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About the Kannan-Bachem algorithm

Francis Sergeraert

Abstract

The Smith reduction is a basic tool when analyzing integer matrices up to equivalence, and the Kannan-Bachem (KB) algorithm is the first polynomial algorithm computing such a reduction. Using this algorithm in complicated situations where the rank of the studied matrix is not maximal revealed an unexpected obstacle in the algorithm. This difficulty is described, analyzed, a simple solution is given to overcome it, finally leading to a general organization of the KB algorithm, simpler than the original one, efficient and having a general scope.

About the Kannan-Bachem algorithm

Abstract

The Smith reduction is a basic tool when analyzing integer matrices up to equivalence, and the Kannan-Bachem (KB) algorithm is the first polynomial algorithm computing such a reduction. Using this algorithm in complicated situations where the rank of the studied matrix is not maximal revealed an unexpected obstacle in the algorithm. This difficulty is described, analyzed, a simple solution is given to overcome it, finally leading to a general organization of the KB algorithm, simpler than the original one, efficient and having a general scope.

Paper Structure

This paper contains 28 sections, 1 theorem, 25 equations.

Table of Contents

  1. Introduction.
  2. Bézout matrices.
  3. The KB algorithm.
  4. Cancelling the entries above the diagonal.
  5. Canceling the entries below the diagonal.
  6. Phase 1.
  7. Phase 2.
  8. Iteration.
  9. Obvious extensions.
  10. First adaptation.
  11. Second adaptation.
  12. End of the computation.
  13. Computing a constructive homology group.
  14. A problem of Algebraic Topology.
  15. An accident, a solution.
  16. ...and 13 more sections

Key Result

Theorem 1

After a finite number $\nu$ of steps, the matrix $M_\nu$ obtained is diagonal.

Theorems & Definitions (1)

  • Theorem 1