On Efficient Sparse Integer Matrix Smith Normal Form Computations

Journal of Symbolic Computation

Volumn 32, Issue 1-2, 2001, Pages 71-99

  Dumas, Jean Guillaume ^a   David Saunders, B ^b   Villard, Gilles ^c  

^a INP Grenoble ENSIMAG   (France)

^b Institute University of Delaware Newark   (United States)

^c UNIV LYON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0005426657     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.2001.0451     Document Type: Article

References (42)

1. Babson E., Bjöerner A., Linusson S., Shareshian J., Welker V. Complexes of not i -connected graphs. Topology. 38:1999;271-299.
2. Bachem A., Kannan R. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8:1979;499-507.
3. Bjöerner A., Lovász L., Vrećica S. T., Živaljević R. T. Chessboard complexes and matching complexes. J. London Math. Soc. 49:1994;25-39.
4. Björner A., Welker V. Complexes of directed graphs. SIAM J. Discrete Math. 12:1999;413-424.
5. Brauer A. Limits for the characteristic roots of a matrix. I. Duke Math. J. 13:1946;387-395.
6. Brauer A. Limits for the characteristic roots of a matrix. II. Duke Math. J. 14:1947;21-26.
7. Brualdi R. A., Mellendorf S. Regions in the complex plane containing the eigenvalues of a matrix. Am. Math. Monthly. 101:1994;975-985.
8. Dixon J. D. Exact solution of linear equations using p-adic expansions. Numer. Math. 40:1982;137-141.
9. Dumas J.-G. Calcul parallèle du polynôme minimal entier en Athapascan-1 et Linbox. RenPar'2000. Actes des Douzièmes Rencontres Francophones du Parallélisme. 2000;Besançon, France.
10. J.-G. Dumas, F. Heckenbach, B. D. Saunders, V. Welker, Simplicial Homology, a (proposed) share package for GAP.
11. Dumas J.-G., Saunders B. D., Villard G. Traverso (2000) Integer Smith form via the Valence: Experience with Large Sparse Matrices from Homology. 2000;. p. 95-105.
12. P. Dusart, 1998.
13. thprime is greater than k(ln k + lnlnk - 1) for k ≥ 2. Math. Comput. 68:1999;411-415.
14. Eberly W., Giesbrecht M. W., Villard G. Computing the determinant and Smith form of an integer matrix. The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA. 2000.
15. Eberly W., Kaltofen E. On randomized Lánczos algorithms. Küchlin W. ISSAC'97: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Maui Hawaii, July 1997). 1997;ACM Press, New York.
16. Gantmacher F. R. The Theory of Matrices. 1959;Chelsea, New York.
17. von zur Gathen J., Gerhard J. Modern Computer Algebra. 1999;Cambridge University Press, New York, NY, USA.
18. Giesbrecht M. W. Probabilistic Computation of the Smith Normal Form of a Sparse Integer Matrix. 1996;. p. 173-186.
19. Giesbrecht M. W. Efficient parallel solution of sparse systems of linear diophantine equations. Parallel Symbolic Computation (PASCO'97), Maui, Hawaii. 1997;. p. 1-10.
20. Golub G. H., Van Loan C. F. Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences. 1996;The Johns Hopkins University Press, Baltimore, MD, USA.
21. 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. SIAM J. Comput. 18:1989;658-669.
22. Kaltofen E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64:1995;777-806.
23. Kaltofen E., Krishnamoorthy M. S., Saunders B. D. Parallel algorithms for matrix normal forms. Linear Algebr. Appl. 136:1990;189-208.
24. Kaltofen E., Lee W.-S., Lobo A. A. Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm. Traverso C. ISSAC 2000: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (St. Andrews, Scotland, August 2000). 2000;192-201 ACM Press, New York.
25. Kaltofen E., Saunders B. D. On Wiedemann's method of solving sparse linear systems. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC '91). 1991;. p. 29-38.
26. Lenstra H. Factoring integers with elliptic curves. The Annals of Mathematics. 1987;. p. 649-673.
27. R. H. Lewis.
28. Massias J.-P., Robin G. Bornes effectives pour certaines fonctions concernant les nombres premiers. J. Théorie Nombres Bordeaux. 8:1996;213-238.
29. Mignotte M. Mathématiques pour le Calcul Formel. 1989;Presses Universitaires Françaises.
30. Mulders T., Storjohann A. Diophantine linear system solving. International Symposium on Symbolic and Algebraic Computation (ISSAC 99), Vancouver, BC, Canada. 1999;. p. 181-188.
31. Munkres J. R. The computability of homology groups, advanced book program. Elements of Algebraic Topology. 1994;Addison-Wesley Publishing Company, California. p. 53-61.
32. P. Ozello, 1987.
33. Rayward-Smith V. J. On computing the Smith normal form of an integer matrix. ACM Trans. Math. Softw. 5:1979;451-456.
34. Reeds J. A., Sloane N. J. A. Shift-register synthesis (modulo m). SIAM J. Comput. 14:1985;505-513.
35. Sibert E., Mattson H. F., Jackson P. Finite field arithmetic using the connection machine. Zippel R. Computer Algebra and Parallelism, Proceedings of the Second International Workshop on Parallel Algebraic Computation,LNCS 584. 1990;51-61 Springer Verlag.
36. Storjohann A. Near optimal algorithms for computing Smith normal forms of integer matrices. Lakshman Y. N. ISSAC '96: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, July 24-26, 1996, Zurich, Switzerland. 1996;267-274 ACM Press, New York, NY 10036, USA.
37. A. Storjohann, 2000.
38. Taussky O. Bounds for characteristic roots of matrices. Duke Math. J. 15:1948;1043-1044.
39. Varga R. S. Matrix Iterative Analysis, Number 27 of Springer Series in Computational Mathematics. 2000;Springer-Verlag.
40. Villard G. Further analysis of Coppersmith's block Wiedemann algorithm for the solution of sparse linear systems. Küchlin W. ISSAC'97: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Maui, Hawaii). 1997;32-39 ACM Press, New York.
41. Wiedemann D. H. Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory. 32:1986;54-62.
42. Zippel R. Zero equivalence testing. Effective Polynomial Computation. 1993;Kluwer Academic Publishers, Boston. p. 189-206.