Lin, M. S. Boudel-lioua and L. Xu, “On the Equivalence and Factorization of Multivariate Polynomial Matrices,” Proceedings of the 2006 International Symposium of Circuits and Systems, Island of Kos, 21-24 May 2006, p. 4914.
[11] M. S. Boudellioua and A. Quadrat, “Serre’s Reduction of Linear Functional Systems,” Mathematics in Computer Science, Vol. 4, No. 2, 2010, pp. 289-312. doi:10.1007/s11786-010-0057-y

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.