Abstract
We present an algorithm for computing the Hermite normal form of a polynomial matrix and an unimodular transformation matrix on a distributed computer network. We provide an algorithm for reducing the off-diagonal entries which is a combination of the standard algorithm and the reduce off-diagonal algorithm given by Chou and Collins. This algorithm is parametrised by an integer variable.
We provide a technique for producing small multiplier matrices if the input matrix is not full-rank, and give an upper bound for the degrees of the entries in the multiplier matrix.
The author is very grateful to the GMD, Sankt Augustin, Germany for offering CPU time on their IBM SP2 installation. Special thanks go to George Havas for his corrections and suggestions.
Chapter PDF
References
W. A. Blankinship. A new version of the Euclidian algorithm. Amer. Math. Monthly, 70(9):742–745, 1963.
W. A. Blankinship. Matrix triangulation with integer arithmetic. Comm. ACM, 9(7):513, 1966.
G. H. Bradley. Algorithms and bound for the greatest common divisor of n integers and multipliers. Comm. ACM, 13:447, 1970.
G. H. Bradley. Algorithms for Hermite and Smith normal matrices and linear Diophantine equations. Math. Comp., 25(116):897–907, 1971.
T. W. J. Chou and G. E. Collins. Algorithms for the solution of systems of linear Diophantine equations. SIAM J. Comput., 11(4):687–708, 1982.
G. D. Forney. Convolutional codes I: Algebraic structure. IEEE Trans. Inform. Theory, IT-16(6):720–738, 1970.
G. Havas and B. S. Majewski. Extended gcd calculation. Congr. Numer., 111:104–114, 1995.
G. Havas, B. S. Majewski, and K. R. Matthews. Extended gcd algorithms. Technical Report TR0302, The University of Queensland, Brisbane, 1995.
G. Havas and C. Wagner. Matrix reduction algorithms for euclidean rings. In Proc. of the 3rd Asian Symposium Computer Mathematics, to appear.
E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Parallel algorithms for matrix normal forms. Linear Algebra Appl., 136:189–208, 1990.
R. Kannan and A. Bachem. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput., 8(4):499–507, 1979.
B. S. Majewski and G. Havas. A solution to the extended gcd problem. In ISSAC’95 (Proc. 1995 Internat. Sympos. Symbolic Algebraic Comput.), pages 248–253. ACM Press, 1995.
C. C. Sims. Computation with Finitely Presented Groups. Cambridge University Press, 1994.
C. Wagner. Normalformberechnung von Matrizen über euklidischen Ringen. PhD thesis, Institut für Experimentelle Mathematik, Universität/GH Essen, 1997. Published by Shaker-Verlag, 52013 Aachen/Germany, 1998.
C. Wagner. Hermite normal form computation over Euclidean rings. Interner Bericht 71, Rechenzentrum der Universität Karlsruhe, 1998. submitted for publication.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wagner, C. (1998). Fast parallel Hermite normal form computation of matrices over \(\mathbb{F}[x]\) . In: Pritchard, D., Reeve, J. (eds) Euro-Par’98 Parallel Processing. Euro-Par 1998. Lecture Notes in Computer Science, vol 1470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0057936
Download citation
DOI: https://doi.org/10.1007/BFb0057936
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64952-6
Online ISBN: 978-3-540-49920-6
eBook Packages: Springer Book Archive