Algorithms for Large Integer Matrix Problems
- First Online:
New algorithms are described and analysed for solving various problems associated with a large integer matrix: computing the Hermite form, computing a kernel basis, and solving a system of linear diophantine equations. The algorithms are space-efficient and for certain types of input matrices — for example, those arising during the computation of class groups and regulators — are faster than previous methods. Experiments with a prototype implementation support the running time analyses.
Unable to display preview. Download preview PDF.
- 1.H. Cohen and H. Lenstra, Jr. Heuristics on class groups of number fields. In Number Theory, Lecture notes in Math., volume 1068, pages 33–62. Springer-Verlag, New York, 1983.Google Scholar
- 7.M. J. Jacobson, Jr. Subexponential Class Group Computation in Quadratic Orders. PhD thesis, Technischen Universität Darmstadt, 1999.Google Scholar
- 8.F. Lübeck. On the computation of elementary divisors of integer matrices. Journal of Symbolic Computation, 2001. To appear.Google Scholar
- 9.T. Mulders and A. Storjohann. Diophantine linear system solving. In S. Dooley, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC’ 99, pages 281–288. ACM Press, 1999.Google Scholar
- 10.T. Mulders and A. Storjohann. Rational solutions of singular linear systems. In C. Traverso, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC’ 00, pages 242–249. ACM Press, 2000.Google Scholar
- 11.A. Storjohann. A solution to the extended gcd problem with applications. In W. W. Küchlin, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC’ 97, pages 109–116. ACM Press, 1997.Google Scholar
- 12.A. Storjohann. Algorithms for Matrix Canonical Forms. PhD thesis, ETH-Swiss Federal Institute of Technology, 2000.Google Scholar
- 13.J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999.Google Scholar