The shifted number system for fast linear algebra on integer matrices

Journal of Complexity

Volumn 21, Issue 4, 2005, Pages 609-650

  Storjohann, Arne ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Bit complexity; Integer matrix; Las Vegas algorithms; Linear system solving; Matrix determinant; Matrix multiplication

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; LINEAR ALGEBRA; MATRIX ALGEBRA; NUMBER THEORY; POLYNOMIALS; THEOREM PROVING;

BIT COMPLEXITY; INTEGER MATRIX; LAS VEGAS ALGORITHMS; LINEAR SYSTEM SOLVING; MATRIX DETERMINANT; MATRIX MULTIPLICATION;

COMPUTATIONAL COMPLEXITY;

EID: 21844458687     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2005.04.002     Document Type: Article

References (45)

1. J. Abbott M. Bronstein T. Mulders Fast deterministic computation of determinants of dense matrices In: S. Dooley Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation: ISSAC '99 1999 ACM Press New York 197-204
2. A.V. Aho J.E. Hopcroft J.D. Ullman The Design and Analysis of Computer Algorithms 1974 Addison-Wesley Reading, MA
3. H. Brönnimann M. Yvinec Efficient exact evaluation of signs of determinants Algorithmica 21 2000 21-56
4. P. Bürgisser M. Clausen M.A. Shokrollahi Algebraic Complexity Theory, Grundlehren der mathematischen Wissenschaften vol. 315 1996 Springer Berlin
5. Z. Chen, A BLAS based C library for exact linear algebra on integer matrices, Master's thesis, School of Computer Science, University of Waterloo, 2005.
6. Z. Chen A. Storjohann A BLAS based C library for exact linear algebra on integer matrices In: M. Kauers Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation: ISSAC '05 2005 ACM Press New York
7. K.L. Clarkson Safe and efficient determinant evaluation Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, Los Alamitos, CA 1992 IEEE Computer Society Press Silver Spring, MO 387-395
8. D. Coppersmith Rectangular matrix multiplication revisited J. Complexity 13 1997 42-49
9. D. Coppersmith S. Winograd Matrix multiplication via arithmetic progressions J. Symbolic Computation 9 1990 251-280
10. T.H. Cormen C.E. Leiserson R.L. Rivest C. Stein Introduction to Algorithms second ed. 2001 MIT Press, McGraw-Hill Cambridge, MA, New York
11. J.D. Dixon Exact solution of linear equations using p-adic expansions Numer. Math. 40 1982 137-141
12. W. Eberly, Asymptotically efficient algorithms for the Frobenius form, Technical Report, Department of Computer Science, University of Calgary, 2000.
13. W. Eberly, M. Giesbrecht, G. Villard, Computing the determinant and Smith form of an integer matrix, in: Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 675-685.
14. I.Z. Emiris, V.Y. Pan, Improved algorithms for computing determinants and resultants, J. Complexity 21 (1) (2005) 43-71.
15. J. von zur Gathen J. Gerhard Modern Computer Algebra second ed. 2003 Cambridge University Press Cambridge
16. K.O. Geddes S.R. Czapor G. Labahn Algorithms for Computer Algebra 1992 Kluwer Boston, MA
17. M. Giesbrecht, Nearly optimal algorithms for canonical matrix forms, Ph.D. thesis, University of Toronto, 1993.
18. M. Giesbrecht Nearly optimal algorithms for canonical matrix forms SIAM J. Comput. 24 1995 948-969
19. M. Giesbrecht Fast computation of the Smith form of a sparse integer matrix Comput. Complexity 10 1 2001 41-69
20. T. Granlund, The GNU multiple precision arithmetic library, 2004, Edition 4.1.4. http://www.swox.com/gmp.
21. J.L. Hafner K.S. McCurley Asymptotically fast triangularization of matrices over rings SIAM J. Comput. 20 6 1991 1068-1083
