#### DMCA

## Recognizing badly presented Z-modules (1993)

Venue: | LINEAR ALGEBRA APPL |

Citations: | 11 - 2 self |

### Citations

157 | Computation with finitely presented groups - Sims - 1994 |

118 |
The elimination form of the inverse and its application to linear programming,
- Markowitz
- 1957
(Show Context)
Citation Context ...jC¡k = j(a1;t; : : : ; as¡1;t; as+1;t; : : : ; am;t)jk 6 Thus these are the corresponding metrics of the row and column containing v, possibly excluding v itself. For numerical applications Markowitz =-=[26]-=- introduced a heuristic where, in our terms, the pivot v is chosen to minimize jvjR¡0 £ jvj C¡ 0 . Suitably modi¯ed for stability, this strategy remains a recommended one for general unsymmetric matri... |

114 | Integer programming and network flows. - Hu - 1969 |

89 | Solving large sparse linear systems over finite fields. - LaMacchia, Odlyzko - 1990 |

74 |
Asymptotically fast triangularization of matrices over rings.
- Hafner, McCurley
- 1991
(Show Context)
Citation Context ...ppendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley =-=[12]-=-. In this paper we discuss both integer based methods and modular techniques. We ¯rst describe the relevant methods and then present examples of their performance. 2 Gauss-Jordan elimination Super¯cia... |

61 | Factoring polynomials with rational coefficients, - Lovasz - 1982 |

58 |
On systems of linear indeterminate equations and congruences
- Smith
- 1861
(Show Context)
Citation Context ...principal ideal domains and their canonical forms in Chapter 7, where they give an algorithm for the computation of a canonical form. For integer matrices (Z-modules) these results date back to Smith =-=[33]-=-. Our motivation is the solution of group-theoretic problems, so we cast our description in terms of abelian groups. Since Z-modules are no more nor less than abelian groups, the principles are releva... |

55 | Some integer factorization algorithms using elliptic curves
- Brent
- 1986
(Show Context)
Citation Context ...2 £ 3, and modular diagonalization reveals that N=N 0 »= 8C2 © 2C4. Note that factorization of numbers like these determinants (say di) is very hard. An elliptic curve method program of Richard Brent =-=[1]-=- reveals: d1 = 2 12:3:7:11:71:109:1459:185914338563:c151; d2 = 2 12:3:13:17:797:857:c162; d3 = 2 12:3:11:89:52501:67153:303302071:283687:p̂123; 10 d4 = 2 12:3:1233653:p14:p21:c131; d5 = 2 12:3:181:284... |

47 |
Computational Methods for General Sparse Matrices,
- Zlatev
- 1991
(Show Context)
Citation Context ...ize of entries. 5 Sparse matrices and pivoting strategies In numerical analysis various techniques exist for handling sparse matrices. These have been much studied, and one recent reference is Zlatev =-=[34]-=-. Some applications of these kinds of techniques to exact matrices appear in Donald and Chang [8] and LaMacchia and Odlyzko [24], where some progress is made. In our context trying to ¯nd sparse initi... |

46 |
Hermite normal form computation using modulo determinant arithmetic.
- Domich, Kannan, et al.
- 1987
(Show Context)
Citation Context ...arious contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter =-=[7]-=-, Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modular techniques. We ¯rst describe the relevant methods ... |

33 |
Algorithms for the solution of systems of linear diophantine equations
- Chou, Collins
- 1982
(Show Context)
Citation Context ...e investigated this problem in various contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins =-=[5]-=-, Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modular techniques. We ¯rs... |

21 |
A modification of the LLL reduction algorithm
- Pohst
- 1987
(Show Context)
Citation Context ...orithm which requires O(X4 log(x)) operations on numbers of length O(X log(x)) and guarantees the quality of the reduced basis. The LLL algorithm was described for square matrices of full rank. Pohst =-=[30]-=- extended it to handle general rectangular matrices producing a modi¯ed algorithm, MLLL, with analogous complexity. MLLL produces a reduced basis from (possibly) linearly dependent vectors. So, to get... |

20 | On the complexity of computing the homology type of a triangulation
- Donald, Chang
- 1991
(Show Context)
Citation Context ...s and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang =-=[8]-=- and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modular techniques. We ¯rst describe the relevant methods and then present examples of their performance. 2 Gauss... |

15 |
Polynominal time algorithms in the theory of linear diophantine equations
- Frumkin
- 1977
(Show Context)
Citation Context ...ing [18] and Sims [32]. Many other researchers have investigated this problem in various contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin =-=[9, 10]-=-, Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both intege... |

15 | Proving a group infinite - Newman - 1990 |

13 | A graphics system for displaying finite quotients of finitely presented groups”, Groups and Computation - Holt, Rees - 1993 |

