# Fast computation of Smith forms of sparse matrices over local rings

```@inproceedings{Elsheikh2012FastCO,
title={Fast computation of Smith forms of sparse matrices over local rings},
author={Mustafa Elsheikh and Mark Giesbrecht and Andrew Novocin and B. David Saunders},
booktitle={ISSAC},
year={2012}
}```
• Published in ISSAC 25 January 2012
• Computer Science
We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the <i>black-box</i> model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an <i>n</i> x <i>n</i> matrix <i>A</i> over the ring F[<i>z</i>]/(<i>f</i><sup><i>e</i></sup>), where <i>f</i><sup…
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