computational complexity

, Volume 10, Issue 1, pp 41–69

# Fast computation of the Smith form of a sparse integer matrix

• M. Giesbrecht

## Abstract.

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix $$A \in {\Bbb Z}^{m \times n}$$. The algorithm treats A as a “black box”—A is only used to compute matrix-vector products and we do not access individual entries in A directly. The algorithm requires about $$O(m^2 {\rm log} \parallel A \parallel)$$ black box evaluations $$w \mapsto Aw\,{\rm mod}\,p$$ for word-sized primes p and $$w \in {\Bbb Z}^{n \times 1}_p$$, plus $$O(m^2 n\,{\rm log} \parallel A \parallel +\,m^3\,{\rm log^2} \parallel A \parallel)$$ additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. We also present an asymptotically fast algorithm for dense matrices which requires about $$O(n \cdot {\rm MM}(m)\,{\rm log} \parallel A \parallel +\,m^3\,{\rm log^2} \parallel A \parallel)$$ bit operations, where O(MM(m)) operations are sufficient to multiply two $$m \times m$$ matrices over a field. Both algorithms are probabilistic of the Monte Carlo type — on any input they return the correct answer with a controllable, exponentially small probability of error.

Keywords. Sparse integer matrix, Smith form, probabilistic algorithms.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Birkhäuser Verlag, Basel 2001

## Authors and Affiliations

• M. Giesbrecht
• 1
1. 1.Department of Computer Science, University of Western Ontario, London, Ontario, Canada N6A 5B7, e-mail: mwg@csd.uwo.ca, http://www.csd.uwo.ca/~mwgCA