Abstract
We have provided a parallel solution for the well-known Smith normal form problem. Our method employs randomization as a tool to remove the iterations along the main diagonal in the classical sequential algorithms, and as such might be useful in similar settings, as well as may speed the sequential methods themselves.
Extended Abstract
This material is based upon work supported by the National Science Foundation under Grant No. DCR-85-04391 (first author), Grant No. MCS-83-14600 (second and third author), and by an IBM Faculty Development Award (first author).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Chou, T. J. and Collins, G. E., “Algorithms for the solution of systems of diophantine linear equations,” SIAM J. Comp., vol. 11, pp. 687–708, 1982.
Cook, S. A., “A taxonomy of problems with fast parallel algorithms,” Inf. Control, vol. 64, pp. 2–22, 1985.
Iliopoulos, C. S., “Worst-case complexity bounds on algorithms for computing the canonical structure of finite Abelian groups and the Hermite and Smith normal forms of an integer matrix,” Manuscript, Purdue Univ., 1986.
Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D., “Fast parallel algorithms for similarity of matrices,” Proc. 1986 ACM Symp. Symbolic Algebraic Comp., pp. 65–70, 1986.
Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D., “Fast parallel computation of Hermite and Smith forms of polynomial matrices,” SIAM J. Alg. Discrete Meth., vol. 8, pp. 683–690, 1987.
Kannan, R., “Polynomial-time algorithms for solving systems of linear equations over polynomials,” Theoretical Comp. Sci., vol. 39, pp. 69–88, 1985.
Kannan, R. and Bachem, A., “Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix,” SIAM J. Comp., vol. 8, pp. 499–507, 1981.
MacDuffee, C. C., Vectors and Matrices, Math. Assoc. America, 1943.
Mulmuley, K., “A fast parallel algorithm to compute the rank of a matrix over an arbitrary field,” Combinatorica, vol. 7, pp. 101–104, 1987.
Schwartz, J. T., “Fast probabilistic algorithms for verification of polynomial identities,” J. ACM, vol. 27, pp. 701–717, 1980.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kaltofen, E., Krishnamoorthy, M.S., Saunders, B.D. (1989). Mr. Smith goes to Las Vegas: Randomized parallel computation of the Smith Normal form of polynomial matrices. In: Davenport, J.H. (eds) Eurocal '87. EUROCAL 1987. Lecture Notes in Computer Science, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51517-8_134
Download citation
DOI: https://doi.org/10.1007/3-540-51517-8_134
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51517-3
Online ISBN: 978-3-540-48207-9
eBook Packages: Springer Book Archive