22. O. Ibarra S. Moran R. Hui A generalization of the fast LUP matrix decomposition algorithm and applications J. Algorithms 3 1982 45-56
23. E. Kaltofen On computing determinants of matrices without divisions In: P.S. Wang Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation: ISSAC '92 1992 ACM Press New York 342-349
24. E. Kaltofen An output-sensitive variant of the baby steps/giant steps determinant algorithm In: T. Mora Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation: ISSAC '02 2002 ACM Press New York 138-144
25. E. Kaltofen M.S. Krishnamoorthy B.D. Saunders Parallel algorithms for matrix normal forms Linear Algebra Appl. 136 1990 189-208
26. E. Kaltofen G. Villard Computing the sign or the value of the determinant of an integer matrix a complexity survey, J. Comput. Appl. Math. 162 1 2004 133-146 Special issue: Proceedings of the International Conference on Linear Algebra and Arithmetic 2001, Rabat, Morocco, 28-31 May 2001, S. El Hajji, N. Revol, P. Van dooren (guest eds.).
27. E. Kaltofen G. Villard On the complexity of computing determinants Comput. Complexity 13 3-4 2004 91-130
28. D.E. Knuth The Art of Computer Programming, vol. 2, Seminumerical Algorithms second ed. 1981 Addison-Wesley Reading, MA
29. W. Krandick J. Johnson Efficient multiprecision floating point multiplication with optimal directional rounding 11th IEEE Symposium on Computer Arithmetic 1993 IEEE Computer Society Press Silver Spring, MD 228-233
30. R.T. Moenck, J.H. Carter, Approximate algorithms to derive exact solutions to systems of linear equations, in: Proceedings EUROSAM '79, Lecture Notes in Compute Science, vol. 72, Springer, Berlin, Heidelberg, New York, 1979, pp. 65-72.
31. T. Mulders A. Storjohann Certified dense linear system solving J. Symbolic Comput. 37 4 2004 485-510
32. M. Newman Integral Matrices 1972 Academic Press New York
33. V.Y. Pan Computing the determinant and the characteristic polynomial of a matrix via solving linear systems of equations Inf. Proc. Lett. 28 1988 71-75
34. V.Y. Pan Randomized acceleration of fundamental matrix computations Proceedings STACS 2002 vol. 2285 2002 Springer Heidelberg, Germany 215-226
35. V.Y. Pan X. Wang On rational number reconstruction and approximation SIAM J. Comput. 33 2 2004 502-503
36. J.B. Rosser L. Schoenfeld Approximate formulas for some functions of prime numbers Ill. J. Math. 6 1962 64-94
37. D. Saunders A. Storjohann G. Villard Matrix rank certification Electronic J. Linear Algebra 11 2004 16-23
38. A. Schönhage Schnelle Berechnung von Kettenbruchentwicklungen Acta Inform. 1 1971 139-144
39. A. Schönhage V. Strassen Schnelle Multiplikation grosser Zahlen Computing 7 1971 281-292
40. A. Storjohann Near optimal algorithms for computing Smith normal forms of integer matrices In: Y.N. Lakshman Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation: ISSAC '96 1996 ACM Press New York 267-274
41. A. Storjohann, Algorithms for Matrix Canonical Forms, Ph.D. Thesis, Swiss Federal Institute of Technology, ETH-Zurich, 2000.
42. A. Storjohann High-order lifting, extended abstract In: T. Mora Ed. Proceedings of the International Symposium on Symbolic and Algebraic Computation; ISSAC '02 2002 ACM Press New York 246-254
43. A. Storjohann High-order lifting and integrality certification J. Symbolic Comput. 36 3-4 2003 613-648
44. V. Strassen Gaussian elimination is not optimal Numer. Math. 13 1969 354-356
45. R.C. Whaley A. Petitet J.J. Dongarra Automated empirical optimization of software and the ATLAS project Parallel Comput. 27 1-2 2001