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## 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 (1989)

Venue: | SIAM J. Comput |

Citations: | 48 - 0 self |

### Citations

989 |
Matrix multiplication via arithmetic progressions.
- COPPERSMITH, WINOQRA
- 1990
(Show Context)
Citation Context ...plication, requiring even fewer multiplications. Currently the best upper bound on the number of multiplications required for matrix multiplication is the following. THEOREM 1.4 (Coppersmith-Winograd =-=[3]-=-). There exists an algorithmfor multiplying two n x n matrices using O(n) multiplications, where 2.376. In the sequel, when the time required for the "computation of the determinant" is not the bottle... |

125 |
Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix
- KANNAN, BACHEM
- 1979
(Show Context)
Citation Context ... normal form, integer matrices, computational complexity AMS(MOS) subject classifications. 15A21, 20K01, 20K05 1. Introduction. Recently Chou and Collins [2] improved the results of Kannan and Bachem =-=[6]-=- on the computation ofthe Hermite and Smith normal form (abbreviated HNF and SNF, respectively) of an integer matrix. Reduction to the HNF and SNF can be done via integer row-column operations (abbrev... |

104 | Matrizentheorie [Matrix theory]. - Gantmacher - 1986 |

33 |
Algorithms for the solution of systems of linear diophantine equations
- Chou, Collins
- 1982
(Show Context)
Citation Context ...87-708]. Key words. Smith normal form, Hermite normal form, integer matrices, computational complexity AMS(MOS) subject classifications. 15A21, 20K01, 20K05 1. Introduction. Recently Chou and Collins =-=[2]-=- improved the results of Kannan and Bachem [6] on the computation ofthe Hermite and Smith normal form (abbreviated HNF and SNF, respectively) of an integer matrix. Reduction to the HNF and SNF can be ... |

26 |
Sur l’introduction des variables continues dans la théorie des nombres
- Hermite
(Show Context)
Citation Context ...3.4. If a finite Abelian group is represented by a matrix of size s, then we can compute its canonical structure in 0(s5M(s2)) elementary operations. 4. Hermite normal form. THEOREM 4.1 (Hermite, see =-=[5]-=-). Given a nonsingular n x n integer matrix A, there exists an n x n unimodular matrix M such that MA T is upper triangular with positive diagonal elements. Further, each off-diagonal element of T is ... |

6 |
Gaussian elimination is not optimal, Numer
- Strassen
- 1969
(Show Context)
Citation Context ...). ] The problems "computing the determinant of a square matrix," "computing the inverse of a nonsingular matrix," and "matrix multiplication" are computationally equivalent (for details see Strassen =-=[12]-=-, Winograd [13], or Aho et al. [1]). In 1969 Strassen showed in [12] that matrix multiplication of 2 x 2 matrices can be done with seven multiplications instead of eight and, in general, the multiplic... |

4 |
Seminumerical Algorithms, 2nd Edition (The
- Knuth
- 1981
(Show Context)
Citation Context ...d algorithm is not practical, due to the very large hidden constant in the upper bound on the number of multiplications required. The following theorem yields an upper bound on Knuth’s algorithm (see =-=[7]-=-) for computing the greatest common divisor of two integers. This upper bound was proved by Sch6nhage in [9]. 660 COSTAS S. ILIOPOULOS THF.ORF.M 1.5 (Extended Euclidean Algorithm (EEA)). There exists ... |

3 | The Design and Analysis qf Computer Algorithms - AHO, HOPCROFT, et al. - 1974 |

1 |
The influence of computers in algebra
- SIM
- 1974
(Show Context)
Citation Context ...Hermite and Smith normal form (abbreviated HNF and SNF, respectively) of an integer matrix. Reduction to the HNF and SNF can be done via integer row-column operations (abbreviated IRC operations; see =-=[8]-=-). A closely related problem is the computation of the canonical structure [8] of a finite Abelian group G represented by a set of defining relations that is associated with an integer matrix. We can ... |

1 |
Schnelle Berechnung vor Kettenbruchentwicklungen
- SCH
- 1971
(Show Context)
Citation Context ...iplications required. The following theorem yields an upper bound on Knuth’s algorithm (see [7]) for computing the greatest common divisor of two integers. This upper bound was proved by Sch6nhage in =-=[9]-=-. 660 COSTAS S. ILIOPOULOS THF.ORF.M 1.5 (Extended Euclidean Algorithm (EEA)). There exists an algorithm for computing the greatest common divisor r of two n bit integers a and a2 and two integers x a... |

1 | Schnelle Multiplikation Grosset Zahlen - SCHtNHAGE, STRASSEN - 1971 |