13 |
Algorithmic Aspects of Vertex Elimination of Directed Graphs,
- Rose, Tarjan
- 1978
(Show Context)
Citation Context ...y says \choose a nonzero entry (the pivot)". We investigate various strategies in practice and ¯nd some which are not expensive to implement and which have good performance. Note that Rose and Tarjan =-=[31]-=- have shown that it is an NP-complete problem to ¯nd a strategy which minimizes ¯ll-in. This indicates that ¯nding optimal pivoting strategies for our purposes is likely to be very di±cult. Thus we co... |

12 | modules and linear algebra - Hartley, Hawkes, et al. - 1970 |

12 |
A Reidemeister-Schreier program.
- Havas
- 1973
(Show Context)
Citation Context ...d in the sense that they are very distant from the canonical presentation given by the Smith normal form. Thus, the abelian group presentations arising from a Reidemeister-Schreier process (see Havas =-=[14]-=-, Havas and Sterling [18] and NeubÄuser [27]) have large numbers of generators and relations, with a substantial number of trivial modules in the canonical form. We want e±cient algorithms for recogni... |

12 |
case complexity bounds on algorithms for computing the canonical structure of in¯nite abelian groups and solving systems linear Diophantine equations
- Iliopoulos, Worst
- 1989
(Show Context)
Citation Context ...rences not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos =-=[21, 22]-=-, Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modular techniques. We ¯rst describe the relevant methods and then present examples of thei... |

9 |
Residual Hermite normal form computations
- Domich
- 1989
(Show Context)
Citation Context ...xts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich =-=[6]-=-, Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modular techniques. We ¯rst describe the relevant methods and then pre... |

9 |
A Tietze transformation program
- Havas, Kenne, et al.
- 1984
(Show Context)
Citation Context ...representation before converting to a standard array representation. During the sparse matrix phase we follow some Tietze transformation program principles (see Havas, Kenne, Richardson and Robertson =-=[15]-=-). Thus we perform short eliminations and (abelianized) substring searching till no further improvement is possible. The short eliminations correspond to well-selected pivoting operations (a pivot v i... |

8 | The last of the Fibonacci groups
- Havas, Richardson, et al.
- 1979
(Show Context)
Citation Context ...ertain subgroups. This technique has led to e®ective understanding of some groups whose structure was not well-enough known, including the Fibonacci group F (2; 9) (see Havas, Richardson and Sterling =-=[17]-=- and Newman [29]) and certain knot groups (see Havas and Kovµacs [16]). A common thread in these applications is the discovery of the module presentation by use of a Reidemeister-Schreier algorithm fo... |

7 |
Congruence techniques for the exact solution of integer systems of linear equations
- Cabay, Lam
- 1977
(Show Context)
Citation Context ...compute matrix ranks and determinants. In such cases these techniques are fast and avoid entry explosion, with calculations being done in prime ¯elds, Zp instead of Z (see, for example, Cabay and Lam =-=[2]-=-). For Smith normal form calculation the situation is somewhat more complicated. Underlying theory for modular techniques may be found in Gerstein [11]. Perhaps the ¯rst algorithm description, though ... |

5 |
Integer Matrices and Abelian Groups,
- Havas, Sterling
- 1979
(Show Context)
Citation Context ...are very distant from the canonical presentation given by the Smith normal form. Thus, the abelian group presentations arising from a Reidemeister-Schreier process (see Havas [14], Havas and Sterling =-=[18]-=- and NeubÄuser [27]) have large numbers of generators and relations, with a substantial number of trivial modules in the canonical form. We want e±cient algorithms for recognizing the badly presented ... |

5 |
Solving large sparse linear systems over ¯nite ¯elds
- LaMacchia, Odlyzko
- 1990
(Show Context)
Citation Context ...rices. These have been much studied, and one recent reference is Zlatev [34]. Some applications of these kinds of techniques to exact matrices appear in Donald and Chang [8] and LaMacchia and Odlyzko =-=[24]-=-, where some progress is made. In our context trying to ¯nd sparse initial relation matrices helps. Then use of pivoting strategies which both maintain sparsity and reduce entry explosion has given or... |

5 |
Factoring polynomials with rational coe±cients
- Lov¶asz
- 1982
(Show Context)
Citation Context ...orresponding routine can also be applied to columns of course, but it does not seem to be worthwhile doing both. In their very important paper on computational number theory Lenstra, Lenstra, Lov¶asz =-=[25]-=- included a new basis reduction algorithm which requires O(X4 log(x)) operations on numbers of length O(X log(x)) and guarantees the quality of the reduced basis. The LLL algorithm was described for s... |

4 |
A handbook of Cayley functions
- Cannon, Bosma
- 1991
(Show Context)
Citation Context ...agnitude of the Kannan and Bachem algorithm was a 13 decimal digit number, 9330076432385. 6.3 Modular methods The modular techniques described here are built into the computer algebra language Cayley =-=[3]-=-. When applied to ¯nding the abelian quotient of the index 152 subgroup of F (2; 9) Cayley proceeds as follows. Cayley uses primes with about 15 bits, so that all required computations can be convenie... |

