Integer matrices and Abelian groups (invited)

  title={Integer matrices and Abelian groups (invited)},
  author={George Havas and Leon Sterling},
Practical methods for computing equivalent forms of integer matrices are presented. Both heuristic and modular techniques are used to overcome integer overflow problems, and have successfully handled matrices with hundreds of rows and columns. Applications to finding the structure of finitely presented abelian groups are described. 
Computing with Groups and Their Character Tables
In this survey an attempt is made to give some impression of the capabilities of currently available programs for computations with finitely generated groups and their representations.
Computing with groups and their character tables
In this survey an attempt is made to give some impression of the capabilities of currently available programs for computations with finitely generated groups and their representations.
Algorithms for Groups
The basic ideas behind some of the more important algorithms for flnitely presented groups and permutation groups are examined. Expand
A comparative study of algorithms for computing the Smith normal form of an integer matrix
Several algorithms have been studied for computing the Smith normal form of an integer matrix. The algorithms are analysed and their run-time performances are compared. The algorithms thatExpand
Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating theExpand
Computation of integral solutions of a special type of systems of quadratic equations
  • M. Pohst
  • Mathematics, Computer Science
  • 1983
A method to decide whether a solution X∈ℤn×n of XtrSX=T exists and, if the answer is affirmative, an algorithm for the computation of X. Expand
Application of Computational Tools for Finitely Presented Groups
  • G. Havas, E. Robertson
  • Mathematics, Computer Science
  • Computational Support for Discrete Mathematics
  • 1992
Under suitable circumstances a finitely presented group can be shown to be soluble and its complete derived series can be determined, using what is in effect a soluble quotient algorithm. Expand
Recognizing badly presented Z-modules
Finitely generated Z-modules have canonical decompositions. When such modules are given in a finitely presented form, there is a classical algorithm for computing a canonical decomposition. This isExpand
Asymptotically fast triangulation of matrices over rings
This paper shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations to lead to an algorithm for triangularizing integer matrices that has a faster running time than the known Hermite normal form algorithms. Expand
Abelian actions on compact bordered Klein surfaces
We state necessary and sufficient conditions for a finite abelian group to act as a group of automorphisms of some compact bordered Klein surface of algebraic genus $$p>1$$p>1. This result provides aExpand


Congruence Techniques for the Exact Solution of Integer Systems of Linear Equations
A new Fortran code, ESOLVE, is discussed for the exact solution of systems of linear equations with multiple-precision integer coefficients by congruence techniques. The code runs significantlyExpand
Algorithms for Hermite and Smith normal matrices and linear Diophantine equations
New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to aExpand
A Reidemeister-Schreier Program
The Reidemeister-Schreier method yields a presentation for a subgroup H of a group G when H is of finite index in G and G is finitely presented. This paper describes the implementation andExpand
Exact solution of linear equations
The congruential method of obtaining the exact solution of a system of linear equations with integral coefficients is critically reviewed. A new and efficient test for checking that a sequence ofExpand
The last of the Fibonacci groups
All the Fibonacci groups in the family F (2, n ) have been either fully identified or determined to be infinite, bar one, namely F (2, 9). Using computer-aided techniques it is shown that F (2, 9)Expand
Exact solutions of linear equations with rational coefficients by congruence techniques
1. Introduction. Sometimes one is interested in exact solutions of linear equations and cannot tolerate any errors at all, be it round-off errors, truncation errors or otherwise. In such situationsExpand
XV. On systems of linear indeterminate equations and congruences
  • H. J. S. Smith
  • Engineering
  • Philosophical Transactions of the Royal Society of London
  • 1861
The theory of the solution, in positive or negative integral numbers, of systems of linear indeterminate equations, requires the consideration of rectangular matrices, the constituents of which areExpand
A method of computing exact inverses of matrices with integer coefficients
In t heory, t he problem of computing t he exac t inverse of a matrix A with integer coefficients is completely solved by solving exact ly the simultaneous equations Ax=y, in which both x and 11 areExpand
Algorithm 524: MP, A Fortran Multiple-Precision Arithmetic Package [A1]
  • R. Brent
  • Mathematics, Computer Science
  • TOMS
  • 1978
A collection of ANSI Standard Fortran subroutines for performing multiple-precision floatingpoint arithmetic and evaluating elementary and special functions is given. The subroutines are machineExpand
Algorithm 287: matrix triangulation with integer arithmetic [F1]
The preceding properties are the basis of the claims for the procedure SOLVEINTEGER, which calls this procedure N T R A N K is designed to minimize the likelihood of overflow, the detection of which is left to the user. Expand