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Black box Frobenius decompositions over small fields

Author:

Wayne EberlyAuthors Info & Claims

ISSAC '00: Proceedings of the 2000 international symposium on Symbolic and algebraic computation

Pages 106 - 113

https://doi.org/10.1145/345542.345596

Published: 01 July 2000 Publication History

Abstract

A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n × n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best known deterministic algorithmic for this problem, recently given by Storjohann and Villard [16]. The new algorithm is based on the evaluation of Krylov spaces, rather than an climination technique, and may therefore be superior when applied to sparse or structured matrices with a small number of invariant factors.

References

[1]

D. Augot and P. Camion. On the computation of minimal polynomials, cyclic vectors, and Frobenius forms. Linear Algebra and its Applications, 260:61-94, 1997.

[2]

E. R. Berlekamp. Algebraic Coding Theory. McGraw Hill, 1968.

[3]

D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9:251-280, 1990.

Digital Library

[4]

W. Eberly. Asymptotically efficient algorithms for the Frobenius form. Manuscript, 2000; available at the author's web site.

[5]

F. R. Gantmacher. The Theory of Matrices, volume one. Chelsea Publishing Company, second edition, 1959.

[6]

M. Giesbrecht. Nearly optimal algorithms for canonical matrix forms. Technical Report 268/93, Department of Computer Science, University of Toronto, 1993.

[7]

M. Giesbrecht. Nearly optimal algorithms for canonical matrix forms. SIAM Journal on Computing, 24:948-969, 1995.

Digital Library

[8]

E. Kaltofen. Challenges of symbolic computation: My favorite open problems. Journal of Symbolic Computation, 2000. To appear; with an additional open problem by R. M. Corless and D. J. Jeffrey.

Digital Library

[9]

R. Lambert. Computational Aspects of Discrete Logarithms. PhD thesis, Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada, 1996.

Digital Library

[10]

H. L/ineburg. On Rational Normal Form of Endomorphisms: A Primer to Constructive Algebra. Wissenschaftsverlag, 1987.

[11]

J. L. Massey. Step by step decoding of the Bose-Chaudhuri-Hocquenghem codes. IEEE Transactions on Information Theory, IT-11:580-585, 1965.

Digital Library

[12]

J. L. Massey. Shift-register synthesis and BCH decoding. IEEE Transactions on Information Theory, IT-15:122-127, 1969.

Digital Library

[13]

P. Ozello. Calcul Exact des Formes de Jordan et de Frobenius d'une Matrice. PhD thesis, Universit~ Scientifique Technologique et Medicale de Grenoble, 1987.

[14]

A. Steel. A new algorithm for the computation of canonical forms of matrices over fields. Journal of Symbolic Computation, 24:409-432, 1997.

Digital Library

[15]

A. Storjohann. An O(n3) algorithm for the Frobenius normal form. In Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, pages 101-104, 1998.

Digital Library

[16]

A. Storjohann and G. Villard. Algorithms for similarity transforms. In Seventh Rhine Workshop on Computer Algebra, Bregenz, Austria, March 2000.

[17]

V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 14:354-356, 1969.

Digital Library

[18]

G. Villard. Computing the Frobenius normal form of a sparse matrix. Manuscript, 2000.

[19]

J. yon zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999.

Digital Library

[20]

D. H. Wiedemann. Solving sparse linear equations over finite fields. IEEE Transactions on Information Theory, IT-32:54-62, 1986.

Digital Library

Cited By

Sankowski PWęgrzycki K(2019)Improved Distance Queries and Cycle Counting by Frobenius Normal FormTheory of Computing Systems10.1007/s00224-018-9894-x63:5(1049-1067)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/s00224-018-9894-x
Eberly WAbramov SZima EGao X(2016)Selecting Algorithms for Black Box MatricesProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930894(207-214)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930894
Harrison GJohnson JSaunders B(2016)Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computationJournal of Symbolic Computation10.1016/j.jsc.2015.06.00574:C(55-69)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.06.005
Show More Cited By

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Black box Frobenius decompositions over small fields

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cover image ACM Conferences

ISSAC '00: Proceedings of the 2000 international symposium on Symbolic and algebraic computation

July 2000

309 pages

ISBN:1581132182

DOI:10.1145/345542

Editor:
Carlo Traverso
Univ. di Pisa

Copyright © 2000 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 July 2000

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ISSAC '00 Paper Acceptance Rate 40 of 98 submissions, 41%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Sankowski PWęgrzycki K(2019)Improved Distance Queries and Cycle Counting by Frobenius Normal FormTheory of Computing Systems10.1007/s00224-018-9894-x63:5(1049-1067)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/s00224-018-9894-x
Eberly WAbramov SZima EGao X(2016)Selecting Algorithms for Black Box MatricesProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930894(207-214)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930894
Harrison GJohnson JSaunders B(2016)Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computationJournal of Symbolic Computation10.1016/j.jsc.2015.06.00574:C(55-69)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.06.005
Mullen GPanario D(2013)CryptographyHandbook of Finite Fields10.1201/b15006-22(777-860)Online publication date: 17-Jun-2013
https://doi.org/10.1201/b15006-22
De Feo LSchost E(2010)transalpyneACM Communications in Computer Algebra10.1145/1838599.183862444:1/2(59-71)Online publication date: 29-Jul-2010
https://dl.acm.org/doi/10.1145/1838599.1838624
Dumas JPernet CSaunders BJohnson JPark HKaltofen E(2009)On finding multiplicities of characteristic polynomial factors of black-box matricesProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576723(135-142)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576723
Eberly WSchicho J(2004)Reliable Krylov-based algorithms for matrix null space and rankProceedings of the 2004 international symposium on Symbolic and algebraic computation10.1145/1005285.1005305(127-134)Online publication date: 4-Jul-2004
https://dl.acm.org/doi/10.1145/1005285.1005305
Storjohann A(2001)Deterministic computation of the Frobenius formProceedings 42nd IEEE Symposium on Foundations of Computer Science10.1109/SFCS.2001.959911(368-377)Online publication date: 2001
https://doi.org/10.1109/SFCS.2001.959911

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