4 |
Algorithms for groups, Austral
- Cannon, Havas
- 1992
(Show Context)
Citation Context ...ation in natural ways. Examples include subgroup presentation by Reidemeister-Schreier processes, cohomology calculations, and as part of soluble quotient computation algorithms (see Cannon and Havas =-=[4]-=- for an overview and references). Our aim is to identify such a ¯nitely presented G using e®ective algorithms. In principle it is easy, but it can be di±cult if n and m are large. The fundamental theo... |

4 |
Polynomial Time algorithms for computing Smith and Hermite normal forms of an integer matrix
- Kannan, Bachem
- 1979
(Show Context)
Citation Context ...y other researchers have investigated this problem in various contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin [9, 10], Kannan and Bachem =-=[23]-=-, Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both integer based methods and modu... |

3 |
An application of modular arithmetic to the construction of algorithms for solving systems of linear equations
- Frumkin
- 1976
(Show Context)
Citation Context ...ing [18] and Sims [32]. Many other researchers have investigated this problem in various contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein [11], Frumkin =-=[9, 10]-=-, Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we discuss both intege... |

3 | Some computational approaches to groups given by finite presentations, Mathematica Universitaria 7: 77–120 (Rio de Janeiro - Neubüser, Sidki - 1988 |

2 |
Distinguishing eleven crossing knots
- Havas, Kovµacs
- 1984
(Show Context)
Citation Context ... some groups whose structure was not well-enough known, including the Fibonacci group F (2; 9) (see Havas, Richardson and Sterling [17] and Newman [29]) and certain knot groups (see Havas and Kovµacs =-=[16]-=-). A common thread in these applications is the discovery of the module presentation by use of a Reidemeister-Schreier algorithm followed by abelianization. Analysis of this process reveals aspects of... |

2 |
A local approach to matrix equivalence
- Gerstein
- 1977
(Show Context)
Citation Context ...avas and Sterling [18] and Sims [32]. Many other researchers have investigated this problem in various contexts. Useful references not included by Havas and Sterling are Hu [20, Appendix A], Gerstein =-=[11]-=-, Frumkin [9, 10], Kannan and Bachem [23], Chou and Collins [5], Domich, Kannan and Trotter [7], Domich [6], Iliopoulos [21, 22], Donald and Chang [8] and Hafner and McCurley [12]. In this paper we di... |

2 |
Computation with ¯nitely presented groups
- Sims
(Show Context)
Citation Context ...f group-theoretic problems, so we cast our description in terms of abelian groups. Since Z-modules are no more nor less than abelian groups, the principles are relevant for Z-modules in general. Sims =-=[32]-=- includes a signi¯cant chapter on abelian groups in his book on computational group theory. ¤Partially supported by Australian Research Council Grant A49030651. 1 A ¯nitely presented abelian group G m... |

1 |
A graphics system for displaying ¯nite quotients of ¯nitely presented groups
- Holt, Rees
- 1991
(Show Context)
Citation Context ...tions for the index 152 subgroup. Choosing n = 2 minimizes the size of the relation matrix. However it also reduces the sparsity. Our algorithms are included in the quotpic package (see Holt and Rees =-=[19]-=-). In that context we initially use a sparse matrix representation before converting to a standard array representation. During the sparse matrix phase we follow some Tietze transformation program pri... |

1 | Integer programming and network °ows - Hu - 1969 |

1 |
An elementary introduction to coset-table methods in computational group theory
- NeubÄuser
- 1984
(Show Context)
Citation Context ...om the canonical presentation given by the Smith normal form. Thus, the abelian group presentations arising from a Reidemeister-Schreier process (see Havas [14], Havas and Sterling [18] and NeubÄuser =-=[27]-=-) have large numbers of generators and relations, with a substantial number of trivial modules in the canonical form. We want e±cient algorithms for recognizing the badly presented Z-modules which ari... |

1 |
Some computational approaches to groups given by ¯nite presentations, Mathematica Universitaria 7: 77{120 (Rio de Janeiro
- NeubÄuser, Sidki
- 1988
(Show Context)
Citation Context ...526 935 938 648 402 030 177 709 499 839 183 308. The gcd of these is 12, from which we can comfortably proceed. A more spectacular example is provided by the Heineken group G (see NeubÄuser and Sidki =-=[28]-=-), a group whose structure is still not well-enough understood. G = hx; y; z j [x; [x; y]] = z; [y; [y; z]] = x; [z; [z; x]] = yi. We study sections to try to understand the group. Consider N = ha5; b... |

1 |
Proving a group in¯nite
- Newman
- 1990
(Show Context)
Citation Context .... This technique has led to e®ective understanding of some groups whose structure was not well-enough known, including the Fibonacci group F (2; 9) (see Havas, Richardson and Sterling [17] and Newman =-=[29]-=-) and certain knot groups (see Havas and Kovµacs [16]). A common thread in these applications is the discovery of the module presentation by use of a Reidemeister-Schreier algorithm followed by abelia... |