Abstract
We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ ℤm×n. The algorithm treats A as a “black-box”; A is only used to compute matrix-vector products and we don't access individual entries in A directly. The algorithm requires about O(m 2 log ∥A∥) such black-box evaluations reduced modulo word-sized primes p on vectors in ℤ n×1p , plus O(m 2 n log ∥A∥) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. For example, on an n×n integer matrix A with O(n log n) non-zero entries, only about O(n 3 log2 ∥A∥) bit operations are required to find the Smith form using standard integer arithmetic. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. The algorithm is probabilistic of the Monte Carlo type — on any input it returns the correct answer with a controllable, exponentially small probability of error.
Research was supported in part by Natural Sciences and Engineering Research Council of Canada research grant OGP0155376, and University of Manitoba research grant 431-1725-80.
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© 1996 Springer-Verlag Berlin Heidelberg
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Giesbrecht, M. (1996). Probabilistic computation of the Smith normal form of a sparse integer matrix. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61581-4_53
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DOI: https://doi.org/10.1007/3-540-61581-4_53
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