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 is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith-normal-form computation, and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups. We investigate such issues as congruential techniques, sparsity considerations, pivoting strategies for Gauss-Jordan elimination, lattice basis reduction, and computational complexity. Our results, which are primarily empirical, show dramatically improved performance on previous methods.

[1]  S. Cabay,et al.  Congruence Techniques for the Exact Solution of Integer Systems of Linear Equations , 1977, TOMS.

[2]  A local approach to matrix equivalence , 1977 .

[3]  Paul D. Domich,et al.  Residual hermite normal form computations , 1989, TOMS.

[4]  George Havas,et al.  The last of the Fibonacci groups , 1979, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Leslie E. Trotter,et al.  Hermite Normal Form Computation Using Modulo Determinant Arithmetic , 1987, Math. Oper. Res..

[6]  Henry John Stephen Smith.  The Collected Mathematical Papers of Henry John Stephen Smith , 1965 .

[7]  J. Neubüser,et al.  Groups – St Andrews 1981: An elementary introduction to coset table methods in computational group theory , 1982 .

[8]  B.R. Donald,et al.  On the complexity of computing the homology type of a triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[9]  George Havas,et al.  Algorithms for Groups , 1992, Aust. Comput. J..

[10]  T. Hawkes,et al.  Rings, Modules and Linear Algebra. , 1972 .

[11]  Henry J. Stephen Smith,et al.  XV. On systems of linear indeterminate equations and congruences , 1861, Philosophical Transactions of the Royal Society of London.

[12]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[13]  Michael E. Pohst,et al.  A Modification of the LLL Reduction Algorithm , 1987, J. Symb. Comput..

[14]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[15]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[16]  Costas S. Iliopoulos,et al.  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, SIAM J. Comput..

[17]  James Lee Hafner,et al.  Asymptotically fast triangulation of matrices over rings , 1991, SODA '90.

[18]  Michael A. Frumkin,et al.  Polynomial Time Algorithms in the Theory of Linear Diophantine Equations , 1977, FCT.

[19]  George Havas,et al.  Integer matrices and Abelian groups (invited) , 1979, EUROSAM.

[20]  Andrew M. Odlyzko,et al.  Solving Large Sparse Linear Systems over Finite Fields , 1990, CRYPTO.

[21]  Derek F. Holt,et al.  A Graphics System for Displaying Finite Quotients of Finitely Presented Groups 113 , 1991, Groups And Computation.

[22]  Z. Zlatev.  Computational Methods for General Sparse Matrices , 1991 .

[23]  Costas S. Iliopoulos.  Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Infinite Abelian Groups and Solving Systems of Linear Diophantine Equations , 1989, SIAM J. Comput..

[24]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

[25]  George E. Collins,et al.  Algorithms for the Solution of Systems of Linear Diophantine Equations , 1982, SIAM J. Comput..

[26]  D. Rose,et al.  Algorithmic aspects of vertex elimination on directed graphs. , 1975 .

[27]  George Havas.  A Reidemeister-Schreier Program , 1974 .

[28]  M. F. Newman.  Proving a group infinite , 1990 .

[29]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..